mathematics teachers’ conceptions of proof: implications for educational research

YI-YIN KO

MATHEMATICS TEACHERS’ CONCEPTIONS OF PROOF:IMPLICATIONS FOR EDUCATIONAL RESEARCH

Received: 8 August 2008; Accepted: 9 August 2010

ABSTRACT. Current mathematics education reforms devoted to reasoning and proofhighlight its importance in pre-kindergarten through grade 12. In order to provide studentswith opportunities to experience and understand proof, mathematics teachers must have asolid understanding of proofs themselves. In light of this challenge, a growing number ofresearchers around the world have started to investigate mathematics teachers’conceptions of proof; however, much more needs to be done. Drawing on lessonslearned from research and curricular recommendations from around the world, the mainpurpose of this paper is to review the literature on elementary and secondary mathematicsteachers’ conceptions of proof and discuss international implications.

KEY WORDS: educational research, mathematics teachers’ conceptions of proof, proof,teacher education

INTRODUCTION

Proof has been considered central to the discipline of mathematics formore than a century; however, its role in school mathematics within theUnited States has been limited to high school geometry, except for a briefperiod of “New Math” in the 1960s. Recent reform recommendations(National Council of Teachers of Mathematics [NCTM], 2000) suggestthat students in pre-kindergarten through grade 12 should regularly studymathematical proof. Although the main purpose of proof is to verify thetruth of mathematical conjectures, proof plays other roles in themathematics community, including “explanation, systematization, dis-covery, communication, construction of empirical theory, exploration ofdefinition and of the consequences of assumptions, and incorporation of awell-known fact into a new framework” (Yackel & Hanna, 2003, p. 228).

Current reforms, which place significant demands on teachers andstudents, emphasize the social nature of proof. In the classroom, a proof isa mathematical argument which integrates three characteristics:

1. It uses statements accepted by the classroom community (set ofaccepted statements) that are true and available without furtherjustification;

International Journal of Science and Mathematics Education (2010) 8: 1109Y1129# National Science Council, Taiwan (2010)

2. It employs forms of reasoning (modes of argumentation) that are validand known to or within the conceptual reach of the classroomcommunity; and

3. It is communicated with forms of expression (modes of argumentrepresentation) that are appropriate and known to or within theconceptual reach of the classroom community (Stylianides, 2007, p. 291).

The definition of proof provided by Stylianides (2007) incorporates theimportant ideas of sets of accepted statements, modes of argumentation,and modes of argument representation. In this definition, proof serves asa means to communicate thoughts with learners in the mathematicscommunity. Aligned with this view, proof activities in the classroomshould provide opportunities for students “to recognize reasoning andproof as fundamental aspects of mathematics, make and investigatemathematical conjectures, develop and evaluate mathematical argumentsand proofs, and select and use various types of reasoning and methods ofproof” (NCTM, 2000, p. 56). Alibert & Thomas (1991) recommended“establishing an environment in which students may see and experiencefirst-hand what is necessary for them to convince others of the truth orfalsehood of propositions” (p. 60). Taken together, proof is the primaryform of community sense-making because students can present individualarguments and determine acceptable justifications in their classroom.

While Stylianides’ (2007) definition of proof in school mathematics isaccepting of many mathematically valid forms, empirical studies havepointed out that many pre-college students tend to favor example-basedarguments as proof (e.g. Bieda, Holden & Knuth, 2006; Chazan, 1993;Knuth, Choppin, Slaughter & Sutherland, 2002) and some mathematicsteachers1 regard particular examples as a valid proof (e.g. Barkai, Tsamir,Tirosh & Dreyfus, 2002; Knuth, 2002a, b; Martin & Harel, 1989; Morris,2002). If mathematics teachers regard empirical examples as valid proofs,is it any surprise that students also hold such beliefs? If mathematicsteachers have an inadequate understanding of proof, should it come as nosurprise that students also have similar misunderstandings?

Given the importance of teachers’ role in the classroom, coupled withtheir frequent misconceptions about proof, more research should be doneto investigate what factors influence mathematics teachers’ conceptions ofproof. By “conceptions of proof,” I mean teachers’ subject matter contentknowledge and beliefs about strategies for proving in mathematics andproof in different domains as well as what constitutes proof and the roleof proof in mathematics (Knuth, 1999). The main purpose of this paper isto review existing empirical studies focused on mathematics teachers’

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conceptions of proof in order to highlight the gaps in mathematicsteachers’ understandings and inform future research and professionaldevelopment efforts. Since most research on this topic has beenconducted in North America, I rely upon that literature but incorporateresearch from Canada, Cyprus, Israel, and the United Kingdom. I beginby outlining the role of proof in mathematics and the importance of proofin learning mathematics. Next, I review relevant research studies thatprovide evidence of teachers’ understandings and misunderstandings withrespect to mathematical proof. Finally, I address recommendationsprovided by existing literature for improving mathematics teachers’conceptions of proof and identify possible implications for educationalresearch in the area of proof.

THE ROLE OF PROOF IN MATHEMATICS

Perhaps themost obvious role of proof inmathematics is to establish whethera statement is true or false. Proof provides certainty in all cases and allowsthe results to be applied when solving problems. But it goes beyond that, asmathematicians are more concerned about why a mathematical statement istrue than whether it is true (Hersh, 1993). Hanna (1995) pointed out that “thebest proof is one which also helps mathematicians understand themeaning ofthe theorem being proved: to see not only that it is true, but also why it istrue” (p. 47). Likewise, Gleason (2001; cited in Yandell, 2002) indicated thata proof does not just convince the reader of the result but shows him or herwhy it is true. In short, proof is an important tool to foster individualunderstanding of why a conjecture is true.

Because proof is a social process (Hanna, 1991), it also serves as ameans of communicating mathematical knowledge, understanding, andresults through social interaction among members in a community(Alibert & Thomas, 1991). Harel & Sowder (2007) identified twocommunication processes: (a) ascertaining, where an individual con-vinces himself or herself, and (b) persuading, where an individualconvinces others. In addition, De Villiers (1999) noted that proof is “ameans of exploring, analyzing, discovering and inventing new results” (p. 5);thus, proof also plays an important role in the discovery of newmathematicalknowledge which has been developed through the process of deductiveproof (De Villiers, 1999).

Proof also systematizes results “into a deductive system of axioms,major concepts and theorems” (Bell, 1976, p. 24). In this role, proof isused to “exhibit the logical structure of ideas and to make deductive

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chains of reasoning explicit” (Coe & Ruthven, 1994, p. 43) and organizeconcepts, statements, definitions, axioms, and theorems from establishedresults (De Villiers, 1999; Harel & Sowder, 2007). Additionally, proof isa means of intellectual challenge (De Villiers, 1999), as an individual canobtain self-realization and fulfillment from constructing a proof (DeVilliers, 1999; Harel & Sowder, 2007). The roles of proof in mathematicsare summarized in Table 1.

THE IMPORTANCE OF PROOF IN LEARNING MATHEMATICS

A mathematical proof requires that sets of accepted statements, modes ofargumentation, and modes of argument representation in the classroomcommunity (Stylianides, 2007) are used to “deduce the truth of onestatement from another” (Tall, 1989, p. 30), helping people understandthe logic behind a statement and gain “insight into how and why it works”(Tall, 1992, p. 506). In other words, mathematical proofs can help peoplenot only see why statements are true but also understand their meanings.In mathematics classrooms, proof is a vehicle to enhance students’understanding of mathematical concepts and promote their mathematicalproficiency and reasoning (Hanna, 2000) as proof is “involved in allsituations where conclusions are to be reached and decisions to be made”(Fawcett, 1938, p.120).

Even though elementary school students are just beginning to learnmathematics, studies by Lampert (1990), Maher & Martino (1996), andCarpenter & Franke (2001) suggest that they can understand proofs. For

TABLE 1

The role of proof

Role Description

Verification A means of verifying the truth of a statementExplanation A means of explaining why a statement is trueCommunication A means of communicating the importance of mathematical

knowledge offered by the proof produced among membersin the community

Discovery A means of discovering new mathematical knowledgeSystemization A means of systemizing of various results into deductive

system of definitions, axioms, and theoremsIntellectual challenge A means of deriving the self-realization and fulfillment

from constructing a proof

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example, Maher & Martino (1996) traced one student’s understanding ofjustification from first to fifth grade as she worked on combinatoricstasks. This student, Stephanie, initially used a trial-and-error method toorganize towers four cubes tall and explained that she kept buildingtowers until she did not find any different from those she already built. Infifth grade, Stephanie used proof by cases to justify all possible towersthree cubes tall. Maher & Martino concluded that the classroomenvironment provided Stephanie many opportunities to use multiplerepresentations, reexamine earlier ideas, and explain and justify herthinking with her own language, which allowed her thinking about proofto progress.

Yet, research shows that not all pre-college students become better atproducing proofs over time. Existing empirical studies have documentedthat some secondary students have difficulty with mathematical proof inschool mathematics (e.g. Balacheff, 1991; Chazan, 1993; Healy &Hoyles, 2000; Knuth et al., 2002; Lin, Yang & Chen, 2004; Senk, 1985;Thompson, 1996), perhaps due to their mathematics teachers’ inappropriateinstruction or task selection (Herbst, 2002). Even though, by the end ofsecondary school, students “should be able to understand and producemathematical proofs” (NCTM, p. 56), studies examining undergraduatestudents’ conceptions of proof document that they seem to possess anunsatisfactory understanding of mathematical concepts (e.g. Hart, 1994;Moore, 1994; Selden & Selden, 2003; Weber, 2001) or lack understandingof the necessary mathematical language (e.g. Dreyfus, 1999; Ko & Knuth,2009; Moore, 1994; Selden & Selden, 1995). These findings do not come asa surprise because undergraduates do not encounter proof again until takinghigher-level courses, such as advanced calculus and abstract algebra.

The above discussion points to the necessity of K-12 students havingearly and appropriate opportunities to incorporate the mathematicalconcepts and language of proof into their work. This can only be doneif their mathematics teachers have a deep understanding of proof. In thenext section, I review many empirical studies on mathematics teachers’conceptions of proof in order to gain more insight into teacher performancein this domain.

MATHEMATICS TEACHERS’ CONCEPTIONS OF PROOF

Most existing studies have focused specifically on pre-service mathe-matics teachers’ conceptions of proof (e.g. Jones, 1997; Martin & Harel,1989; Morris, 2002; Stylianides, Stylianides & Philippou, 2004, 2007;


Varghese, 2007). Fewer studies have considered the conceptions of proofheld by in-service mathematics teachers (Barkai et al., 2002; Dickerson,2008; and Knuth, 1999, 2002a, b are exceptions). Several studies havefocused on mathematics teachers’ conceptions of proof by askingparticipants to select which argument they regard as a valid mathematicalproof from a set of choices (e.g. Martin & Harel, 1989) and to explaintheir thinking (e.g. Knuth, 2002a; Stylianides et al., 2004, 2007) asopposed to asking them to produce proofs. In this section, I reviewrelevant research studies of pre-service and in-service elementary andsecondary school mathematics teachers’ conceptions of proof. Thesestudies included a focus on teachers’ conceptions of the role of proof,logical principles of proof by contraposition and induction, and whatconstitutes and counts as a mathematical proof in the domains of numbertheory and geometry.

Teachers’ Conceptions of the Role of Proof

Studies investigating mathematics teachers’ conceptions of the role ofproof show that some teachers possess a limited understanding of the rolethat proof can play in mathematics. Goetting (1995) interviewed 11 USelementary mathematics majors and 16 US secondary mathematics majorsfrom a class focused on writing mathematical proofs. She found thatelementary mathematics majors believed that proof supported argumentsand secondary mathematics majors believed that proof served as a meansof explanation. Mingus & Grassl (1999) surveyed 30 US pre-serviceelementary teachers enrolled in a mathematics content course and 21 USpre-service secondary mathematics teachers enrolled in an abstractalgebra course. The majority of their participants only indicated thatproofs helped explain mathematical concepts used in arguments.

Jones (1997) investigated the conceptions of mathematical proof heldby secondary student teachers with degrees in mathematics in the UnitedKingdom. He did this by asking his participants to list key terms relatedto mathematical proof through group brainstorming sessions and then toindividually produce a concept map to represent their conceptions ofproof by linking these key terms. He reported that the most mathemati-cally able student teacher did not provide the greatest number ofrelationships among key terms on his or her concept map of mathematicalproof. He concluded that majoring in mathematics did not guarantee thatteachers were equipped with adequate subject knowledge for teaching.

Because pre-service mathematics teachers do not have teachingexperience yet, one might expect that in-service mathematics teachers

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might have more sophisticated conceptions of the role of proof. Thelimited number of research studies on experienced mathematics teachers’conceptions of the role of proof document that they still possess aninadequate understanding (Knuth, 2002a, b). Sixteen US in-servicesecondary mathematics teachers in Knuth’s (2002a) study were asked aseries of questions focused on their conceptions of the role of proof (e.g.“Do proofs ever become invalid?,” p. 383). All teachers claimed that themain role of mathematical proof is to establish a true proposition.However, four of them indicated that proofs were not influenced bycontradictory evidence, six of them believed that a proof was possiblycontradictory to a counterexample, and the remaining teachers believedthat proofs might become invalid in the new axiomatic system. Threeteachers indicated that proof could explain why a mathematicalproposition was true; however, none of the teachers brought up theexplanatory role that proof could play in promoting mathematicalunderstanding. Twelve teachers believed that proof could be used tocommunicate mathematical ideas in order to accept or refute the claims ofothers, and eight teachers viewed proofs as “the creation of mathematicalknowledge and in its systematization” (p. 390).

When asked their beliefs about what constituted proof in secondaryschool mathematics, 11 US in-service secondary mathematics teachersin Knuth’s (2002b) study indicated that proof as a deductive argumentguaranteed the certainty of statements and the remaining perceived“proof as a convincing argument” (p. 71). Thirteen teachers consideredthe major role of proof in secondary school mathematics to be todevelop students’ logical and reasoning skills, findings which wereechoed by Dickerson’s (2008) study with 17 US in-service secondarymathematics teachers. Knuth also found that 10 teachers perceivedproofs as convincing and accepted arguments in the classroomcommunity, four viewed proofs as opportunities for students todemonstrate their thinking, and seven saw proofs as a means to assiststudents in understanding the truth of statements. Even though fourteachers indicated that proofs enabled students to be independentthinkers, none of them mentioned that proofs served as a means ofexplaining underlying mathematical concepts.

In short, the above studies suggest that teachers do largely understandthat proof serves as a means of verifying the truth of a statement. Only afew teachers stated that proof serves the functions of communicatingmathematics, helping students make discoveries, and systemizing results,and none of the teachers mentioned proof as a means of providingintellectual challenge.


Teachers’ Conceptions of Proof by Contraposition

Findings from existing literature show that, even though teachers havesome basic knowledge of proof by contraposition, they are unable toassociate a direct argument with an indirect argument of an equivalentstatement when proving by contraposition. More specifically, teachers donot seem to understand the logical principle used in the proof( p ) q � � q ) � p). Riley (2003) conducted a study of 23 US pre-service secondary mathematics teachers’ abilities to write a mathematicalproof for the proposition “Let x be an integer. If x3 is even, then x iseven” (p. 175). All of the participants had completed at least two calculuscourses, an introductory proof class and a college geometry course; however,only nine participants provided a valid proof. The responses of threeparticipants were judged to be invalid with some reasoning demonstrated,and the justifications of 11 participants were judged to be lacking enoughreasoning to construct a valid proof. Moreover, two participants whosearguments were judged to be invalid proofs used examples as justifications.

Stylianides et al. (2004) study involved 70 prospective elementaryschool teachers who had taken several relevant mathematics courses and25 prospective secondary school mathematics teachers who weremathematics majors in Cyprus. All participants were asked to select“true” or “false” for a simple proof by contraposition. Sixty-four percentof prospective secondary school mathematics teachers and 20% ofprospective elementary school teachers considered a proof by contra-position to be a valid mathematical proof, while 28% of prospectivesecondary school mathematics teachers and 47.1% of prospectiveelementary school teachers thought that proof by contraposition was notacceptable. The remaining teachers believed that the statement proved bycontraposition was true in some cases, or they had no opinion at all.Given a proof by contraposition displayed in Figure 1, Knuth (2002a)

Figure 1. From Knuth (2002a, p. 393). Copyright 2002 by the National Council ofTeachers of Mathematics. Adapted with permission of the author

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found that 10 US in-service secondary school teachers considered thisargument to be a mathematical proof because of an easy-to-understandalgebraic procedure. His results indicated that in-service mathematicsteachers only paid attention to the correctness of the algebraic processesrather than to the logic of the proof of the converse.

While the aforementioned studies show that some mathematicsteachers regarded proof by contraposition as a correct proof or were ableto provide a valid proof by contraposition, there were still considerablenumbers of teachers who did not understand the logic behind thecontraposition equivalence rule for proving by contraposition.

Teachers’ Conceptions of Proof by Mathematical Induction

Existing literature documents that some mathematics teachers have consid-erable difficulty understanding the base step and inductive steps or themeaning of the possible domain of an infinite set to support mathematicalpropositions proved true by mathematical induction. Movshovitz-Hadar(1993) reported fragile knowledge of mathematical induction held by 24prospective Israeli high school teachers in a problem-solving course. Mostparticipants (75– 100%) predicted that they were able to construe a correctproof by mathematical induction; however, only two participants demon-strated complete understanding of the principles of mathematical induction.Movshovitz-Hadar also found that all students completed the final step of aproof by induction and accepted a false proof by mathematical induction ascorrect because “[a] gap [exists] between their procedural and theirconceptual understanding of [mathematical induction]” (p. 260).

Teachers’ difficulty with proof by mathematical induction has also beenreported by Stylianides et al. (2007), Dickerson (2006), and Knuth (1999).Stylianides et al. (2007) used two items to examine 70 prospectiveelementary and 25 prospective secondary school mathematics teachers’conceptions of proof by mathematical induction in Cyprus. On one item, “[f]or every n ∈ N the following is true: 1þ 3þ 5þ . . .þ 2n� 1ð Þ ¼ n2 þ 3”(p. 151), the majority of the prospective secondary school mathematicsteachers (88%) and almost a quarter of the prospective elementary schoolteachers (24%) claimed that the provided argument was an invalid proof bymathematical induction, but their written responses were limited to explain-ing the missing first step instead of showing the importance of this processon the first task. Another item, “[f]or every natural number n≥ 5 thefollowing is true: 1 • 2 • … • (n − 1) • n 9 2n” (p. 152), the majorityof the prospective secondary school teachers (92%) and just over half ofthe prospective elementary school teachers (54%) claimed that the


provided argument was a valid proof by mathematical induction.However, many prospective elementary school teachers argued that thisproof was only true in some cases because a valid proof by mathematicalinduction should support a true mathematical statement for all naturalnumbers.

In Knuth’s (1999) study, 18 US in-service secondary schoolmathematics teachers were asked to rate the validity of providedarguments for the statement: “The sum, S(n), of the first n positiveintegers is equal to n(n+ 1)/2” (pp. 194 – 198). Knuth found that themajority of the teachers accepted an argument by mathematical inductionas a valid proof although they either did not understand or feltunconvinced by this proof method. Dickerson (2006) confirmed Knuth’s(1999) results, finding that at least half of 10 US pre-service mathematicteachers were convinced by proof by mathematical induction on the basisof the superficial form of the argument.

To sum up, the above studies suggest that mathematics teacherspossess inadequate understanding of the principles of proof by mathe-matical induction; in some cases, finding proof by induction convincingbased on the presented format of the argument.

Teachers’ Conceptions of Proof in the Domain of Number Theory

While many pre-service teachers have learned quite abstract mathematics inundergraduate mathematics courses, many of them accept empiricalexamples or a combination of empirical examples and deductive argumentsas valid mathematical proofs in number theory. Martin & Harel (1989)examined the conceptions of proof held by 101 US pre-service elementaryteachers enrolled in a mathematics course focused on proofs by asking themto rate whether inductive and deductive justifications of a statement countedas valid mathematical proofs in number theory. They found that more thanone half of the participants accepted either an inductive argument or a falsedeductive argument as a valid mathematical proof, regardless of theirfamiliarity with the mathematical content. Additionally, more than 33% ofparticipants accepted both inductive and deductive arguments about thesame statement as acceptable mathematical proofs. Morris (2002) confirmedMartin & Harel’s (1989) results, finding that 40% of 30 US pre-serviceelementary and middle school teachers believed that both deductive andinductive arguments established mathematical truth and failed to distinguishbetween the two forms of arguments. Goetting (1995) reported a similarfinding that up to approximately eight US elementary mathematics majorsaccepted empirical evidence as a valid mathematical proof. Gholamazad,

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Liljedahl & Zazkis (2004), in a study of 75 Canadian pre-service elementaryteachers who enrolled in a class emphasizing conceptual understanding ofmathematics, found that the majority of participants accepted all numericalcases in the finite set and some cases in the infinite set as valid proofs andbelieved that two or three particular examples were sufficient to prove a trueclaim.

Findings from research on in-service mathematics teachers’ conceptionsof proof, however, also point out that many teachers tend to favor empiricalexamples as valid proof in number theory. Barkai et al. (2002) examined theknowledge of 27 in-service Israeli elementary school teachers who enrolledin a 3-year professional development program focused on mathematicaltopics. They were asked to decide which given proposition in number theorythey believed to be true or false and to provide a justification of their claims.More than 50% of participants stated correctly the true or false condition ofall the propositions. However, many experienced teachers claimed that theirexplanations with supportive examples were mathematical proofs. Some ofthem even used counterexamples to refute mathematical propositions thatwere actually true. Additionally, some participants who provided counter-examples to disprove a conjecture still regarded the counterexamples asmathematical proofs.

When asked to rate an argument’s validity shown in Figure 2 using a four-point scale, Knuth (2002a) found that four US in-service secondary schoolmathematics teachers considered this argument to be a valid mathematicalproof for the general case. Six teachers rated this argument as a proof only forthe particular case, and the remaining six rated this argument as an invalidproof because it was not general. One teacher explained this was amathematical proof due to the true statement, although it was only used ina specific case. Another teacher indicated that he or she replaced x, y, and zfor 7, 5, and 6, respectively, and got the same result. Thus, the research

Figure 2. From Knuth (2002a, p. 394). Copyright 2002 by the National Council ofTeachers of Mathematics. Adapted with permission of the author


studies above point out teachers’misunderstanding of algebraic proofs, suchas viewing particular examples and counterexamples as valid proofs, andteachers’ difficulties in constructing algebraic proofs, such as misinterpretingthe problem and lacking algebraic language.

Teachers’ Conceptions of Proof in the Domain of Geometry

Given that K-12 proof has traditionally been limited to a single highschool geometry class, it is of paramount importance for teachers topossess adequate conceptions of geometry proof. Varghese (2007) asked17 Canadian student teachers with mathematics majors to write amathematical proof for the following proposition: “Prove that the sumof the exterior angles of a polygon is 360°” (p. 58). Of the 14 responsesprovided, only two students constructed a correct proof in diagrammaticor verbal form and two students produced an almost correct proof in thesymbolic form. The remaining students gave incorrect proofs in the formof a diagram, verbal or symbolic statements with little information abouttheir thinking, or partially correct understandings related to this task.

Existing literature also documents that teachers have a narrow under-standing of what constitutes proof and perceive empirical arguments as validproofs. The results of Goetting’s (1995) study indicated that more US pre-service elementary mathematics teachers were inclined to accept drawnexamples in geometry as a valid proof. Given the empirical argument that“the sum of the measures of the interior angles of any triangle is equal to180°” (p. 392), Knuth (2002a) found that five US teachers rated thisexample-based argument as a valid proof. Although two teachers did notthink the empirical argument was a mathematical proof because they foundfaults, one teacher indicated that such proof was valid but not formal.Dickerson (2008), who studied 17 US in-service high school mathematicsteachers’ understanding of proof, reported that 16 of his participants were notconvinced by a visual proof in geometry due to unacceptable format of theargument in high school or insufficient detail presented in the argument. Insum, the above studies suggest that some teachers have difficulty with proofsin geometry due to their limited mathematical knowledge.

RECOMMENDATIONS FOR IMPROVING MATHEMATICS TEACHERS’ CONCEPTIONS

OF PROOF

The majority of the studies discussed above come from North America,and many are based on only a small sample of participants and/or

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assessment items. In some cases, conclusions were drawn on the basis ofresults from only one or two items. While the discussed findings mightnot be generalizable for all teachers around the world, such resultsprovide valuable information about mathematics teachers’ conceptions ofproof. The research studies indicate that some mathematics teachers onlyregard proof as a means of verification and explanation. Such findingsshould not come as a surprise as proof in undergraduate mathematicscourses focuses on providing convincing arguments to support truestatements and explain mathematical knowledge used in proofs (Mingus& Grassl, 1999) and such courses only teach students how instead of whyto write a mathematical proof (Dickerson, 2008). In order to betterprepare mathematics teachers, Jones (1997) and Knuth (2002a) suggestthat undergraduate mathematics and teacher education courses shouldfocus more on the various roles of proof in mathematics.

The aforementioned studies document that some mathematics teachersaccept particular examples or both empirical and deductive arguments asvalid mathematical proofs. The former misconception might come frommathematics teachers’ previous experiences learning proof (Martin &Harel, 1989; Morris, 2002), and the latter suggests that mathematicsteachers might construct the inductive frame earlier than the deductiveframe (Martin & Harel, 1989). Proof at the elementary school levelfocuses primarily on informal arguments on the grounds that elementaryschool students are just learning mathematics. It should, therefore, notcome as a surprise that researchers have not found different methods of proof(e.g. proof by contraposition, proof by mathematical induction) used byelementary mathematics teachers. On the other hand, teaching proof at theelementary school level is challenging work because teachers need toestablish a mathematical foundation for young children while using languagethat they can understand (Barkai et al., 2002; Gholamazad et al., 2004).Given the important role played by elementary school mathematics teachers,it is necessary to provide more opportunities for them to validate purportedarguments (Gholamazad et al., 2004;Martin &Harel, 1989), to evaluate theirown proof productions (Stylianides & Stylianides, 2009), and to use algebraas a tool to express students’ conjectures (Barkai et al., 2002).

Common sense also suggests that secondary school mathematicsteachers in many countries have taken more undergraduate mathematicscourses and, therefore, are expected to have an adequate understanding ofproof and be able to produce proofs. However, the above studies showthat some secondary school mathematics teachers still tend to favorspecific examples as valid proofs and possess considerable difficulty withvarious types of proof techniques as they focus on the superficial format of


the argument rather than on the correctness of reasoning presented.Undergraduate students who are pursuing mathematics or other sciencemajors (e.g. engineering, computer science) may have opportunities toeducate school mathematics teachers, and those who are mathematicseducation majors will become mathematics teachers. Much more attentionshould thus be given to the preparation of students with mathematics or otherscience majors as well as mathematics education majors in order toimplement prevailing reform suggestions with respect to proof. Supportingfuture secondary mathematics teachers to learn proof with understanding isclearly important work; however, most of the studies discussed above onlyfocus on secondary mathematics teachers’ conceptions of proof in thedomains of number theory and geometry. One question raised in this paper isif it is necessary for secondary mathematics teachers to learn about proof inall subject areas of mathematics or if it is enough for them to get a deepunderstanding of proof only by learning number theory and geometry in theirundergraduate mathematics courses.

Indeed, mathematics teachers’ conceptions of proof are usuallyaffected by undergraduate mathematics courses in which they learn thecontext of proof in mathematics (Blanton, Stylianou & David, 2009;Knuth, 2002b; Smith, Nichols, Yoo & Oehler, 2009). In such courses,most students spend much time observing and passively taking notes asprofessors present proofs (Weber, 2004) rather than writing proofs ontheir own. So it follows that, because they just copy the final productions,they are unlikely to understand proof techniques and mathematical conceptsneeded to construct proofs. While the limited research has suggested thatbuilding a classroom environment in which undergraduate students canengage in problem-based activities (Smith et al., 2009) and participate inclassroom discourses (Blanton et al., 2009) are effective teaching strategiesfor enhancing students’ conceptions of proof, we do not yet fully know howto better prepare pre-service as well as in-service elementary and secondarymathematics teachers to have adequate knowledge in the area of proof. Asthese school mathematics teachers play important roles in educating K-12students, much more research is needed on nuanced teaching strategies forfostering the growth and improvement of mathematics teachers’ reasoningand proving about mathematics.

IMPLICATIONS FOR EDUCATIONAL RESEARCH

Several researchers suggest that collaboration between mathematicseducation and mathematics professors might improve teachers’ concep-

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tions of proof (e.g. Jones, 1997; Knuth, 1999, 2002a). While this is not anew idea, it is still unclear what a proof course for school mathematicsteachers designed by mathematics education and mathematics professorsmight look like. To date, little is known about what kind of mathematicalproof competence as well as mathematics educational competence schoolmathematics teachers should have with respect to teaching proof. In orderto move forward with this idea, it is necessary to develop a framework ofwhat mathematical knowledge and what competence in mathematics aswell as in mathematics education is needed for elementary and/orsecondary school mathematics teachers to teach proof in their classroomsas a guide to enhance content preparation. The ideal of such a contentcourse is to promote school mathematics teachers’ conceptions of proofand develop their competence for teaching proof. However, currently,many graduate students in the mathematics and mathematics educationdepartment are hired as instructors to teach required content and methodcourses for prospective mathematics teachers. Therefore, it is worthconsidering if each instructor has satisfactory conceptions of proof aboutK-12 mathematics as well as, perhaps, the ability to teach students thoseconcepts. As teaching proof involves mathematical ideas, terms, reason-ing, and skills, mathematics teachers should have “the mathematicalknowledge needed to carry out the work of teaching mathematics” (Ball,Thames & Phelps, 2008, p. 395). In order to address this concern, it isalso important to develop a framework of what mathematical knowledgeas well as competence is needed for instructors to teach proof for theirprospective elementary and/or secondary school mathematics teachers.

Research has suggested that the important work of enhancingmathematics teachers’ conceptions of proof requires developing teachers’appreciation of the role of proof “as a tool for studying and understandingmathematics” (Knuth, 1999, p. 166). To accomplish this goal, teachersshould be provided more opportunities to engage in proof, evaluate theirown proof productions, and validate mathematical arguments in teachereducation courses (Jones, 1997; Gholamazad et al., 2004; Knuth, 2002a;Martin & Harel, 1989; Stylianides & Stylianides, 2009). If such activitiesare to be successful in increasing mathematics teachers’ development ofrobust conceptions of proof, more research is needed on what curriculartasks better promote elementary and secondary school mathematicsteachers’ understanding of proof and provide rich opportunities for theirstudents to experience proof.

While strong understanding of proof is vital to implementing currentreform recommendations suggested by NCTM (2000), little is knownabout in-service mathematics teachers’ conceptions of proof (although


Barkai et al., 2002; Dickerson, 2008; and Knuth, 2002a, b areexceptions). It is surprising that so little attention has been paid tomathematics teachers’ instruction of proof (although Herbst, 2002 andMartin, McCrone, Bower & Dindyal, 2005 are exceptions). Furtherresearch is needed to examine the relationships between (1) in-serviceteachers’ mathematical proof competence and their pedagogy of provingwith respect to teaching proof and (2) their conceptions of proof andinstructional practices, as well as their students’ mathematics performancein order to prepare teachers to better implement proof throughout K-12mathematics.

CONCLUSIONS

The above review highlighted the importance of proof in the K-12classroom, difficulties demonstrated by some mathematics teachers invarious countries in this domain, and the importance of teachers having arobust understanding of proof. It also discussed implications for educa-tional research concerning teachers’ conceptions of proof. Mathematicsteachers in many countries around the world receive a bachelor’s ormaster’s degree in mathematics and are thus expected to have a solidunderstanding of proof and to have mastered the skills required toconstruct proofs. However, evidence from the reviewed studies has shownthat some teachers with mathematics backgrounds can still possessinadequate knowledge of proof. Such results illustrate a need foreducators and researchers to conduct more studies that examinemathematics teachers’ conceptions of proof by asking them to describethe role of proof, evaluate purported arguments, and produce proofs forpropositions. Observations of teaching practices in classrooms are alsoneeded in order to gain more insight into teachers’ conceptions of proof.In doing so, we will have a better understanding of how to supportmathematics teachers in learning proof meaningfully in their under-graduate mathematics courses and in teaching proof effectively in theirclassrooms.

Since mathematics teachers’ conceptions of mathematical proofinfluence not only the experiences they provide for their students butalso the expectations they hold for their students in learning proof (Knuth& Elliott, 1997), having a robust understanding of proof is important forteachers. Moreover, teaching proof demands understanding of the contentas mathematics teachers should select tasks, show students how toconstruct proofs, explain their reasoning to students, respond to students’

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questions, identify students’ errors, evaluate students’ written work, andprovide students with feedback (Ball et al., 2008; Weber, 2008).However, we still do not yet know specifically what teaching strategiesused by mathematics instructors are effective to enhance mathematicsteachers’ competence with respect to learning and teaching proof andwhat knowledge and competence is needed for those school mathematicsteachers to teach proof. In order to make proof meaningful to mathematicsteachers so that they can teach it effectively to their students, much moreresearch is needed about how to best develop their mathematicalknowledge and instructional practices in the area of proof.

ACKNOWLEDGEMENTS

The author would like to thank Dr. Eric Knuth, Dr. Ana Stephens, BethGodbee, Clara Burke, Caroline C. Williams, and anonymous reviewersfor their helpful comments on earlier versions of this paper.

NOTE

1 Mathematics teachers in this paper refer to pre-service and in-service K-12mathematics teachers.

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