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Proof Construction Perspectives: Structure, Sequences of
Actions, and Local Memory
John Selden, Annie Selden
New Mexico State University
This theoretical paper considers several perspectives for understanding and teaching university
students’ autonomous proof construction. We describe the logical structure of statements, the formal-rhetorical part of a proof text, and proof frameworks. We view proof construction as a sequence of actions, and consider actions in the proving process, both situation-action pairs and behavioral schemas. We introduce the concept of local memory – subsets of memory that are partly activated during prolonged consideration of proofs and facilitate success.
Introduction
The question
If one wants students to learn to construct proofs, what should one teach them? One answer is: The content of some subfields of mathematics, such as linear algebra or real analysis. That is, theorems, explanations of their proofs, plus some intuition about those subfields, with student proving relegated mainly to homework and tests. We suspect this answer is close to how many mathematicians themselves were taught, and this itself is evidence that such teaching is sometimes, perhaps often, effective. There is another reason the above answer might be favored. We recall proving a nice result in field A using a result from an “unrelated” seminar in field B. This serendipitous proof experience probably happens often enough to suggest it is valuable for students to take courses covering a wide variety of mathematical content. However, students just learning to construct proofs are not in a position to use such serendipitous experiences. Thus for many students the teaching of mathematical content contains too little practice to be adequate for developing beginning proof construction skills.
Another answer might come from observing students’ proof construction attempts and asking, what can prevent them from succeeding? Of course, mistakes can, but what else? The perspectives that follow may suggest answers that might contribute to a “content of proof construction” that will provide a new approach to teaching and learning proving. Before describing the perspectives, we describe the course from which some of the ideas developed.
The course
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We see autonomous proof construction as an activity, like learning a sport, that is mastered largely through doing it, perhaps with some coaching. This requires a supply of theorems whose proofs the students have not yet seen or even received advice about. In order to ensure this, we taught from our own notes, rather than a book, with students proving all the theorems.
Structures
The logical structure of statements
Statements, such as theorems or definitions, have a logical structure that can be described as formal or informal. A statement is formal if the variables are named; quantifiers are expressed explicitly and typically written first; and logical operators are just the most commonly used ones: and, or, not, if-then, and if-and-only-if. In addition, a formal statement should not be logically reducible to a shorter one. Otherwise, a statement is informal; examples are: “Differentiable functions are continuous”, and in a semigroup context, “A group has no proper left ideals”. A formal version of the later is: “For all semigroups S and for all left ideals
L of S, if S is a group, then L=S”, which removes the hidden double negative, “no proper”.
Informal statements are often used to state theorems, perhaps because they are memorable and psychologically combine easily with other information. However, we have found that beginning university students of proof construction are not likely to be able to reliably unpack them into formal statements (Selden & Selden, 1995). Such unpacking is important for both proof construction and validation (Selden & Selden, 2003). Thus, to build student self-efficacy, it is better at the beginning of a proof construction course to state theorems formally.
The formal-rhetorical part of a proof
A completed proof text can be divided into a formal-rhetorical part and, its complement, a problem-centered part. The formal-rhetorical part is the part that depends only on the logical structure of the statement of the theorem, earlier results, and associated definitions. It does not depend greatly on intuition about, or a deeper understanding of, the concepts involved or genuine problem solving in the sense of Schoenfeld (1985, p. 74). The problem-centered part does depend on problem solving, intuition, heuristics, and a deeper conceptual understanding (Selden & Selden, 2011). We suggest that beginning university students of proof construction are likely to benefit most from constructing proofs that have large formal-rhetorical parts and more advanced university mathematics students are likely to benefit most from those that have large problem-centered parts.
Proof frameworks
A major structure that can contribute to construction of the formal-rhetorical part of a proof is a proof framework (Selden & Selden, 1995). Here is an example. Suppose the statement of a theorem has the form “For all x∊X,there is a y∊Y so that if P(x,y) then Q(x,y).” Then the proof
framework starts by “introducing the variables” this way: “Let x∊X and y∊Y be …” where the
ellipsis constitutes a blank space to be filled in as needed later in the proof. It is important not to change the order of the variables with “mixed” quantifiers as this can change the meaning of the theorem. The framework continues “Suppose P(x,y). … Therefore Q(x,y).” Again the
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proof. In many cases, a (second-level) framework can be constructed for the proof of Q(x,y) and placed in the blank space of the first framework. In this way, a proof framework is constructed from the top and bottom towards the middle. The “Let x∊X” above means x will be treated as an arbitrary constant so that the proof construction will depend only on propositional calculus, rather than the harder predicate calculus. Students may not feel that doing this is appropriate for some time. (See the case of Mary, described in Selden, McKee, and Selden, 2010, p. 209).
Sequences of actions
Action, situations, and warrants
We treat actions as being physical (e.g., writing a line of a proof or drawing a sketch) or mental (e.g., attempting to recall a theorem or changing one’s focus from hypothesis to conclusion). During proving, actions arise from situations in the partly completed construction and are often justified by a warrant. If in constructing proofs, several similar situations are paired with similar actions, associative learning takes over and the situation-action pair no longer requires a warrant.
Behavioral schemas
If the above process continues beyond the forming of the original association, overlearning occurs and the relationship becomes automated. We then call the pair a behavioral schema
(Selden, McKee, & Selden, 2010). We think this process can occur both implicitly and explicitly; for example, by a teacher explicitly arranging appropriate practice problems, perhaps proof fragments for students to add a bit onto. Situation-action pairs and behavioral schemas can be either beneficial or detrimental, with the later unwanted and often occurring implicitly. We believe that both the situation and the effect of the action must be at least partly conscious, which means neither behavioral schemas nor situation-action pairs can be chained together entirely outside of consciousness (Selden, McKee, Selden, 2010). The creation of beneficial situation-action pairs and behavioral schemas greatly reduces the burden on working memory during proof construction. For an example of a behavioral schema in a non-proof situation, consider driving a car and suddenly coming upon a small child in the street. One does not need to think about stopping, one just does it. Such triggering situations do not need to have a name. Much of what we have described can be viewed as an adoption/extension of social psychology results (e.g., Bargh, 1997) to proof construction.
Proof construction as a sequence of actions
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Local memory
One might think from the above that proof construction, and deductive reasoning generally, consists entirely of communication with others or oneself using speech, vision, etc., or their inner versions. That is, it is all conscious. (For this perspective, see Sfard, 2010).
We take a different view. There appears to be a very large amount of memory maintained outside of consciousness. Conscious information can sometimes influence the activation of related information in memory and sometimes cannot do so (Selden, Selden, Mason, & Hauk, 2000). In constructing a proof, often much more relevant information can be activated than can be simultaneously held in mind. When information that cannot be held in mind is lost from consciousness, it seems not to be returned to its original state, but to a state of partial activation, and hence, and can be easily recalled. Often, in attempting a long and difficult proof, a considerable amount of information is generated and partially activated. We call such partially activated information local memory. We have found that we can easily recover such local memory even a day or so after putting aside a proof construction session provided we have not engaged in some other cognition similar to proving. That is, we can easily “pick up where we have left off”.
Memory activation can itself provide a useful flexibility. When information is activated and then returned to memory, it may unintentionally be slightly altered. After the same information is activated several times, somewhat different information may, seemingly serendipitously, be activated in the next iteration, and consequently, produce new proof ideas.
We hope others will extend and improve these ideas, perhaps including unemphasized ideas such as improving System 2 cognition by intentionally converting some of it to System 1 cognition (Selden & Selden, 2011).
References
Bargh, J. A. (1997). The automaticity of everyday life. In R. S. Wyer, Jr. (Ed.), The automaticity of everyday life (pp. 1-62). Mahwah, NJ; Lawrence Erlbaum Associates.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.
Selden, A., McKee, K., & Selden, J. (2010) Affect, behavioural schemas, and the proving process. International Journal of Mathematical Education in Science and Technology, 41(2), 199-215.
Selden, A., & Selden, J. (2014). The roles of behavioral schemas,persistence, and self-efficacy in proof construction. In. B. Ubuz, C. Hasar, & M. A. Mariotti (Eds.), Proceedings of CERME-8 (pp. 246-255). Ankara, Turkey: Middle East Technical University.
Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Jour. for Research in Mathematics Education, 34(1), 4-36.
Selden, J., & Selden, A. (2011). The role of procedural knowledge in mathematical reasoning. In B. Ubuz (Ed.), Proceedings of the 35th Confer3ence of the International Group for the Psychology of
Mathematics Education, Vol. 4 (pp. 145-152) Ankara, Turkey: Middle East Technical University.
Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29(2), 123-151.
Selden, J., Selden, A., Mason, A., & Hauk, S. (2000). Why can’t calculus students access their knowledge to solve nonroutine problems? Research in Collegiate Mathematics Education, IV, CBMS Series Issues in Mathematics Education, Vol. 8, 2000, 128-153.
