GULCIN TAN-SISMAN and MERAL AKSU

THE LENGTH MEASUREMENT IN THE TURKISHMATHEMATICS CURRICULUM: ITS POTENTIALTO CONTRIBUTE TO STUDENTS’ LEARNING

Received: 4 March 2010; Accepted: 26 May 2011

ABSTRACT. “Knowing and doing measurement” is a fundamental competency in reallife since measuring attributes of objects in appropriate units and using measuring toolsassist students to quantify and understand the world. For this reason, the study ofmeasurement has a special place in every mathematics curriculum. Among the domains ofmeasurement, length measurement is extremely important for students to understand basicideas about measurement and to construct bases for more advance topics as area andvolume measurement. However, research on students’ understanding of lengthmeasurement reveals serious difficulties varying from incorrect alignment with a ruler toconfusion of perimeter with area. One of the reasons for students’ struggles inmeasurement is considered as weaknesses in the intended (written) curriculum. In thisrespect, this study was designed to investigate the content of length measurement in theTurkish mathematics curriculum (1st–fifth grade) in terms of its potential to supportstudents’ understanding. For this purpose, the Turkish Mathematics Curriculum Guidewas carefully analyzed by considering main components of the curriculum (e.g. learningobjectives). In spite of some weaknesses, the length measurement content in thecurriculum seems to provide meaningful opportunities for students to develop theconcepts and skills involved in length measurement.

KEY WORDS: curriculum analysis, length measurement, teaching and learningmeasurement, Turkish Elementary Mathematics Curriculum

INTRODUCTION

Measurement is one of the subjects in mathematics that can be consideredas a top agent for connecting mathematics not only with everyday life butalso with other areas (e.g. geometry, statistics). Because of its well-knownimportance, the study of measurement has received special attention bycurriculum developers, and it has a special place in every mathematicscurriculum. Considering the measurement topics, length measurement,which is the starting point of almost all mathematics curricula, is criticallyimportant for the construction of students’ basic ideas about measurement.

Some part of this study was presented at the European Conference of Educational Research (ECER),Vienna, 28–30 September 2009.

International Journal of Science and Mathematics Education (2012) 10: 363Y385# National Science Council, Taiwan (2011)

Research studies point out the importance of students’ genuine under-standing of length measurement that it is not only knowing “how tomeasure” but also knowing “what it means to measure” as well asunderstanding the relationship between them, to gain more complexmeasurement concepts and skills such as area and volume.

In order to make sense of linear measurement as well as establishing basesfor area and volume where the idea of one-dimensional unit is broadened intotwo and three dimensions, students need to develop the vital concepts asindicated in Figure 1 (Stephan&Clements, 2003; Lehrer, 2003). The skills oflinear measurement include direct and visual comparison, visual estimation,measuring with nonstandard and standard units, measuring with rulers, tapemeasurements, performing conversions, knowing the abbreviations of lengthunits, and using the formula for perimeter.

Research on students’ understanding of length measurement revealsthat they have serious difficulties while learning length measurement aswell as other domains of measurement. Considering the large-scale test

Figure 1. The conceptual bases of length measurement

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results, fourth grade US students’ performance on one of the NAEP 2003released item “finding the length of a toothpick placed on the broken-ruler” indicated poor understanding of length measurement as only 20%of them found the correct answer (Blume, Galindo & Walcott, 2007).Similarly, only 41% of eighth grade students responded correctly for theTrends in International Mathematics Study (TIMSS) 1999 item asking tofind the length of a curved string placed on a ruler (TIMSS InternationalStudy Center, 2000). One explanation for this weak performance is that manyelementary and secondary school students struggle to understand the meaningof hash marks and/or numbers on a ruler and the relationship betweennumbers and marks on a ruler (Mullis, Dossey, Owen & Phillips, 1991).

Furthermore, small-scale classroom studies have also yielded consis-tent results with large-scale researches. Bragg & Outhred (2000)conducted a research study with 120 students from grades 1 – 5 toinvestigate students’ knowledge of length measurement. They observedthat most of the students used procedural strategies (e.g. counting unitsand/or marks) to measure length with a ruler. Although many of thestudents could measure and draw lines correctly by grade 5, they failed tocomplete the tasks accurately which required the understanding of scale(e.g. the concept of zero point). The results also indicated that a smallnumber of students could use informal units to construct a ruler andunderstand the meaning of numerals on a ruler. In addition to children’misconceptions about a ruler, a significant number of students in the agegroup of 9 – 13 years could not conceptualize that while making iterationswith a unit, the quantity being measured must be covered withoutoverlaps or gaps (Hiebert, 1981).

As far as the concept of perimeter concerned, Tan-Sisman & Aksu(2009) reported that most of the seventh grade students confused not onlythe concept of perimeter with the concept of area but also confused theformula for perimeter with area. They also found that many students havedifficulties in units of length and area measurement. Similarly, Kidman &Cooper (1997) found that most of the middle grade students were good atarea and perimeter questions that call for use of formula, yet theyconfused the area concept with the concept of perimeter.

The mostly reported students’ mistakes while learning linear measure-ment are as follows: (a) starting from 1 rather than 0 (e.g. Ellis & Siegler,1995; Lehrer, Jacobson, Thoyre, Kemeny, Strom, Horvath, Gance et al.,1998a), (b) iterating units by leaving gaps or overlapping units (e.g.Lehrer, Jacobson, Kemeny & Strom, 1999; Hiebert, 1981), (c) incorrectalignment with a ruler (e.g. Stephan & Clements, 2003; Bragg & Outhred,2000), (d) counting hash marks or numbers on a ruler/scale instead of

LENGTH MEASUREMENT IN THE TURKISH MATHEMATICS CURRICULUM 365

intervals (e.g. Stephan & Clements, 2003; Bragg & Outhred, 2000;Boulton-Lewis, Wilss & Mutch, 1996), (e) focusing on end point whilemeasuring with a ruler (e.g. Lehrer, 2003; Bragg & Outhred, 2000), (f)mixing units of length with other units of measurement (e.g. Szilagyi,2007), and (g) confusing the concept of perimeter with area (Kidman andCooper, 1997; Tan-Sisman & Aksu, 2009).

Furthermore, Smith, Tan-Sisman, Figueras, Lee, Dietiker & Lehrer(2008a) and Smith, Tan-Sisman, Dietiker, Figueras, Males, Lee, Mosier etal. (2008b) identified the six factors associated with the students’ weaklearning of measurement as given in Figure 2. All of these factors eitherdirectly or interactively affect students’ learning of spatial measurement.

Considering the fact that a written curriculum includes at least threeanswers for the questions of “what should be taught, how it should betaught, how it should be evaluated?,” there is a close and clearrelationship between what students learn and how a curriculum organized

Figure 2. Six factors associated with the students’ weak learning of measurement

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in terms of learning objectives, content, implementation, and assessmentprocess. Emphasizing on the relationship between students’ learning anda mathematics curriculum, the National Council of Teachers ofMathematics’ (NCTM) document entitled Principles and Standards forSchool Mathematics (2000) states that “A school mathematics curriculumis a strong determinant of what students have an opportunity to learn andwhat they do learn.” (p. 15). In this respect, a well-developed curriculummay improve students’ mathematical understanding. For instance, Lehreret al. (1998a) and Lehrer, Jenkins & Osana (1998b) conducted teachingexperiments to investigate elementary students’ understanding of geom-etry and space. The results indicated that in a well-designed learningenvironment, students went beyond procedural competence to understandthe underpinnings of measurement.

Another study conducted by Clements, Battista, Sarama, Swaminathan& McMillen (1997) investigated students’ development of linearmeasurement concepts within a mathematics curriculum called Inves-tigations in Number, Data and Space in which a computer softwareprogram, Geo-Logo, was embedded. They found that making use of suchsoftware program as Geo-Logo facilitated students’ construction of linksbetween their own number and spatial schemes and provided meaningfullearning tasks for students. They also suggested that the connectionsbetween numbers and spatial sense should be highlighted throughout themathematics curriculum and the classroom tasks should engage learnersto explore, use, and communicate mathematical ideas. In a similar vein,Stephan, Bowers, Cobb & Gravemeijer (2003) claimed that students wereable to do and understand measurement in a carefully designed learningenvironments. Thus, if a written curriculum has weaknesses in cultivatingstudents’ understanding of measurement (e.g. not helping studentsrecognize the fact that any point on a ruler can serve as the startingpoint), superficial understanding in measurement will never end. In lightof the above explanations, it can be concluded that in a well-designedmathematics curriculum, students can learn and understand measure-ment. Although there have been many studies on students’ difficultieswith learning measurement, there is a limited number of studiesconcerned with the problems or weaknesses of written curriculum thatmight be the cause of insufficient understanding of measurement (e.g.Smith et al., 2008a, b).

In addition, the educational system in Turkey, which is the maincontext of this study, is centrally structured, and the Ministry of NationalEducation (MONE) is the main body for planning, programming,executing, monitoring, and controlling all educational services from

LENGTH MEASUREMENT IN THE TURKISH MATHEMATICS CURRICULUM 367

development of school curricula to the appointment of teachers at primaryand secondary school levels (Binbasioglu, 1995). Since the foundation ofthe Turkish Republic in 1923, education has always been conceived as ameans of keeping up with the modern age and its world view. MustafaKemal Atatürk, the founder of the Turkish Republic, invited John Deweyto Turkey and requested that he assisted to establish a democraticeducational system in the country and in educating teachers for thedesired democratic society (Ata, 2001; Onkol & Tan, 2005; Turan, 2000;Wolf-Gazo, 1996). The ultimate goal of the Turkish educational system isstated as follows:

…to raise students as highly skillful individuals who have improved thinking, perceivingand problem solving abilities, are open to universal values and new ideas, have a strongnational culture, personal feelings of responsibility and social sensitivity, are committed toAtatürk's nationalism and Atatürk's principles and revolution and are inclined toproduction of technology and science (Ministry of National Education [MONE], 2001).

In Turkey, there have been continuous attempts and reform movementsundertaken to realize the ultimate goal in the field of education. Certainly,being a candidate member of European Union (EU) has also acceleratedTurkey’s educational reform movements. Over the decades, dramaticincreases have been observed in the number of schools, enrolment rate ofstudents, number of universities, and expenditure on education (Ministryof National Education [MONE], 2009a). Furthermore, the 8-year basiccompulsory education started to be implemented in 1997.

As far as curricular changes and revisions concerned, the majorchanges made in Turkish Elementary Mathematics Curriculum (TEMC)are dated as 1924, 1936, 1948, 1962, 1968, 1982, 1993, and 1998 as aresult of continuous revision movements and of changes in theeconomical, political, and social structures of the Turkish society (Delil& Güleş, 2007; Koc, Isiksal & Bulut, 2007). During the 2004 – 2005school year, the last revision made on elementary school curriculumincluding the subject of mathematics, science, social science, life science,and Turkish was used as one of the attempts to improve the quality ofeducation. The nation-wide implementation of the new elementary schoolcurricula (first–fifth grades) was begun in the 2005 – 2006 school year,after pilot tests were conducted for a whole academic year in ninedifferent provinces in Turkey.

The new curriculum was developed in line with the national needs andvalues, contemporary scientific studies, national and internationalevaluation reports, and also current developments in the field of education(Ministry of National Education [MONE], 2009b). Moreover, Turkey’s

GULCIN TAN-SISMAN AND MERAL AKSU368

candidacy for EU accession and the poor performance of Turkishstudents’ both in national and international studies have also beeninfluential in developmental actions (Aksit, 2007; Erbas & Ulubay, 2008;Koc et al., 2007). It is aimed to create student-centered classroomenvironments in which active participation of students, use of differentinstructional methods and strategies, and integration of different topics,subjects, and disciplines are required for the establishment of construc-tivist way of learning (Ministry of National Education [MONE], 2009b).Overall, the striking shifts of the new curriculum can be explained asmoving from subject-centered curriculum and behaviorist way ofteaching to a more learner-centered curriculum and to a moreconstructivist way of learning.

THE PURPOSE OF THE STUDY

The main purpose of this study is to explore how the length measurementin the written Turkish mathematics curriculum (first–fifth grade) contrib-utes to students’ understanding by addressing the following researchquestions:

(a) How is the content of length measurement organized in terms ofscope and sequence?

(b) To what extent do the suggested instructional activities for lengthmeasurement have potential to provide opportunities for meaningfullearning?

(c) How is the conceptual and procedural knowledge elements of lengthmeasurement embedded in the curriculum?

METHOD

This study was designed to examine the length measurement content inthe written elementary mathematics curriculum (first–fifth grade) in termsof its potential to support students’ understanding. For this purpose, datawere collected through document analyses. The main data collectionsource of this study was the Turkish Elementary MathematicsCurriculum Guide for first–fifth grades published by the NationalBoard of Education. The full text of the guide is also available foreveryone on the website of the MONE.

LENGTH MEASUREMENT IN THE TURKISH MATHEMATICS CURRICULUM 369

The guide has 12 main chapters including vision, main principles,aims, common skills of the curriculum, and the suggested assessmentmethods and techniques. Furthermore, in separate chapters, each learningarea (strand) of mathematics by grades is presented according to thetopics, learning objectives, suggested ways of instruction, and the specificexplanations including cautions for teachers, extracurricular activities,connections with other mathematical strands as well as other subjectareas, and examples for assessment of students’ learning.

Figure 3 is one of the pages for a 3rd Grade Mathematics Curriculumtaken from the guide. It includes the learning expectations for the concept ofperimeter and describes how to teach the topic. Explanations are given in thelast column. It might provide a better picture for the structure of the guide.

The curriculum guide was examined carefully to find out the extent towhich the length measurement topic supports students’ understanding.The following steps were followed throughout the study: (1) locatingrelevant data, (2) developing a framework, (3) analyzing data, and (4)interpreting the results. During the first step, the measurement learningarea in each grade level was reviewed to distinguish the topics for length

Sub-topic Learning Objectives

Suggested activities for Instruction Explanations

“Calculate the perimeter of planar shapes.”

Using geo-board, dot or square papers, have students find the perimeter of square, rectangle, or the shapes formed by joining square and rectangle.

The perimeter of the shape above is 14 units.

The perimeters of planar shapes are calculated.

Measuring the side lengths of such objects as picture, blackboard, window, etc., have students generate a table. Then, adding the side lengths of each object together, have students calculate the perimeters.

[!] Any kinds of planar shapes can be used for calculating the perimeter. But, these shapes are not introduced to students as “Polygon”.

3rd grade

P

E

R

I

M

E

T

E

R

Abbreviations: : Classroom

Activity

[!] Caution

Connections with other subject areas

“Solving and generating problems related to the perimeters of planar shapes.”

Have students find the perimeter of planar shapes drawn on isometric paper.

[!] Do not ask such problems as “giving the perimeter of a planar shape and asking what the side length is.

Turkish Lesson- “Writing” Learning Expectation 5

Figure 3. A page from the curriculum guide for the concept of perimeter

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measurement. Then, each page of length measurement including thelearning objectives, suggested activities for instruction, example problemsfor student assessment, etc. was drawn from the core content ofmeasurement. Following this step, the framework was developed byconsidering the existing research on students’ understanding of lengthmeasurement, the theoretical bases of linear measurement, and theresearch questions of the study. It specifically includes the concepts andthe skills related to linear measurement (e.g. unit iteration, perimeter,measuring with a ruler, etc.) and common student mistakes/misconcep-tions as listed previously. The document analysis was done in line withthe framework. Lastly, data collected were categorized by taking intoconsideration the research questions of the study. Before presenting theresults, it is believed that the general information not only about theTurkish Elementary Mathematics Curriculum but also MeasurementLearning Area might be useful starting points for a better understanding.

The Turkish Elementary Mathematics Curriculum

The TEMC is based on the principle that “Every child can learnmathematics.” In the TEMC, special attention is given to the developmentof students’ conceptual knowledge and problem solving skills, alongwith procedural knowledge. Furthermore, relationship between con-ceptual and procedural knowledge is strongly underlined. In thisconceptually driven approach, “learning mathematics with understanding” isthe primary goal to be achieved by all students (Ministry of NationalEducation [MONE], 2009b).

In the TEMC, creating such learning environments in which studentslearn mathematics through investigations, discoveries, problem solvingactivities, sharing, and discussing their ideas is emphasized. It is alsoimportant for students to see the esthetical and enjoyable aspect oflearning mathematics. While teaching the concepts and skills ofmathematics, students’ previous knowledge and experiences are consideredas the starting point for their current and future learning. Thus, providingopportunities for students to construct their own knowledge based on whatthey knew and experienced is very important in the TEMC (Ministry ofNational Education [MONE], 2009b). The connections betweenmathematics and daily life are also taken into consideration to makemathematics more meaningful for the student.

With regard to the subjects, the TEMC covers four main learningareas, namely numbers, geometry, data, and measurement. Likewise,problem solving, communication, reasoning, and connections are

LENGTH MEASUREMENT IN THE TURKISH MATHEMATICS CURRICULUM 371

determined as the basic skills embedded in the main learning areas asindicated in Figure 4.

The perspective of assessment of students’ learning is also changed in away that students are measured not only in terms of cognitive outcomes butalso affective and psychomotor outcomes. Therefore, the TEMC offers useof various assessment methods such as portfolios, projects, anecdotes, selfevaluation reports, multiple choice tests, performance assessments, rubrics,mathematics diaries, observations, and attitude scales. In this respect, theassessment techniques offered are used for both formative and summativepurposes. Overall, the constructivist way of learning is reflected in everyaspect of the TEMC (Babadoğan & Olkun, 2006).

Measurement Learning Area in the Turkish Elementary MathematicsCurriculum

Measurement is one of the main learning areas in the TEMC. The studyof measurement is begun to be taught from first grade to eighth grade.

Figure 4. Learning areas and basic skill of TEMC

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Considering the proportional distribution of learning areas included in theTEMC (first–fifth grade), measurement is ranked second among otherlearning areas. As seen clearly from Table 1, “numbers” covers 59.2% ofthe mathematics curriculum, “measurement” is 18.8%, “geometry” is17%, and “data” only constitutes only 5% of the curriculum content.

The general aims of measurement stated in the TEMC are as follows:At the end of 5-year schooling, students are expected to (a) develop anawareness of the need for standard units, (b) use standard (metric) andnonstandard units to estimate measurements and compare estimate to theactual measurement, and (c) to appreciate the importance of measurement ineveryday life. Measurement learning area covers (1) length measurement,(2) money, (3) time measurement, (4) weight, (5) liquid measurement, (6)perimeter, (7) area measurement, and (8) volume measurement. Table 2indicates the sub-topics of measurement in terms of learning objectives,allocated time, and proportion.

As clearly seen in Table 2, among the sub-topics of measurement, onlylength and time measurement start to be taught from first to fifth gradestudents In particular, there are notable differences between the lengthmeasurement and the other sub-topics of measurement in terms of thetotal number of learning objectives, the allocated time, and proportion.

RESULTS

The findings gathered from the content analysis of length measurement inthe written Turkish Mathematics Curriculum for grades first–fifth arepresented in line with the research questions.

The Scope and Sequence of Length Measurement

In order to examine the scope and sequence of the length measurement,both the learning objectives and the suggested activities for instruction foreach grade levels were carefully analyzed by taking into consideration theframework. The study of length measurement, at first grade, focuses onmeasuring and estimating with nonstandard units and comparing andordering objects by length. Children first learn to directly compare objects(max. 3) such as pencils, rulers, etc. on the basis of their intuitivejudgments. While making comparisons, such conversations about lengthof objects as “Longer than” and “The same length as” “Tallest,”“Longest,” “Shorter,” “Tall,” and “Shortest” should be emphasized sothat children can recognize the attributes of length. Moreover, first

LENGTH MEASUREMENT IN THE TURKISH MATHEMATICS CURRICULUM 373

TABLE1

The

prop

ortio

naldistribu

tionof

learning

areas

Grade

levels

Con

tent

Num

bers

Geometry

Measurement

Data

1st

#LO

27LH

100

%Prop71

#LO

7LH

22%Prop15

#LO

9LH

18%Prop13

#LO

1LH

2%Prop1

2nd

#LO

33LH

88%Prop60

#LO

10LH

20%Prop14

#LO

13LH

28%Prop20

#LO

3LH

8%Prop6

3rd

#LO

28LH

73%Prop51

#LO

19LH

30%Prop19

#LO

18LH

35%Prop25

#LO

4LH

6%Prop5

4th

#LO

40LH

79%Prop55

#LO

17LH

27%Prop19

#LO

24LH

32%Prop22

#LO

4LH

6%Prop4

5th

#LO

46LH

85%Prop59

#LO

23LH

27%Prop18

#LO

16LH

22%Prop14

#LO

9LH

10%Prop9

LO

learning

objective,

LH

lesson

hour,Propproportio

n

GULCIN TAN-SISMAN AND MERAL AKSU374

graders order objects (max. 5) according to the attribute of length.Measuring with nonstandard units, including bodily units such as handspan, pace, and arm span, is also introduced to children at first grade. The

TABLE 2

The sub-topics of measurement

Topics of measurementNumber of learningobjectives

Time devoted (a lessonhour, 40 min) Proportion (%)

Length measurement(1st–5th grade)

4 6 46 11 85 10 76 6 4

Perimeter (3rd–5th grade) 2 3 23 5 44 6 44 5 334 LO 52 lessons 36

Time measurement(1st–5th grade)

2 4 33 6 43 5 44 4 31 3 213 LO 22 lessons 16

Weight (1st–4th grade) 2 4 32 5 42 4 23 4 39 LO 17 lessons 12

Liquid measurement(2nd–5th grade)

1 3 23 5 44 6 42 3 210 LO 17 lessons 12

Area measurement(3rd–5th grade)

1 3 23 6 45 5 39 LO 14 lessons 9

Money (1st–3rd grade) 1 4 31 3 21 3 23 LO 10 lessons 7

Volume measurement(5th grade)

2 3 22 LO 3 lessons 2

LO learning objectives

LENGTH MEASUREMENT IN THE TURKISH MATHEMATICS CURRICULUM 375

importance of “reporting a measure with the unit used to measure” isexplicitly stated in the curriculum.

When children come to second grade, they learn to use differentnonstandard units together to measure objects. For example, whilemeasuring the length of a book, students use both their hand span andfingers; they report the measurement result as “The length of a book is 2hand spans and 3 fingers.” Furthermore, the important aspects of lengthmeasurement in second grade includes understanding of the need formeasuring with standard tools (e.g. ruler, tape measure, etc.), measuringwith standard tools, introduction of length units (only meters andcentimeters), estimating and measuring the length of objects in metersand centimeters, the concept of unit iteration, the concept of zero point,and the concept of conservation. Although the concept of partitioning isincluded in this grade level, it is implicitly embedded in one of theactivity related to ruler construction. As emphasized in first grade, theimportance of “reporting a measure with the unit used to measure” isagain explicitly stated.

In grade 3, the study of length measurement focuses on the relationshipbetween meter and centimeter, simple unit conversions from centimetersto meters and vice versa, and use of a ruler. Not only the concept of zeropoint but also reporting a measure with the unit used is highlighted onceagain. Abbreviations for meter and centimeter are introduced in this gradelevel. Just as emphasized in previous grades, the development ofestimation skills is also continued in third grade. In addition, studentsencounter the concept of perimeter by third grade. Making use of string,ribbon, etc., students enclose/frame different objects such as pictures,postcards, etc. and then measure the length of a string/ribbon. In additionto studying on plane/regular shapes (e.g. postcard), students work onirregular shapes such as apple, potato, and carrot which are crosscut andenclosed by string to determine their perimeters. Moreover, students aretaught to calculate the perimeter of planar shapes by measuring theirsides’ lengths and then adding them together.

By fourth grade, more emphasis is placed on the standard units oflength measurement especially kilometers and millimeters. In the contextof real-life examples (e.g. building bridges, designing aircrafts, determin-ing the distance between two cities, etc.), the importance of recording anaccurate measurement and the need for measuring in millimeters andkilometers are stressed. Students also explore the relationship betweenmillimeter and centimeter, centimeter and meter, and meter and kilometer.Abbreviations for kilometer and millimeter are also introduced. Like inprevious grades, the development of students’ estimation skills is also

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continued in fourth grade. Another important issue for understanding oflength measurement taught to students by fourth grade is expressinglength measurements in equivalent forms (e.g. expressing 237 cm as 2 mand 37 cm or 372 mm as 37 cm and 2 mm).

Furthermore, fourth grade students learn about finding the perimeter ofplanar shapes through their side lengths without any usage of the formula.It is explicitly stated in the curriculum guide that while finding theperimeters of different planar shapes, neither the planar shapes are namedas polygons nor the formula for perimeter is used/provided. However,working on the relationship between the side lengths of rectangles andsquares and their perimeters, students discover the following notations:(a) The perimeter of a square = 4 × side of a square, (b) The perimeter ofa square is four times the length of one side of the square, and (c) “Theperimeter of a rectangle = (2 × long side of a rectangle) + (2 × short sideof a rectangle)”. Moreover, the inclusion of another crucial notion thatdifferent shapes may have the same perimeter expands students’ under-standing of the perimeter.

The study of length measurement in fifth grade covers unit conversionsamong meters–kilometers and meters–centimeters–millimeters and findingthe perimeter of squares, rectangles, triangles, parallelograms, trapezoids,and the circumference of circles. In addition, putting emphasize on therelationships between the circumference of circle, radius, and diameter,the number (pi) and the circumference of a circle are also taught tostudents by grade 5.

It is also worth to note that solving and generating problems related tothe topic of length measurement is highly emphasized in each grade level.Both the complexity and diversity of the problems gradually increases asgrade level increases. Below are the example problems for lengthmeasurement taken from the curriculum guide.

The Suggested Instructional Activities for Length Measurement

In the guide, teaching and learning activities for each learning area areprovided by grade levels. Since the focus of this study is on lengthmeasurement, the results presented here were collected from the section

Problem for 2nd Grade: What is the length between two arrows in centimeters?

Problem for 3rd Grade: What is the length between two arrows in centimeters?

LENGTH MEASUREMENT IN THE TURKISH MATHEMATICS CURRICULUM 377

of suggested activities for linear measurement so as to find out the extentto which these activities provide opportunities for meaningful learning.

As stated previously, the TEMC adopts a constructivist approach toteaching and learning mathematics. As a natural consequence ofconstructivist approach, the curriculum puts emphasis on students’ activeparticipation in the construction of their own mathematical knowledge.Moreover, it is essential for a teacher to design a learning environment inwhich students are engaged with worthwhile tasks that not only sparktheir curiosity but also motivate them to think, discuss, inquiry, andquestion mathematical ideas.

Considering length measurement, the suggested instructional approachmoves from conceptual underpinnings of the topic to proceduralcompetencies. For instance, students are expected to discover therelationship between meters and centimeters, before learning how toconvert centimeters into meters. Likewise, instead of directly giving theformulas for the perimeter of parallelograms and trapezoids, students areexpected to generate the formulas for parallelograms and trapezoids usingwhat they have previously learned about how to find the perimeter of arectangle, square, and triangle.

Furthermore, most of the teaching and learning activities suggested inthe guide require for use of different manipulatives and materials. A paperclip, pencil, toothpick, etc. are introduced as nonstandard units of measureand used for measuring the length of objects. After starting to measurewith standard units of length, standardized tools, especially rulers, gainmore importance in students’ learning of linear measurement. Studentsare asked, for instance, to construct a ruler, to use the ruler for finding theperimeter of shapes as well as for measuring length of objects.Another suggested activity requiring use of standardized tools isasking students to prepare a 10-cm-long cardboard and then to findhow many 10-cm-long cardboard in a meter stick. Calculators are usedas a means for highlighting the numerical pattern behind the number π(pi). While studying on the circumference and diameter of manycircular objects, students are asked to use a calculator to divide thecircumference measurements by diameter measurements in order toderive the number π.

In the curriculum guide, the close relationship between the study ofmeasurement and real life is acknowledged. So, suggested teaching andlearning activities for length measurement include various kinds ofexamples, questions, and connections from real-world experiences ofstudents. For instance, who is the tallest student in our classroom? Whatis the length of teacher’s desk? In addition, while learning abbreviations

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for meters and centimeters, the widely used abbreviations in the Turkishsociety are also mentioned. Similarly, the understanding of differencebetween meters and centimeters and development of estimation skills areembedded in the challenging question of “What would you do if you buya pair of pants without having a chance to try it on?”

The Conceptual and Procedural Knowledge Elements of LengthMeasurement Embedded in the Curriculum Guide

The major focus of content analysis here was to identify all instances ofconceptual and procedural knowledge of length measurement in thewritten curriculum. Hence, each aspect of length measurement in thecurriculum guide (e.g. the learning objectives, suggested instructionalactivities, sample problems for assessing students’ learning, etc.) wasanalyzed according to the framework. The conceptual knowledgeelements refer to the conceptual underpinnings of length measurementthat rationalize the procedures involved in measuring length. In a similarvein, the procedural knowledge elements refer to techniques, tools, andformulas used for measuring length.

With regard to the learning objectives, there are a total 34 objectivesfor the topic of linear measurement from first grade to fifth grade. Onlyone of them in fourth grade is related to adoption of the metric system inthe years of 1925 – 1931. Table 3 indicates the learning objectives in adetailed manner.

Considering the conceptual and procedural knowledge elementsembedded in the objectives, it was observed that most of them is

TABLE 3

The learning objectives of length measurement

Grade levelNumber of learningobjectives Time devoted to LM Proportion of LM

1st grade 4 6 lessons (40 min each) 4%2nd grade 6 11 lessons 8%3rd grade 8 (5 LM + 3 P) 15 lessons (10 LM + 5 P) 11% (7% LM + 4% P)4th grade 10 (6 LM + 4 P) 12 lessons (6 LM + 6 P) 8% (4% LM + 4 P)5th grade 6 (2 LM + 4 P) 8 lessons (3 LM + 5 P) 5% (2 LM + 3 P)Total 34 learning objectives 52 lessons 36%

LM length measurement, P perimeter

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procedurally dominated. The learning objectives classified as proce-durally dominated include direct comparison, visual comparison,visual estimation, measuring with nonstandard and standard units,carrying out conversions, and using formula for calculating perimeter.Conversely, conceptually dominated learning objectives include theunderstanding of the need for standard units of length measurement,understanding the relationship between units of length measurement(meter–centimeter; kilometer–millimeter; centimeter–millimeter;meter–kilometer), the concept of perimeter as a distance around anobject, the relationship between side lengths and perimeter of ashape, and the notion that different shapes may have the sameperimeter. Among six vital concepts of linear measurement, asexplained earlier, none of them was mentioned explicitly in thelearning objectives.

As far as the suggested learning and teaching activities concerned, itwas observed that conceptual underpinnings of length measurement aremore emphasized than the procedural ones. In early grades, theimportance of reporting a measure with the unit used to measure isexplicitly stated in the curriculum. Indeed, paying attention to units inearly grades might be considered as a useful starting point for thedevelopment of students’ understanding of relationship betweennumber and measurement. Further, the concept of unit iteration whichis at the heart of any understanding of measurement is introduced tostudents in second grade. Under the section of explanations, it is statedthat when a unit is iterated, there must be no gaps or overlaps.Conservation of length, another key concept, is also included in thecontent. In addition, there are suggested activities that explicitlyunderline the concept of zero point yet implicitly introduce the conceptof partitioning. Moreover, the suggested way for teaching the conceptof perimeter is dominated with the conceptual knowledge. In otherwords, the meaning of perimeter rather than the formula is the focus ofinstruction in third and fourth grade.

Although the suggested learning and teaching activities for lengthmeasurement in the guide are more conceptually oriented, there aresufficient number of activities cultivating necessary procedures formeasuring length and formula for perimeter. Measuring with nonstandardand standard units and tools, estimating, visual and direct comparison,calculating the perimeter of shapes by using the formula, and performingconversion are the procedural knowledge elements embedded in thecurriculum. Measuring with a ruler and estimating are particularly themain emphasized skills.

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CONCLUSION AND DISCUSSION

The world we live in today is becoming a more and more mathematicallysurrounded place. We are frequently encountering with mathematicalsituations requiring for mathematical decisions. In this respect, measure-ment is one of the mathematical strands filling up our everyday life.Besides, among the domains of measurement, length measurement is anextremely important topic for building the concepts and skills of area andvolume measurement. Thus, the concepts and skills involved in lengthmeasurement as well as in other domains should be learned withunderstanding.

What is more, curricula are usually embodied in curriculum guides, orsyllabuses describing the goals, objectives/expectations, content, timeallocation, implementation, and assessment process. As declared by Schmidtet al. (2002; as cited in Hook, 2004), “Specifically the curriculum itself -whatis taught- makes a huge difference” in students’ achievement (p. 1).

In this respect, the present study was designed to examine theadequacy of the length measurement in the written Turkish mathematicscurriculum related to the development of student’s meaningful learning.For this purpose, the mathematics curriculum guide was analyzed by theresearchers in a detailed manner. In general, it can be concluded that thelength measurement in the written curriculum seems to be well-organizedin terms of the learning objectives, the scope and sequence, and theinstructional approaches to teaching and learning process.

It was found that in a gradual process, the study of length measurementstarted with comparing lengths and moves progressively from measuringwith nonstandard units, to standard units, to measuring with a ruler andother standardized tools, and to the concept of perimeter. The sequence oflinear measurement in the TEMC is considered as the mostly advisedapproach to teaching of linear measurement (Clements, 1999; Kamii &Clark, 1997; National Council of Teachers of Mathematics [NCTM],2000; Stephan & Clements, 2003).

Concerning the scope of length measurement in the TEMC, the contentcovers the essential knowledge and skills identified in the Principles andStandards for School Mathematics (National Council of Teachers ofMathematics [NCTM], 2000) as well as in the mathematics educationliterature. Nonetheless, focusing only on the mostly used units of lengthmeasurement in real life (kilometers, meters, centimeters, millimeters) mightlimit a student’s understanding of the metric system and might hindermeaningful learning of carrying out conversions without connection withdecimalized system of measurement (Albayrak, Isik & ve Ipek, 2006).

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As far as six key concepts in linear measurement are concerned, theresults of the study indicated that the concept of unit iteration, zero point,and conservation are the concepts mentioned explicitly in the guide. Apartfrom the concept of partitioning which is embedded in the context of rulerconstruction task implicitly, we have not seen any implicit/explicitstatements related to other important concepts. The absence of transitivity,accumulation of distance, and relationship between measurement andnumber is considered as the deficiency of the TEMC, since these conceptsare essential for a complete understanding of linear measurement.

Further, the content devoted to the understanding of ruler structureseems to be quite narrow. To illustrate, the figure below taken from thecurriculum guide indicates how second grade students are expected toconstruct their centimeter rulers. Indeed, there is no opportunity forstudents to understand how a ruler works, apart from the one presented inthe Figure 5 and the example problems mentioned previously (see p. 15).

Considering conceptual and procedural knowledge elements includedin the length measurement content, although the way the contentdelivered is dominated with conceptual knowledge elements, the learningobjectives which cultivate students’ procedural competencies in linearmeasurement are outnumbered to those objectives which developstudents’ understanding of the concepts. In other words, the lengthmeasurement in the TEMC employs conceptually oriented instruction inorder to reach procedurally dominated learning expectations. Overall, itcan be concluded that in spite of some weaknesses, the length measurementcontent in the Turkish Elementary Mathematics Curriculum seems to havepotential to provide learning opportunities for young children to develop theconcepts and skills involved in length measurement.

Finally, several suggestions can be drawn from the study. First andforemost, the six key concepts of length measurement should be includedexplicitly in the content in a spiral manner. Considering the fact that pre-school education is not compulsory in Turkey, the increased emphasis onthese important concepts is needed. Indeed, it will be better, if these

Figure 5. The suggested way of a centimeter ruler construction

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important concepts become learning objectives. Secondly, the learningand teaching activities for facilitating students’ understanding of measur-ing processes built into rulers seems to be superficial and inadequate.Therefore, detailed information on how a teacher guides his/her studentsto understand the underpinnings of a ruler should be provided. Inaddition, different activities in a variety of contexts (e.g. working on abroken ruler) might be included in the curriculum. Thirdly, the curriculumguide should inform the teacher about common student misconceptionsand difficulties in linear measurement as well as research findings. Thiskind of information might be valuable input for removing the barriers thatblocks a student from learning. As a closing remark, it must be noted thata well-written mathematics curriculum is not the only factor influencingstudents’ understanding.

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Gulcin Tan-Sisman

Department of Educational Sciences,Faculty of EducationAkdeniz UniversityAntalya, TurkeyE-mail: gulcintans@gmail.com

Gulcin Tan-SismanE-mail: gtan@akdeniz.edu.tr

Meral Aksu

Middle East Technical UniversityAnkara, Turkey

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