Keywords

Definition

Argumentation, reasoning, and proof are concepts with ill-defined boundaries. More precisely, they are words that different people use in different ways. What one can perhaps say is that reasoning is the concept with the widest compass. Logic is usually taken to mean a more structured form of reasoning, with its own subset, formal logic, which is logic in its most rigidly structured form. Though people most closely associate logic with mathematics, all forms of reasoning have had, and continue to have, valuable roles in mathematical practice. For that reason and, perhaps even more important, because of their usefulness in teaching, the many forms of reasoning have also found their place in the mathematics curriculum.

Characteristics

This entry will explore in more detail the concepts of argumentation, reasoning, and proof as understood by mathematicians and educators and present some of their implications for mathematics education. It will go on to describe some more recent thinking in mathematics education and in the field of mathematics itself.

Mathematical Proof

Mathematics curricula worldwide aim at teaching students to understand and produce proofs, both to reflect proof’s central position in mathematics and to reap its many educational benefits. Most documents addressed to teachers, such as that by the National Council of Teachers of Mathematics (NCTM 2000), give the following reasons for teaching proof: (1) to establish certainty; (2) to gain understanding; (3) to communicate ideas; (4) to meet an intellectual challenge; (5) to create something elegant, surprising, or insightful; and (6) to construct a larger mathematical theory.

This list encompasses not only justification but also considerations of understanding, insight, and aesthetics and in so doing further reflects mathematics itself. These additional considerations are important not only in the classroom but in mathematical practice as well: for mathematicians, too, a proof is much more than a sequence of logical steps that justifies an assertion.

Proof also plays other significant roles in mathematical practice. Proof can serve to present new methods and demonstrate their value, to inspire new hypotheses, and to show connections between different parts of mathematics. For practicing mathematicians, these too are valuable aspects of proof; yet the mathematics curricula, by and large, have failed to explore their educational potential.

Proof pervades all mathematical work. Unless it is considered an axiom, a mathematical assertion without a proof must remain a conjecture. To justify an assertion is the role of a proof. In the purest sense, a mathematical proof is a logical derivation of a given statement from axioms through an explicit chain of inferences obeying accepted rules of deduction. A “formal proof” will employ formal notation, syntax, and rules of inference (“axiomatic method”). Thus, strictly formal derivations will consist of unambiguous strings of symbols and conform to a mechanical procedure that will permit the correctness of the proof to be checked. Such proofs are considered highly reliable.

However, proofs in mathematical journals rarely conform to this pattern. As Rav (1999) pointed out, mathematicians express “ordinary” proofs in a mixture of natural and formal language, employing passages of explicit formal deductions only where appropriate. They bridge between these passages of formal deduction using passages of informal language in which they provide only the direction of the proof, by making reference to accepted chains of deduction. Consequently, most mathematicians would characterize ordinary proofs as informal arguments or “proof sketches.”

Nevertheless, these ordinary informal proofs do provide a very high level of reliability, because the bridges are “derivation indicators” that are easily recognized by other mathematicians and provide enough detail to allow easy detection and repair of errors (Azzouni 2004). In this way, the social process by which such proofs are scrutinized and ultimately accepted improves their validity. In fact, most accepted mathematical proofs consist of valid arguments that may not lend themselves to easy formalization (Hanna 2000; Manin 1998; Thurston 1994).

To reflect mathematical practice, then, a mathematics curriculum has to present both formal and informal modes of proof. If they wish to teach students how to follow and evaluate a mathematical argument, make and test a conjecture, and develop and justify their own mathematical arguments and proofs, educators have to provide the students with the entire gamut of mathematical tools, including both the formal and informal ones. Without this important double approach, students will lack the body of mathematical knowledge that enables practicing mathematicians to communicate effectively by using “derivation indicators” and other mathematical shorthand (cf. Hanna and de Villiers 2012).

Reasoning and Proof

Most mathematics curricula recognize that reasoning and proof are fundamental aspects of mathematics. In fact, much of the literature on mathematics teaching refers to them as one entity called “reasoning and proof.”

We may take reasoning, in the broadest sense, to mean the common human ability to make inferences, deductive or otherwise. As Fischbein (1999) noted, everyday reasoning may differ from explicit mathematical reasoning in both process and result. In everyday reasoning, for example, we may even accept a statement without any type of proof at all, because we judge it to be self-evident or intuitively plausible, or at least more plausible than its contradiction. However, in many realms, including mathematics, such everyday reasoning provides little help (e.g., it is not intuitively clear that the sum of the angles in any triangle is always 180°). In all such cases we would need defined rules of reasoning in order to reach a valid conclusion. We would need to construct a correct chain of inference – that is, to construct a proof.

Thus, all mathematics educators aim to teach students the rules of reasoning. In the Western tradition, the rules of reasoning are derived from classical mathematics and philosophy and include, for example, the syllogism and such elementary rules as modus ponens, modus tollens, and reductio ad absurdum. Students typically first encounter these basic concepts of logic in the axiomatic proofs of Euclidean geometry.

Here the teacher’s role is crucial. In addition to concepts specific to the mathematical topic, the teacher must make the students familiar with rules of reasoning, patterns of argumentation, and appropriate terms (e.g., assumption, conjecture, example, refutation, theorem, and axiom). How students actually learn these concepts is unfortunately a question of cognition that educators have yet to resolve, though researchers investigating this issue have proposed a number of promising models of cognition. One such model, the “cognitive development of proof,” combines three worlds of mathematics: the conceptual/embodied, the proceptual/symbolic, and the axiomatic/formal (cf. Tall et al., chapter 2 in Hanna and de Villiers 2012). Another, based on extensive observations of college-level students learning mathematics, uses a psychological framework of “proof schemes” (Harel and Sowder 1998). Yet another (Balacheff 2010) aims at analyzing the learning of proof by considering how three “dimensions” – the subject, the milieu, and the problem – can be used to build a bridge between knowing and proving. Duval (2007) model stresses that the cognitive processes needed to understand and devise a proof depend on students’ learning “how proof really works” (learning its syntactic and deductive elements) and “how to be convinced by proof.” Stylianides (2008) proposes that the processes of reasoning and proving encompass three “components” – mathematical, psychological, and pedagogical – while Reid and Knipping (2010) discuss still other variations.

Argumentation and Proof

Many researchers in mathematics education have chosen to use the term “argumentation,” which encompasses the various approaches to logical disputation, such as heuristics, plausible, and diagrammatic reasoning, and other arguments of widely differing degrees of formality (e.g., inductive, probabilistic, visual, intuitive, and empirical). Essentially, argumentation includes any technique that aims at persuading others that one’s reasoning is right. As used by its proponents, the concept also implies exchange and cooperation in forming and criticizing arguments so as to arrive at the best conclusion despite imperfect knowledge. Evidently, the broad concept of argumentation encompasses mathematical proof as a special case.

In recent years, however, mathematics educators have been accustomed to use “argumentation” to mean “not yet proof” and “proof” to mean “mathematical proof.” Consequently, opinion remains divided on the usefulness of encouraging students to engage in “argumentation” as a step in learning proof. Boero (in La lettre de la preuve 1999) and others see a great benefit in having students engage in conjecturing and argumentation as they develop an understanding of mathematical proof. Others take a quite different view, claiming that argumentation, because it aims only to establish plausibility, can never be more than a distraction from the task of teaching proof (e.g., Balacheff 1999; Duval – in La lettre de la preuve 1999). Despite these differences of opinion, however, the practice of teaching students the techniques of argumentation has recently been gaining ground in the classroom.

Durand-Guerrier et al. (Chapter 15 in Hanna and de Villiers 2012) reported on over 100 recent studies on argumentation in mathematics education that discuss the complex relationships between argumentation and proof from various mathematical and educational perspectives. Most of these studies reported that students can benefit from argumentation’s openness of exploration and flexible validation rules as a prelude to the stricter uses of rules and symbols essential in constructing a mathematical proof. They also showed that appropriate learning environments can facilitate both argumentation and proof in mathematics classes.

Furthermore, some studies provided evidence that students who initially embarked upon heuristic argumentation in the classroom were nevertheless capable of going on to construct a valid mathematical proof. By way of explanation, Garuti et al. (1996) introduced the notion of “cognitive unity,” referring to the potential continuity between producing a conjecture through argumentation and constructing its proof. Several other researchers have provided support for this idea and for other benefits or limitations of argumentation, particularly argumentation based on Toulmin’s (1958) model of argument.

Toulmin’s model, the one now most commonly used in mathematics education, proposes that an argument is best seen as comprising six elements: the Claim (C), which is the statement to be proved as a theorem or the conclusion of the argument; the Data (D), the premises; the Warrant (W) or justification, which is the connection between the Claim and the Data; the Backing (B), which gives authority to the Warrant; the Qualifier (Q), which indicates the strength of the Warrant by terms such as “necessarily,” “presumably,” “most,” “usually,” “always,” and so on; and the Rebuttal (R), which specifies conditions that preclude the Claim (e.g., if the Warrant is not convincing).

Clearly, Toulmin’s model reflects practical and plausible reasoning. It includes several types of inferences, admits of both inductive and deductive reasoning, and makes explicit both the premises and the conclusion, as well as the support that led from premises to conclusion. It is particularly relevant to mathematical proof in that it can include formal derivations of theorems by logical inference.

Practical Classroom Approaches

In addition to argumentation, a number of other approaches have been investigated for their value in teaching mathematical reasoning. Educators have debated, for example, whether the study of symbolic logic, more particularly the propositional calculus, would help students understand and produce proofs. Durand-Guerrier et al. (Chapter 16 in Hanna and de Villiers 2012) have examined this question and provide some evidence for the value of integrating techniques of symbolic logic into the teaching of proof.

Visualization, and diagrammatic reasoning in particular, is another technique whose value in teaching mathematics, and especially proof, has been discussed extensively in the literature and in conferences, albeit inconclusively. After examining numerous research findings, Dreyfus et al. (Chapter 8 in Hanna and de Villiers 2012) concluded that the issue required further research; in fact, both philosophers of mathematics and mathematics educators are still debating the contribution of visualization to the production of proofs. Current computing technologies have offered mathematicians an array of powerful tools for experiments, explorations, and visual displays that can enhance mathematical reasoning and limit mathematical error. These techniques have classroom potential as well. Borwein (Chapter 4 in Hanna and de Villiers 2012) sees several roles for computer-assisted exploration, many of them related to proof: graphing to expose mathematical facts, rigorously testing (and especially falsifying) conjectures, exploring a possible result to see whether it merits formal proof, and suggesting approaches to formal proof. Considerable research has demonstrated that the judicious use of dynamic geometry software can foster an understanding of proof at the school level (de Villiers 2003; Jones et al. 2000).

Physical artifacts (such as abaci, rulers, and other ancient and modern tools) provide another technique for facilitating the teaching of proof. Arzarello et al. (Chapter 5 in Hanna and de Villiers 2012) demonstrate how using such material aids can help students make the transition from exploring to proving. In particular, they show that students who use the artifacts improve their ability to understand mathematical concepts, engage in productive explorations, make conjectures, and come up with successful proofs.

Trends in Proof

In mathematical practice, as we have seen, ordinary informal proofs are considered appropriate and suitable for publication. Still, mathematicians would like to have access to a higher level of certainty than those informal proofs afford. For this reason, contemporary mathematical practice is trending toward the production of proofs much more rigorous and formal than those of a century ago (Wiedijk 2008). In practice, however, one cannot write out in full any formal proof that is not trivial, because it encompasses far too many logical inferences and calculations.

The last 20 years have seen the advent of several computer programs known as “automatic proof checkers” or “proof assistants.” Because computers are better than humans at checking conformance to formal rules and making massive calculations, these new programs can check the correctness of a proof to a level no human can match. According to Wiedijk (2008), such programs have been successful in confirming the validity of several well-known theorems, such as the Fundamental Theorem of Algebra (2000) and the Prime Number Theorem (2008).

Mathematics educators and students have already benefitted greatly from educational software packages in areas other than proof, such as Dynamic Geometric Software (DGS) and Computer Algebra Systems (CAS), and researchers are working on advanced proof software specifically for mathematics education. For example, there is now a fully functional version of Theorem-Prover System (TPS) appropriate for the school and undergraduate levels, named eduTPS (Maric and Neuper 2011). The role of Artificial Intelligence in mathematics education, and in particular that of automated proof assistants, has already been the subject of several doctoral dissertations. Unfortunately, mathematics educators have not yet tested the proof software or tried it in the classroom, so its usefulness for teaching mathematics has not yet been firmly established.

Cross-References