Explanations and justifications in Israeli mathematics textbooks and ...

Thesis for the degree

Doctor of Philosophy

Submitted to the Scientific Council of the

Weizmann Institute of Science

Rehovot, Israel

עבודת גמר )תזה( לתואר

דוקטור לפילוסופיה

למועצה המדעית שלמוגשת

מכון ויצמן למדע

רחובות, ישראל

By

Boaz Silverman

מאת

בועז זילברמן

בספרי לימוד במתמטיקה בישראל הסברים והצדקות

ותרומת ספר הלימוד לעיצוב הלמידה בכיתה

Explanations and justifications in Israeli mathematics textbooks

and the textbook's contribution to shaping classroom learning

Advisor:

Prof. Ruhama Even

מנחה:

פרופ' רוחמה אבן

July 2017 זע"שהת תמוז

Acknowledgements

To Prof. Ruhama Even, for your guidance, assistance, patience, and endless

devotion. Your precise observations have often left me in awe and pushed me to

improve myself and my work.

To Prof. Tommy Dreyfus and Prof. Edit Yerushalmy, for your ideas, support, and

thought-provoking criticism every step of the way.

To Dr. Michal Ayalon, Dr. Shai Olsher, Ayelet Gottlieb, and Anna Hoffman, for

being there with me (and for me) through and through.

To my families, near and far, past and present, blood and otherwise, for pushing me

forward, giving me love, and believing in me. I needed that.

And to my beloved Sydney Rachel, for being my partner, my equal, and my everything.

Table of Contents

List of Figures .................................................................................................................. i

List of Tables .................................................................................................................. v

Abstract .......................................................................................................................... 1

3 ............................................................................................................................... תקציר

1. Introduction .............................................................................................................. 5

2. Theoretical background ........................................................................................... 9

2.1. Justification and explanation in school mathematics ............................................. 9

2.1.1. Justification and explanation in mathematics education ............................... 9

2.1.2. Perceptions about justification and explanation in mathematics education 17

2.2. The role of textbooks in mathematics education .................................................. 19

2.2.1. The importance of textbooks in mathematics education ............................ 20

2.2.2. Justification and explanation in mathematics textbooks ............................. 21

3. Research questions ................................................................................................. 27

4. Methodology ........................................................................................................... 29

4.1. Part I: Justifications and explanations in Israeli 7th grade textbooks ................... 29

4.1.1. Sample selection ......................................................................................... 29

4.1.2. Data sources ................................................................................................ 30

4.1.3. Data analysis ............................................................................................... 31

4.2. Part II: The contribution of the textbook, teacher, and students to shaping

classroom justifications and explanations ............................................................ 34

4.2.1. Research design and participants ................................................................ 34

4.2.2. Data sources ................................................................................................ 37

4.2.3. Data analysis ............................................................................................... 39

5. Types of justification in the textbooks .................................................................. 43

5.1. The types of justification offered ......................................................................... 44

5.2. Types of justification across textbooks ................................................................ 48

5.3. Types of justification across mathematical statements ........................................ 50

5.4. Sequences of types of justification ....................................................................... 52

6. Justification strategies in the textbooks ............................................................... 55

6.1. Equivalent expressions ......................................................................................... 55

6.2. Division by zero ................................................................................................... 59

6.3. Distributive law .................................................................................................... 62

6.4. Equivalent equations ............................................................................................ 66

6.5. Product of negatives ............................................................................................. 71

6.6. Area of a trapezium .............................................................................................. 76

6.7. Area of a disk ....................................................................................................... 79

6.8. Vertical angles ...................................................................................................... 82

6.9. Corresponding angles ........................................................................................... 85

6.10. Angle sum of a triangle ........................................................................................ 89

6.11. Summary .............................................................................................................. 94

7. Paths of justification in Lena's classes ................................................................. 98

7.1. Equivalent equations ............................................................................................ 99

7.2. Area of a trapezium ............................................................................................ 102

7.3. Vertical angles .................................................................................................... 108

7.4. Angle sum of a triangle ...................................................................................... 112

7.5. Area of a disk ..................................................................................................... 115

7.6. Summary ............................................................................................................ 120

8. Paths of justification in Millie's classes .............................................................. 123

8.1. Equivalent equations .......................................................................................... 124

8.2. Product of negatives ........................................................................................... 126

8.3. Vertical angles .................................................................................................... 131

8.4. Corresponding angles ......................................................................................... 134

8.5. Area of a disk ..................................................................................................... 137

8.6. Summary ............................................................................................................ 141

9. Discussion .............................................................................................................. 143

References .................................................................................................................. 155

Appendix – Interview items ..................................................................................... 172

List of publications .................................................................................................... 177

Declaration................................................................................................................. 178

| i

List of Figures

Figure 1. Instances of justification – area of a trapezium (Textbook B, v.2, pp. 195-196) .................. 31

Figure 2. Paths of justification for the area of a trapezium (in Textbooks B and F) ............................ 33

Figure 3. The research design for part II of this study .......................................................................... 34

Figure 4. A set of paths of justification for product of negatives (Interview item)............................... 39

Figure 5. Outline of the teaching sequence for area of a trapezium in Lena's classes ......................... 40

Figure 6. Experimental demonstration by Dissection (Screenshot in Class L1) .................................. 41

Figure 7. Concordance of a rule with a model by Dissection (Screenshot in Class L2) ...................... 41

Figure 8. Appeal to authority (adapted from Textbook G, vol 2, pp. 61-62) ........................................ 44

Figure 9. Experimental demontration (Textbook B, vol 3, p.161)........................................................ 45

Figure 10. Concordance of a rule with a model (Textbook D, vol 2, p. 186) ....................................... 45

Figure 11. Deduction using a model (Textbook H, vol 3, p. 55) .......................................................... 46

Figure 12. Deduction using a specific case (Textbook B, vol 2, p. 195) .............................................. 46

Figure 13. Deduction using a general case (Textbook B, vol 2, p. 196) .............................................. 47

Figure 14. Relative frequencies of the types of justification in the textbooks, by category ................. 48

Figure 15. Frequencies of types of justification by topic and textbook ................................................ 50

Figure 16. Deduction using a specific case by Rules & conventions (Textbook D, vol 1, p. 43) ......... 55

Figure 17. Deduction using a specific case by Rules & conventions (Textbook A, vol 1, p. 188) ....... 56

Figure 18. Experimental demonstration by Substitution (Textbook A, vol 1, p. 26) ............................ 56

Figure 19. Deduction using a model by Description equivalence (Textbook C, vol 1, p. 57) .............. 57

Figure 20. Equivalent expressions – Paths of justification ................................................................... 58

Figure 21. Deduction using a specific case by Inverse of multiplication (Textbook B, vol 1, p. 80) ... 59

Figure 22. Deduction using a specific case by Inverse of multiplication (Textbook F, vol 1, p. 31) ... 60

Figure 23. Deduction using a a model by Repeated subtraction (Textbook A, vol 1, p. 64) ............... 60

Figure 24. Division by zero – Paths of justification .............................................................................. 61

Figure 25. Deduction using a model by Area (Textbook D, vol 1, p. 75) ............................................. 62

Figure 26. Deduction using a model by Array (Textbook E, vol 1, p. 96) ........................................... 63

Figure 27. Concordance of a rule with a model by Arith. conventions (Textbook E, vol 1, p. 97) ...... 63

Figure 28. Concordance of a rule with a model by Arith. conventions (Textbook B, vol 1, p. 72) ...... 63

Figure 29. Distributive law – Paths of justification .............................................................................. 65

Figure 30. Deduction using a model by Balance model (Textbook H, vol 3, p. 55)............................. 66

Figure 31. Experimental demonstration by Undoing (adapted from Textbook B, vol 2, p. 17) ........... 67

Figure 32. Concordanceof a rule w/ model by intuition (adapted from Textbook F, vol 2, p. 135) ..... 68

ii |

Figure 33. Experimental demonstration by intuition (adapted from Textbook D, vol 2, p. 15) ........... 68

Figure 34. Deduction using a model by Segment model (Textbook A, vol 1, p. 214) .......................... 69

Figure 35. Equivalent equations – Paths of justification. ..................................................................... 70

Figure 36. Deduction using a specific case by Discovering patterns (Textbook C, vol 2, p. 42) ........ 71

Figure 37. Deduction using a specific case by Discovering patterns (Textbook H, vol 2, p. 46) ........ 72

Figure 38. Appeal to authority by Discov. patterns (adapted from Textbook G, v. 2, pp. 61-62) ........ 72

Figure 39. Deduction using a specific case by Extension of properties (Textbook F, vol 2, p. 48) ..... 73

Figure 40. Deduction using a model by Line model (adapted from Textbook A, vol 2, p. 392) .......... 73

Figure 41. Deduction using a model by Line model (Textbook A, vol 2, p. 395) ................................ 73

Figure 42. A mnemonic relying on Double negation (Textbook A, vol 2, p. 394) ............................... 74

Figure 43. Product of negatives – Paths of justification ....................................................................... 75

Figure 44. Experimental demonstration by Dissection (Textbook D, vol 2, p.184) ............................. 76

Figure 45. Deduction using a general case by Construction (Textbook G, vol 3, p.93) ...................... 77

Figure 46. Area of a trapezium – Paths of justification ........................................................................ 78

Figure 47. Deduction using a general case by Dissection into sectors (Textbook G, part 3, p. 118) .. 79

Figure 48. Deduction using a general case by Dissection into rings (Textbook F, vol 2, p. 223) ....... 80

Figure 49. Area of a disk – Paths of justification .................................................................................. 81

Figure 50. Deduction using a general case by Supplementary angles (Textbook F, vol 3, p. 136) ..... 82

Figure 51. Experimental demonstration by Measurement (Textbook F, vol 3, p. 136) ........................ 83

Figure 52. Vertical angles – Paths of justification ................................................................................ 84

Figure 53. Experimental demonstration by Measurement (Textbook G, vol 2, p. 182) ....................... 85

Figure 54. Experimental demonstration by Measurement (adapted from Textbook C, vol 2, p. 189) . 86

Figure 55. Deduction using a specific case by Alternate angles (Textbook H, vol 2, p. 231) .............. 86

Figure 56. Corresponding angles – Paths of justification..................................................................... 88

Figure 57. Deduction using a general case by Parallel line (Textbook C, vol 3, p. 173) .................... 89

Figure 58. Experimental demonstration by Angle rearrangement (Textbook B, vol 3, p. 161) ........... 90

Figure 59. Experimental demonstration by Angle rearrangement (Textbook B, vol 3, p. 161) ........... 90

Figure 60. Deduction using a general case by Right triangle (Textbook D, vol 3, p. 135).................. 90

Figure 61. Deduction using a general case by Right triangle (Textbook A, vol 3, p. 647).................. 91

Figure 62. Experimental demonstration by Measurement (Textbook H, vol 3, p. 167) ....................... 91

Figure 63. Experimental demonstration by Measurement (adapted from Textbook H, vol 3, p. 167) . 91

Figure 64. Deduction using a general case by Parallel line & extension (Textbook A, vol 3, p. 621) 92

Figure 65. Angle sum of a triangle – Paths of justification ................................................................... 93

Figure 66. Common sequences of justification strategies in the textbooks and the Israeli curriculum 97

| iii

Figure 67. Paths of justification in Lena's classes for Equivalent equations ........................................ 99

Figure 68. Balance model (Textbook C, vol 3, p. 54) ......................................................................... 100

Figure 69. Balance model – limitations (Textbook C, vol 3, p. 58) .................................................... 100

Figure 70. Balance model (screenshots in Class L1 [a] and Class L2 [b]) ......................................... 101

Figure 71. Paths of justification in Lena's classes for Area of a trapezium ........................................ 103

Figure 72. Dissection (Textbook C, vol 2, p. 112) .............................................................................. 103

Figure 73. Dissection and Construction (Textbook C, vol 2, p. 112) ................................................. 103

Figure 74. Deduction using a general case (Textbook C, vol 2, p. 112) ............................................ 104

Figure 75. Dissection (screenshots in Class L1 [a] and Class L2 [b]) ................................................ 104

Figure 76. Derivation of the area formula of a trapezium in a specific case ...................................... 105

Figure 77. The deductive process in Class L2 [a], replaced by giving the rule [b] ............................ 105

Figure 78. Concordance of a rule with a model (screenshot in Class L2) .......................................... 106

Figure 79. Derivation of the area formula of a trapezium in a specific case ...................................... 106

Figure 80. Deduction using a specific case (screenshots in Class L1 [a] and Class L2 [b])............... 107

Figure 81. Paths of justification in Lena's classes for Vertical angles ................................................ 109

Figure 82. Vertical angles – deduction using a specific case (Textbook C, vol 2, p. 184) ................. 109

Figure 83. Vertical angles – deduction using a general case (Textbook C, vol 2, p. 184) ................. 109

Figure 84. Vertical angles (screenshot in Class L1) ........................................................................... 110

Figure 85. A set of paths of justification for vertical angles (interview item). ................................... 111

Figure 86. Paths of justification in Lena's classes for Angle sum of a triangle ................................... 112

Figure 87. Angle sum of a triangle – Angle rearrangement (Textbook C, vol 3, p. 172) ................... 113

Figure 88. Angle sum of a triangle – Parallel line (adapted from Textbook C, vol 3, p. 173) ............ 113

Figure 89. Angle sum of a triangle – Parallel line (Textbook C, vol 3, p. 173) .................................. 113

Figure 90. Paths of justification in Lena's classes for Area of a disk .................................................. 116

Figure 91. Area of a disk – Dissection into sectors (Textbook C, part 3, p. 156) ............................... 116

Figure 92. Area of a disk – the circumference and the area (Textbook C, part 3, p. 156) .................. 117

Figure 93. A set of paths of justification for area of a disk (interview item) ..................................... 118

Figure 94. Paths of justification in Lena's classes, by statement ........................................................ 120

Figure 95. Paths of justification in Millie's classes for Equivalent equations .................................... 124

Figure 96. Balance model (screenshot in Class M2) .......................................................................... 125

Figure 97. Paths of justification in Millie's classes for Product of negatives ..................................... 127

Figure 98. Product of negatives – Discovering patterns (Textbook C, vol 2, pp. 42-43) ................... 127

Figure 99. Product of negatives – Extension of properties (Textbook C, vol 2, p. 44) ...................... 128

Figure 100. Discovering patterns (screenshots in Class M1 [a] and Class M2 [b]) ........................... 129

iv |

Figure 101. A set of paths of justification for product of negatives (Interview item)......................... 130

Figure 102. Paths of justification in Millie's classes for Vertical angles ............................................ 132

Figure 103. Vertical angles – supplementary angles (screenshot in Class M1) ................................. 132

Figure 104. Paths of justification in Millie's classes for Corresponding angles ................................. 135

Figure 105. Corresponding angles – measurement (Textbook C, vol 2, p. 189) ................................ 135

Figure 106. Corresponding angles – measurement (adapted from Textbook C, vol 2, p. 189) .......... 135

Figure 107. Measurement (screenshot in Class M1 [a] and Class M2 [b]) ......................................... 136

Figure 108. Paths of justification in Millie's classes for Area of a disk .............................................. 138

Figure 109. Area of a disk – Dissection (screenshot in Class M1) ..................................................... 139

Figure 110. Area of a disk – Dissection (screenshot in Class M2) ..................................................... 139

Figure 111. A set of paths of justification for area of a disk (interview item). .................................. 140

Figure 112. Grid with Concordance of a rule with a model (screenshots in Class M1) ..................... 140

Figure 113. Paths of justification in Millie's classes, by statement ..................................................... 141

| v

List of Tables

Table 1. General characteristics of the analysed textbooks .................................................................. 29

Table 2. Types of justification, by categories (adapted from: Stacey & Vincent, 2009). ..................... 32

Table 3. Duration of observed lesson sections by classroom (in minutes and percentages) ................. 36

Table 4. Number of lesson observations, by classroom and mathematical statement .......................... 37

Table 5. Frequencies of instances of justifications, by textbook section .............................................. 43

Table 6. Frequencies of types of justification, by textbook .................................................................. 49

Table 7. Average frequencies for types of justification, by textbook scope ......................................... 49

Table 8. Frequencies of types of justification, by mathematical statement .......................................... 51

Table 9. Paths of justification formed by types of justification, by textbook and statement. ............... 53

Table 10. Average path lengths, by statement and textbook scope....................................................... 54

Table 11. Equivalent expressions – frequencies of justification strategies, by textbook ...................... 57

Table 12. Division by zero – frequencies of justification strategies, by textbook. ................................ 60

Table 13. Distributive law – frequencies of justification strategies, by textbook. ................................ 64

Table 14. Equivalent equations – frequencies of justification strategies, by textbook ......................... 70

Table 15. Product of negatives – frequencies of justification strategies, by textbook .......................... 74

Table 16. Area of a trapezium – frequencies of justification strategies, by textbook ........................... 77

Table 17. Area of a disk – frequencies of justification strategies, by textbook .................................... 81

Table 18. Vertical angles – frequencies of justification strategies, by textbook .................................. 83

Table 19. Corresponding angles – frequencies of justification strategies, by textbook ....................... 87

Table 20. Angle sum of a triangle – frequencies of justification strategies, by textbook ..................... 92

Table 21. Number of justification strategies per path, by statement and textbook. .............................. 94

Table 22. Frequencies of justification strategies, by justification types ............................................... 95

1 | Abstract

Abstract

The study deals with explanation and justification in Israeli mathematics textbooks and with

the textbook’s contribution to shaping their learning in the classroom. The study comprises

two parts. The first part examines the opportunities offered in 7th grade Israeli textbooks for

students to learn how to explain and justify mathematical statements. The second examines

the ways in which the textbook, in conjunction with the teacher and the students, shape the

opportunities offered in 7th grade Israeli classrooms to learn how to explain and justify.

Part I of the study investigates the explanations and justifications offered in 7th grade

mathematics textbooks and the paths of justification – i.e., the sequences of justifications

each textbook offers for each mathematical statement. The data sources include the textbook

chapters introducing ten key mathematical statements, in eight Israeli 7th grade mathematics

textbooks (two of limited scope, intended for students with low achievements; six of

standard/expanded scope, intended for the general student population). Comparative analyses

of the paths of justification, by textbook and by mathematical statement, focused on three

attributes: (1) path length – the number of instances of justification offered in a textbook for

each mathematical statement; (2) characteristics of the instances comprising each path –

justification types (a meta-level characteristic, following Stacey & Vincent, 2009) and

justification strategies (a content-specific characteristic, dealing with the specific warrant);

and (3) sequencing – the order in which justifications were offered in the textbook.

Part II of the study investigates the ways in which textbooks, in conjunction with the teacher

and the students, shape the opportunities offered in 7th grade mathematics classrooms. This

part of the study revolves around two case studies, each focused on a mathematics teacher

who uses the same textbook in two 7th grade classes. The data sources include 11-14 lesson

observations in each of the four classes (a total of 49 lessons) and teacher interviews.

Comparative analyses of the paths of justification, by teacher, by classroom, and by topic,

focused on three attributes: (1) path length, (2) characteristics, and (3) sequencing.

The findings of Part I reveal that the analyzed textbooks provided justification for all

analyzed statements (all but one statement in one textbook). Path lengths varied considerably

– where some textbooks offered long paths for a mathematical statement, other textbooks

offered rather short paths. Paths of justification typically comprised either deductive or

empirical justification types, whereas external types were extremely rare.

Abstract | 2

Three justification types were especially common – an empirical type (Experimental

demonstration) and two deductive types (Deduction using a specific/general case). However,

these types were distributed differently among mathematical topics and among textbooks of

different scopes: (1) Deduction using a general case, the justification type closest to a formal

proof, was included solely in geometry, while Deduction using a specific case (i.e., generic

examples) was used mostly in statements relying on an algebraic derivation; and (2)

Deduction using a general case was roughly three times more common in textbooks of

standard/expanded scope compared with textbooks of limited scope. The three most common

justification types were often similarly sequenced across textbooks and topics: paths that

involved both experimentation and a deductive process tended to offer the empirical type

before the deductive, and paths that involved deduction using both a generic example and the

general case tended to offer the specific case before the general.

Justification strategies were associated with justification types, yet the correspondence was

not one-to-one. In algebra statements, almost every justification strategy corresponded to a

single type across textbooks. In geometry, however, justification strategies often occurred

several times in paths of justification, with various justification types.

The findings of Part II suggest that the textbook contributed greatly to shaping the paths of

justification and was the main source for justifications in all observed classrooms. Paths of

justification in every class were generally similar to, yet typically shorter than, the paths

offered in the textbook – both in their characteristics and in their sequencing.

Additionally, the findings reveal that the teachers' perception of their students' abilities was

instrumental in constructing the paths of justifications in the classes. Instances of justification

were excluded if the teacher regarded them as too difficult in two cases: (1) the justification

type was deduction using a general case, and (2) the textbook marked them as intended for

high-achieving students. Additionally, noisy classroom environment interrupted several

discussions and contributed to altering a justification type from deductive to empirical.

This study focuses on textbooks that are currently in use and brings to light certain nontrivial

aspects: It maps the paths of justification offered for mathematical statements and

characterizes both the justification strategies and the corresponding justification types.

Additionally, the study discusses the contribution of the textbook to shaping classroom

learning of explanation and justification. This information is important for researchers,

educators, textbook authors, curriculum developers, and decision makers.

3 | Abstract

תקציר

הסברים בספרי מתמטיקה בישראל ובתרומת ספר הלימוד לעיצוב ההזדמנויות ללמידתם בהמחקר עוסק בהצדקות ו

בכיתה. הוא כולל שני חלקים. החלק הראשון בוחן את ההזדמנויות המוצעות לתלמידים בספרי לימוד לכיתה ז'

ת. החלק השני בוחן באילו אופנים ספר הלימוד, יחד עם בישראל ללמוד כיצד להצדיק ולהסביר אמירות מתמטיו

להצדיק ולהסביר.כיצד המורה והתלמידים, מעצבים את ההזדמנויות המוצעות בכיתות ז' בישראל ללמוד

האופנים – מהלכי ההצדקהחוקר את ההצדקות וההסברים המוצעים בספרי לימוד במתמטיקה לכיתה ז' ואת Iחלק

של אמירה מתמטית בספרי הלימוד. מקורות הנתונים כוללים את פרקי ותכל ההצדק ותומאורגנ ותשבהם מסודר

ספרי הלימוד המציגים עשר אמירות מתמטיות מרכזיות, בשמונה ספרי לימוד במתמטיקה לכיתה ז' )שניים בהיקף

למידים(. מצומצם, המיועדים לתלמידים בעלי הישגים נמוכים; שישה בהיקף רגיל/מורחב, המיועדים לכלל הת

ניתוחים השוואתיים של מהלכי ההצדקה, לפי ספר הלימוד ולפי האמירה המתמטית, התמקדו בשלושה היבטים:

בכל הצדקות( מאפייני ה2המוצעים בספר לימוד לכל אמירה מתמטית; ) ותמספר ההצדק –( אורך המהלך 1)

אסטרטגיות ההצדקה( וStacey & Vincent, 2009על, לפי המסגרת של -)מאפיין ברמת סוגי ההצדקה –מהלך

בספר הלימוד. ותסדר הצגת ההצדק –( רצף 3)-תוכן ותלוי אמירה(; ו-)מאפיין תלוי

חוקר את האופנים שבהם ספרי לימוד, יחד עם המורה והתלמידים, מעצבים את ההזדמנויות המוצעות IIחלק

אחד מהם מלמדת . חלק זה מורכב משני חקרי מקרה, שבכלללמוד להסביר ולהצדיק בשיעורי מתמטיקה בכיתות ז'

11-11-מורה מסוימת מתמטיקה בשתי כיתות ז' בעזרת אותו ספר לימוד. מקורות הנתונים כוללים תצפיות ב

שיעורים( וראיונות עם כל מורה. ניתוחים 14שיעורים בכל אחת מארבע הכיתות לאורך שנת הלימודים )סה"כ

טית, התמקדו בשלושה היבטים: אורך המהלך, השוואתיים של מהלכי ההצדקה, לפי מורה, כיתה, ואמירה מתמ

רצף.הבכל מהלך, ו הצדקותמאפייני ה

מראים כי ספרי הלימוד שנותחו מציעים הצדקות לכל האמירות שנותחו )פרט לאמירה אחת Iהממצאים של חלק

ת, חלק מספרי הלימוד הציעו מהלכים ארוכים לאמירה מתמטי –בספר אחד(. נמצאו הבדלים באורכי המהלכים

בעוד ספרי לימוד אחרים הציעו מהלכים קצרים למדי. בנוסף, מהלכי ההצדקה לרוב הכילו סוגי הצדקה אמפיריים

מתמטיים היו נדירים מאד. -או דדוקטיביים, בעוד סוגי הצדקה חוץ

רת הסקה בעז( ושני סוגים דדוקטיביים )הדגמה בהתנסותסוג אמפירי ) –שלושה סוגי הצדקה היו נפוצים במיוחד

(. עם זאת, נמצא הבדל בשכיחות של סוגים אלו בין נושאים מתמטיים שונים וכן בין ספרי לימוד כללי/ מקרה פרטי

, סוג ההצדקה הקרוב ביותר להוכחה פורמלית, הוצע הסקה בעזרת מקרה כללי( 1המיועדים לאוכלוסיות שונות: )

מא גנרית( היה נפוץ בעיקר באמירות המערבות )משמע, דוג הסקה בעזרת מקרה פרטיאך ורק בגיאומטריה, בעוד

בספרי לימוד בהיקף –בערך פי שלושה –היה שכיח יותר הסקה בעזרת מקרה כללי( 2)-מניפולציות אלגבריות; ו

רגיל/מורחב לעומת ספרי לימוד בהיקף מצומצם. סדר ההופעה של שלושת סוגי ההצדקה הנפוצים היה דומה בין

הלכים שכללו הן התנסות והן תהליך דדוקטיבי נטו להציע את הסוג האמפירי לפני הספרים ברוב האמירות: מ

גנרית והן בעזרת המקרה הכללי נטו להציע את -הדדוקטיבי, ומהלכים שכללו הסקה הן בעזרת דוגמה פרטית

המקרה הפרטי לפני הכללי.

Abstract | 4

ערכית. באלגברה, כמעט כל -חד-אסטרטגיות ההצדקה נמצאו קשורות לסוגי ההצדקה, אם כי ההתאמה לא היתה חד

אחת מאסטרטגיות ההצדקה התאימה לסוג הצדקה יחיד בכל ספרי הלימוד שנותחו. בגיאומטריה, לעומת זאת,

אסטרטגיות הצדקה הופיעו לעתים קרובות מספר פעמים במהלכי ההצדקה, בכל פעם עם סוגי הצדקה שונים.

הלימוד לעיצוב מהלכי ההצדקה. ספר הלימוד היה המקור מצביעים על תרומה רבה של ספר IIהממצאים של חלק

הן במאפייני –בכל הכיתות שנצפו, ומהלכי ההצדקה היו דומים למדי למהלכים שבספר הלימוד ותהמרכזי להצדק

אם כי היו קצרים יותר מאשר בספר. –כל הצדקה והן בסדר שלהם

תלמידיהן מילא תפקיד משמעותי בבניית מהלכי הממצאים מראים גם כי האופן שבו תפשו המורות את יכולות

הסקה ( סוג ההצדקה היה 1שנתפשו על ידי המורה כקשים מדי אם: ) ותההצדקה בכיתות. המורות לא כללו הצדק

לתלמידים מתקדמים. בנוסף, סביבת לימודים תבספר הלימוד כמיועד ההצדקה סומנה( 2)-, ובעזרת מקרה כללי

רועשת בכיתות הקשתה על קיום דיונים ואף תרמה לשינוי סוג הצדקה מדדוקטיבי לאמפירי.

מורים מלמדים בעזרתם. המחקר מציף ומציג מאפיינים שהמחקר מתמקד בספרי לימוד הנמצאים כעת בשימוש ו

המוצעים בספרי הלימוד לאמירות מתמטיות, ומאפיין הן שאינם מובנים מאליהם: הוא ממפה את מהלכי ההצדקה

את האמצעים המתמטיים שבהם נעשה שימוש והן את סוגי ההצדקה המתאימים. בנוסף, המחקר דן בתרומה של

ספר הלימוד לעיצוב הלמידה של הצדקות והסברים בכיתות. מידע זה חשוב לחוקרים, למורים, למחברי ספרי

ראה, ולמקבלי החלטות.לימוד, למפתחי חומרי הו

5 | Introduction

1. Introduction

In this chapter, I focus on three issues: (i) The background for this study, (ii) the research

goals, and (iii) the structure of this thesis.

1.1. Background

Explaining and justifying are the bread and butter of mathematics – the essential, sustaining

element. They are considered by many to be central components of doing and learning

mathematics (e.g., M. Ayalon & Even, 2010; Ball & Bass, 2003; Cabassut, 2005; Chazan,

1993; Schwarz, Hershkowitz, & Prusak, 2010; Yackel & Hanna, 2003). Their importance in

school mathematics is emphasized in the Israeli national junior high school curriculum (Israel

Ministry of Education, 2009), as well as in other curricula worldwide (e.g., Australian

Education Council, 1991; Common Core State Standards Initiative, 2010; Department of

Education, 2010; NCTM - National Council of Teachers of Mathematics, 2000).

Israel has a centralized educational system. School curricula are developed and regulated by

the Israeli Ministry of Education, and textbooks are published under authorization of the

Ministry. Approval of textbooks and learning resources to be used in schools involves several

necessary conditions: procedural (e.g., submission of proper documentation), technical (e.g.,

maximal page count), and pedagogical (e.g., curriculum adequacy and correspondence to

scientific-pedagogical quality criteria) (Israel Ministry of Education, 2015, sec. 6). The

adequacy of a textbook to the Israeli curriculum must be complete, and correspond to the

number of teaching hours that are specified in the curriculum, or the textbook is categorically

rejected and returned to its developers.

In 2006 and 2009 the Israeli Ministry of Education launched new national elementary and

junior high school mathematics curricula (respectively). The junior high school curriculum,

intended for grades 7-9, underwent revisions since it was launched and was finally approved

in 2013. The new national high school curriculum is currently under development.

The Israeli junior high school curriculum (henceforth: Israeli curriculum) comprises three

strands: arithmetic, algebra, and geometry (Israel Ministry of Education, 2009). It maintains

and builds on the emphases in the elementary school national curriculum, and explicitly

emphasizes justification, explanation, and proof, as well as investigation, exploration,

problem solving, and the generation of mathematical conjectures. Moreover, it stresses these

aspects for both algebra and geometry.

Introduction | 6

In response to the introduction of the new school curriculum, several teams began developing

experimental curriculum materials (e.g., textbook series, teacher's guides, and online learning

resources). Eight parallel mathematics textbooks were approved in 2012 by the Ministry of

Education for use in Hebrew speaking 7th grade schools: six intended for the general student

population (standard/expanded scope) and two for students with low achievements (limited

scope). These 7th grade textbooks have been used in schools since the 2012/2013 school year,

8th grade textbooks were approved for use in the 2014/2015 school year, and 9th grade

textbooks were approved for use in the 2015/2016 school year.

A recent study investigated aspects of justification in the experimental edition of the new

mathematics textbooks. Dolev and Even (2013) examined the requests made by six 7th grade

textbooks for students to explain and justify their mathematical work, in two central topics in

the Israeli curriculum: equation solving in the algebra strand, and triangle properties in the

geometry strand. The findings revealed that larger percentages of these tasks were included in

the geometric topic, compared with the algebraic topic, in all six textbooks. In an M.Sc.

thesis, Dolev (2011) further analyzed the justifications offered for these mathematical topics

in the textbooks and found that in some cases, textbooks justified a statement using several

instances of justification.

Research suggests that in Israel and in several other countries, the textbooks used in class

considerably influence students’ opportunities to learn mathematics in general (e.g.,

Eisenmann & Even, 2011; Haggarty & Pepin, 2002; Shield & Dole, 2013; Van den Heuvel-

Panhuizen, 2000), and to explain and justify in particular (e.g., Ayalon & Even, 2016).

Accordingly, an increasing number of studies focus on the opportunities offered in

mathematics textbooks to learn to justify. Two lines of research are common in the literature:

(1) the justifications for mathematical statements presented in textbooks (e.g., Dolev, 2011;

Stacey & Vincent, 2009), and (2) the opportunities for students to explain and justify their

own mathematical work (e.g., Dolev & Even, 2013; Stylianides, 2009). The current study

belongs to the first of these two lines.

Textbook studies reveal a variety of explanations and justifications offered for mathematical

statements (e.g., Stacey & Vincent, 2009; Stylianides, 2009). Furthermore, in some cases,

textbooks offered multiple instances of justification for a single statement: either more than

one type of justification (e.g., both empirical and deductive types) or more than one instance

of the same type (Dolev, 2011; Stacey & Vincent, 2009).

7 | Introduction

This exposure to a multitude of explanations and justification for a single statement is likely

to have an additive effect on students: Some instances of justification may invoke previous

knowledge or experience, thereby serving a didactic goal (e.g., Sierpinska, 1994; Stacey &

Vincent, 2009); other instances may function as a catalyst or a trigger for making a conjecture

or for providing a more formal justification (e.g., J. D. Davis, Smith, Roy, & Bilgic, 2014;

Stylianides, 2009); and yet others might strengthen the conceptual basis for the mathematical

statement that is being explained or justified (Sierpinska, 1994). Therefore, in addition to

examining each individual instance of justification separately, it is important to attend to the

paths of justification – the ways in which instances of justification of one statement are

arranged, structured, and sequenced – an aspect that receives little attention in the literature.

Part I of this study examines the paths of justification for mathematical statements that are

offered in Israeli 7th grade mathematics textbooks. It focuses on the 7th grade because, as

described earlier, for this grade level: (1) the Israeli curriculum emphasizes justification and

explanation; and (2) in many Israeli schools, the 7th grade marks a transition from elementary

to junior high school, which involves a more deductive approach.

Previous studies suggest that textbooks are but one of several factors involved in shaping the

opportunities for students to learn how to explain and justify mathematical statements in the

classroom, such as teachers, students, and mathematical topics (e.g., M. Ayalon & Even,

2015; Even & Kvatinsky, 2010). Part II of this study examines this issue and focuses on the

contribution of the textbook, together with the teacher and the students, to learning to explain

and justify in class.

Two working hypotheses underlie this study: (i) Textbooks are instrumental in shaping Israeli

students' opportunities to learn to explain and justify (e.g., M. Ayalon & Even, 2016); and (ii)

These opportunities are shaped by the characteristics of several additional factors, such as the

teacher and the students (e.g., M. Ayalon & Even, 2015; Even & Kvatinsky, 2010).

1.2. Research goals

The study has two central research goals. First, by using a novel approach to textbook

justifications, which focuses on the paths of justification, this study aims to characterize the

opportunities offered in Israeli textbooks for students to learn how to explain and justify

mathematical statements. Second, in light of the joint contribution of several factors to

shaping these opportunities in classrooms, and by relying on the textbook analysis described

above, this study aims to examine the ways in which the textbook, in conjunction with the

Introduction | 8

teacher and the students, shape the opportunities offered in Israeli classrooms for students to

learn how to explain and justify.

1.3. Structure of the thesis

The dissertation comprises nine chapters and one appendix.

Chapter 2 reviews the literature relevant to the current study. The chapter comprises two

sections. The first section reviews the literature on the teaching and learning of school

mathematics, with a particular focus on explanation and justification. The second section

reviews the literature on the role of textbooks in mathematics education.

Chapter 3 presents the research questions for this study.

Chapter 4 describes the methodology for each of the two parts of the study, including the

research design, the sample selection, the participants in part II of the study (Lena and

Millie), the data sources, the conceptual framework, and the stages of data analysis.

Chapter 5 and Chapter 6 focus on findings related to my first research goal, dealing with the

opportunities offered to students in 7th grade Israeli mathematics textbooks to learn how to

explain and justify mathematical statements. Chapter 5 deals with results regarding the types

of justification offered in the textbooks, and Chapter 6 describes results regarding the

justification strategies offered in the textbooks.

Chapter 7 and Chapter 8 focus on findings related to my second research goal, dealing with

the ways in which the textbook, together with the teacher and the students, shape these

opportunities in the classroom. Each chapter focuses on one case study: Chapter 7 focuses on

Lena's classes and Chapter 8 on Millie's classes.

In Chapter 9 I discuss the results achieved in this study and present my conclusions based on

the findings. In addition, I discuss the implications of the study and its limitations.

9 | Theoretical background

2. Theoretical background

This study deals with justification and explanation (J&E) in Israeli mathematics textbooks

and with the contribution of the textbook to shaping the learning of J&E in the classroom.

Accordingly, this literature review comprises two parts.

The first part deals with J&E in school mathematics. It discusses two issues: J&E in

mathematics education and perceptions about J&E in mathematics education. The second part

is concerned with the role of textbooks in mathematics education and discusses two issues:

the importance of textbooks in mathematics education and J&E in mathematics textbooks.

2.1. Justification and explanation in school mathematics

2.1.1. Justification and explanation in mathematics education

Asking "Why is it true?" is a fundamental part of learning and doing mathematics, and is a

gateway to activities such as identifying patterns, generating conjectures, finding evidence

supporting or refuting the conjectures, evaluating arguments, and formally proving or

disproving conjectures (e.g., Hanna, Jahnke, & Pulte, 2010; Lakatos, 1976; Pólya, 1954).

A great emphasis is given to these concepts in the mathematics education literature. Some

studies have approached them theoretically (Blum & Kirsch, 1991; Cai & Cirillo, 2014;

Conner, Singletary, Smith, Wagner, & Francisco, 2014; Hanna, 2014; Harel & Sowder, 2007;

Kidron & Dreyfus, 2009; Peirce, 1878/1998; Schwarz, 2009; Sierpinska, 1994; Toulmin,

1958/2003; Werndl, 2009), some empirically (e.g., Bieda, Ji, Drwencke, & Picard, 2014;

Chazan, 1993; B. Davis & Simmt, 2006; J. D. Davis et al., 2014; Dolev, 2011; Dolev &

Even, 2013; Haggarty & Pepin, 2002; Senk, Thompson, & Johnson, 2008; Thompson, Senk,

& Johnson, 2012), and some combined the two approaches (e.g., Stacey & Vincent, 2009;

Stylianides, 2009).

Four issues are discussed below. First, the definitions used in these studies for J&E are not

consistent, and different authors assign different meanings to these concepts. Therefore, this

section begins with a short review of terminology. The second issue is a detailed review of

several conceptual frameworks suggested in the literature. The third issue involves the

different roles and types of J&E in mathematics education and in mathematics. The fourth

issue concerns J&E in the mathematics school curriculum.

Section 2.1 – Justification and explanation in school mathematics | 10

2.1.1.1. Background and review of terminology

A plethora of terms is used in the literature in the discussion of justification and explanation

activities. Among them are argumentation, warrant, reasoning, and proof. These terms are

close in meaning, yet their boundaries are ill-defined (Hanna, 2014).

Argumentation is a very inclusive concept, frequently used in the science teaching literature.

Some authors define it as a dialectical discourse, which involves putting forward propositions

in order to justify or refute a certain standpoint (van Eemeren, Grootendorst, & Henkemans,

1996). Others define it as a social process involving an adjustment of one's interpretations by

presenting their rationale verbally (Banegas, 2013; Krummheuer, 1995). Yet others define it

as involving skills such as making conjectures, testing their plausibility, justifying claims,

constructing a connected sequence of assertions (i.e., an argument), and evaluating arguments

(J. D. Davis et al., 2014; Hanna, 2014; Pólya, 1954; Umland & Sriraman, 2014).

A warrant is an essential part of an argument, which justifies the claim with given data.

Toulmin's (1958/2003) widely cited Argument Model identifies six elements as constituting

arguments: (1) Claim – a challenged statement; (2) Grounds – the premises, examples, data,

and facts used to support the claim; (3) Warrant – the (often implicit) logical connection

between the data and the claim; (4) Backing – an additional support, authority, and validation

to the warrant; (5) Qualifier – the certainty level of the claim (e.g., quite likely, presumably);

and (6) Rebuttal – circumstances under which the warrant is not valid (e.g., counterexamples

and special cases). When applied to mathematics education, several researchers modify the

argument by omitting or combining elements (e.g., Krummheuer, 1995; Prusak, Hershkowitz,

& Schwarz, 2012; Yackel, 2002), yet a growing number of authors argue for using the full

model (e.g., Conner et al., 2014; Inglis, Mejia-Ramos, & Simpson, 2007; Simpson, 2015).

Reasoning is closely related to argumentation, yet is more common in the mathematics

education literature. Reasoning has been defined by many authors. Some claim that the two

concepts – reasoning and argumentation – are one and the same when individual activities are

considered (e.g., Conner et al., 2014). Hanna (2014) defines it as the common human ability

to make inferences, Stylianides (2008a) regards it as a set of activities involved in the process

of sense-making and establishing mathematical knowledge, and Johnson-Laird (1999) as a

thought process that yields a conclusion from percepts and assertions.


The literature distinguishes several kinds of reasoning. Peirce (1878/1998) discerned three

types of reasoning according to their underlying process: (1) deduction of a result from a rule

and an occurrence of a case (e.g., "The angle sum of any [planar] triangle is 180o"+"T is a

triangle" → "The angle sum of T is 180o"); (2) induction of a rule from the occurrence of one

or more cases and a certain result, by means of probable deduction (i.e., the occurrences are

viewed as randomly selected, e.g., "T1, T2,… are arbitrary triangles" + "The angle sum of T1,

T2,… is 180o each" → "The angle sum of any triangle is 180o"); and (3) abduction – the

inference of a particular case from a general rule and a probable result of an application of the

rule to the case (i.e., similar to Sherlock Holmes's famous method of reasoning, e.g., "The

angle sum of any triangle is 180o"+"Hmm, the angle sum of T is 180o" → "T is a triangle").

Modern authors (Conner et al., 2014; English, 1998; Harel & Sowder, 1998, 2007; Knipping

& Reid, 2013; Yopp & Ely, 2016; Zazkis, Weber, & Mejía-Ramos, 2016) identify several

additional kinds of reasoning, among them are: generic example – specific examples are used

as representing a general class; empirical proof – specific examples are used but are not

representative; analogical reasoning – reliance on an analogy between two systems; and

external conviction – reliance on an external authority or on visual attributes of the argument.

Generic examples are regarded as a step that can help students generalize and achieve formal

abstraction (Harel & Tall, 1991)

Deductive reasoning is a kind of reasoning which pertains to the process of using rules and

syllogisms of formal logic to infer valid conclusions from known information. It is highly

regarded by many (Luria, 1976), especially in mathematics (M. Ayalon & Even, 2008;

Hanna, 1990; Johnson-Laird, 1999; Mariotti, 2006; Yackel & Hanna, 2003), whereas

empirical reasoning is typically predominant in other sciences.

Mathematical proof is a special type of deductive reasoning. Formal proof refers to a

sequence of propositions (i.e., axioms and results of rules of inference applied to previous

formulae in the sequence) terminating in the theorem that is proved. However, the

mathematician John Dawson (2006, 2015) considers these proofs to be "abstractions from

mathematical practice that fail to capture many important aspects of that practice" (2006, p.

270), and irrelevant to certain areas of mathematics that have not been formally based on

axioms. Placing rigor over logical formality, Dawson defines proof as "an informal argument

whose purpose is to convince those who endeavor to follow it that a certain mathematical

statement is true" (2006, p. 270). In recent years, other types of proof are gaining acceptance,

such as visual and non-verbal proof (e.g., Nelsen, 1993, 2000).


In the course of the past few decades, research has often explored the purpose of proof in

mathematics education (e.g., Bell, 1976; de Villiers, 1990; Hanna, 2000). Contemporary

views of the forms of proof (Dreyfus, Nardi, & Leikin, 2012) and its functions in

mathematics and in mathematics education (Hanna, 2000; Hanna et al., 2010; Kidron &

Dreyfus, 2009, 2010; Mejia-Ramos & Weber, 2014; NCTM, 2000; Weber & Mejia-Ramos,

2011) distinguish among a multitude of aspects and purposes of proofs: (1) establishing

certainty, (2) gaining understanding, (3) communicating ideas, (4) meeting an intellectual

challenge, (5) discovering new results, (6) presenting new methods, (7) exploring the

implications of a statement, (8) showing connections between different parts of mathematics,

and (9) constructing a greater mathematical theory.

Justification in mathematics is focused on what makes a certain proposition true. It can vary

in formality, from intuitive and informal to rigorous and deductive (e.g., Blum & Kirsch,

1991; Harel & Sowder, 2007; Miyazaki, 2000; Sierpinska, 1994). Regardless of formality,

justifications can have an explanatory role (i.e., focusing on why a proposition is true) and

contribute to the understanding of a proposition. Hanna (1990) differentiates between proofs

that prove and proofs that explain – both show that a statement is true, yet the former

provides only substantiation while the latter additionally provides a rationale showing why it

is true. On the other hand, Balacheff (2010) does not make this distinction and argues that

every proof starts as an explanation, which later undergoes a process of validation by the

appropriate community.

In this study, I use the terms justification and justification and explanation interchangeably.

They denote elements of language, be it spoken, written, or gestured, that can explain and/or

justify a mathematical statement, regardless of their formality. Such elements may include

formal proofs, empirical experimentations, proofs without words, and elements that serve

didactical purposes only (e.g., invoking students' intuition, or affirming the statement).

In the following section, I discuss several conceptual frameworks proposed in the literature to

discuss and classify justification.

2.1.1.2. Conceptual frameworks

Several classification systems for kinds of justification and explanation were suggested by

mathematics education researchers (e.g., Balacheff, 1988; Bell, 1976; Blum & Kirsch, 1991;

Branford, 1908; Harel & Sowder, 2007; Miyazaki, 2000; Sierpinska, 1994; Stacey &

Vincent, 2009). These frameworks establish levels of justification as based on different


dimensions and criteria, such as levels of formality (Blum & Kirsch, 1991; Branford, 1908),

strength of the deduction (Bell, 1976), and the nature of the reasoning involved (Balacheff,

1988; Harel & Sowder, 1998, 2007; Stacey & Vincent, 2009). Despite different viewpoints,

many frameworks agree on three classes of justification.

One such class is a valid proof, based on logical inferences and transformations, with or

without relying directly on axioms. This class has been denoted by many names over the

years, among them are scientific explanation (Branford, 1908; Sierpinska, 1994), deductive /

analytical proof scheme (Harel & Sowder, 1998, 2007), deduction using a general case

(Stacey & Vincent, 2009), conceptual proof (Balacheff, 1988), strong deduction (Bell, 1976),

proof A (Miyazaki, 2000), and formal proof (Blum & Kirsch, 1991). Some authors

distinguish generic examples from this class (e.g., Balacheff, 1988; Stacey & Vincent, 2009).

A second class is an empirical verification of the result, based on a finite number of special

cases. This class is denoted in the literature in many ways, such as experimental argument

(Blum & Kirsch, 1991; Branford, 1908), empirical proof scheme (Harel & Sowder, 1998,

2007), experimental demonstration (Stacey & Vincent, 2009), weak deduction (Bell, 1976),

proof C (Miyazaki, 2000), and pragmatic proof (Balacheff, 1988).

A third class is based on intuitive and non-formal arguments. This class is denoted as

intuitional proof (Branford, 1908), pre-formal proof (Blum & Kirsch, 1991), and appeal to

authority and qualitative analogy (Stacey & Vincent, 2009). Harel and Sowder (1998, 2007)

refer to it as an external conviction proof scheme, which includes reliance on the authority of

a book or of a person (e.g., a teacher or a mathematician), or on certain visual and ritualistic

attributes of the argument (e.g., a two column argument might be perceived as a proof).

Sierpinska (1994) suggested a distinction of a different nature, based on the target audience to

be convinced: (1) didactic – a justification which aims to provide a sense of familiarity in a

school setting (e.g., an example or a model); and (2) scientific – a proof sufficient to satisfy

the mathematical community.

The distinction between mathematical proof and other types of justification, made in the

conceptual frameworks mentioned above, reflects certain inherent differences between the

mathematical community and in mathematics education community. The following section

focuses on the comparison of justification in these two communities.


2.1.1.3. Justification in mathematics education and in mathematics

The literature identifies several differences in justification in mathematics education and in

mathematics, such as the nature of acceptable justification, its purposes, its process of

construction, and the topics in which justification is expected.

First, there is a difference in the very nature of justification acceptable in each community.

Traditionally, justification in mathematics is identified with a rigorous, logical, and deductive

proof (e.g., P. J. Davis & Hersh, 1981; Eves, 1972; Hanna, 2014; Sriraman & Umland, 2014;

Umland & Sriraman, 2014; Weber, 2008), whereas in mathematics education other types of

justification are acceptable. Moreover, these other types are valued and encouraged for

pedagogical purposes, such as catering to students' level of mathematical knowledge and

understanding (Ball & Bass, 2003; Reid, 2005; Sierpinska, 1994; Yackel & Hanna, 2003).

These differences might be related to a difference in maturation of cognitive abilities and

mental structures related to proof (Tall, 2014; Tall et al., 2012), different norms (Bass & Ball,

2014; A. Watson, 2008; Yackel & Cobb, 1996), and beliefs (Furinghetti & Morselli, 2011).

Second, justification serves somewhat different purposes and functions in each community.

As discussed above, proof serves a multitude of roles in mathematics, such as verification,

explanation, and discovery (e.g., de Villiers, 1990; Hanna et al., 2010). While some of these

functions exist in school mathematics, such as explanation and validation, most functions are

different (Cabassut, 2005; Zaslavsky, Nickerson, Stylianides, Kidron, & Winicki-Landman,

2012). For example, proving in order to discover new mathematical results is generally not

part of school mathematics (Herbst & Brach, 2006).

Furthermore, multiple justifications for the same mathematical statement serve different roles

in the two communities. For mathematicians, multiple proofs may serve practical, personal,

or aesthetic goals (Dawson, 2006). Such goals include remedying flaws in earlier arguments

or simplifying them, providing multiple perspectives, deepening understanding of concepts,

helping discover new techniques, connecting various mathematical concepts, demonstrating

the power of various methodologies, or presenting a particularly elegant and ingenious

strategy (Dawson, 2006; Rav, 1999; Siu, 2008). In mathematics education, multiple

justifications often serve pedagogical purposes, such as catering to students with different

levels of cognitive ability (Tall et al., 2012), demonstrating the power of dynamic geometry

software (Sigler, Segal, & Stupel, 2016), encouraging flexibility and creativity (Leikin, 2009,


2014), deepening understanding (Sierpinska, 1994), and developing an appreciation of the

value of justification, explanation, and proof in mathematics (Polya, 1945/1973).

Third, several studies focused on the processes of construction of justifications. These studies

suggest that while both learners (Bieda & Lepak, 2012; Kidron & Dreyfus, 2009) and

practicing mathematicians (Kidron & Dreyfus, 2010; Wilkerson-Jerde & Wilensky, 2011)

explore specific examples when making sense of a what they are required to justify, learners

tend to use examples unproductively (Bieda & Lepak, 2012). Moreover, many students rely

(incorrectly) on empirical evidence as justification for general assertions (Chazan, 1993;

Healy & Hoyles, 2000), even at the tertiary level (e.g., Selden & Selden, 2008).

Fourth, justification is expected for different topics in each community. Until recent years,

the teaching of deductive justification was limited to geometry, and algebra was viewed as a

domain suitable only for generalizing computational processes (Arcavi, Drijvers, & Stacey,

2017; Sfard, 1995). Near the end of the twentieth century, other goals for teaching

justification began emerging, such as an opportunity for students to experience the work of

doing mathematics (González & Herbst, 2006). Additionally, the introduction of modern

technologies allowed students to engage in a wider range of reasoning activities, such as

generating conjectures and investigating them, thus allowing students to better understand

mathematics and the need for a mathematical proof to justify claims (e.g., Dreyfus & Hadas,

1996; Tabach, Hershkowitz, & Dreyfus, 2013; Verzosa, Guzon, & De las peñas, 2014;

Yerushalmy & Chazan, 1987). Still, students see little value in justification (Healy & Hoyles,

2000), and there is a gap between the views of mathematicians and mathematics educators in

this respect (Weber, 2014).

2.1.1.4. Justification in the mathematics school curriculum

Contemporary school mathematics curricula worldwide attribute a central role to developing

students' ability to reason and prove at all grade levels and content domains (e.g., Australian

Education Council, 1991; Common Core State Standards Initiative, 2015; Department for

Education - England, 2014; Israel Ministry of Education, 2009; NCTM, 2000).

The Israeli mathematics school curriculum for grades 7-9 explicitly emphasizes justification,

explanation, and proof, as well as investigation, exploration, problem solving, and the

generation of mathematical conjectures (Israel Ministry of Education, 2009). These aspects

are emphasized in three parts of the curriculum document: (i) as its main highlights, (ii) as its

goals, and (iii) for each content domain.


The Israeli curriculum highlights the "development of mathematical ways of thinking

throughout the learning process: teaching not only facts and procedures, but also ways of

discovering mathematical phenomena, searching for ways to explain them, and finding

connections among them." (2009, p. 1). More specifically, the geometry curriculum involves

"an earlier introduction of the teaching of geometry, starting at the 7th grade," instead of at the

8th grade, which was the policy prior to the current curriculum, "in order to: "(1) ensure a

smooth transition between the elementary school curriculum and the junior high school

curriculum, and (2) prepare the students for a 'soft entrance' to deductive geometry studies"

(2009, p. 2). Furthermore, the geometry curriculum aims at "establishing and expanding the

knowledge of geometric facts, which are familiar from elementary school, learning

applicative aspects of geometry, and learning reasoning and conventions of a type that

comprises proofs, before learning the organized deductive structure." (2009, p. 2).

The goals of the Israeli curriculum explicitly stress explanation and justification for both

algebra and geometry: "Understanding the essence of algebra as a mathematical branch

dealing with processes of generalization, generating conjectures and justifying them;

Developing argumentative discourse: ways to explain or prove algebraic properties and rules;

… Giving explanation and proof for geometric properties" (2009, p. 3). In addition, the

Israeli curriculum aims to "connect and integrate algebra and geometry; for examples: by

giving an algebraic proof for a problem in geometry and vice versa" (p. 3).

In algebra, the Israeli curriculum emphasizes "thinking processes: constructing concepts and

their definitions, exploration and discovering phenomena, generating conjectures,

generalization and justification." (p. 6). It suggests linking algebra and geometry in proofs

which rely on the general case: "the general proof [for the congruence of vertically opposite

angles] shows an interesting combination between algebra and geometry." (p. 10)

In geometry, proof is described as "a key concept in deductive geometry." (p. 10), and the

curriculum aims to introduce it in a gradual process over three years, "through experimenting

with reasoning and informal justification of claims, understanding proofs, and eventually

even constructing proofs." The curriculum further suggests offering different types of

justification to different students, according to the students' level: "In teaching proofs there

are several steps towards writing a full proof in formal language, and it is necessary to adjust

the requirements [on formalism] to students' abilities." (p. 10). Specifically, "Phrasing the

rationale of the proof non-formally, compared with phrasing it in a mathematically rigorous

form, defines different levels of difficulty."


Other curricula worldwide have similar emphases and goals. For example, in the United

States, the Common Core emphasizes mathematical practices such as "reason abstractly and

quantitatively" (Common Core State Standards Initiative, 2010, p. 47). Similarly, the British

curriculum emphasizes that "teachers should develop pupils' numeracy and mathematical

reasoning in all subjects" (Department of Education, 2010, p. 9), to ensure that all pupils

"reason mathematically by following a line of enquiry, conjecturing relationships and

generalizations, and developing an argument, justification or proof using mathematical

language" (p. 37).

2.1.2. Perceptions about justification and explanation in mathematics education

Research has demonstrated repeatedly that students face fundamental difficulties with

activities that involve justification skills, and especially with proving (Balacheff, 1991; Bell,

1976; Chazan, 1993; Dreyfus, 1999; Harel & Sowder, 1998; Herbst & Brach, 2006;

Schoenfeld, 1985; Senk, 1985; Weber, 2001), even high level students (Healy & Hoyles,

2000; Selden & Selden, 2008) and pre-service teachers (Crespo & Nicol, 2006).

In order to better understand the role of justification in mathematics education, it is important

to examine what is known about the perceptions and beliefs of students and of teachers –

about mathematics and about the nature of mathematics teaching and learning.

2.1.2.1. Student perceptions

A large body of evidence regarding students’ performance with justification reveals

difficulties in reading or constructing formal justification, in understanding its nature and

purpose, in differentiating deductive justification from other types, and in deriving additional

results (e.g., Balacheff, 1988; Chazan, 1993; Harel & Sowder, 1998).

Studies on students' beliefs and conceptions about proof suggest that students hold

counterproductive beliefs regarding the nature of justification in mathematics (Harel &

Sowder, 1998; Martin & Harel, 1989; Senk, 1985). For example, many students view

empirical evidence as sufficient justification for a general claim, yet not as a valid proof

(Balacheff, 1988; Healy & Hoyles, 2000). This ability to determine whether an argument

suffices as proof might be related to age, or at least to experience with teachers’ expectations

(Bieda, Holden, & Knuth, 2006). It may also be connected to another student belief,

according to which an argument must follow a two-column structure to be a valid

mathematical proof (e.g., Harel & Sowder, 1998).


A common challenge for students in constructing justifications is deciding which theorems

and facts to apply at each step. Some students attempt symbol manipulation, which may

involve mathematically nonsensical operations (Harel & Sowder, 1998). Weber (2001, 2004)

suggested that students might not be familiar with powerful proof techniques and may not

know how to determine whether and when symbol manipulation might be applicable.

As a result, students often consider proof to be a meaningless ritual (Ball, Hoyles, Jahnke, &

Movshovitz-Hadar, 2002). Furthermore, this notion is strengthened by an underlying belief

regarding the certainty of mathematical knowledge and the absolute truth of mathematically

proven claims. These beliefs have been shaken several times in the history of mathematics

yet are still common: The introduction of non-Euclidean geometries shook the belief in a

single system of axioms, Russell's Paradox shook the belief in the infallibility of derivation

from axioms, and certain proofs of mathematical theorems (e.g., Andrew Weyl's proof of

Fermat's last theorem) were found to be inaccurate or incorrect (Rott, Leuders, & Stahl,

2015). Additionally, students do not see a reason for providing more than one justification for

a mathematical claim, and often believe that counterexamples for a general claim are merely

exceptions, and may exist alongside a general proof (Balacheff, 1988; Bell, 1976).

2.1.2.2. Teacher perceptions

Teachers play several roles in shaping students' opportunities to learn to explain and justify:

establishing a discourse in which students can justify their claims and evaluate claims made

by others (e.g., Sherin, 2002; Wood, 1999), introducing students to acceptable justification

strategies (Mariotti, 2006; Stylianides, 2008b), offering examples allowing pattern detection

(Bieda & Lepak, 2012; Zodik & Zaslavsky, 2008), and adapting written curriculum materials

(Ziebarth et al., 2009). Additionally, research suggests that teachers' beliefs about the role of

argumentation (M. Ayalon, 2011), their intentions (Nie et al., 2013) and gestures (Weinberg,

Fukawa-Connelly, & Wiesner, 2015) shape the nature and opportunities for justification.

Research suggests that while teachers recognize formal proofs, many hold limited views of

what constitutes mathematical proof and regard other types of justification (e.g., verbal,

visual, or using a generic example) as insufficient (Dreyfus, 2000; Knuth, 2002). Moreover,

studies suggest that teachers can name a variety of proof’s functions in mathematics, yet not

many recognized proof as a tool for learning mathematics (Knuth, 2002). This might suggest

that teachers focus on justification as a process of justifying, rather than the product of a

process (Furinghetti & Morselli, 2011). Even when viewed as a process, proofs can convey

mathematical methods, strategies, and techniques to students (Rav, 1999).


Studies suggest that in-service secondary school teachers focus less on the validity of a

justification and more on checking whether it contained redundant (albeit correct) statements

(Tsamir, Tirosh, Dreyfus, Barkai, & Tabach, 2009), and verifying the correctness of the

algebraic manipulations (Knuth, 2002). Even more so, teachers traditionally view

justification as representing high-order thinking, and therefore not suitable for students with

low achievements (Zohar, Degani, & Vaaknin, 2001). This was found to be related to a view

of learning as a hierarchical process which progresses from simple skills to more complex

ones. Zohar and Dori (2003) suggested that students with low achievements actually gain

much from learning higher order thinking skills. Moreover, a meta-analysis by Baker,

Gersten, and Lee (2002) suggests that students with low achievements gain most when

concepts, skills, and procedures are provided explicitly.

2.2. The role of textbooks in mathematics education

An increasing number of textbook studies have been conducted since the turn of the century,

most of which are textbook analysis and comparison (Fan, Zhu, & Miao, 2013). These studies

examined various aspects, such as national trends and educational approaches (e.g., Jones et

al., 2009; Xu, 2013), the content presented in textbooks (e.g., Sun & Kulm, 2010), the level

of the cognitive demand required by tasks (e.g., da Ponte, 2007; D. L. Jones & Tarr, 2007;

Niv, 2011), the nature of proof (e.g., Miyakawa, 2012, 2017), and opportunities offered in

textbooks for students to learn to justify (e.g., Dolev, 2011; Dolev & Even, 2013). Several

research studies have focused on variations in different grade levels, topics, and textbook

series (e.g., D. L. Jones & Tarr, 2007; Stylianides, 2009).

Accumulating research suggests that textbooks serve an important role in shaping students’

opportunities to learn mathematics (J. A. Newton, 2012; Stein, Remillard, & Smith, 2007).

Additional factors shape these opportunities, such as teacher and student characteristics,

beliefs, and expectations, as well as classroom norms and learning environments (e.g.,

Chazan, 2000; Tarr et al., 2008), yet not much is known about the interplay of these factors.

Fan et al. (2013) identify four categories of mathematics textbook studies based on their

focus: (i) the role of textbooks in teaching and learning mathematics, (ii) the ways in which

textbooks shape teaching and learning, (iii) analysis of features of mathematics textbooks and

(in some cases) comparison of textbook series, and (iv) miscellaneous. This section focuses

on two issues: (1) the importance and contribution of textbooks in mathematics education,

and (2) justification and explanation in mathematics textbooks.

Section 2.2 – The role of textbooks in mathematics education | 20

2.2.1. The importance of textbooks in mathematics education

Several models were suggested to identify and map levels of curriculum within an education

system (e.g., Goodlad, Tye, & Klein, 1979; Schmidt et al., 1996; Valverde, Bianchi, Wolfe,

Schmidt, & Houang, 2002). For example, Schmidt et al.'s (1996) model comprises three

levels – intended, implemented, and attained curricula. Studies focusing on the role of

textbooks, such as the Third Trends in International Mathematics and Science Study, suggest

that textbook constitute a fourth level (Valverde et al., 2002). This level mediates between the

intentions of curriculum documents designers and the implementation and enactment of

policy by teachers. Specifically, textbooks can translate abstract guidelines into activities and

content accessible to teachers and students (Howson, 2013).

Studies focusing on the ways in which textbooks shape teaching and learning indicate that the

influence of textbooks is considerable, for teachers and students alike (Eisenmann & Even,

2009, 2011; Haggarty & Pepin, 2002; Mesa & Griffiths, 2012; Remillard, 2009; Rezat, 2012;

Shield & Dole, 2013; Trgalová & Jahn, 2013; Usiskin, 2013). However, even though

mathematics textbooks are commonly addressed at students (Remillard, 2012), studies on

students' use of textbook are rare.

The literature suggests that textbooks are a major learning resource for students when solving

tasks in class and when completing homework problems (Mesa & Griffiths, 2012; Rezat,

2012). Analysis of the textbook sections used by students reveals that introductory texts were

infrequently read by students when solving homework assignments (Mesa & Griffiths, 2012),

yet they have the potential to change a teacher's lesson plans (Rezat, 2012).

Teachers use textbooks as one of the main sources for content and activities to include in

their lesson plans (Eisenmann & Even, 2009, 2011; Haggarty & Pepin, 2002), as well as for

teaching strategies (Fan, 2013; Fan et al., 2013; Son & Senk, 2010), sometimes with other

curriculum materials (Remillard, 2009). Moreover, textbooks influence how teachers portray

mathematical topics and implement their understanding of students’ learning trajectories in a

classroom (Valverde et al., 2002).

Together with the textbook, additional factors shape mathematics teaching and learning. For

example, national culture and school setting were revealed to influence the mathematics

available to students (e.g., number and type of examples) as well as student access to

textbooks (Haggarty & Pepin, 2002). Additionally, Rezat (2012) demonstrated how the

interaction between teacher and students actively shapes the enacted curriculum.


Studies have shown empirically that when using the same textbook, different teachers use it

in different manners and with different emphases (Eisenmann & Even, 2009, 2011; Even &

Kvatinsky, 2010; Thompson & Senk, 2014; Tirosh, Even, & Robinson, 1998). For example,

students are given different opportunities to learn to justify (Even & Kvatinsky, 2010;

Haggarty & Pepin, 2002; Knuth, 2002; Mesa & Griffiths, 2012) and thus even if

opportunities for justification are provided in a textbook, they might not become available to

students (Mesa & Griffiths, 2012; Thompson & Senk, 2014).

The research program Same Teacher – Different Classrooms, which focuses on teachers who

teach mathematics in more than one classroom while using the same textbook or syllabus,

was developed to study the interplay of factors shaping students' opportunities to learn

mathematics (Even, 2008, 2014). Studies that belong to this research program have revealed

detailed information about the role of the classroom and teacher characteristics in shaping

these opportunities (e.g., Ayalon & Even, 2015; Eisenmann & Even, 2009, 2011, Even &

Kvatinsky, 2009, 2010). These studies show differences in the mathematics taught in the

classrooms between classes of different teachers, between classes of the same teacher who

use the same textbook, and between mathematical topics.

In sharp contrast to the relative abundance of textbook studies, studies that focus on

justification were very rare in the previous century (Hanna & de Bruyn, 1999), and their

number has been slowly growing in the past decade.

2.2.2. Justification and explanation in mathematics textbooks

Textbook studies focusing on justification are not common in the mathematics education

literature. However, in recent years a growing number of studies have been conducted around

the world. Several studies centered on the notion of proof only, yet some examined

justification without restricting themselves to proof-related aspects. Generally, these studies

report a small number of opportunities for students to learn to explain and justify, especially

outside geometry. This number of opportunities varies among textbooks, but it is ultimately

low. However, research on textbook justification relies on several very different research

frameworks – conceptual and analytic. Therefore, comparison across studies is problematic.

This section briefly reviews these studies, their methodologies, and their main findings.

2.2.2.1. Frameworks for textbook analysis of justification

Much like a scaffold encloses a building and allows access to otherwise unreachable areas, a

research framework provides a basic structure of relevant features of a phenomenon and


relationships among them (Lester, 2005). Thus, it allows data interpretation, crystallization of

ideas, and conceptualization and design of research studies. I have identified in the literature

four research frameworks for textbook analysis of justification.

Stylianides (2008a, 2009) proposed the Reasoning-and-Proving analytic framework, which

emphasizes the integrated nature of reasoning and proof. The framework comprises two main

components. One involves making a generalization, by identifying a pattern and making a

conjecture. Another involves providing support to claims, either by providing a proof (i.e.,

creating a valid argument based on axioms, generic examples, and other truths taken as

shared by the community) or providing a non-proof argument (i.e., justifying a statement by

using either empirical reasoning or giving the general rationale).

Stylianides’s framework has directly influenced a growing number of textbook studies at all

grade levels, from elementary school level (Bieda et al., 2014; McCrory & Stylianides, 2014),

through middle school (Fujita & Jones, 2014) and to high school level (J. D. Davis et al.,

2014; Otten, Males, & Gilbertson, 2014). The following framework was developed in parallel

to Stylianides's, and "has many similarities to it" (Thompson et al., 2012, p. 258).

Thompson, Senk, and Johnson (2012) developed an analytic framework in order to identify

opportunities to learn reasoning and proof in high school mathematics textbooks. The

framework comprises two main components. One involves student activities such as making

and investigating conjectures, developing and evaluating arguments, and providing

counterexamples. The other discerns two types of textbook justification offered in

explanatory texts: a valid general proof and a generic example. If no justification is given, it

distinguishes between two cases: either it is explicitly left for the student to complete, or is

indeed missing. The framework was adapted and refined by several researchers (Bergwall &

Hemmi, 2017; Otten, Gilbertson, Males, & Clark, 2014; Otten, Males, et al., 2014).

Stacey and Vincent (2009) proposed a conceptual framework following a textbook analysis

study. Their conceptual framework refines Harel and Sowder's (2007) categories and

comprises seven modes of reasoning in textbook justifications. Two external conviction

types: (1) Appeal to authority – reliance on the authority of a person or of the textbook itself,

when no justification is given for a statement; and (2) Qualitative analogy – a surface

similarity between a claim and a non-mathematical situation (e.g., 'fruit salad algebra', in

which pronumerals are used to represent objects rather than numbers). Two empirical types:

(3) Concordance of a rule with a model - a rule and a model yield the same results for the


selected examples; and (4) Experimental demonstration – an observed regularity of results

obtained by using special cases. And three deductive types: (5) Deduction using a model – a

model which serves to illustrate a general claim (i.e., as opposed to specific cases); (6)

Deduction using a specific case – a generic example, given as a chain of logical deduction

conducted in terms of a special case; and (7) Deduction using a general case – either a chain

of reasoning backed by previously established knowledge, or a generic example followed by

a generalization of the process.

Ronda and Adler (2017) suggested an analytic framework aimed at the mathematics made

possible in a textbook lesson. Their framework extends the Mathematical Discourse in

Instruction (MDI) analytic tool, originally developed for classroom lesson analysis (Adler &

Ronda, 2015). One of its four components focuses on justification. It discerns three kinds of

Legitimations: (1) no justification is offered (appeal to authority), (2) specific examples are

given and used to explain a statement (substantiation by example), and (3) using

counterexamples and previously established definitions, principles, and procedures

(substantiation by general case). This framework draws from Stacey and Vincent's (2009)

framework and simplifies it "for a relatively simple categorization within each of our

elements of MDI" (Ronda & Adler, 2017, p. 9).

2.2.2.2. Methodologies for textbook analysis of justification

The limited research on justification in mathematics textbooks may be related to relatively

underdeveloped methodological techniques, compared with techniques for researching

classroom practice (Stylianides, 2014). Three methodological issues await researchers

coming to design a textbook analysis study: (1) Locating justifications, (2) Types of data

sources to use, and (3) Choosing perspective/s for analysis.

The first issue is related to the location of justifications. Justification is relevant across all age

groups, and researchers may focus on any group: elementary school (Bieda et al., 2014; D. P.

Newton & Newton, 2007), middle school (Dolev & Even, 2013; Stylianides, 2009), high

school (Nordström & Löfwall, 2005; Thompson et al., 2012), and tertiary school (J. D. Davis,

2009; McCrory & Stylianides, 2014).

In addition, justification is relevant across mathematical topics, which necessitates a choice of

which textbooks to analyze. Many focused on Geometry (Fujita & Jones, 2014; Miyakawa,

2012, 2017; Otten, Gilbertson, et al., 2014; Otten, Males, et al., 2014), others on Algebra (J.

D. Davis et al., 2014; Thompson et al., 2012), and yet others on a combination of several


topics, including algebra, geometry, functions, and/or trigonometry (Dolev & Even, 2013;

Hanna & de Bruyn, 1999; Stacey & Vincent, 2009).

Moreover, even after selecting mathematical topics for analysis, the question of which

textbook section to analyze arises. Generally, mathematics textbook chapters can be broken

into two distinct types of sections: (1) explanatory texts, intended for the entire class

population (e.g., to introduce, exemplify, and justify new concepts and results); and (2) task

pools, comprising activities intended for student work – individually or in small groups.

Some studies analyzing the explanatory texts in textbooks (e.g., Cabassut, 2005; Stacey &

Vincent, 2009); other studies focused only on analyzing the task pools (e.g., Dolev & Even,

2013; Stylianides, 2009); and yet other studies focused on both sections (e.g., Bergwall &

Hemmi, 2017; Hanna & de Bruyn, 1999; Otten, Males, et al., 2014; Thompson et al., 2012).

The second methodological issue relates to the types of data sources used for analysis. For

example, many textbook studies referred to the corresponding teacher’s guide (e.g., Bieda et

al., 2014; J. D. Davis et al., 2014; Dolev & Even, 2013; Fujita & Jones, 2014; Otten,

Gilbertson, et al., 2014). Consulting the teacher’s guide for data triangulation can shed light

on the goals for textbook items with information provided by the textbook’s authors. A small

number of studies combined textbook analysis with classroom observations and/or interviews

(e.g., Haggarty & Pepin, 2002; Sears & Chávez, 2014).

The third methodological issue relates to choosing perspective/s for phrasing a research

question. Many studies choose a student perspective and focus on opportunities offered for

students to learn to explain and justify (Dolev & Even, 2013; Stacey & Vincent, 2009;

Stylianides, 2009). Some chose a mathematical perspective, focusing on the mathematical

potential of textbook items (e.g., Stylianides, 2009). Other perspectives include a teacher

perspective and a textbook author perspective (Stylianides, 2014).

2.2.2.3. Findings from textbook analysis studies of justification

Two lines of research are common in studies which focus on the opportunities offered for

students in mathematics textbooks to learn to explain and justify: (1) studies focusing on

opportunities for students to engage in tasks that involve justification (e.g., Dolev & Even,

2013; Sidenvall, Lithner, & Jäder, 2015; Stylianides, 2009); and (2) studies focusing on

opportunities for students to read justifications in introductory sections in textbooks (e.g.,

Bergwall & Hemmi, 2017; Dolev, 2011; Hanna & de Bruyn, 1999; Stacey & Vincent, 2009).


Studies of the opportunities for students to engage in explaining and justifying in textbook

tasks have often revealed a very low ratio of such tasks, both in Israel (Dolev & Even, 2013)

and in other countries (J. D. Davis, 2012; Hanna & de Bruyn, 1999; Nordström & Löfwall,

2005; Stylianides, 2009; Thompson et al., 2012). For example, Stylianides (2009) analyzed

textbooks for grades 6-8 and found a low ratio of tasks calling for empirical arguments and

generic examples, together with a relatively high ratio of tasks calling for rationales – a type

of valid yet incomplete argument that does not qualify as proof.

The low ratio of justification-related tasks is evident at all age groups – from elementary

school (Bieda et al., 2014), through lower-secondary (Stylianides, 2009) and higher-

secondary schools (J. D. Davis, 2012; Hanna & de Bruyn, 1999; Nordström & Löfwall, 2005;

Thompson et al., 2012), and up to prospective teachers (McCrory & Stylianides, 2014).

Moreover, this low ratio is discernable for most mathematical topics, and especially outside

geometry (Bergwall & Hemmi, 2017; Dolev & Even, 2013; Hanna & de Bruyn, 1999). For

example, Dolev and Even (2013) analyzed six Israeli 7th grade textbooks (experimental

version) and found a low ratio of justification tasks in algebra and a moderately high ratio in

geometry. Furthermore, Davis (2012) examined the use of technological tools (e.g., graphing

calculators and Computer Algebra System) in textbook tasks and found that CAS was used

mainly for pattern identification but was not used for tasks that involve constructing an

argument. These findings are consistent with a traditional view of geometry as a domain

suitable for teaching students how to prove.

Studies of the opportunities for students to read justifications in student textbooks show that

mathematical claims are justified in various ways – deductive and empirical (Cabassut, 2005;

Dolev, 2011; Otten, Gilbertson, et al., 2014; Stacey & Vincent, 2009; Thompson et al., 2012).

However, the differences between the mathematical validity of these justifications are not

always made visible to students (Nordström & Löfwall, 2005). Moreover, some claims are

left unjustified or rely on non-justification strategies, such as the textbook authors’ authority

(Cabassut, 2005; Otten, Gilbertson, et al., 2014; Ronda & Adler, 2017; Stacey & Vincent,

2009). For example, Stacey and Vincent (2009) analyzed the justifications offered for seven

mathematical statements in nine 8th grade Australian textbooks. Their results indicate that

17% of the justifications were coded as external.

Studies that focused on proof-related reasoning revealed that deductive justifications, and in

particular valid proofs, are rare, especially outside geometry. This pattern is evident in


textbooks in many countries – such as Canada (Hanna & de Bruyn, 1999), the United States

(J. D. Davis et al., 2014; McCrory & Stylianides, 2014; Otten, Gilbertson, et al., 2014;

Thompson et al., 2012), Ireland (J. D. Davis, 2013), and Sweden (Bergwall & Hemmi, 2017)

– but not every country (e.g., Bergwall & Hemmi, 2017; Fujita & Jones, 2014).

Using a conceptual framework which refined Harel and Sowder's (2007) categories of proof

schemes used by students, Stacey and Vincent (2009) found that the textbooks employed

several types of justification when justifying mathematical statements. In some cases,

textbooks justified a statement using more than one type of justification or one type more

than once. Dolev (2011) used this framework to analyze the justifications offered for three

mathematical statements in Israeli 7th grade textbooks and obtained similar results.

This use of several justifications for one mathematical claim may reflect a complex

argumentation structure, in which several reasons supporting a standpoint are offered either

as alternative unrelated defenses or as a chain of mutually reinforcing reasons (Van Eemeren

& Grootendorst, 2004). Such structure is likely to have an additive effect (Sierpinska, 1994),

and could serve didactical and pedagogical purposes. For example, a properly constructed

sequence of arguments can prevent reasoning gaps by attending to both conceptual and

logical aspects (Triantafillou, Spiliotopoulou, & Potari, 2016), cater to students with different

levels of cognitive ability (Tall et al., 2012), reinforce and extend understanding even if the

justification is not considered sufficient in the mathematical community (Sierpinska, 1994),

and develop an appreciation of the value of justification, explanation, and proof in

mathematics (Polya, 1945/1973).

Furthermore, the inclusion of more than one justification for one statement indicates that in

addition to characterizing each textbook justification separately, it is important to attend also

to the Paths of Justification – the ways in which justifications of one statement are arranged,

structured, and sequenced. This aspect has received little attention in the literature. That is the

focus of this study. It examines the paths of justification offered in Israeli 7th grade textbooks

for key mathematical statements, and the contribution of the textbook, together with

additional factors, to shaping the paths of justification in the classroom.

27 | Research questions

3. Research questions

This study focuses on explanations and justifications in 7th grade mathematics textbooks, and

the role of the textbook in shaping the opportunities offered in 7th grade Israeli classrooms to

learn how to explain and justify.

As the literature review shows, textbook analysis is an emerging subject of research.

Research suggests that textbooks offer a variety of explanations and justifications for

mathematical statements. Moreover, studies reveal that some textbooks offer more than one

justification per mathematical statement. This use of several justifications for one statement is

likely to have an additive effect, such as serving a didactic goal of reinforcing and extending

students’ understanding. Therefore, it is important to attend not only to each instance of

justification as an independent unit, but also to the paths of justification, i.e., to the ways

justifications of one statement are arranged and structured – an aspect that receives little

attention in the literature.

Moreover, the literature review suggests that several factors (e.g., teachers, students, and

mathematical topics) are involved in shaping the opportunities offered for students to learn

how to explain and justify mathematical statements in mathematics classrooms (e.g., M.

Ayalon & Even, 2013; Even & Kvatinsky, 2010).

The study builds on these results, in order to characterize the opportunities offered in 7th

grade Israeli textbooks for students to learn how to explain and justify, and the ways in which

the textbook, in conjunction with the teacher and the students, shape these opportunities.

Two research questions were derived from these goals. The first research question focuses on

mathematics textbooks, and the second focuses on the contribution of the textbook to learning

to justify in class, alongside other factors – the teacher and the students. Each question

comprises two parts:

1. What opportunities to learn how to explain and justify mathematical statements are

offered to students in Israeli 7th grade mathematics textbooks?

a) What characterizes the instances of justification for mathematical statements in 7th

grade mathematics textbooks?

b) What characterizes the paths of justification for mathematical statements in 7th

grade mathematics textbooks?

Research questions | 28

2. How does the textbook, together with the teacher and the students, shape students'

opportunities to learn how to explain and justify mathematical statements in 7th grade

mathematics classrooms?

a) How does the textbook, together with the teacher and the students, shape the

instances of justification for mathematical statements in 7th grade mathematics

classrooms?

b) How does the textbook, together with the teacher and the students, shape the paths

of justification for mathematical statements in 7th grade mathematics classrooms?

29 | Methodology

4. Methodology

4.1. Part I: Justifications and explanations in Israeli 7th grade textbooks

In this section I present the methodology related to my first research question. The question

focuses on the opportunities offered in 7th grade Israeli mathematics textbooks for students to

learn how to explain and justify mathematical statements. First, I describe the sample

selection of textbooks and mathematical statements. Then I describe the data sources. Last, I

describe the methods of analysis.

4.1.1. Sample selection

Analysis included all eight approved Israeli 7th grade mathematics textbooks for Hebrew

speakers (at time of analysis). Table 1 summarizes general characteristics of the textbooks.

Each textbook is split into three volumes. The first volume is intended for all students, and

the other two volumes are either of standard/expanded scope, intended for the general student

population (six textbooks, labelled A-F in this study), or of limited scope, intended for

students with low achievements (two textbooks, labelled G-H). Two pairs of textbooks have a

common first volume – Textbooks B and G, and Textbooks C and H.

Three textbooks are published by commercial publishers: Textbook A (Luzon, Amoyal,

Cooperman, Bamberger, & Ginsburg, 2012), Textbook E (Yekuel & Bloomenkrantz, 2012),

and Textbook F (Shalev & Ozeri, 2012). One textbook is published by a non-profit

organization – Textbook D (Hershkovitz & Gilad, 2012). Four textbooks are published by

academic institutions: Textbook B (Zaslavsky et al., 2012), Textbook C (Ozrusso-Hagiag et

al., 2012), Textbook G (T. Ayalon, 2012), and Textbook H (Bouhadana et al., 2014).

Table 1. General characteristics of the analysed textbooks

Label Textbook Publishing Press Pages Textbook scope

(A) 10 BaRibua Commercial 665 Standard/Expanded

(B) Efshar Gam Aheret Academia 708 Standard/Expanded

(C) Mathematica Meshulevet: Blue Academia 639 Standard/Expanded

(D) Mathematica LeHatab Non-profit organization 704 Standard/Expanded

(E) Mathematica LeKita 7 Commercial 777 Standard/Expanded

(F) Zameret – Mathematica LeKita 7 Commercial 693 Standard/Expanded

(G) Kfiza LaGova Academia 654 Limited

(H) Mathematica Meshulevet: Green Academia 672 Limited

Section 4.1 – Part I: Justifications and explanations in Israeli 7th grade textbooks | 30

Ten key statements were selected for analysis from the Israeli 7th grade school mathematics

curriculum, five in algebra and five in geometry. Each selected statement is considered to be

central in the Israeli curriculum, and has received attention in the mathematics education

literature (see Chapter 6 – Justification strategies ). Statements were selected both in algebra

and in geometry in light of the historic bias in school mathematics, in which proof is reserved

for geometry statements. This was done in order to examine whether different justification is

offered for statements in different topics. The formulation of each statement was similar

across the analyzed textbooks. The 10 statements, presented by order of appearance in the

Israeli curriculum, are:

Algebra:

1. Two algebraic expressions are equivalent if one expression can be transformed into the

other by performing a sequence of valid operations for a common domain of numbers.

2. Division by zero is undefined.

3. The distributive property: a*(b + c) = ab + ac for any three numbers a, b, c.

4. Performing valid operations on both sides of an equation yields an equivalent equation.

5. The product of two negative numbers is a positive number.

Geometry:

6. The area formula for a trapezium with bases a, b and altitude h is (a + b)*h/2.

7. The area formula for a disk with radius r is πr2.

8. Vertically opposite angles are congruent.

9. The corresponding angles between parallel lines are equal.

10. The angle sum of a triangle is 180o.

The term mathematical statement is used rather than claim because two statements (i.e.,

division by zero and product of negatives) are historically regarded as mathematical

conventions, defined in order to ensure that certain principles (e.g., the distributive property)

remain consistent upon extending the concept of number beyond the set of natural numbers.

4.1.2. Data sources

For each statement, the data sources included the textbook chapters introducing it – a total of

889 textbook pages (3-58 pages per statement per textbook). I analysed each chapter

attending both to the introductory sections (e.g., introductory activities, narrative blocks,

definitions, and worked examples) and the related collections of tasks. I believe that analysis

of both sections is necessary to allow a complete and coherent picture of the opportunities to

learn how to explain and justify. Task pools may involve the students actively in providing

explanation and justification, and introductory sections involve multiple opportunities, such

as reading the textbook authors’ justifications and engaging in teacher-mediated justification.

31 | Methodology

Each chapter was analysed exhaustively, in order to identify every instance of justification

and fully characterize the path of justification offered in the textbook for that mathematical

statement. In addition, I analysed the relevant sections in each textbook's teacher's guide in

order to better understand and interpret the justifications offered in the textbooks.

4.1.3. Data analysis

Approximately 80% of the data were coded by 1-4 additional researchers, all familiar with

the conceptual framework used in this study (Krippendorf’s alpha was 0.79). The coding was

discussed among the coders in one of two ways: a) each member coded a path of justification

separately (46% of the paths), or b) I presented my coding of a path of justification to my

colleagues (33% of the paths). For each path, one code was consensually decided.

Analysis comprised four stages:

1. Identifying instances of justification for each statement, in each textbook.

(i) Segmenting each chapter into blocks (e.g., narrative blocks and introductory activities,

following Valverde, Bianchi, Wolfe, Schmidt, & Houang, 2002), and parsing each block

into separate elements (e.g., individual activities and text boxes).

(ii) Analysing the teaching aim of each element, with help from the teacher's guide.

(iii) Compiling a list of elements that explain and/or justify the mathematical statement –

including elements that may serve didactical purposes only (e.g., invoking students'

intuition, or affirming the statement) and elements that may serve as a precursor for

conjecture (e.g., an activity that aims at an identification of a pattern). If none were

offered in the textbook for a certain statement, it was coded as ‘no justification’.

Figure 1 illustrates two instances of justification for area of a trapezium, which were offered

in the introductory sections in Textbook B.

(a) (b)

Figure 1. Instances of justification – area of a trapezium (Textbook B, v.2, pp. 195-196)

Section 4.1 – Part I: Justifications and explanations in Israeli 7th grade textbooks | 32

2. Coding each instance of justification for two attributes: (1) the justification strategy, and

(2) the justification types (following the conceptual framework by Stacey & Vincent,

2009). In the following I describe each attribute.

The first attribute, the justification strategy, focuses on a content-specific characteristic

of each item. This characteristic deals with the warrants offered in justification of each

mathematical statement (following the argument model in Toulmin, 1958/2003). For

each item, I deconstructed it, identified its underlying strategy, and searched the

mathematics education literature for a mention of a similar warrant.

The second attribute, the type of justification, focuses on a meta-level characteristic of

each item. In my analysis of the types of justification I relied on Stacey and Vincent's

(2009) conceptual framework. I chose it due to its robustness and its proven usefulness in

analysing Israeli textbook justifications (e.g., Dolev, 2011). Table 2 lists the types of

justification, grouped into three categories: Deductive, Empirical, and External

justification (following Harel & Sowder, 2007). In addition, it presents the definitions I

used in this study. A more detailed presentation is given in Section 5.1 – The types of

justification offered.

I relied on the original definitions, with one exception. Stacey and Vincent defined two

variants of appeal to authority – either a reliance on an external figure of authority (e.g.,

mathematicians, a teacher, or a calculator) or no justification was given (i.e., appealing to

the authority of the textbook). I defined appeal to authority strictly as a reliance on an

Table 2. Types of justification, by categories (adapted from: Stacey & Vincent, 2009).

Type of justification Definition in this study

External

Appeal to authority Reliance on external sources of authority.

Qualitative analogy A surface similarity to non-mathematical situations.

Empirical

Experimental demonstration A pattern formed after checking specific examples.

Concordance of a rule with a model Matching specific results of a rule and a model.

Deductive

Deduction using a model A model illustrating a mathematical structure.

Deduction using a specific case An inference process relating to a generic example.

Deduction using a general case An inference process relating to the general case.

33 | Methodology

external figure of authority, because I believe that the absence of justification does not

necessarily imply a reliance on the textbook's authority. For example, it might imply an

assumption that students are already familiar with the justification from previous years.

The item in Figure 1(a) was coded as a deduction using a specific case, and the item in Figure

1(b) was coded as a deduction using a general case. Both items were coded as relying on the

same justification strategy – Dissection (i.e., representing the area of a given trapezium as a

sum of areas of shapes, each with an area formula known to the students).

3. Visually representing the paths of justification for each statement, in each textbook (80

paths in total). Figure 2 presents the paths for area of a trapezium in Textbooks B and F.

Each step represents a single instance of justification by order of appearance in the

textbook, and both the justification strategy and type are presented.

For example, the path in Textbook B comprised five items. The justification strategy in

the first three items was dissection, relying on three types of justification: experimental

demonstration, deduction using a specific case, and then deduction using a general case.

The last two items relied on deduction using a general case, by using two justification

strategies: Dissection and Construction (i.e., arranging several congruent trapeziums

such that they form a shape with an area formula known to the students).

Textbook Paths of justification

B

→

→

→

→

F

→

→

→

→

e=experimental demonstration; s/g=deduction using a specific/general case.

Figure 2. Paths of justification for the area of a trapezium (in Textbooks B and F)

4. Performing comparative analyses, by textbook and by mathematical statement, of the paths

of justification. Analysis focused on two aspects: (i) the characteristics of each instance

of justification (i.e., justification strategies and types), and (ii) the order of instances in

the textbook. Special attention was given to comparison between topics (i.e., geometry

and algebra, in light of the historic bias towards justification in geometry) and

comparison between textbook scopes (i.e., limited and standard / expanded, in light of

the accumulated body of research regarding the nature of the opportunities offered to

students with low achievements).

Dissection

e

Dissection

s

Dissection

g

Construction

g

Dissection

g

Dissection

e

Dissection

e

Dissection

e

Dissection

g

Construction

g

Section 4.2 – Part II: Textbooks, teachers, and students | 34

4.2. Part II: The contribution of the textbook, teacher, and students to

shaping classroom justifications and explanations

In this section I present the methodology related to my second research question. The

question focuses on the ways in which textbooks, in conjunction with the teacher and the

students, shape students' opportunities to learn how to explain and justify mathematical

statements. First, I describe the research design and participants. Then I describe the data

sources. Last, I describe the methods of analysis.

4.2.1. Research design and participants

Part II of the study utilized the "Same teacher – different classrooms" research design (Even,

2008, 2014): two case studies, each focused on a mathematics teacher who uses the same

textbook in two 7th grade classes (see Figure 3). Several studies using this design were

conducted by Ruhama Even's research group (M. Ayalon & Even, 2015, 2016, Eisenmann &

Even, 2009, 2011, Even & Kvatinsky, 2009, 2010). These studies consistently show different

opportunities to learn mathematics in different classrooms of the same teacher, even in cases

in which the same curriculum materials were used. This design was chosen for the current

study because it was shown to provide valuable data about the interplay between the teacher,

the students, and the curriculum materials used in the classroom. Specifically, it has the

potential to help better understand the contribution of the textbook in each classroom.

Figure 3. The research design for part II of this study

I started recruiting teachers in July 2015. My criteria were: 7th grade mathematics teachers

who use the same textbook in more than one classroom. By December 2015 I recruited two

teachers, both using Textbook C. In the following I describe each teacher and her classes.

35 | Methodology

4.2.1.1. Lena and her classes

Lena (pseudonym) received a B.Ed. from a teachers' college, and majored in mathematics. At

the beginning of the year of data collection she had five years of experience teaching

mathematics, all at the junior-high school level.

The school was a secular elementary school (grades 1-8), whose students came mostly from

four neighboring communal or cooperative settlements. The year of data collection was

Lena's first year teaching at this school. The school was categorized by the Israeli Ministry of

Education to be in the upper 30th percentile of the Socio-Economic Status index (SES). There

were two 7th grade classes in the school.

Lena taught both 7th grade classes: Class L1 with 29 students and Class L2 with 24.

Observations suggest that the student ability level in both classes was diverse, and there were

frequent disciplinary issues. Once a week, a supplementary teacher aided Lena during

geometry lessons by taking twelve advanced students out of the classroom and teaching them

in parallel. The lessons given by the supplementary teacher were not documented due to

research constraints.

On average, roughly 34% of lesson time in Lena's classes was dedicated to whole-class work

and the rest to student-work (see Table 3). During the time allotted for student-work, Lena

commonly attended to individual students or to small group of students.

The year of data collection was the first year Lena had used Textbook C. In the concluding

teacher interview Lena conveyed that she planned her lessons by reviewing Textbook C and

its teacher's guide: "I read through the teacher's guide in order to see what, to first see 'why',

how the lesson opens, what the aim of that opening is, and then I read all the explanations on

the side [e.g., relating to the level of difficulty of each task]". However, Lena revealed that

she thinks the textbook was unsuitable at times for students with low achievements: "I felt

that my class was not at its level, because it's a textbook that attends to students that are

already at a certain level". Indeed, Lena occasionally used other textbooks as a source of

introductory exercises and activities.


4.2.1.2. Millie and her classes

Millie (pseudonym) received a college bachelor's degree in business management, and later

received her teaching credentials in mathematics from a teachers' college, in a teaching

certification program for those holding an academic degree. At the beginning of the year of

data collection Millie had three years of experience teaching math, all at the junior-high level.

The school was a secular junior-high school (grades 7-9), located in a town. The year of data

collection was Millie's fourth year teaching at this school. The school was categorized by the

Israeli Ministry of Education to be in the upper 30th percentile of the Socio-Economic Status

index (SES). When observations began in December 2015, there were eight 7th grade classes

in the school and two classes for lower-track students (Mitzuy). Each lower-track class

comprised students from four classrooms. In late March 2016, the lower-track classes were

disassembled, and students rejoined their original classes.

Millie taught two 7th grade classes: When observations began, Class M1 with 32 students

(main stream), and Class M2 with 20 students (lower-track). After the lower-track classes

were disassembled: Class M1 with 35 students (32 stayed from the original M1), and Class

M2 with 30 students (5 stayed from the original M2). Observations suggest that most of

Millie's students actively participated in the classroom and there were very few disciplinary

issues. On average, roughly 86% of lesson time was dedicated to whole-class work and the

rest to student work (see Table 3).

The year of data collection was the fourth year Millie had used Textbook C. In the concluding

teacher interview Millie conveyed that she liked the way the textbook presented mathematics

to students by using real-life examples, and that she relied mostly on it when planning her

lessons: "I might use other textbooks, like Mishbetzet [i.e., Textbook E] for additional

exercises, but not for lesson planning – for the lesson, I [used] solely the textbook. I really,

really love it".

Table 3. Duration of observed lesson sections by classroom (in minutes and percentages)

Section Class Total

L1 L2 M1 M2 Lena Millie

Whole-class work 168 (40%) 140 (29%) 403 (85%) 507 (88%) 308 (34%) 469 (86%)

Student work 250 (60%) 347 (71%) 83 (15%) 69 (12%) 597 (66%) 152 (14%)

Total (100%) 418 487 544 576 905 1120

37 | Methodology

4.2.2. Data sources

Data sources included lesson observations and teacher interviews.

Data from lessons introducing each of the selected mathematical statements were recorded via

videotaped observations, audio recordings, and field notes, taking the role of a non-participant

observer (Sabar Ben-Yehoshua, 2001). A single video camera was used. The camera was

operated by the researcher and was positioned such that it will record the whiteboard and the

teacher for the entire duration of the lesson. An audio recorder was placed on the teacher's

desk, in order to improve the voice recording quality of both the teacher and the students.

Student individual work was recorded intermittently in field notes.

Due to research constraints, only three mathematical statements (out of the 10 analyzed in

Part I of the study) could be observed in all four classes. Four additional statements were

observed in at least two classes (of the same teacher), a total of 49 lesson observations (11-14

per class, lesson unit length is 45 minutes, see Table 4).

Table 4. Number of lesson observations, by classroom and mathematical statement

Mathematical statement Class

L1 L2 M1 M2

Performing valid operations on an equation yields an equivalent equation. 4 4 4 4

The area formula for a disk with radius r is πr2. 1 2 4 4

Vertically opposite angles are congruent. 2 2 1 2

Additional observations:

The product of two negative numbers is a positive number. 0 0 2 2

The corresponding angles between parallel lines are equal. 0 0 1 2

The angle sum of a triangle is 180o. 2 2 0 0

The area formula for a trapezium with bases a,b and altitude h is (a + b).h/2. 2 2 0(*) 0(*)

Total 11 12 12 14

(*)This mathematical statement was analyzed to a limited extent, based on the documents Millie's

students received as part of a remote learning activity.

Semi-structured interviews were held during the summer vacation following the final

classroom observations and recorded via audio recordings (see Appendix for a copy of the

interview questions, in Hebrew). The interview questions were developed in collaboration

with 1-3 additional colleagues, and the aims of each section and of each question were

discussed. Additionally, a pilot interview was conducted. Following it, slight phrasing

adjustments were made in order to make certain items sound more natural.


The interviews dealt with the ways in which each teacher perceived aspects of explanation

and justification, and her ways of addressing these aspects in her two classes. The interview

questions comprised three sections: (i) background, (ii) teacher-textbook-classroom

dynamics, and (iii) paths of justification. In the following I describe each part.

The first section focused on the teacher's background. The questions dealt with the teacher's

academic studies, teaching experience, and motivation for becoming a teacher.

The second section focused on the teacher-textbook-classroom dynamics. The questions dealt

with the teacher's perspectives on the similarities and differences between her two classes, her

use of the textbook and of other resources when she planned lessons (e.g., colleagues, the

internet, or other textbooks), and the ways in which teaching a lesson in one class influenced

the planned lesson in her other classroom.

The third and last section focused on the paths of justification. The section started by

discussing the teacher's approach to explanation and justification in the seventh grade. The

questions dealt with whether, why, and when it is necessary to explain why mathematical

statements are true. In addition, the questions involved the issue of justification for topics that

were taught in elementary school, and whether justification in seventh grade should be the

same as in higher grades.

Then, the teachers were shown sets of paths of justifications for three statements – vertical

angles, product of negatives, and area of a disk. A set of three paths per mathematical

statements was shown. Each set illustrated existing paths, either in the analyzed textbooks or

in the observed classes (see a set of three paths for product of negatives in Figure 4). The sets

were introduced as illustrating paths of justification suggested by other teachers who

participated in the study – each row in the figure represents a full path of justification by one

teacher, and is independent from the other rows. The teachers were asked to describe what

they like and dislike about each path, to construct an "ideal path of justification" for that

statement, and to describe how such a path might fare in a classroom.

39 | Methodology

Figure 4. A set of paths of justification for product of negatives (Interview item)

4.2.3. Data analysis

Approximately 75% of the data were coded by 1-4 additional researchers, all familiar with the

conceptual framework used in this study (Krippendorf’s alpha was 0.86). The coding was

discussed among the coders in one of two ways: a) each member coded a path of justification

separately (35% of the paths), or b) I presented my coding of a path of justification to my

colleagues (40% of the paths). For each path, one code was consensually decided.

Analysis was based on the methods described in Part I and comprised five stages:

1. Outlining the teaching sequence in each observed lesson.

(i) Watching the video recording of each classroom observation several times.

(ii) Segmenting each lesson into blocks (e.g., general assembly, student work), and

parsing each block into separate sections based on the topic at hand.

(iii) Transcribing the parts of each lesson dealing with the mathematical statement, and

translating the parts of the transcript that are cited in this dissertation.


Figure 5 shows an outline of the teaching sequence for area of a trapezium in Lena's classes. It

describes the main sections of the lesson, along with graphic representations of the duration

of each section in each class.

Class L1 Class L2 Activity (student grouping)

9 minutes

9 minutes

Administration

Non-academic activities (e.g., management, announcements, discipline)

6 minutes

5 minutes

Recap and Defining a trapezium (Whole class)

Lesson goal – finding the area of a trapezium by using known area formulae.

Recap of the area formulae for rectangle, square, triangle, and parallelogram.

Defining a trapezium as a quadrilateral with two parallel sides.

–

3 minutes

Trapezium dissections (Whole class)

In L2: Lena asks for ways to dissect a trapezium into the known shapes.

Several options are suggested [ , , , ], and Lena discusses the

potential efficiency of the dissections.

10 minutes

10 minutes

Calculating the area of a trapezium numerically (Whole class)

Lena sketches a trapezium dissected into a rectangle and two triangles, and

provides the measures for each segment. Lena instructs the class to calculate

the area of each part separately and add them together.

21 minutes

23 minutes

Calculating the area of a trapezium (Individual student work)

Students work on textbook tasks: calculating the area of a trapezium by using

known area formulae, and identifying trapezia by their definition.

9 minutes

11 minutes

Justifying the area formula for a trapezium (Whole class)

In L2: Lena relies on the intermediate results achieved earlier by the students

and begins89 min a process of rearranging these results in order to derive the

area formula. However, this process is not completed, and instead Lena writes

the general area formula.

In both classes: Lena then draws a diagonal to dissect a given trapezium into

two triangles, and obtains the general area formula: altitude*(short base + long

base)/2.

26 minutes

28 minutes


Students continue working on textbook tasks by using the area formula.

81 minutes 89 minutes

Figure 5. Outline of the teaching sequence for area of a trapezium in Lena's classes

2. Identifying instances of justification for each statement, in each classroom. I Compiled a

list of elements that explain and/or justify the mathematical statement – including

41 | Methodology

elements that may serve didactical purposes only (e.g., offering mnemonics, invoking

students' intuition, or affirming the statement) and elements that may serve as a precursor

for conjecture (e.g., an activity that aims at an identification of a pattern). Figure 6 and

Figure 7 present two instances for area of a trapezium, both in Lena's classes.

Figure 6. Experimental demonstration by Dissection (Screenshot in Class L1)

Figure 7. Concordance of a rule with a model by Dissection (Screenshot in Class L2)

3. Coding each instance of justification for two attributes: the justification strategy and the

justification type (following Stacey & Vincent, 2009). For example, the justification

strategy underlying both the item in Figure 6 and the item in Figure 7 was Dissection.

However, the item in Figure 6 was coded as an experimental demonstration, whereas the

item in Figure 7 was coded as a Concordance of a rule with a model.

4. Visually representing the paths of justification for each statement, in each observed

classroom (22 paths in total). This stage of analysis is identical to the stage described in

Part I of this study.


5. Comparative analyses of two aspects of the paths of justification: (1) the characteristics of

the instances of justification (i.e., justification strategies and types) and (2) the order of

the instances of justification. In order to address the textbook-teacher-classroom

dynamics, two comparisons were made: (i) between classes of the same teacher, and (ii)

between Textbook C and the classes of a single teacher. Due to research constrains,

comparison between teachers involved limiting the analysis to the three mathematical

statements that were observed in all four classes: Equivalent equations, Area of a disk,

and Vertical angles. The other two comparisons were made by using the entire data pool.

In the following chapters I present my results.

Chapter 5 and Chapter 6 focus on results related to my first research question, dealing with

the opportunities offered to students in 7th grade Israeli mathematics textbooks to learn how to

explain and justify mathematical statements. More specifically, Chapter 5 focuses on types of

justification, and Chapter 6 focuses on justification strategies.

Chapter 7 and Chapter 8 focus on results related to my second research question, dealing with

the ways in which the textbook, together with the teacher and the students, shape these

opportunities in the classroom. More specifically, Chapter 7 focuses on Lena's classes, and

Chapter 8 focuses on Millie's classes.

In Chapter 9 I discuss my findings and their implications.

43 | Types of justification in the textbooks

5. Types of justification in the textbooks

In this chapter I focus on a meta-level characteristic of the justifications offered in the

textbooks – the types of justification. I address four aspects regarding the types of

justification in the analyzed textbooks: (1) The types of justification offered, (2) comparison

of the types of justification across textbooks, (3) comparison of the types of justification

across mathematical statements, (4) The paths of justification in the textbook, focusing on the

types of justification.

Analysis reveals that Israeli 7th grade mathematics textbooks provided justifications for all

analysed statements – all but one statement in one textbook. A total of 183 instances of

justification were found for the ten analysed mathematical statements.

Table 5 presents the frequencies of instances of justification by textbook section. Analysis

reveals that justifications for the analysed statements were typically included in the

introductory sections and in tasks intended for a general-assembly class discussion (89%),

and seldom in tasks intended for student individual or small-group work. However, both

textbook sections offered many opportunities for justification of other mathematical claims,

which were not analyzed in this study. A similar pattern was found in all textbooks (except

textbook A), regardless of the target student population, and across all analyzed statements.

Table 5. Frequencies of instances of justifications, by textbook section

Textbook Section Textbook Total

A B C D E F G H

Introductory sections 19 20 21 18 22 26 18 19 163

Student work 12 2 0 6 0 0 0 0 20

Total 31 22 21 24 22 26 18 19 183

Section 5.1 – The types of justification offered | 44

5.1. The types of justification offered

Six out of the seven types of justification in Stacey and Vincent’s (2009) framework were

identified in the Israeli textbooks – all but Qualitative analogy. In the following I provide

examples for each type.

5.1.1. Appeal to authority: This type of justification was defined as a reliance on external

sources of authority (e.g., a mathematician or a calculator). Figure 8 illustrates an instance of

this type in justification of the product of negatives. Students calculated an assortment of

products of directed numbers by using a calculator, examined the results for patterns, and

summarized their observations. The justification relies both on the students' ability to find a

pattern based on a limited number of specific examples and on the authority of the calculator

in order to determine that the product of two negative numbers is positive. However, I coded

this as an appeal to authority because of the reliance on the calculator to carry out the

multiplications of negative numbers.

Figure 8. Appeal to authority (adapted from Textbook G, vol 2, pp. 61-62)

5.1.2. Qualitative analogy: This type of justification was defined as an analogy that relies on

a superficial similarity between a mathematical concept and a non-mathematical situation.

This analogy does not reflect the underlying mathematical principle and cannot qualify as a

model. No instances of this type of justification were found in the textbook.


5.1.3. Experimental Demonstration: This type of justification was defined as a pattern that

emerges after checking either one or more specific examples. Figure 9 illustrates an instance

of this type in justification of the angle sum of a triangle. The students were instructed to cut

triangles out of a piece of paper, tear these triangles such that each piece contains one angle

of the triangle, and rearrange the three angles, in order to convince themselves that the angle

sum in a triangle is a straight angle (see Figure 9).

Figure 9. Experimental demontration (Textbook B, vol 3, p.161)

5.1.4. Concordance of a rule with a model: This type of justification was defined as a result

of a comparison of the results obtained in two ways – by using a rule and by using a model.

Figure 10 illustrates an instance of this type in justification of the area of a trapezium. The

areas of several trapeziums were computed in two ways – with and without using the formula

that was previously derived. The essence of the justification is in the correspondence of the

answers obtained in two ways – by using a visual model and by using the formula (the rule).

Figure 10. Concordance of a rule with a model (Textbook D, vol 2, p. 186)

Section 5.1 – The types of justification offered | 46

5.1.5. Deduction using a model: This type of justification was defined as a model which is

used to illustrate the mathematical structure underlying the justification. Figure 11 illustrates

an instance of this type in justification of equivalent equations. A given equation of the form

Ax+B=Cx+D was translated into a balance scale model, and pairs of objects of either known

or unknown weights were repeatedly removed from both sides of the scales, thus maintaining

equilibrium. This justification is based on a structural similarity between the scales (the

model) and the balanced equations (the mathematics involved).

Figure 11. Deduction using a model (Textbook H, vol 3, p. 55)

5.1.6. Deduction using a specific case: This type of justification was defined as a process of

inference which is based on a generic example. Figure 12 illustrates an instance of this type

in justification of the area of a trapezium. The area formula of a trapezium was justified by

forming a chain of reasoning, in which each step was logically deduced from previous steps.

The given measures were intended as a generic case (i.e., the specific given values can be

replaced without loss of generality).

Figure 12. Deduction using a specific case (Textbook B, vol 2, p. 195)


5.1.7. Deduction using a general case: This type of justification was defined as a process of

inference which is based on the general case. Figure 13 illustrates an instance of this type in

justification of the area of a trapezium. The area formula of a trapezium was justified by

forming a chain of reasoning, in which each step was logically deduced from previous steps.

Pronumerals were used to denote the measures of the bases of the trapezium and its altitude.

Figure 13. Deduction using a general case (Textbook B, vol 2, p. 196)

Section 5.2 – Types of justification across textbooks | 48

5.2. Types of justification across textbooks

Figure 14 presents the relative frequencies of the types of justification in the textbooks,

grouped in three categories – Deductive, Empirical, and External justification (following

Harel & Sowder, 1998, 2007). The relative frequencies of deductive justifications ranged

between 39%-81%, yet were similar across six of the textbooks, comprising roughly two-

thirds of the instances of justification. External types of justification were rare, accounting for

less than 1% of all instances of justification in the textbooks.

Figure 14. Relative frequencies of the types of justification in the textbooks, by category

A further analysis involved a breakdown of the categories by using the framework suggested

by Stacey and Vincent (2009). Table 6 presents the frequencies for each type of justification,

by textbook. As can be seen, the total number of instances of justification was between 18-31

instances per textbook. Noticeable variation was found in the frequencies of three types – one

empirical and two deductive: Experimental demonstration, Deduction using a specific case,

and Deduction using a general case. Nevertheless, these three justification types constituted

most of the instances of justification in the analyzed textbooks. For example, the two

deductive types accounted for 46-67% of the instances in seven of the textbooks and 33% in

Textbook G. The two external types of justification – Appeal to authority and Qualitative

analogy – were either extremely rare or entirely absent in the textbooks.


Table 7 presents the average number of instances of justification, by type and textbook scope.

The comparison focused on seven out of the ten mathematical statements due to the structure

of textbooks of limited scope. The other three statements were introduced in the first volume

of each textbook, and including them would have prevented comparison between the pairs of

textbooks that have a common first volume – textbooks B and G, and textbooks C and H.

While no significant differences were found for justification types across the textbooks, a

noticeable difference was found for deduction using a general case. Textbooks of limited

scope (Textbooks G-H) offered fewer instances of justification involving deduction using a

general case compared with textbooks of standard/expanded scope (Textbooks A-F) –

roughly one-third the number. Comparison between Textbooks B and G, and between

Textbooks C and H, revealed similar ratios (7:2 and 2:1, respectively).

Table 6. Frequencies of types of justification, by textbook

Type of justification Textbook Total (%)

A B C D E F G H

External

Appeal to authority - - - - - - 1 - 1 (0%)

Qualitative analogy - - - - - - - - 0 (0%)

Empirical

Experimental demonstration 6 9 4 9 4 9 9 5 55 (30%)

Concordance of a rule with a model 2 1 - 2 3 3 1 - 12 (7%)

Deductive

Deduction using a model 7 1 3 2 3 2 1 3 22 (12%)

Deduction using a specific case 8 4 10 5 10 5 4 9 55 (30%)

Deduction using a general case 8 7 4 6 2 7 2 2 38 (21%)

Total 31 22 21 24 22 26 18 19 183 (100%)

Table 7. Average number of instances for types of justification, by textbook scope

Type of justification Textbook scope Total (S.D.)

Standard Scope Limited Scope

External

Appeal to authority 0 0.5 0.1 (0.3)

Qualitative analogy 0 0 0 (0)

Empirical

Experimental demonstration 6 6.5 6.1 (2)

Concordance of a rule with a model 0.7 0 0.5 (0.5)

Deductive

Deduction using a model 1.2 0.5 1 (1.2)

Deduction using a specific case 3.7 3.5 3.6 (2.2)

Deduction using a general case 5.7 2 4.8 (2.4)

Section 5.3 – Types of justification across mathematical statements | 50

5.3. Types of justification across mathematical statements

Table 8 presents the frequencies of types of justification, by mathematical statement. As can

be seen, the number of instances varied greatly across the statements, between 8-39 instances

per statement. Furthermore, there was a great variation in the frequencies of almost every

type (all but the rarely used one) across the statements.

Analysis of the types of justification offered in the textbooks suggests that they were used to

different extents for algebra and geometry statements (see Figure 15): Statements involving

algebra were typically justified by two deductive types – either deduction using a specific

case or deduction using a model (Figure 15(a)); Statements involving geometry were usually

justified by an empirical type – experimental demonstration as well as by two deductive types

– either deduction using a general case or deduction using a specific case (Figure 15(b)).

Algebra Geometry

(a) (b)

Figure 15. Frequencies of types of justification by topic and textbook

Additional analysis focused on the three most commonly used types of justification (i.e.,

experimental demonstration and deduction using a specific/general case). This analysis

further suggests there were differences based on the content topic (see Table 8): Deduction

using a general case was offered exclusively for geometry statements, whereas deduction

using a specific case was offered mainly for statements involving algebra (i.e., including area

of a trapezium). In contrast, Experimental demonstration was used in justification of

statements both in algebra and in geometry.


a=

appea

l to a

uth

ority; q

=qualita

tive analo

gy; e=

experim

enta

l dem

onstra

tion

; r=co

nco

rdan

ce of a

rule w

ith a

model; m

=ded

uctio

n u

sing

a

model; s=

ded

uctio

n u

sing a

specific ca

se; g=

ded

uctio

n u

sing a

gen

eral ca

se.

Tota

l

g s

m

Ded

uctiv

e

r e

Em

pirical

q

a

Extern

al

Justification

type

Tab

le 8. F

requen

cies of ty

pes o

f justificatio

n, b

y m

athem

atical statemen

t

20

-

10

4 - 6 - -

Equiv

alent

expressio

ns

Alg

ebra S

tatemen

ts

17

-

16

1 - - - -

Div

ision

by zero

17

- - 9 8 - - -

Distrib

utiv

e

law

12

- - 6 1

5 - -

Equiv

alent

equatio

ns

15

- 5

2 - 7 - 1

Pro

duct o

f

neg

atives

39

10

14

- 3

12

- -

Area o

f a

trapeziu

m

Geo

metry

Statem

ents

8

8 - - - - - -

Area o

f a

disk

16

7

5 - - 4 - -

Vertical

angles

13

2

2 - - 9 - -

Corresp

.

angles

26

11

3 - -

12

- -

Angle su

m o

f

a triangle

183

38

55

22

12

55

0

1

Tota

l

Section 5.4 – Sequences of types of justification | 52

5.4. Sequences of types of justification

Analysis of the paths of justification focused on three attributes: (1) path length – the number

of instances of justification offered in a textbook for each mathematical statement; (2)

characteristics – the types of justification included in each path; and (3) sequencing – the

order in which types of justification were offered in the textbook. Table 9 presents the paths

of justification for each mathematical statement, by textbook, focusing on justification types.

Analysis of path lengths revealed that the number of instances of justification offered in a

textbook for each mathematical statement varied considerably, between one and six instances

of justification per path. As can be seen in Table 9, path lengths varied both for different

statements in the same textbook, and for the same statement across textbooks. For example,

in Textbook A, the path for the product of negatives included four instances, but only one for

the area of a disk; In Textbook F the path for the product of negatives included one instance.

Table 10 presents the average lengths of the paths of justification for each mathematical

statement, by textbook scope. The average path length was 2.29 instances (1-4.88 instances

per path). Short paths were thus defined as paths including either one or two instances of

justification, and long paths were defined as including three or more instances. As can be

seen, 52 paths were coded as short, 27 as long, and one as neither – no justification was given

for the area of a disk in Textbook E. For most statements, average paths lengths were

generally slightly shorter in textbooks of limited scope, compared with textbooks of

standard/expanded scope (7 of 10 statements). No differences were found in path lengths

between algebra and geometry statements.

Analysis of the characteristics of the instances of justification revealed a strong preference to

deductive types of justification. Paths of justification often included more than one type of

justification (45 of 80) – typically both empirical and deductive (38 of 45), and occasionally

only deductive types (7 of 45). Paths that included just one justification type (34 of 80) were

usually deductive (22 of 34). Further analysis compared short and long paths. Purely

deductive paths were more common in short paths (26 of 52) compared with long paths (3 of

27). Long paths typically included both empirical and deductive justification types (24 of 27).


a=

appea

l to

auth

ority;

e=exp

erimen

tal

dem

onstra

tion

; r=

conco

rdance

of

a

rule

with

a

model;

m=

ded

uctio

n

usin

g

a

model;

s=ded

uctio

n u

sing a

specific ca

se; g=

ded

uctio

n u

sing a

gen

eral ca

se.

H

G

F

E

D

C

B

A Textbook

Tab

le 9. P

aths o

f justificatio

n fo

rmed

by ty

pes o

f justificatio

n, b

y tex

tbook an

d statem

ent.

m,s

e,s

s,e,s

m,e,s

e,s

m,s

e,s

e,s,m,s

Equiv

alent

expressio

ns

Alg

ebra S

tatemen

ts

s,s,s

s s,s

s,s

s,s,s

s,s,s

s s,m

Div

ision

by zero

m

m,r

m,r,r

m,r,r

m,m

,r

m

m,r

m,r

Distrib

utiv

e

law

m

e m,e,r

m,e

e m

e m,m

Equiv

alent

equatio

ns

e a s e,s

e,s

e,s

e,e

m,s,e,m

Pro

duct o

f

neg

atives

e,s,s,s

e,e,g,e

e,e,e,g,g

s,s,s,r

e,e,e,g,r,g

e,s,s,s,g

e,s,g,g

,g

s,g,r,s,s,s

Area o

f a

trapeziu

m

Geo

metry

Statem

ents

g

g

g,g

–

g

g

g

g

Area o

f a

disk

s,g

e,s

e,g

s,g

g

s,g

s,g

e,e,g

Vertical

angles

s,e

e,e

e s e,e

e e e,g,g

Corresp

.

angles

e,e

e,s

e,e,g,g

e,s,g

e,g,g

e,s,g

e,e,e,g,g

e,g,g

,g

Angle su

m o

f

a triangle

Section 5.4 – Sequences of types of justification | 54

Table 10. Average path lengths, by statement and textbook scope.

Mathematical statement Textbook scope Total

Standard Scope Limited Scope

Equivalent expressions 2.67 2 2.5

Division by zero 2.17 2 2.13

Distributive law 2.3 1.5 2.13

Equivalent equations 1.67 1 1.5

Product of negatives 2.17 1 1.88

Area of a trapezium 5.17 4 4.88

Area of a disk 1 1 1

Vertical angles 2 2 2

Corresponding angles 1.5 2 1.63

Angle sum of a triangle 3.67 2 3.25

Total 2.43 1.85 2.29

Analysis of the sequencing in the paths of justification focused on the three most commonly

used types (see Table 6) – an empirical type (Experimental demonstration) and two deductive

types (Deduction using a specific/general case). The analysis suggests that in almost all paths

that included the empirical type and either deductive type, Experimental demonstration

preceded the deductive types (25 of 28). Similarly, in paths that included both Deduction

using a specific case and Deduction using a general case, a generic example always preceded

the general case (9 of 9).

Further analysis focused on sequences involving Deduction using a general case (27 of 80

paths, all in geometry). The analysis reveals that the general case was not always preceded by

a generic example – only a relatively small fraction of these paths involved Deduction using

a specific case (33%, 9 of 27). Other paths involved either Experimental demonstration (10

of 27) or just one type of justification (8 of 27).

55 | Justification strategies in the textbooks

6. Justification strategies in the textbooks

In this chapter I focus on a content-specific characteristic of the instances of justification –

the justification strategies offered in the textbooks. I describe and illustrate the strategies

offered in paths of justification for the 10 analysed statements.

6.1. Equivalent expressions

Three justification strategies were used in the analyzed textbooks for justifying why two

algebraic expressions are equivalent if one expression can be transformed into the other by

performing a sequence of valid operations for a common domain of numbers. All three

justification strategies are addressed and reviewed in the mathematics education literature: (i)

Rules and conventions (Kieran, 1992, 2006), (ii) Substitution (Kieran & Sfard, 1999; Tabach

& Friedlander, 2008; Tirosh et al., 1998), and (iii) Description equivalence (Kieran & Sfard,

1999; Tabach & Friedlander, 2008; Zwetzschler & Prediger, 2013). Additional warrants are

mentioned in the literature yet were not found in the analyzed textbooks (e.g., relying on a

functional approach and comparing the graphs of functions described by each of the algebraic

expressions, see Kieran & Sfard, 1999). In this section I address two aspects: (1) The

justification strategies offered, in descending order of frequency in the textbooks; and (2) The

paths of justification.

The justification strategies offered

6.1.1. Rules and conventions: This justification strategy is based on an extension of

properties of arithmetic operations from numbers to pronumerals, by relying on students'

elementary school knowledge. Instances of this justification strategy in the textbooks

involved various arithmetic facts to justify equivalence of algebraic equivalence, such as the

definition of multiplication as repeated addition (e.g., Figure 16) and the associative and

commutative properties (e.g., Figure 17).

Figure 16. Deduction using a specific case by Rules & conventions (Textbook D, vol 1, p. 43)

Section 6.1 – Equivalent expressions | 56

Figure 17. Deduction using a specific case by Rules & conventions (Textbook A, vol 1, p. 188)

6.1.2. Substitution: This justification strategy is based on an inductive process, in which

selected values are substituted for the variable in two algebraic expressions. The equivalence

of the two expressions is validated if identical results are achieved in both expressions for

each of the selected examples (e.g., Figure 18).

Figure 18. Experimental demonstration by Substitution (Textbook A, vol 1, p. 26)

This justification strategy is generally not a valid method of justifying the equivalence of two

algebraic expressions, except for when it is possible to exhaust the domain. Underlying this

warrant is an alternative meaning for the equivalence of algebraic expressions. This definition

states that two expressions are equivalent if the substitution of all numbers in the expressions

will produce equal results for their common domain (Tabach & Friedlander, 2008;

Zwetzschler & Prediger, 2013).

6.1.3. Description equivalence: This justification strategy is based on modelling a given

situation in two distinct ways in order to construct two algebraic expressions. Instances of

this justification strategy in the textbooks involved either relying on two viewpoints or using

two counting methods (e.g., calculating the perimeter of a polygon by grouping different

sides or by working in different order, see Figure 19).


Figure 19. Deduction using a model by Description equivalence (Textbook C, vol 1, p. 57)

The algebraic transformation (or sequence of transformations) between the two expressions

can be established by validating each viewpoint or counting method. Underlying this warrant

is an alternative meaning for the equivalence of algebraic expressions. This definition states

that two expressions are equivalent if they describe the same phenomenon – the same

geometric pattern, the same situation, or the same object (Tabach & Friedlander, 2008;

Zwetzschler & Prediger, 2013).

The Paths of justification

Table 11 summarizes the frequencies of the justification strategies offered in Israeli 7th grade

mathematics textbooks for the mathematical statement equivalent expressions. As can be

seen, justification was offered in every textbook. Moreover, every textbook offered a path

based on two or three justification strategies, one of which being rules and conventions,

which is often considered to be the more mathematically sound strategy.

Figure 20 presents the paths of justification offered in the textbooks for this mathematical

statement. As can be seen, paths opened in any of the three justification strategies. However,

description equivalence was commonly offered at the beginning of the paths (whenever it

was offered), and rules and conventions was always the last justification strategy offered.

Further analysis of the paths, focused on both characteristics of each instance of justification

Table 11. Equivalent expressions – frequencies of justification strategies, by textbook

Justification strategy Textbook Total

A B C D E F G H

Rules and conventions 2 1 1 1 1 2 1 1 10

Substitution 1 1 . 1 1 1 1 . 6

Description equivalence 1 . 1 . 1 . . 1 4

Total 4 2 2 2 3 3 2 2 20

Section 6.1 – Equivalent expressions | 58

(i.e., justification strategy and type), suggests that most paths offered a shift from concrete to

abstract. Seven paths began either with the empirical experimental demonstration or with a

visual strategy relying on a deduction using a model, and all paths ended with generic

examples, relying on the more formal type of justification deduction using a specific case.

In justification of equivalent expressions, the Israeli school curriculum for grades 7-9

suggests one justification strategy: rules and conventions (Israel Ministry of Education,

2009). As Figure 20 shows, this justification strategy was offered in every textbook – at least

once and at the beginning of the path.

Textbook Path of justification

A

B

C

D

E

F

G

H

e=experimental demonstration; m/s=deduction using a model / a specific case.

Figure 20. Equivalent expressions – Paths of justification

Substitution

e

Conventions

s

Description

m

Conventions

s

Substitution

e

Conventions

s

Description

m

Conventions

s

Substitution

e

Conventions

s

Description

m

Substitution

e

Conventions

s

Conventions

s

Substitution

e

Conventions

s

Substitution

e

Conventions

s

Description

m

Conventions

s


6.2. Division by zero

Two justification strategies were used in the analyzed textbooks for justifying why Division

by zero is undefined. Both justification strategies are addressed and reviewed in the

mathematics education literature (Crespo & Nicol, 2006; Kim, 2007; Knifong & Burton,

1980; Tsamir & Sheffer, 2000; J. M. Watson, 1991): (i) The inverse of multiplication, and (ii)

Repeated Subtraction. Additional warrants are mentioned in the literature yet were not found

in the analyzed textbooks (e.g., relying on an intuitive notion of limit of the sequence of

reciprocals of the natural numbers, see Tsamir & Sheffer, 2000; Watson, 1991). In this

section I address two aspects: (1) The justification strategies offered, in descending order of

frequency in the textbooks; and (2) The paths of justification.


6.2.1. The inverse of multiplication: This justification strategy is based on treating division

as the inverse of multiplication. Instances of this justification strategy in the textbooks

involved either the case a:0 (a≠0), by relying on the fact that zero has no multiplicative

inverse (e.g., Figure 21), or the case 0:0, by relying on the definition of division as an

operation with a unique result (e.g., Figure 22).

Figure 21. Deduction using a specific case by Inverse of multiplication (Textbook B, vol 1, p. 80)

Section 6.2 – Division by zero | 60

Figure 22. Deduction using a specific case by Inverse of multiplication (Textbook F, vol 1, p. 31)

6.2.2. Repeated subtraction: This justification strategy is based on modelling division as

repeated subtraction. First, a defined case is introduced (e.g., 20:4) and the question "How

many times should one subtract 4 from 20 to reach 0?" is discussed. Then, an undefined case

is introduced (e.g., 20:0) and the question "How many times should one subtract 0 from 20 to

reach 0?" is discussed. Since there is no real number that answers this question, this case

must be left undefined (e.g., Figure 23).

Figure 23. Deduction using a a model by Repeated subtraction (Textbook A, vol 1, p. 64)



mathematics textbooks for the mathematical statement division by zero. As can be seen,

justifications were offered in every textbook, commonly more than one instance of

justification per path. Every textbook offered the inverse of multiplication, which is often

seen as a mathematically sound strategy. One textbook offered repeated subtraction as well.

Table 12. Division by zero – frequencies of justification strategies, by textbook.


A B C D E F G H

The inverse of multiplication 1 1 3 3 2 2 1 3 16

Repeated subtraction 1 . . . . . . . 1

Total 2 1 3 3 2 2 1 3 17



statement. As can be seen, all textbooks offered a path based on the inverse of multiplication,

dealing with the case a:0 (a≠0). Five textbooks proceeded to attend to the case 0:0 as well.

One textbook offered both justification strategies as equals, and requested the student to

decide which of the two strategies is more convincing.

Further analysis of the paths, focused on both characteristics of each instance of justification

(i.e., justification strategy and type), suggests that the inverse of multiplication was associated

in each instance with deduction using a specific case, thus relying on a generic example to

justify why division by zero is undefined.

In justification of division by zero, the Israeli school curriculum for grades 7-9 suggests one

justification strategy: the inverse of multiplication, and attends to the case 0:0 as well (Israel

Ministry of Education, 2009). As Figure 24 shows, this justification strategy was offered in

every textbook – at least once and at the beginning of the path.


A

B

C

D

E

F

G

H

m=deduction using a model; s=deduction using a specific case.

Figure 24. Division by zero – Paths of justification

Inverse a:0

s

Subtraction

m

Inverse a:0

s

Inverse a:0

s

Inverse a:0

s

Inverse 0:0

s

Inverse a:0

s

Inverse a:0

s

Inverse 0:0

s

Inverse a:0

s

Inverse 0:0

s

Inverse a:0

s

Inverse 0:0

s

Inverse a:0

s

Inverse a:0

s

Inverse a:0

s

Inverse 0:0

s

Section 6.3 – Distributive law | 62

6.3. Distributive law

Two justification strategies were used in the analyzed textbooks for justifying why

a*(b+c)=a*b+a*c for any three numbers a, b, c. Both justification strategies are addressed

and reviewed in the mathematics education literature: (i) Area / array (B. Davis & Simmt,

2006; Ding & Li, 2014; Lampert, 1986; Wu, 1999), and (ii) Arithmetic conventions (Lampert,

1986). Additional warrants are mentioned in the literature yet were not found in the analyzed

textbooks (e.g., relying on grid-based multiplication, see Davis & Simmt, 2006). In this




6.3.1. Area / Array: This justification strategy is based on modelling multiplication of

positive numbers as the area of a rectangle. Instances of this justification strategy in the

textbooks involved calculating either the area of a rectangle with sides "a+b" and "c" (e.g.,

Figure 25), or the number of items in a rectangular array (e.g., Figure 26). In either case, the

result is calculated twice – once as the product of the rectangle's sides, and once as the sum of

the areas of each box. The two algebraic expressions represent different ways of looking at

the same object and are therefore equivalent.

Figure 25. Deduction using a model by Area (Textbook D, vol 1, p. 75)


Figure 26. Deduction using a model by Array (Textbook E, vol 1, p. 96)

6.3.2. Arithmetic conventions: This justification strategy is based on identifying a

correspondence between the results obtained by using the distributive property and by using

PEMDAS rule (i.e., order of operations: Parentheses; Exponents; Multiplication or Division

(left to right); and Addition or Subtraction (left to right)). Instances of this justification

strategy in the textbooks involved either an abstract setting (e.g., Figure 27) or a word-

problem setting (e.g., Figure 28).

Figure 27. Concordance of a rule with a model by Arith. conventions (Textbook E, vol 1, p. 97)

Figure 28. Concordance of a rule with a model by Arith. conventions (Textbook B, vol 1, p. 72)

Section 6.3 – Distributive law | 64



mathematics textbooks for the mathematical statement the distributive law. As can be seen,


justification per path. Every textbook offered the more mathematically sound strategy,

area/array, and six textbooks offered arithmetic conventions as well.


statement. As can be seen, area/array was offered before arithmetic conventions in every

textbook. Further analysis of the paths, focused on both characteristics of each instance of

justification (i.e., justification strategy and type), suggests that area/array was associated in

each instance with deduction using a model, thus providing a visual model to allow deduction

of the distributive property. In addition, every instance of arithmetic conventions was

associated with concordance of a rule and a model, and was used mainly in order to affirm

the rule which was presented beforehand by using area/array.

In justification of the distributive law, the Israeli school curriculum for grades 7-9 suggests

two justification strategy: area/array, followed by arithmetic conventions, and attends to the

case 0:0 as well (Israel Ministry of Education, 2009). As Figure 29 shows, these justification

strategies were offered in most textbooks – in the same order.

Table 13. Distributive law – frequencies of justification strategies, by textbook.


A B C D E F G H

Area / Array 1 1 1 2 1 1 1 1 9

Arithmetic conventions 1 1 . 1 2 2 1 . 8

Total 2 2 1 3 3 3 2 1 17



A

B

C

D

E

F

G

H

r=concordance of a rule with a model; m=deduction using a model.

Figure 29. Distributive law – Paths of justification

Area/Array

m

Conventions

r

Area/Array

m

Conventions

r

Area/Array

m

Area/Array

m

Area/Array

m

Conventions

r

Area/Array

m

Conventions

r

Conventions

r

Area/Array

m

Conventions

r

Conventions

r

Area/Array

m

Conventions

r

Area/Array

m

Section 6.4 – Equivalent equations | 66

6.4. Equivalent equations

Four justification strategies were used in the analyzed textbooks for justifying why

performing valid operations on both sides of an equation yields an equivalent equation. All

four justification strategies are addressed and reviewed in the mathematics education

literature: (i) Balance model (e.g., Filloy & Rojano, 1989; Linchevski & Herscovics, 1996;

Vlassis, 2002), (ii) Undoing (e.g., Bernard & Cohen, 1988; Kieran, 1992), (iii) Segment

model (e.g., Dickinson & Eade, 2004); and (iv) Intuitive comparison (e.g., Bernard & Cohen,

1988; Kieran, 1992, 2006). Additional warrants are mentioned in the literature yet were not

found in the analyzed textbooks (e.g., an area model, in which equivalence of equations

corresponds with equal areas, see Filloy & Rojano, 1989). In this section I address two

aspects: (1) The justification strategies offered, in descending order of frequency in the

textbooks; and (2) The paths of justification.


6.4.1. Balance model: This justification strategy is based on modelling an equation as two

sides of a balance scale, with known and unknown weights on each side, representing

constants and variables (respectively). A valid operation on an equation is modelled as a

manipulation on the weights, such that the scales remain balanced (e.g., addition is modelled

as adding identical weights on both sides of the scale). Instances of this justification strategy

in the textbooks involved discussing concrete manipulations that can be performed on a

balance scale without disrupting the balance (e.g., Figure 30).

Figure 30. Deduction using a model by Balance model (Textbook H, vol 3, p. 55)


The model has several limitations. For example, it relies on a type of scales with which many

junior-high school students are no longer familiar. More importantly, it is valid only for

equations of the form Ax+B=Cx+D, where A,B,C,D are all non-negative numbers. This

limitation stems from the association between numbers and weights, because the magnitude

of weights is non-negative in nature. This limitation was discussed explicitly in the teacher's

guides for three textbooks, and one of these textbooks explicitly addressed this limitation in a

task intended for a whole-class discussion.

6.4.2. Undoing: This justification strategy is based on analyzing an equation as a sequence of

reversible steps that have been applied to an unknown number “x”. In other words, each

invertible operation on an equation is undoing an opposite (inverse) operation that was

performed on the variable. Instances of this justification strategy in the textbooks involved

identifying equal elements in both sides of the equation (e.g., the addition of one, see Figure

31). Each step relies on the students' elementary school experience in recalling number facts

relating to addition, subtraction, multiplication, and division.

Figure 31. Experimental demonstration by Undoing (adapted from Textbook B, vol 2, p. 17)

The process of "undoing" an equation relies on certain students' abilities, such as the ability to

see an algebraic expression as an entity and the ability to find the optimal manipulation to

perform in order to divide an algebraic expression into sub-structures. These abilities are

commonly grouped under the term structure sense in the literature (Hoch & Dreyfus, 2004;

Linchevski & Livneh, 1999; Novotná & Hoch, 2008).

6.4.3. Intuitive comparison: This justification strategy is based on a comparison of results

with an intuitive method of equation solving. Instances of this justification strategy relied on

the cover-up method (e.g., Figure 32) or on unspecified previously taught methods for

solving equations (e.g., Figure 33). The cover-up method involves breaking down an equation

of the form Ax±B=C in a sequence of steps. In each step, the equation is verbalized, and an


algebraic expression is covered up and is replaced with the question "what is the number for

which". In each step, this question is answered by using elementary arithmetic considerations

(e.g., Figure 32(a)). Then, the same equation is solved by operating on both sides of the

equation, and the results are compared (e.g., the equation 2x-7=15 is solved in two ways in

Figure 32(b), and both results are compared to the result obtained in Figure 32(a)).

(b) (a) Figure 32. Concordanceof a rule w/ model by intuition (adapted from Textbook F, vol 2, p. 135)

Figure 33. Experimental demonstration by intuition (adapted from Textbook D, vol 2, p. 15)

The cover-up method was often employed in the Israeli textbooks in teaching how to solve

equations, along with several other methods. However, the relationship between these

methods and the formal procedure of performing the same operation on both sides of an


equation was scarcely made explicit. In most cases, the justification strategies were offered

implicitly, by treating each step in the solution as having the same set of solutions as the

original equation but without mentioning explicitly that the two equations are equivalent.

6.4.4. Segment model: This justification strategy is based on modelling an equation as two

segments of equal length, each partitioned into pieces of known and unknown lengths,

representing constants and variables (respectively). A valid operation on an equation is

modelled as a manipulation on the pieces such that the segments remain of equal measures

(e.g., subtraction is modelled as removing pieces of equal lengths from both segments).

Instances of this justification strategy in the textbooks involved using two separate segments

(e.g., Figure 34).

Figure 34. Deduction using a model by Segment model (Textbook A, vol 1, p. 214)

This model was used in some textbooks as a method for solving equations. Such occurrences

were not counted as instances of justification for this mathematical statement.



mathematics textbooks for the mathematical statement equivalent equations. As can be seen,


justification per path. Four textbooks offered a path based on a combination of several

justification strategies.



statement. As can be seen, paths commonly opened by relying on balance model, and

involved a single type of justification.

In justification of equivalent equations, the Israeli school curriculum for grades 7-9 does not

explicitly offer any justification strategy, yet hints at relying on balance model (Israel

Ministry of Education, 2009). As Figure 35 shows, this justification strategy was offered in

five textbooks – at the beginning of the path.


A

B

C

D

E

F

G

H

e=experimental demonstr.; r=concordance of a rule with a model; m=deduction w/ a model.

Figure 35. Equivalent equations – Paths of justification.

Table 14. Equivalent equations – frequencies of justification strategies, by textbook


A B C D E F G H

Balance model 1 . 1 . 1 1 . 1 5

Undoing . 1 . . 1 . 1 . 3

Intuitive comparison . . . 1 . 2 . . 3

Segment model 1 . . . . . . . 1

Total 2 1 1 1 2 3 1 1 12

Balance

m

Segment

m

Undoing

e

Balance

m

Intuition

e

Balance

m

Undoing

e

Balance

m

Intuition

e

Intuition

r

Undoing

e

Balance

m


6.5. Product of negatives

Three justification strategies were used in the analyzed textbooks for justifying why the

product of two negative numbers is a positive number. All three justification strategies are

addressed and reviewed in the mathematics education literature: (i) Discovering patterns

(e.g., Arcavi & Bruckheimer, 1981; Askey, 1999; Cable, 1971; J. C. Peterson, 1972); (ii)

Extension of properties (e.g., Arcavi & Bruckheimer, 1981; Brown, 1969; Crowley & Dunn,

1985; Hefendehl-Hebeker, 1991; Rapke, 2008; Sfard, 2000); and (iii) Number-line models

(Arcavi & Bruckheimer, 1981; Brown, 1969; Cable, 1971; Crowley & Dunn, 1985).

Additional warrants are mentioned in the literature yet were not found in the analyzed

textbooks (e.g., a formal axiomatic warrant, involving formally defining integers as ordered

pairs and defining their multiplication, see Arcavi & Bruckheimer, 1981; Brown, 1969; J. C.

Peterson, 1972). In this section I address two aspects: (1) The justification strategies offered,

in descending order of frequency in the textbooks; and (2) The paths of justification.


6.5.1. Discovering patterns: This justification strategy is based on discovery and

extrapolation of patterns. First, patterns are discovered in a sequence of calculations of

products of natural numbers. Then, these patterns are extended from natural numbers to

negative integers. Instances of this justification strategy in the textbooks typically involved

either completing a multiplication table (e.g., Figure 36) or continuing a sequence of

calculations (e.g., Figure 37). Additionally, one textbook involved simply checking several

unrelated specific examples (e.g., Figure 38).

Figure 36. Deduction using a specific case by Discovering patterns (Textbook C, vol 2, p. 42)

Section 6.5 – Product of negatives | 72

Figure 37. Deduction using a specific case by Discovering patterns (Textbook H, vol 2, p. 46)

Figure 38. Appeal to authority by Discov. patterns (adapted from Textbook G, v. 2, pp. 61-62)

A common condition for this extrapolation of traits is that each sequence of products is an

arithmetic sequence (i.e., with a constant difference between successive values in the

sequence). This condition was not made explicit in any of the analyzed textbooks. For

example, in Figure 37, students are expected to complete the sequence of products -6,-4,-2,0

with the numbers 2,4,6, yet no information is given to rule out other possibilities.

6.5.2. Extension of properties: This justification strategy is based on the assumption that

certain properties of natural numbers (e.g., the distributive law, the commutative law, and the

multiplicative property of zero) can be extended to the set of integers and be preserved.

Instances of justification involving this justification strategy involved the use of the

multiplicative property of zero and of the distributive property of multiplication over addition

(e.g., Figure 39). This justification strategy is often considered in the literature to be of

greater mathematical validity, or at least the most sophisticated, compared with the other

three kinds I found in the textbooks (e.g., Arcavi & Bruckheimer, 1981; J. C. Peterson, 1972).


Figure 39. Deduction using a specific case by Extension of properties (Textbook F, vol 2, p. 48)

6.5.3. Number-line models: This justification strategy is based on modelling signed numbers

in a way that deals both with the sign and with the absolute value of the product. Instances in

the textbooks involved either movement forward and backward in time on a number line

(Figure 40), or a change in weight over time (Figure 41).

Figure 40. Deduction using a model by Line model (adapted from Textbook A, vol 2, p. 392)

Figure 41. Deduction using a model by Line model (Textbook A, vol 2, p. 395)


In these models, the product a.b is modelled such that the multiplicand b represents a signed

rate of change of a certain quantity (e.g., weight or distance) and the multiplier a represents a

change in time, prior or later from a given moment. For example, the product (-4)*(-3) can be

modelled either as "Where was the ant four minutes ago, if it moved against the direction of

the arrow, and its speed is three units per minute?" (e.g., Figure 40), or as "How much did the

whale weigh four weeks ago, if it loses 3 kilograms per week?" (e.g., Figure 41).



mathematics textbooks for the mathematical statement the product of negatives. As can be

seen, justifications were offered in every textbook, commonly more than one instance of

justification per path. Four textbooks offered paths based on a combination of several

justification strategies, commonly including discovering patterns.


statement. As can be seen, paths commonly opened with discovering patterns and ended with

the more formal justification strategy – extension of properties. One textbook (Textbook A)

offered a path based on four justification strategies, including Number-line models. As a

curiosity, Textbook A provided a mnemonic for the sign rule which is based on the logical

principle stating that double negation is an affirmation. The mnemonic was offered after

properly establishing the sign rule, and relied on a superficial similarity to the proverb 'the

enemy of my enemy is my friend' (see Figure 42).

Figure 42. A mnemonic relying on Double negation (Textbook A, vol 2, p. 394)

Table 15. Product of negatives – frequencies of justification strategies, by textbook


A B C D E F G H

Discovering patterns 1 2 1 1 1 . 1 1 8

Extension of properties 1 . 1 1 1 1 . . 5

Number-line models 2 . . . . . . . 2

Total 4 2 2 2 2 1 1 1 15


Discovering patterns was associated with two types of justification. Further analysis suggests

that this is a result of two methods underlying these instances of justification – either

detecting the pattern numerically by observing a sequence of numbers, or detecting a pattern

by observing a set of results achieved by using a calculator.

In justification of the product of negatives, the Israeli school curriculum for grades 7-9

suggests one justification strategy: extension of properties (Israel Ministry of Education,

2009). As Figure 43 shows, this justification strategy was common in the textbooks (five

textbooks).


A

B

C

D

E

F

G

H

a=appeal to authority; e=experimental demonstration; m=deduction using a model;

s=deduction using a specific case.

Figure 43. Product of negatives – Paths of justification

Number-line

m

Properties

s

Patterns

e

Number-line

m

Patterns

e

Patterns

e

Patterns

e

Properties

s

Patterns

e

Properties

s

Patterns

e

Properties

s

Properties

s

Patterns

a

Patterns

e

Section 6.6 – Area of a trapezium | 76

6.6. Area of a trapezium

Two justification strategies were used in the analyzed textbooks for justifying why the area

formula for a trapezium with bases a, b and altitude h is (a + b).h/2. Both justification

strategies are addressed and reviewed in the mathematics education literature: (i) Dissection,

and (ii) Construction (e.g., Hoosain, 2010; L. L. Peterson & Saul, 1990; Usnick, Lamphere,

& Bright, 1992). Additional warrants are mentioned in the literature yet were not found in the

analyzed textbooks (e.g., relying on Pick's rule for computing the area of a simple polygon by

using the number of lattice points in its interior, see Hirsch, 1974; Hoosain, 2010). In this




6.6.1. Dissection: This justification strategy is based on representing the area of a given

trapezium as a sum of areas of shapes, each with an area formula known to the students.

Instances of this justification strategy in the textbooks involved six variations, suggesting

various methods of dissection (e.g., Figure 44): (1) dissection into a triangle and a

parallelogram by drawing a segment parallel to one of the sides of the trapezium, (2)

dissection into two triangles by drawing a diagonal, (3) dissection into two triangles and a

rectangle by drawing two altitudes, (4) dissection into three triangles by drawing a segment

from each of the vertices of the small base to the long base, (5) dissection by drawing a

segment from a vertex of the small base to the midpoint of one of the sides of the trapezium

and rearranging the two areas to form a triangle, and (6) dissection by using a mid-segment to

dissect a trapezium into two trapeziums and rearranging the two areas to form a

parallelogram.

Figure 44. Experimental demonstration by Dissection (Textbook D, vol 2, p.184)


6.6.2. Construction: This justification strategy is based on the arrangement of several

congruent trapeziums such that they form a shape with an area formula known to the

students. Instances of this justification strategy involved construction of a parallelogram, the

area of which is represented as a sum of areas of two trapeziums (e.g., Figure 45).

Figure 45. Deduction using a general case by Construction (Textbook G, vol 3, p.93)

A common implicit assumption for this construction is that the shape created by rearranging

the congruent trapeziums is indeed a parallelogram. This assumption was not justified in any

of the analyzed textbooks, yet several teacher's guides mentioned it as a caveat. In spite of

this limitation, this construction was frequently offered in the textbooks.


Table 16 summarizes the frequencies for the justification strategies offered in Israeli 7th grade

mathematics textbooks for the mathematical statement area of a trapezium. As can be seen,

instances of justification were offered in every textbook, commonly more than four instances

per textbook. Seven textbooks offered a path based on a combination of both justification

strategies, often offering several forms of dissections in each path.

Table 16. Area of a trapezium – frequencies of justification strategies, by textbook


A B C D E F G H

Dissection 3 4 3 6 3 4 2 3 28

Construction 3 1 2 . 1 1 2 1 11

Total 6 5 5 6 4 5 4 4 39



statement. As can be seen, paths typically opened with several forms of dissection, followed

by construction (of a parallelogram). However, two textbooks opened by construction.

According to the teacher's guide for Textbook A, this decision was deliberate, and aimed at

helping students remember the need to divide by two as part of the calculation by using the

formula. Textbook E's teacher's guide does not provide information regarding this decision.

Further analysis of the paths of justification focused on both characteristics of each instance

of justification (i.e., justification strategy and type). The analysis revealed that several types

of justification were associated with each of the justification strategies – both empirical and

deductive types. However, construction was commonly associated with deductive types of

justification, possibly for the same didactic power mentioned in Textbook A's teacher's guide.

In justification of area of a trapezium, the Israeli school curriculum for grades 7-9 suggests

two justification strategies: dissection followed by construction of a parallelogram (Israel

Ministry of Education, 2009). As Figure 46 shows, textbooks commonly offered this sequence.


A

B

C

D

E

F

G

H

e=experimental demonstration; r=concordance of a rule with a model; s/g=deduction using a

specific/general case.

Figure 46. Area of a trapezium – Paths of justification

Construction

s

Construction

g

Construction

r

Dissection

s

Dissection

s

Dissection

s

Dissection

e

Dissection

s

Dissection

g

Construction

g

Dissection

g

Dissection

e

Dissection

s

Dissection

s

Construction

s

Construction

g

Dissection

e

Dissection

e

Dissection

e

Dissection

g

Dissection

r

Dissection

g

Construction

s

Dissection

s

Dissection

s

Dissection

r

Dissection

e

Dissection

e

Dissection

e

Dissection

g

Construction

g

Dissection

e

Construction

e

Construction

g

Dissection

e

Dissection

e

Dissection

s

Dissection

s

Construction

s


6.7. Area of a disk

One justification strategy was used in the analyzed textbooks for justifying why the area

formula for a disk with radius r is πr2. This justification strategy is addressed and reviewed in

the mathematics education literature: Dissection (e.g., Borko et al., 2000; Moore, 2013).


textbooks (e.g., overlaying a grid paper on the disk, counting squares that coincide with the

disk, and estimating the magnitude of the parts of the squares that lie near the edge, see

Moore, 2013). In this section I address two aspects: (1) The justification strategies offered, in

descending order of frequency in the textbooks; and (2) The paths of justification.


Dissection: This justification strategy is based on the method of dissection and

rearrangement. The area of a disk is represented as a sum of areas and rearranged to resemble

a shape with a known area formula (as the number of parts tends to infinity). Instances of this

justification strategy commonly involved a rearrangement of sectors to form an approximate

parallelogram (e.g., Figure 47). One textbook offered a rearrangement of "opened out"

concentric rings to form an approximate right-angled triangle (e.g., Figure 48).

Figure 47. Deduction using a general case by Dissection into sectors (Textbook G, part 3, p. 118)

Section 6.7 – Area of a disk | 80

There are two major differences between dissection of a trapezium and of a disk: (1) number

of steps, and (2) previous familiarity. First, while dissection of a trapezium is a finite process

which terminates after a small number of moves, dissection of a disk involves an infinite

number of cuts and rearrangements. This fundamental difference hints at the concept of limit,

even in an unrefined way. It is therefore less intuitive and involves a higher cognitive load.

Second, Israeli students first encounter the area formula in elementary school, without

justification. This familiarity, combined with the complexity level of the explanation, might

deter textbook authors from allocating time and resources to formulating a path of

justification with multiple types of justification.



mathematics textbooks for the mathematical statement area of a disk. As can be seen,

instances of justification were offered in seven textbooks, commonly only one instance per

textbook. Textbook E presented this mathematical statement as a reminder, and did not offer

any justification or explanation. The textbooks offered a path based on one justification

strategy. Textbook F offered three instances of this justification strategy, including both

variants (i.e., dissection into sectors and into concentric circles).

Figure 48. Deduction using a general case by Dissection into rings (Textbook F, vol 2, p. 223)



statement. As can be seen, one justification strategy was offered in the textbooks. Further

analysis of the paths of justification focused on both characteristics of each instance of

justification (i.e., justification strategy and type). This analysis reveals that as a rule,

dissection was associated with one type of justification – deduction using a general case.

This almost absolute correlation is very different from the diverse assortment of types of

justification that was associated with dissection in the case of the area of a trapezium, and

might stem from the inherent differences discussed above.

In justification of the area of a disk, the Israeli school curriculum for grades 7-9 suggests one

justification strategy: dissection (Israel Ministry of Education, 2009). As Figure 49 shows, this

justification strategy was found in seven textbooks.


A

B

C

D

E –

F

G

H

e=experimental demonstration; g=deduction using a general case.

Figure 49. Area of a disk – Paths of justification

Table 17. Area of a disk – frequencies of justification strategies, by textbook


A B C D E F G H

Dissection 1 1 1 1 . 2 1 1 8

Total 1 1 1 1 0 2 1 1 8

Dissection

g

Dissection

g

Dissection

g

Dissection

g

Dissection

g

Dissection

g

Dissection

g

Dissection

g

Section 6.8 – Vertical angles | 82

6.8. Vertical angles

Two justification strategies were used in the analyzed textbooks for justifying why vertically

opposite angles are congruent. Both justification strategies are addressed and reviewed in the

mathematics education literature: (i) Supplementary angles (Host, Baynham, & McMaster,

2015; Reid, 1997), and (ii) Measurement (J. C. Chen, 2006; Host et al., 2015; Howard, 1919).


textbooks (e.g., by using other means to compare the measures of the angles, such as paper

folding or dynamic geometry software, see C. L. Chen & Herbst, 2013; Olson, 1975). In this




6.8.1. Supplementary angles: This justification strategy is based on the fact that angles on a

straight line add up to a straight angle. Given two vertically opposite angles, a third angle is

identified as an angle that supplements each of the angles. It follows that the two vertically

opposite angles are congruent (e.g., Figure 50).

Figure 50. Deduction using a general case by Supplementary angles (Textbook F, vol 3, p. 136)


6.8.2. Measurement: This justification strategy is based on the fact that angles with the same

measure are congruent. Instances of this justification strategy in the textbooks involved direct

comparison of two vertically opposite angles, in one or more special cases, by using a

protractor (e.g., Figure 51).



mathematics textbooks for the mathematical statement vertical angles. As can be seen,

instances of justification were offered in every textbook, commonly two instances per

textbook. Three textbooks offered a path based on a combination of both justification

strategies.

Figure 51. Experimental demonstration by Measurement (Textbook F, vol 3, p. 136)

Table 18. Vertical angles – frequencies of justification strategies, by textbook


A B C D E F G H

Supplementary angles 1 2 2 1 2 1 1 2 12

Measurement 2 . . . . 1 1 . 4

Total 3 2 2 1 2 2 2 2 16



statement. As can be seen, measurement always preceded supplementary angles. Five

textbooks offered only one strategy, relying on supplementary angles.

Further analysis of the paths of justification, focused on both characteristics of each instance

of justification (i.e., justification strategy and type), reveals that supplementary angles was

associated with two types of justification – deduction using a specific case, deduction using a

general case. Three types of paths leading to deduction using a general case were found in

the textbooks: (1) immediately following deduction using a specific case, (2) immediately

following an experimental demonstration relying on measurement, and (3) on its own.

In justification of vertical angles, the Israeli school curriculum for grades 7-9 suggests one

justification strategy: measurement followed by supplementary angles (Israel Ministry of

Education, 2009). As Figure 52 shows, this sequence was offered in three textbooks, and most

books relied only on supplementary angles.


A

B

C

D

E

F

G

H


Figure 52. Vertical angles – Paths of justification

Measurement

e

Measurement

e

Supplementar

g

Supplementar

s

Supplementar

g

Supplementar

s

Supplementar

g

Supplementar

g

Supplementar

s

Supplementar

g

Measurement

e

Supplementar

g

Measurement

e

Supplementar

s

Supplementar

s

Supplementar

g


6.9. Corresponding angles

Two justification strategies were used in the analyzed textbooks for justifying why the

corresponding angles between parallel lines are equal. Both justification strategies are

addressed and reviewed in the mathematics education literature: (i) Measurement (C. L. Chen

& Herbst, 2013; Weaver & Quinn, 1999), and (ii) Alternate angles. Additional warrants are

mentioned in the literature yet were not found in the analyzed textbooks (e.g., relying on

paper folding to demonstrate the congruence of the corresponding angles, see Olson, 1975).

In this section I address two aspects: (1) The justification strategies offered, in descending

order of frequency in the textbooks; and (2) The paths of justification.


6.9.1. Measurement: This justification strategy is based on direct comparison of the

corresponding angles by using a measuring tool. Instances of this justification strategy

involved several measuring tools, either physical (e.g., a protractor or a transparent sheet, see

Figure 53), or a dynamic geometry software (e.g., Figure 54).

Figure 53. Experimental demonstration by Measurement (Textbook G, vol 2, p. 182)

Section 6.9 – Corresponding angles | 86

Figure 54. Experimental demonstration by Measurement (adapted from Textbook C, vol 2, p. 189)

6.9.2. Alternate angles: This justification strategy is based on two theorems – the Alternate

Interior Angle Theorem (i.e., Alternate angles between parallels are congruent) and the

Vertical Angles Theorem. Given two parallel lines and a transversal, two corresponding

angles can be shown to be congruent to a third angle by using these theorems in either order.

It follows that the corresponding angles are congruent to each other (e.g., Figure 55).

Figure 55. Deduction using a specific case by Alternate angles (Textbook H, vol 2, p. 231)

Instances of this justification strategy in the textbooks involved a specific order of

presentation of topics – the congruence of alternate angles between parallel lines must be

established beforehand, in order to serve as a warrant for the congruence of the corresponding

angles.




mathematics textbooks for the mathematical statement corresponding angles. Instances of

justification were offered in every textbook, typically more than one instance per textbook.

Two textbooks offered a path based on a combination of both justification strategies. Five

textbooks offered only one strategy, commonly relying on measurement.


statement. As can be seen, paths that included alternate angles typically opened with it, and

offered measurement as an affirmation of the result.

In justification of corresponding angles, the Israeli school curriculum for grades 7-9 suggests

two justification strategies: measurement followed by alternate angles (Israel Ministry of

Education, 2009). The Israeli curriculum suggests beginning by justifying the congruence of

alternate angles between parallel lines by using measurement and paper folding, and only

then to rely on this congruence in order to deduce the congruence of corresponding angles

between parallel lines. As Figure 56 shows, this sequence was offered in three textbooks, and

most books relied only on measurement.

Table 19. Corresponding angles – frequencies of justification strategies, by textbook


A B C D E F G H

Measurement 1 1 1 2 . 1 2 1 9

Alternate angles 2 . . . 1 . . 1 4

Total 3 1 1 2 1 1 2 2 13



A

B

C

D

E

F

G

H


Figure 56. Corresponding angles – Paths of justification

Measurement

e

Altern. Angles

g

Altern. Angles

g

Measurement

e

Measurement

e

Measurement

e

Measurement

e

Altern. Angles

s

Measurement

e

Measurement

e

Measurement

e

Altern. Angles

s

Measurement

e


6.10. Angle sum of a triangle

Five justification strategies were used in the analyzed textbooks for justifying why the angle

sum of a triangle is 180o. All five justification strategies are addressed and reviewed in the

mathematics education literature: (i) Parallel line (Chazan, 1993; Knuth, 2002; Reiss &

Renkl, 2002; Tall et al., 2012); (ii) Angle rearrangement (Knuth, 2002; Olson, 1975; Reiss &

Renkl, 2002; Tall et al., 2012); (iii) Measurement (Reiss & Renkl, 2002; Tall et al., 2012;

Weaver & Quinn, 1999); (iv) Right triangle (Harel & Sowder, 1998; Knuth, 2002); and (v)

Parallel line and extension (Tall et al., 2012). Additional warrants are mentioned in the

literature yet were not found in the analyzed textbooks (e.g., by arranging congruent triangles

such that they tile a plane, see Reiss & Renkl, 2002). In this section I address two aspects: (1)

The justification strategies offered, in descending order of frequency in the textbooks; and (2)

The paths of justification.


6.10.1. Parallel line: This justification strategy is based on the Alternate Interior Angle

theorem. An auxiliary line, parallel to one of the sides of the triangle, is drawn through the

opposite vertex. The AIA theorem is then applied twice, to find three angles that form a

straight line and thus their sum is 180o (e.g., Figure 57).

Figure 57. Deduction using a general case by Parallel line (Textbook C, vol 3, p. 173)

6.10.2. Angle rearrangement: This justification strategy is based on a physical experiment.

The interior angles of the triangle are rearranged to show that they form a straight line.

Instances of this justification strategy involved either folding a triangle cut out of paper (e.g.,

Figure 58) or tearing it (e.g., Figure 59).

Section 6.10 – Angle sum of a triangle | 90

Figure 58. Experimental demonstration by Angle rearrangement (Textbook B, vol 3, p. 161)

Figure 59. Experimental demonstration by Angle rearrangement (Textbook B, vol 3, p. 161)

6.10.3. Right triangle: This justification strategy is based on properties of right triangles.

Instances of this justification strategy involved relying either on a previously established fact

regarding the sum of the two acute angles of a right triangle (e.g., Figure 60) or on the

property of a rectangle as a quadrilateral with four right angles (e.g., Figure 61).

Figure 60. Deduction using a general case by Right triangle (Textbook D, vol 3, p. 135)


Figure 61. Deduction using a general case by Right triangle (Textbook A, vol 3, p. 647)

6.10.4. Measurement: This justification strategy is based on a practical measurement. The

angles of a triangle are measured to check that their sum is equal to 180o. Instances of this

justification strategy involved using either a protractor (e.g., Figure 62) or dynamic geometry

software (e.g., Figure 63).

Figure 62. Experimental demonstration by Measurement (Textbook H, vol 3, p. 167)

Figure 63. Experimental demonstration by Measurement (adapted from Textbook H, vol 3, p. 167)


6.10.5. Parallel line and extension: This justification strategy is based on relying both on the

Alternate Interior Angle theorem and the Corresponding Angles theorem. An auxiliary line,

parallel to one of the sides of the triangle, is drawn through the opposite vertex, and a ray

extends another side of the triangle. The AIA and CA theorems are then applied once each, to

get three angles that form a straight line and thus their sum is 180o (e.g., Figure 64)

Figure 64. Deduction using a general case by Parallel line & extension (Textbook A, vol 3, p. 621)



mathematics textbooks for the mathematical statement angle sum of a triangle. As can be

seen, justifications were offered in every textbook, commonly more than two instances of

justification per path. All eight textbooks offered a path based on a combination of several

justification strategies, usually parallel line and angle rearrangement.


statement. As can be seen, paths typically opened with a concrete demonstration, either by

angle rearrangement or by measurement. Paths that included both strategies opened with the

more familiar measurement. Following the concrete justification, more abstract justification

strategies were offered, typically relying on parallel line. Four textbooks offered an

additional strategy, relying on properties of right triangle.

Table 20. Angle sum of a triangle – frequencies of justification strategies, by textbook


A B C D E F G H

Parallel line 1 1 2 1 2 1 1 . 9

Angle rearrangement 1 2 1 1 1 1 1 1 9

Right triangle 1 1 . 1 . 1 . . 4

Measurement . 1 . . . 1 . 1 3

Parallel line and extension 1 . . . . . . . 1

Total 4 5 3 3 3 4 2 2 26


In justification of the angle sum of a triangle, the Israeli school curriculum for grades 7-9

suggests two justification strategy: angle rearrangement followed by parallel line (Israel

Ministry of Education, 2009). As Figure 65 shows, this sequence was offered in five

textbooks.


A

B

C

D

E

F

G

H


Figure 65. Angle sum of a triangle – Paths of justification

Rearranging

e

Parallel line

g

Parallel&ext

g

Rt. triangle

g

Measuring

e

Rearranging

e

Rearranging

e

Parallel line

g

Rt. triangle

g

Rearranging

e

Parallel line

s

Parallel line

g

Rearranging

e

Rt. triangle

g

Parallel line

g

Rearranging

e

Parallel line

s

Parallel line

g

Measuring

e

Rearranging

e

Rt. triangle

g

Parallel line

g

Rearranging

e

Parallel line

s

Measuring

e

Rearranging

e

Section 6.11 – Summary | 94

6.11. Summary

Analysis of the paths of justification focused on three attributes: (1) characteristics – the

justification strategies in each path; (2) sequencing – the order in which justification

strategies were offered in the textbook, and (3) the ways in which the specific characteristics

(i.e., justification strategy) relate to the meta-level characteristics (i.e., justification type).

Table 21 presents the number of justification strategies per path of justification, by

mathematical statement and textbook. A total of 26 distinct justification strategies were

offered in the textbooks for the 10 analysed statements, commonly two or more per statement

across the textbooks. Half of the paths included more than one strategy (42 of 80). Most

textbooks (all but Textbook A) offered a similar number of strategies, regardless of the scope

of the textbook (i.e., limited or standard/expanded scope). Additionally, a similar number of

strategies were offered in algebra and in geometry.

Table 21. Number of justification strategies per path, by statement and textbook.

Mathematical statement Textbook Total

A

B

2.3

B

2.17

2

C D E F G H Distinct

Equivalent expressions 3 2 2 2 3 2 2 2 3

Division by zero 2 1 1 1 1 1 1 1 2

Distributive law 2 2 1 2 2 2 2 1 2

Equivalent equations 2 1 1 1 2 2 1 1 4

Product of negatives 3 1 2 2 2 1 1 1 3

Area of a trapezium 2 2 2 1 2 2 2 2 2

Area of a disk 1 1 1 1 0 1 1 1 1

Vertical angles 2 1 1 1 1 2 2 1 2

Corresponding angles 2 1 1 1 1 1 1 2 2

Angle sum of a triangle 4 4 2 3 2 4 2 2 5

Total 23 16 14 15 16 18 15 14 26

Out of the 26 justification strategies identified in the textbooks, 23 were used more than once

for a specific statement across the textbooks. Table 22 presents the frequencies of types of

justification in the textbooks, by justification strategy. As can be seen, justification strategies

often corresponded to a single type of justification (70%, 16 of 23). For certain justification

strategies, as many as 16 instances of justification all involved a single type of justification.

For example, in justification of the area of a disk, 8 instances of justification relied on

dissection, and all eight involved a single type of justification – deduction using a general

case. This suggests the existence of a one-sided dependency between the two characteristics

of instances of justification.


Table 22. Frequencies of justification strategies, by justification types

Justification strategy Justification type Total

a q r e m s g

Equivalent expressions

Rules and conventions . . . . . 10 . 10

Substitution . . . 6 . . . 6

Description equivalence . . . . 4 . . 4

Division by zero

Inverse of multiplication . . . . . 16 . 16

Repeated subtraction . . . . 1 . . 1

Distributive law

Area / array . . . . 9 . . 9

Arithmetic conventions . . 8 . . . . 8

Equivalent equations

Balance model . . . . 5 . . 5

Undoing . . . 3 . . . 3

Intuitive comparison . . 1 2 . . . 3

Segment model . . . . 1 . . 1

Product of negatives

Discovering patterns 1 . . 7 . . . 8

Extension of properties . . . . . 5 . 5

Number-line models . . . . 2 . . 2

Area of a trapezium

Dissection . . 2 11 . 10 5 28

Construction . . 1 1 . 4 5 11

Area of a disk

Dissection . . . . . . 9 8

Vertical angles

Supplementary angles . . . . . 5 7 12

Measurement . . . 1 . . . 4

Corresponding angles


Alternate angles . . . . . 2 2 4

Angle sum of a triangle

Parallel line . . . . . 3 6 9

Angle rearrangement . . . 4 . . . 9

Right triangle . . . . . . 4 4


Parallel line and extension . . . . . . 1 1

a=appeal to authority; q=qualitative analogy; r=concordance of a rule with a model;

e=experimental demonstration; m/s/g=deduction using a model/ specific case/general case.


Further analysis focused on the mathematical topic – algebra and geometry. Again, analysis

focused on mathematical means that were used more than once for each statement across the

textbooks. This analysis reveals that the association between the two characteristics was

much stronger for algebra statements (83%, 10 of 12) compared with geometry statements

(55%, 6 of 11). In other words, paths of justification in geometry offered a greater number of

justification strategies which were associated with multiple types of justification. Analysis of

these paths revealed two distinct patterns. First, for justification strategies that involved both

empirical and deductive types, the empirical preceded the deductive types (2 of 2). Second,

strategies that involved two deductive types of justification were offered in a particular

sequence: deduction using a specific case followed by deduction using a general case (5 of 5).

For each mathematical statement I analyzed the paths of justification for sequences of

justification strategies (i.e., all combinations of one or more justification strategies offered

sequentially in each path). Figure 66 presents the most common segment of justification

strategies for each statement across the textbooks, the textbooks that offer that segment, and

the sequence suggested in the Israeli school curriculum for grades 7-9 (Israel Ministry of

Education, 2009). As can be seen, for most statements (8 of 10), at least five textbooks

offered similar sequences of justification strategies. Moreover, for most statements (8 of 10),

the segments were very similar to the sequences suggested in the Israeli curriculum – either

identical or included in these sequences. For two statements, two segments were equally

common in the textbooks, and the longer segment is illustrated here. Segments of three or

more justification strategies were very rarely found in more than one path (only once, for

angle sum of a triangle).

Further analysis of the sequences of justification strategies focused on the associated types of

justification. This analysis suggests that the order in which justification strategies were

offered in the analyzed textbooks might be directly related to the sequences of justification

types. As described in Section 5.4, Experimental demonstration almost always preceded

deductive type(s), and Deduction using a specific case always preceded Deduction using a

general case. Similarly, paths of justification commonly offered justification strategies

associated with experimentation were before strategies associated with deduction using either

a generic example or a general case. Figure 66 exemplifies this tendency for the most common

segments of these sequences.


Statement Most common segment (Textbooks) Israeli curriculum

Equivalent expressions

(A,B,D,E,F,G)

Division by zero

(B,C,D,E,F,G,H)

Distributive law

(A,B,D,E,F,G)

Equivalent equations

(C,H)*

Product of negatives

(C,D,E)*

Area of a trapezium

(B,C,F,G,H)

Area of a disk

(A,B,C,D,F,G,H)

Vertical angles

(B,C,D,E,H)

Corresponding angle

(B,C,D,F,G)

Angle sum of a triangle

(A,B,C,E,G)

(*) indicates that two segments were equally common in the textbooks for that statement.

Figure 66. Common sequences of justification strategies in the textbooks and the Israeli curriculum

Substitution

e

Conventions

s

Conventions

Inverse a:0 s

Inverse a:0

Inverse 0:0

Area/Array

m

Conventions

r

Area/Array

Conventions

Balance

m

Balance

Patterns

e

Properties

s

Properties

Dissection

e/s/g

Construction

e/s/g

Dissection

Construction

Dissection

g

Dissection

Supplementar

s/g

Measurement

Supplementar

Measurement

e

Measurement

Altern. Angles

Rearranging

e

Parallel line

g

Rearranging

Parallel line

Paths of justification in Lena's classes – background | 98

7. Paths of justification in Lena's classes

In this chapter I examine the ways in which the textbook, together with the teacher and the

students, shape the opportunities offered for students to learn to explain and justify. The

current chapter focuses on findings for one case study: Lena and her classes. Chapter 8 deals

with Millie and her classes.

Lena received a B.Ed. from a teachers' college, and majored in mathematics. She had five

years of experience teaching mathematics, all at the junior-high school level. However, the

year of data collection was Lena's first year teaching at that school.

There were two 7th grade classes in the school, and Lena taught both: Class L1 with 29

students and Class L2 with 24. Observations suggest that the student ability level in both

classes was diverse, and there were frequent disciplinary issues. Once a week, a

supplementary teacher aided Lena during geometry lessons by taking twelve advanced

students out of the classroom and teaching them in parallel. The lessons given by the

supplementary teacher were not documented due to research constraints.

The year of data collection was the first year Lena had used Textbook C. In the concluding

teacher interview Lena conveyed that she planned her lessons by reviewing Textbook C and

its teacher's guide. However, she revealed that she thought the textbook was unsuitable at

times for students with low achievements. Indeed, Lena occasionally used other textbooks as

a source of introductory exercises and activities.

Observations in Lena's classes were made for five mathematical statements, one in algebra

and four in geometry: Equivalent equations, Area of a trapezium, Area of a disk, Vertical

angles, and Angle sum of a triangle. For each statement, lessons in both classes took place on

the same day – the lesson in Class L2 was immediately followed by a lesson in Class L1.

This chapter comprises six sections. The first five sections deal with the paths of justification

in the observed classrooms for each statement. Each section begins with lesson graphs which

provide a general outline of the observed lessons in each class. The graphs describe the main

sections of the observed lesson, along with a graphic representation of the duration of each

section in each class. The graphs are followed by a description of the paths of justification in

the textbook and in the classes. The sixth and final section summarizes the case study.

99 | Paths of justification in Lena's classes


7.1.1. Lesson graphs


14 minutes

8 minutes

Administration


30 minutes

20 minutes

Equation solving (Whole class)

Lesson goal – revisiting equations, an emphasis on understanding.

Launching task (see Figure 68) – finding the weight of a ball on a

balanced scale. Lena discusses the relation between operations on the

scale and operations on a corresponding equation.

Reading a general-knowledge textbook segment about balance scales.

36 minutes

59 minutes

Equation solving (Individual student work)

Students work on textbook tasks: solving equations with and without

the balance model, and attending to the limitations of the model.

After 20-25 minutes of student work, Lena sits with two-three students,

brings a physical model of a balance scale, and uses it to explain the

relation between operations on it and equations. During that time the

rest of the students keep working.


7.1.2. The paths of justification

Figure 67 presents the paths of justification for the statement Performing valid operations on

both sides of an equation yields an equivalent equation, by class and textbook. Each step

represents a single instance of justification by order of appearance in the textbook/classroom

lesson, and both the justification strategy and type are presented.

Statement Class L1 Class L2 Textbook C

Equivalent

equations

m =deduction using a model.

Figure 67. Paths of justification in Lena's classes for Equivalent equations

Balance

m

Balance

m

Balance

m


7.1.2.1. The path of justification in the textbook

The path of justification in Textbook C comprised one instance of justification. The textbook

began by discussing several intuitive methods for solving equations (e.g., cover-up). One

such method was introduced as performing operations on both sides of an equation, yet

without justification. In the following volume of the textbook, several chapters later, the

subject of equation solving was addressed again. The mathematical statement was justified by

using a Balance model (i.e., modelling an equation as two sides of a balance scale) in a

deduction using a model. The textbook discussed the relationship between operations on the

model and on the original equation (see Figure 68), as well as certain limitations of the model

(e.g., not every equation can be modelled as a balance scale, see Figure 69).

Figure 68. Balance model (Textbook C, vol 3, p. 54)

Figure 69. Balance model – limitations (Textbook C, vol 3, p. 58)


7.1.2.2. The paths of justification in the classrooms

The paths of justification in Lena's classes were very similar to the textbook path and to each

other. In both classes, the paths included one instance of justification, relying on balance

model in a deduction using a model (see Figure 70 (a) and (b)). Lena solved a textbook task

that involved finding the weight of a ball on a balanced scale by performing operations on a

balance model. Then, she made connected between operations on the model and operations

on an equation (e.g., subtracting a number from both sides of an equation is modelled as

removing that number of weights from both sides of the scale).

This connection was exemplified differently in each class. In Class L2 Lena mentioned the

corresponding equation, whereas later that day in Class L1, Lena explicitly wrote down that

equation and solved it (see Figure 70 (a)). This difference appears to be didactical in nature

and might stem from the experience Lena gained between lessons.

(a) (b) Figure 70. Balance model (screenshots in Class L1 [a] and Class L2 [b])

In both classes, the flow of the lesson was interrupted frequently in order to discipline the

students. These interruptions cut short several conversations between Lena and her students,

some of them pertaining to the justification of the mathematical statement.


7.2. Area of a trapezium



9 minutes

9 minutes

Administration


6 minutes

5 minutes

Recap and Defining a trapezium (Whole class)

Lesson goal – finding the area of a trapezium by using known area

formulae.

Recap of the area formulae for rectangle, square, triangle, and

parallelogram.

Defining a trapezium as a quadrilateral with two parallel sides.

–

3 minutes

Trapezium dissections (Whole class)

In L2: Lena asks for ways to dissect a trapezium into the known shapes.

Several options are suggested [ , , , ], and Lena discusses

the potential efficiency of the dissections.

10 minutes

10 minutes

Calculating the area of a trapezium numerically (Whole class)

Lena sketches a trapezium dissected into a rectangle and two triangles,

and provides the measures for each segment. Lena instructs the class to

calculate the area of each part separately and add them together.

21 minutes

23 minutes


Students work on textbook tasks: calculating the area of a trapezium by

using known area formulae, and identifying trapezia by their definition.

9 minutes

11 minutes

Justifying the area formula for a trapezium (Whole class)

In L2: Lena relies on the intermediate results achieved earlier by the

students and begins89 min a process of rearranging these results in order

to derive the area formula. However, this process is not completed, and

instead Lena writes the general area formula.

In both classes: Lena then draws a diagonal to dissect a given trapezium

into two triangles, and obtains the general area formula: altitude*(short

base + long base)/2.

26 minutes

28 minutes


Students continue working on textbook tasks by using the area formula.




Figure 71 presents the paths of justification for the statement The area formula for a

trapezium with bases a, b and altitude h is (a + b)*h/2, by class and textbook.


Area of a

trapezium

e=experimental demonstration; r=concordance of a rule with a model; s/g =deduction using

a specific case/ a general case; D=Dissection, C=Construction.

Figure 71. Paths of justification in Lena's classes for Area of a trapezium


The path of justification in Textbook C comprised five instances of justification. Two

justification strategies were offered: dissection (i.e., the area is represented as a sum of areas

of shapes with known area formulae), followed by construction (i.e., arranging congruent

trapeziums to form a shape with a known area formula).

Dissection involved a transition from an empirical type of justification (experimental

demonstration, see Figure 72) to a deductive type (deduction using a specific case, see Figure

73). Construction involved a transition from a special case (deduction using a specific case,

see Figure 73) to the general (deduction using a general case, see Figure 74).

Figure 72. Dissection (Textbook C, vol 2, p. 112)

Figure 73. Dissection and Construction (Textbook C, vol 2, p. 112)

D s

C s

D s

D s

D e

D s

D r

C s

D s

D s

D e

C g

C s

D s

D s

D e


Figure 74. Deduction using a general case (Textbook C, vol 2, p. 112)


The paths of justification in Lena's classes were very similar to the textbook path. Lena

started with experimental demonstration by dissection both in Class L1 (see Figure 75(a)) and

in Class L2 (see Figure 75(b)). Lena dissected a trapezium ABCD with given measures into a

rectangle ABEH and two right triangles – ADH and BCE, and instructed the students to find

the area of each shape and add them together.

(a) (b) Figure 75. Dissection (screenshots in Class L1 [a] and Class L2 [b])

Then Lena instructed the students to work individually on two textbook tasks. These tasks

offered four additional instances of justification – one relying on experimental demonstration

(see Figure 72) and three relying on deduction using a specific case (see Figure 73). The

students reported experiencing difficulties during this task.

Following the individual student work, Lena stated their goal of finding an area formula

instead of dissecting it each time. In both classes, Lena began a process of dissection by

deduction using a specific case. However, in Class L2 Lena altered the type of justification

mid-process.


The deductive process that began in Class L2 can be described in four steps (see Figure 76):

(1) Dissecting Trapezium ABCD into a rectangle ABEH and two right triangles – ADH and

BCE, (2) Finding the area of each shape, (3) rewriting the sum of areas in order to have a

single denominator, and (4) rewriting the sum by identifying the lengths of each base. The

third step requires careful arithmetic manipulation – multiplying and dividing the area of the

rectangle by the same number.

1 2 3 43 10 10 2 4 15 10 43 4 2 4

10 42 2 2 2ABCD ADH ABEH BCES S S S

Figure 76. Derivation of the area formula of a trapezium in a specific case

In Class L2, Lena relied on the intermediate results achieved earlier by the students (see

Figure 75(b)) for the first two steps of the process. She began the third step (see Figure 77(a)),

but stopped it following numerous interjections from the students (e.g., suggestions, many of

which incorrect). Instead, she declared they were going to "sum it up", and dictated the area

formula for a trapezium (see Figure 77(b)). Finally, Lena verified the formula by showing a

correspondence between two results – the result achieved by using the rule and the result

achieved earlier by the students (Figure 78).

(a) (b) Figure 77. The deductive process in Class L2 [a], replaced by giving the rule [b]


Figure 78. Concordance of a rule with a model (screenshot in Class L2)

Then, Lena emphasized that the students were required to remember only the area formula,

and not a specific method of dissection. To strengthen her point, Lena dissected the trapezium

into two triangles by drawing a diagonal. The deduction process can be described in three

steps (see Figure 79): (1) Dissecting Trapezium ABCD into two triangles – ABC and ACD,

(2) Finding the area of each shape, (3) rewriting the sum to have a single denominator.

Here the arithmetic manipulation is much simpler, yet finding the area of triangle ABC might

be a demanding task for students due to need to extend one side and draw an exterior altitude.

As Lena commented in Class L2: "By the way, in this dissection we can see better, so I will

do it for you so it will be better for you". Lena followed the three steps of the deductive

process mentioned above and completed this justification, both in Class L1 (see Figure 80(a))

and in Class L2 (see Figure 80(b)).

1 2 3 15 10 410 4 15 4

2 2 2ABCD ABC ACDS S S

Figure 79. Derivation of the area formula of a trapezium in a specific case


(a) (b) Figure 80. Deduction using a specific case (screenshots in Class L1 [a] and Class L2 [b])

The differences between the two lessons are likely a result of the experience Lena gained by

teaching in Class L2. It is not likely that these differences stem from time constraints,

because in both classes, approximately 50 minutes were allotted to student work following

the derivation of the area formula as a whole class activity.





6 minutes

3 minutes

Administration


7 minutes

6 minutes

Defining supplementary angles (Whole class)

Lena defines supplementary angles as a pair of angles whose sum is

180o.

Lena draws a line and the students identify two straight angles. Lena

adds an additional line and the students identify four pairs of

supplementary angles.

10 minutes

9 minutes

Defining vertical angles + phrasing the rule (Whole class)

Lena directs attention to a pair of vertical angles, names them, and asks

what can be said about the magnitude of vertically opposite angles.

In L1: several students reply that they are equal and Lena approves.

In L2: Lena says they are equal.

Lena writes the rule, provides a numerical example, and defines

vertical angles: 'a pair of angles with a common vertex that are not

supplementary angles'.

39 minutes

65 minutes

Naming and calculating angles (Individual student work)

Students work on textbook tasks: identifying vertical and

supplementary angles, and using the rule to calculate angle measures.

Students work on tasks from photocopied from a different textbook

(Textbook B): naming angles by three letters.

3 minutes

–

Naming angles (Whole class)

In L1: Due to student difficulties, Lena interrupts the individual work

for a short review on naming an angle by using three letters.

17 minutes

Naming and calculating angles (Individual student work)

Students continue working on tasks photocopied from Textbook B:

naming angles by using three letters.




Figure 81 presents the paths of justification for the statement Vertically opposite angles are

congruent, by class and textbook.


Vertical

angles

– –

s/g =deduction using a specific case/ a general case.

Figure 81. Paths of justification in Lena's classes for Vertical angles


The path of justification in Textbook C comprised two instances of justification. Both relied

on the same justification strategy: supplementary angles (i.e., a third angle is identified as an

angle that supplements each of the vertically opposite angles). Additionally, both instances

relied on deductive types of justification: First by deduction using a specific case (see Figure

82) and then by deduction using a general case (see Figure 83).

Figure 82. Vertical angles – deduction using a specific case (Textbook C, vol 2, p. 184)

Figure 83. Vertical angles – deduction using a general case (Textbook C, vol 2, p. 184)

Supplementary

s

Supplementary

g



Lena presented the rule but did not offer any instances of justification in either class.

In each class, Lena started by drawing a pair of intersecting lines and instructed the students

to identify pairs of supplementary angles and vertically opposite angles. Then Lena inquired

whether any student knew what property these angles have. After a student gave the correct

answer, Lena confirmed and dictated the rule. For example, in Class L1:

Lena: [points at angles 2 and 4, see Figure 84] what can you say about their magnitude?

Student: They are obtuse.

Lena: […] other than their being obtuse, what else?

Same student: They are equal.

Lena: They are equal, well done

Figure 84. Vertical angles (screenshot in Class L1)

Classroom observations reveal that justification was neither requested nor given in Lena's

classes, either by Lena or by the students. After stating the rule, Lena instructed the students

to solve a textbook task by using the rule (see Figure 82). However, Lena expressed a

different approach to the need for justification for this statement in the concluding interview.

As part of the interview, Lena was shown a set of three paths of justification for this

mathematical statement (see Figure 85). In this set, one teacher offers a path similar to the

path offered in Textbook C (i.e., a generic example, followed by derivation in the general

case, and then followed by a statement of the rule), a second teacher offers the same path but

in reverse-order, and a third teacher offers a path that is based on a direct comparison of the

vertically opposite angles by using a protractor.


Figure 85. A set of paths of justification for vertical angles (interview item).

Lena described each path as a variation she had used in the past in her classes, based on the

level of achievements of her students – either low (Teacher B), intermediate (Teacher A), or

high (Teacher C). Moreover, Lena expressed her disdain for Teacher B for stating the rule

without justifying it:

"I have taught these three explanations [i.e., paths of justification] before, depending on the

class. First of all, Teacher B – I give that to the weakest students. I come, give them the rule,

and prove it for them. Giving it as, as a proof. […] I don't like Teacher B, coming and saying

'okay, this is the rule, learn it by heart and that is all there is', because they [the students] don't

remember much [that way]."


7.4. Angle sum of a triangle



4 minutes

8 minutes

Administration


9 minutes

10 minutes

The angle sum of a triangle (Whole class)

Lena describes a process in which a triangle is dissected and its angles

are rearranged (see Figure 87).

In L1: Lena asks what the outcome of the process is and students

suggest various options. When a student suggests that the angles form a

straight angle, Lena approves and states that this is the proof.

In L2: Lena reveals that the angles form a straight angle, and states that

this is the proof they need to remember.

Students solve a textbook task involving calculating the third angle of a

triangle (see Figure 88). Lena asks for two justifications – by using the

rule and by using alternate angles, and regards them as equivalent.

Lena dictates the rule: ‘the angle sum in a triangle is 180’.

64 minutes

65 minutes

The angle sum of a triangle (Individual student work)

Students work on textbook tasks: calculating angles in triangles by

using the rule, numerically and with variables.



Figure 86 presents the paths of justification for the statement The angle sum of a triangle is

180o, by class and textbook.


Angle sum

of a ∆

e=experimental demonstrations/g =deduction using a specific case/ a general case.

Figure 86. Paths of justification in Lena's classes for Angle sum of a triangle

Rearranging e

Rearranging e

Rearranging e

Parallel line s

Parallel line g



The path of justification in Textbook C comprised three instances of justification. First, the

textbook offered experimental demonstration by angle rearrangement (i.e., the interior

angles are rearranged to form a straight line, see Figure 87). Based on the physical

experiment, students were asked to make a conjecture regarding the angle sum of a triangle,

and then the textbook stated that the conjecture will be properly verified. Two items were

offered involving parallel line (i.e., relying on the Alternate Interior Angle theorem twice to

find three angles that form a straight line): first by deduction using a specific case (see Figure

88), and then by deduction using a general case (see Figure 89).

Figure 87. Angle sum of a triangle – Angle rearrangement (Textbook C, vol 3, p. 172)

Figure 88. Angle sum of a triangle – Parallel line (adapted from Textbook C, vol 3, p. 173)

Figure 89. Angle sum of a triangle – Parallel line (Textbook C, vol 3, p. 173)



The paths of justification in Lena's classes were very similar to each other and relied on the

textbook. In both classes, the paths included one instance of justification, relying on

experimental demonstration by angle rearrangement.

In both classes, Lena read the textbook task (see Figure 87) and used gestures to describe how

the angles of the triangle were rearranged. In both classes, Lena was the provider of the

justification, and the students helped phrase the rule. For example, in Class L2:

Lena: Pay attention to the task, we are not going to do it […] Look at [page] 172, the

way they tore it. They had ABC triangle and they wanted to know, without a protractor

[…] They had a triangle, do you see it?

Students: Yes.

Lena: They tore it in some manner, to know what the angle sum is, and they added A, B,

and C. Can you see the illustration below? [points at Figure 87] And we see that if we add

the three angles of the triangle, what do we get?

Student: 180 degrees.

Lena: 180, straight angle. […] This is the proof you need to remember, why for the three

angles inside a triangle, their sum is 180. You can do the experiment. You'll tear it and

try to connect it [gestures with her hands].

Similarly, in Class L1, Lena explicitly referred to this instance of justification as a proof that

the angle sum of a triangle is a straight angle.

Lena: The proof is like so – if you add all three, you get a straight angle, and you know

that a straight angle is 180. Therefore, any triangle, if I add all three, what do I get?

Students: 180, a straight angle.

Then, in both classes, Lena instructed her students to solve a textbook task (see Figure 88) in

two ways: (1) by relying on the rule, and (2) by relying on the congruence of alternate angles

between parallel lines. Lena then described these two ways as equivalent and sufficient for

exam questions. This was not regarded in class as a justification for the rule and accordingly

was not coded as an instance of justification.


7.5. Area of a disk



2 minutes

– Administration


8 minutes

8 minutes

Defining terminology (Whole class)

Lena discusses the following terms: circle, disk, radius, chord, and

diameter. The discussion involves asking students what each term is,

defining the term, and the appropriate mathematical notation.

6 minutes

5 minutes

The perimeter of a circle (Whole class)

Lena tells the student a story about how the constant ratio between the

perimeter of a circle and its diameter was discovered.

Lena deduces that in order to calculate the perimeter of a circle, one has

to multiply its diameter by π.

In L2: Lena demonstrates solving a numerical example by using the

rule, and emphasizes writing an exact answer and not an approximation

(e.g., the perimeter of a circle with radius 3 is 6π and not 18.84).

5 minutes

7 minutes

The area of a disk (Whole class)

Lena inquires whether any student remembers how to calculate the area

of a disk. Students make suggestions, among them is π times radius

squared. Lena approves and dictates the rule:

In L1: ‘S = radius*radius*π = radius2*π’.

In L2: ‘S = r*r*π = r2*π’.

Lena demonstrates solving a numerical example by using the rule.

– 47 minutes

The area of a disk (Individual student work)

In L2: Students work on textbook tasks: calculating the radius and

perimeter of a circle, and the area of a disk, all by using the rules.

18 minutes

9 minutes

Perimeter and area (Whole class)

Lena solves a question involving the area and perimeter of a circle. The

question was taken from an exam similar to an upcoming regional

exam.




Figure 90 presents the paths of justification for the statement The area formula for a disk with

radius r is πr2, by class and textbook.


Area of a

disk

– –

g =deduction using a general case.

Figure 90. Paths of justification in Lena's classes for Area of a disk


The path of justification in Textbook C comprised one instance of justification. The textbook

offered a process of deduction using a general case by dissection (i.e., dissecting a disk into

sectors and rearranging them to form an approximate rectangle, see Figure 91). Students were

given a disk with unspecified measures and were required to estimate the measures of the

sides of the approximate rectangle. However, neither the textbook nor the teacher's guide

specified how to determine these measures (e.g., numerically or with pronumerals). In

addition, the relation between the circumference formula and the area was explicitly stated

only after the task (see Figure 92).

Figure 91. Area of a disk – Dissection into sectors (Textbook C, part 3, p. 156)

Dissection

g


Figure 92. Area of a disk – the circumference and the area (Textbook C, part 3, p. 156)


Lena presented the rule but did not offer any instances of justification in either class.

In both classes, Lena prefaced by describing the lesson as a review of a topic they had learned

in the preceding year (i.e., in elementary school) and had not discussed since. Then, Lena

inquired whether any of the students remembered the area formula of a disk. Several students

made an attempt at answering it, calling out several suggestions, until one student gave the

correct formula. Lena confirmed and dictated the rule. For example, in Class L1:

Lena: How do I calculate the area, anybody remembers?

Student1: Me, me, me! Radius times Pi!

Student2: Pi times the diameter! Diameter times 3.14!

Student3: Radius squared

Student4: I know! Radius plus radius- [Lena signals to start over] radius times radius

[Lena signals to continue] times Pi.

Lena: Well done, radius times radius […] times Pi.

Classroom observations reveal that justification was neither requested nor given in Lena's

classes, either by Lena or by the students. After stating the rule, Lena instructed the students

to solve a task by using the rule. However, Lena expressed a different approach to the need

for justification for this statement in the concluding interview.

Lena spoke about the importance she sees in showing 7th grade students why things they

learned in elementary school are true. Lena described the difference between proving a rule

in the seventh grade and merely stating it at the elementary school level:

The bar is raised. In third and fourth grades you just state the formula and demonstrate in

one, two, three, four ways. You don't prove. There's no proof in sixth and fifth grades.

[…] In higher grades you prove. It is different, because then you explain why.


Additionally, Lena was shown a set of three paths of justification for this statement (see

Figure 93). In this set, one teacher states the rule without offering any justification, claiming

she's pressed for time and that most students are already familiar with the formula; the second

teacher offers a path with two instances of justification, relying on two types of dissection –

into sectors and into concentric rings; and the third teacher offers a path with two instances of

justification, relying first on grid approximation (i.e., counting squares that coincide with the

disk) and then on dissection into sectors.

Figure 93. A set of paths of justification for area of a disk (interview item)

Lena claimed that unlike the first teacher, none of her students ever remembers the area

formula on their own. Furthermore, Lena described her own path of justification as similar to

the third teacher yet comprising only the first item (i.e., relying on the grid approximation

justification strategy). When asked to elaborate, Lena's answer suggested a limited

understanding of the reasoning of the strategy:


Lena: I put a disk inside a square, and then we count the squares – one, two, three, four,

this can be completed to five [pointing at the top-middle square], and this to six

[pointing at the bottom-middle square].

Interviewer: [pointing at one corner] what about this here?

Lena: That's it, the sides are pi. That's a supplement.

Interviewer: Wait, do you count squares?

Lena: I count the squares, yes. […] We can see that the area of the disk is less than nine

[…] Less than nine, more than five. How can I know? So I tell them that the

supplements are pi. You need to multiply by pi. That's why, if we have a square, it's

three times three, so it's three squared, so I multiply by pi, because I have those little

bits. And then they [i.e., the students] remember.


7.6. Summary

In this section I discuss the contribution of the textbook, together with the teacher and the

students to shaping students' opportunities to learn how to explain and justify mathematical

statements in Lena's classes. Figure 94 presents the paths of justification for each of the five

observed statements, by class.


Equivalent

equations

Area of a

trapezium

Vertical

angles

– –

Angle sum

of a ∆

Area of a

disk

– –

e=experimental demonstration; r=concordance of a rule with a model; m/s/g =deduction using

a model/ a specific case/ a general case; D=Dissection, C=Construction.

Figure 94. Paths of justification in Lena's classes, by statement

The textbook was the main source for instances of justification in Lena's classes. Analysis of

the paths of justification revealed that for each mathematical statement, every justification

strategy and almost every justification type that Lena offered in her classes, were offered in

the textbook as well and in the same order.

Based on the results of the analysis conducted in Part I of this study, it appears that instances

of justification from other textbooks were not found in the observed lessons. Lena

occasionally used other sources (e.g., other textbooks and worksheets), merely as a resource

for additional tasks and activities. Lena described this use as a result of her impression of

Textbook C as containing an insufficient number of basic-level exercises.

Balance

m

Balance

m

Balance

m

D

s

C

s

D

s

D

s

D

e

D

s

D

r

C

s

D

s

D

s

D

e

C

g

C

s

D

s

D

s

D

e

Supplementary

s

Supplementary

g

Rearranging e

Rearranging e

Rearranging e

Parallel line s

Parallel line g

Dissection

g


Near-identical paths were offered in both classes. In the concluding teacher interview Lena

implied that the similarity may be intentional, and may reflect a similarity in her perception

of the level of the students: "The level of each class was essentially the same […] so the

worksheets and the lesson plans were almost identical in both classes."

The general similarity between the paths of justification in the textbook and in the classes

suggests that Lena often followed teaching sequences suggested by the textbook. However,

despite these similarities, several differences were found – pertaining mostly to deductive

types of justification.

First, several instances of justification were offered in the textbook but not in Lena's classes.

Most of these excluded instances were coded as deduction using a general case – the type of

justification closest to a formal proof. Moreover, for area of a trapezium and angle sum of a

triangle, Lena followed the path of justification offered in the textbook, but excluded

instances of justification which relied on deduction using a general case. Lena did not offer

justification for the other two statements.

The absence of the deductive justification type in Lena's classes is likely intentional. Lena

described justification and proof as important for remembering the material, instead of rote

learning, yet not suitable for all students:

It depends on the kid's level. Some children are not ready, it does not matter whether you

explain to them – they will not understand. […] However, for those with [mathematical]

thinking, that they're good at it, it was important to me, yes, because then they know the

origin, the 'why', and then it is easier to remember the formula. They don't memorize.

Furthermore, Lena described the extensive preparation required in classes with mixed ability

before proof is given: "If the [ability] level is mixed, in geometry, I cannot just give a proof

directly without taking into consideration those who do not understand." As an interim

solution, her advanced students were given the option to study geometry separately, in a

small group with a supporting teacher, focusing on proof and proving: "Our intention was

that as soon as seventh grade geometry lessons begin, she [the supporting teacher] will teach

them […] what proof is, how to write it, why we do it."

Lena introduced two mathematical statements in class without any justification. In both cases,

neither Lena nor any student asked for a justification to be provided. Classroom observations

suggest that Lena had considered these statements as either self-explanatory (e.g., vertical

angles) or as known results (e.g., area of a disk).


Additionally, Lena made a few alterations to the paths of justification offered in the textbook.

For example, for angle sum of a triangle, Lena regarded an empirical experiment as a

sufficient proof for the statement, whereas the textbook path of justification regarded it as an

opportunity to make a conjecture before providing a deductive justification. For area of a

trapezium, Lena altered the type of justification of one instance of justification from

deductive to empirical. The change was likely related to student difficulties.

Moreover, Lena was the sole initiator of instances of justification in her classes. Lena

commonly requested the students to make a claim (e.g., cite a fact or generate a conjecture)

regarding the mathematical statement in question. Then, one or more students attempted at

phrasing the claim to the teacher's satisfaction. Once the claim was properly phrased, either

Lena provided one or more instances of justification for it, during which students participated

by answering specific questions, or the claim was left without any justification. The students

were not requested to provide additional justifications for any of the observed claims, even

when no justification was offered by the teacher or when mistakes were made in the process.

The classroom atmosphere in both classes was typically noisy. As a result, the flow of most

lessons was interrupted frequently for disciplinary purposes in light of excessive noise and

behavioural issues. These interruptions contributed to shaping the paths of justification by

cutting short several classroom discussions, some of them pertaining to instances of

justification.

123 | Paths of justification in Millie's classes

8. Paths of justification in Millie's classes

This chapter further examines the ways in which the textbook, together with the teacher and

the students, shape the opportunities offered for students to learn to explain and justify. It

focuses on one case study – Millie and her classes. Chapter 7 dealt with Lena and her classes.

Millie received a college bachelor's degree in business management, and later received her

teaching credentials in mathematics from a teachers' college, in a teaching certification

program for those holding an academic degree. She had three years of prior experience

teaching mathematics, all at the same junior-high school.

When observations began in December 2015, there were eight 7th grade classes in the school

and two classes for lower-track students (Mitzuy). Each lower-track class comprised students

from four classes. In late March 2016 the lower-track was cancelled and students rejoined

their original classes.

Millie taught two 7th grade classes. When observations began: Class M1 with 32 students

(main stream), and Class M2 with 20 students (lower-track). After the lower-track classes

were disassembled: Class M1 with 35 students (32 stayed from the original M1), and Class

M2 with 30 students (5 stayed from the original M2). Observations suggest that most students

actively participated in the classroom and there were very few disciplinary issues.

The year of data collection was the fourth year Millie had used Textbook C. In the interview

Millie conveyed that she was fond of the way the textbook presented mathematics to students

by using real-life examples, and that she relied mostly on it when planning her lessons. In

class, Millie projected the textbook on the whiteboard and interacted with it.

Observations in Millie's classrooms were made for five mathematical statements, two in

algebra and three in geometry: Equivalent equations, Product of negatives, Vertical angles,

Corresponding angles, and Area of a disk. For each statement, the lesson in Class M1 took

place before the lesson in Class M2, commonly within the same week.

This chapter comprises six sections. The first five sections deal with the paths of justification

in the observed classrooms for each statement. Each section begins with lesson graphs which

provide a general outline of the observed lessons in each class. The graphs describe the main

sections of the observed lesson, along with a graphic representation of the duration of each

section in each class. The graphs are followed by a description of the paths of justification in

the textbook and in the classes. The sixth and final section summarizes the case study.




Class M1 Class M2 Activity (grouping)

10 minutes

8 minutes

Administration


28 minutes

35 minutes

Equation solving (Whole class)

Lesson goal – learning to solve a new type of equations.

Launching task – finding the weight of a ball on a balanced scale. Millie

discussed the relation between operations on the scale and operations on

a corresponding equation.

6 minutes

4 minutes

Equation solving (Individual student work)

Students work on textbook tasks: solving equations with and without the

balance model, and attending to the limitations of the model.

42 minutes

31 minutes

Word problems (Whole class)

Millie provides an algorithm for approaching and solving word problems.



Figure 95 presents the paths of justification for the statement Performing valid operations on

both sides of an equation yields an equivalent equation, by class and textbook. Each step

represents a single instance of justification by order of appearance in the textbook/classroom

lesson, and both the justification strategy and type are presented.

Statement Class M1 Class M2 Textbook C

Equivalent

equations

m = deduction using a model.

Figure 95. Paths of justification in Millie's classes for Equivalent equations


This path was described in discussion of Lena's classes (see Section 7.1.2).

Balance

m

Balance

m

Balance

m



The paths of justification in Millie's classes were very similar to the textbook path and to

each other. In both classes, the paths included one instance of justification, relying on

balance model in a deduction using a model (see Figure 96). Millie solved a task that involved

finding the weight of a ball on a balanced scale by performing operations on a balance model.

Then, she made connected between operations on the model and operations on an equation

(e.g., subtracting a number from both sides of an equation is modelled as removing that

number of weights from both sides of the scale).

Figure 96. Balance model (screenshot in Class M2)

In addition, Millie briefly referred in both classes to the limitations of the balance model.

After describing the importance of performing the same operation on both sides of a balanced

scale, Millie mentioned one type of operation – removal of items. In each class, a student

commented that there are other operations (e.g., addition and nullifying). Millie confirmed

and added that the balance model helps only with positive numbers, not with negatives. No

further discussion of these limitations was offered or requested in the observed lessons.


8.2. Product of negatives


Class M1 Class M2 Activity (student grouping)

7 minutes

1 minute

Administration


20 minutes

25 minutes

Completing a multiplication table (Whole class)

Launching task – students multiply numbers by zero and calculate

products of positive integers (see Figure 100).

Millie directs the students to observe the pattern formed in the rows and

columns of the table and complete it row by row.

Millie dictates the sign law and provides a mnemonic for it.

A student comments that in order to multiply two signed numbers they

need more than just the sign. Millie responds that they will learn by

example.

22 minutes

11 minutes

Using the sign law (Whole class + Individual student work)

In M1: Millie writes 10 multiplication exercises, demonstrates how to

solve the first two, and then students work on the list (6 minutes). Millie

solves the tasks together with the students (13 minutes).

In M2: Millie writes 6 multiplication exercises on the board, and

demonstrates how to solve each one.

18 minutes

38 minutes

Multiplying more than two numbers (Whole class)

Launching task – students calculate products with varying number of

negative factors, in order to find a pattern for the sign of the product.

Millie dictates a rule: ‘when multiplying more than two numbers: (1) if

the numbers of negative factors is odd, the product is negative. (2) if the

numbers of negative factors is even, the product is positive.’

4 minutes

–

Multiplying more than two numbers (Individual student work)

Students solve a textbook task: determining the sign of a product. Millie

instructs the students to count the number of negative factors.

6 minutes

11 minutes

Exponents with negative bases (Whole class)

Millie demonstrates a calculation of powers of negative numbers and

dictates a rule:

In M1: ‘if the power is an odd number, the result is negative, if it is an

even number, the result is positive.’

In M2: ‘if the base is negative, we look at the power. Even -> the result

is positive, odd -> the result is negative.’




Figure 97 presents the paths of justification for the statement The product of two negative

numbers is a positive number, by class and textbook.


Product of

negatives

s =deduction using a specific case.

Figure 97. Paths of justification in Millie's classes for Product of negatives


The path of justification in Textbook C for this statement was based on two items, one of

which was intended for advanced students.

The first item relied on discovering patterns (i.e., identifying patterns and extrapolating them,

see Figure 98) by deduction using a specific case. Students were required to complete a

multiplication table of numbers ranging from -3 to 3, by following a set of guided steps. The

steps began with using the property of zero and prior knowledge regarding multiplication of

positive integers, and ended with determining the product of negative numbers. The last steps

involved making a conjecture regarding the outcome of the product of negatives and testing it

by continuing a sequence of products. However, the textbook did not specify why it is

allowed to assume that there is a constant difference between each two consecutive elements

in the sequence, or discuss the problem in determining the rule of a sequence based on a finite

number of elements.

(a) (b) Figure 98. Product of negatives – Discovering patterns (Textbook C, vol 2, pp. 42-43)

Patterns

s

Patterns

s

Patterns

s

Extension

s


The second textbook item relied on deduction using a specific case, by extension of

properties (i.e., extending arithmetical properties from natural numbers to negatives, see

Figure 99). Students were required to copy each stage of the calculation and justify it.

Justification involved the use of the multiplicative property of zero and of the distributive

property of multiplication over addition. Following the calculation of the generic example,

students were requested to give the rule. According to the teacher's guide, this item was

intended for advanced students who are interested in the structure of mathematics.

Figure 99. Product of negatives – Extension of properties (Textbook C, vol 2, p. 44)

Historically, the product of negatives was defined to be positive in order to allow a consistent

extension of the arithmetical laws from the natural numbers to signed numbers. However,

such an explanation was not found in any of the textbooks.


Paths of justification in Millie's classes comprised one instance of justification: discovering

patterns by deduction using a specific case (see Figure 100(a)). In both classes, Millie began

by relying on known facts: the product of positive numbers is positive, and the special

properties of multiplication by one and by zero. Then, she instructed her students to pay

attention to the patterns formed in each row and column in the table. By viewing each row

and column as a sequence of numbers, Millie highlighted the constant difference between

successive values in each sequence. First, Millie attended to the two quadrants dealing with

the product of a negative and a positive, and then to the quadrant involving the product of

negative numbers (see Figure 100(b)).


(a) (b) Figure 100. Discovering patterns (screenshots in Class M1 [a] and Class M2 [b])

In both classes, students actively participated in completing the table. However, it is not clear

whether they were convinced by it. In M1, after a student exclaimed she didn't understand the

lesson, several students attempted to assist her by relying on the mnemonic rather than on the

table. However, Millie quickly intervened and reminded the class that the justification for this

statement was based on the multiplication table.

Student1: I don't get it. Millie, I don't understand anything. How can a minus times a

minus equal a min-, equal a plus?

Student2: Because they are not negative together!

Student3: Let's say you have two black socks, they match, right? And a white sock and a

black sock don't match.

Student4: It's like not-not.

Millie: We discovered by using the table, by way of patterns, that negative numbers

times a negative, a negative times a negative actually gives a positive result, okay?

During the concluding interview, Millie was asked whether there are cases in which it is

unnecessary or not important to explain in the seventh grade why certain things are true. In

response, Millie stated that (1) she didn't know why the product of negatives is positive, and

(2) she doesn't dwell on this statement in class:

There are many things that cannot really be explained, I don't think they can, or I don't

know enough because I, myself, still feel that there are many things I need to learn […]

For example with the plus and minus. Why minus times minus is a plus. It really seems

to me like spending too much time. Truth is, I myself don't know and I need to find that

out. […] but I will not dwell on it, because exactly for that reason there are the nice and

easy ways to know that.


Additionally, Millie claimed that the item that relied on extension of properties was too

difficult – but did not specify whether it was difficult for her or for students. Millie was

shown a set of three paths of justification for this mathematical statement (see Figure 101). In

that set, one teacher offers a path similar to Textbook C (i.e., the same instances of

justification and in the same order), a second teacher provides only a mnemonic for the sign

law – without any justification, and the third teacher offers a path comprising two items,

based on a number-line model (i.e., modelling signed numbers as movement forward and

backward in time on a number line) and then on discovering patterns. Millie described the

first teacher’s path of justification as the closest to her own preferred way of justifying the

statement, yet clarified that she related only to the first item.

Figure 101. A set of paths of justification for product of negatives (Interview item).





6 minutes

5 minutes

Administration


–

15 minutes

Recap: Equation solving (Whole class)

In M2: Millie demonstrates homework tasks involving equations.

–

11 minutes

Recap: Naming angles (Whole class)

In M2: Millie reviews angle addition and naming angles by 3 letters.

20 minutes

32 minutes

Defining supplementary angles (Whole class)

Launch – students identify pairs of angles which form a straight angle.

Millie defines supplementary angles as a pair of angles, formed by a

ray extended from a point on a line, whose sum is 180o.

22 minutes

13 minutes

Calculating angles (Individual student work + Whole class)

Students work on calculating angle measures (3-5 minutes).

Millie solves the tasks together with the students (10-17 minutes).

18 minutes

9 minutes

Defining vertical angles + phrasing the rule (Whole class)

Launching task – students identify pairs of vertical angles.

Millie defines vertical angles: ‘a pair of angles, formed by the

intersection of two lines, which are not supplementary angles’.

In M1: Millie sketches two intersecting lines, sets value to one angle,

and instructs the class to find the measures of the other angles. Students

approach the board in turns, calculate each angle and explain their

reasoning. A student suggests: ‘angles that are opposite are equal’.

Millie confirms and dictates: ‘vertically opposite angles are equal’.

In M2: Millie tells the class that vertical angles have a special property

which will be made clear later on – they are congruent. Millie gives an

example and dictates the rule: ‘vertically opposite angles are equal’.

23 minutes

–

Defining corresponding angles + phrasing the rule (Whole class)

Millie explains the term ‘corresponding’ by telling a story about

students switching tables but moving to their corresponding seats.

Millie copies an angle to a transparent sheet, places it on its

corresponding angle, and asks the class what can be seen. A student

responds that the angles are equal, and Millie approves and dictates the

rule: ‘every pair of corresponding angles is congruent’.




Figure 102 presents the paths of justification for the statement Vertically opposite angles are

congruent, by class and textbook.


Vertical

angles

–

s/g =deduction using a specific case/ a general case.

Figure 102. Paths of justification in Millie's classes for Vertical angles




The lessons in both classes took place a few days before a high-stakes exam. The exam was

given to the entire cohort and covered the Vertically Opposite Angles Theorem. Each lesson

dealt with several subjects in preparation for the exam (e.g., equation solving and pairs of

angles – supplementary, vertical, corresponding, and alternate).

In Class M1, The path of justification comprised one instance, relying on deduction using a

specific case by supplementary angles (i.e., a third angle is identified as an angle that

supplements each of the vertically opposite angles). Millie sketched two intersecting lines,

marked one angle, and hinted that the students were permitted to apply their knowledge of

other types of angles. Students approached the board, determined the measures of the other

angles, and justified their assertion with the Supplementary Angles Theorem (see Figure

103). After the fourth angle was found, a student suggested that vertically opposite angles are

congruent. Millie confirmed, and summarized the activity.

Figure 103. Vertical angles – supplementary angles (screenshot in Class M1)

Supplement

s

Supplement

s

Supplement

g


In Class M2, no justification was given in the observed lessons. Millie started discussing

vertical angles less than 10 minutes before the end of the lesson. Millie dictated the definition

and stated the rule along with the definition. She commented that justification will be given at

a later point in time: "I will find that there is an explanation for it, for vertical angles, I will

tell you now, we will do it in the future as well, we will find out they are also equal."

Analysis suggests that Millie introduced the rule in order to prepare her students for a

standardized exam that was held later that week, even if it meant postponing the justification.

Classroom observations were held in the following lesson, yet no justification was given at

that time either. At the beginning of that lesson, Millie summarized the definitions and rules

regarding two types of angles – supplementary angles and vertically opposite angles, without

justification. Justification was neither requested nor given by the students. The rest of the

lesson focused on corresponding angles and on alternate angles.

In the concluding interview, Millie described the differences she sees in justification for this

statement by ability grouping. Millie discussed the number of numerical examples each group

might need before they can figure out the rule, described the use of pronumerals as suitable

only in some classes, and emphasized that she refrains altogether from using letters in lower-

track classes: "In lower-track classes I would provide additional numerical examples, and let

them figure it on their own. I would completely avoid letters." Whereas in high-achieving

classes "one numerical example will suffice and then immediately the rule."


8.4. Corresponding angles



6 minutes

4 minutes

Administration


–

7 minutes

Recap: Supplementary and vertical angles (Whole class)

In M2: Millie summarizes the definitions and rules regarding two types

of angles – supplementary angles and vertically opposite angles.

20 minutes

– Defining supplementary angles (Whole class)

In M1: Launch – students identify pairs of angles which form a straight

angle. Millie defines supplementary angles as a pair of angles formed

by a ray extended from a point on a line, whose sum is 180o.

22 minutes

– Calculating angles (Individual student work + Whole class)

In M1: Students work on calculating angle measures (5 minutes). Millie

solves the tasks together with the students (17 minutes).

18 minutes

– Defining vertical angles + phrasing the rule (Whole class)

In M1: Launching task – students identify pairs of vertical angles.

Millie defines vertical angles: ‘a pair of angles formed by the

intersection of two lines, that are not supplementary angles’, sketches

two intersecting lines, sets value to one angle, and instructs the class to

find the measures of the other angles. Students approach, calculate

angles and explain. A student suggests: ‘opposite angles are equal’.

Millie approves and dictates: ‘vertically opposite angles are equal’.

23 minutes

48 minutes

Defining corresponding angles + phrasing the rule (Whole class)

Millie explains the term ‘corresponding’ by telling a story: "students

switch tables but move to their corresponding seats". Millie copies an

angle to a transparent sheet and places it on its corresponding angle.

In M1: Millie asks the class what can be seen. A student responds that

the angles are equal, Millie approves and dictates the rule.

In M2: Millie tells the class that the angles are equal, dictates the rule:

‘every pair of corresponding angles is congruent’, and solves examples.

–

27 minutes

Defining alternate angles + phrasing the rule (Whole class)

In M2: Millie explains the term ‘alternate’ by tweaking the story about

a group of students switching tables, copies an angle to a transparent

sheet, places it on its alternate angle, and tells the class that the angles

are equal. Millie dictates the rule: ‘alternate angles between parallel

lines are congruent’, and solves examples.




Figure 104 presents the paths of justification for the statement The corresponding angles

between parallel lines are equal, by class and textbook.


Corresp.

angles

e=experimental demonstration.

Figure 104. Paths of justification in Millie's classes for Corresponding angles


The path of justification in Textbook C comprised one instance of justification, relying on

experimental demonstration by measurement (i.e., direct comparison of the corresponding

angles by using a measuring tool). The textbook offered a choice of measuring tools, either a

protractor (see Figure 105), or a dynamic geometry software (see Figure 106).

Figure 105. Corresponding angles – measurement (Textbook C, vol 2, p. 189)

Figure 106. Corresponding angles – measurement (adapted from Textbook C, vol 2, p. 189)

Measuring

e

Measuring

e

Measuring

e



The paths of justification in both classes were very similar to the textbook. Millie offered one

instance of justification, relying on experimental demonstration by measurement both in

Class M1 (see Figure 107(a)) and in Class M2 (see Figure 107(b)). The students' involvement

was only in determining whether the two corresponding angles indeed appear congruent.

(a) (b) Figure 107. Measurement (screenshot in Class M1 [a] and Class M2 [b])

In both classes, Millie used a physical tool different from the one suggested in the textbook.

While the textbook suggests using a protractor, Millie reminded her students that she

promised them they will not be using a protractor anymore and instead used a sheet of paper

to copy the measures of the compared angles. In the concluding interview, Millie elaborated

that her intention was to convey to the students the importance of relying on what is explicitly

given, even if the sketch seems to imply different or additional information. However, Millie

did not seem aware that the justification she offered in class relied on measurement

nonetheless.

I have no intention of letting them measure. Measurement ended at the sixth grade. […]

Later, in ninth grade geometry, they need to understand that what is given to them is all

that is given. Even if something appears to be an isosceles triangle, if that is not given,

then it is not isosceles.


8.5. Area of a disk



17 minutes

7 minutes

Administration


38 minutes

26 minutes

Defining terminology (Whole class)

Millie discusses terminology: circle, disk, radius, chord, and diameter.

Millie defines each term (in M1: asks for the name of the term), and

provides mnemonics.

Millie defines central angle and arc, dictates a formula connecting these

terms, and demonstrates how to use it in a textbook task.

15 minutes

23 minutes

Defining terminology (Individual student work + Whole class)

Students calculate arc and central angles (5-6 minutes).


10 minutes

23 minutes

The perimeter of a circle (Whole class)

Millie solves textbook tasks involving estimation of the perimeter of a

circle by inscribing it in a square and circumscribing it about a regular

hexagon. Millie deduces that the perimeter of a circle is between the

perimeters of these polygons, and denotes it 2πR.

7 minutes

11 minutes


Millie shows a short video depicting an approximation of the area of a

disk by overlaying a grid and counting the coinciding squares.

In M2: Millie dissects a disk into a number of sectors.


16 minutes

25 minutes

Administration


12 minutes

22 minutes


Millie dissects a disk into a number of sectors and rearranges them as a

shape tending to a rectangle, yet incorrectly justifies why one of the

measures is rπ. Millie then dictates the rule: ‘the area of a disk is r2*π’.

21 minutes

34 minutes

The area of a disk (Individual student work + Whole class)

Students calculate area of disks (8-9 minutes).


40 minutes

8 minutes

Arcs and regions (Whole class)

Millie solves textbook tasks involving arcs, sectors, and segments.




Figure 108 presents the paths of justification for the statement The area formula for a disk

with radius r is πr2, by class and textbook.


Area of a

disk

g =deduction using a general case.

Figure 108. Paths of justification in Millie's classes for Area of a disk




The paths of justification in both classes were very similar to the textbook. Millie offered one

instance of justification, relying on deduction using a general case by dissection (i.e.,

dissecting a disk into sectors and rearranging them to form an approximate rectangle).

Millie projected a textbook item on the whiteboard – a dissected disk with radius r – both in

Class M1 (see Figure 109) and in Class M2 (see Figure 110). She discussed the dimensions of

the resulting rectangular shape, correctly determined that the lengths are r and rπ, but

justified incorrectly why the measure of the longer edge was rπ. Millie claimed that rπ was

the remainder of a process in which the radius is subtracted from the circumference of the

circle, instead of asserting that rπ represents one half of the circumference. Millie repeated

this erroneous reasoning several times in each class. The students accepted the justification

without challenge. For example, in Class M1:

Millie: The length of the rectangle is exactly like what?

Students: The radius.

Millie: The length of the radius. So what remains here? [points at the long edge of the

approximate rectangle] […] If this is my radius [points at the short edge], what is left

for the circumference? We were talking about the entire circumference of the circle.

The circumference is two radii and pi. Two radii times pi. If this is a radius, what is

left? The other radius and pi, right? That means, what is left for me is to complete it,

radius times pi.

Dissection

g

Dissection

g

Dissection

g


Figure 109. Area of a disk – Dissection (screenshot in Class M1)

Figure 110. Area of a disk – Dissection (screenshot in Class M2)

As part of the concluding interview, Millie was shown a set of three paths of justification for

this statement (see Figure 111). In this set, one teacher states the rule without offering any

justification, claiming she's pressed for time and that most students are already familiar with

the formula; the second teacher offers a path with two instances of justification, relying on

two types of dissection – into sectors and into concentric rings; and the third teacher offers a

path with two instances of justification, relying first on grid approximation (i.e., counting

squares that coincide with the disk) and then on dissection into sectors.

Millie discussed the importance she sees in justifying this statement in class. She identified

with the first teacher, and claimed that 7th grade students cannot fully understand every step

of the proof, especially given the time pressure. Furthermore, she claimed it is unnecessary

because students are not required to know why or how the formula was found. Nevertheless,

Millie stated that her ideal path of justification will be the one described above.


Figure 111. A set of paths of justification for area of a disk (interview item).

A methodological note: in each class, Millie began by presenting a short video. The video

demonstrated grid approximation (see Figure 112(a)), and showed its accuracy by calculating

the area in two ways: by approximation and by using the area formula (see Figure 112(b)).

However, Millie instructed her students to attend mostly to a segment showing a decimal

representation of the mathematical constant π. Moreover, in the concluding interview, Millie

referred to this video as a way to excite the students. This was not coded as an instance of

justification.

(a) (b) Figure 112. Grid with Concordance of a rule with a model (screenshots in Class M1)


8.6. Summary

In this section I discuss the contribution of the textbook, together with the teacher and the

students to shaping students' opportunities to learn how to explain and justify mathematical

statements in Millie's classes. Figure 113 presents the paths of justification for each of the five

observed statements, by class.


Equivalent

equations

Product of

negatives

Vertical

angles

–

Corresp.

angles

Area of a

disk

e=experimental demonstration; m/s/g =deduction using a model/ a specific case/ a general case.

Figure 113. Paths of justification in Millie's classes, by statement

The textbook was the main source for instances of justification in Millie's classes. Analysis of

the paths of justification revealed that for each mathematical statement, every justification

strategy and almost every justification type that Millie offered in her classes, were offered in

the textbook as well and in the same order. Moreover, for each statement, almost every

instance of justification in the textbook was offered in Millie's classes as well.

The textbook received a special place in Millie's classes. During the lessons, Millie projected

the textbook on the whiteboard and interacted with it for solving tasks and activities.

Moreover, based on the results of the analysis conducted in Part I of this study, it appears that

other textbooks were not used as sources for instances of justification.

Balance

m

Balance

m

Balance

m

Patterns

s

Patterns

s

Patterns

s

Extension

s

Supplement

s

Supplement

s

Supplement

g

Measuring

e

Measuring

e

Measuring

e

Dissection

g

Dissection

g

Dissection

g


Near-identical paths of justification were offered in both classes. This is noteworthy

especially because of differences in student ability levels between the classes (i.e., lower-

track students in Class M2 and the rest in Class M1). In the concluding interview, Millie

discussed her belief in teaching all students in a similar way, as basic as possible, regardless

of their ability. According to Millie, the level of ability dictates a different scope and pace: "I

like using the same methods. The way I see it, any method that helps lower-track students can

help those whose level is good and even support them, so essentially I use the same teaching

methods and materials."

The great similarity between the paths of justification in the textbook and in the classes

suggests that Millie often followed teaching sequences suggested by the textbook. However,

despite these similarities, two main differences were found.

First, for two statements – area of a disk and product of negatives – Millie followed the path

of justification offered in the textbook, yet the inference process presented in her classes was

flawed. For example, the key claim in the inference process for product of negatives in her

classes was left unwarranted. Additionally, Millie described the justification for both

statements as difficult. This is likely related to the absence of explicit explanation in the

textbook, but might reflect gaps in Millie's knowledge as well.

Second, for vertical angles, Millie offered justification only in Class M1 and did not justify it

in Class M2. Millie stated the rule and commented that justification will be provided at a later

point in time. Additional observations were conducted in Class M2 in subsequent lessons, yet

no justification was observed.

Millie was the sole initiator of instances of justification in her classes. Millie commonly

requested the students to make a claim (e.g., cite a fact or generate a conjecture) regarding the

mathematical statement in question. Then, one or more students attempted at phrasing the

claim to the teacher's satisfaction. Once the claim was properly phrased, Millie typically

provided one or more instances of justification for it, while engaging the students by asking

questions (except in one case, in which the claim was left without any justification or

explanation). The students were not requested to provide additional justifications for any of

the observed claims, and they neither challenged the teacher's justification nor opposed it,

even when mistakes were made in the process.

143 | Discussion

9. Discussion

The study deals with justification and explanation. It comprises two parts: The first part

examined the opportunities offered in 7th grade Israeli textbooks for students to learn how to

justify mathematical statements; the second part examined the ways in which the textbook, in

conjunction with the teacher and the students, shape the opportunities offered in 7th grade

Israeli classrooms to learn how to justify.

In this chapter I present a summary of my results and discuss my main findings. The

discussion addresses each part separately and the work as a whole. Last, I address the

contributions of this study to practice and the research community, discuss its limitations, and

suggest directions for future research.

9.1. Opportunities in the textbooks to learn to justify

Part I of this study joins a line of research that focuses on the justifications offered in

textbooks for mathematical claims and statements (e.g., Bergwall & Hemmi, 2017; Dolev,

2011; Hanna & de Bruyn, 1999; Stacey & Vincent, 2009). Previous textbook studies regarded

each instance of justification as an independent unit. This study suggests a novel approach,

which views textbook justification as the complete sequence of instances of justification that

are offered for a mathematical statement – its path of justification. This shift of focus was

intended to better reflect the nature of textbook justifications and thus to better characterize

the opportunities that are offered to students to learn to justify.

The study investigated the paths of justification for ten mathematical statements in eight

Israeli 7th grade mathematics textbooks – a total of 80 paths. Analysis focused on three

attributes: (1) path length – the number of instances of justification offered in a textbook for

each mathematical statement; (2) characteristics of the instances comprising the path –

justification strategies and types; and (3) sequencing – the order in which instances of

justification were offered in the textbook.

The main findings are as follows: (1) For most statements, paths of justification involved

similar sequences across textbooks; (2) Paths of justification in algebra and in geometry were

structured differently and involved different justification types; and (3) Paths of justification

in textbooks of different scopes (i.e., limited and standard/expanded) were structured

similarly – except for instances of the type of justification closest to a formal proof. In this

section I discuss these findings. I focus on three aspects: the paths of justification,

justification in algebra and in geometry, and justification for students with low achievements.

Discussion – Opportunities in the textbooks to learn to justify | 144


The lengths of the paths of justification varied greatly among the analyzed textbooks: where

some textbooks offered long paths for a mathematical claim or statement, other textbooks

offered rather short paths – with one or two instances of justification. Moreover, long paths

involved an assortment of justification strategies and types. In other words, the inclusion of

several justification strategies, or of multiple types of justification for certain justification

strategies, had contributed to extending these paths of justification. These differences in path

lengths imply a great variety among textbooks in students’ opportunities to learn to justify.

Long paths were found to offer students a variety of opportunities, which in turn are likely to

have an additive effect: They provide different ways in which students can be convinced of a

mathematical truth, even if the justification is not considered sufficient in mathematics

(Sierpinska, 1994); Additionally, long paths can teach students that there is more than one

"right" way of justifying (Lampert, 1990; Schoenfeld, 1992).

Paths of justification typically comprised deductive and/or empirical types of justification,

while external justifications were near absence. Empirical and deductive types are generally

considered desirable in mathematics education (e.g., Harel & Sowder, 2007; Stylianides,

2009). Similar results were found in the literature – both regarding the abundance of

empirical and deductive types in textbook justifications, (e.g., Bergwall & Hemmi, 2017;

Dolev, 2011; Hanna & de Bruyn, 1999; Otten, Males, & Gilbertson, 2014; Thompson, Senk,

& Johnson, 2012) and the low ratio of external types (e.g., Cabassut, 2005; Otten, Gilbertson,

et al., 2014; Ronda & Adler, 2017; Stacey & Vincent, 2009).

Despite the differences in lengths, the order and sequencing of most paths were similar across

the analyzed textbooks. In almost all paths that included experimentation and deduction using

either a specific or a general case, the empiric type preceded the deductive type/s. Similarly,

in paths that included both Deduction using a specific case and Deduction using a general

case, a generic example always preceded the general case. However, the general case was not

always preceded by a generic example – it did so only in a third of the paths that involved the

general case. The frequent sequences found in this study are very common in mathematics

education and are in accordance with theories of learning and cognitive development, which

involve moving from concrete to abstract (e.g., Piaget, 1964). The low frequency of a specific

case before the general case is somewhat surprising, given the didactical value of generic

examples for introducing the deductive structure of mathematics and as a step that can help

students generalize and achieve formal abstraction (e.g., Harel & Tall, 1991).

145 | Discussion

For most mathematical statements, almost every justification strategy corresponded to a

single type of justification across the textbooks. For example, for the statement Division by

zero, all 16 instances of the justification strategy inverse of multiplication used the same

justification type – deduction using a specific case, and for Area of a disk, all eight instances

of the justification strategy dissection relied on deduction using a general case. As a result,

the order in which justification strategies were offered in the analyzed textbooks was directly

related to the sequences of justification types described above. This association between the

two characteristics of the instances of justification may shed light on the structure of the paths

of justification, as well as on sequences of justification strategies that may benefit students in

learning how to justify by relying on knowledge on cognitive development.

The reported similarities across textbooks may reflect shared views among textbook authors

about potentially useful principles for helping students learn to justify in mathematics. The

findings suggest at least three shared principles: (1) deductive and empirical justification

types are preferred, (2) paths of justification should progress from empirical to deductive

justifications, and (3) these sequences are preferred regardless of path length.

The preference of deductive and empirical justification types is in accordance with the Israeli

school curriculum for grades 7-9 (Israel Ministry of Education, 2009). The Israeli curriculum

explicitly emphasizes justification, explanation, and proof, for both algebra and geometry:

Understanding the essence of algebra as a mathematical branch that deals with processes

of generalization, generating conjectures, and justifying them; Developing argumentative

discourse: ways to explain or prove algebraic properties and rules; … Discovery of

attributes of geometric figures and geometric facts and understanding the deductive

relations among them; … Giving explanation and proof for geometric properties (p. 3).

The Israeli curriculum emphasizes proof instead of "checking multiple cases in which the

claim is true." (p. 10). Justifications that are pre-deductive in nature are favored in the 7th

grade: "geometry studies in the 7th and 8th grades can be referred to as 'pre-deductive'." (p.

28). The deductive structure of geometry is first introduced in the middle of the 8th grade for

all students, in part because "it is important to know the deductive structure of geometry as

part of the general human culture." (p. 9). This emphasis on justification exemplifies the

intention of the Israeli curriculum developers to address elements which are considered by

many to be central components of doing and learning mathematics (e.g., Ayalon & Even,

2010; Ball & Bass, 2003; Cabassut, 2005; Chazan, 1993; Yackel & Hanna, 2003). Similarly,


many school mathematics curricula around the world attribute a central role to developing

students' ability to justify, at all grade levels and across content domains (e.g., Australian

Education Council, 1991; Common Core State Standards Initiative, 2010; Department of

Education, 2010; Finnish National Board of Education, 2003; NCTM - National Council of

Teachers of Mathematics, 2000; Swedish National Agency for Education, 2011).

The progression of sequences from empirical to deductive justifications is in line with the

Israeli curriculum as well. For most of the statements analyzed in the study, sequences (or

segments of sequences) which were very common in the analyzed textbooks involved

instances of justification similar to the instances suggested in the Israeli curriculum, and in

the same order. The strict requirements for curriculum adequacy might further explain both

these similarities and the other similarities found among the textbooks. Studies in several

countries found that textbooks often attempt to reflect national curriculum documents (e.g.,

Fan, Zhu, & Miao, 2013; Jones & Fujita, 2013; Valverde, Bianchi, Wolfe, Schmidt, &

Houang, 2002). The current findings might indicate a similar phenomenon in Israel.

However, the contribution of a national curriculum to shaping the types of justification

offered in textbooks might not necessarily be direct. For example, Bergwall and Hemmi

(2017) focused on integral calculus and found differences among textbooks from Sweden and

Finland in the justification types. They found that Finnish textbooks offered a greater number

of general proofs, compared with generic cases and non-proof, whereas in Swedish textbooks

the situation was reversed. However, as Hemmi, Lepik, and Viholainen (2013) found, proof

and deductive reasoning were emphasized strongly in Swedish upper secondary curriculum

documents, whereas Finnish upper secondary curriculum documents emphasize mathematical

thinking but do not mention proof at all.

9.1.2. Justification in algebra and in geometry

Paths of justification in algebra and in geometry were of similar lengths, yet some differences

were found in their structure. Specifically, different justification types were often involved in

each topic. Deduction using a general case was included solely in paths for geometry

statements, and Deduction using a specific case was used mostly either in paths for algebra

statements or in statements relying on an algebraic derivation (i.e., Area of a trapezium).

These differences may represent different opportunities for students to learn to justify.

Reserving the type of justification closest to a formal proof mainly for geometry statements

might convey to students that proof is a part of doing mathematics only in geometry and not

147 | Discussion

in algebra, where one could use “softer” ways of justification. Be that as it may, Deduction

using a specific case may allow students who are newcomers to algebra to experience an

inference process with a lower risk of ‘getting lost’ in algebraic manipulations.

The association between justification strategies and justification types, reported above, was

much stronger for algebra statements compared with geometry statements. In algebra, almost

every justification strategy corresponded to a single type of justification across the textbooks

for that mathematical statement. In geometry, however, several justification strategies

occurred multiple times in paths of justification, associated with different justification types.

For example, for Area of a trapezium, the 11 instances of the justification strategy

construction relied on four justification types, and for Angle sum of a triangle, the nine

instances of the justification strategy parallel line relied on two justification types. These

sequences often involved a process of generalization and a progression from empirical types

of justification to deductive ones. By repeating a certain justification strategy several times,

and offering a transition from an empirical type to deductive type, textbook authors may aim

to provide a soft entrance and a gradual transition from elementary school to the more

deductive justifications which are expected in junior high school.

The topic-based distinction between justification types is in line with the traditional

asymmetric emphasis on proof and proving in the teaching of geometry and algebra in school

(P. J. Davis & Hersh, 1981; Harel & Sowder, 1998). Historically, geometry was viewed as

the most appropriate domain for teaching proof and for developing students’ ability to reason

logically (M. Ayalon & Even, 2010; González & Herbst, 2006), whereas algebra was viewed

as a domain “concerned with generalized computational processes” (Sfard, 1995, p. 17).

Furthermore, this distinction is in line with previous textbook studies. Several studies report

that justification outside of geometry involves arguments about a specific case significantly

more often than general arguments, whereas geometry statements offer a greater number of

general arguments (e.g., Dolev, 2011; Stacey & Vincent, 2009; Thompson et al., 2012).

The topic-based distinction between justification types is not in accordance with the Israeli

school curriculum for grades 7-9 (Israel Ministry of Education, 2009). The Israeli curriculum

emphasizes justification for both algebra and geometry, and proof is regarded as a tool for

linking algebra and geometry (e.g., "connect and integrate algebra and geometry; for

examples: by giving an algebraic proof for a problem in geometry and vice versa" (p. 3)).


9.1.3. Justification for students with low achievements

Several similarities were found among paths of justification for each mathematical statement;

Paths comprised similar instances of justification and similar sequences across textbooks –

both in textbooks of limited scope, intended for students with low achievements, and in

textbooks of standard/expanded scope, intended for the general student population. Five out

of the six types of justification that were identified in the analyzed textbooks were offered in

textbooks of both scopes. Additionally, for each mathematical statement, justification

strategies were generally offered in textbooks of both scopes. Textbooks of both scopes

offered similar sequences, which translate to opportunities for students to transition from

empirical and inductive justification to a deductive proof scheme (Harel & Sowder, 2007).

However, two differences were found among the analyzed textbooks: in path characteristics

and in path lengths. Paths of justification in textbooks of standard/expanded scope involved a

greater number of instances of deduction using a general case compared with textbooks of

limited scope – roughly three times as much. This difference was not found to be statistically

significant, yet it is pronounced and noticeable. Additionally, path lengths in textbooks of

limited scope were consistently slightly shorter for most mathematical statements, compared

with textbooks of standard/expanded scope.

The difference in prevalence of deduction using a general case is in accordance with the

literature on teaching proof in classes of different achievement levels (e.g., Raudenbush,

Rowan, & Cheong, 1993). Generally, the literature demonstrates that mathematics teachers

differentiate their teaching based on certain characteristics of the students, such as low-

achieving students compared with high-achieving students (e.g., Even & Kvatinsky, 2009).

Specifically, students with low achievements are often given fewer and different

opportunities to learn to justify (e.g., Even & Kvatinsky, 2009; Zohar & Dori, 2003).

Moreover, justification is traditionally viewed as representing high-order thinking, and

therefore not suitable for students with low achievements (Zohar et al., 2001). Differences in

teaching approaches and the mathematics addressed in classes of different achievement levels

are well documented in the literature (e.g., Even & Kvatinsky, 2009, 2010; Hollingsworth,

McCrae, & Lokan, 2003; Metz, 1979; Oakes, 1985; Page, 1991; Wiliam, 1998; Zohar,

Degani, & Vaaknin, 2001; Zohar & Dori, 2003).

Additionally, this difference in justification types is in accordance with the Israeli school

curriculum for grades 7-9 (Israel Ministry of Education, 2009). The Israeli curriculum

149 | Discussion

suggests offering different types of justification to different students: "In teaching proofs

there are several steps towards writing a full proof in formal language, and it is necessary to

adjust the requirements [on formalism] to students' abilities." (p. 10). This is done in order to

accommodate different student levels, and also to "prevent a feeling of failure and make

[mathematics] more likeable for students, for example by differential teaching, accustomed to

the variability among students in regards to reasoning in geometry." (2009, p. 3).

9.2. Textbook, teacher, and students

Part I of the study focused on the opportunities which Israeli mathematics textbooks offer for

students to learn to explain and justify. However, several additional factors contribute to

shaping classroom teaching and learning, such as teacher and student characteristics, beliefs,

and expectations (M. Ayalon & Even, 2016; Chazan, 2000; Eisenmann & Even, 2011; Even

& Kvatinsky, 2010; Tarr et al., 2008). Part II examined the ways in which the textbook,

together with the teacher and the students, shape these opportunities.

The study utilized the "Same teacher – different classrooms" research design (e.g., Ayalon &

Even, 2015; Eisenmann & Even, 2009, 2011; Even, 2014; Even & Kvatinsky, 2009, 2010):

two case studies, each focused on a mathematics teacher who uses the same textbook in two

7th grade classes. For each teacher, the study investigated the paths of justification for five

mathematical statements. Comparative analyses of the paths of justification, by teacher, by

classroom, and by topic, focused on three attributes: path length, characteristics (i.e.,

justification strategies and types), and sequencing.

The main findings suggest that the textbook contributed greatly to shaping the paths of

justification and was the main source for instances of justification in all observed classrooms.

Paths of justification in every class were generally similar to the paths offered in the textbook

– both in the characteristics of each instance of justification and in their sequencing. Yet, the

teachers' perception of the abilities of their students was also instrumental in constructing the

paths of justifications in the classes. For example, instances of deduction using a general case

were excluded from classes for which proof was regarded as unsuitable, and instances of

justification that were marked in the textbook as intended for advanced students were

excluded if considered too difficult. Additionally, a noisy classroom environment interrupted

several discussions and contributed to altering a type of justification from deductive to

empirical. In this section I discuss these findings, focusing on the contribution of the

textbook, the teacher, and the students to shaping the opportunities to learn to justify in class.

Discussion – Textbook, teacher, and students | 150

The textbook was found to be the main source for instances of justification in the observed

lessons. Paths of justification in the observed classes were generally similar to the paths

offered in the textbook: every justification strategy and almost every justification type that

was offered in class was offered in the textbook as well and in the same order. Based on the

results of the analysis conducted in Part I of this study, instances of justification from other

textbooks were not found in the observed lessons. One teacher – Lena – occasionally used

other sources (e.g., other textbooks and worksheets), yet merely as a resource for additional

tasks and activities and not for justification of mathematical statements.

This great contribution of the textbook to shaping the paths of justification in class is in line

with the literature. Previous studies suggest that in many countries, textbooks serve an

important role in shaping students’ opportunities to learn mathematics (J. A. Newton, 2012;

Stein et al., 2007), and that teachers use textbooks as a main source for content and activities

to include in their lesson plans, and often follow teaching sequences suggested by textbooks

(Eisenmann & Even, 2009, 2011; Haggarty & Pepin, 2002).

However, as mentioned before, the textbook alone does not tell the entire story. As can be

expected, the classroom lessons were not an exact copy of the corresponding textbook

lessons, and for each teacher noteworthy differences were found between the justifications

offered in the textbook and in their classes. Most notable was the exclusion of instances of

deduction using a general case and those marked as intended for advanced students.

Instances of deduction using a general case were excluded from classes for which proof was

judged as unsuitable. The interviews suggested that the declared aim of the teachers in

teaching students to justify was to ensure that students would remember the studied topics.

Furthermore, Lena described justification as important yet not suitable for all students:

It depends on the kid's level. Some children are not ready, it does not matter whether you

explain to them – they will not understand. […] However, for those with [mathematical]

thinking, that they're good at it, it was important to me, yes, because then they know the

origin, the 'why', and then it is easier to remember the formula. They don't memorize.

This perception of justification led Lena to refrain from incorporating proof in her classes.

However, in the year of observations Lena taught in two classes with mixed levels of ability,

in which some students were considered high-achievers. As an interim solution, Lena's

advanced students were given the option to study geometry separately, in a small group with

a supporting teacher, focusing on proof and proving.

151 | Discussion

The absence of the type of justification closest to a formal proof may impede some students

from learning about the role and importance of proof in mathematics. On the other hand, this

absence is in accordance with the pre-deductive approach recommended for the 7th grade in

the Israeli curriculum.

The findings further suggest that instances of justification that were marked in the textbook as

intended for high-achieving students, were excluded from the paths of justification in the

observed classrooms. In the concluding teacher interview, Millie described certain instances

of justification as too difficult and discussed her belief in teaching all students in a similar

way, as basic as possible, regardless of their ability: "I like using the same methods. The way

I see it, any method that helps lower-track students can help those whose level is good and

even support them, so essentially I use the same teaching methods and materials."

Moreover, the exclusion of instances of justification that were marked as suitable for

mathematically-inclined students was related to the teacher's mathematical background. For

example, Millie commented about her lack of knowledge of the underlying reasoning for

several topics, such as the positive product of negative numbers, and claimed that certain

instances of justification (i.e., for the product of negatives and for area of a disk) were too

difficult. Furthermore, for two of the observed mathematical statements, despite the fact that

Millie followed the paths of justification offered in the textbook, she presented a flawed

inference process in both of her classes.

Additionally, the classroom atmosphere was instrumental in shaping the paths of justification.

For example, the flow of most of Lena's lessons was interrupted frequently for disciplinary

purposes in light of excessive noise and behavioural issues. These interruptions contributed to

shaping the paths of justification by cutting short several classroom discussions, some of

them pertaining to instances of justification. Specifically, a noisy classroom environment

contributed to altering a type of justification from deductive to empirical.

The contribution of the teacher and the students to shaping their opportunities to engage in

argumentative activities is documented in the literature (e.g., Ayalon & Even, 2016;

Eisenmann & Even, 2011; Even & Kvatinsky, 2009). The near-identical paths of justification

each teacher offered in her two classes, even when facing disparate student levels (i.e., low-

achieving students in one class and mainstream and high-achieving students in the other),

reflect an invariant teaching approach to justification. Ayalon and Even (2016) reported a

Discussion – Textbook, teacher, and students | 152

similar "fixed" approach to argumentation in a study that used a research design similar to the

design of Part II of this study ("Same teacher – different classrooms", Even, 2008, 2014).

9.3. Implications

The findings of this study have several practical and theoretical implications, which are

discussed in this section. Closing this section is a discussion of caveats and issues for future

research raised by the study.

9.3.1. Contribution to practice

This study focused on textbooks that were published in accordance to the current Israeli

junior high school mathematics curriculum, and are in use by in-service teachers. It brings to

light aspects regarding the justifications offered in the textbooks – both in the introductory

sections and the related collections of tasks. The study mapped the paths of justification and

characterized the justification strategies and types offered for the analyzed mathematical

statements. These aspects are nontrivial, and are the result of an in-depth textbook analysis.

Therefore, this study provides useful information to teachers and subject coordinators, for

deciding which textbook to choose and use in class.

Teachers can rely on this analysis to become acquainted with a multitude of justification

strategies, and construct paths of justification that include instances of justification that are

better suited for different types of learners, by catering to individual students' ways of

thinking. In addition, these paths can include the use of a certain justification strategy several

times, each using a different type of justification, to allow generalization of results and

promote understanding.

Additionally, this study might be of interest to textbook authors, in order to better understand

common principles which underlie Israeli 7th grade textbooks and the opportunities each

textbook provides for learning to explain and justify. Thus, future editions of their textbooks

might offer paths of justification inspired by this study.

This study provides detailed information for Israeli educators, curriculum developers, and

decision makers regarding the opportunities for justification that are offered in Israeli 7th

grade mathematics textbooks, both by topic (i.e., algebra and geometry) and by the intended

student population (i.e., students with low achievements and the general population). The

findings regarding the different distribution of types of justification in different topics and in

textbooks of different scope may raise points for discussion regarding the extent and the

desirability of these phenomena in mathematics education.

153 | Discussion

Furthermore, this study analyzed the ways in which the textbook, together with the teacher

and the students, contribute to the teaching of justification in class. The findings revealed that

instances of justification that were offered in the textbook were the main source for instances

of justification in the classroom, yet two kinds of instances were typically excluded: items

marked as suitable for high-achievers and/or of the type of justification closest to a formal

proof. This information can help facilitators design professional development courses that

focus on the instances of justification offered in different textbooks and on various paths of

justification that may be suitable in different classrooms.

9.3.2. Contribution to the research community

This comprehensive textbook analysis provides researchers in mathematics education with

ways of better understanding the opportunities offered for students to learn how to explain

and justify. It shows the great value in combining a framework that attends to meta-level

aspects with elements that attend to the justification strategy in each instance of justification.

This has the potential to improve research methods for future studies that analyze

mathematics textbooks or examine classroom instruction and student learning.

Three methodological notes are in order. First, this study shows that the conceptual

framework suggested by Stacey and Vincent (2009) provides valuable information about the

nature of justification, not only in textbooks but also in classrooms. Analysis by using this

framework revealed differences that were not visible when using the more general categories

described in Harel and Sowder's (2007) proof schemes. For example, it revealed differences

in the frequencies of the type of justification closest to a formal proof, both among textbooks

of different scopes and between the textbook and the observed classed. These differences

give merit to using a framework with a higher resolution and a wider scope in analysis of

explanations and justifications for mathematical statements.

A second note about methodology: In analysis of the part of the lesson in which instances of

justification were offered, the study showed that most of the justifications for the analysed

statements were included either in the introductory sections or in tasks intended for classroom

discussion. However, some instances of justification were embedded in tasks intended for

student individual or small-group work. While the inclusion of these instances did not affect

the emerging patterns, analyzing all parts of the lesson – both those intended for a whole

class grouping and those intended for individual student work – may provide a more

complete picture of the opportunities offered for students to learn how to justify.

Discussion – Implications | 154

A third note deals with the research design which was utilized in this study – "Same teacher –

different classrooms" (Even, 2008, 2014). This design is extremely useful because

implementation of curriculum materials is a dynamic process with several participants, and

isolating the contribution of each factor for shaping mathematics teaching and learning is not

possible (Lloyd, 1999). On the other hand, while the methodological choice to focus on two

case studies can provide rich and valuable data, it cannot lead to generalizations.

9.3.3. Caveats and opportunities for future research

In this study, analysis of the instances of justification focused on two characteristics –

justification strategies and types. These attributes shed light on important aspects of the paths

of justification regarding the reasoning involved for each mathematical statement. However,

these two attributes tell an incomplete story. One such missing aspect is the (possible)

intended purpose of each instance of justification – both in the textbook and in the classroom.

For example, some instances of justification may have served didactical purposes only (e.g.,

invoking students' intuition, or affirming the statement), or served as a precursor for

conjecture (e.g., an activity that aims at an identification of a pattern). Future research of

justification should take all three attributes into consideration – justification strategies and

types, and the intended purposes of the instances of justification.

Additionally, in my discussion of the findings in the textbooks I made certain assumptions

regarding the textbook authors' intentions, views, and opinions regarding justification. These

assumptions are merely educated speculation based on the results in the comparative analysis

of the paths of justification in the Israeli textbooks and of my review of the literature. In order

to gain a better understanding of the textbook authors' intentions, it is recommended to

present the authors with the findings of this study and interview each author.

In Part II of this study, I intended to compare paths of justification for certain mathematical

statements between teachers. This comparison involved limiting the analysis to mathematical

statements that were observed in all four classes. However, due to research constraints,

insufficient data were collected to enable a meaningful analysis. It is therefore recommended

to conduct additional classroom observations and to include more teachers in future studies.

Last, this study focused on 7th grade textbooks. As Thompson (2014) noted, similarities and

differences which were identified among textbooks for a particular grade level might change

in the course of that textbook series. Additional research is needed to characterize the paths of

justification in Israeli textbooks intended for higher grades.

155 | References

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The analyzed textbooks

-מתמטיקה משולבת (. 2102תעיזי, נ. )… ג., בוהדנה, ר., הדס, נ., פורמן, ט., פרידלנדר, א., קירו, ש., ’, ג’חג-אוזרוסו

.. רחובות: מכון ויצמן למדע, המחלקה להוראת המדעיםמסלול כחול

.. יבנה: כ. בונוס הפצות בע"מקפ"ל -קפיצה לגובה (. 2102אילון, ט. )

. מסלול ירוק -מתמטיקה משולבת (. 2104, ט., פרידלנדר, א., רובינזון, נ., & תעיזי, נ. )בוהדנה, ר., גולדנברג, ג., פורמן

.רחובות: מכון ויצמן למדע, המחלקה להוראת המדעים

.המרכז לטכנולוגיה חינוכית -. תל אביב: מטח מתמטיקה לחטיבת הביניים(. 2102הרשקוביץ, ש., & גלעד, ש. )

. יבנה: כ. אפשר גם אחרת(. 2102רון, ג. )… בסקי, ל., אילון, ט., בוכבינדר, א., גורביץ, א., זודיק, א., ’זסלבסקי, א., לינצ

.בונוס הפצות בע"מ

.. קרית טבעון: משבצת’מתמטיקה לכיתה ז(. 2102יקואל, ג., & בלומנקרנץ, ר. )

. ירושלים: למדא יוזמות חינוכיות עשר בריבוע(. 2102גינזברג, ר. ) לוזון, ד., אמויאל, א., קופרמן, ש., במברגר, ל., &

.בע"מ

.. אשדוד: לוני כהן בע"ממתמטיקה לכיתה ז -צמרת (. 2102שלו, י., & עוזרי, א. )

Appendix – Interview items | 172

Appendix – Interview items

שאלות לראיון - לנה

דקות( 5-3) חלק א': רקע

כיתות וזה היה מאד מעניין. אני רוצה היום לשאול אותך על תיבמהלך השנה האחרונה ישבתי אצלך בשיעורים בש

כמה דברים שמעניינים אותי במיוחד, אבל לפני כן אשמח קודם כל להשלים כמה פרטים:

אילו תארים/תעודות יש לך? –בחינת השכלה מ א. .1

איפה למדת ומתי? ב.

איך הגעת להיות מורה למתמטיקה? ג.

?לימדת עד כה רמות לימודכיתות ובאילו באילו -מבחינת ניסיון בהוראה א. .2

אילו כיתות ואילו רמות את בדרך כלל מלמדת? ב.

?עד כה ימדתהתלמידים שלמאיזה רקע סוציואקונומי היו ג.

דקות( 22-51) מורה, כיתה, וספר לימוד –חלק ב'

במהלך השנה לימדת בשתי כיתות ז' במקביל. תוכלי לספר לי מה היה דומה ומה שונה בללמד בשתי א. .3

הכיתות?

איך היה ללמד בשתי הכיתות בשיעורים שבהם יצאו תלמידים אל רון או אירנה? ב.

חלטתם כמה ואילו תלמידים להוציא?הלפי מה ג.

איך את מתכננת את השיעורים שלך? למשל, האם ואיך את נעזרת בספר הלימוד, במקורות אחרים, א. .1

בצוות בית הספר, במדריכה, וכדומה?

(?2: האם השתמשת בספר הלימוד? איך... )התייחסה לספר הלימודלא אם ( 1) ב.

תוכלי לתת דוגמה? ?כשבנית את השיעור ספר הלימודהשתמשת ביך א הלימוד:( אם התייחסה לספר 2)

(?2: האם השתמשת במקורות נוספים מעבר לספר? איך... )מקורות נוספיםל( אם לא התייחסה 1) ג.

אפשר דוגמא? רות נוספים מעבר לספר הלימוד?השתמשת במקו איך :( אם התייחסה למקורות נוספים2)

(?2: האם העבודה עם צוות ביה"ס השפיעה על התכנון? איך... )התייחסה לאנשים אחריםלא אם ( 1) ד.

ך העבודה עם כל אחד השפיעה על תכנון השיעורים? תוכלי להדגים?אי :( אם התייחסה לאחרים2)

דוגמא? אפשרזה השפיע על תכנון השיעור בכיתה האחרת? האםשהעברת שיעור מסוים בכיתה אחת, אחרי .5

173 | Appendix – Interview items

[דקות לכל שאלה 7-כדקות: 12] חלק ג': מהלכי ההצדקה

גם הסברים פרונטליים וגם הסברים במשימות –אני רוצה עכשיו לעבור ולהתמקד בהסברים שניתנים בכיתה ז'

לתלמידים.

למשל, שזוויות מתאימות בין מקבילים שוות זו לזו –דברים למההאם לדעתך צריך להראות בכיתה ז' א. .6

דברים כאלה הם נכונים? למההאם חשוב להסביר –

?למה דברים הם נכונים למה חשוב להראות בכיתה ז' ב.

האם יש מקרים שבהם אין צורך, או לא כדאי להסביר? אפשר דוגמא? ג.

בכיתות גבוהות?צריך להסביר באותו אופן כמו כיתה ז' האם ב ד.

חלק מהנושאים שנלמדים בכיתה ז' מוכרים לתלמידים מביה"ס היסודי, למשל שאסור לחלק באפס, או א. .7

הדברים שהם למדו ביסודי הם למהראות לתלמידים כמה חשוב בעינייך להעד הנוסחה לשטח עיגול.

נכונים?

ב. האם נדרש סוג שונה של הסברים ביסודי ובחטיבת הביניים? : אם חשוב

תוכלי לפרט מדוע חשוב להראות זאת בכיתה ז'? אפשר דוגמא? .ג

ד. האם יש מקרים שבהם לא צריך להראות למה דברים מהיסודי נכונים? למה זה לא חשוב?

? האם תוכלי להרחיב? סביר למה הדברים נכוניםחשוב לה: ב. באילו מקרים אם תלוי בנושא

לא חשוב?זה ? מדוע סבירמתי לא חשוב לה. ג

: ב. תוכלי לפרט מדוע?אם לא חשוב

.שמוע ממך קצת לגביהםלאני אשמח ה מהנושאים שלימדת השנה, ושלושלדבר בצורה ספציפית על עכשיו עבור נ

נתחיל בזוויות קודקודיות. .9

מורות לגבי האם ואיך להראות למה זוויות קודקודיות שוות זו לזו. אלו במסגרת המחקר יצא לי לדבר עם

חלק מהתשובות שקיבלתי )לעבור לדף רלוונטי(

.כפל מספרים מכווניםנעבור ל .4

גם כאן, אשמח שנעבור על כמה תשובות שקיבלתי לגבי האם ואיך להראות למה מספר שלילי כפול מספר

רלוונטי(שלילי זה מספר חיובי )לעבור לדף

.שטח עיגולהנוסחה להנושא האחרון שבו אני רוצה להתמקד הוא .12

הנוסחה למה שוב, אשמח אם נוכל לעבור על כמה תשובות שקיבלתי ממורות לגבי האם ואיך להראות

)לעבור לדף רלוונטי(2rהיא rלשטח עיגול שרדיוסו

בחלק מההסברים שעליהם עברנו ראינו שהמורה הביאה את הכלל בסוף ההסבר, לפעמים באמצעו, ולפעמים .11

)האם תוכלי להרחיב? במה זה ?. מה לדעתך עדיףאותו המקיואז נ –אולי כהשערה – את הכלל ההציג

תלוי? האם תוכלי להדגים?(

תודה רבה! לספר לי?האם יש עוד משהו שהיית רוצה פחות או יותר סיימנו. .12


נתחיל בזוויות קודקודיות.. 8

. אלו חלק למה זוויות קודקודיות שוות זו לזויצא לי לדבר עם מורות לגבי האם ואיך להראות במסגרת המחקר

:מהתשובות שקיבלתי

:מורה א'

I אני בדרך כלל מתחילה עם .

דוגמה מספרית:עבודה בכיתה על

II אותיות:. ואחר כך עובדים עם III ואז אני נותנת את הכלל .

כמסקנה:

מדוע .,חשבו את גודלן של

הן שוות?

אילו זוויות הן קודקודיות? מדוע הן

ת?ושו

מסקנה: זוויות קודקודיות שוות זו

לזו.

:מורה ב'

I .:אני קודם כל נותנת את הכלל II:ואז מתחילה בעבודה עם אותיות . III עבודה בכיתה על . ורק אחר כך

דוגמה מספרית:

זוויות קודקודיות שוות זו לזו.

אילו זוויות הן קודקודיות? מדוע

הן שוות?

.ושל חשבו את גודלן של

מדוע הן שוות?

:מורה ג'

I אני נותנת לתלמידים למדוד .

:זווית-זוויות עם מד

II:ואז נותנת את הכלל כמסקנה .

אילו הן? האם קיבלתם זוויות שוות?

שוות זו מסקנה: זוויות קודקודיות

.לזו

מוצא חן בעינייך אצל כל מורה, ומה את לא אוהבת?מה .א

נסי לבנות את המהלך האולטימטיבי מבחינתך כדי להסביר למה זוויות קודקודיות שוות זו לזו. .ב

את יכולה להעזר בהסברים שנתנו המורות, לשנות בהם מה שאת רוצה, או להוסיף משל עצמך.

זה שהצעת עובד בכיתה?איך מהלך כמו .ג

175 | Appendix – Interview items

. כפל מספרים מכוונים. נעבור ל4

למה מספר שלילי כפול מספר גם כאן, אשמח שנעבור על כמה תשובות שקיבלתי לגבי האם ואיך להראות

:שלילי זה מספר חיובי

:מורה ד'

I אני משלימה עם התלמידים לוח כפל .

של מספרים מכוונים, לפי חוקיות

בשורות ובעמודות:

II. אחר כך אני

אומרת שמתוך לוח

הכפל עולה השערה:

III:ואז רושמת על הלוח חישוב בשלבים .

השערה:

שלילי כפול

שלילי =

.מספר חיובי

)סכום נגדיים = אפס( 3 4 4 0

)לפי חוק הפילוג( 3 4 3 4 0

)פתיחת סוגריים( 3 4 12 0

)סכום נגדיים = אפס( 3 4 12

:מורה ה'

I כל ה"הסברים" שאני מכירה לכלל .

הזה הם לא מספקים מבחינה מתמטית, או

קשים מדי בכיתה ז', אז אני רק נותנת את

:הכלל

II ואז נותנת להם .

טיפ כדי לזכור את

הכלל:

.שלילי כפול שלילי = מספר חיובי

טיפ: "האויב

של האויב שלי

הוא חבר שלי".

:מורה ו'

I ציר הליכה על עםשאלה ב. אני מתחילה

רגיל לגבי הת מספרים, 4 3 ? :

II ואז נותנת את .

המסקנה:III ואחר כך אני מבקשת מהם לזהות את .

ולהשלים בעזרתה סדרת תרגילים:החוקיות

החיפושית הלכה בניגוד לכיוון החץ והגיעה

חיפושית ה. בכל דקה עברה 0למספר

שלושה קטעי יחידה. באיזו נקודה היא היתה

דקות? 4לפני

מסקנה: שלילי

כפול שלילי =

.מספר חיובי

בעינייך אצל כל מורה, ומה את לא אוהבת?מוצא חן מה .א

נסי לבנות את המהלך האולטימטיבי מבחינתך כדי להסביר למה שלילי כפול שלילי זה מספר .ב

חיובי. את יכולה להעזר בהסברים שנתנו המורות, לשנות בהם מה שאת רוצה, או להוסיף משל

עצמך.

איך מהלך כמו זה שהצעת עובד בכיתה? .ג

.שטח עיגולהנוסחה להנושא האחרון שבו אני רוצה להתמקד הוא . 12


הנוסחה לשטח עיגול למה שוב, אשמח אם נוכל לעבור על כמה תשובות שקיבלתי ממורות לגבי האם ואיך להראות

:2rהיא rשרדיוסו

:מורה ז'

Iכל . מהניסיון שלי, כמעט

התלמידים בכיתה ז' מכירים את

הנוסחה. בשלב הזה של השנה

אני כבר לחוצה בזמן, אז אני רק

מזכירה את הכלל:

שטח העיגול הוא 2r

:מורה ח'

I איך מקבלים את . אני מראה

הנוסחה על ידי חלוקה של

העיגול לגזרות והרכבה של

"מלבן":

II . אני מראה איך מקבלים את אח"כ

הנוסחה על ידי חלוקה של העיגול לפסים

והרכבה של "משולש":

III רושמת על הלוח את . ואז

:המסקנה

מסקנה: שטח העיגול הוא

2r

:מורה ט'

I . אני מזכירה לתלמידים את

הנוסחה:II . אח"כ מראה שהנוסחה נותנת תוצאה

שקרובה מאד לתוצאה בחישוב

:אינטואיטיבי

III ואז מראה איך מקבלים את .

הנוסחה על ידי חלוקה של

העיגול לגזרות והרכבה של

"מלבן":

שטח העיגול הוא 2r

ס"מ. מה שטחו? 3נתון מעגל שקוטרו

דרך א': 2 2

1.5 1.5 3.14 7.069

-דרך ב': שטח העיגול הוא כ7

משטח הריבוע, 9

לכן הוא בקירוב7 2

9 3 7 .

מוצא חן בעינייך אצל כל מורה, ומה את לא אוהבת?מה .א

. את יכולה 2rנסי לבנות את המהלך האולטימטיבי מבחינתך כדי להסביר למה שטח עיגול זה .ב

להעזר בהסברים שנתנו המורות, לשנות בהם מה שאת רוצה, או להוסיף משל עצמך.

איך מהלך כמו זה שהצעת עובד בכיתה? .ג

177 | List of publications

List of publications

Refereed Conference Proceedings

Silverman, B., & Even, R. (2014). Modes of reasoning in Israeli 7th grade mathematics

textbook explanations. In K. Jones, C. Bokhove, G. Howson, & L. Fan (Eds.),

Proceedings of the International Conference on Mathematics Textbook Research and

Development (ICMT-2014) (pp. 427–432).

Silverman, B., & Even, R. (2015). Textbook explanations: Modes of reasoning in 7th grade

Israeli mathematics textbooks. In K. Krainer & N. Vondrová (Eds.), CERME 9 - Ninth

Congress of the European Society for Research in Mathematics Education (pp. 205–

212). Prague, Czech Republic: Charles University in Prague, Faculty of Education

and ERME. Retrieved from https://hal.archives-ouvertes.fr/hal-01281094

Silverman, B., & Even, R. (2016a). Paths of justification in Israeli 7th grade mathematics

textbooks. In E. Naftaliev & N. Adin (Eds.), Proceedings of the 4th Jerusalem

Conference on Research in Mathematics Education (pp. 52–54). Jerusalem, Israel.

(Hebrew version of 2016b).

Silverman, B., & Even, R. (2016b). Paths of Justification in Israeli 7th grade mathematics

textbooks. In C. Csíkos, A. Rausch, & J. Szitányi (Eds.), Proceedings of the 40th


(Vol. 4, pp. 203–210). Szeged, Hungary: PME.

Declaration | 178

Declaration

I declare that the thesis summarizes my independent research.

Explanations and justifications in Israeli mathematics textbooks and ...

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