DEPARTMENT OF MATHEMATICSTECHNICAL REPORT

TRANSITIONS AND PROOF ANDPROVING AT TERTIARY LEVEL

ANNIE SELDEN

JULY 2010

No. 2010-3

TENNESSEE TECHNOLOGICAL UNIVERSITYCookeville, TN 38505

Transitions and Proof and Proving at Tertiary Level1

Annie Selden2

By tertiary level, in this chapter, we will be referring to undergraduate students

majoring in mathematics, including preservice secondary mathematics teachers. Also, in so far as there is information, this chapter will also deal with undergraduate students who major in other subjects and take courses having proofs, such as engineering and computer science majors. It will also consider inservice secondary teachers, as well as other professionals such as engineers, who take additional mathematics courses that deal with proof. In addition, tertiary level will be interpreted to include Masters and Ph.D. students in mathematics and mathematics education.

After discussing in Section 1 how proof is different at the tertiary level, we present an overview of research, resources, and ideas about transitions, and proof and proving at the tertiary level. In Section 2, we present information and research on transitions from secondary level to undergraduate level and from undergraduate study to graduate study in mathematics. In Section 3, we present research on how tertiary students deal with various aspects of proof and proving including logical reasoning, understanding and using definitions and theorems, selecting helpful representations, and knowing how to read and check proofs. In Section 4, we give information, research, and resources on teaching proof and proving at the tertiary level. In Section 5, we pose questions for future research.

1. How is proof different at the tertiary level?

At the tertiary level, proofs involve considerable creativity and insight as well as both understanding and using formal definitions and previously established theorems. Proofs tend to be longer, more complex, and more rigorous than those at earlier educational levels. To understand and construct such proofs involves a major transition for students but one that is sometimes supported by relatively little explicit instruction.

Proofs that tertiary students are expected to study, and to construct, are more formal than those expected of students at primary or secondary level. In general, mathematicians and aspiring mathematicians, such as university mathematics majors, prove theorems, whereas children or novices tend to justify through less formal forms of argumentation. Informal arguments and examples can still be used to think things through initially, and intuitive reasoning is valued at this stage, but eventually such reasoning must be made more formal for communication purposes (Hanna, Jahnke, & Pulte, 2009), whether this is for assessments, theses, dissertations, or publications.

Proofs at tertiary level tend to be more precise, more concise, and have a more complex structure than those expected of students at primary or secondary level. If one compares typical secondary school geometry proofs with proofs in real analysis, linear algebra, abstract algebra, or topology, one sees that the objects in geometry are 1 This is a draft of a chapter to appear in an upcoming International Commission on Mathematical Instruction (ICMI) Study Volume based upon the 19th ICMI Study Conference on Proof and Proving in Mathematics Education, held in Taiwan in May 2009. 2 Annie Selden is Professor Emerita of Mathematics from Tennessee Technological University and Adjunct Professor at New Mexico State University.

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idealizations of real things (points, lines, planes), whereas the objects in real analysis, linear algebra, abstract algebra, or topology (functions, vector spaces, groups, topological spaces) are abstract reifications.

Furthermore, proofs at tertiary level require a larger knowledge base. For example, a much deeper knowledge of the real number system is required for real analysis proofs than is required for secondary school algebra or geometry proofs. Also, tertiary teachers increasingly use students' original proof constructions, not just the reproduction of textbook or lecture proofs and also not just the use of textbook theorems to solve applied problems, as a means of assessing their students’ content understanding. In addition, there is an increasing emphasis on fostering and assessing students’ creativity within mathematics, especially at the Masters and Ph.D. level. At that level, professors expect students to conjecture interesting new results and to prove and publish them.

2. The character of transitions at tertiary level There are many changes in the didactic contract (Brousseau, 1997) when moving from secondary school to undergraduate study in mathematics or from undergraduate to graduate study in mathematics. Beginning university students often face “a difficult transition, from a position where concepts have an intuitive basis founded on experience, to one where they [the concepts] are specified by formal definitions and their properties reconstructed through logical definitions” (Tall, 1992). In addition, the nature of proofs and proving at tertiary level, with its increased demands for rigor, seems to constitute a major hurdle for many beginning university students. These changes can be seen as a shift from elementary to advanced mathematical thinking (Selden & Selden, 2005; Tall 1991), and as going “from describing to defining, from convincing to proving in a logical manner based on definitions” (Tall, 1991, p. 20). 2.1 The secondary-tertiary transition The problem of the passage from secondary to tertiary mathematics is not new. In the first volume of the UNESCO series, New Trends in Mathematics Teaching (ICMI, 1966), there is a conference report devoted to this issue. More recently, a substantial amount has been written about the various mathematical, social, and cultural difficulties involved in this transition (Clark & Lovric, 2009; Gueudet, 2008; Guzman, M. de, Hodgson, B. R., Robert, A., & Villani, V., 1998; Hourigan & O’Donoghue, 2007; Kajander & Loveric, 2005; Zazkis & Holton, 2005). In particular, it has been noted that in many countries, upon entry to the university, the emphasis changes from a more computational, problem-solving approach to a more proof-based approach to mathematics.

In the U.S., a similar transition occurs when university mathematics students transition from lower-division courses, such as calculus, to upper-division courses, such as real analysis and abstract algebra. At many U.S. universities, there is an attempt to alleviate this transition by offering “bridge” or “transition-to-proof” courses that cover such topics as logic, sets, functions, equivalence relations, mathematical induction, and simple proofs (Moore, 1994). (For more details, see Section 4.1.1) 2.1.1 Looking at the transition from an anthropological point of view

Clark and Lovric (2008, 2009) regard the secondary-tertiary transition as a “modern-day rite of passage”. They have considered the role of the university mathematical community in this transition, the change in the didactic contract with its

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new expectations of student autonomy, the introduction of more abstract concepts, the strong student emotional reactions to these changes and expectations, the long time such transitions take, and the new responsibilities assumed of students and their teachers.

In regard to proof and proving, Clark and Lovric (2008, pp. 763-764) noted that there is a change in the nature of the mathematical objects considered at university, a change in the kinds of reasoning done about those objects, and a passage from informal to formal language – all of which can give students difficulties. In addition, professors often don’t really know what is taught in secondary school and mistakenly assume that high school students are familiar with elements of logic and can work with implications and quantifiers. Students must learn to reason from formal definitions, understand what theorems say, apply theorems correctly, and make connections between concepts. To ease the transition, Lovric (2005) suggesed using slightly older peer tutors who can empathize with entering university students’ experiences and can act as undergraduate teaching assistants, as is done at McMaster University. 2.1.2 The secondary-tertiary transition in various countries

Gueudet (2008) in her survey article discussed many aspects of the secondary-tertiary transition. She noted that studies from many different countries “have shown that only a minority of students are able to build consistent proofs at the end of high school” (Gueudet, 2008, p. 243). At university, in addition to expecting the production of rigorous proofs, professors expect students to use symbols, especially quantifiers, properly and to exhibit autonomy and flexibility of thinking. There are also new expectations about what requires justification and what does not. For example, in French secondary school, when students are asked to show a family of vectors is orthogonal, there is no additional need to show the family forms an independent (or non-collinear) set as this can be taken for granted, whereas in a university linear algebra course a proof is expected (Gueudet, 2008, p. 246). Such expectations, while not often explicitly addressed by professors, constitute a vast change in the didactic contract.

In 1998, elementary number theory was introduced into the French secondary school curriculum to promote students’ mathematical reasoning in an area additional to geometry. Topics include divisibility, Euclid’s algorithm, relatively prime numbers, the existence and uniqueness of prime factorization, least common multiples, Bézout’s identity, and Gauss’ theorem. While investigating the secondary-tertiary transition within number theory, Battie (2009) proposed two complementary, and closely intertwined, epistemological dimensions. The organizing dimension includes selecting the global, logical structure of the proof and associated techniques, such as contradiction, induction, and reduction to a finite number of cases. The operative dimension includes working effectively with specific techniques in the implementation of those global choices. This means performing actions, such as using key theorems and properties and selecting appropriate symbolic representations and algebraic transformations. Battie (2010) found that, due to traditional teaching, French secondary students are given autonomy mainly in the operative dimension, whereas first-year university students are also expected to be responsible for the organizing dimension of proofs and to be fluent in both dimensions, and that this was one source of their difficulties with proving.

According to Praslon (as cited in Gueudet, 2008, pp. 247-248), who akso researched the secondary-tertiary transition in France, proofs at university play a new role. Some results establish methods or provide useful tools, while others serve as

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intermediate results on a path toward an important theorem. In studying tasks related to the derivative in secondary and university textbooks, Praslon observed that secondary school tasks are split into simpler subtasks, the derivative is a tool for studying function behavior rather than being an object of study, and there is no need for students to develop real mathematical autonomy. In contrast, at university, student autonomy and flexibility between mathematical registers (e.g., algebraic, graphical, or natural language) are required.

Similarly in Spain, Bosch, Fonseca, and Gascon (as cited in Gueudet, 2008, pp. 246-247) found that novice university students knew only a single technique for a given task, and after applying it, were unable to interpret the result obtained. They found this corresponded to secondary school textbooks where a single technique is proposed; and in most cases, interpreting the result is not required, and working with mathematical models is rare. The focus in secondary school textbooks is on narrow tasks with rigid solution methods; also topics are poorly connected to one another. However, university teachers often mistakenly assume that entering students come to them with well organized knowledge.

A case study of two typical Irish senior secondary school mathematics classes (Hourigan & O’Donoghue, 2007) found that the (implicit) didactic contract included the following: The terminal exam should be the central aim of the class. The teacher should not depart for the set lesson routine of going over homework quickly, introducing a new topic through a few worked examples, and having students practice similar exercises. The teacher should not ask pupils complicated questions. The teacher should provide pupils a step-by-step breakdown of problem-solving techniques. Pupils should not interrupt the lesson unnecessarily. The teacher should not ask questions that require thought and reflection (pp. 471-472). While the authors concentrated on problem solving, and did not specifically mention proof and proving, they concluded that “mathematics-intensive courses at tertiary level need independent learners possessing [the] conceptual and transferable skills required to solve unfamiliar problems [but] the development of these essential skills is not fostered within the classrooms studied” (p. 473). 2.1.3 Epistemological and cognitive difficulties of the secondary-tertiary transition

In considering some epistemological and cognitive difficulties of the secondary-tertiary transition, Guzman et al.(1998) noted that “existence proofs are notoriously difficult for students” as “it is not easy for them [the students] to recognize their need, as this type of situation is rarely raised in secondary mathematics”. When given a problem, high school students can (almost) always take it for granted that it has a solution. Proofs of sufficiency are also difficult. “Sometimes, a proof requires not only to apply directly a theorem in a particular case, but also to adapt or even to transform a theorem before recognizing and/or using it.” Tertiary students need to learn to distinguish between mathematical knowledge and meta-mathematical knowledge of correctness, relevance, and elegance of a piece of mathematics. They need to take responsibility for their own mathematical learning and seek to establish links between concepts. 2.2 Concepts learned at secondary school may no longer apply 2.2.1 The influence of secondary school geometry There are concepts learned at secondary school that are not easily generalizable to tertiary mathematics. For example, in secondary school geometry, the tangent is that unique line that touches the circle at exactly one point and is perpendicular to the radius

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at that point. However, upon coming to calculus, the tangent to a function at a point is defined as the limit of approximating secant lines, and somewhat later, as the line whose slope is given by the value of the derivative at that point. Research in France by Castela (as reported in Artigue, 1992, pp. 209-210) found that many of the 372 secondary students questioned, after having studied analysis (i.e., calculus) for a year, had great difficulty determining from a graph whether a given line was tangent to a particular function at an inflection point or a cusp. This new way of viewing tangents often requires a difficult reconstruction of mathematical knowledge. Further, Biza, Christou, & Zachariades (2008) noted that in Euclidean geometry the characteristics of tangency are global (e.g., the tangent touches at one pt, the curve lies entirely in one half plane), whereas in analysis the characteristics are local (e.g., a linear approximation to the curve at a point and the derivative gives the slope of the curve at this point). Consequently, students need to reconstruct their concept images as they transition from geometry to analytic geometry to analysis. Secondary teachers of geometry, especially in the U.S. where two-column proofs are customary, adhere to two norms “(a) that students are expected not to make assumptions while proving and (b) every statement must be followed immediately by a reason”. The latter “refers to the structure of the proof and the timing of writing each part of the proof. It is not sufficient to write a reason; it should be written immediately after the related statement has been written” (Nachlieli & Herbst, 2009, p. 433). Since these norms are foreign to mathematicians, who often make assumptions to see where they might lead, it would be interesting to find out whether students carry over such norms to their first proof courses at university and whether this has unfortunate consequences. 2.2.2 The influence of secondary school algebra Another example is provided by the treatment of equality in analysis. Whereas in secondary school algebra and trigonometry, students have grown used to proving that two expressions are equal by transforming one into the other using known equivalences, in analysis one can prove two numbers a and b are equal by showing that for every 0ε > , one has | |a b ε− < . That this idea is not an easy one is indicated by a French study in which more than 40% of entering university students thought that if two numbers a and b are less than 1/N for every positive integer N, then they are not equal, only infinitely close (Artigue, 1999, p. 1379). 2.3 Transition from undergraduate to graduate study in mathematics

There are also transitions from undergraduate to Masters’ and Ph.D. level study in mathematics (Duffin & Simpson, 2002, 2006; Geraniou, 2010; Herzig, 2002). 2.3.1 The transition to a research degree in the U.K.

Geraniou (2010) noted that, in the U.K., “the transition from a taught degree to a research degree involves significant changes in the way students deal with the subject.” The importance of persistence, interest and confidence, problem-solving skills, as well as the role of the thesis/dissertation supervisor are often cited as crucial issues for the successful transition to graduate studies. Geraniou (2010) described Ph.D. students as going through three motivational transition stages. The first stage occurs in the first year; it is the adjustment stage during which students come to terms with what a pure research Ph.D. degree is and develop survival strategies. The second stage occurs in the first half of the second year and continues until a student’s research it complete; it is the expertise stage during which time the student must research a problem with an unknown solution,

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for which that student initially may have no idea of a direction in which to proceed. In addition, many students still lack appropriate problem-solving skills due to having depended on memorization to succeed at the undergraduate level. The third and final stage is the articulation stage during which time the Ph.D. dissertation is actually written in final form.

In regard to U.K. graduate students, Duffin and Simpson (2006) noted that students come from lecture classes of 100 or more where syllabuses are clearly defined, the pace of new material is set, and the majority of the problems to solve have predetermined solutions, whereas

there is no formal requirement to attend or pass taught courses during the period of doctoral study and almost all of a student’s three years of study is spent in independent research towards the production of a single, substantial dissertation … [which is] required to be the result of original research, to show an awareness of the relationship of the research to a wider field of knowledge and, potentially, to be publishable. (p. 236) In a small study, Duffin and Simpson (2002, 2006) uncovered three distinct

learning styles of U.K. Ph.D. students. There were natural learners whose tendency for independent thinking, rather than memorization, no longer proved a hindrance and who found “the movement to graduate study seemed quite smooth”. There were alien learners who had succeeded in the past by reproducing proofs and who had previously accepted new ideas without being “overly concerned” about their meanings. Their emphasis on procedural techniques no longer sufficed, and consequently, they found the transition to graduate study far from smooth. Lastly, there was one student with a flexible learning style who adjusted his learning style according to what he wished to gain from the material. 2.3.2 The transition to graduate study in the U.S.

In the U.S. doctoral students spend the first three years taking courses and passing a set of qualifying/comprehensive examinations. Only after that, do they begin their research. Even so, many entering graduate students feel unprepared. Herzig (2002) found in one large Ph.D-granting university that students were discouraged from filling in gaps by taking Masters’ level courses, felt they could not ask questions of their professors, and wanted more feedback in their courses. “Rather than the coursework building on students’ enthusiasm for mathematics and involving them in authentic mathematical work, coursework distanced students from mathematics making it more difficult for them to learn what they needed to learn to participate effectively in mathematical practice” (Herzog, 2002, p. 192). Even after students had “proved themselves” by passing their qualifying/ comprehensive examinations, they felt they wanted more interaction with professors.

To succeed in doctoral study, U.S. students first need to pass content courses that require students to construct original proofs, and later for a dissertation, they need to conjecture and prove interesting, new results. Seeing the limited ability of many entering U.S. graduate students to construct proofs as a major stumbling block to their success in their doctoral content courses, Selden and Selden (2009b) designed a one-semester course to alleviate this problem. The course consists of having students construct a variety of different kinds of proofs, present them at the blackboard, and receive extensive criticism and advice. The Seldens see proving as consisting of a sequence of mental and physical

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actions and have divided proofs into their formal-rhetorical parts and their problem-centered parts. One of their aims is to have students automate the construction of the formal-rhetorical parts so that scarce working memory resources can be devoted to the problem-centered parts of proofs (Selden, McKee, & Selden, 2010). The formal-rhetorical part of a proof is similar to what Battie (2009) has called the organizing dimension, and the problem-centered part is similar to what she has called the operative dimension. (See Section 2.1.2 above.)

3. How tertiary students deal with various aspects of proof and proving

Both learning to understand and construct proofs and teaching tertiary students about proof and proving are daunting tasks -- ones that are not easily approached by lecture alone. The following sections provide information on tertiary students’ documented proving difficulties, along with some suggestions from the literature on helping them to avoid or overcome them. 3.1 Logical reasoning It is important that students know the difference between the pragmatic everyday use of logical terms and their mathematical use. For example, in everyday language “some” means “some, but not all”, as it would be impolite to say less than one knew, an example of Gricean implicature. Thus, in everyday speech, one would not deduce from “all X have property Y”, that “some (but possibly not all) X have property Y.” Some university students find it challenging to keep the everyday and the mathematics registers separate (Lee & Smith, 2009). Also, students sometimes confuse “limit” with its everyday meaning of a bound as in “speed limit” (Cornu, 1991). Even students for whom this is not a problem, can have other difficulties with mathematical language. 3.1.1 Logical connectives

Discerning the logical structure of informally stated theorems, that is, theorems that omit specific mention of some variables or depart from the use of for all, there exists, and, or, not, if-then, and if-and-only-if in a significant way can be difficult for beginning university students. For example, the statement, Differentiable functions are continuous, is informal because a universal quantifier and the associated variable are understood by convention, but not explicitly indicated. Similarly, A function is continuous whenever it is differentiable is informal because it departs from the familiar if-then expression of the conditional, as well as by not explicitly specifying the universal quantifier and variable.

Knowing the logical structure of a statement can help one recognize how one might begin and end a direct proof of it. When asked to unpack the logical structure of informally worded statements, but not to prove them, U.S. undergraduate mathematics students, many in their third or fourth year, did so correctly just 8.5% of the time. Especially difficult for them was the correct interpretation of the order of the existential and universal quantifiers in statements such as: For a b< , there is a c so that ( )f c y= whenever ( )f a < y

and 3( )y f b< (Selden & Selden, 1995). Rather than presenting students with truth tables or having them formally

construct valid arguments, Epp (2003) suggested stressing the difference between

3 If f were continuous, this would be the Intermediate Value Theorem as stated in most beginning calculus textbooks.

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everyday and mathematical language and emphasizing “exercises where students apply logical principles to a mix of carefully chosen natural language and mathematical statements” because “simply memorizing abstract, logical formulas and learning to apply them mechanically has little impact on students’ broader reasoning powers”. 3.1.2 Quantifiers Students have difficulties understanding the importance of, and the implications of, the order of existential and universal quantifiers, as well as knowing their often implied scope (Dubinsky & Yiparaki, 2000; Epp, 2009). This is especially true when evaluating or constructing Nε − orε δ− proofs in real analysis (Roh, 2009).

Undergraduate students often consider the effect of an interchange of existential and universal quantifiers as a mere rewording. For example, when given the two statements:

• For every positive number there exists a positive number such that b . a b a<• There exists a positive number b such that for every positive number . a b a<

44% of undergraduates and 33% of graduate students in the U.S. said they were "the same" or were merely reworded (Dubinsky & Yiparaki, 2000).

Roh (2009) gave several groups of introductory real analysis students in the U.S. two arguments, one purporting to show that the sequence {1/n} converges to 0; the other purporting to show that it did not converge to 0. Both were fallacious because ε was chosen to depend on the positive integer N. However, because the conclusion to the first argument was correct, the students immediately accepted it. Also, they were convinced the second argument was flawed because the conclusion was false and tried to determine why. They first had trouble negating the definition of sequence convergence. They then puzzled over pieces of the definition and its negation wondering whether to consider

or | 0n N> |na ε− < and whether one had to show the inequality for all or just some n. None noticed the dependency of

n N>ε on N, that is, that the order of the universal

and existential quantifiers had been reversed. In general, students have difficulty with proofs of universally quantified

statements, in which one customarily considers an arbitrary, but fixed element, as inε δ− analysis proofs. They often wonder why the result has been proved for all elements, such as ε , rather than only for the specific arbitrary, but fixed element selected. Selden, McKee, and Selden (2010) gave the example of Mary, a teacher returning for graduate study in mathematics, who did not quite comprehend the underlying logic of such proofs for at least half a semester in a real analysis course and only later became comfortable with such proofs. 3.1.3 Proof by contradiction and its link with negation It is well know that students sometimes find proofs by contradiction difficult to understand and construct (Epp, 2003; Reid & Dobbin, 1998). One reason may be the difficulties students have with the formulation of negations in mathematics. Antonini (2001) found students have three ways, or schemes, for dealing with negation, often taken over from natural language. The first scheme was that of the opposite. For example, first-year, and even one fourth-year Italian physics student, said that the negation of “f is increasing” is “f is decreasing” and insisted that is how to begin a proof by contradiction.

The second scheme, which was more prevalent amongst students, is that of considering possibilities. For example, the negation of “g is strictly decreasing” was seen by first-year Italian calculus students as having several possibilities, namely, g can be increasing, constant, or decreasing but not strictly decreasing. The third scheme, that of

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properties, is the one used in mathematical reasoning. For example, to deny that “f is increasing” one says “there exist x and y such that x<y and ( ) ( )f x f y≥ ” which expresses the negation in a positive way using a property common to all non-increasing functions. Doing so allows one to proceed with a proof by contradiction. Antonini (2001) conjectured that one reason for the strength of students’ scheme of possibilities, namely, that in natural language one can often only express the negation of p as “not p”. For example, the statement “I did not travel by train” has no simple positive version. Because of this, Antonini suggested that the tendency to express negation in terms of possibilities may require specific pedagogical attention.

Even if students do not use the scheme of possibilities, there may be additional difficulties in forming a negation when there are mixed universal and existential quantifiers, as seen by Roh’s (2009) sequence convergence example (discussed in Section 3.1.2 above). However, in the case of “f increasing”, the definition only has universal quantifiers, namely, “for all x and y, if x<y , then ( ) ( )f x f y≤ ”. So in forming its negation, one need not be aware of the order of the quantifiers; however, one must know how to negate the implication as p → q p q∧¬ .

Antonini (2003) has suggested one way students might be encouraged to produce a conjecture, and a proof thereof, by contradiction. Namely, given an open-ended question of the form “given A what can you deduce?”, students can be led to make conjectures. After this, they often can go on to use indirect argumentation “… if it were not so, it would happen that …” to generate a non-example that can later be elaborated into a proof by contradiction. 3.2 Students’ views of proof 3.2.1 Non-standard views of proof

Harel and Sowder (1998) investigated undergraduate students’ views of proof. They defined a person’s proof scheme to be that which “constitutes ascertaining and persuading for that person” (p. 244). They categorized proof schemes into three major categories: external conviction proof schemes, empirical proof schemes, and analytical proof schemes. While the first two categories are somewhat idiosyncratic and nonstandard, the latter are like proofs that mathematicians would accept. These three major categories were further divided into subcategories such as a symbolic proof scheme in which symbols take on a life of their own with no reference to underlying meanings, an empirical proof scheme in which a student is convinced by a number of examples, a generic proof scheme in which a student proves the theorem in a specific context, and an axiomatic proof scheme in which a student understands that in principle a mathematical justification should start with undefined terms and axioms. We wonder whether such nonstandard proof schemes really present an obstacle when attempting to teach tertiary students the deductive proof scheme.

In the 10th week of a sophomore-level mathematics course, 101 preservice elementary teachers, were asked to judge verifications of a familiar result, if the sum of the digits of a whole number is divisible by 3, then the number is divisible by 3, and an unfamiliar result, if a divides b and b divides c, then a divides c. For each of these, students were given, in randomized order, inductive arguments based on examples, patterns, and specific large numbers, and deductive arguments -- a general proof, a false "proof," and a particular (or generic) proof. These students, who had met the idea of proof in their high school geometry courses and in their current course, rated these

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arguments on a four-point scale, where 4 indicated they considered the argument to be a mathematical proof and 1 indicated it was not a proof. The results showed that both inductive and deductive arguments were acceptable to the students. Apparently their current course that had given "extensive and explicit instruction about the nature of proof and verification in mathematics" had not achieved its goal. In particular, each of the inductive arguments was rated high (3 or 4) by more than 50% of the students. For both familiar and unfamiliar contexts, 80% gave a high rating (3 or 4) to a least one inductive argument, and over 50% gave a very high rating (4) to at least one inductive argument. Also, while over 60% accepted a correct deductive argument as a valid mathematical proof, 52% also accepted an incorrect deductive argument (Martin & Harel, 1989).

Tertiary students sometimes mistakenly think proofs are constructed from the “top down” because this is the way they have seen them presented in lecture. Selden and Selden (2009a) report a mathematics education graduate student who thought this based on her undergraduate experience in a real analysis course and was very surprised to learn this was not so. Such ideas may also be a result, at least in the U.S, of being required to construct two-column proofs from the “top down” in secondary school geometry. The normative way to construct proofs, as seen by U.S. secondary school geometry teachers, is to always give a reason for each statement in a two-column proof before continuing, that is, immediately after the statement has been made (Weiss, Herbst & Chen, 2009). Further, U.S. secondary school teachers view it as unnatural to let a student make an assumption in the middle of a proof in order to see whether the proof can be completed that way, and then come back later to reconsider the assumption (Nachlieli & Herbst, 2009). Perhaps encouraging a less rigid style of proving at secondary school and university would help alleviate such “top down” views of constructing proofs.

Another suggestion is to have students first write the formal-rhetorical part of a proof, that is, the hypotheses and the conclusion and then unpack and examine the conclusion in order to structure the proof. Doing so “exposes the real problem” to be solved and allows one to concentrate on the problem-centered part of a proof (Selden & Selden, 2009a). For further information, see Section 2.3.2 above. 3.3 Knowing, using, and understanding other constituents of proofs and proving Although proofs at tertiary level are rigorous deductive arguments and reasoning using correct logic is necessary, there is more to be learned in order to become a successful prover. 3.3.1 Understanding formal definitions and using them in proving

Too often tertiary mathematics students, can recite definitions, yet fail to use them when asked to solve problems or prove theorems. Alcock and Simpson (1999) reported on some real analysis students who argued from particular examples that they viewed as prototypical of a class of objects rather than from the definition. In particular, they responded to the task "prove that xn is convergent" not by showing that xn satisfies the definition of sequence convergence, but rather by arguing from particular examples of convergent sequences that they seemed to view as prototypical of the set of all convergent sequences.

Definitions need to become operable for an individual. For example, given a definition of set equality as A = B means A and B have the same elements, a student needs to realize this means that to show A = B one must show that if x is an element of A, then x is an element of B, and vice versa. Bills and Tall (1998) considered a definition to

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be formally operable for a student if that student “is able to use it in creating or (meaningfully) reproducing a formal argument [proof]” (p. 104).

In addition, as Edwards and Ward (2004) pointed out, the distinction between mathematical definitions (also referred to as stipulated or analytic definitions) and many everyday or dictionary definitions (also referred to as descriptive, extracted, or synthetic definitions) needs to be made clear. Students may not be aware of, or may not make, this distinction. One could help them become aware of this distinction by discussing it with them and by engaging them in the act of defining. For example, when using Henderson's (2001) investigational geometry text, one can begin with a definition of triangle initially useful in the Euclidean plane, on the sphere, and on the hyperbolic plane, but eventually students will notice that the usual Side-Angle-Side Theorem (SAS) is not true for all triangles on the sphere. At this point, students can be brought to see the need for, and to participate in developing, a definition of “small triangle” for which SAS remains true on the sphere. Thus, they would come to see that the original Euclidean definition does not carry over from one geometry to another and the need for caution when employing definitions.

However, if one defines parallel lines as lines that do not intersect, this does carry over from Euclidean geometry to Poincaré geometry on the half-plane. The following problem (adapted from Neto, Breda, Costa, & Godinho [sic], 2009) can be used to engender a discussion of the meaning of parallelism: Consider the hyperbolic lines (l, m, n, and k) defined on the Poincaré half-plan: l: 2 2( 7) 1x y 6− + = and y>0; m: and y>0; n: 2 2( 6.5) 6.25x y− + = 2 2( 3)x y 1− + = and y>0; k: 11x = and y>0. Which lines are parallel and which are not? Justify your answer. (See Neto et al., 2009 for details.) Furthermore, according to Lakatos (1961), there can be a dialectical interplay between concept formation, definition construction, and proof. To bring this interplay to students' awareness, one needs to design problem situations whose resolution requires definition construction. Following Lakatos (1961), Ouvrier-Buffet (2004) considered: zero-definitions, that is, tentative or "working definitions" emerging at the start of an investigation; and proof-generated definitions, linked directly to problem situations and attempts at proof. In particular, in order to engage university students in zero-definition construction, Ouvrier-Buffet (2002) selected the mathematical concept of tree (giving it a neutral name), partly because there are many equivalent definitions (she lists seven), and in France formal definitions of tree are not presented in secondary school. She first gave several groups of students four examples and two non-examples of trees, asking them to infer a definition. This was followed by a problem situation, namely, a request to prove: Let G be a connected graph (i.e., a collection of vertices and edges, so that between any two vertices there is a path, that is, a sequence of vertices each joined to the next by an edge). Prove that G contains a spanning tree (i.e., a tree that is a subgraph and has the same vertex set as G), in order to allow students to return to, and possibly reconstruct, their original zero-definitions. Ouvrier-Buffet (2002) reported that attempting the proof sometimes, but not always, resulted in revised definitions. 3.3.2 Understanding the statement of a theorem to be proved At tertiary level, the pace of lecture courses like abstract algebra or real analysis is often very fast. Formal definitions are introduced one after another, followed by theorems

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proved by the professor or the textbook. Subsequent assessments often consist of requests to prove (moderately) original theorems using the newly introduced definitions and theorems. While it may be more effective to ask students to explore situations and generate their own conjectures and proofs, as some mathematics educators have suggested, there is often little time in university lecture courses to do so. Upon being given a statement to prove, a student’s first job is to understand the structure of the statement, as well as its content. For example, if a theorem is stated informally, that is, if the wording departs from a natural language version of predicate calculus, then students must first ascertain which are the hypotheses and which is the conclusion. For example, students need to know that “whenever” is usually a synonym for “if”, and hence, introduces one of the hypotheses. While one can attempt to teach such linguistic variations, it would probably be easier if professors, at least initially, stated theorems using the familiar if-then, and if-and-only-if forms. Having identified the hypotheses and the conclusion, a student’s next job is to unpack the conclusion in order to understand it. This can entail looking up and unpacking the definitions of unfamiliar terms. For example, a real analysis student might have recently encountered the concept of uniform continuity and soon thereafter be asked to prove: Let , and and be uniformly continuous. If f and g are bounded on D, then is uniformly continuous. Carefully unpacking the definition of uniformly continuity would tell the student to begin the proof by supposing

D ⊆ R :f D → R :g D → R:f g D⋅ → R

0ε > and then attempting to find a 0δ > so that the relevant inequality can be made less than ε . It is often the case that, instead of focusing on and unpacking the conclusion, students will focus on the hypotheses and just deduce anything and never arrive at the conclusion (Selden, McKee, & Selden, 2010). Some methods of trying to understand the content of a theorem or question are more successful than others. When Hazzan (1999) observed Israeli computer science students in an abstract algebra course, she found them reducing the level of abstraction to familiar groups such as the real numbers. For example, when asked whether Z3 is a group under multiplication mod 3, one student said “the inverse” of 2 is ½, and since ½ is not an element of Z3, it is not a group. This erroneous interpretation may have come to the student automatically, and hence, resulted in misunderstanding what was being asked. Students need to learn to read very carefully and precisely. 3.3.3 Interpreting and using previous theorems and definitions in proving Undergraduate students sometimes fail to use or interpret relevant theorems correctly or they fail to verify that the conditions of the hypotheses of the theorem are satisfied. The Fundamental Theorem of Arithmetic, guaranteeing a unique prime decomposition of integers, is part of the core mathematics curriculum for preservice elementary teachers, but in practice some of these students appear to deny the uniqueness. Instead of applying this theorem, when asked whether 173 was a square number or whether K = 16,199 = 97 x 167 could have 13 as a divisor, given both 97 and 167 are primes, these students took out their calculators -- in the first instance, to multiply out and extract the square root, and in the second instance, to divide by 13. When asked to determine (and explain) whether 2 23 5 7M = × × was divisible by 2, 5, 7, 9, 11, 15, or 63, a majority (29 of 54) stated that 3, 5, 7 were divisors since those were among the factors

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in the prime decomposition. However, sixteen were unable to apply similar reasoning to 2 and 11, noting instead that “M is an odd number” so “2 can’t go into it” or resorting to calculations (like the above) for 11. In addition, many of these students believed that prime decomposition means decomposition into small primes (Zazkis & Campbell, 1996; Zazkis & Liljedahl, 2004). Undergraduate students often ignore relevant hypotheses or apply the converse when it does not hold. A well-know instance is the use, by Calculus II students, of the converse of: If converges, then na∑ lim 0nn

a→∞

= , as an easy, but incorrect, test for

convergence. Some calculus books go on to point out that this theorem provides a Test for Divergence. But, perhaps it would be better to explicitly state the contrapositive, If

, then diverges. lim 0nna

→∞≠ na∑

Sometimes undergraduate students use theorems, especially theorems with names, as vague "slogans" that can be easily retrieved from memory, especially when they are asked to answer questions to which the theorems seemingly apply. For example, twenty-three abstract algebra students were asked: True or false? Please justify your answer. "In S7 there is no element of order 8." It was expected that students would check whether there was a permutation in S7 having 8 as the least common multiple of the lengths of its cycles. Instead, 12 of the 16 students who gave incorrect answers invoked Lagrange's Theorem4 or its converse. Seven of them incorrectly invoked Lagrange's Theorem to say the statement was false -- there is such an element since 8 divides 5040. Another two students, inappropriately invoked a contrapositive form of Lagrange's Theorem to say the statement was true because 8 doesn't divide 7 (Hazzan & Leron, 1996). The authors go on to point out that students often think Lagrange's Theorem is an existence theorem, although its contrapositive shows that it is a non-existence theorem: If k doesn't divide o(G), then there doesn't exist a subgroup of order k. Perhaps it would be good to state this version explicitly for students. While the above examples refer to students misuse of theorems when asked to solve specific problems, for example, determine whether a number is prime or a series converges, or decide whether a group has an element of order 8, it is not hard to imagine similar difficulties when students attempt to use theorems in constructing their own proofs.

Students sometimes have difficulty with substitution in the statement of a theorem to be used or to be proved. This is more likely when the substitution involves a compound variable, such as x+h in the definition of the derivative as the limit of the difference quotient. Or, as reported by Selden, McKee, and Selden (2010, p. 212), this occurred when Sofia, a beginning mathematics graduate student, was trying to prove from the open set definition that the relative topology on a subset Y of a topological space X satisfies the defining properties of a topology. She needed to show { |U Y U∅∈ ∩ ∈ U }, but did not know how. It turned out this was so because she did not know how to show an object is in a set when the defining variable is compound, for example, U Y∩ . 3.3.4 Knowing and using relevant concepts in proving

4 Lagrange's Theorem states: Let G be a finite group. If H is a subgroup of G, then o(H) divides o(G). Here o(H) stands for the order of the group H, i.e., its cardinality.

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Constructing all but the most straightforward of proofs involves a good deal of persistence and problem solving to put together relevant concepts. In order to use a concept flexibly, it is important to have a rich concept image, that is, a lot of examples, non-examples, facts, properties, relationships, diagrams, images, and visualizations, that one associates with that concept.5

In many upper-level mathematics courses, students are given abstract definitions together with a few examples, after which they are expected to use these definitions reasonably flexibly. To do this, students may need to find additional examples and non-examples and to prove or disprove related conjectures, more or less without guidance. One can think of these activities as helping to build students' concept images. How does one go about building a rich concept image for a newly introduced concept? Do undergraduate students actively try to enhance their concept images, for instance, by actively constructing examples and non-examples? (See Section 3.3.5 below.) In one study, (Dahlberg & Housman, 1997) presented upper-level U.S. undergraduates with the following formal definition. A function is called fine if it has a root (zero) at each integer. They were asked to study the definition, to generate examples and nonexamples, to determine which of several functions were fine, and lastly to determine the truth of four conjectures. Four basic learning strategies were used by the students: example generation, reformulation, decomposition and synthesis, and memorization. Of these, example generation, together with reflection, elicited the most powerful "learning events", that is, instances where the authors thought students made real progress in understanding the newly introduced concept. Those who employed memorization or decomposition and synthesis often misinterpreted the definition.

Thus, it seems that while students are often reluctant, or unable, to generate examples and counterexamples, doing so helps enrich their concept images and enables them to judge the probable truth of conjectures. 3.3.5 Having a repertoire of examples and using them appropriately To disprove a conjecture, it only takes one counterexample, so having a repertoire of examples is useful. In addition, examples are helpful in “making sense” of the statements in a proof and in understanding newly defined concepts. Whitely (2009), a mathematician, claims he understands a theorem better if he has both a proof and a few key models. Further, examples and counterexamples can be used to probe a theorem or a conjecture to see which assumptions are necessary. In addition, students who can generate examples exhibit the greatest understanding when encountering a newly defined concept and in evaluating conjectures (Dahlberg & Housman, 1997).

But generating examples is not easy for students. When Watson and Mason (2002) asked postgraduate students for an example of a continuous function which is not everywhere differentiable, they all responded | |x . When asked for a second example, most suggested a translate of | |x . When asked for a third example, they commented that a vast number of examples could be generated from | |x , but had trouble coming up with different examples.

Weber, Alcock, and Radu (2005) examined transition-to-proof course students’ use of examples in proving. While some never used examples, those who did, genuinely 5 The idea of concept image was introduced by Vinner and Hershkowitz (1980), elaborated by Tall and Vinner (1981), and is now a much used notion in mathematics education research.

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tried to use them to understand a statement, to evaluate whether an assertion was correct, and to construct a counterexample. However, those attempts were largely ineffective; only one in nine attempts produced a valid or mostly valid proof. Frequently, the students in this study selected inappropriate examples and did not check to see whether the examples satisfied the relevant definitions or the conditions of the statement they were trying to prove.

In coming to terms with a newly defined concept, it is useful to have both examples and non-examples to clarify the difference. “The ideal examples to use in teaching are those that are only just examples, and the ideal non-examples are those that are very nearly examples” (Askey & Wiliam, 1995, p. iii). 3.3.6 Selecting helpful representations when proving Another aspect of understanding and using a concept is knowing which symbolic representations are likely to be appropriate in certain situations. Concepts can have several (easily manipulated) symbolic representations or none at all. For example, prime numbers have no such representation; they are sometimes defined as those positive integers having exactly two factors or being divisible only by 1 and themselves. It has been argued that the lack of an (easily manipulated) symbolic representation makes understanding prime numbers especially difficult, in particular, for preservice teachers (Zazkis & Liljedahl, 2004). Similarly, irrational numbers have no such representation; thus, in proving results, such as 2 is irrational or the sum of a rational and an irrational is irrational, one is led to consider proofs by contradiction -- something that is often difficult for beginning university students. (See Section 3.1.3 above.) Some symbolic representations can make certain features transparent and others opaque.6 For example, if one wants to prove a multiplicative property of complex numbers, it is often better to use the representation ire θ , rather than x iy+ , and if one wants to prove certain results in linear algebra, it may be better to use linear transformations, T, rather than matrices. Students often lack the experience to know when a given representation is likely to be useful. Moving flexibly between representations (e.g., of functions given symbolically or as a graph) is an indication of the richness of one's understanding of a concept (Even, 1998). However, conversion between representations can be cognitively complex and not symmetric, that is, it can be easier to go from an algebraic to a graphical representation than vice versa, and in linear algebra, such conversions are particularly difficult (Duval, 2006). 3.3.7 Knowing, but not using, factual knowledge In two companion studies, first-year U.S. university calculus students and U.S. second-year differential equations students, were asked to solve five moderately non-routine first calculus problems, that is, problems somewhat, but not very, different from what they had been taught. Immediately afterwards, they took a short routine test, covering the resources needed to solve the non-routine problems. Two-thirds of the calculus students failed to solve a single problem completely and more than 40% did not make substantial progress on a single problem. Also, more than half of the differential equations students were unable to solve even one problem and more than a third made no 6 Representations can be transparent or opaque with respect to certain features. For example, representing 784 as 282 makes the property of being a perfect square transparent and the property of being divisible by 98 opaque. For more details, see Zazkis and Liljedahl (2004, pp. 165-166).

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substantial progress toward a solution. On the problems for which the students had full factual knowledge, just 18% of the calculus students' solutions and 24% of the differential equations students' solutions were completely correct. (Selden, Selden, Hauk, & Mason, 2000; Selden, Selden, & Mason, 1994). The authors conjectured that, in studying and doing homework, the students had mainly followed worked examples from their textbooks and had thus never needed to consider various different ways to attempt problems. Thus, they had no experience at bringing their assorted resources to mind. How does one think of bringing the appropriate knowledge to bear at the right time? To date, mathematics education research has had only a little to say about the difficult question of how an idea, formula, definition, or theorem comes to mind when it would be particularly helpful, and probably there are several ways. In their study of problem solving, Carlson and Bloom (2005) found that mathematicians frequently did not access the most useful information at the right time, suggesting how difficult it is to draw from even a vast reservoir of facts, concepts, and heuristics when attempting to solve a problem or to prove a theorem. Instead, the authors found that mathematicians’ progress was dependent on their approach, that is, on such things as their ability to persist in making and testing various conjectures. It seems very likely that similar phenomena could occur in attempting to prove theorems. 3.3.8 Knowing which theorems are important and useful

Seeing the relevance and usefulness of one's knowledge and bringing it to bear on a problem, or a proof, is not easy. Schoenfeld (1985, pp. 36-42) provides an example of two beginning university students who had completed a year of high school geometry and were asked: You are given two intersecting straight lines and a point P marked on one of them. Show how to construct, using straightedge and compass, a circle that is tangent to both lines and that has the point P as its point of tangency to one of the lines. They made rough sketches and conjectures, and tested their conjectures by making constructions. When asked why their constructions ought, or ought not, to work, they responded in terms of the mechanics of construction, but did not provide any mathematical justification. Yet the next day they were able to give the proof of two relevant geometric theorems. Apparently, these students simply did not see the relevance of these theorems at the time. In another study, four undergraduates who had completed a first abstract algebra course and four doctoral students working on algebraic topics were observed as they attempted to prove or disprove whether specific pairs of groups, such as Zn and Sn,

Q and Z, and Zp x Zq and Zpq (where p and q are coprime), are isomorphic. The

undergraduates first looked to see if the groups had the same cardinality; after which they attempted unsuccessfully to construct an isomorphism between the groups. They rarely considered properties preserved under isomorphism, despite knowing them, as was subsequently ascertained. For example, they all knew Z is cyclic, Q is not, and a cyclic

group cannot be isomorphic to a non-cyclic group, but they did not use these facts at the time. These facts did not seem to come to mind spontaneously, or in reaction to this kind of question. In contrast, the doctoral students rarely considered the definition of isomorphic groups. Instead, they examined properties preserved under isomorphism (Weber & Alcock, 2004).

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In a somewhat similar study of undergraduates who had just completed abstract algebra and doctoral students, who were writing dissertations on algebraic topics, it was found that the doctoral students had knowledge of which theorems were important when considering homomorphisms. For example, in considering the proposition: Let G and H be groups. G has order pq (where p and q are prime). f is a surjective homomorphism from G to H. Show that G is isomorphic to H or H is abelian, all four doctoral students recalled the First Isomorphism Theorem within 90 seconds. In contrast, the undergraduates did not invoke the theorem. When the doctoral students were asked why they used such sophisticated techniques, a typical response was, "Because this is such a fundamental and crucial fact that it's one of the first things you turn to” (Weber, 2001). In yet another study, four undergraduates, who had recently completed their second course in abstract algebra, and four mathematics professors, who regularly used group-theoretic concepts in their research, were interviewed about isomorphism and proof. They were asked for the ways they think about and represent groups, for the formal definition and intuitive descriptions of isomorphism, and about how to prove or disprove two groups are isomorphic. The algebraists thought about groups in terms of group multiplication tables and also in terms of generators and relations, as well as having representations that applied only to specific groups, such as matrix groups. To decide whether two groups are isomorphic, they would consider properties preserved by isomorphism. In contrast, none of the undergraduates could provide a single intuitive description of a group; for them, it was a structure that satisfies a list of axioms. None could provide an intuitive description. To decide whether two groups are isomorphic, they would first compare the order (i.e., the cardinality) of the groups. If the groups were of the same order, they would look for bijective maps between them and check whether these maps were isomorphisms (Weber & Alcock, 2004). It may be that undergraduates mainly study completed proofs and focus on their details, rather than noticing the importance of certain results and how they fit together. That is, they may not come to see some theorems as particularly important or useful. The mathematics education research literature contains few specific teaching suggestions on how to help students come to know which theorems are likely to be important in various situations. However, it might be helpful to discuss with students: a) which theorems and properties you (the teacher) think are important and why, b) your own intuitive, or informal ideas, regarding concepts, and c) the advantages and disadvantages of various representations. Knowing the most effective way to proceed in proving a theorem, whether syntactically or semantically, is important. Weber & Alcock (2004) consider a semantic reasoning strategy as one in which the prover uses examples to gain insight and translates that into an argument based on appropriate definitions and theorems. They consider a syntactic reasoning strategy as one in which the prover uses the statement of the theorem to structure a proof and draws logical inferences from associated definitions and theorems.

Alcock (2009c) described the contrasting perspectives of two professors who teach transition-to-proof courses in the U.S. Both found their students unprepared for the reasoning expected in their courses, but had different responses. One used a semantic approach and emphasized meaning and the generation of examples. The other used a

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syntactic approach and emphasized precision in notation and structural thinking. Yet both were preparing students for later proof-based courses.

Iannone (2009) explored U.K. mathematicians’ views on this. They said that there are proofs about concepts without an initial pictorial representation for which resorting to syntactic knowledge is an effective tool or the only suitable approach, such as showing a sequence does, or does not, converge. There are proofs for which syntactic knowledge can be used, but only ineffectively, or cannot be used at all due to the complexity of the syntactic representation, such as: For an n n× matrix A with det( ) 0A ≠ , prove that

. The mathematicians in this study considered knowing how to tackle a proof, whether syntactically or semantically, a meta-mathematical ability that students need to acquire. How they are to acquire it was apparently not discussed and remains an open question.

( 2)adj(adj( )) (det( )) nA A −= A

3.3.9 Knowing How to Read and Check Proofs An integral part of the proving process is the ability to tell whether one’s argument is correct and proves the theorem it was intended to prove. Selden and Selden (2003) conducted a study of how undergraduates from the beginning of a transition-to-proof course validated, that is, evaluated and judged the correctness of, four student-generated "proofs" of a very elementary number theory theorem: For any positive integer

, if is a multiple of 3 , then n is a multiple of 3 . The students made judgments regarding the correctness of each purported proof four times. The first time, these judgments were just 46% correct, whereas the last time they were 81% were correct. The authors attributed the difference to students having reflected on the purported proofs several times. Most of the proving errors detected were of a local/detailed nature rather than a global/structural nature. When asked how they read proofs, the students said they attempted careful line-by-line checks to see whether each mathematical assertion followed from previous statements, checked to make sure the steps were logical, and looked to see whether any computations were left out. But what students say seems to be a poor indicator of whether they can actually validate proofs with reasonable reliability. On the other hand, even without explicit instruction, reflection and reconsideration yielded more correct judgments, suggesting that explicit instruction in validation could be effective. Indeed, several transition-to-proof textbooks include purported proofs to validate, but it would probably also be helpful to have students validate actual student-generated proofs.

n 2n

In another study, sixteen secondary school mathematics teachers, with from 3 to 20 years' teaching experience, some with master's degrees, who were participating in a professional development program, were interviewed on their conceptions of proof and its place in secondary school mathematics (Knuth, 2000). All professed the view that a proof establishes the truth of a conclusion, but several also thought it might be possible to find some contradictory evidence to refute a proof. In the context of secondary school, the teachers distinguished formal proofs, less formal proofs, and informal proofs. For some, two-column geometry proofs were the epitome of formal proofs. All considered proof as appropriate only for those students in advanced mathematics classes and those intending to pursue mathematics-related majors at university.

The teachers were given five sets of statements with 3 to 5 arguments purporting to justify them; in all, there were 13 arguments that were proofs and 8 that were not. In general, the teachers were successful in recognizing proofs, with 93% of the proofs rated

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as such. However, a third of the nonproofs were also rated as proofs. In fact, every teacher rated at least one of the eight nonproofs as a proof, and ten teachers considered an

argument demonstrating the converse of the statement, If , then 0x > 1 2xx

+ ≥ , to be a

proof of it. These teachers seemed to focus on the correctness of the algebraic manipulations, rather than on the validity of the argument. 4. Teaching proof and proving at the tertiary level There are at least two different classroom contexts in which tertiary students learn about proof and proving: a) small classes, where in addition to, or instead of, listening to lectures, students may be given the opportunity to present their proofs at the blackboard and receive critiques; or where students may be encouraged to work in small groups with the teacher acting as a resource/coach; or where teachers and students jointly participate in a community of practice to develop the mathematics; and b) large lecture classes of 40 to 100 (or more) where such individual attention is not possible and other suggestions and resources might be of help. 4.1 Courses that teach proving 4.1.1 Transition-to-proof courses

Perhaps because U.S. undergraduate mathematics majors spend the first two years in computationally taught courses like calculus, beginning differential equations, and elementary linear algebra before going on to proof-based courses, such as real analysis and abstract algebra, many U.S. universities have instituted transition-to-proof or bridge courses (Moore, 1994), while others use linear algebra, number theory (Smith, 2006) or a discrete mathematics course (Epp, 2004) for this purpose. More than 25 transition-to-proof course textbooks have been written by mathematicians, for example, Velleman’s (1994) How to Prove it: A Structured Approach and Fendel and Resek’s (1990) Foundations of HIGHER Mathematics: Exploration And Proof. While the content of such courses and books may vary, they often begin with a decontextualized treatment of logic emphasizing truth tables and valid arguments, followed by a discussion of direct and indirect proofs, sets, equivalence relations, functions, and mathematical induction. After that, the books vary, selecting specific mathematical areas, such as graph theory or number theory, in which students are to practice proving.

These courses are often relatively small, having perhaps 15-40 students and the teaching methods can vary widely. Small group learning in class, student presentations at the blackboard, and lecture, or a mix of these are used. However, not many studies of the effectiveness of such courses have been conducted. Marty (1991) examined the later success of every student in his introductory proof course (i.e., bridge course) over a ten-year period. He compared students who received his instruction with students taught in a traditional lecture format. He found his students two to three times more likely to pass their subsequent courses in real analysis and four times as likely to continue their studies of advanced mathematics.

A number of studies, other than for effectiveness, have been conducted on transition-to-proof courses. For example, Baker and Campbell (2004) identified problems transition-to-proof course students experience in proof writing. Weber, Alcock, and Radu, (2005) examined how transition-to-proof course students use examples. (See Section 3.3.5 above.)

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4.1.2 Proving in a community of practice In a second university course in plane geometry for Columbian preservice secondary teachers, Perry, Samper, Camargo, Echeverry, and Molina (2008) developed a methodology based in part on a reconstructive approach (Human & Nel, 1984) that itself harks back to Freudenthal’s (1973) ideas. The content is not from a textbook nor is it presented by the teacher. Instead, the teacher develops a community of practice in which students actively participate in the construction and development of a reduced Euclidean axiom system involving points, lines, planes, angles, triangles, and quadrilaterals. Teacher and students jointly define geometric objects, empirically explore problems, formulate and verify conjectures, and write deductive arguments. In effect, students participate in a kind of local axiomatization, and as a consequence, know which basic elements they can use to justify statements. The teacher orchestrates the process through a carefully developed set of questions and tasks (Perry, Camargo, Samper, Molina, & Echeverry, 2009). What makes this development possible are: the carefully selected tasks, the use of a dynamic geometry program (Cabri) for developing conjectures, and the teacher’s role as expert, along with careful management of the class. The approach provides preservice teachers with experiences that they can use with their future pupils 4.1.3 Moore Method courses

This distinctive method of teaching at tertiary level developed out of the teaching experiences of a single accomplished U.S. mathematician, R. L. Moore, and has been continued by his students and their mathematical descendants (Mahavier, 1999; Coppin, Mahavier, May, & Parker, 2009). It has been remarkably successful in Ph.D. level courses, but has also been used at the undergraduate level in small clases of 15 to 30. In many versions, students are provided with a set of notes containing definitions and statements of theorems or conjectures and asked to prove them or provide counterexamples without significant help from anyone (except perhaps the teacher). The teacher provides the structuring of the material and critiques the students' efforts. Students just begin. This sounds a little like throwing someone into a lake in order that the person learn to swim. However, once a student proves the first small theorem (stays afloat mathematically), it is often possible for that student to make very rapid progress. Such courses have been researched very little (Smith, 2006), and could provide interesting opportunities for research in mathematics education. 4.1.4 Co-construction of proofs This method has been implemented in a small (at most 10 students) supplementary voluntary proving class for undergraduate real analysis (McKee, Savic, Selden, & Selden, 2010). It was intended for students who felt unsure how to proceed in constructing proofs. A theorem entirely new to them, but similar to a theorem they were to prove for homework was written on the board by the supplement teacher after which the students themselves, or a supplement teacher if need be, offered suggestions about what to do next. For each suggested action, such as writing an appropriate definition, drawing a sketch, or introducing cases, one student was asked to carry out the action at the blackboard. This process was introduced with the intention that the students would reflect on what had occurred and later perform this or a similar action autonomously on their assigned homework. All students were encouraged to participate in co-constructing the proof.

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For example, if the students were to prove a sequence 1{ }n na ∞= converges to A ,

they would begin by writing the hypotheses, leave a space, and write the conclusion. After unpacking the conclusion, they would write “Let 0ε > ” immediately after the hypotheses, leave a space for the determination of , write “Let n ,” leave another space, and finally write “Then |

N N≥|na A ε− < ” prior to the conclusion at the bottom of their

proof. This brought them to the problem-centered part of the proof (Selden & Selden, 2009a), where some “exploration” or “brainstorming” on the side board would ensue. The entire co-construction process, and accompanying discussions, was a slow one – so slow that only one theorem was proved and discussed in detail in each 75-minute supplement class period. However, students reported that they enjoyed this method of learning and the real analysis teacher reported seeing improvement is students’ proof writing.

4.1.5 The method of scientific debate Daniel Alibert and colleagues at the University of Grenoble designed the method

of scientific debate in which first-year university students are encouraged to become part of a classroom mathematical community that constructs its own concepts, understands the need for proof, and senses a need for mathematics. The class consisted of about 100 students and was conducted as follows. First, the teacher got students to make conjectures which were written on the blackboard without an immediate evaluation. Then the students discussed these, supporting their views by various arguments – a proof, a refutation, or a counterexample. The conjectures that were proved became theorems. There rest became “false-statements” with a corresponding counterexample. A typical debate might start with the following: Let I be an interval on the reals, a be a fixed

element of I, and x be an element of I. For f integrable over I, set . The

teacher might then ask: Can you make some conjectures of the form: If f …, then F …? Students would generate conjectures such as: If f is increasing, then F is increasing too, and then debate whether this is true or not. It is not. (Alibert & Thomas, 1991).

( ) ( )x

aF x f t dt= ∫

For the method of scientific debate to be successful, renegotiation of the didactic contract (Brousseau, 1997) is necessary so that students come to understand and accept their responsibilities. Also, the teacher needs to refrain from revealing her/his opinions, allow time for students to develop their arguments, and encourage maximum student participation. 4.2 Alternative ways of presenting proofs While a traditional definition-theorem-proof style of lecture presentation may convey the content in the most efficient way, there are other ways of presenting proofs that may enable students to gain more insight. 4.2.1 Generic proofs For certain theorems, a teacher can go through a (suitable) proof using a generic example (particular case) that is neither too trivial nor too complicated. Gauss' proof that the sum of the first n integers is n(n+1)/2, done for n = 100 is one such generic proof. Done with care, going over a generic proof interactively with students can enable them to "see" for themselves the general argument embedded in the particular case. If the theorem involves a property about primes, 13 and 19 are often suitable, provided the proof is constructive and that prime (e.g., 13) can be "tracked" through the stages of the argument. A generic proof, but not the standard one, can be given for Wilson's Theorem: For all

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primes p, . That argument for p = 13 involves pairing each integer from 2 to 11 with its (distinct) multiplicative inverse mod 13, noting the product of each pair is congruent to 1(mod 13), and concluding that 12

( 1)! 1(modp p− ≡ − )p

! 1 1 12(mod13)≡ × × .7 There is one caveat; there is some danger that students will not understand the generic character of the proof. In an attempt to avoid this, one can subsequently have them write out the general proof (Rowland, 2002). Generic proofs have also been called transparent pseudo-proofs or TPP. If a professor wishes to construct a TTP for use in lecture, that professor needs to a) select an appropriate formal proof which the TPP is to reflect; b) choose an appropriate value that is neither too small nor too large for the particular case; c) make sure nothing specific to the particular case enters into the proof; d) consider the “level of transparency” to suit the mathematical background of the target audience; and e) consider the “level of generality” to suit the instructor's goals (Malek & Movshovitz-Hadar, 2009, 2010). In addition, Rowland (2002) suggested that one may need to emphasize aspects of the argument that are invariant with regard to the particular case so that students can perceive the invariant parts of the argument. One advantage of generic proofs is that “they enable students to engage with the main ideas of the complete proof in an intuitive and familiar context, temporarily suspending the formidable issues of full generality, formalism, and symbolism”, making them more accessible to students (Leron & Zaslavsky, 2009). 4.2.2 Structuring Mathematical Proofs Instead of presenting a generic proof and letting students write out the general proof, one might try presenting a proof differently. Proofs, as normally presented in lecture or advanced textbooks, tend to be presented in a step-by-step linear fashion, which is well suited for checking the proof’s validity, but is not as good for communicating the main ideas of the proof. Leron (1983) has suggested that one might arrange, and present, a proof in levels, proceeding from the top down, where the top level gives in general, but precise, terms the main ideas of the proof. The second level elaborates on the top level ideas, supplying proofs for as yet unsubstantiated statements and more details, including the construction of objects whose existence had merely been asserted before. If the second level is complicated, one may give only a top-level description there, and push the details down to lower levels. The top level is normally short and free of technical details, while the bottom level is quite detailed, resembling a standard linear proof. Leron (1983) gives three sample structured proofs: one from number theory, one about the algebra of limits in calculus, and one about an orthonormal basis in a vector space. Structured proofs take longer to present, but are easier for students to comprehend. There are several ways an instructor might use structured proofs: a) to present the higher levels of the proof and let the students complete the lower levels; b) to have students take a standard textbook proof and find its structure; c) to give students two similar theorems and have them determine to what level the similarity extends, counting from the top down. 4.2.3 Stimulating-response presentations

7 For details of this and some other number-theoretic generic proofs, along with a description of how they were used with Cambridge University undergraduates, see Rowland (2002).

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Movshovitz-Hadar (1988) has suggested that incorporating surprises into one’s proof presentations will be more motivating for students and create more positive attitudes towards mathematics. She gives an example using Sundaram’s Sieve: Consider the following infinite table:

4 7 10 13 16 19 22 … 7 12 17 22 27 32 37 … 10 17 24 31 38 45 52 … 13 22 31 40 49 58 67 … 16 27 38 49 60 71 82 …

It has the property that: If N occurs in the table, then 2N+1 is not a prime number; if N does not occur in the table, then 2N+1 is a prime number. This is a surprising result because it relates arithmetic progressions (an additive property) to primes (a multiplicative property); the former display regularity whereas the latter do not. The proof proceeds by various easily understood reformulations of the table until one obtains a table whose first row and first column are the consecutive odd integers, which contain all primes except 2, and the remaining rows and columns consist of the above table, which contains only odd composite integers and no primes, so the result is proved. (For details, see Movshovitz-Hadar, 1988, pp. 14-15.) 4.3 When students construct proofs of their own 4.3.1 Make clear what is expected of students

Sometimes students do not know what they should produce when asked to “Explain”, “Demonstrate”, “Show, “Justify” or “Prove”. This may be because teachers and textbooks do not make this clear, and themselves use different modes of argumentation (visual, intuitive, etc.) at different times, leaving students confused about which behavior to imitate. However, sometimes it is because students don’t have facility with mathematical language. For example, a first-year linear algebra student was asked to determine whether the following statement was true or false and explain. If {v1, v2, v3, v4} is a linearly independent set, then {v1, v2, v3} is also a linearly independent set. The student’s answer was “True because taking down a vector does not help linear dependence.” It is possible the student understood, but using “taking down” and “help” points to a lack of linguistic capability. A teacher is left to speculate on the student’s understanding (Dreyfus, 1999, p. 88). On the other hand, sometimes students give a step-by-step account, somewhat like a travelogue, of how the problem was solved or the proof generated, which is not what most teachers want. Students also give redundant explanations.

Expectations for what is to be produced needs to be specified by the teacher or emerge from ongoing classroom practices. What counts as an explanation or a proof is a sociomathematical norm (Yackel & Cobb, 1996; Yackel, Rasmussen, & King, 2000). The distinction between previously accepted explanations and argumentation and the kind of proof expected at tertiary level, is that proof is more formal and concise and is based on deductive reasoning. Still it would provide clarity for students if teachers provided samples of what is expected when given the directions “Explain”, “Demonstrate”, “Show”, “Justify”, or “Prove”. It would also provide clarity for the students if they were informed of the axioms, definitions, and facts that they are allowed to assume and base their reasoning on.

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Duval (as summarized in Dreyfus, 1999) distinguishes explanation, argumentation, and proof. The function of explanation is descriptive – to answer why something is so. Both arguments and proofs provide reasons that must be free of contradictions. In argumentation, the semantic content of the reasons is important. For proofs, the structure, as well as the content, determine the truth of the claim. Doeuk (1999), commenting on Duval’s distinctions, augments his description by saying that, in argumentation, one is allowed a wider range of reasoning than in proof. Not only deduction but also metaphor and analogy can be employed.

In general, professors should avoid “dumbing down” their assessments by asking routine questions that can be answered by mimicking. One needs to modify the “didactic contract” in order to achieve this; otherwise, questions requiring genuine problem-solving and proving will be considered “unfair”. Professors also need to be exemplary in their use of quantifiers. Chellougui (2004) observed that many different formulations of quantification are used in the same book or by the same teacher of first-year university students. Sometimes quantifiers remain implicit; sometimes they are used very strictly; sometimes they are used incorrectly as shorthand, even by teachers who do not accept such usage from their students. 4.4 Alternative methods of assessing students’ comprehension of proofs 4.4.1 Writing to learn about proofs and proving Writing to learn mathematics is not a new idea (Ganguli & Henry, 1994). Kasman (2006), a mathematician, has used the following technique in both modern algebra and introductory proofs courses. Students are given a fictitious scenario of two students discussing how they are going to construct a proof, along with the final written proof. Students are to write a short expository reaction paper stating whether the fictional proof is valid, identify any errors in it, comment on whether the fictitious students have the right idea but are not expressing it well, decide which fictional student has a better understanding, and give justifications for their reactions. Kasman (2006) provides sample scenarios, as well as advice for writing more scenarios, suggestions for specific directions to give to students in order to get the kind of thoughtful reactions one wants, and suggestions for grading. 4.4.2 Testing proof comprehension via a variety of questions Conradie and Frith (2000) point out that proof comprehension tests have been used successfully on mid-term tests and final examination given by the Mathematics Department of the University of Cape Town for at least nine years. Students are given a proof of a theorem that has already been proven in class, such as: 20 is irrational. This is followed by questions like: What method of proof is used here? How is 20 defined? When is a real number irrational? Why may we assume m and n (used in the proof) have no common factors? Variations of this suggestion include: having students fill in gaps in a proof or asking how the proof would be affected if condition X were replaced by condition Y. An advantage of such tests is that it gives students incentive to understand the proofs presented in the course, not just to memorize them. The authors claim such tests give a clear evaluation of a student’s understanding. Furthermore, students can be prepared for such comprehension tests by giving them exercises of this type as homework. In addition, students preferred this type of testing to the usual request for the reproduction of proofs.

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The above suggestions come from mathematicians, but as reported by Mejia-Ramos & Inglis (2009), there has been little mathematics education research on the comprehension of proofs, as compared with the amount of research on the construction and validation of proofs. 4.5 Videos about proofs and proving to use with students

Raman, Sandefur, Birky, Campbell, and Somers (2009) are producing videos of students engaged in proving theorems that many students find difficult, along with materials to help teachers use those videos. The intention is for students to learn to reflect on their own thinking by discussing and reflecting on the thinking of the students in the videos. These materials have been tested at four U.S. universities in transition-to-proof courses. (See Section 4.1 for a description of such courses.) In the process of creating these videos, Raman et al. (2009) have identified three significant “moments” in constructing a proof: the moment where one gets a key idea that gives one a sense of why the theorem might be true; the discovery of a technical handle needed to translate that key idea into a written proof; and putting the proof in final standard form with a level of rigor appropriate for the intended audience. Successfully negotiating one or two of these moments does not guarantee completion of the other(s).

For example, one of the videos shows two students proving: If , then > . Within two minutes they come up with the key idea that a cubic function grows

faster than a quadratic. Rather than pursuing this, they manipulate the conclusion algebraically to get

3n ≥ 3n2( 1)n +

1( 2)( 1)n n n− + > . The students notice that if , the term on the

left is a positive integer and

3n ≥

1n is between 0 and 1. However, despite being pleased with

this, the students wondered whether this was a proof. Rather than writing it up formally, they tried to prove the contrapositive, but instead formed the converse, tried several examples, and got confused. While a simple reordering of their original steps would have constituted a proof, the students did not recognize this.

While such videos could be useful in relatively small classes where discussion is possible, this is not realistic in large classes. Alcock (2009a), who routinely teaches real analysis to classes of over 100, has developed an alternate video technique, called e-Proofs. Eight analysis theorems also proved in lecture were elucidated by giving line-by-line explanations of the reasoning used, as well as a breakdown into large-scale structural sections. Students could replay particular sections of the video as often as they wished. The aim was to improve students’ comprehension of theorems proved in class. While the first versions of e-Proofs were designed by Alcock specifically for her own classes, she and others are now working on an e-Proofs authoring tool called ExPOUND, a project of JISC (Joint Information Systems Committee) which is part of the U.K. Higher Education Funding Council. Information can be found online. 4.6 Resources about proof and proving for mathematicians 4.6.1 A DVD showing students constructing proofs

Alcock (2009b) has a DVD, Students and Proof: Bounds and Functions, designed to provide mathematicians with an opportunity to watch undergraduate mathematics students attempting to construct two proofs, one on upper bounds of sets of real numbers, the other on increasing functions and maxima. The students’ proof attempts are broken up into excerpts, with prompts at the end of each excerpt designed to facilitate reflection or

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group discussion. The DVD also contains additional material for those wishing to follow up these ideas in the mathematics education research literature. 4.6.2 Two booklets incorporating ideas from mathematics education research 4.6.2.1 A booklet on students’ proving. Nardi and Iannone (2006) have a concise 52-page booklet, How to Prove It: A Brief Guide to Teaching Proof To Year 1 Mathematics Undergraduates, designed for teachers of first-year U.K. undergraduates. There are sections on conceptualizing formal reasoning, “proof” by example, proof by counterexample, proof by mathematical induction, and proof by contradiction. Each section contains examples of students’ written responses, some relevant mathematics education research findings, some suggested pedagogical practices, and references to the research literature. This booklet can be downloaded without charge from http://mathstore.ac.uk/publications/Proof%20Guide.pdf . 4.6.2.2 A more general booklet for mathematicians. Another somewhat longer booklet by Alcock and Simpson (2009), Ideas from Mathematics Education: An Introduction for Mathematicians, contains chapters on the distinction between a student’s concept image and the formal concept definition; on the difference between thinking of a mathematical notion, such as function, as a process or as an object; and on the distinction between semantic and syntactic reasoning strategies in proving. While both strategies are useful in proving, some students seem to prefer one over the other. (See Section 3.3.8 above.) This booklet can be downloaded for free from http://www.mathstore.ac.uk/index.php?pid=257 . 4.6.3 Other resources The Mathematical Association of American’s Special Interest Group on Research in Undergraduate Mathematics Education has produced a “research into practice” volume, in which researchers describe their research in an expository way for mathematicians (Carlson & Rasmussen, 2008). The volume contains twenty-three chapters, of which five are on proving theorems (pp. 93-164). Other attempts to bring such information to mathematicians include sessions at professional meetings, such as those of the American Mathematical Society and the Mathematical Association of America.

The extent to which these resources have been consulted by, and are found to be useful by, those teaching proof and proving to tertiary students needs more investigation. 5. Questions for future research on proof and proving at tertiary level

Teachers at tertiary level increasingly use students' original proof constructions as a means of assessing their students’ understanding. However, many questions remain about how students at the tertiary level come to understand and construct proofs. Here are some questions that could be, but have not yet been, the object of much research.

How are instructors’ expectations about students’ performance in proof-based mathematics courses different from those in courses students have previously experienced? To what extent do tertiary instructors use the ability to construct proofs as a measure of a student’s understanding? What is known about the effectiveness of instructors’ proof presentations? That is, what do students “take away” from lectures? Are there effective ways to use technology, such as clickers and computers, to help students learn to construct proofs that are acceptable to their instructors? How much, and what kinds of, conceptual knowledge is directly useful in making proofs in subjects such

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as abstract algebra and real analysis? How do tertiary students conceive of theorems, proofs, axioms, and definitions, and the relationships among them? How does this change over time?

Should teaching proof to future mathematics teachers differ from teaching proof to future mathematicians or students of other mathematics oriented careers? How? How can we design opportunities for secondary preservice teachers to acquire the knowledge, skills, understandings, and dispositions necessary to provide effective instruction about argumentation, proof, and proving to their pupils?

Is learning to prove partly, or even mainly, a matter of enculturation into the practices of mathematicians (Nickerson & Rasmussen, 2009; Perry, Samper, Camargo, Molina, & Echeverry, 2009)? If so, how is this achieved? What are tertiary students' views of proof and how are their views influenced by their university experiences with proving? What are the roles of problem solving, heuristics, intuition, visualization, procedural and conceptual knowledge, logic, and validation in learning to construct proofs? What previous positive experiences have students had with argumentation and proof that tertiary teachers can and should take into consideration? While there has been some research on the above, for example, on tertiary students’ views of proof and proving, there could be much more. Acknowledgement. Special thanks to the members of Working Group 6 for their input: Lara Alcock, Véronique Battie, Geoffrey Birky, Ana Breda, Connie Campbell, Paolo Iannone, Yi-Yin Ko, Nitsa Movshovitz-Hadar, Teresa Neto, Kyeong Hah Roh, Carmen Samper, John Selden, and Victor Tan. Thanks also to Kerry McKee for her input on two-column proofs. References Alcock, L. (2009a). E-Proofs: Student experience of online resources to aid

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