Research Sampler

8. Students' Difficulties with Proof

June 23, 2003

by Keith Weber

Proof is a notoriously difficult mathematical concept for students. Empirical studies have shown that many students emerge from proof-oriented courses such as high school geometry [Senk, 1985], introduction to proof [Moore, 1994], real analysis [Bills and Tall, 1998], and abstract algebra [Weber, 2001] unable to construct anything beyond very trivial proofs. Furthermore, most university students do not know what constitutes a proof [Recio and Godino, 2001] and cannot determine whether a purported proof is valid [Selden and Selden, 2003].

What is proof and what is its role in mathematics?

Fields Medalist William Thurston [1994] argues that it is important to distinguish between formal proofs and proofs that mathematicians actually construct. In the latter case, many routine calculations and logical manipulations are suppressed. Such omissions are not due to carelessness; rather this is done because proofs would be impossibly long if every logical detail were included. Renz [1981] has estimated that if one proceeded to prove the Pythagorean theorem using only the axioms and rules of inference allowed in Euclid's Elements, the proof would be over 80 pages long!

Davis and Hersh [1981] argue that it is probably impossible to define precisely what type of argument will be accepted as a valid proof by the mathematical community. Of course, there are some aspects of proof that distinguish it from other types of arguments. For example, proofs about a concept must use the concept's definition and must proceed deductively, as opposed to examining prototypical cases or giving an intuitive argument. And if a result is incorporated in a proof, that result must be accepted by the mathematical community [Tall, 1989]. Beyond this, some mathematics educators argue that whether or not an argument is accepted as a proof depends not only on its logical structure, but also on how convincing the argument is [Hanna , 1991].

At different places in the mathematics education literature, a proof has been defined as an argument that convinces an enemy [Mason, Burton, and Stacey, 1982], an argument that convinces a mathematician who knows the subject [Davis and Hersh , 1981], or an argument that suffices to convince a reasonable skeptic [Volmink, 1990]. Others, who focus on the social and contextual nature of proof, offer the following relativist descriptions: “We call proof an explanation accepted by a given community at a given time” [Balacheff, 1987, translated from French]. “An argument becomes a proof after the social act of accepting it as a proof ” [Manin, 1977]. Many mathematics educators believe that focusing exclusively on the logical nature of proof can be harmful to students' development. Such a narrow view leads students to focus on logical manipulations rather than on forming and understanding convincing explanations for why a statement is true [Alibert and Thomas, 1991].

Mathematics educators and mathematicians believe that establishing the veracity of a statement is only one of many reasons for constructing or presenting a proof. Besides convincing, mathematics educators have proposed a number of alternative purposes of proof. For example,

• Explanation. By examining a proof, a reader can understand why a certain statement is true. Many mathematics educators argue that explanation should be the primary purpose of proof in the mathematics classroom [e.g., Hanna, 1990; Hersh, 1993].
• Systemization. One can use proofs to organize previously disparate results into a unified whole. By organizing a system deductively, one can also uncover arguments that may be fallacious, circular, or incomplete [de Villiers, 1990].
• Communication. The language of proof can be used to communicate and debate ideas with other students and mathematicians [e.g., de Villiers, 1990; Knuth, 2002].
• Discovery of new results. By exploring the logical consequences of definitions and an axiomatic system, new models or theories can be developed [de Villiers, 1990].
• Justification of a definition. One can show that a definition is adequate to capture the intuitive essence of a concept by proving that all of the concept's essential properties can be derived from the proposed definition [Weber, 2002a].
• Developing intuition. By examining the logical entailments of a concept's definition, one can sometimes develop a conceptual and intuitive understanding of the concept that one is studying [Pinto and Tall, 1999].
• Providing autonomy. Teaching students how to prove can allow them to independently construct and validate new mathematical knowledge [Yackel and Cobb, 1996].

What difficulties do students have with proofs?

Variations of this “proof ” are surprisingly common. They illustrate one of students' most ubiquitous difficulties with the concept of proof: Students often believe that non-deductive arguments constitute a proof. Below are some common student beliefs about what constitutes a mathematical proof. A comprehensive taxonomy of such beliefs is given in Harel and Sowder [1998].

• Ritual. An argument is a proof if and only if it is in accordance with a specific mathematical convention. For instance, many pre-service teachers believe a geometry argument must be in a two-column format to be a proof [Martin and Harel, 1989].
• Authoritative. An argument is a proof if it is presented by or approved by an established authority, such as a teacher or a famous mathematician.
• Inductive. Checking that a general statement holds for one example, or perhaps several examples, is sufficient to demonstrate its veracity. The above faulty proof is an example.
• Perceptual. By way of an appropriate diagram, one can visually demonstrate that a certain property holds [Harel and Sowder, 1998]. For instance, to prove that the sequence 1/n converges, some students will draw a diagram illustrating that as n grows large, the terms of 1/n will become arbitrarily close to zero.

Recently, research has focused on why students may possess these invalid beliefs about proof. Recio and Godino [2001] note that many such invalid proof techniques would be appropriate in non-mathematical domains. For instance, drawing a general conclusion by examining many specific cases is entirely appropriate in the social sciences. Alcock and Simpson [2002] observe that reasoning about a concept using a prototypical example is common in our everyday experience, but in the realm of formal mathematics, one must reason using the concept's definition.

Inadequate cognitive development

Level 0: Visualization. Students can recognize a geometric figure as an entity (e.g., a square), but cannot recognize properties of this figure (e.g., a right angle).

Level 1: Analysis. Students can recognize components and properties of a figure. However, students cannot see relationships between properties and figures, nor can they define a figure in terms of its properties. For instance, students at this stage may observe that all rectangles have four right angles, but they would not realize that this entailed a square was a rectangle.

Level 2: Informal deduction. Students can recognize interrelationships between figures and properties and they can justify these relationships informally. Such students might recognize that a square is a rectangle since it had all the properties of a rectangle, but be unable to produce arguments starting with unfamiliar premises. For example, such students could not construct trivial proofs about objects that were unfamiliar to them even if they knew how that object was defined.

Level 3: Deduction. Students can reason about geometric objects using their defined properties in a deductive pattern. They can use an axiom system to construct proofs. Students at this stage could construct the types of proofs that one would find in a typical high school geometry course (e.g., isosceles triangles have two congruent angles).

Level 4: Rigor. Students can compare different axiom systems. Geometry is seen as an abstract rigorous field. [See Teppo, 1991].

The van Hieles postulated that (a) students must progress through each of these stages, i.e. they cannot “skip” a level, and (b) instruction targeted for students at a higher level will not be comprehensible for students at a lower level. Senk [1989] found strong empirical support for the van Hieles' claim. In a large scale study, Senk demonstrated that by determining a student's van Hiele level at the beginning of a high school geometry course, one could very accurately predict that students' proof-writing ability at the end of the course. Senk also found that many students enter high school geometry with a low Van Hiele level of understanding, and suggests that this may be why geometry gives high school students so much difficulty [Senk 1985, 1989].

A lack of cognitive development may also prevent college students from understanding the concept of proof. Piaget claims that students will be unable to discern or construct deductive arguments until they have reached what he calls a formal operational stage of cognitive development. Ausubel and his colleagues [1968] investigated college students' stages of cognitive development and found that just 22% of college students had achieved a formal operational stage of development. These findings have led Tall [1991] and others to observe that many university students may (at least initially) be unable to understand deductive proofs.

Notational difficulties

Sociomathematical norms

Yackel and Cobb [1996] have coined the term sociomathematical norms to discuss how environmental influences, such as students' textbooks, teachers' comments, and their feedback on assignments, determine students' mathematical beliefs and subsequent behavior. Dreyfus [1999] claims that “What counts as an acceptable mathematical justification” is one example of a sociomathematical norm. As illustrated above, at different times in a student's academic career, different types of justification are required. However, which type of justification is required is rarely explicated to the student. It is often the case that the student receives mixed messages. For instance, many mathematical textbooks will offer an intuitive explanation of one statement, an example to justify another statement, and a rigorous (formal) proof of another, yet the transition between intuitive, empirical, and rigorous thought is not clearly marked [Dreyfus, 1999]. Researchers suggest that this may lead students to acquire undesirable mathematical beliefs about rigor, explanation, and proof, and may partially explain why students will submit informal arguments as proofs in advanced courses [e.g., Dreyfus, 1999; Raman , 2002].

Poor conceptual understanding and ineffective proof strategies

When writing a proof, there are many valid inferences one could draw. Clearly one is unlikely to construct a proof by deriving inferences in a haphazard manner. To illustrate, Newell and Simon [1972] demonstrated how a breadth-first automated theorem prover would need to examine over 10¹⁰⁰⁰ proofs to prove some of the theorems in Whitehead and Russell's classic logic text, Principia Mathematica [1935]. In order to construct non-trivial proofs, undergraduates need strategies and heuristics to help them to decide how they should attack problems. In two studies, Weber [2001, 2002b] observed eight undergraduates who had completed an abstract algebra course constructing non-trivial proofs about group homomorphisms and isomorphisms. The studies considered only those cases in which the undergraduates were aware of the facts and theorems needed to prove a statement and could construct a proof when specifically told which facts to use. Even in these cases, the undergraduates failed to construct a proof without prompting 68% of the time. Examination of these undergraduates' behaviors revealed that their strategies for constructing proofs were ineffective and crude. For instance, to prove a statement B, these undergraduates would often try to find any theorem of the form “A implies B” and try to prove A, even when the antecedent was implausible.

How can the concept of proof be taught effectively?

For the sake of brevity, there are important issues (e.g., grading) that are not discussed here. A more complete description of the Moore method is given in Jones [1977]. A recent article discussing how the Moore method can be used specifically in undergraduate mathematics is given by Mahavier [1999].

Scientific debate

“By establishing an environment in which students may see and experience first-hand what is necessary for them to convince others of the truth or falsehood of propositions, proof becomes an instrument of personal value which they will be happier to use (or teach) in the future” [Alibert and Thomas, 1991, p. 230].

They create such an environment in a novel real analysis course through the use of “scientific debate.” To use this method, the instructor first presents students with a mathematical situation. Such a situation might be:

Suppose f (x) is a real-valued function that is integrable over the real numbers. Suppose that,

Marty's introductory proof course

• The purpose of the course was to cover a set of proof techniques (e.g., direct proof, proof by contradiction). Each proof technique was focused on one at a time and in great detail.
• Students encountered each proof technique in a variety of mathematical settings.
• Little attention was paid to mathematical content. Content was covered only deep enough that meaningful theorems could be presented and proved.
• Students were asked to apply proof techniques in a variety of settings and to present their proofs on the board. The problems the students were asked to solve were basic and did not involve cleverness or tricks.
• Alternative proofs to an already proved statement were encouraged.

To illustrate the success of his methods empirically, Marty examined the future mathematical success of every student who completed the introductory proof course at his university over a ten year period. Marty compared the students who received his instruction with students who were taught in a traditional lecture format. He found that students in his class were two to three times more likely to pass their subsequent course in real analysis and four times as likely to continue their studies of advanced mathematics [Marty, 1990].

Proof in the high school classroom

Considering the difficulty that mathematics majors have with proof, these goals are certainly ambitious. To assess the plausibility of these goals, Knuth [2002] interviewed 16 qualified in-service high school teachers, some with a master's degree, to investigate their conceptions of mathematical proof. His findings were that many of these teachers' beliefs about proof were rather naïve. For instance, when asked if a valid proof could ever become invalid, six teachers answered that contradictory evidence to a statement would invalidate that statement's proof. When asked about the role of proof, only three teachers indicated that proofs could be used to explain why statements were true and none responded that proofs could be used to promote understanding. Knuth concluded that many of these teachers would be unable to effectively meet the NCTM Standards.

Knuth and others [e.g., Alibert and Thomas, 1991] have suggested that students' and teachers' conceptions about proof are likely formed during their undergraduate mathematics courses. If one wants to improve the proof abilities of high school students, perhaps the best place for mathematics educators and mathematicians to look is toward the proof-oriented college courses for pre-service mathematics teachers.

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