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What is proof and what is its role in mathematics?
What difficulties do students have with proofs?
How can the concept of proof be taught effectively?
References
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].
Many mathematicians and mathematics teachers would consider the answers to this question straightforward. The traditional view is that ’a mathematical proof is a formal and logical line of reasoning that begins with a set of axioms and moves through logical steps to a conclusionâ? [Griffiths, 2000, p. 2]. And, the purpose of proving a theorem is to establish its mathematical certainty. ’A proof confirms truth for a mathematician the way experiment or observation does for the natural scientistâ? [Griffiths, 2000, p. 2]. Such views are commonly held by mathematics teachers and are passed along to our students. However, many mathematics educators and some mathematicians believe that proofs are much more than this.
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 timeâ? [Balacheff, 1987, translated from French]. ’An argument becomes a proof after the social act of accepting it as a proof â? [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,
Students who believe that proofs are used solely to establish the certainty of mathematical statements find proofs of seemingly obvious results (e.g., 1+1 = 2 using the Peano postulates) to be pedantic [Harel, 1998; Weber, 2002a]. On the other hand, an awareness of the varied uses of proof may improve students' appreciation for the role of proof in mathematics.
Variations of this ’proof â? 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].
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.
Two Dutch researchers, Dina and Pierre van Hiele, proposed a learning cycle (now known as the van Hiele levels) through which students may progress as they learn Euclidean geometry:
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 ’skipâ? 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.
Many proofs, especially those in advanced undergraduate courses, require the use of formal notation. Students find many aspects of this notation, particularly the use of multiple quantifiers, to be troublesome. Selden and Selden [1995] asked 61 students in introductory proof courses to translate informal mathematical statements into the language of predicate calculus. They found that students were successful at this task less than 10% of the time. For instance, not one of the 20 students asked could express the statement ’A function f is increasing on an interval I means that for any numbers x_{1} and x_{2 } in I, if x_{1} x_{2}, then f (x_{1}) f (x_{2} ).â? as a logical sentence. Other research has illustrated how extracting meaning from a quantified logical statement is a very difficult and complex process [e.g., Dubinsky, Eltermann, and Gong, 1988]. Some students' reasoning about multiply quantified statements involves only the predicate part of the statement, while ignoring how the variables are quantified [e.g., Pinto and Tall, 1999].
Consider a student who is writing a proof and wants to use the following fact: ’If f (x) = x(1 + x) , then f â?² (x) = 2x + 1.â? What type of justification would be expected of such a student? The answer is, it depends. If the student was in a traditional introductory calculus course, he or she might be expected to perform a mathematical calculation. In a reform-oriented calculus course, numerical approximations or an inspection of the graph of f might suffice. In an undergraduate real analysis course, an argument using the definition of derivative would probably be required. In an advanced graduate course, this fact would be considered so obvious that an explanation would be unnecessary.
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 justificationâ? 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].
Even if students are ’logically capableâ?-- that is, they know what constitutes a proof and they can reason deductively, recite and manipulate definitions, and draw valid inferences -- this does not guarantee that they can construct anything beyond very trivial proofs. Knowing logical rules and the definition of a concept does not ensure that students can reason about that concept. Students often require an intuitive (conceptual) understanding of the concept that they are working with before they can construct proofs. Moore [1994] closely observed five undergraduates as they progressed through an introductory proof course. He found that students could sometimes state a concept's definition while having little understanding of the concept. These students could not, for instance, describe the concept in their own words or generate a single example of this concept. When these students were asked to write proofs about this concept, they did not know how to begin.
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 10^{1000} 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 Bâ? and try to prove A, even when the antecedent was implausible.
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].
Alibert and his colleagues at Grenoble University in France note that many mathematics majors and future high school mathematics teachers view proof as pedantic and divorced from the rest of mathematics [Alibert, 1988; Alibert and Thomas, 1991]. This, they suspect, is due to the context in which these students encounter proof. They believe that:
’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 futureâ? [Alibert and Thomas, 1991, p. 230].
They create such an environment in a novel real analysis course through the use of ’scientific debate.â? 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,
Students are then invited to conjecture propositions that may or may not be true. Such a proposition for the above situation might be ’If f (x) is increasing, then F (x) is increasing tooâ?. (Of course, this proposition is false). The students at this stage are asked to put forth arguments to convince their classmates of the conjecture's truth or falsehood. Alibert claims that as the semester progresses, the students recognize the need for precise definitions, clear arguments, and rigorous proofs as a means of deciding whether conjectures are true or false. Thus, the students come to see proof as an instrumental tool used to settle scientific debates. Alibert found that 75% of students responding to a questionnaire preferred the method of scientific debate to traditional instruction, while only 10% rejected the method as being inaccessible or not sufficiently organized [Alibert, 1988].
Marty [1986] believes that a shortcoming of introductory proof courses is that they focus on content (i.e. the meaning of the statements that are proven or the proof itself), rather than on techniques. To address this, Marty created an introductory proof course where proof techniques were emphasized and the importance of mathematical content was minimized. The following summarize the principles that Marty used in his course:
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].
One reason that university students find proof so difficult is that their experience with constructing proofs is typically limited to high school geometry [Moore, 1994]. To rectify this, the NCTM Standards [2000] recommend that proof be introduced early in the mathematics curriculum, arguing that ’reasoning and proof should be a consistent part of students' mathematical experience in pre-kindergarten through grade 12â? [p. 56]. Further, the NCTM Standards and others [e.g., Schoenfeld, 1994] argue that proof should not be seen as a distinct mathematical topic, but rather as a way of thinking that can be applied to any mathematical topic. Finally, the NCTM Standards argue that by the time students complete 12^{th} grade, they should recognize proof as fundamental to mathematics, be comfortable with constructing proofs, and be able to determine whether a given argument is a proof.
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 naÃ¯ve. 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.
Keith Weber
Rutgers University
Graduate School of Education
10 Seminary Place
New Brunswick, NJ 08901
Email: weber26@mac.com
(732) 932-7496