Proof construction

Of the four aspects, proof construction has received the most attention (Inglis & Mejia- Ramos, 2009). Early work in proof construction research focused on students’ difficulties while writing proofs. Some of the difficulties reported include:

• Difficulties with the underlying mathematical logic (e.g., Knipping, 2008; Moore, 1994; Stavrou, 2014);

• Difficulties understanding the mathematical concepts within the proof (e.g., Mej´ıa- Ramos et al., 2015; Moore, 1994; Stavrou, 2014);

– Induction (e.g., Andrew, 2007; J. D. Baker, 1996; Dubinsky, 1986, 1989; Dubinsky & Lewin, 1986; Harel, 2002) and

– Contradiction (e.g., Antonini & Mariotti, 2009; Barnard & Tall, 1997; Harel & Sowder, 1998; Reid & Dobbin, 1998);

• Difficulties knowing how to approach proving statements, i.e. lack of proof-writing strategies (e.g., Hanna, 2000; Hoyles, 1997; Moore, 1994; Weber, 2001, 2004); and

• Difficulties with quantified statements (e.g., Barnard, 1995; Dubinsky, 1989; Dubinsky & Yiparaki, 2000; Piatek-Jimenez, 2010).

Of note for proof by contradiction are the last two difficulties: knowing how to approach statements and quantified statements. For the former, students must realize they need to write a proof by contradiction and have some general strategies to complete the proof before they can successfully do so. For the latter, it is commonly required that students negate a quantified statement when writing a proof by contradiction. The rest of this subsection will describe prominent literature on these two difficulties.

Many researchers and educators have noted that in some situations, students may be aware of the theorems necessary to complete a proof and yet cannot do so. To uncover why this may occur, Weber (2001) conducted interviews with four undergraduate and four graduate students in which participants were asked to prove statements in abstract algebra. While the undergraduate students were aware of the facts necessary to complete the proof, as well as the proof methods to use, they were unable to complete the proofs. Unlike the undergraduates, the graduate students were able to complete the proofs using strategic knowledge, defined as: “knowledge of how to choose which facts and theorems to apply” (Weber, 2001, p. 101). The author hypothesized that there are four types of strategic knowledge: knowledge of the domain’s proof techniques, knowledge of which theorems are important and when they will be useful, knowledge of when and when not to use syntactic

strategies1_{, and procedural}2 _strategies.

Weber (2003) expanded on this research with a focus on six undergraduates’ use of procedural strategies in understanding proof. From this study and his previous study, We- ber (2004) presented a framework for describing undergraduate proof construction processes based on the observations of 176 undergraduate students’ proofs over multiple studies. This framework classified the types of proofs produced as one of the following: procedural, syn- tactic, or semantic.

In a proof using procedural methods, “one attempts to construct a proof by applying a procedure, i.e., a prescribed set of specific steps, that he or she believes will yield a valid proof” (Weber, 2004, p. 426). The procedure can either be an algorithm or a process.

Algorithms are characterized as external and highly mechanical to the student, whereas a

process is internal and flexible. Note the close relation of algorithms in proof construction to Actions in APOS Theory; both are external and require step-by-step instruction. Also note the close relation of processes in proof construction to Processes in APOS Theory; both are described as internal and more flexible than the step-by-step nature of algorithms/Actions. In a proof using syntactic methods, “one attempts to write a proof by manipulating correctly stated definitions and other relevant facts in a logically permissible way” (Weber, 2004, p. 428). Proofs of this form are no more than unpacking definitions and using tau- tologies to manipulate symbols in order to reach the desired conclusion. Students using this method do not need to consider the meaning of their syntactic statements. In terms of APOS Theory, syntactic methods may be either an Action (e.g. if the students have memorized which definitions to unpack for specific problems), or a Process (e.g. if students have a general rule to “unpack definitions” when necessary).

In a proof using semantic methods, “one first attempts to understand why a statement is true by examining representations (e.g., diagrams) of relevant mathematical objects and then uses this intuitive argument as a basis for constructing a formal proof” (Weber, 2004,

1_{Proof by simply manipulating symbols.}

p. 429). Very few undergraduate research subjects, if any, attempted semantic proofs; 0 of 56 proofs in abstract algebra and 17 of 120 proofs in real analysis. In terms of APOS Theory, semantic methods seem to correlate with an Object level of understanding as both require the individual to understand the statement in totality. A lack of students attempting semantic proofs may indicate a lack of understanding with either the mathematical content or proof itself.

Another difficulty students may have when writing a proof is with the quantification of the statement (Dubinsky, 1989; Dubinsky & Yiparaki, 2000; Piatek-Jimenez, 2010). Student difficulties with quantification can be viewed in three categories: single-level quantification (e.g., for all . . . ), multiple-level quantification (e.g., for all . . . , there exists . . . such that . . . ), and the negation of any quantification. In particular, both Dubinsky and Yiparaki (2000) and Piatek-Jimenez (2010) focus on student difficulties with AE (for all . . . , there exists . . . ) and EA (there exists . . . for all. . . ) statements, concluding that students struggle to understand the mathematical meaning behind multiple-level quantification. Dubinsky (1989) had difficulties isolating examples of students negating single-level quantification, though quoted an unpublished observation in which 49 of 52 sophomore mathematics majors could not negate the statement “Every member of my family is unemployed.”

Quantification in proof by contradiction normally appears in one of three forms: nonex- istence, uniqueness, and infinitely many. Nonexistence claims such as “There is no odd integer that can be expressed in the form 4j −1 and in the form 4k+ 1 for integers j and

k” are commonly proved by assuming there does exist an object (in this example, an odd integer) that has the desired property and deriving a contradiction from this assumption. Uniqueness claims such as “For every non-zero rational number, there exists a unique multi- plicative reciprocal” are commonly proved by assuming either there are no objects or there are two distinct objects with the desired quality (in this example, multiplicative reciprocals of a non-zero rational number), showing at least one object exists with the desired quality, and finally showing that two distinct objects are actually the same, i.e. deriving a contradiction. Infinity claims such as “The set of primes is infinite” are commonly proved by assuming the

set is finite and constructing a new element not in the set, thus deriving a contradiction to the original assumption. In each of these three cases, the student must negate a single level of quantification before proceeding with the proof by contradiction. Therefore, students’ un- derstanding of quantification will be considered when studying their understanding of proof by contradiction.