High school mathematics teachers’ perspectives on the purposes of mathematical proof in school mathematics

ORIGINAL ARTICLE

High school mathematics teachers’ perspectiveson the purposes of mathematical proofin school mathematics

David S. Dickerson & Helen M. Doerr

Received: 3 May 2012 /Revised: 23 May 2013 /Accepted: 9 December 2013# Mathematics Education Research Group of Australasia, Inc. 2014

Abstract Proof serves many purposes in mathematics. In this qualitative study of 17high school mathematics teachers, we found that these teachers perceived that two ofthe most important purposes for proof in school mathematics were (a) to enhancestudents’ mathematical understanding and (b) to develop generalized thinking skillsthat were transferable to other fields of endeavor. We found teachers were divided onthe characteristics (or features) of proofs that would serve these purposes. Teachers withless experience tended to believe that proofs in the high school should adhere to strictstandards of language and reasoning while teachers with more experience tended tobelieve that proofs based on concrete or visual features were well suited for high schoolmathematics. This study has implications for teacher preparation because it appears thatthere is a wide variation in how teachers think about proof. It seems likely that studentswould experience proof very differently merely because they were seated in differentclassrooms.

Keywords Mathematics education .Mathematical proof . Advancedmathematicalthinking . Secondary mathematics

Introduction

Over the past three decades, there has been a growing body of literature concerned withproof and justification in school mathematics. Much of the research into mathematicalproof identifies difficulties and misconceptions students have regarding mathematicalproof and reasoning (e.g., Williams 1979; Fischbein 1982; Ernest 1984; Schoenfeld1985; Dubinsky 1986; Balacheff 1991; Chazan 1993; Selden and Selden 1995; Harel

Math Ed Res JDOI 10.1007/s13394-013-0091-6

D. S. Dickerson (*)State University of New York College at Cortland, Cortland, NY, USAe-mail: [email protected]

H. M. DoerrSyracuse University, Syracuse, NY, USA

and Sowder 1998; Stylianides et al. 2004). Since the late 1980s, some researchers havefocused on the purposes that proof has in school mathematics (e.g., Hanna 1989; deVilliers 1990; Lowenthal and Eisenberg 1992; Knuth 2002a, b). In this study, weexamined high school mathematics teachers’ perspectives on the pedagogical purposesof proof.

Related literature

Some researchers have investigated teachers’ understandings of proof and have foundthat some teachers exhibit poor or incomplete understandings of proof. Other re-searchers have focused on how teachers positioned proof in relation to their students.Knuth (2002b) reported on 15 high school mathematics teachers’ perceptions of proofin the context of school mathematics. He found that teachers had three broad schemesthat covered most student-generated proofs: (1) formal proofs, (2) less formal proofs,and (3) informal proofs. Formal proofs conformed to rigid standards of form (usuallytwo-column) and language. Less formal proofs were essentially correct general argu-ments that were perceived to be lacking in form or vocabulary. Informal proofs werenot really proofs at all but arguments based on the testing of cases. These distinctionsare important because 14 of the 15 teachers did not consider formal proof to be anappropriate subject of study for all high school mathematics students. Although thelanguage included in the National Council of Teachers of Mathematics standards(NCTM 1989, 2000) states that proof and reasoning should be taught at all gradelevels, some of the teachers in Knuth’s study believed that NCTM could not havemeant that all students should be taught either formal or less formal proofs. Similarly,Varghese (2009) reported that the participants in his study (17 preservice teachers)believed that proof was a waste of time for most students in the high school mathe-matics classroom.

Several researchers have written about the purposes of proof. Combining the worksof Davis and Hersh (1981), Hanna (2000), and Rodd (2000) results in a lengthy list ofpurposes including verification, explanation, communication, entertainment, ritual,conviction, exploration, discovery, systematization, and celebration of mind. All ofthese purposes have the discipline of mathematics as a primary context, yet proof hasother purposes that relate primarily to the teaching of mathematics of which theenhancement of student understanding is perhaps the most important (Hanna 1989).Knuth (2002b) reports that the teachers in his study indicated that in the context ofschool mathematics, enhancing students’ understanding is an important purpose forproof; specifically, they said proofs could be used to develop logical thinking skills thatare applicable outside of mathematics classrooms and to free students from a relianceon external authorities such as the textbook or the teacher. These purposes have littlemeaning in the context of the discipline of mathematics but were perceived by some ofthe teachers in Knuth’s study to be important in the context of school mathematics.

Dickerson and Doerr (2008) report that some teachers perceive that certain validproofs are not acceptable in high school mathematics because they deviate from anintended line of reasoning and thus fail to require a student to demonstrate facility witha prescribed bit of content or reasoning. Herbst (2002) argues that teachers may findthemselves in a double bind with respect to the purposes of proof; on the one hand,

D. S. Dickerson, H. M. Doerr

teachers want their students to find a justification for a claim, but on the other, theyneed students to produce the proof, that is, producing any valid justification for aparticular claim and demonstrating the ability to write a detailed proof for that claimalong a prescribed line of reasoning. These competing demands may influence whatproof tasks are chosen and to what purposes proofs are actually used in classrooms.While mathematics education researchers may agree that enhancing students’ under-standing is an important purpose of proof in school mathematics, we do not know agreat deal about what purposes of proof teachers perceive to be important in schoolmathematics. These perceptions along with navigating the double bind would neces-sarily affect how proof is treated in mathematics classrooms.

The correctness of argumentation in mathematical proof is of utmost important forsome purposes but less so in others. Convincing students of the truth of mathematicalstatements for example is easy and may require no more than a handful of examples. Aformal and correct proof does not necessarily lead to a personal conviction as evenprofessional mathematicians are sometimes unconvinced by their own proofs (Segal2000). Instead, mathematicians often rely on factors other than rigorous proof whenconvincing themselves of the truth of a mathematical statement. Currently, there arehundreds of thousands of new proofs published worldwide each year of which only ahandful of them ever get closely scrutinized (Hanna 1991). Proofs today are oftenextremely long (running hundreds of pages), and it may take months for a mathema-tician to validate a new proof. In the light of these new realities, there are several criteriaemployed by mathematicians to determine when a mathematical argument becomes aproof of a theorem. Hanna has compiled a list of criteria that mathematicians employ incombination to determine whether an argument has been proved.

& They understand the theorem, the concepts embodied in it, its logical antecedents,and its implications. There is nothing to suggest it is not true;

& The theorem is significant enough to have implications in one or more branches ofmathematics (and is thus important and useful enough to warrant detailed study andanalysis);

& The theorem is consistent with the body of accepted mathematical results;& The author has an unimpeachable reputation as an expert in the subject matter of the

theorem;& There is a convincing mathematical argument for it (rigorous or otherwise), of a

type they have encountered before.

If there is a rank order of criteria for admissibility, then these five criteria all rankhigher than rigorous proof (Hanna 1991, p. 58). So, while nobody disputes the need forcorrect and careful reasoning, both students and mathematicians sometimes rely onfactors other than reading a proof to convince them of the truth of mathematicalstatements.

Furthermore, proofs that deviate from a reliance on precise language, explicit logic,or even careful and correct reasoning can provide insight into mathematical ideas thatperhaps a formal, rigorous proof might not. Hanna and Sidoli (2007) summarize thewide range of philosophical perspectives on visual argumentation in mathematics.Some mathematicians believe that “visual representations can never be more thanuseful adjuncts to proof” (p. 74), but others believe that a diagram can be an entireproof requiring no further explanation. Between these two extremes, there is a wide

High school mathematics teachers’ perspectives on the purposes

variety of positions, and the mathematical community is “still far from a consensus onthe potential roles of visualization in mathematics and mathematics education, and inparticular on its role in proof” (p. 77). Davis (1993) argued that visual argumentation(as opposed to symbolic argumentation) is reasonable and that heuristic reasoningbased on diagrams can complement rigorous argumentation and vice versa with thegoal of achieving a deeper understanding of the underlying mathematics. Dickerson(2006) found that some preservice teachers in advanced undergraduate mathematicscourses preferred visual reasoning to rigorous argumentation because diagrams oftenilluminate the underlying mathematics that rigorous argumentation sometimes ob-scures. Similarly, Weber (2010) reported that advanced undergraduate mathematicsmajors were content with certain valid proofs based on diagrammatic arguments, evenindicating that they were rigorous.

The purpose of our study was to understand how teachers thought about the purposesof proof in the context of school mathematics. While we know from Knuth and othersthat teachers indicate that the enhancement of student understanding is an importantpurpose for proof, we wanted to probe more deeply into what teachers meant by this.Furthermore, we were interested in what deviations from the traditional two-columnproofs teachers believed were useful to serve these purposes. Our research questions forthis study are: (1)What do high school mathematics teachers perceive to be the purposeof proof in school mathematics? and (2) What are the characteristics (or features) ofproofs that best serve high school mathematics teachers’ perceived purposes?

Methodology

Participants

The 17 participants were recruited from ten different high schools in northeastern UnitedStates on the basis of their willingness to participate in the study. We recruited experi-enced teachers because we were interested in teachers’ perceptions of the pedagogicalaspects of mathematical proof, and it seemed likely that experienced teachersmight havebetter developed ideas regarding many pedagogical issues surrounding mathematicalproof. The least experienced teachers were nearing the end of their fifth year of teaching,and the most experienced teachers had been teaching for more than 35 years. Theparticipants included ten women and seven men: 14 from public schools, three fromprivate schools, with nine teachers with between 4 and 10 years of experience and eightteachers with 20 or more years of experience. Collectively, the participants had taught atall levels of high school from remedial mathematics to AP Calculus. Those withbetween 4 and 10 years of experience, we call the experienced teachers, and those withmore than 20 years of experience, we call the veteran teachers.

Data collection

The data for this study come from three, hour-long interviews with each of theparticipants. Each set of interviews was conducted over about a 3-week period. Thefirst interview was semi-structured and focused on participants’ personal and mathe-matical histories, and their pedagogical conceptions of mathematical proof. During this


interview, participants discussed what a proof was and what kinds of student-generatedarguments met or failed to meet that definition, the certainty of the knowledge gainedthrough a proof and whether two valid proofs could contradict each other, the value ofteaching it at the high school level, and the role of visual representations in proofs.

The second interview was task-based (Goldin 2000) in which participants wereasked to evaluate a series of 15 mathematical arguments. There were several factors thatinformed our selection of the proof tasks. Each of the tasks was intended to elicit adiscussion regarding the nature and purposes of proof. The tasks were supposed to beunderstandable to high school mathematics teachers, to span a wide range of rigor, andto represent a variety of written formats and logical structures. The tasks includedarguments written in natural language, bullets, two-column format, and argumentsbased on visual reasoning. The logical structures of the tasks included direct proofs,indirect proofs, inductive proofs, and algebraic derivations of formulae. Participantswere asked to discuss whether each argument conformed to what they perceived aproof should be and in what contexts the argument was sufficient. The Appendixcontains seven of the proof tasks used in this study.

Some of the proof tasks were selected because they would likely be familiar to teachers(see Interior Angles I, and Quadratic Formula in Appendix), and some were selectedbecause they would likely be unfamiliar (see Interior Angles II andCompleting the Squarein Appendix). The validity of some of the proof tasks was dubious. Proof tasks that wereclearly invalid were labeled empirical, and participants were not asked to pass judgmenton the validity of the proof tasks. Rather, they were asked if each of the proof tasks wereclear, convincing, and in what circumstances they were acceptable arguments. All inter-views were audio-recorded and transcribed. The participants’ responses to the 15 tasks andtheir discussions during the semi-structured interviews were used to illuminate theirperceptions regarding the purposes of proof in school mathematics.

The third interview was semi-structured and focused on the participants’ perceptionsof the mathematical purposes of proof such as verification, explanation, and commu-nication. Participants discussed rigor and its relationship to these purposes, the level ofcertainty guaranteed by a proof, features of proofs that made for better explanations andclearer communication, and what it means to be convinced by a mathematical argu-ment. While the topics of the interviews were different from each other, during thesecond and especially the third interviews, participants frequently drew from previousquestions and conversations to illustrate their thinking. Because participants’ responseswere at times colored by two or more hours of previous conversations, these interviewsread more like one long 3-hour interview than like three separate interviews overdistinct, non-overlapping topics. Taken together, the interviews enabled us to under-stand what the participants believed were the most significant purposes of proof andwhat characteristics (or features) of proofs best served these purposes.

Data analysis

The data were coded using both internal and external coding schemes. In the internalcoding scheme, the data were coded inductively (that is without an a priori set of codes); thecodes were emergent and grounded in the data. In the external coding scheme, the codeswere researcher-generated and based on our theoretical framework regarding the roles ofproof in school mathematics. The internal coding was done first, and codes (e.g., proofs


develop metacognitive thinking skills, proofs demonstrate to students that math is a humanendeavor) emerged and were refined. The external coding was done second using codesfrom our a priori coding scheme (e.g., the role of precise language in a proof in proofwriting, the role of diagrams in a proof) but now informed and refined somewhat by theemergent codes from the first phase of coding. Excerpts from the interviewswere organizedby code and by participant to see how prevalent all the codes were among each participant’sinterviews and how prevalent each code was among all the participants’ interviews. At theend of this process, the codes themselves were analyzed, and similar or related codes werethen grouped together into larger, superordinate codes we created to capture the largermeaning of the group of codes.

Table 1 illustrates four somewhat related codes that were grouped together under thesuperordinate code of Enhancement of Student Understanding. The idea of using proof forenhancing students’ understanding was one of the most pervasive ideas discussed byteachers as a reason for continuing to teach proof in school even though students oftenstruggle with it. We found more than 130 instances where the participants discussed usingproof as a way to enhance students’ understanding in ways such as helping students todevelop mathematical thinking skills, or helping students to develop other kinds ofthinking skills that are of primary usefulness outside of mathematics classrooms.

Results

There are two main findings of this study. First, high school mathematics teachersbroadly identified enhancement of student understanding as a primary purpose of proof.Their discussions regarding student understanding tended to fit into two categories: thedevelopment of mathematical thinking skills and the development of generalized,

Table 1 Codes and data excerpts

Code Excerpt

Proofs develop transferablethinking skills

OK. So, in mathematics I see a solid reason to teach proof. [I’m] …hoping togive them a strategy that could then be transferable to other areas …Should everybody be able to …follow a proof [line by line]? I don’t knowbut the reasoning skills we use to get there are valuable for just aboutanything. [Brenda Int. #2 lines 710–726]

Proofs reify mathematicalknowledge

I think by teaching proof, hopefully it solidifies some of the concepts abouttriangles, quadrilaterals, and circles …When they are constantly forced touse the same properties over and over …in proofs, I think it might help tosolidify that knowledge. [Cory Int. #1 lines 444–456]

Proof develops mathematicalunderstanding

I think [proof] is important…It helps train their thought processes …It helpsthem think logically …[By] actually seeing the proof, and following theproof, they get a better understanding of what the problem is about, andwhere it came from, and how to use it [Melanie Int. #1 lines 340–346].

Proof develops metacognitivethinking skills

If you ask, they may not understand their thinking. So you begin to ask themdifferent questions and you start working backwards: “Well, what madeyou think that it was five to begin with?” Or “How do you know that it’snot seven?” …You ask them some of these other types of questions and as[do], their thinking process usually becomes more clear to them [BartInt. #1 lines 28–32]


transferable thinking skills. Second, there was some disagreement among the partici-pants as to the importance of certain features of proofs. For use in classrooms, some ofthe participants preferred detailed proofs that adhere to strict standards of preciselanguage and careful, explicit logic, while others welcomed proofs that rely on commonsense, shared assumptions, or diagrams. These preferences were fairly evenly dividedwith experienced teachers were more likely to be in the first group and the veteranteachers were more likely to be in the second group.

Enhancement of student understanding

Sixteen of the 17 participants indicated that the enhancement of student understandingwas one of the most significant purposes for proof in school mathematics. The partici-pants indicated that proof could help students develop two kinds of thinking skills.First, proof was perceived to provide deeper insight into mathematics content and tohelp students to reason mathematically. These skills and insights are used primarilywithin the context of the discipline of mathematics and are the hallmark of someonewho is good at mathematics. Second, proof was perceived to develop a set ofgeneralized thinking skills that had little to do with mathematics. These includedlogical, critical, and metacognitive thinking skills most useful in contexts outside ofmathematics. The participants believed these thinking skills could be developed inmathematics classrooms to be used in contexts outside of mathematics. We will discusseach of these purposes in turn.

Development of mathematical understanding

Using proof to develop students’mathematical thinking skills was mentioned by 14 of the17 participants. These participants believed that proofs could be used to accomplish this inone of two ways. Proofs could provide deeper insight into the mathematicalmaterial at hand, and second proofs could help students to develop mathemat-ical reasoning skills.

Proofs provide deeper insight into mathematical content Proofs can be used toexplain mathematics. By revealing an underlying system of reasoning, a proofcan provide a deeper insight into the mathematics under consideration. Severalof the teachers in this study talked about using proofs to show students “whereit all comes from.” Kelly, for example said, “My goal in showing them theproof or doing the proof with them is [because] they’re just using [the fact that]without any idea of what it means. So, if I were to go through and show themthe proof …they can develop an understanding of where the stuff comes from”[Kelly Int. #1 lines 468–473]. Kelly believed that it is not necessary to have adeep understanding of formulas in order to use them effectively but that there issome value in showing students where formulas come from. Similarly, Lestersaid

I would go through this to show themwhere [we got] the quadratic formula from but[also] where we could then take that and get the sum of the roots and the product ofthe roots from the quadratic formula…I want to enhance their knowledge because…


a lot of times what we do is we just use formulas for computation and we don’tunderstand…where it comes from. [Lester Int. #2 lines 1281–1292]

Like Kelly, Lester wanted to enhance his students’ mathematical knowledge andbelieved that an understanding of where formulas came from had value to high schoolmathematics students.

In evaluating the proof task Interior Angles I (see Appendix), Melanie said its valuelay in showing students “why a triangle absolutely, positively has to add up to 180°.They can…see that [it] is 180°” [Melanie Int. #2 lines 1218–1228]. She discussed areaformulas similarly: “If you develop the area formula for a triangle from a parallelo-gram…then the kids can understand where it came from and it makes it clear as to whythey’re…doing the base times height, why they have to divide it by two” [Melanie Int.#2 lines 1378–1382].Whether she was proving something new to students or supportingthe mathematical content knowledge her students already had, she believed that showingthe proof helped students to get a deeper understanding into the content. “When you’retrying to get them to understand the material, they really need to know the logic behind it…Then the logic behind it is what leads to the proof and then by doing the proof andputting the logic all together you’re getting back to understanding what the material istrying to say” [Melanie Int. #1 lines 234–256]. Melanie believed that understanding theunderlying logic is important in making sense of mathematical content. By revealingthat underlying system, proofs are a way to provide students deeper insight intomathematical content such as rules and formulas. Later in the same interview, she said:

I think by actually seeing the proof and following the proof they get a betterunderstanding of…where it came from…I mean…if they forget sine squared pluscosine squared equals one they can always go back and draw the picture and getto it. [It’s the] same thing with some of the area formulas that they learn ingeometry and the lower grades. [Melanie Int. #1 lines 344–352]

In Melanie’s view, proofs could be used to provide students with a deepunderstanding of mathematical content enabling them to derive formulas forthemselves.

Monty discussed why proof should be part of the high school curriculum:“The way we have been teaching it, these kids are already coming in withknowledge …My job is to say, ‘Yes, we know that but here is why’” [MontyInt. #1 lines 18–21]. Middle school students, Monty claimed, come to highschool with a large array of rules and formulas that they have learned inprevious years and proofs helped deepen their understanding of them.

Some of the teachers in this study believed that using proof to reveal the mathe-matics underlying rules and formulas was valuable to students. Whether it was indeveloping new formulas from old, or in reifying students’ prior mathematical knowl-edge, they believed that teachers could use proof as a way to enhance their students’understanding of mathematics by providing deeper insight into mathematical contentsuch as rules and formulas.

Proofs help students develop mathematical reasoning Using proof as a way ofdemonstrating or of developing reasoning skills was discussed by each of theparticipants. Tracy felt that proof was important for helping students to


develop mathematical reasoning skills. Tracy said, “I honestly think that [learn-ing to develop a line of reasoning] is the most important part of proofs. It’s notthe ‘Can you do a geometry proof?’ [It’s] ‘Do you understand the rules ofgeometry?’” [Tracy Int. #1 lines 319–321]. Tracy regarded many of the proof taskswe presented to her as valid yet unacceptable from students nonetheless. To her, proofsare an opportunity for students to demonstrate an understanding of an intended line ofreasoning. For example, she said that she would not take Completing the Square from astudent because it was a “use-it-once” strategy and a “shortcut” that bypassed the intendedtask, which in this case was (in her opinion) to prove it by algebraic manipulation. Validproofs that deviate from prescribed lines of reasoning might not require the student todemonstrate the very reasoning skills she was trying to teach, leaving students unable tocomplete similar tasks in the future. In her view, a proof task was not a merely a claim thatneeded to be justified, it was an opportunity for a student to demonstrate a facility with anintended set of reasoning skills.

Melanie discussed using proof as a way to teach mathematical reasoning saying thatproof “helps train their thought processes …It helps them think logically. You can’t justshake the bag and everything falls out nice and neat, you know, you gotta know how to putpieces together” [Melanie Int. #1 lines 340–344]. Melanie believed that studying proofdeveloped both students’ understanding ofmathematical content and students’mathematicalreasoning skills. In other words, proof helps students to develop a deep understanding ofeach of “the pieces” and an understanding of how these pieces might fit together to create awhole.

Caleb discussed using proof as a way for students and teachers alike to demonstratemathematical reasoning. “We’re not necessarily trying to prove something as much asteach them how to represent their thinking” [Caleb Int. #3 lines 1911–1912]. Caleb saidthat the reasoning process is often why a proof is demonstrated or assigned in schooland that teachers use proof to as a way to teach students how to demonstrate theirthinking.

Bart believed that proofs were useful in helping students grasp the overall structure ofmathematics saying that math teachers want their students to “realize that you’re creatingnew truths from old truths by a…logical method and that the new truths are as valid as theold truths fromwhich they came” [Bart Int. #3 lines 2070–2072]. Bart wanted to use proof insuch a way that his students would appreciate the logical method and to have a deeperunderstanding of the structure of mathematics.

All of the participants in this study indicated that proof was connected with developingstudents’ mathematical reasoning skills. At the high school level, many of the claims inquestion have been verified facts for centuries and so verifying them again and again isredundant and may seem hollow, but using proof as a means to enhance students’understanding of mathematics (either by providing deeper insight into mathematical contentor by developing students mathematical reasoning skills) is a purpose that renews itself witheach successive generation of students.

Development of generalized thinking skills

One purpose for proof mentioned by 14 participants is that the learning of proofdevelops thinking skills that are transferable to other fields of intellectual endeavor.


Proof helps students develop students’ minds so that they can be used for otherpurposes rather than merely helping students to develop their understanding of math-ematics. In this sense, proof is taught in service of developing logical, critical, andmetacognitive thinking skills.

Proofs help students develop logical thinking skills Several teachers indicated that thebenefits associated with studying proof included that the study of mathematical proofmade students more able to learn other, unrelated things at a later date. Vance said, “A lotof the purpose behind the proof is that we’re trying to teach them how to reason, and youknow how to think their way through things” [Vance Int. #1 lines 127–132]. Caleb said ofteaching proof, “I think it’s just a maturity and a part of developing an analytic, maturemind for way beyond just a math classroom, for any situations there are in life” [CalebInt. #1 lines 275–278]. Similarly, Brenda said, “We’re doing everything in abstract so itcan transfer over to any realm of endeavor in the world” [Brenda Int. #1 lines 278–280].These teachers’ assertions suggest that proofs help students develop reasoning skillsthat will be useful in analyzing problems outside the mathematics classroom. Theabstract nature of this reasoning is seen to help to develop a set of generalizedthinking skills that is transferable to other situations or endeavors.

Several teachers claimed that much of school mathematics is not directly relevant tostudents’ lives or future careers but that learning the systematic logic of proof would helpwith the development of students’minds. For example, when asked why we teach proofin the high school, Lester answered, “‘Why dowe teach half the stuff that we teach kids inhigh school mathematics?’ is the fundamental question…It teaches them an approach toanalyzing the problem, that they don’t get any other way” [Lester Int. #1 lines 385–389].He went on to say, “I think that kind [proof based] of logic is very important for braindevelopment” [Lester Int. #1 lines 410–416]. When Tracy discussed the average studentin her classes, she said “Generally speaking you don’t give them an understanding ofgeometry but I think all kids can benefit from doing geometry proofs because it developsthinking skills …I mean some kids are never going to use the thoughts of geometryproofs but if they develop the ability to think then proof itself was helpful” [Tracy Int. #1lines 52–58]. She later said of proofs in school, “I say this in all my math courses, ‘If younever use it again, at least you developed your ability to think’” [Tracy Int. #1 lines 420–433]. Like Lester, Tracy believed that much of the mathematics curriculum had limitedvalue outside of school but that proof tasks helped develop the students’ minds: “I don’tthink the point of having the kids do proof is having kids do proof. I think it’s developinga line of thinking that will help you later in life because again I think very few of our kidsare going to continue on in [mathematics] and do proofs” [Tracy Int. #2 lines 1021–1025]. Proof, as an advanced topic in mathematics, is seen to be of limited direct value tothe lives of students outside of school but that the development of the mind required forsuccess in the mathematics classroom would be a lifelong benefit for the student.

At least four participants gave specific contexts unrelated to mathematics to whichthese thinking skills would be useful. Gina said, “Give me a lawyer who hasn’t studiedlogic and I’ll give you a bad lawyer” [Gina Int. #1 lines 719–720]. Caleb said,

The world’s filled with sheep and shepherds…There’s the kid that stands [behindthe register] at [the local grocery store] and when something goes wrong he hits


the light because he can’t figure it out. The cash register isn’t doing what it’ssupposed to do. And there’s the person that comes over and analyzes the situationand in a minute says “Bing-bing-bing-bing-bing. Here, it works again.” Whichone are you gonna be? [Caleb Int. #1 lines 295–301]

Caleb gave another situation of an auto mechanic diagnosing engine trouble. Lesterbelieved that learning to use your head was the biggest benefit to teaching proof inschool.

I think what’s most important is for kids to be able to think…I mean I even wantthe kid who mows my lawn to be able to know how low he should cut it …If heknows it’s going to be dry for the next 4 days [and] it’s going to be hot thatperhaps he shouldn’t cut it as low, exposing the roots to that much more heat.[Lester Int. #1 lines 445–540]

These vignettes illustrate the participants’ belief that proof-like logical thinkingskills are transferable to situations not directly related to mathematics.

Proofs help students develop critical thinking skills Rather than logical thinking skills,two teachers stressed the importance of critical thinking skills. For example, Montyemphasized the need for students to investigate facts for themselves and to take acritical stance: “I want [them] to know that, don’t just assume that everything you hear[is true] …Sometimes you have to do a little investigation yourself’” [Monty Int. #1lines 265–270]. He added, “I mostly want the students to leave my classroom and beable to think critically outside of the classroom on other things. I want them to …make up their own beliefs …through their thought process and what theygained in my classroom” [Monty Int. #1 lines 363–368]. Similar to Monty’sstress of critical thinking skills, Kelly stressed decision-making skills. She said,

I think we teach kids to prove things to be able to develop that understanding ofevidence, being able to decide to make conclusions on any kind of idea or topicand to be able to take information and reason with it and justify with it, categorizeit and to be able to draw conclusions from the information whether it’s mathe-matical or not. [Kelly Int. #1 lines 379–385]

Unlike the participants who wanted students to gain from exposure to proofslogical problem-solving skills and to be able to apply them in contexts outsideof class, both Monty and Kelly voiced the belief that the benefit of teachingproof is to help students to develop the critical decision-making skills in reallife situations.

Proofs help students develop metacognitive thinking skills One participant had quitea different view of the value of studying mathematical proof. Unlike the otherswho emphasized the development of the mature mind, or the acquisition oflogical or critical thinking skills, Bart believed that studying proof developsstudents’ ability to analyze their own thinking and the thinking of others and asa result to understand the nature of knowledge. He said that students may notunderstand their own thinking, and this kind of argumentation helps themunderstand how one comes to know or to believe.


If you ask, they may not understand their thinking. So you begin to ask themdifferent questions and you start working backwards: “Well, what made you thinkthat it was five to begin with?” Or “How do you know that it’s not seven?”…Youask them some of these other types of questions and as [do], their thinkingprocess usually becomes more clear to them [Bart Int. #1 lines 28–32]

Bart believed that teenagers need to acquire the skills to analyze their own baselineassumptions and beliefs. He gave a non-mathematical context in which students aregiven the opportunity to analyze their assumptions in a characteristically mathematicalway.

[The purpose of proof] is also to teach the students an awareness of the fact thatconclusions that you reach in any area really depend on beginning assumptionsthat you make. And that we need to learn to analyze what are the assumptions thatwe make going into a situation so that [you can know] how true are thoseassumptions to help us understand how accurate [are] the conclusions we’rereaching. [For example] a kid comes in and says, “That guy’s a jerk,” and yousay, “Well, how do you know that?”…Then you say, “Well, what assumptions areyou making? What does it mean to be a jerk?…What observations about theperson that match well? How many of the parts of being a jerk does a person haveto satisfy before he can be classified as one?” [Bart Int. #1 lines 225–241]

Bart believed that understanding one’s own assumptions is key to understandingone’s own thinking. Bart believed that once students understand their own thinking,they can better understand the thinking of others.

I want them to understand the process so that if someone stands up and says “Ihave come to this conclusion,” or “I’ve proven this theorem,” that they under-stand and in some general way the process of thinking that that person wentthrough to arrive at their conclusion. [Bart Int. #1 lines 488–492]

Bart mentioned this idea of confronting students with their own thinking severaltimes over the course of the interviews. The question of how one comes to knowledgein any field was very important to him and could be analyzed in a mathematicalfashion. Helping students to develop metacognitive skills was the main reason he gavefor teaching proof.

We found that some of the participants viewed the teaching of proof to be valuableto students in developing generalized thinking skills, and that these skills are transfer-able to other, everyday areas of life outside of school mathematics. The participantsdiscussed using proof as a means to teach skills such as logical thinking, criticalthinking, and analysis of one’s own thinking and the thinking of others. In thisrespect, the reason for teaching proof can be seen as a service to activities notdirectly related to doing mathematics or understanding particular mathematicalcontent. Rather, the teachers’ comments indicate that they felt that these skillsare the benefits of learning proof that last long after the particular mathematicalknowledge is either forgotten or deemed irrelevant. Using proof to teach thesethinking skills was an important reason for teaching proof to high schoolstudents. The enhancement of student understanding in one form or another


was a major topic among 16 of the 17 participants of this study with 14discussing mathematical thinking skills and 14 discussing generalized thinkingskills. The participants in this study believed that teaching proof is a way todeepen students’ knowledge and understanding of mathematics and/or to helpthem to develop the logical and critical thinking skills that could aid them laterin life.

Important features of mathematical proofs

There were a wide variety of features of proofs that the participants thought wereimportant. These features included details, step-wise justifications, precise vocabulary,diagrams, and appeals to common sense. Some teachers thought that certain of thesewere more important than others. While all participants agreed that some level of rigorwas important, some felt that emphasis on details, vocabulary, and justification werenecessary at the high school level while others believed that these sometimes impededstudents’ understanding of the mathematics.

There were differences in how the experienced and the veteran teachers discussedproofs in school mathematics. We found these differences were most pronounced whendiscussing the value and necessity of (1) explicit reasoning, (2) precise language andcareful logic, (3) sufficient and careful details, and the use of (4) visual and concretefeatures. In each case, the experienced teachers tended to describe proof as a formalway of doing and writing mathematics and to place a high value on each of thesefeatures while the veteran teachers were more likely to place a high value on proofs thatexplained the underlying mathematics even at the expense of these three features.

Explicit reasoning

Step-by-step justifications were considered to be important or necessary bysome of the teacher in the study. While none doubted the validity of solvingequations or deriving formulas by algebraic manipulation, some teachers be-lieved these were not to be considered proofs unless they were accompanied byexplicit step-by-step reasoning. This view was particularly prevalent among theexperienced teachers in the study.

Seven out of nine (78 %) of the experienced teachers indicated that a proof requiredexplicit, step-by-step justifications as opposed to only two (25 %) of the veteranteachers. Tracy (experienced) said that in order to prove that a quadrilateral is arhombus it wasn’t enough to show that all four sides are equal. “I would think that aproof has to have some statement to say what you did and why you did it …You haveto show it and [then] say ‘a rhombus is something that has all four sides the same, andI’ve proved it [by] getting all four sides the same’” [Tracy Int. #1 lines 139–159]. Cory(experienced) said that solving an equation as simple as x – 2=0 might be considered aproof if the sole step was explicitly justified. “To call it a proof, I think I would want tosee more than just x – 2=0 and x=0. Maybe even a statement like ‘Because when youadd 2 to both sides’ …Something like that” [Cory Int. #1 lines 150–154]. Kelly(experienced) too, in describing the process of solving a linear equation indicated thatshe wanted each step justified if it were to be considered a proof. “I would say, ‘How


can you add 1 to both sides?’ That would be the proof…if I’m justifying why I am ableto do that” [Kelly Int. #1 lines 248–252]. In order for a mathematical argument to beconsidered a proof, these teachers felt that each step had to be explicitly justified. Coryand Kelly both believed that an algebraic manipulation could be considered a proof butonly if each step along the way was accompanied with a justification—even if thatjustification was merely a statement that one is allowed to add the same number to bothsides. Similarly, Burgess (experienced) did not believe that the Quadratic Formula task(see Appendix) was a real proof. “I have to admit, I thought about it a lot after we talkedlast time. Why do I think that that’s not a proof? I don’t know that I’ve come up withanything beyond the fact that you know we don’t have these justifications” [BurgessInt. #2 lines 1007–1010]. Bailey (experienced) also believed that Quadratic Formulaneeded more justifications. “[If] it’s going to be a formal proof, I would assume thatyou need steps, you need explanation why to do [each] step” [Bailey Int. #2 lines 1028–1029]. Without explicit step-by-step justifications, neither Burgess nor Baily wereprepared to accept the familiar derivation of the quadratic formula as a proof. Themajority of the experienced teachers felt that a proof was different from algebraicmanipulations and derivations in that a proof had to be explicit in its justifications.

By contrast, only two (25 %) of the veteran teachers discussed the need for explicitstep-by-step justifications. Bart’s (veteran) views contrasted with the views of themajority of the experienced teachers. He said, often times, the tasks given to studentsinvolve “a procedure that you’ve gone through a hundred times and it’s become anaccepted procedure,” and so it can be taken as common knowledge [Bart Int. #1 lines193–195]. He said, “To me, a proof would be that they would say ‘I know it’s truebecause …’ and be able to give you a more in depth [explanation, rather] than …[justifying] the steps they took as they went through the solving of the quadraticequation” [Bart Int. #1 lines 196–199]. Giving another example, Bart said that theverification of each step of a trigonometric identity doesn’t have to be given in order forit to be a proof. “Even though you don’t ask them to list what they did at each step,usually by the time they’ve worked on those…most of them are capable of giving sortof a verbal …‘I replaced it with this because …’” [Bart Int. #1 lines 213–218]. Unlikethe experienced teachers, Bart was much less interested in explicit step-by-step justi-fications than in larger-scale statements of how one came to know.

Precise language and careful logic

Given the difference between the two groups on their beliefs about what aproof should look like, it should not be surprising that they differed on whatthey looked for in a proof. Six (67 %) of the experienced teachers indicatedthat they looked for precise language, and careful and explicit logic whengrading student-generated proofs. Proper vocabulary in students’ work wasparticularly important to Tracy (experienced). “I think [vocabulary] showsunderstanding. If you don’t have vocab, if you say ‘fold it’ versus ‘symmetry’…you haven’t learned anything …or you don’t understand why it’s folding andworking …If you don’t say ‘supplementary’ …you haven’t learned anything”[Tracy Int. #2 lines 783–789]. Vance (experienced) described himself as adetail-oriented person and said, “Personally, I need the details to make it clearto me most of the time” [Vance Int. #3 lines 2083–2084]. It should then be no


surprise that he looked for logical order, proper use of theorems, and wanted toknow “Do they even know how to state the theorems?” [Vance Int. #1 lines395–396]. The use of precise language was a gauge that experienced teachersused to measure their students’ understanding of mathematics.

Only three (38 %) of the veteran teachers expressed concern for preciselanguage and careful logic when grading student-generated proofs. In contrastto the experienced teachers above, Brenda (veteran) was more interested in keyideas than in precise language and explicit logic. “We used to be very particularabout trivial things. You had to say ‘Measure of angle ABC equals the measureof angle DEF’, then you had to say ‘Angle ABC is congruent to angle DEF’…That is not the case anymore” [Brenda Int. #1 lines 510–516]. She went onto say that focusing on key ideas rather than on technical and very preciselanguage is “better in a sense of getting students to understand a little bit moreabout proof. But we don’t have the precision we used to have” [Brenda Int. #1,lines 527–530]. While all the participants were looking for evidence of studentthinking when reading student-generated proofs, the experienced teachers weremore concerned than the veteran teachers about the students’ use of preciselanguage and careful logic.

Sufficient and careful details

Given the experienced teachers’ greater focus on explicit reasoning, preciselanguage, and careful logic, it was not surprising that they felt that an importantfeature of proofs that made them convincing was sufficient detail. Whendiscussing features that made proofs convincing, eight (89 %) of the experi-enced teachers cited sufficient detail, while only four (50 %) of the veteranteachers did so. Cameron (experienced) said that proofs that were detailed weremore convincing. He said that convincing means:

That when I’m done reading it, I feel like it addressed everything that couldpossibly be addressed so I’m convinced…Casual [arguments] might make sense[but] I say to myself, ‘You know? I don’t know if this really accounts for someother things’ …I may not even know what those things are but if it’s not formal, itjust might leave that kind of questioning feeling behind and I’m not whollyconvinced. [Cameron Int. #2 lines 1866–1873]

For Cameron, more formal reasoning meant a more convincing proof. He indicatedthat arguments such as the Chocolate Bar task (see Appendix) are not quite formalenough for him to be 100 % convinced.

Veteran teachers tended to be more interested in the overall structure of arguments.Bart (veteran) felt that too many details can get in the way of clear communication.Unlike Cory, Kelly, and Burgess (experienced, mentioned above), Bart felt that solvingan equation was a proof, even without stepwise justifications.

If you’re to be very careful about solving an equation, be sure that you used theproper commutative property and associative property and all those things in thesteps as well as all the steps of the additive [property of] equality and all of thatwhich you normally don’t do with kids. You don’t even talk about such things


as…replacing subtraction with additive inverses and all of those definitionsbecause that’s…so minute and you’re taking every step so meticulously…thatyou’re trying to talk about the trees when all they really need to see is the forest.[Bart Int. #2, 2009–2022]

Bart felt that putting in too many details (such as the stepwise justificationsfor solving an equation) tended to focus attention on the details rather than onthe overall process of solving an equation and by extension of proving atheorem. Some teachers felt that very detailed proofs were clearer, moreconvincing, and showed deeper understanding of mathematics, and some feltthat insisting on excessive details was not helpful and even impeded studentunderstanding. While there were both experienced and veteran teachers on eachside of this argument, the veteran teachers were less likely to be concernedwith details in student-generated proofs and, in Bart’s words, were moreinterested in the forest than in the trees.

Visual and concrete features

Four (50 %) of the veteran teachers said that concrete or visual features made a proofclearer or more convincing while only two (22 %) of the experienced teachers did so.Caleb (veteran) felt more convinced by the less formal Interior Angles II task (seeAppendix) than by the very formal Interior Angles I (see Appendix) indicating the lackof formal language and abstract nature make Interior Angles II more convincing thanInterior Angles I.

Because you’re not lost in all the language…Math [is] so abstract and you’rethinking…“If these were numbers, what would happen in the real world?”Whereas this [Interior Angles II] is the real world…There it is. There’s yourstraight line. There are your angles forming a straight line. It takes the abstractnessaway. [Caleb Int. #2 lines 1004–1016]

Caleb indicated that the abstract nature of some mathematical argumentsmakes them difficult to understand and seeing arguments based on moretangible ideas makes them more accessible and more convincing. Similarly,Gwen (veteran) said that Odd Squares II is more convincing than OddSquares I (see Appendix) because, “the avoidance of the algebra makes it moreconvincing because you can’t really argue with that [sequence, but] the algebrahas that abstract quality to it” [Gwen Int. #2 lines 990–993]. Caleb and Gwenboth believed that visual arguments cut through a lot of confusing language.Using a diagram as the primary tool in an argument and not as merely asupplemental to an argument based on algebraic symbols appealed to half ofthe veteran teachers.

As mentioned above, the majority of the experienced teachers believed thatproofs should conform to strict standards of language and logic and thatsufficient mathematical detail made proofs clearer and more convincing, andas a result, found the visual arguments lacking. The veteran teachers tended toview proof as more integrated into everyday mathematics rather than as aseparate and formal way of doing mathematics. Five (63 %) of the veteran


teachers indicated that if two valid proofs existed for the same claim, one couldnot be inherently better than the other while only one (11 %) of the experi-enced teachers voiced a similar thought. Teachers with less than 10 years ofexperience tended to value detailed proofs that contain precise language andexplicit step-by-step justifications whereas teachers with more than 20 years ofexperience tended to value proofs that were less formal, focused on the overallstructure rather than on the details, and based on concrete or visual features.

Discussion and conclusion

The teachers in this study perceived that proofs served pedagogical purposes in schoolmathematics such as the development of both mathematical understanding (e.g., beingable to think mathematically, or providing deeper insight into mathematics) andgeneralized (e.g., logical, critical, and metacognitive) thinking skills. They indicatedthat most of their students were not going to take many mathematics courses aftergraduating from high school and, confirming and expanding on the results of Knuth(2002b), indicated that one of the most significant reasons for teaching proof at the highschool level was to help students to develop the thinking skills most useful outside ofmathematics classrooms. Some of the participants indicated that helping students todevelop critical and metacognitive thinking skills was an important purpose for proof inthe mathematics classroom. Vocations such as lawn mowers, cashiers, and lawyerswere listed as requiring skills similar to those required in writing mathematical proofs.For these teachers, one reason for learning to write proofs in high school has less to dowith learning mathematics but much to do with becoming a thinking person. It wastaken as common knowledge by these participants that mathematics in general (butproof in particular) develops thinking skills that can be transferred to other endeavorsoutside of mathematics classrooms.

There was agreement among the participants as to the importance and usefulness ofusing proof to enhance students’ understanding but disagreement about what kinds ofproofs best accomplish this. Participants with more than 20 years of teaching experi-ence were more likely to be satisfied by the arguments that were less formal, ordiverged from an anticipated line of reasoning. Their reasons for accepting sucharguments typically included that the argument in question showed sufficient studentunderstanding. Those with less than 10 years of experience frequently stressed theimportance of details in proof writing. They wanted their students to justify each step ofa proof and looked for careful details in student-generated proofs believing that studentswho show details demonstrate understanding. These differences indicate that theexperienced teachers may have had a more formal view of proof. The reason for thismay lie in the differences in their conceptions of proof and mathematics. The experi-enced teachers’ had taken their last college-level abstract mathematics course muchmore recently than had the veteran teachers, and so their perceptions may have beenmore strongly influenced by this more recent experience. There may also be a reasonrooted in pedagogical experience for the differences described above. It may be that,after decades of experience in trying to communicate mathematics to high schoolstudents, the veteran teachers were more willing or able than the experienced teachersto communicate at their students’ level and to accept arguments that were less formal


yet more meaningful to individual learners. These are merely conjectures and are inneed of further investigation.

A deep and robust understanding of proof has to include more than one’s abilityto write valid proofs; it must also depend on one’s conceptions regarding multiplepurposes of proof. It is particularly important for teachers who have to have notonly a robust understanding of the subject material but have to have a deepunderstanding of pedagogy as well. For this reason, the pedagogical purposes forproof (some of which would hardly matter to a professional mathematician) aresome of the more important aspects of a teacher’s understanding of proof. Manyteachers have taken a transition to abstract mathematics course as part of theirundergraduate career, but these courses often focus on how to write proofs, notwhy to write proofs or how to best use proofs in a high school classroom. Thus,teachers have had to construct their own ideas regarding proof, its purposes, andits application in school mathematics in the absence of coursework designed toaddress these aspects of proof. Given the centrality of proof to mathematics, and tothe mathematics curriculum as recommended by NCTM (2000), and given theknowledge that teachers’ levels of understanding matter, it is important to findways to support teachers’ development along these lines. Determining what fea-tures of proof are most important and useful for educating high school studentsand creating ways to support teachers’ development toward a more robust under-standing would be a potentially valuable investment of time and energy.

Appendix

Odd Squares I

Claim: If a2 is odd, then a is odd as well.Proof: Wewill prove the claim by proving the contrapositive: If a is even, then a2 is even.

Let a=2n, then a2=4n2, and a2=2(2a2). So, a2 is even.Therefore if a is even, then a2 is even and if a2 is odd, then a is odd.

Odd Squares II

Claim: If a2 is odd, then a is odd as well.Proof: Wewill prove the claim by proving the contrapositive: If a is even, then a2 is even.

Even numbers end in 0, 2, 4, 6, or 8.The square of any number ending in 0 ends in 0.The square of any number ending in 2 ends in 4.The square of any number ending in 4 ends in 6.The square of any number ending in 6 ends in 6.The square of any number ending in 8 ends in 4.

These squares are then all even. Therefore, the square of an evennumber is even.

So if an odd number is a square, then its square root is also odd.


Interior Angles I

Claim: The sum of the degree measures of the interior angles of a triangle is 180.Proof:1

Statements Reasons

1. Through B, draw DE⟷

AC⟷�� : 1. Through a point not on a given line, there exists one and

only one line parallel to the given line.

2. ∠1 ≅ ∠4, ∠3 ≅ ∠5 2. If two parallel lines are cut by a transversal, then each pairof the alternate interior angles is congruent.

3. ∠4 and ∠ABE are supplementary. 3. Supplement axiom

4. m∠4 + m∠ABE = 180 4. Definition of supplementary angles

5. m∠ABE = m∠2 + m∠5 5. Angle addition axiom

6. m∠4 + m∠2 + m∠5 = 180 6. Substitution axiom

7. m∠1 + m∠4, m∠3 =m∠5 7. Definition of congruent angles

8. m∠1 + m∠2 + m∠3 = 180 8. Substitution axiom

Interior Angles II

Claim: The sum of the interior angles of a triangle is 180˚.Proof:2 Cut a triangle from a piece of paper. Arrange it so that its longest side is the base.

Fold the triangle on a line parallel to the base so that the top vertex lies onthe base as shown below.

1 Proof adapted from Klutch et al. (1991), p. 154.2 Proof adapted from http://www.math.csusb.edu/courses/m129/tri180.html. Retrieved in the spring of 2006but at the time of this writing, this link was dead.

D

A

BD E

1 3

524

C


http://www.math.csusb.edu/courses/m129/tri180.html

Now fold the triangle so that the right vertex touches the top vertex (in its foldeddown position). The two parts of the right side of the triangle should line up, andthe right part of the bottom side should be folded along itself as shown below.

Fold in the left vertex in the same way.

You should see the three interior angles of the triangle meeting at a point,forming an 180˚ angle with no overlapping.

Quadratic Formula

Claim:The solution to ax2 + bx + c = 0 is x ¼ −b�

ffiffiffiffiffiffiffiffiffiffib2−4ac

p2a :

Proof: Solve: ax2 + bx + c = 0If: ax2 + bx + c = 0Then: x2 þ b

a xþ ca ¼ 0

By completing the square, we get:

xþ b

2a

� �2

−b

2a

� �2

þ c

a¼ 0

Solving this for x, we get the following sequence of steps:

xþ b

2a

� �2

¼ b

2a

� �2

−c

a

x þ b

2a¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib

2a

� �2

−c

a

s

x ¼ −b

2a�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib

2a

� �2

−c

a

s

x ¼ −b

2a�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2

4a2−c

a

s

x ¼ −b

2a�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2

4a2−4ac

4a2

s

x ¼ −b

2a�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffib2−4ac

p

2a


And so the solution to the quadratic equation ax2 + bx + c = 0 is:

x ¼ −b�ffiffiffiffiffiffiffiffiffiffiffiffiffiffib2−4ac

p

2aCompleting the Square

Claim: x2 þ ax ¼ xþ a2

� �2− a2

� �2Proof:3

x

x

a

+ == =+

Chocolate Bar

You have a rectangular chocolate bar marked intom × n squares, and you wish to breakup the bar into its constituent squares. At each step, you may pick up one piece andbreak it along any of its marked vertical or horizontal lines.

Claim: Every method finishes in the same number of stepsProof:4 Each time you break the bar, you end up with one more piece than you had

before. So it will take mn – 1 breaks (no matter how you proceed) to go fromone large piece to mn little pieces.

3 Proof adapted from Nelsen (1993), p. 19.4 Problem adapted from Winkler (2004), p. 82.


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http://www.k-12prep.math.ttu.edu/

Documents

High school mathematics teachers’ perspectives on the purposes of mathematical proof in school mathematics