Proof validation

Proof validation is rarely taught explicitly in transition-to-proof courses and yet may be critical in developing a complete understanding of proof (Ko & Knuth, 2013; A. Selden & Selden, 1999). As mentioned in Chapter 1, reflection is necessary for developing an understanding of a concept. In this way, proof validation is key in developing a student’s understanding of proof. Yet, very little research has focused on proof validation and so the extent to which validation and comprehension are linked is unknown (Ko & Knuth, 2013). In addition, there is no single set of validation techniques accepted by the mathematical community (Inglis, Mejia-Ramos, Weber, & Alcock, 2013), which may be a reason why proof validation is not explicitly taught in transition-to-proof courses. What is known, however, is that much of what constitutes convincing for undergraduate students (e.g., authority, appeals to physical objects, truth value of the statement) is not considered convincing for practicing mathematicians (Harel & Sowder, 1998; Weber & Alcock, 2005). The following subsection will thus be devoted to the most common ways in which undergraduate students validate proofs.

proof schemes3_{, or that which convinces the individual a statement is true}4 _{(Harel & Sowder,}

1998). This classification of consists of three categories: external conviction, empirical, and analytical, as well as many subcategories. A brief explanation of the overarching categories follows.

Anexternal conviction proof schemeis characterized by students removing doubt by “the ritual of the argument presentation, the word of an authority, or the symbolic form of the argument” (Harel & Sowder, 1998, p. 246). The authors claimed that students are prone to accept false proof verifications on the basis of ritual and form when formality in mathematics is emphasized prematurely, emphasizing that “accepting false-proof verifications on the basis of their appearance is a severe deficiency in one’s mathematical education” (Harel & Sowder, 1998, p. 246), though claimed that proofs by authority have some value. Indeed, Weber and Mejia-Ramos (2013) found that if a paper was published in a highly reputable journal or another highly reputable source, proofs were accepted as true without justification. This practical acceptance of proofs by authority show that external proof schemes are not always harmful. An empirical proof scheme is one in which “conjectures are validated, impugned, or subverted by appeals to physical facts or sensory experiences” (Harel & Sowder, 1998, p. 252). Research preceding Harel and Sowder (1998) validates the dominance of empirical proof schemes among novice and advanced students alike (e.g., Chazan, 1993; Goetting, 1995; Yerushalmy, 1986). An analytical proof scheme is “one that validates conjectures by means of logical deduction” (Harel & Sowder, 1998, p. 258). This scheme can be thought of in two parts: the transformation of mental images by deduction (transformational) and justification based on axioms (axiomatic). The authors state “Although the authoritarian and empirical schemes have value, we feel that mathematics majors in particular should also eventually show evidence of the analytical proof schemes” (Harel & Sowder, 1998, p. 277). In other words, mathematics majors should transition to an analytic proof scheme at some

3_{Note this is not the same as a proof Schema, which is the collection of Actions, Processes, Objects, and}

other Schemas linked by some general principal to form a framework of what proof is to an individual.

4_{Recall that the truth value of statements within a proof is connected to local comprehension of a proof}

point in their undergraduate studies.

Harel and Sowder’s (1998) classification of proof schemes has been used in several stud- ies (e.g., Hadas, Hershkowitz, & Schwarz, 2000; Housman & Porter, 2003) and has been useful in describing how students validate specific lines of proofs during proof construction. For example, both Hadas et al. (2000) and Housman and Porter (2003) found that gener- ating examples aided students in validating specific statements (i.e., local comprehension). However, their classification of proof schemes is not useful when considering how students validate a proof (as opposed to a statement). What follows is a brief overview of two studies that analyzed how students validate a given proof of a statement.

A. Selden and Selden (2003) evaluated how eight undergraduate students read and validate purported proofs of a single theorem. While students professed they “check proofs step by step, follow arguments logically, generate examples, and make sure the ideas in a proof make sense” (A. Selden & Selden, 2003, p. 27) in order to validate the proof, the authors found little evidence that students actually do so based on their judgment results. That is, students purported to validate proofs by considering local and holistic aspects of a proof, yet their results suggested they did not actually do so. These results indicate that there is more to validating proofs than simply knowing useful validation strategies.

In a more narrow study on proof validation, Weber and Alcock (2005) conducted an in-depth theoretical analysis of students’ use of warranted implications to understand and validate proofs, wherewarranted implications are understood as “good reason to believe that each statement follows from the preceding statements or from other accepted knowledge” (Weber & Alcock, 2005, p. 125). The authors conclude that the emphasis on the truth value of the precedent and antecedent over whether the implication was warranted lends to student difficulty validating proofs. This is especially relevant to proof by contradiction as students may focus on the truth value of the statement being negated over whether the implication is warranted.