Providing Written Feedback on Students' Mathematical Arguments: Proof Validations of Prospective...

Providing written feedback on students’ mathematicalarguments: proof validations of prospective secondarymathematics teachers

Sarah K. Bleiler • Denisse R. Thompson • Mile Krajcevski

Published online: 1 September 2013� Springer Science+Business Media Dordrecht 2013

Abstract Mathematics teachers play a unique role as experts who provide opportunities

for students to engage in the practices of the mathematics community. Proof is a tool

essential to the practice of mathematics, and therefore, if teachers are to provide adequate

opportunities for students to engage with this tool, they must be able to validate student

arguments and provide feedback to students based on those validations. Prior research has

demonstrated several weaknesses teachers have with respect to proof validation, but little

research has investigated instructional sequences aimed to improve this skill. In this article,

we present the results from the implementation of such an instructional sequence. A sample

of 34 prospective secondary mathematics teachers (PSMTs) validated twelve mathematical

arguments written by high school students. They provided a numeric score as well as a

short paragraph of written feedback, indicating the strengths and weaknesses of each

argument. The results provide insight into the errors to which PSMTs attend when vali-

dating mathematical arguments. In particular, PSMTs’ written feedback indicated that they

were aware of the limitations of inductive argumentation. However, PSMTs had a

superficial understanding of the ‘‘proof by contradiction’’ mode of argumentation, and their

attendance to particular errors seemed to be mediated by the mode of argument

Electronic supplementary material The online version of this article (doi:10.1007/s10857-013-9248-1)contains supplementary material, which is available to authorized users.

S. K. Bleiler (&)Department of Mathematical Sciences, Middle Tennessee State University, MTSU Box 34,Murfreesboro, TN 37132, USAe-mail: [email protected]

D. R. ThompsonCollege of Education, University of South Florida, 4202 E. Fowler Ave., STOP EDU105,Tampa, FL 33620, USA

M. KrajcevskiDepartment of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave., CMC342, Tampa, FL 33620, USA

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J Math Teacher Educ (2014) 17:105–127DOI 10.1007/s10857-013-9248-1

http://dx.doi.org/10.1007/s10857-013-9248-1

representation (e.g., symbolic, verbal). We discuss implications of these findings for

mathematics teacher education.

Keywords Proof and reasoning � Validation � College/university � Content

knowledge � Pedagogical knowledge � Secondary mathematics teachers

Introduction

Proving has been a major focal point within mathematics education for some time.

Notably, Principles and Standards for School Mathematics [National Council of Teachers

of Mathematics (NCTM) 2000] recommends ‘‘Reasoning and proof should be a consistent

part of students’ mathematical experience in pre-kindergarten through grade 12. Reasoning

mathematically is a habit of mind, and … must be developed through consistent use’’ (p.

56). Regular employment of proof in school mathematics may provide consistency and

coherence to a curriculum most students perceive as disjointed (NCTM 2009; Stylianides

and Ball 2008; Stylianou et al. 2009) and can help students be successful in the proof-

intensive curriculum of university mathematics (Howe 2001; Stylianou et al. 2009).

Despite national recommendations, proving is not common practice in most US

mathematics classrooms (Stylianou et al. 2009). Rather, it is often introduced within the

isolated setting of high school geometry instead of being a natural extension of previous

encounters with mathematics (Stylianides 2007; Stylianou et al. 2009). Consequently, it is

not surprising that students demonstrate difficulty constructing and understanding proof

(Chazan 1993; Harel and Sowder 2007; Healy and Hoyles 2000).

Why does proof not permeate the curriculum? In addition to lack of opportunities within

curriculum materials (Johnson et al. 2010; Thompson et al. 2012), researchers have shown

that teachers have difficulty constructing proofs (Harel and Sowder 2007; Morris 2002) as

well as validating others’ proofs (Knuth 2002a; Martin and Harel 1989; Morris 2002, 2007;

Selden and Selden 2003), skills necessary to integrate proof and reasoning into one’s

instruction.

In this article, we describe results of an exploratory study examining prospective sec-

ondary mathematics teachers’ (PSMTs’) validation of high school students’ arguments

before and after implementation of a structured set of activities in a mathematics methods

course. We describe our theoretical perspective on proof, overview literature on teachers’

validation of mathematical proof, and argue for instructional sequences that aid teachers’

development in this area. We conclude by discussing recommendations for future research

and implications for teacher education.

Theoretical perspective

Our theoretical perspective of proof and proving is based on the works of Hemmi (2006),

Pfeiffer (2011), Selden and Selden (2003), and Stylianides (2007). We follow Hemmi

(2006) and Pfeiffer (2011) who view proof as an artifact of the mathematics community of

practice, where proof is conceived as a tool to accomplish purposes such as verification,

explanation, systematization, discovery, communication, and intellectual challenge (De

Villiers 1999, as cited in Pfeiffer 2011). Thus, we believe it is important to understand the

strengths and weaknesses of PSMTs’ ability to validate and provide written feedback on

106 S. K. Bleiler et al.

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student arguments. Written feedback is one window through which teachers provide stu-

dents with insight into the accepted practices of the mathematics community. Through

their feedback, teachers attend to the needs of students as mathematical learners while also

maintaining an honest representation of mathematics.

We worked from Stylianides’ (2007) framework of proof, which honors students as learners

in the context of school mathematics while remaining true to mathematics as a discipline:

Proof is a mathematical argument, a connected sequence of assertions for or against a

mathematical claim, with the following characteristics:

1. It uses statements accepted by the classroom community (set of accepted statements)

that are true and available without further justification;

2. It employs forms of reasoning (modes of argumentation) that are valid and known to,

or within the conceptual reach of, the classroom community; and

3. It is communicated with forms of expression (modes of argument representation) that

are appropriate and known to, or within the conceptual reach of, the classroom

community. (p. 291)

Stylianides (2007) describes definitions, axioms, and theorems as examples of those

objects constituting the set of accepted statements. Modes of argumentation include

‘‘application of logical rules of inference (such as modus ponens…), use of definitions to

derive general statements, systematic enumeration of all cases…, construction of coun-

terexamples, development of a reasoning that shows that acceptance of a statement leads to

a contradiction’’ (p. 292). Modes of argument representation include linguistic/narrative,

diagrammatic/pictorial, and symbolic/algebraic. We follow Selden and Selden (2003) in

their conceptualization of proof validations as the ‘‘readings of, and reflections on, proofs

to determine their correctness’’ (p. 5). However, we recognize that validation is informed

by more than a simple identification of errors. With respect to Stylianides’ (2007)

framework, we approached this research assuming PSMTs’ validations would be influ-

enced by (a) the context in which students constructed the arguments (relying on what

PSMTs perceive as accepted statements in a high school context), (b) the mode of argu-

mentation employed or suggested by the instructions of each proof task, and (c) the mode

of argument representation utilized in each argument.

Literature on teachers and validation of proof

We designed our instructional activities from a review of the literature that indicated two

critical areas where teachers and undergraduates experience difficulties with respect to

proof validation: (a) focusing on (local) specifics of an argument but overlooking the

(global) logical structure of an argument and (b) holding an empirical inductive proof

scheme, described by Harel and Sowder (2007) as one in which a person offers or accepts

evidence from one or more examples in order to prove a mathematical proposition. We

discuss this literature as presented by the primary authors but recognize the ‘‘correctness’’

of participants’ validations may not be as clear as portrayed by some authors and is

dependent on the social context/community (Inglis and Alcock 2012; Thurston 1994).

Focusing on local components of an argument

When teachers validate arguments, researchers have found that they tend to focus on local

(e.g., line by line) components of the argument and neglect more global (e.g., mode of

Proof validations of prospective secondary mathematics teachers 107

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argumentation) components. For example, Knuth (2002a), in his study of 16 practicing

secondary mathematics teachers, found that they successfully identified valid proofs but

had greater difficulty identifying invalid arguments. Most errors occurred when teachers

focused on the validity of each step of an argument (e.g., the algebraic manipulation from

step 2 to step 3) instead of focusing on the reasoning and logic sustaining the argument as a

whole. For instance, one argument actually proved the converse of a given proposition; ten

of the 16 teachers decided this argument was a valid proof of the given statement, most

likely because they focused on the local details in the proof instead of the overall reasoning

and assumptions.

Similarly, Selden and Selden (2003) interviewed eight undergraduates (four PSMTs and

four mathematics majors) about the validity of four arguments. After the initial reading of

the arguments, less than half of the 32 validations were aligned with the researchers’

judgments of the validity of the arguments. Selden and Selden attributed this poor per-

formance to the students’ focus on the local details in a proof, noting that few students

made comments about the structural elements of the argument.

Weber (2010), in his study of 28 undergraduate mathematics majors who had completed

a transition to proof course, found that the students regarded four invalid arguments as

valid proofs in 60 % of their ratings. When incorrect judgments about the validity could

not be attributed to gaps in content knowledge or oversight when reading, the most fre-

quent causes of incorrect judgments were (a) students’ tendency to focus on the validity of

the proof from step to step rather than as a holistic entity and (b) their neglect of checking

that assumptions and conclusions of the argument were aligned with assumptions and

conclusions of the given proposition.

Holding the empirical inductive proof scheme

A key element of teachers’ knowledge related to proof is their ‘‘ability to understand and

distinguish between empirical and deductive forms of argument’’ (Stylianides and Ball

2008, p. 310). Researchers have shown that students, and some samples of teachers, use

and are convinced by inductive arguments to prove mathematical assertions. For example,

Martin and Harel (1989) studied 101 prospective elementary school teachers; for five of

seven inductive arguments, a significantly larger number of prospective teachers rated

those arguments as valid mathematical proofs (i.e., score of 3 or 4) than as invalid

mathematical arguments (i.e., score of 1 or 2). The researchers also found that prospective

teachers’ acceptance of inductive and deductive arguments was not mutually exclusive,

with over 46 % of participants simultaneously giving a high rating for both the ‘‘general

proof’’ deductive argument and at least one of the inductive arguments.

Knuth (2002a), in his interviews with practicing secondary teachers, found some rated

an argument of a particular case as a valid proof. Although many teachers voiced the

limitations of inductive argumentation, in practice they expected and frequently accepted

inductive arguments from their students (Knuth 2002b). Moreover, teachers often found

empirical arguments to be the most convincing of the modes of argumentation presented.

Weber (2010) investigated whether the mathematics majors in his study held an

empirical proof scheme. Of his 28 participants, 26 successfully identified an inductive

argument as invalid, suggesting that by the time mathematics majors have sufficiently

progressed in their education, they may no longer hold the empirical proof scheme.

However, his study included only one empirical argument; therefore, Weber called for

replication of the result to strengthen its reliability.


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Similarly, Pfeiffer (2011) showed that first-year honors mathematics students did not

hold the empirical proof scheme. These students recognized that an argument consisting

only of several examples did not qualify as mathematical proof. However, students did find

the inclusion of examples provided clarity and explanatory power.

The need for instruction on proof validation

Many researchers have called for explicit instruction related to proof validation to ameliorate

common difficulties related to accepting inductive arguments or focusing on local rather than

global elements of an argument (Knuth 2002a, b; Martin and Harel 1989; Morris 2002, 2007;

Pfeiffer 2011; Selden and Selden 2003). As explained by Selden and Selden (2003),

Validation of proofs is part of the implicit curriculum, but it is a largely invisible

mental process. Few university teachers try to teach it explicitly, although some may

admonish students to ‘‘read with pencil and paper in hand’’ …. This advice is at best

descriptive, but certainly not usefully prescriptive, so students tend to interpret such

vague directions idiosyncratically, spotting mainly local notational and computa-

tional difficulties. (p. 28)

At present, 45 of 50 US states have adopted the Common Core State Standards for

Mathematics (CCSSM) [Common Core State Standards Initiative (CCSSI) 2010]. The

CCSSM outlines eight standards for mathematical practice, including that students ‘‘construct

viable arguments and critique the reasoning of others’’ (CCSSI 2010, p. 6). Therefore, at least

in the USA, validation of mathematical arguments warrants increased attention in teacher

preparation programs not only because it is important mathematical knowledge for teaching,

but also because it is a practice in which students will increasingly be expected to engage.

Despite calls for explicit instruction related to validation, little research has explored the

effectiveness of such instruction or how such instruction might be structured. We explore

the findings from a short instructional sequence aimed at increasing PSMTs’ awareness of

the limitations of empirical reasoning and the importance of considering a mathematical

argument as a holistic entity.

Research questions

The purpose of this study was twofold: (a) to determine the effectiveness of an instruc-

tional sequence designed based on two major findings from the literature and (b) to gain an

understanding of PSMTs’ strengths and weaknesses when validating mathematical argu-

ments. The following questions guided our inquiry:

• In what ways do PSMTs’ validations of students’ written proofs differ before and after

the implementation of a structured set of class activities?

• To which errors in authentic samples of high school students’ written mathematical

arguments do PSMTs attend when engaging in proof validation?

Method

We designed and implemented a series of five activities to raise PSMTs’ awareness of and

improve their skill in validation of mathematical arguments (in general) and pay greater


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attention to notions of inductive/deductive reasoning and local/global elements of math-

ematical arguments (in particular). We created these activities primarily in response to

Selden and Selden (2003); after the students in their study conducted two full readings of

mathematical arguments and engaged in reflective discourse with the interviewer about the

arguments they read, their validations improved. The students’ improvement as a result of

time and reflection suggests explicit instruction related to validation of mathematical

proofs may aid in one’s development of this skill. Throughout our discussions, we focused

largely on each argument’s mode of argumentation and mode of argument representation,

and how these characteristics of an argument influenced PSMTs’ validations. Also, we

framed discussions in the context of secondary-level mathematics, attending to PSMTs’

perceptions of what might be considered an appropriate set of accepted statements.

Table 1 summarizes the design and objectives for each activity in our instructional

sequence. The ‘‘curriculum’’ employed many of the arguments used as research devices in

prior studies. These examples offered information from the respective studies about

potential misconceptions and difficulties in relation to validation. Thus, we had baseline

information against which to guide class discussion relative to validation issues.

Sample

We implemented this sequence of activities during three semesters (Fall 2009, Fall 2010,

and Fall 2011) within a university methods course called Reading the Language of

Mathematics, a required course for PSMTs at the University of South Florida that satisfies

the state’s ‘‘reading in the content area’’ requirements and is uniquely designed for future

teachers of mathematics. Throughout the course, PSMTs considered many issues related to

mathematical literacy, including reading and writing in mathematics. Hence, a focus on

reading and understanding proofs was a natural extension of other communication activ-

ities. The class met once per week for 3 hours throughout the 15-week semester. Activities

1–3 occurred during one class session; activities 4–5 occurred during the subsequent class

session. Activities 1 and 5 (i.e., the pre- and post-assessment) form the basis of our

analysis.

All students enrolled in this course were preparing to become secondary mathematics

teachers; the majority were in their last semester of coursework before their final intern-

ship. We conducted all activities as a normal part of class during the final weeks of the

semester. Most activities focused on argumentation in Number Theory, which was a

required course in the mathematics education major. Most PSMTs completed, or were near

completion of, this course. At the end of activity 5, we asked for consent to use responses

to activities 1 and 5 for purposes of this study. Students were assured their data would have

no identifiers, and that the instructor of record would have no access to their consent

decision prior to submission of grades. Only two students (one from 2009 and one from

2011) declined to provide consent.

Our data analysis is based on validations by the 5 PSMTs from Fall 2009, 16 from Fall

2010, and 13 from Fall 2011 who were in attendance during both class sessions and fully

completed activities 1 and 5; several students in Fall 2009 did not have complete data. The

same five activities were conducted in all three semesters and were implemented (in so far

as possible) the same way during each semester; therefore, we combine the results into a

larger group of 34 PSMTs. The instructor of record in 2009 and 2010 was the second

author of this article, and in 2011 was the first author. However, during all class sessions in

which this instructional sequence was conducted, the first author implemented the activities

while the second author took observation notes.


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Table 1 Description of instructional sequence

Activity

(duration)

Design of activity Objectives Insights from class discussion

1 (30 min) Individually, PSMTs assign a

score (from 1 to 4) to six

arguments written by high

school students.

PSMTs provide written feedback

to the student authors of the six

arguments, highlighting

strengths and weaknesses of

each argument.

Assess PSMTs’ validation of

mathematical arguments

before the implementation of

activities 2–4.

2 (30 min) PSMTs read a vignette about a

7th grade classroom in which

the teacher encounters a

student conjecture based on

inductive reasoning and needs

to decide his next instructional

move (vignette modified from

Stylianides and Stylianides

2010, p. 162).

Motivate a context in which

teachers need to employ

knowledge of inductive/

deductive arguments.

Instructor led discussion by asking:

If you were the teacher, what

would you do in this situation?

What would be your next actions?

PSMTs indicated they might ask

students to give them a day or two

to attempt an argument.

PSMTs suggested introducing the

term ‘‘conjecture.’’

The terms inductive/deductive were

not used during the discussion,

but PSMTs began to think about

what qualifies as valid

mathematical argumentation.

The instructor proved the given

conjecture in ways that honor

students as 7th grade learners as

well as mathematics as a

discipline.

3 (90 min) 3A. PSMTs work in groups to

discuss and validate four

arguments provided by Morris

(2002), pp. 115–116.

3B. PSMTs work in groups to

discuss and validate four

additional arguments provided

by Morris (2002), pp. 116–118.

PSMTs identify arguments as

inductive or deductive.

3C. PSMTs read sample

arguments from Martin and

Harel (1989), pp. 44–45 and

decide whether each is

inductive/deductive and valid/

invalid.

Encourage PSMTs to

(a) recognize the difference

between inductive and

deductive argumentation,

(b) understand the value of

inductive reasoning for initial

understanding and conjecture

formation, (c) recognize

limitations of inductive

reasoning as a form of valid

generalized mathematical

argumentation, and

(d) understand the distinction

between the dual concepts of

inductive/deductive and valid/

invalid argumentation.

After 3A: Instructor provides

definitions and examples of

inductive and deductive reasoning

in the form of verbal syllogisms.

Instructor and students return to 3A

and discuss which arguments are

inductive/deductive. Repeat for

arguments in 3B. PSMTs justify

decisions.

Before 3B: Instructor initiates class

discussion about the role

inductive argumentation has in

initial reasoning and conjecture

formation, but highlights the

limitations of this type of

argument as valid (generalized)

mathematical proof.

After 3B: Instructor initiates class

discussion about the validity of

various types of arguments by

posing questions such as: Can an

inductive argument be a valid

mathematical proof? How about a

deductive argument? Are all

deductive arguments valid

mathematical proofs? (Why or

why not?) When would a

deductive argument not be a valid

mathematical proof?


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The pre- and post-assessment (activities 1 and 5)

We provided PSMTs with the instructions in Fig. 1 for activities 1 and 5. PSMTs had

previously worked on a unit related to rubrics, were familiar with this 4-point holistic

rubric described by Thompson and Senk (1998), and had applied it to middle and high

school students’ responses on items not focusing on proof. During both activities 1 and 5,

PSMTs assigned a score of 1–4 for six authentic student arguments and provided written

feedback to the student authors about the arguments’ strengths and weaknesses. In a similar

way, Pfeiffer (2011) asked first-year college students to validate purported proofs (with a

score out of five) and ‘‘give a line of advice to the student.’’ Pfeiffer found it rare that

students would provide information about the strengths of the arguments. We asked

PSMTs to point to both strengths and weaknesses of the arguments they validated, which

seemed to alleviate this problem.

We did not discuss the arguments, or the feedback PSMTs provided, after activity 1.

Instead, we proceeded directly to activity 2. Also, we did not provide PSMTs with an

indication of what constitutes a good or valid proof before activity 1. Therefore, this

research offers insight into what PSMTs perceive as valid argumentation within the context

of high school mathematics.

We refer to the six arguments in activity 1 as 1a–1f and to those in activity 5 as 5a–5f.

All twelve were written by high school juniors and seniors participating in an evaluation of

a curriculum focused on precalculus and discrete mathematics (Thompson 1992). Students

wrote the arguments as part of posttest assessments that were not necessarily a part of their

course grade. We selected the arguments purposefully to assess PSMTs’ validation skills in

relation to inductive/deductive and local/global issues, choosing arguments so the errors in

1a–1f paralleled the errors in 5a–5f as closely as possible. If argument 1a employed an

Table 1 continued

Activity

(duration)

Design of activity Objectives Insights from class discussion

4 (60 min) PSMTs work in groups to discuss

and validate four arguments

provided by Selden and Selden

(2003), p. 33.

Instructor asks PSMTs to

‘‘identify as many errors as

possible in the four given

arguments.’’

Encourage PSMTs to focus on

the global errors in a proof’s

structure (e.g., an incorrect

mode of argumentation) in

addition to local errors.

Instructor leads a class discussion

about all of the errors in the

arguments, making note of

unidentified errors and explicitly

referencing the ‘‘global’’ errors

not identified.

In discussing one of the arguments,

the instructor highlights the

difference between proof by

contraposition and proof by

contradiction

Instructor helps PSMTs focus on

the logical non-equivalence of a

proposition and its converse.

5 (30 min) Individually, PSMTs assign a

score (from 1 to 4) to six

arguments written by high

school students.

PSMTs provide written feedback

to the student authors of the six

arguments, highlighting

strengths and weaknesses of

each argument.

Assess PSMTs’ validation of

mathematical arguments after

the implementation of

activities 2–4.


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incorrect mode of argumentation, argument 5a employed the same incorrect mode of

argumentation. Moreover, the propositions being ‘‘proven’’ in corresponding arguments

were identical.

We presented the arguments in the original high school students’ handwriting to create

an authentic validation context. Here, we have typed the student arguments word for word

(see Table 2) but include the handwritten versions in Online Resource 1. We aligned

corresponding arguments from activities 1 and 5 to highlight similarities/differences in the

arguments’ structures and associated errors.

Data analysis

To investigate the ways PSMTs’ validations of students’ written proofs differed before and

after implementation of our activities (Research Question #1), we determined whether they

made a ‘‘correct judgment’’ on their validation rating based on our expert consensus. For

example, we agreed 1a should be considered invalid as a mathematical proof. Therefore, if

a PSMT rated 1a with a rubric score of 1 or 2, we considered their rating a ‘‘correct

judgment,’’ but if the PSMT rated 1a with a rubric score of 3 or 4, we considered their

rating an ‘‘incorrect judgment.’’ For each matched pre- and post-argument, we assigned

PSMTs’ ratings into one of four categories (CC, IC, CI, and II), as defined in Table 3.

Next, we investigated the errors to which PSMTs attended, as evidenced by their written

feedback (Research Question #2). The three authors of this article, two mathematics

educators and one mathematician, independently created a list of all errors in the six

matched pairs of arguments. When at least two of us identified a similar error, we created a

‘‘composite’’ statement to describe the error. Overall, 92 % of the errors were identified by

at least two of us, indicating a high level of reliability.

Moreover, we independently identified each of the arguments as ‘‘valid’’ or ‘‘invalid’’ as

mathematical proofs. Among the three of us, there were no disagreements on the

You are a mathematics teacher at Smith High School. This year you have decided to focus on students’ proof abilities in all of your classes. On the following pages are several proof tasks that you gave to your students. Now you need to determine if the student responses are valid or

Successful response: 4- The student’s argument is a valid mathematical proof. 3- The student’s argument would be a valid mathematical proof after the correction of some minor clerical mistakes.

Unsuccessful response: 2- The student’s argument is an invalid mathematical proof that starts in the proper direction but contains major conceptual errors.

invalid as mathematical proofs. The following rubric will be used to grade the student responses.

1- The student’s argument is an invalid mathematical proof that makes little meaningful progress.

For each student response, you should:

(1) Provide a score based on the general rubric above. (2) Write a short paragraph as feedback to the student about why you gave them that score.

If the student’s response was unsuccessful, help the student understand what they did wrong and how they might improve their response. If the student’s response was successful, help the student understand the strong points of their argument.

Fig. 1 Instructions for activity 1 and 5


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Table 2 Authentic student arguments constituting activities 1 and 5

Reproduced student arguments: activity 1 (pre-assessment)

Reproduced student arguments: activity 5 (post-assessment)

Prove the following theorem: For all integers m, n and p, if m is a factor of n and m is a factor of p,then m is a factor of n 1 p.

1a (Invalid empirical argument): There exists integersm, n, and p, such that, if m is a factor of n and m is afactor of p, then m is a factor of n ? p

For example, m = 2, n = 6, and p = 12nm¼ 6

2¼ 3 p

m¼ 12

2¼ 6 nþp

m¼ 18

2¼ 9

Therefore, by the factor sum theorem, this is true.

5a (Invalid empirical argument): Suppose m,n and p are real integers. If m is a factor ofn and a factor of p, then it must be a factor ofn ? p

Suppose m = 3, n = 9, and p = 15nþ p ¼ 9þ 15 ¼ 24� 3 ¼ 8So m is a factor of n ? p

Prove the following theorem: If m is any odd integer and n is any even integer, then m – n is an oddinteger.

1b (Valid proof): Let m be an odd integer and n be aneven integer. Then, m = 2x ? 1 and n = 2y for somex, y [ Z

m� n ¼ 2xþ 1ð Þ � ð2yÞ ¼ 2x� 2yþ 1 ¼ 2 x� yð Þ þ 1Because x and y are both integers, x – y is an integer, call

it k. Then, m – n = 2(x - y) ? 1 = 2k ? l. Thus, m –n is odd.

5b (Valid proof): We can write m = 2k ? 1 forsome k [ Z, and n = 2l for some l [ Z, then

m� n ¼ 2k þ 1ð Þ � 2l ¼ 2 k � lð Þ þ 1Since k and l are integers, k – l is an integer, and

so 2(k - l) ? 1 is an odd integer.

Use a proof by contradiction to prove the following statement is true: If a is a rational number and b isan irrational number, then a 1 b is an irrational number.

1c (Invalid, direct argument disguised as indirect):Suppose a is rational and b is an irrational number,then a ? b is a rational number.

Because a is rational, it can be written as a fraction (def.of rational)

Because b is irrational, it cannot be written as a fraction(def. of irrational)

a ? b an irrational number because they cannot beadded together rationally (by addition of rational/irrational)

We have a contradiction. ; a ? b is an irrationalnumber.

5c (Invalid, direct argument disguised asindirect): Suppose that if a is a rationalnumber and b is an irrational number, a ? b isa rational number.

a ¼ k; b ¼ xy

by definition of rational and

irrational #s

Then aþ b ¼ k þ xy¼ yk

yþ x

y¼ kyþx

y¼ irrational

Contradicts assumption at beginning of the proofthat a ? b is rational, so original statement istrue.

Is the following statement true or false? If true, prove that it is true. If false, prove that it is false. Forall integers a, b, and c, if a is a factor of b and a is a factor of c, then a is a factor of b�c.

1d (Invalid empirical argument): True, 3, 6, 9; a = 3,b = 6, c = 9

3 is a factor of 6, 3 is a factor of 9, and 3 is a factor of 54If a is a factor of b and c, then it must be a factor of

b�c. They all must be in the same group of multiples; This statement is true

5d (Invalid empirical argument): True. Forany numbers that are multiplied together, theresult of those can always be divided by one ofthe integers multiplied, if all of the numbersinvolved are factors of each other. a = 2,b = 4, c = 6

2 goes into 4, 292 goes into 6, 392 goes into (6 9 4) ? 24, 129

Same proposition as above in 1a/5a

1e (Valid proof, with possible clericalimprovements): If m is a factor of n, then n = c1m

If m is a factor of p, then p = c2mn ? p = c1m ? c2m = m (c1 ? c2) and m is a factor of

m (c1 ? c2)

5e (Valid proof, with possible clericalimprovements): Let n = mk; p = ml;n ? p = mk ? ml = m (k ? l); m is a factorof n ? p


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classification of any of the arguments. Prior research has shown that agreement about what

constitutes a valid mathematical proof (among experts) is not always consistent (Inglis and

Alcock 2012). We believe the consistency across both our ratings of valid/invalid and our

identification of errors within each argument provides an indication of content validity for

the items in this study.

Results

Validations before and after the instructional sequence

Table 4 displays the number of PSMTs whose ratings fell into each of the four categories

(CC, IC, CI, and II) and the percent of PSMTs who ultimately validated the arguments

appropriately as indicated by a correct judgment on Activity 5.

The results in Table 4 suggest that, overall, PSMTs’ validations on the matched

arguments reasonably aligned with the authors’ judgments of the arguments’ validity

(except for 1c/5c). PSMTs provided correct judgments (Category CC) for both pairs of

matched arguments we classified as valid mathematical proofs (1b/5b and 1e/5e). How-

ever, PSMTs demonstrated more variability in their ratings of the four matched arguments

determined to be invalid. Particularly for 1c/5c, more PSMTs moved from correct vali-

dations to incorrect validations (CI) than vice versa (IC), and a substantial subset of PSMTs

persisted in incorrect validations (II). For 1d/5d and 1f/5f, about an equal number of

Table 3 Categories describing correctness of PSMTs’ ratings from pre- to post-assessment

Category Pre-assessment rating (activity 1) Post-assessment rating (activity 5)

CC Correct judgment ? Correct judgment

IC Incorrect judgment ? Correct judgment

CI Correct judgment ? Incorrect judgment

II Incorrect judgment ? Incorrect judgment

Table 2 continued

Reproduced student arguments: activity 1 (pre-assessment)

Reproduced student arguments: activity 5 (post-assessment)

Same proposition as above in 1c/5c

1f (Invalid, empirical reasoning AND directargument disguised as indirect): If a is a rational #and b is an irrational number, then a ? b is a rationalnumber.

If a = 2 and b ¼ffiffiffiffiffi

23p

, then a ? b is rational.

2þffiffiffiffiffi

23p

¼ 6:79583523 (contradiction, this number isnot rational)

Therefore, a ? b is irrational.

5f (Invalid, empirical reasoning AND directargument disguised as indirect): Assume thatif a is a rational number and b is an irrationalnumber, then a ? b is a rational number

a = 1 and b = ia ? b = 1 ? ia ? b is not a rational numbera ? b is irrational; Original proof is true.

The arguments appear here in the same order they appeared on the pre- and post-activities. In parentheses,we give our judgments of the validity of each argument; these determinations were not presented to PSMTs.


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PSMTs moved from correct to incorrect validations (CI) as those who moved from

incorrect to correct (IC).

For each of the 34 PSMTs, we calculated the percent of the six matched arguments for

which PSMTs’ validations were categorized as either CC or IC (i.e., the two categories

resulting in a final correct validation). Figure 2 illustrates the results across the 34 PSMTs.

Eight (23.5 %) of the 34 appropriately validated all six matched arguments by the final

activity, and 17 (50 %) appropriately validated 5 of the 6 matched arguments. The

remainder appropriately validated 4 of the 6.

Identification of errors in the arguments

In the next few sections, we look closer at the errors to which PSMTs attended in their

written feedback for the three matched pairs of arguments that had the greatest variability

in terms of PSMTs’ numeric validation ratings. Tables 5, 6, and 7 contain our composite

list of errors, together with the percentage of PSMTs whose written feedback referred to

those particular errors, for 1c/5c, 1d/5d, and 1f/5f.

Arguments 1c and 5c, direct arguments disguised as indirect

Table 5 reports the errors to which PSMTs attended in 1c and 5c. The errors we identified

in the arguments were similar: (a) student uses imprecise or incorrect definitions of rational

and irrational numbers, (b) student provides an invalid justification, or provides no justi-

fication, for the step in argument purporting to show a ? b is irrational, and (c) student

employs direct, rather than indirect, argumentation. Although the errors in the two

Table 4 Number of PSMTs in categories CC, IC, CI, and II for all matched arguments and percent whorendered a final appropriate validation (CC or IC)

Tasks1a/5a(invalid)

Tasks1b/5b(valid)

Tasks1c/5c(invalid)

Tasks1d/5d(invalid)

Tasks1e/5e(valid)

Tasks 1f/5f(invalid)

# of CCs 30 34 6 24 34 24

# of ICs 4 0 5 4 0 5

# of CIs 0 0 12 6 0 5

# of IIs 0 0 11 0 0 0

Percent of PSMTs who provided acorrect final validation on activity5 (CC or IC)

100 100 32 83 100 85

Italicized columns identify the matched items to be discussed in further detail in this article

50 1000

Fig. 2 Box plot of percent ofmatched arguments validatedappropriately (CC or IC) for thesample of 34 PSMTs


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arguments are similar, their modes of argument representation differ; argument 1c employs

a verbal/linguistic mode while 5c employs a symbolic/algebraic mode.

For both 1c and 5c, less than half of the PSMTs recognized that students used imprecise

or incorrect definitions for rational and irrational numbers, a percentage we would have

expected to be higher. In their written feedback for 1c, most PSMTs who identified this

error focused on encouraging the student to use symbolic notation to communicate defi-

nitions of rational and irrational:

You start off right, but you don’t write a mathematical representation of ‘‘a is

rational’’ which would be a ¼ xy. So you can’t just say a ? b is irrational without

mathematical representation. (1c, PSMT #7, Fall 2010)

This proof is going in the right direction. Instead of saying ‘‘fraction,’’ say ‘‘pq

for

some integers p and q’’ and so on. This will make your argument more clear. (1c,

PSMT #29, Fall 2011)

For 5c, the PSMTs who identified this error focused on helping students recognize the

incorrectness of the symbolic representation in the argument, frequently indicating that

they believed the argument would be valid if only the student corrected the definitions. In

fact, 9 of the 34 PSMTs cited this as the sole error (of the three we identified) in 5c.

Moreover, although these PSMTs identified students’ imprecise or incorrect use of defi-

nition, their feedback suggested that they did not have a complete conception of the

definition themselves:

You are off to a great start but you got the representation of rational and irrational

wrong. A rational number a should be represented by the fraction xy, where x is an

Table 5 Errors and percentage of PSMTs attending to errors in arguments 1c and 5c

Pre-assessment 1c Post-assessment 5c

Errors identified (by authors) Percentage ofPSMTs whosefeedback referredto the error


Definitions for rational/irrational are imprecise.

41 % Definitions for rational/irrational numbers areincorrect.

44 %

Student understands on anintuitive level the nature ofirrationality, but provides novalid argument for whya ? b is irrational.

44 % Student provides no

justification why kyþxy

is

irrational.

18 %

Student does not work from thenegation of the conclusion(i.e., that a ? b is rational) toderive a contradiction. That is,student does not use anindirect proof, but attempts toprove a ? b is irrationaldirectly.

24 % Student does not work from thenegation of the conclusion(i.e., that a ? b is rational) toderive a contradiction. Thatis, student does not use anindirect proof, but attempts toprove a ? b is irrationaldirectly.

6 %


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integer and y is a natural number, and irrational number b should be represented by

k. Once you straighten that out, your proof will work out. (5c, PSMT # 17, Fall 2010)

This is a well-developed proof but I think you might have mixed up the definitions of

rational and irrational. If you changed those in your argument you would be on a

great track. (5c, PSMT #32, Fall 2011)

The most frequently identified error in PSMTs’ feedback for 1c was that the student

provided no valid justification for why a ? b was irrational but instead cited the statement

that was to be proven as the justification for this step. Many PSMTs (44 %) noted this

error, providing feedback such as:

Cannot use the definition of the statement you are attempting to prove as a proof of

the statement. (1c, PSMT #4, Fall 2009)

Far fewer PSMTs (only 18 %) provided feedback for 5c that indicated a justification was

needed to demonstrate why aþ b ¼ kyþxy

is irrational.

Only 24 % of the PSMTs for 1c and 6 % for 5c indicated that the arguments did not

follow the appropriate mode of argumentation. That is, even though the directions

prompted ‘‘use a proof by contradiction,’’ and even though the high school students who

wrote 1c and 5c claimed to come to a contradiction, few PSMTs recognized that the

student arguments were actually attempts at direct rather than indirect argumentation.

Because we asked the PSMTs to provide feedback that indicated both strengths and

weaknesses of the arguments, we found many PSMTs seemed to think that the students’

arguments were strong examples of proof by contradiction:

You understand the concept of contradiction; however, you need to be more explicit

in your writing and use of definitions. If a can be written as a fraction, then show

a equal to a fraction and how the addition of an irrational number and rational

number gives an irrational number. (1c, PSMT #26, Fall 2011)

You need to specify what k, x, y represent. What if y divides x? You need to make

sure to state that y does not divide x. Everything else seems to work. Your contra-

diction is correct. (5c, PSMT #16, Fall 2010)

One emphasis in PSMTs’ feedback on 1c and 5c was their indication that the arguments

started in the right direction, but because the definitions were imprecise or incorrect, the

proof could not be completed successfully. PSMTs focused on the negation of the con-

clusion of the proposition, where the student authors changed ‘‘a ? b is irrational’’ to

‘‘a ? b is rational’’ in the introductory lines of the arguments and took this as evidence that

the students understood proof by contradiction.

Began correctly with stating the contradiction, but you need to use your definitions to

prove the statements. (1c, PSMT #18, Fall 2010)

The student starts off in the right direction by assuming a ? b is rational, but the

definitions the student provided for rational or irrational are incorrect. (5c, PSMT

#19, Fall 2010)

Several PSMTs recognized that the arguments did not successfully employ an indirect

mode of argumentation. However, most who identified this error did so for 1c and

neglected to address this error in their feedback for 5c. For example, the feedback from one

PSMT for 1c and 5c reads:


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In order to reach a contradiction, the proof needs to start with ‘‘suppose not.’’ That is,

suppose a ? b is a rational number. From this you would conclude a and b would both

have to be rational, which contradicts the given condition. (1c, PSMT #34, Fall 2011)

The logical flow of this argument is great! You have provided the correct justifi-

cations for your arguments. The only thing is there needs to be a statement

explaining what k, x, y represent and what kind of numbers they are. (5c, PSMT #34,

Fall 2011)

In this case, the PSMT commented on the mode of argumentation for 1c and appeared to

understand the underlying issue. However, for 5c, not only did the PSMT not address the

argument form, but it is not clear he or she even recognized the error, especially given the

comment that the ‘‘flow of this argument is great.’’ It appears that the different modes of

argument representation (i.e., verbal vs. symbolic) were instrumental in PSMTs’ recog-

nition of errors in 1c and 5c, with fewer errors identified in the argument employing a

symbolic mode. No matter what mode of argument representation was used, few PSMTs

recognized the direct nature of 1c and 5c (i.e., the mode of argumentation).

Arguments 1d and 5d, empirical arguments

Table 6 reports the errors to which PSMTs attended in 1d and 5d. The errors we identified

in these arguments were similar: (a) student provides a proof by example, employing

empirical/inductive reasoning, and (b) student includes a narrative element that has little

clear direction and does not add to the argument’s generality.

The PSMTs were successful at identifying the major empirical reasoning error in 1d and

5d. In 1d, 91 % provided feedback that suggested they understood the limitations of

inductive argumentation. Similarly, in 5d, 85 % identified this error. The following are

examples of typical feedback:

Table 6 Errors and percentage of PSMTs attending to errors in arguments 1d and 5d

Pre-assessment 1d Post-assessment 5d



Student uses empiricalreasoning.

91 % Student uses empiricalreasoning.

85 %

After providing an example,student attempts to generalizeby stating ‘‘they all must be inthe same group of multiples,’’which leaves the readerunclear about meaning, anddoes not successfully add to ageneral argument.

21 % The beginning narrative ofstudent’s argument provides astatement that while correct isnot equivalent to the theoremto be proved. Student does notprogress to any furthergeneralized reasoning andlacks clarity in direction anddefinitions as demonstrated byconfusion about the symmetryof the relation ‘‘factor.’’

38 %


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An example doesn’t give enough reason to prove the statement. You need to prove in

a more generalized notion. Although the example is correct, you need to prove your

examples using a, b, and c. (1d, PSMT #1, Fall 2009)

One example doesn’t necessarily prove anything. You understand what the statement

is asking, but need to explain why it works for all cases. (5d, PSMT #9, Fall 2010)

The PSMTs frequently provided feedback for 1d and 5d that encouraged students to use

‘‘variables’’ or provide a more ‘‘mathematical’’ representation in their argument. Similar to

that found by Pfeiffer (2011), our PSMTs seemed to use the term ‘‘mathematical’’ to refer

to formal, algebraic language. The feedback for these arguments again revealed PSMTs’

preference for symbolic modes of argument representation:

When starting, think of how you might write ‘‘a is a factor of b’’ mathematically. (1d,

PSMT #7, Fall 2010)

This is not successful because you did not actually prove anything, you just gave an

example. When you do proofs, you should not assign numbers to variables, keep

them general. For example, if a is a factor of b, b can be written b = ak; k an integer.

(5d, PSMT #6, Fall 2010)

Overall, our PSMTs demonstrated understanding of the limitations of empirical rea-

soning. However, they did not discount the importance of examples for understanding

mathematical propositions. The feedback from PSMTs, on both 1d and 5d, indicated that

examples are an important stepping stone for students to make sense of the propositions

and deduce a pattern that may aid in the formation of a deductive argument. For instance,

Examples are a great way to help you understand what the question is asking you but

it is not enough to build a proof on. If you go back to the question it states FOR ALL

integers, and you only provided an example. Try to use your example to generalize

the information so that you prove it for all integers. (1d, PSMT #27, Fall 2011)

Arguments 1f and 5f, empirical reasoning AND direct arguments disguised as indirect

Table 7 reports the errors to which PSMTs attended in 1f and 5f. Two of the errors we

identified were similar: (a) student provides a proof by example, employing empirical/

inductive reasoning, and (b) student employs direct, rather than indirect, argumentation.

Thus, these arguments build on what we learned from 1c/5c and 1d/5d. The other errors in

1f/5f were specific, and local in nature, and will not be discussed at length.

As with 1d and 5d, the error PSMTs attended to with greatest frequency in 1f and 5f was

the student authors’ inappropriate use of one example to demonstrate the validity of the

argument. For 1f, 88 % of the PSMTs provided feedback that referred to the limitations of

the student’s inductive argument; for 5f, 68 % provided such feedback:

Where is the generalization for any rational or irrational number? Student fails to

expand on why a ? b is rational. Uses a good example but needs to make gener-

alizations for proving the case for any such a: a is rational and b: b is irrational. (1f,

PSMT #5, Fall 2009)

Examples help to support your proof; however, they are specific examples. Try to use

variables so that your proof can be applied to all numbers and not specific ones (5f,

PSMT #31, Fall 2011)


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The rate of identification of this error for 5f was likely lower than for 1f because the

example used in 5f was an unclear application of the theorem, whereas the example used in

1f was a correct application. Some PSMTs only addressed the unclear nature of the

example in 5f and did not proceed in their feedback to discuss the limitations of empirical

reasoning. In fact, seven of the PSMTs cited the students’ possible confusion between

irrational and imaginary numbers as the sole error (of the four we identified) in their

feedback for 5f. This result aligns with findings from Ko and Knuth (2013) in which

mathematics majors did not proceed with further validation of an argument once they

noticed an initial error that invalidated the argument.

In 1f and 5f, the student authors first negated the conclusion (i.e., changed ‘‘a ? b is

irrational’’ to ‘‘a ? b is rational’’) and then used one specific example that provided a

counterexample to the statement ‘‘a ? b is rational,’’ seemingly in an attempt to serve as

their contradiction. In essence, 1f and 5f are direct arguments demonstrating the validity of

the proposition for one example (and in the case of 5f, an unclear example). Only 12 % of

the PSMTs for 1f and 0 % for 5f suggested that the arguments did not follow an indirect

mode of argumentation. In fact, many PSMTs (eight for 1f, nine for 5f) provided feedback

that specifically mentioned the students’ strength in understanding proof by contradiction:

You started in the right direction- you showed that you understand how to prove by

contradiction, but one example is not a proof! You need to show that this holds for

every rational number a and every irrational number b. (1f, PSMT #33, Fall 2011)

Table 7 Errors and percentage of PSMTs attending to errors in arguments 1f and 5f

Pre-assessment 1f Post-assessment 5f



Student uses empiricalreasoning.

88 % Student uses empiricalreasoning.

68 %

The student attempts to

demonstrate that 2þffiffiffiffiffi

23p

isan irrational number byequating the rational number

6.79583523 with 2þffiffiffiffiffi

23p

3 %

Student possibly confusedirrational with imaginarynumbers.

35 %

Student states that 1 ? i isirrational, but provides noevidence for why it isirrational.

9 %

Student does not work from thenegation of the conclusion(i.e., that a ? b is rational) toderive a contradiction. Thatis, student does not use anindirect proof, but attempts toprove a ? b is irrationaldirectly.

12 % Student does not work from thenegation of the conclusion(i.e., that a ? b is rational) toderive a contradiction. Thatis, student does not use anindirect proof, but attempts toprove that a ? b is irrationaldirectly.

0 %


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The whole form of the proof is correct, but it is specific. Just needs to be more

general. (5f, PSMT #3, Fall 2009)

From 1f/5f, we see repeated the results from our investigation of 1c/5c and 1d/5d. In

particular, the PSMTs successfully recognized the limitations of empirical reasoning, but

neglected to recognize the direct nature of the arguments that purported to be proofs by

contradiction.

Discussion

This exploratory study led to several insights related to PSMTs and their validation of

mathematical arguments. We review the findings for our two research questions and dis-

cuss implications for research and practice.

Validations before and after the instructional activities

Our first research question focused on the ways PSMTs’ validations differed before and

after implementation of the instructional sequence. Our aim was to increase PSMTs’

awareness of (a) the limitations of empirical reasoning and (b) the importance of attending

to both the global structure of an argument and the local line-by-line elements. We planned

the instruction in response to Selden and Selden’s (2003) question, ‘‘What is a good way to

teach validation? Is it a concept like proof… which is perhaps best learned through

experiences, such as finding and discussing errors in actual student ‘proofs’?’’ (p. 30).

Although short in duration, our sequence provided PSMTs with multiple opportunities to

read student arguments, reflect on the strengths and weaknesses of the arguments in small-

group analyses, and discuss as a class ideas related to validation of arguments. Class

discussions paid particular attention to modes of argumentation and modes of argument

representation as well as the hypothetical/perceived set of accepted statements that might

be appropriate in a secondary classroom context.

Overall, PSMTs were quite successful at correctly judging an argument’s validity (both

before and after the instructional sequence). Aligned with findings from prior research

(Knuth 2002a; Ko and Knuth 2013; Weber 2010), all PSMTs rated correctly the two pairs

of matched arguments we judged to be valid mathematical proofs. Moreover, with the

exception of one set of invalid matched arguments (1c/5c), the majority were successful at

identifying invalid arguments. Of the final validations made on the four invalid arguments,

25 % were incorrect judgments.

Weber (2010) found a larger percentage of incorrect validations on the four ‘‘invalid

deductive proofs’’ presented to his undergraduate mathematics majors. To explain this

discrepancy, three of our four invalid matched arguments (1a/5a, 1d/5d, 1f/5f) employed

invalid empirical reasoning. Invalid matched arguments 1c/5c were the only ones that

could be considered ‘‘invalid deductive proofs’’ as defined by Weber. For this pair of

matched arguments, 68 % of final validations were incorrectly rated as valid, aligning with

the findings from Weber.

PSMTs assigned more incorrect ratings for 5c than for 1c. The mode of argument

representation in these two arguments was different and may help explain the poorer

performance post-instruction. In particular, 1c is a verbal argument and employs no

symbolic notation, whereas 5c contains significant symbolic notation. Previous research

has demonstrated that secondary teachers ‘‘may overvalue the generality of symbolic mode


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of representation and under-value the generality of verbal ones’’ (Tabach et al. 2011,

p. 465).

Activity 4 was intended to help PSMTs focus on the global elements/errors of an

argument. One argument we discussed [from Selden and Selden (2003)] dealt with proof

by contradiction. This argument was structured in such a way that it was unclear whether

the intent of the author was to provide an indirect argument in the form of a ‘‘proof by

contradiction’’ or a direct argument of the ‘‘contrapositive.’’ We structured the class dis-

cussion with the assumption that PSMTs had prior knowledge of indirect argumentation

and attempted to increase their awareness of and ability to identify the overall structure and

mode of argumentation in a proof, rather than teach, for example, what is proof by

contradiction. Our discussion seemed to indicate that PSMTs understood the distinction

between proof by contraposition and contradiction within the context of the Selden and

Selden (2003) example. Nevertheless, this understanding of indirect argumentation did not

translate when PSMTs provided feedback on the student arguments. Both before and after

the instructional sequence, PSMTs’ feedback suggested that they hold a superficial

understanding of proof by contradiction. Although we emphasized the importance of

attending to all errors as a way to provide students with the most informative feedback, on

the pre- and post- assessments, PSMTs were simply asked to indicate strengths and

weaknesses of the student arguments and were not asked to indicate all posible error-

s. Also, if we had discussed the errors in the student arguments from the pre-assessment, it

is possible that PSMTs would have focused their responses on the posttest arguments

differently.

The errors to which PSMTs attend

Our second research question focused on investigating the errors to which PSMTs attend

when engaged in validation. We gained several insights. First, we replicated the results of

Weber (2010) and Pfeiffer (2011) by showing that by the time undergraduate mathematics

students (in our case, PSMTs) progress sufficiently in their study of mathematics, most no

longer hold the empirical proof scheme. When inductive reasoning was evident in an

argument, the majority of PSMTs provided feedback to indicate its limitations, frequently

suggesting that students should employ a symbolic mode of argument representation to

promote generality. In arguments 1f/5f, PSMTs seemed to place an overemphasis on

students’ use of inductive reasoning, neglecting to address other errors in their feedback or

believing it unnecessary to do so. In addition, the feedback from PSMTs suggested that

they recognized the importance of inductive reasoning for initial conjecture formation and

understanding of a proposition.

Second, our data suggest that PSMTs had a superficial understanding of proof by

contradiction. Few PSMTs’ feedback indicated awareness of the direct nature of arguments

1c/5c and 1f/5f. Moreover, by asking PSMTs to comment on the strengths and weaknesses

of arguments, we deduced that many believed the student authors of 1c/5c and 1f/5f

provided evidence of understanding the indirect mode of argumentation.

Prior research on indirect proof has described proof by contradiction as one of the most

difficult modes of argumentation to learn (Thompson 1996). In a proof by contradiction,

one typically provides a direct argument of a ‘‘secondary statement’’ in order to derive a

contradiction and prove the truth of the ‘‘principal statement’’ (Antonini and Mariotti

2008). As an example, for matched arguments 1c/5c and 1f/5f, the principal statement is

‘‘If a is a rational number and b is an irrational number, then a ? b is an irrational

number.’’ A successful proof by contradiction of this principal statement relies on a direct


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proof of a secondary statement, such as ‘‘If a ? b is a rational number and a is rational,

then b is rational.’’ However, proving this secondary statement provides little explanatory

insight into why the principal statement is true. One of the primary functions of proof is

explanation, but the mystery inherent in the logical equivalence between principal and

secondary statement can deter from students’ ability to feel convinced by an argument

(Antonini and Mariotti 2008; Leron 1985).

In the arguments related to proof by contradiction, although the student authors misused

direct reasoning, their direct reasoning provides some insight into why the propositions are

true. We hypothesize that PSMTs were satisfied with the overall structure of these argu-

ments because the student authors negated the conclusion of the proposition [satisfying the

first requirement of an indirect proof (Thompson 1996)] and proceeded with direct rea-

soning that provided explanatory power for why the statement must be true. PSMTs did not

have to engage in the cognitive dissonance that typically accompanies a movement from

proof of the secondary statement to logical equivalence with the principal statement.

PSMTs seemed to be satisfied by the ‘‘proof framework’’ (Mejia-Ramos et al. 2012; Selden

and Selden 1995) and could simultaneously feel convinced of the veracity of the

proposition.

Finally, we found PSMTs’ attendance to particular errors and neglect of others was

mediated by the mode of argument representation employed by the student authors. For

instance, in 1c (a verbal argument), 24 % of PSMTs explicitly cited the students’ inap-

propriate use of direct reasoning, whereas in 5c (a symbolic argument) only 6 % identified

the same error. In addition, for 1f and 5f (both numeric/symbolic), few PSMTs recognized

the direct nature of the argumentation. Moreover, the feedback provided by PSMTs often

contained advice to the ‘‘student’’ to use symbolic notation as a means of expressing

generality. This aligns with prior research discussing students’ and teachers’ preference for

the symbolic mode of argumentation (e.g., Healy and Hoyles 2000).

Implications for research and practice

The results from this investigation into PSMTs’ proof validations provide insights into

needed modifications for our instructional sequence and can benefit others involved in the

professional development of mathematics teachers. Two-thirds of the time spent on

activities 2-4 was devoted to helping PSMTs with issues related to empirical reasoning

(e.g., inductive/deductive distinction, importance of empirical reasoning for conjecture

formation, inadequacy of empirical reasoning as mathematical proof). The validations and

feedback provided by PSMTs suggest that they had a good understanding of these topics.

However, of note was PSMTs’ neglect of other errors in arguments once the empirical

reasoning error had been identified. In future instruction, we plan to emphasize the

importance of providing feedback beyond the identification of the first and perhaps most

‘‘obvious’’ error.

Weber (2010) and Pfeiffer (2011) were published after we had created our instructional

sequence in 2009. Based on their work, and what we found, we would not spend as much

instructional time on issues related to the inductive empirical proof scheme in future

implementations. We did not conduct a systematic analysis of the data in this study until

after the third implementation, basing our instruction largely on what we found in the

research and on our intuition about PSMTs’ progression through the activities. In retro-

spect, a closer analysis of the pre- and post-assessments after each implementation could

have facilitated modifications of the sequence to focus more directly on the other diffi-

culties demonstrated by our PSMTs. This was an important takeaway for us; often as


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practitioners, we may overrely on intuition when systematic analysis of student work could

provide valuable ideas for improving instruction.

Our results suggest that PSMTs may need more explicit focus in their teacher prepa-

ration program on the logical inferences within indirect modes of argumentation. PSMTs

would benefit from instruction that helps them move beyond a superficial understanding of

the ‘‘proof framework’’ (Mejia-Ramos et al. 2012) of a contradiction argument. One

potential means of addressing this issue may be drawing PSMTs’ attention to arguments

such as those used in our pre- and post-assessment. Asking PSMTs to validate those

arguments, and then drawing their attention to the logical elements of the arguments that

were not attended to in their feedback, may help heighten their awareness of the logical

inferences within an indirect argument. Because we were collecting this information for

research purposes, we decided not to have such reflective conversations about the PSMTs’

validations, but recognize their pedagogical potential.

This research also suggests that teacher educators should provide PSMTs with oppor-

tunities to engage in validating arguments written in various modes of argument repre-

sentation (e.g., symbolic, verbal, and pictorial). PSMTs seemed to overvalue the symbolic

mode, and therefore, explicit discussions about the validity of arguments in varying for-

mats may help PSMTs become less restrictive in their conception of proof. This is espe-

cially important so teachers do not limit students’ opportunities to engage with proof due to

a belief that ‘‘proof’’ must be structured in a formal symbolic mode of argumentation (see

Knuth 2002b).

Most students have limited interactions with proof in the K-12 setting. Because this is

the reality, future research should investigate teachers’ proof validation related to standard

high school content. We used samples of student arguments from a high school course in

precalculus and discrete mathematics. Although our PSMTs should have been familiar

with the content, we acknowledge that such content is not widespread in the K-12 cur-

riculum. Replication of this study using arguments from more common content areas, such

as algebra or geometry, could provide insight into validation practices in a context more

closely linked to teachers’ practice.

Ko and Knuth (2013) found undergraduate mathematics majors’ validation strategies

and success at validation varied based on the mathematical content area of the arguments

and perhaps based on their familiarity with the content. PSMTs’ content knowledge related

to rational and irrational numbers seemed to be influential in their validations. Several of

our PSMTs appeared to have insufficient knowledge of the content of rational and irra-

tional numbers. Although they could identify numbers as rational or irrational, they were

unable to articulate appropriate definitions. Their preference for the symbolic mode of

argument representation may have hindered their ability to provide a clear definition of

irrational number (as irrational is typically defined as ‘‘not rational’’ instead of in an

explicit symbolic format).

An unintended benefit of such instructional activities is that our PSMTs began to

comment on different means to engage their own students in proof activities. In particular,

they highlighted the following strategies: (1) have students provide justifications; (2) have

students critique proofs; (3) have students explain what they are doing regularly; and (4)

ensure that assessment with proof concepts matches instruction related to proof concepts.

In this study, we focused primarily on the errors to which PSMTs attended in their

feedback. We did not discuss the quality of their feedback in terms of instructional value.

In our analysis, we noticed variability in the quality of feedback and believed that this

should be an additional focus of future instructional sequences. If teachers cannot com-

municate clearly the strengths and weaknesses of a student’s argument, then it matters little


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whether they can identify the error. In the future, we envision having PSMTs read and

reflect on samples of teacher feedback; they would not only determine if the teacher

feedback correctly identifies the errors in the student argument, but also judge the quality

of the feedback in terms of potential facilitation of student learning.

Our work here suggests one means of addressing validation to scale. Most previous

research on validation has used interviews as the research methodology (e.g., Selden and

Selden 2003; Weber 2010), which can limit the size of the sample and is time intensive.

Having PSMTs make validation judgments and write about strengths and weaknesses

enables instructors to assess validation skills for a larger sample and a larger number of

arguments at one time. With the suggested revisions, such as peer discussions of the

comments to students, as well as asking PSMTs to list all strengths and weaknesses, even

more information might be obtainable.

Although our instructional sequence was conducted in a content-specific pedagogy

course, we believe such sequences would also be appropriate in content courses, partic-

ularly in those taken primarily by prospective teachers. As demonstrated here, the ability of

PSMTs to identify errors and articulate them provides insight into content as well as

pedagogical knowledge.

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