A Perspective for University Students’ Proof Construction
John Selden Annie Selden
New Mexico State University New Mexico State University
This theoretical paper suggests a perspective for understanding university students’ proof construction. It is based on the ideas of conceptual and procedural knowledge, explicit and implicit learning, cognitive feelings and beliefs, behavioral schemas, automaticity, working memory, consciousness, and System 1 and System 2 cognition. In particular, we discuss proving actions, such as the construction of proof frameworks that could be automated, thereby reducing the burden on working memory and enabling university students to devote more resources to the truly hard parts of proofs.
Key words: proof construction, behavioral schemas, automaticity, consciousness, System 1 and System 2 cognition
Introduction
In this theoretical paper we suggest a perspective for understanding university mathematics students’ proof constructions and for improving and facilitating their abilities and skills for constructing proofs We are interested in how various types of knowledge (e.g., implicit, explicit, procedural, conceptual) are used during proof construction, in how such knowledge can be constructed, and in how one can control and direct one’s own thinking. If that were better understood, then it might be possible to better facilitate university students’ learning through doing, that is, through proof construction experiences. Although one can learn some things from lectures, this is almost certainly not the most effective, or efficient, way to learn proof construction, which is a kind of activity. Indeed, inquiry-based transition-to-proof courses seem to be more effective than lecture-based courses (e.g., Smith, 2006). In this paper, we are referring just to inquiry into proof construction, not into theorem or definition generation, although these are also interesting areas of study. These ideas emerged from an ongoing sequence of design experiment courses meant to teach proof construction in a medium-sized US PhD-granting university.
The Courses
sets, functions, real analysis, and algebra. The graduate course also included some topology. More information on the graduate course can be found in Selden, McKee, and Selden (2010, p. 207).
Introductory Psychological Considerations
Much has been written in the psychological, neuropsychological, and neuroscience literature about ideas of conceptual and procedural knowledge, explicit and implicit learning, automaticity, working memory, consciousness, beliefs and feelings such as self-efficacy, and System 1 (S1) and System 2 (S2) cognition (e.g., Bargh & Chartrand, 2000; Bargh & Morsella, 2008; Bor, 2012; Cleeremans, 1993; Hassin, Bargh, Engell, & McCulloch, 2009; Stanovich, 2009; Stanovich & West, 2000).
In trying to relate these ideas to proof construction, we have discussed procedural knowledge, situation-action links, and behavioral schemas (Selden, McKee, & Selden, 2010; Selden & Selden, 2011). However, more remains to be done in order to weave these ideas into a coherent perspective.
In doing this, two key ideas are working memory and the roles that S1 and S2 cognition can play in proof construction. Working memory includes the central executive, the phonological loop, the visuospatial sketchpad, and an episodic buffer (Baddeley, 2000) and makes cognition possible. It is related to learning and attention and has a limited capacity which varies across individuals. When working memory capacity is exceeded, errors and oversights are likely to occur. The idea behind S1 and S2 cognition is that there are two kinds 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, conscious, effortful, evolutionarily recent, and puts considerable call on working memory (Stanovich & West, 2000). Also, System 2 cognition is thought to monitor System 1 cognition and to sometimes take over cognition when System 1 appears to be going astray. Of the several kinds of consciousness, we are referring to phenomenal consciousness—approximately, awareness of experience.
In both proving and learning to prove, it appears to be important for both teachers and students to understand the various kinds of progress to be made and the kinds of tasks to be performed. We have divided these into two parts, one more directly related to producing a proof text, and the other more related to the psychological influences on cognition and the mind when producing that text. We turn now to the first of the two parts of the proposed perspective.
Part 1: Producing the Proof Text The Genre of Proofs
validity is often seen to be independent of time, place, and author. Details can be found in Selden and Selden (2013).
Structure in Proofs
A proof can be divided into a rhetorical part and a problem-centered part. The formal-rhetorical part is the part of a proof that depends only on unpacking and using the logical structure of the statement of the theorem, associated definitions, and earlier results. In general, this part does not depend on a deep understanding of, or intuition about, the concepts involved or on genuine problem solving in the sense of Schoenfeld (1985, p. 74). Instead it depends on a kind of “technical skill”. We call the remaining part of a proof the problem-centered part. It is the part that does depend on genuine problem solving, intuition, heuristics, and a deeper understanding of the concepts involved (Selden & Selden, 2011).
Perhaps the most common feature of writing the formal-rhetorical part of a proof or subproof is what to do when the theorem statement starts with a universal quantifier, such as, “For all x ϵ X …”. One normally starts its proof with “Let x ϵ X”, meaning that the variable x in the statement will be regarded in the proof as “fixed, but arbitrary”, that is, as a single unspecified constant. This facilitates constructing proofs about infinite sets and changes most logic required from predicate to propositional calculus, which we think is closer to common sense reasoning.
We have noticed informally that a considerable number of our beginning transition-to-proof course students tend not to carry out the above action (of considering x as fixed, but arbitrary). Also, we have reported on an interview with a returning graduate student, Mary, who said that because of her real analysis teacher’s instructions, she (successfully) carried out the action of considering a fixed, but arbitrary ɛ>0 in her proofs and even appended a reason why it was needed. That is, she would write, at the end of her proofs, “Because ɛ was arbitrary, the theorem has been proved for all ɛ.” She reported to us that she did not feel that doing so was appropriate for about half a semester (Selden, McKee, & Selden, 2010, p. 209). Mary’s difficulty suggests that some ideas in beginning proof construction may be adopted only slowly by some students even when they carry out the associated actions and provide appropriate warrants.
Proof Frameworks
A major feature that can help one write the formal-rhetorical part of a proof is what we have called a proof framework,1 of which there are several kinds, and in most cases, both a first-level and a second-level framework. For example, given a theorem of the form “For all real numbers x, if P(x) then Q(x)”, a first-level proof framework would be “Let x be a real number. Suppose P(x). … Therefore Q(x),” with the remainder of the proof ultimately replacing the ellipsis. A second-level framework can often be obtained by “unpacking” the meaning of Q(x) and putting the second-level framework between the lines already written for the first-level framework. Thus, the proof would “grow” from both ends toward the middle, instead of being written from the top down. In case there are subproofs, these can be handled in a similar way. A more detailed explanation with examples can be found in Selden, Benkhalti, and Selden (2014). A proof need not show evidence of a proof framework to be correct. However, we have noticed that use of proof frameworks tends to help novice university mathematics students write correct, well-organized, and easy-to-read proofs (McKee, Savic, Selden, & Selden, 2010).
1
Operable Interpretations
Another feature that can help one write the formal-rhetorical part of a proof is converting definitions and previously proved results into operable interpretations. These interpretations are similar to Bills and Tall’s (1998) idea of operable definitions. For example, in the courses described above, 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 translation into an operable form would be unnecessary or easy, especially because each symbol in { x X | f(x) A} can be translated into a word in a one-to-one way. However, we have found that for some students this is not easy, even when the definition can be consulted. We have also noted instances in which students have had available both a definition and an operable interpretation, but still did not act appropriately. Thus, actually implementing an operable interpretation is separate from knowing that one can implement it.
One can also have operable interpretations for situations in a partly completed proof. For example, when a conclusion is negatively phrased (e.g., a set is empty or a number is irrational), one might early in the proving process attempt a proof by contradiction. Also when the conclusion asserts the equivalence of two statements, or that two sets are equal, often the proof should be divided into two parts, in which there are two implications to prove. Finally, if in a partly completed proof, one has arrived at a statement of the form p or q, the proof can be divided into two cases, one assuming p and the other assuming q.
We suggest that students, or small groups of students, can and should develop some operable interpretations independently of a teacher. However, if or when this should be done in a particular course is a design problem.
Part 2: Psychological Features of Proof Construction
In this second, psychological part of the perspective, we view proof construction as a sequence of actions which can be physical (e.g., writing a line of the proof or drawing a sketch) or mental (e.g., changing one’s focus from the hypothesis to the conclusion or trying to recall a relevant theorem). The sequence of all of the actions that eventually lead to a proof is usually considerably longer than the final proof text itself. This fine-grained action approach appears to facilitate noticing which actions should be taken to write various parts of a proof correctly, which beneficial student proving actions to encourage, and which detrimental student proving actions to discourage.
Situations and Actions
What matters from the point of view of a student, who is learning to construct proofs, is the perceived situation. By a perceived, or inner, situation in proving, we mean a portion of a partly completed proof construction, perhaps including an interpretation drawn from long-term memory that can suggest a further action. The interpretation is likely to depend on recognition of the situation, which is easier than recall, perhaps because fewer brain areas are involved (Cabeza, et al., 1997). An inner situation is unobservable. However, a teacher can often infer an inner situation from the corresponding outer situation, that is, from the, usually written, portion of a partly completed proof.
(Selden & Selden, 2014). We also include “meta-actions” meant to alter one’s own thinking, such as focusing on another part of a developing proof construction.
Examples of Inner Situations
Norton and D’Ambrosio (2008) have described what amounts to an illustration of the distinction between an inner situation and an outer situation for two middle school students, Will and Hillary, who viewed the same external situation involving a fraction such as 2/3. Hillary had (in her knowledge base) a partitive fractional scheme, as well as a part-whole fractional scheme, while Will had only the second scheme. This caused Will and Hillary to “see” the external situation differently, that is, to have differing inner situations, and hence, to act differently. In particular, Hillary was able to solve a problem that Will could not. Will could only solve the problem after he had developed a partitive fractional scheme, and then presumably experienced a richer inner situation.
In the above illustration, Will’s internal view of the external situation could not be enriched by a concept of fraction that was not yet available in his knowledge base. But that is not the only way for two persons to have significantly differing inner situations for the same external situation. Selden, Selden, Hauk and Mason (2000) reported on mid-level undergraduates in a first course on differential equations attempting to solve moderately non-routine beginning calculus problems. Many students did not solve the problems, even though the solutions called on familiar calculus facts, as ascertained by a subsequent routine test. For example, one problem asked: Does x21 + x19 - x -1 + 2 = 0 have any roots between –1 and 0? Why or why not? This problem could not be solved by simple algebraic techniques, but could be solved by noticing that f(-1) > 0 and f ׳(x) > 0 on [-1,0). Selden, Selden, Hauk, and Mason (2000) were able to show that a number of the students could not solve moderately non-routine problems for which they had adequate information in their knowledge bases. Apparently, the students were unable to bring this information to mind, that is, bring it into consciousness, because their knowledge bases lacked certain links between “kinds” of problems and information that might be useful in solving them. Thus, the students were unable to enrich their views of the external situations to create inner situations, including connections to resources such as a function is increasing where its derivative is positive, that might have stimulated the enactment of appropriate problem-solving behaviors.
Situation-Action Links, Automaticity, and Behavioral Schemas
If, during several proof constructions in the past, similar situations have corresponded to similar reasoning 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 intermediate reasoning. Using such situation-action links strengthens them, and after sufficient practice/experience, they can become overlearned, and thus, automated. We call automated situation-action links behavioral schemas.
Features of Automaticity
Regarding skill learning and automaticity, it is known that the neural correlates of novel actions are distinct from those of actions that are overlearned, such as driving or tying one’s shoes. Regions [of the brain] primarily responsible for the control of movements during the early stages of skill acquisition are different from the regions that are activated by overlearned actions. In essence, when an action becomes automatized, there is a ‘gradual shift from cortical to subcortical involvement …’ (p. 13).
Because cognition often involves inner speech, which in turn is connected with the physical control of speech production, the above information on the brain regions involved in physical skill acquisition is at least a hint that forming behavioral schemas not only converts S2 cognition into S1 cognition, but also suggests that different parts of the brain are involved in access and retrieval.
In particular, there may be a shift from cortical to subcortical involvement. Neural activity associated with doing mathematics is generally located in the frontal and parietal lobes (Norton, 2015). Also, more resources (in both the frontal and parietal lobes) have to work in concert when a person is doing tasks with higher cognitive demand because those tasks require greater use of working memory and executive function (Sauseng, Klimensch, Schabus, & Doppelmayr, 2005). Thus, it is important to conserve those resources for working on high cognitive demand tasks such as the truly hard parts of problems or proofs.
Behavioral Schemas as a Kind of Knowledge
We 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”. We suggest that, in using a situation-action link or a behavioral schema, almost always both the situation and the action (or its result) will be at least partly conscious.
Here is an example of one such possible behavioral schema that can 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 large parts of proof construction skill can be automated, that is, that one can facilitate mid-level university students in turning parts of S2 cognition into S1 cognition, and that doing so 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.
The Genesis and Enactment of Behavioral Schemas
enactment of its own behavioral schema that was “triggered” by the action of the preceding schema in the procedure. Multi-digit subtraction of natural numbers is an example of such a compound behavioral schema. When viewed in a large grain-size (mainly compound actions), behavioral schemas might also be regarded as habits of mind (Margolis, 1993). Habits of mind are similar to physical habits, and people are similarly often unaware of, or do not remember, them (as habits).
It appears that consciousness plays an essential role in triggering the enactment of behavioral schemas for doing mathematics. This is reminiscent of the role consciousness plays in reflection. It is hard to see how reflection, treated as selectively re-presenting past experiences, could be possible without first having had the experiences and without each experience triggering the next. We have developed the following six-point theoretical sketch of the genesis and enactment of behavioral schemas (Selden, McKee, & Selden, 2010, pp. 205-206).
1) Within very broad contextual considerations, behavioral schemas are immediately available. They do not normally have to be remembered, that is, searched for and brought to mind before their application. This distinguishes them from most conceptual knowledge and episodic and declarative memory, which generally do have to be recalled or brought to mind before their application.
2) Simple behavioral schemas operate outside of consciousness. One is not aware of doing anything immediately prior to the resulting action – one just does it. Thus, the enactment of a simple behavioral schema that leads to an error is not under conscious control, and we should not expect that merely understanding the origin of the error, or being shown a counterexample, would prevent future reoccurrences. Compound behavioral schemas are also largely not under conscious control.
3) Behavioral schemas tend to produce immediate action, which may lead to subsequent action. One becomes conscious of the action resulting from a behavioral schema as it occurs or immediately after it occurs.
4) Behavioral schemas were once actions arising from situations through warrants that no longer need to be brought to mind. So one might reasonably ask, can several behavioral schemas be “chained together” and act outside of consciousness, as if they were one schema? For most persons, this seems not to be possible. If it were so, one would expect that a person familiar with solving linear equations could start with 3x + 5 = 14, and without bringing anything else to mind, immediately say x = 3. We expect that very few (or no) people can do this, that is, consciousness of the results of enacting the individual schemas is required
5) An action due to a behavioral schema depends on conscious input, at least in large part. In general, a stimulus need not become conscious to influence a person’s actions, but such influence is normally not precise enough for doing mathematics. For example, in many psychological experiments a stimulus-response connection is considered established when its occurrence departs from chance over multiple trials.
Implicit Learning of Behavioral Schemas
It appears that the process of learning a behavioral schema can be implicit, although the situation and the action are in part conscious. That is, a person can acquire a behavioral schema without being aware that it is happening. Indeed, such unintentional, or implicit, learning happens frequently and has been studied by psychologists and neuroscientists (e.g., Cleeremans, 1993; Cleeremans & Jiménez, 2001). In the case of proof construction, we suggest that with the experience of proving a considerable number of theorems in which similar situations occur, an individual might implicitly acquire a number of relevant beneficial behavioral schemas. As a result he or she might simply not have to think quite so deeply as before about certain portions of the proving process, and might, as a consequence of having more working memory available, take fewer “wrong turns”.
Something similar has been described in the psychology literature regarding the automated actions of everyday life. For example, an experienced driver can reliably stop at a traffic light while carrying on a conversation. But not all implicitly learned automated actions are positive. For example, a person can develop stereotypical behavior without being aware of the acquisition process and can even be unaware of its triggering situations (Chen & Bargh, 1997). This suggests that we should consider the possibility of mathematics students implicitly developing similarly unintended negative situation-action links, and the corresponding detrimental behavioral schemas, during mathematics learning, and in particular, during proof construction.
Detrimental Behavioral Schemas
We begin with a simple and perhaps familiar algebraic error. Many teachers can recall having a student write (a2 + b2) = a + b, giving a counterexample to the student, and then having the student make the same error somewhat later. Rather than being a misconception (i.e., believing something that is false), this may well be the result of an implicitly learned detrimental behavioral schema. If so, the student would not have been thinking very deeply about this calculation when writing it. Furthermore, having previously understood the counterexample would also have little effect in the moment. It seems that to weaken/remove this particular detrimental schema, the triggering situation of the form (a2 + b2) should occur a number of times when the student can be prevented from automatically writing “= a + b” in response. However, this might require working with the student individually on a number of examples mixed with nonexamples.
For an example of an apparently implicitly learned detrimental behavioral schema for proving, we turn to Sofia, a first-year graduate student in one of the above mentioned graduate courses. Sofia was a diligent student, but as the course progressed what we came to call an “unreflective guess” schema emerged (Selden, McKee, & Selden, 2010, pp. 211-212). After completing just the formal-rhetorical part of a proof (essentially a proof framework) and realizing there was more to do, Sofia often offered a suggestion that we could not see as being remotely helpful. At first we thought she might be panicking, but on reviewing the videos there was no evidence of that. A first unreflective guess tended to lead to another, and another, and after a while, the proof would not be completed.
considerably improved in her proving ability by the end of the course (Selden, McKee, & Selden, 2010, p. 212).
Feelings and Proof Construction
The word “feeling” is used in a variety of ways in the literature so we will first indicate how we will use it. Often feelings and emotions are used more or less interchangeably--both appear to be conscious reports of unconscious mental states, and each can, but need not, engender the other. However, we will follow Damasio (2003) in separating feelings from emotions because emotions are expressed by observable physical characteristics, such as temperature, facial expression, blood pressure, pulse rate, perspiration, and so forth, while feelings are not. Indeed, Damasio has described a brain operation during which the patient was awake and could report on her state of mind. She experienced both feelings and emotions, but clearly at different times (Damasio, 2003, pp. 67-70).
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 because 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 will 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 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 especially available to be focused on and can directly influence action.
Additional (nonemotional cognitive) feelings, different from a feeling of knowing, are a feeling of familiarity and a feeling of rightness. Mangan (2001) has distinguished these. Of the former, he wrote that the “intensity with which we feel familiarity indicates how often a content now in consciousness has been encountered before”, and this feeling is different from a feeling of rightness. It is rightness, not familiarity, that is “the feeling-of-knowing in implicit cognition”. Rightness is “the core feeling of positive evaluation, of coherence, of meaningfulness, of knowledge”. In regard to a feeling of rightness, Mangan has written that “people are often unable to identify the precise phenomenological basis for their judgments, even though they can make these judgments with consistency and, often, with conviction. To explain this capacity, people talk about ‘gut feelings’, ‘just knowing’, hunches, [and] intuitions”. Often such quick judgments (i.e., the results of S1 cognition) can be correct, but they sometimes need to be checked, that is, S2 cognition needs to “kick in” and override such incorrect quick judgments.
Self-Efficacy
In order to prove harder theorems, ones with a substantial problem-centered part, students need to persist in their efforts, and such persistence is facilitated by a sense of self-efficacy. According to Bandura (1995), self-efficacy is “a person’s belief in his or her ability to succeed in a particular situation” (Bandura, 1995). Of developing a sense of self-efficacy, Bandura (1994) stated that “The most effective way of developing a strong sense of self-efficacy is through mastery experiences,” that performing a task successfully strengthens one’s sense of self -efficacy. Also, according to Bandura, “Seeing people similar to oneself succeed by sustained effort raises observers’ beliefs that they too possess the capabilities to master comparable activities to succeed.”
According to Bandura (1994), individuals with a strong sense of self-efficacy: (1) view challenging problems as tasks to be mastered; (2) develop deeper interest in the activities in which they participate; (3) form a stronger sense of commitment to their interests and activities; and (4) recover quickly from setbacks and disappointments. In contrast, people with a weak sense of self-efficacy: (1) avoid challenging tasks; (2) believe that difficult tasks and situations are beyond their capabilities; (3) focus on personal failings and negative outcomes; and (4) quickly lose confidence in personal abilities.
Bandura’s ideas “ring true” with our past experiences as mathematicians teaching courses by the classical Moore Method (Mahavier, 1999). Classical Moore Method advanced undergraduate or graduate courses are taught from a brief set of notes2 consisting of definitions, a few requests for examples, statements of major results, and those lesser results needed to prove the major ones. Exercises of the sort found in most textbooks are largely omitted. In class meetings, the professor invites individual students to present their original proofs and then only very briefly comments on errors.3 Once students are able to successfully prove their first few theorems, they often progress very rapidly in their proving ability, even without any apparent explicit teaching, and persist even when subsequent proofs are more complex or require creating new mathematical objects or lemmas.
Why should this be? We conjectured then, and also conjecture now, that students obtained a sense of self-efficacy from having proved their first few theorems successfully, and that this sense of self-efficacy grew over time and helped them persist in explorations, re-examinations, and validations when these were needed in proving subsequent difficult theorems.
Seeing Similarities, Searching, and Exploring
How does one recognize situations as similar? Different people see situations as similar depending both upon their past experiences and upon what they choose to, or happen to, focus on. While similarities can sometimes be extracted implicitly (Markman & Gentner, 2005), teachers may occasionally need to direct students’ attention to relevant proving similarities. On the other hand, such direction should probably be as little as possible because the ability to autonomously see similarities can, and should, be learned.
For example, it would be good to have general suggestions for helping students “see”, without being told, that the situations of a set being empty (i.e., having no elements), of a number being irrational (i.e., not rational), and of the primes being infinite (i.e., not finite) are similar.
2
A brief set of notes for the first semester of an undergraduate abstract algebra course taught by the classical Moore Method is provided in the Appendix of Selden and Selden (1978).
3
That is, while the three situations—empty, irrational, and infinite—may not seem similar on the surface, they can be rephrased to expose the existence of a negative definition. And, unless students rephrase these situations, it seems unlikely that they would see this similarity and link these situations (when they occur as conclusions to theorems to prove) to the action of beginning a proof by contradiction.
In addition to automating small portions of the proving process, such as writing proof frameworks, we would also like to enhance students’ searching skills, that is, their tendency to look for helpful previously proved results. We would also like to enhance students’ tendency to “explore” various possibilities when they don’t know what to do next. In a previous paper (Selden & Selden, 2014, p. 250), we discussed the kind of exploring entailed in proving the rather difficult (for students) Theorem: If S is a commutative semigroup with no proper ideals, then S is a group. Well before such a theorem appears in a set of course notes like ours, one might provide students with advice, or better yet, experiences showing the value of exploring what is not obviously useful. For example, one could discuss the usefulness of starting with abba = e, for arbitrary a, b S, when attempting to show commutativity of a semigroup with identity e, having s2 = e, for all s S (as discussed in Selden, Benkhalti, and Selden, 2014).
Using this Perspective to Analyze Students’ Proof Attempts
We hope the following analysis that highlights actions in proofs and proving will provide insights into what might be emphasized when teaching particular groups of students.4
In examining students’ proof attempts, we are not just looking for mistakes or misconceptions, but rather we are looking for possible detrimental actions, possible beneficial actions, and for potential beneficial actions not taken. Below we give two examples of how we have analyzed students’ (incorrect) proof attempts (Selden, Benkhalti, & Selden, 2014).
Example 1. The student had attempted to prove the following on an examination. Theorem: Let S be a semigroup with identity e. If, for all s in S, ss = e, then S is commutative. Here we are examining, and analyzing, the student’s written work using our theoretical perspective of actions. The lines are numbered for convenient reference. The student’s accompanying scratch work consisted of the definitions of identity and commutative. The proof went as follows:
1. Let S be a semigroup with an identity element, e. 2. Let s S such that ss = e.
3. Because e is an identity element, es = se = s. 4. Now, s = se = s(ss).
5. Since S is a semigroup, (ss)s = es = s. 6. Thus es = se.
7. Therefore, S is commutative. QED.
Analysis. Line 2 only hypothesizes a single s and should have been, “Suppose for all s S, ss = e.” With this change, Lines 1, 2, and 7 are the correct first-level framework.5 There is no second-level framework between Lines 2 and 7. This was a beneficial action not taken and should have been: “Let aS and bS. … Then ab = ba.” inserted between Lines 2 and 7.
4
We are not suggesting that this kind of proof analysis be used as a way of grading, or marking, students’ work. Rather we are suggesting that this kind of proof analysis might be helpful for teachers and course designers.
5
Line 3 violates the genre of proof by including a definition easily available outside of the proof. Lines 3, 4, 5, and 6 are not wrong, but do not move the proof forward. Writing these lines may have been detrimental actions that subconsciously primed the student’s feeling that something useful had been accomplished, and thus, may have brought the proving process to a premature close.
Example 2. Next we consider another student’s proof attempt of the following theorem on an examination. Theorem. Let S and T be semigroups and f:S→T be a homomorphism. If G is a subset of S and G is a group with identity e, then f(G) is a group. Here, again, we are examining, and analyzing, the student’s written work using our theoretical perspective of actions. The lines of the student’s proof attempt are numbered for convenient reference.
1. Let S and T be semigroups and f:S→T be a homomorphism. 2. Suppose G S and G is a group with identity e.
3. Since G is a group and it has identity e, then for each element g in G there is an element g' in G such that gg' = g'g = e.
4. Since f is a homomorphism, then for each element x ϵ S and y ϵ S, f(xy)=f(x)f(y). 5. Since G S, then f(gg')=f(g)f(g'). So f(gg') = f(g'g) = f(e).
6. So f(G) has an element f(e) since f is a function. 7. Therefore, f(G) is a group. QED.
Analysis. The student has written the first-level framework, namely Lines 1, 2, and 7, correctly, assuming that Line 7 was written immediately after writing the first two lines. To complete the proof framework, the student should have unpacked the last line and written the second-level framework. That is, the student should have considered f(G) and noted that there are three parts to prove, namely, that f(G) is a subsemigroup, that there is an identity in f(G), and that each element in f(G) has an inverse in f(G). This unpacking of the conclusion is a beneficial action not taken.
Instead, in Line 3, the student wrote into the proof the definition of G being a group, and in Line 4, stated what it means for f to be a homomorphism. These actions are not wrong, but they do not move the proof forward and are detrimental because they can give the student a feeling that something useful has been done. Perhaps, in Lines 5 and 6, the student was trying to show the existence of an identity and inverses in f(G) and was unsuccessful, but one cannot know this. If the second-level proof framework had been written, the proof would have been reduced to three easier parts, each of which also has a proof framework, and this might have been helpful to the student.
Teaching and Research Considerations
Also, helping students interpret formal mathematical definitions so that these become operable might be another place to start. This would be helpful because one often needs to convert a definition into an operable interpretation in order to use it to construct a second-level proof framework. However, eventually students should learn to make such interpretations themselves.
Finally, we believe this particular perspective on proving, using situation-action links and behavioral schemas, together with information from psychology and neuroscience, is mostly new to the field and is likely to lead to additional insights.
References
Baddeley, A. (2000). The episodic buffer: A new component of working memory? Trends in Cognitive Science, 4(11), 417-423.
Bandura, A. (1994). Self-efficacy. In V. S. Ramachaudran (Ed.), Encyclopedia of human behaviour (Vol. 4, pp. 71-81). New York: Academic Press.
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.
Bargh, J. A., & Chartrand, T. L. (2000). Studying the mind in the middle: A practical guide to priming and automaticity research. In H. T. Reid & C. M. Judd (Eds.), Handbook of research methods in social psychology (pp. 253-285). New York: Cambridge University Press.
Bargh, J. A., & Morsella, E. (2008). The unconscious mind. Perspectives on Psychological Science, 3(1), 73-79.
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.
Bor, D. (2012). The ravenous brain. New York: Basic Books.
Cabeza, R., Kapur, S., Craik, F. I. M., McIntosh, A. R., Houle, S., & Tulving, E. (1997). Functional neuroanatomy of recall and recognition: A PET study of episodic memory. Journal of Cognitive Neuroscience, 9, 254-265.
Chen, M., & Bargh, J. A. (1997). Nonconscious behavioral confirmation processes: The self-fulfilling consequences of automatic stereotype activation. Journal of Experimental Social Psychology, 33, 541-560.
Cleeremans, A. (1993). Mechanisms of implicit learning: Connectionist models of sequence processing. Cambridge, MA: MIT Press.
Cleeremans, A., & Jiménez, L. (2001). Implicit learning and consciousness: A graded, dynamic perspective.Retrieved December 1, 2014, from
http://srsc.ulb.ac.be/axcWWW/papers/pdf/01-AXCLJ.pdf.
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.
Hassin, R. R., Bargh, J. A., Engell, A. D., & McCulloch, K. C. (2009). Implicit working memory. Conscious Cognition, 18(3), 665-678.
James, W. (1890). The Principles of Psychology. New York: Holt.
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.
Mahavier, W. S. (1999). What is the Moore Method? PRIMUS, 9, 339-354.
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.
Margolis, H. (1993). Paradigms and barriers: How habits of mind govern scientific beliefs. Chicago: University of Chicago Press.
Markman, A. B., & Gentner, D. (2005). Nonintentional similarity processing. In T. Hassin, J. Bargh, & J. Uleman, The new unconscious (pp. 107-137). New York: Oxford University Press.
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.
McKee, K., Savic, M., Selden, J., and Selden, A. (2010). Making actions in the proving process explicit, visible, and “reflectable”. In Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics Education. Retrieved Feb. 10, 2011, from
http://sigmaa.maa.org/rume/crume2010/Archive/McKee.pdf .
Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27(3), 249-266.
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.
Norton, A. (2015). Neural correlates for action-object theories. Proceedings of the 18th Annual Conference on Research in Undergraduate Mathematics Education. Retrieved February 27, 2015,from http://timsdataserver.goodwin.drexel.edu/RUME-2015/rume18_submission2.pdf. Norton, A., & D’Ambrosio, B. S. (2008). ZPC and ZPD: Zones of teaching and learning. Journal
for Research in Mathematics Education, 39(3), 220-246.
Sauseng, P., Klimensch, W., Schabus, M., & Doppelmayer, M. (2005) Fronto-parietal EEG coherence in theta and upper alpha reflect central executive functions of working memory. International Journal of Psychophysiology, 57, 97-103.
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. A., & Selden, J. (1978). Errors students make in mathematical reasoning. Bosphorus University Journal, 6, 67-87. Available online through www.academia.edu.
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. (2013). The genre of proof. In M. N. Fried & T. Dreyfus (Eds.), Mathematics and mathematics education: Searching for common ground (pp. 248-
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 the Eighth Congress of the European Society for Research in Mathematics Education [CERME8] (pp. 246-255). Ankara, Turkey: Middle East Technical University.
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.
Selden, J., Benkhalti, A., & Selden, A. (2014). An analysis of transition-to-proof course students’ proof constructions with a view towards course redesign. In T. Fukawa-Connolly, G. Karakok, K. Keene, & M. Zandieh (Eds.), Proceedings of the 17th Annual Conference on Research in Undergraduate Mathematics Education (pp. 246-259). Denver, Colorado. Retrieved March 1, 2015 from http://sigmaa.maa.org/rume/RUME17.pdf .
Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29, 123-151.
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 Mathematics Education, Vol. 4 (pp. 124-152). Ankara, Turkey: Middle East Technical University.
Smith, J. C. (2006). A sense-making approach to proof: Strategies of students in traditional and problem-based number theory courses. Journal of Mathematical Behavior, 25, 73-90.
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: Oxford University Press.
