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 call on several ideas from the psychological literature and introduce the concept of local memory – a subset of memory that is partly activated during prolonged consideration of a proof.
Introduction
Question
What should be taught to university students who want to learn proof construction? 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 kind of 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. For many students the teaching of mathematical content contains too little proving practice to be adequate for developing beginning proof construction skills.
A second answer to the above question comes from observing students’ proof construction attempts and noting what prevents them from succeeding. Of course, mistakes can, but what else about the proof construction process, other than the content of mathematics, can prevent success? What follows describes perspectives that elaborate this second answer. These perspectives might contribute to a “content of proof construction” that will provide an alternative approach to teaching and learning proving. These perspectives will often be abstracted above the level of mathematical content--for example, difficulty with definitions, as opposed to difficulty with the meaning of normal subgroup. Before describing the perspectives, we describe the course from which some of the ideas emerged.
The course
activity, like learning a sport, that is mastered largely through doing it, perhaps with some coaching. Thus, in our “proofs course” we maximized student proof construction experiences. We and several mathematics education graduate students collected field notes and videos of these classes and analyzed them. We were looking for ways to help students learn to autonomously construct proofs, and the mathematical content involved was only a means to that end. In order to include a variety of kinds of proofs that students might write in subsequent courses, we included sets, functions, a little real analysis, some abstract algebra (semigroups), and if there was time, some topology. There were no lectures and the 4-10 students, presented their proof attempts at the board in class. For each proof attempt, we provided a, sometimes extensive, critique. Occasionally there were explanatory side comments, such as on logic. For more information, see Selden, McKee and Selden (2010, p. 207).
The proof text
The genre
There are distinctive features that commonly occur in proofs and reduce unnecessary distractions in validation (reading/reflecting proofs to judge their correctness). These features increase the probability that any errors will be found, thereby improving the reliability of the corresponding theorems. Proofs are not reports of the proving process, contain little redundancy, and contain minimal explanations of inferences. They contain only very short overviews or advance organizers and do not quote entire statements of previous theorems or definitions that are available outside of the proof. Symbols are generally introduced in one-to-one correspondence with mathematical objects. For example, one does not say, “Let x ϵ R. Now let y = x.” Finally, proofs are “logically concrete” in the sense that, where possible, they avoid quantifiers, especially universal quantifiers. Their validity is often seen to be independent of time, place, and author. Details can be found in Selden and Selden (2013).
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 in a semigroup context 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”. For more information, see Selden and Selden (in press).
student self-efficacy (Bandura, 1995), it is better at the beginning of a proof construction course to state theorems and definitions formally.
A structure of proof texts
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 of the concepts involved (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 text is a proof framework (Selden & Selden, 1995) of which there are several kinds. A proof framework is roughly the logical parts of the theorem statement placed in the approximate position they would occur in the completed proof text. Here is an example. Suppose the statement of a theorem has the form “For all x∊X, if P(x) then Q(x).” Then a proof framework would start: “Let x∊X. Suppose P(x). … Then Q(x).” where the ellipsis represents a blank space to be filled. In many cases, a (second-level) framework can be constructed for the proof of Q(x) and placed in the blank space of the first framework. In this way, a proof framework is constructed from the top and bottom of a proof towards the middle. The “Let x∊X” above means x will be treated as a fixed, but arbitrary constant, rather than a variable, so that the proof construction will depend only on propositional calculus, rather than the harder predicate calculus. For some time students may not feel that doing this is appropriate. (See the case of Mary, described in Selden, McKee, and Selden, 2010, p. 209).
Operable interpretations
In writing the formal-rhetorical part of a proof, it can be helpful to associate definitions and previously proved results with operable interpretations. These interpretations are similar to Bills and Tall’s (1998) idea of operable definitions. For example, given a function f: X → Y and A Y, we define f -1 (A) = { x ∊ X | f(x) ∊A}. An operable interpretation would say, “If you have b∊ f -1(A), then you can write f(b) ∊A and vice versa.” One might think that this sort of association of a definition with an operable form would be unnecessary. However, we have found that for some students doing so is not easy. Indeed, we have also noted instances in which students have had both a definition and an operable interpretation available, but still did not act appropriately. Apparently, actually implementing an operable interpretation is distinct from knowing that one could implement it.
Psychological considerations
Much of proof construction and its teaching and learning can be explained, or even guided, by psychological considerations. Here are a few ideas/structures we call on. Working memory includes the central executive, the phonological loop, the visuospatial sketchpad, and an episodic buffer (Baddeley, 2000) and makes cognition possible. It is involved in learning and attention and has limited capacity which, when exceeded, produces errors and oversights. There are several kinds of consciousness but we will always mean phenomenal consciousness -- approximately, awareness of experience. There are (at least) two systems of cognition that operate in parallel. S1 cognition is fast, unconscious, automatic, effortless, evolutionarily ancient, and places little burden on working memory. In contrast, S2 cognition is slow, and conscious. It requires attention, is effortful, evolutionarily recent, and burdens working memory (Stanovich & West, 2000). System 2 may monitor System 1 and may sometimes take over. The idea includes that S1 and S2 have some underlying causal structure/mechanism. Furthermore, S1 is probably a system of systems (Stanovich, 2009).
Proof construction as a sequence of actions
Proof construction can be seen as a sequence of actions which can be physical (e.g., writing a line of the proof ) or mental (e.g., changing one’s focus or trying to recall a relevant theorem). A sequence of all of the actions that eventually leads to a proof is usually considerably longer than the final proof text itself and often proceeds in a different order. For an example of how circuitous this can be, see Dr. G s’ ultimately successful proving episode in Selden and Selden (2014). This fine-grained action approach facilitates noticing which beneficial student proving actions to encourage, and which detrimental student proving actions, to discourage.
Each action in a proof construction arises from a situation in the partly completed proof. The situation may be interpreted by the prover by drawing on information from long-term memory and a warrant for the acton may be developed. Interpreted situations are mental states and so are unobservable. However, a teacher can often infer an interpreted situation from observing the partly completed proof.
Behavioral schemas
If, during several proof constructions in the past, similar situations have corresponded to similar reasoning/warrants leading to similar actions, then, just as in classical associative learning (Machamer, 2009), a link may be learned between them, so that another similar situation evokes the corresponding action in future proof constructions without the need for the earlier warrant or intermediate reasoning. Using such situation-action links strengthens them, and after sufficient practice/experience, they can become overlearned, and thus automated (Morsella, 2009). We call automated situation-action links behavioral schemas (Selden, McKee, & Selden, 2010).
We also view behavioral schemas as belonging to a person’s knowledge base. They can be considered as partly conceptual knowledge (recognizing and interpreting the situation) and partly procedural knowledge (the action), and as related to Mason and Spence’s (1999) idea of “knowing-to-act in the moment”. In using a situation-action link or a behavioral schema, both the situation and the action (or its result) seem always to be at least partly conscious.
Here is a hypothetical example of one such possible behavioral schema that could conserve resources. One might be starting to prove a statement having a conclusion of the form p or q. This would be the situation at the beginning of the proof construction. If one had encountered this situation a number of times before, one might readily take an appropriate action, namely, in the written proof assume not p and prove q or vice versa. While this action can be warranted by logic (if not p then q, is equivalent to, p or q), there would no longer be a need to bring the warrant to mind.
It is our contention that, by forming behavioral schemas, large parts of proof construction skill can be automated, that is, that one can facilitate university students in turning what has been regarded as parts of S2 cognition into S1 cognition. Doing this would make more resources, such as working memory, available for such high cognitive demand tasks as the truly hard problems that need to be solved to complete many proofs.
The idea that much of the deductive reasoning that occurs during proof construction could become automated may be counterintuitive because many psychologists (e.g., Schechter, 2012), and (given the terminology) probably many mathematicians, assume that deductive reasoning is largely S2. We think that for successful students this change now sometimes happens naturally and implicitly, but with teaching, could be greatly enhanced.
influence a person’s actions, but such influence is normally not precise enough for doing mathematics. (6) Behavioral schemas are acquired (learned) through (possibly tacit) practice. That is, to acquire a beneficial schema a person should actually carry out the appropriate action correctly a number of times – not just understand its appropriateness. Changing a detrimental behavioral schema requires similar, perhaps longer, practice.
Feelings and proof construction
The words “feelings” and “emotions” are often used more or less interchangeably. Both appear to be conscious reports of unconscious mental states, and each can, but need not, engender the other. We will follow Damasio (2003) in separating feelings from emotions with emotions expressed by physical states, such as temperature, facial expression, blood pressure, pulse rate, perspiration, and so forth, while feelings are not (Damasio, 2003, pp. 67-70). That is, feelins are conscious mental states, rather than physical states. Feelings such as a feeling of knowing can play a considerable role in proof construction (Selden, McKee, & Selden, 2010). For example, one might experience a feeling of knowing that one has seen a theorem useful for constructing a proof, but not be able to bring it to mind at the moment. Such feelings of knowing can guide cognitive actions; for example, they can influence whether one continues a search or aborts it (Clore, 1992, p. 151). We call such feelings that can influence cognition cognitive feelings. When we speak of feelings here, we mean non-emotional cognitive feelings.
For the nature of feelings, we follow Mangan (2001), who has drawn somewhat on William James (1890). Feelings seem to be summative in nature and to pervade one’s whole field of consciousness at any particular moment. For example, to illustrate what it might mean for a feeling to pervade one’s whole field of consciousness, consider a hypothetical student taking a test with several other students in a room with a window. If, at a particular time, the student looks at his test, then towards the other students, and finally out of the window, at each of the three moments he or she perceives external information from only that moment. But if the student feels confident (i.e., has a feeling of knowing) that he or she will do well on the test during one of these moments, then he or she will also feel confident during the other two. This suggests that feelings are widely available to be focused on and can directly influence action.
Finally, we conjecture that feelings may eventually be found to play a larger role in proof construction than they as yet have. They provide a direct link between the conscious mind and the structures and possible actions of the unconscious mind, which has not been well studied in the proving context.
Local memory
One might think that proof construction consists mainly of communication with others or oneself using speech, vision, etc., or their inner versions (Sfard, 2010). That is, it is mainly conscious. We take a somewhat 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 (i.e., bring something to mind) 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 can be easily recalled. Often, in attempting a long 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 may itself provide a useful flexibility. When information is activated and then returned to memory, it may 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, especially those that call on psychology, which we think has developed in ways useful in mathematics education since the cognitive revolution around the mid 20th century.
References
Baddeley, A. (2000). The episodic buffer: A new component of working memory? Trends in Cognitive Science, 4(11), 417-423.
Bandura, A. (1995). Self-efficacy in changing societies. Cambridge: Cambridge University Press.
Bargh, J. A. (1994). The four horseman of automaticity: Awareness, intention, efficiency and control in social cognition. In R. Wyer & T. Srull (Eds.), Handbook of social cognition, Second Edition, Vol. 1 (pp. 1-40). Mahwah, NJ: Lawrence Erlbaum Associates.
Bills, L., & Tall, D. (1998). Operable definitions in advanced mathematics: The case of the least upper bound. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 104-111). Stellenbosch, South Africa: University of Stellenbosch.
Clore, G. L. (1992). Cognitive phenomenology: Feelings and the construction of judgment. In L. L. Martin & A. Tesser (Eds.), The construction of social judgments (pp. 133-162). Hillsdale, NJ: Lawrence Erlbaum Associates.
Damasio, W. (2003). Looking for Spinoza: Joy, sorrow, and the feeling brain. Orlando, FL: Harcourt.
Machamer, P. (2009).Learning, neuroscience, and the return to behaviorism. In J. Bickle (Ed.), The Oxford handbook of philosophy and neurosciences (pp. 166-176). Oxford: Oxford University Press.
Mangan, B. (2001). Sensation’s ghost: The non-sensory ‘fringe’ of consciousness. Psyche, 7(18).
Retrieved September 29, 2009, from http://psyche.cs.monash.edu.au/v7/psyche-7-18-mangan.html.
Mason, J., & Spence, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing-to-act in the moment. Educational Studies in Mathematics, 28(1-3), 135-161.
Morsella, E. (2009). The mechanisms of human action: Introduction and background. In E. Morsella, J. A. Bargh, & P. M. Goldwitzer (Eds.), Oxford handbook of human action (pp. 1-34). Oxford: Oxford University Press.
Schechter, J., (2013). Deductive reasoning. In H. Pashler (Ed.), Encyclopedia of the mind. Los Angeles, CA: SAGE Publications.
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. (in press). An example of a linguistic obstacle to proof construction: Dori and
the hidden double negative. Proceedings of the 19th Annual Conference on Research in Undergraduate Mathematica Education.
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. (2013). The genre of proof. In M. N. Fried & T. Dreyfus (Eds.), Mathematics and mathematics education: Searching for common ground (pp. 248-251). New York: Springer. Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell
whether an argument proves a theorem? Jounal 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 Conference of the International Group for the Psychology of Mathematica Education, Vol. 4 (pp. 145-152) Ankara, Turkey: Middle East Tech. University. Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies
in Mathematics, 29(2), 123-151.
Selden, A., Selden, J., Hauk, S., & Mason, A. (2000). Why can’t calculus students access their knowledge to solve non-routine problems? In A. H. Schoenfeld., J. Kaput, & E. Dubinsky, (Eds.),
Research in collegiate mathematics education. IV. Issues in mathematics education: Vol. 8.
(pp. 128-153). Providence, RI: American Mathematical Society.
Sfard, A. (2010). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge, UK: Cambridge University Press.
Stanovich, K. E. (2009). Distinguishing the reflective, algorithmic, and autonomous minds: Is it time for a tri-process theory? In J. Evans, & K. Frankish (Eds.), In two minds: Dual processes and beyond (pp. 55–88). Oxford, UK: Oxford University.
