AN ANALYSIS OF TRANSITION-TO-PROOF COURSE STUDENTS’ PROOF CONSTRUCTIONS WITH A VIEW TOWARDS COURSE REDESIGN
John Selden Ahmed Benkhalti Annie Selden New Mexico State New Mexico State New Mexico State
The purpose of this study was to gain knowledge about undergraduate transition-to-proof course students’ proving difficulties. We analyzed the final examination papers of students in one such course. Our perspective included drawing inferences about students’ sometimes automated links between situations and mental, as well as physical, actions. We have identified process, rather than mathematical content, difficulties such as not constructing a proof framework, not unpacking the conclusion, and not using definitions correctly. The ultimate goal is to contribute to an understanding of some of these kinds of difficulties as pedagogical content knowledge with which to teach or redesign transition-to-proof courses.
Key words: Transition-to-proof, Proof construction, Pedagogical content knowledge, Actions, Proof framework
This paper presents an analysis of transition-to-proof course students’ final examinations in an effort to describe some of their main proving difficulties. By inferring kinds of difficulties in students’ proof construction processes from their written proof attempts, and by focusing away from specific fields of mathematics, we begin to answer the question: How can the general proving process be taught so that it applies broadly to many fields of mathematics? For example, what knowledge, habits of mind, and self-efficacy will facilitate students’ proof construction processes and might be candidates for explicit teaching in a transition-to-proof course?
Analyzing student examination papers in the service of teaching and course redesign might seem unpromising for a typical, that is, content driven, mathematics course because there is already mathematical terminology to connect student examination difficulties with parts, or precursors, in a way useful in teaching. For example, consider student difficulties in finding d/dx sin x2. The chain rule and composition of functions come immediately to mind as things the student must be able to use.
In contrast, the proving difficulties we examined were mostly about the process of constructing a proof, not mathematical content. Thus new concepts and vocabulary may emerge to connect overall difficulties, for example, not being able to finish constructing a particular complete proof, with contributing parts or precursors, for example, the ability to use abstract definitions, in a way useful for teaching and course design. We turn now to some related prior research.
Literature Review
misconceptions in undergraduate abstract algebra students’ proof attempts. However, the difficulties reported there have little in common with those observed in this study. In addition, Weber’s (2001) study, contrasting undergraduate abstract algebra students with doctoral students in algebra, showed that the latter had strategic (content) knowledge to use in constructing abstract algebra proofs that the undergraduates did not have. Our study, in contrast, gives insight into the proving difficulties of relative beginners, that is, undergraduate students at the end of a to-proof course. We note that Moore (1994) observed a traditionally taught transition-to-proof course and reported seven student proving difficulties, some of which do overlap with our categories, although in general, our categories are more fine-grained. In addition, Baker and Campbell (2004) reported three observations of somewhat less sophisticated transition-to-proof course students. Selden and Selden (1995) did observe process difficulties in unpacking the logic of informal mathematical statements. They reported that informal statements, that is, those that departed from the simplest natural language rendering of predicate and propositional calculus were difficult for students to unpack and hence difficult to prove. This information was indeed used in designing our current course. In it we decided to write mathematical statements in the course notes rather formally so the students would not need to unpack their logical structure and could focus on the rest of the proof construction. This is not because we did not value learning to autonomously unpack the logical structure of statements, but because certain proving actions seem to call for self-efficacy (Selden & Selden, 2014), which can be encouraged by providing students early opportunities to succeed in constructing proofs. Thus, we gave building self-efficacy priority (Selden & Selden, 2014).
There is some additional literature that supports our theoretical perspective, but it is perhaps best understood in that section below.
Theoretical Perspective
First, we will suggest some psychologically-based ideas and then mention a few concepts that have emerged from earlier iterations of the course under consideration. We view the process of proof construction as a sequence of mental (e.g., “unpacking” the meaning of the conclusion in inner speech) or physical (e.g., drawing a diagram) actions. Such a sequence of actions is somewhat related to, and extends, what we have earlier called a “possible construction path” of a proof, illustrated in Selden and Selden (2009). The actions derive from a person’s nonobservable, and sometimes partly nonconscious, inner interpretation of usually outer and observable situations in a partly completed proof construction.
Inner interpretations cannot be observed, but they can be inferred, sometimes very convincingly. Norton and D’Ambrosio (2008, pp. 14-15) provide an illustration of this 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 interpretations, and hence to act differently. Hillary was able to solve the problem, but Will couldn’t. Will could only solve the problem after he had developed a partitive fractional scheme, and presumably then experienced a richer inner interpretation.
Selden, 2010). This appears to be a form of procedural memory, which can be thought of as “knowinghow” as opposed to “knowing that”. A brief discussion of kinds of memory, including procedural, can be found in Ranganath, Libby, and Wong (2012, pp. 121-123). Also, automated actions have been extensively studies by psychologists interested in their occurrence in everyday life (Bargh, 2014, 1997).
Many proving actions appear to be the result of the enactment of small, linked, automated situation-action pairs that we have termed behavioral schemas (Selden, McKee, & Selden, 2010; Selden & Selden, 2008). Automating actions can considerably reduce the burden on working memory, a very limited resource, and thus tends to reduce errors. (Baddeley, 2000) The value of automating actions in proof construction is illustrated in the comments following Sample Corect Proof 3, below.
Nonemotional cognitive feelings and nonconscious priming can influence whether a situation-action link is activated. Nonemotional cognitive feelings, such as the feeling of being, or not being, on the right track, typically are vague conscious states that pervade one’s whole conscious field and can combine with anything being focused upon (Selden, McKee, & Selden, 2010). Nonconscious priming occurs when an individual is unaware of the way a situation is influencing an action. For example, a student attempting to construct a proof might have written several line into a proof which, although true, do not move the proof forward. The student might then wrongly decide the proof was finished without realizing those several lines made the work “look like” something useful had been done. Below we discuss this appearing to occur in one student’s work. (See Sample Incorrect Student Proof Attempt 2.)
Some actions can be meta-actions, that is, actions on one’s own cognition, such as focusing on a particular part of a partly finished proof. Meta-actions should be distinguished from meta-cognition, that is, thinking about one’s own thinking. Retrospective meta-cognition is likely to be a useful addition to understanding one’s own proof construction, but attempting simultaneous meta-cognition could compete for working memory with the cognition that it is observing.
Some actions are beneficial for proof construction and should be initiated or encouraged. (See Sample Incorrect Student Proof Attempt 4 below for a beneficial action, not taken, that should be encouraged.) Other actions can be detrimental and should be eliminated or discouraged.
During proof construction, the partly completed proof and scratchwork are an important part of the construction. They can be used as aids to reflection and to reduce the burden on working memory. For a few psychologists, these might even be seen as an external part of cognition, which is normally seen as entirely mental and inner. In this setting, a proof should be regarded as a text, that is, as something that can be passed between persons. It consists of some of the actions in a proof construction. Indeed, this suggests why it is often hard for a student to mimic the proof of a previous theorem in trying to prove another theorem. If anything could be mimicked, it would be some of the actions taken during the proof construction of the previous theorem, many of which are not available for later viewing.
We now turn away from the more psychological part of our perspective to some more mathematical aspects that have emerged from teaching several earlier iterations of the current course.
We first describe the writing of a proof framework in more detail than can be found in Selden and Selden (1995). Proof frameworks are determined by the logical structure of what is to be proved. The most common form of theorem in our course notes is: some quantified variables; then “if P”, where P is a predicate about those variables; then “then Q”, where Q is another predicate involving some of the variables.
A proof framework starts by introducing the variables. If “for all a ϵ A” occurs in the theorem, one writes in the emerging proof “Let a ϵ A”, in which case a is henceforth regarded as fixed, but unspecified. If “there is a b ϵ B” occurs, one must “find/create” such a b and a space is left to insert or explain that. If the quantifiers are mixed, some “for all” and some “there exist”, then in the proof these should be introduced in the same order as in the statement of the theorem. This avoids inadvertently changing the meaning of the theorem during the proving process. Where the theorem says “if P”, one writes in the proof, “Suppose P” and leaves a space for further parts of the proof. Where the theorem says “then Q”, one writes at the end of the emerging proof “Therefore Q”. This produces the first-level of the proof framework.
At this point the student should focus on Q and “unpack” its meaning, that is, remember or look up its definition, being careful to change the names of its variables to fit the proof at hand. It may happen that the meaning of Q has the same logical form as the original theorem. In that case, one can repeat the above process, providing a second-level of proof framework which is written into the blank space immediately above “Therefore Q”. If in writing the second-level framework, some variables have already been introduced, one does not re-introduce them.
All of this is rather complicated to explain, but much easier to understand in practice, and is illustrated in our sample proofs, below. Also, once students can produce and use a proof framework for the above “if P, then Q” logical structure, it appears to be relatively easy to introduce frameworks for the seven or so other logical structures needed in the course. Finally, we are not claiming that mathematicians write proofs in the way we are describing, but only that doing so will be helpful for students and that mathematicians will accept the results.
We turn now to the idea of operable interpretations of definitions. Consider the following definition: Let f: X Y be a function and A X. Then f(A) = { y | y ϵ Y and there is a ϵ A so that f(a) = y}. To use this, if one knows “q ϵ f(A)” one can say “there is p ϵ A so that f(p) = q”. Also, if one knows “p ϵ A” then one can say “f(p) ϵ f(A)”. One might expect that a beginning transition-to-proof course student would be able to autonomously discover such operable interpretations of definitions, but we have noticed that many cannot. Learning to do so would be a useful skill for anyone wishing to read mathematics or prove theorems independently. However, before this skill is learned, it may be helpful to provide some of the operable interpretations. There are around 30 in our course.
Searching the course notes or one’s own knowledge base may not seem to be a very sophisticated skill, but we find that some students do not do it when some result or definition is called for in constructing a proof. We now try to arrange the course notes so that students can experience the benefits of noticing useful prior results.
a students’ self-efficacy (Selden & Selden, 2014) and for this reason we try to arrange for students to have early proving successes in our course.
We will now borrow a point from the genre of proof, namely, that definitions available outside of a proof are not normally written into it, at least not in proofs published in journals (Selden & Selden, 2013). Quoting an entire definition exactly into a proof can wrongly suggest to a student that something has been done that moves the proof forward. The student may then prematurely stop work on the proof. This should not be confused with using a definition in a proof. For example, in using the fact that f is continuous in a proof, one normally writes, “Because f is continuous, there is a such that …”. This looks rather like, but is not, quoting the definition.
Finally, we have found it helpful to have a, at least crude, gauge of the difficulty of a proof, independent of the ideas in the rest of this perspective. We say proofs are of Type 1, 2, or 3 as follows. A Type 1 proof calls for a student to see the need for a lemma, a subproof that could be proved separately, but could also be found located in the course notes. In a Type 2 proof, the student must articulate a lemma not proved in the notes, but the lemma’s articulation and its proof are straightforward. In a Type 3 proof, either the articulation or proof is not straightforward and may require insight or exploration. (Selden & Selden, 2013b, pp. 319-320).
The Course
The course, from which the data came, was inquiry-based as regards the proofs, but not as regards the mathematical structures or theorems. It was taught entirely from notes with students constructing original proofs and receiving critiques in class. The one-semester three-credit course is meant as a second-year university transition-to-proof course for mathematics and secondary education mathematics majors. It was given at a Southwestern Ph.D.-granting university and was taught in a very modified Moore Method way (Coppin, Mahavier, May, & Parker, 2009; Mahavier, 1999). That is, students were given course notes with definitions, questions, requests for examples, and statements of theorems to prove.
The students in this study proved the theorems outside of class and presented their proofs in class on the blackboard and received extensive critiques. These critiques consisted of careful line-by-line readings and validations of the students’ proof attempts. This was followed by a second reading of the students’ proof attempts, indicating how these might have been written in “better style” to conform to the genre of proofs (Selden & Selden, 2013a). Once these corrections and suggestions had been made, the student, who had made the proof attempt, was asked to write it up carefully, including any corrections and suggestions, for duplication for the entire class. In this way, by the end of the semester, the students had obtained one correct, well-written proof for each theorem in the course notes. Sometimes, if the students seemed to need it, there were mini-lectures on topics such as logic or proof by contradiction.
the teaching aim was to facilitate students’ learning the proof construction process in an embodied way through experience constructing as many different kinds of proofs as possible, especially in abstract algebra and real analysis, and not to learn a particular mathematical content.
Methodology
Guided by our theoretical perspective and our aim for the course, we analyzed all 16 four-proof take-home, and all 16 four-four-proof in-class, final examination papers from the course. Altogether 128 student proof attempts were analyzed in detail through several iterations, using a combination of grounded theory and textual analysis. When we found indications of a difficulty in a student’s attempted proof, we drew inferences about the probable proving action that might have led to that difficulty. We were looking for categories at a level of abstraction above specific mathematical topics so they would reflect process difficulties. For example, we considered a student’s not unpacking a conclusion, as opposed to a student having difficulty with a particular mathematical concept, such as a minimal ideal in a semigroup. We began with no particular categories in mind and made several passes through the data, until we came to an agreement on what we saw in each student’s proof attempt. In this way our categories emerged from our data.
We expected, and found, that when one difficulty occurred in a student’s proof attempt, other difficulties often also occurred. In addition, since our main interest was in finding a few difficulties that we might be able to alleviate with explicit teaching interventions, we did not attempt to search for categories that did not overlap or were not within other categories. Such information might be useful in designing teaching interventions.
Categories: The Most Common Student Difficulties
We have thus far identified the following categories: omitting beneficial actions; taking detrimental actions; inadequate proof framework (e.g., not unpacking the conclusion); mathematical syntax errors; wrong or improperly used definitions; misuse of logic; insufficient warrant; assumption of all or part of the conclusion; extraneous statements; assumption of the negation of a previously established fact; difficulties with proof by contradiction; inappropriately mimicking a prior proof; mathematical syntax errors, failure to use cases when appropriate; incorrect deduction; assertion of an untrue result; and computational errors.
While most categories can be easily understood from their names, there is one sufficiently odd that it might benefit from an illustration. Here is an example of a mathematical syntax error. In an attempt to prove that the split domain function h, defined by h(x) = f(x) if x ≥ a and h(x) = g(x) if x < a, is continuous at a, given that both f and g are continuous at a and f(a) = g(a), one student wrote: “|f(x)-f(a)|< ε/2 – |g(x)-g(a)|< ε/2”. This action, subtracting a statement such as “|g(x)-g(a)|< ε/2”, from another statement, violates normal mathematical syntax. Subtraction is an arithmetic operation used between numbers or variables representing numbers, not a logical operation used between statements.
In our textual analysis below, we illustrate omitting beneficial actions; taking detrimental actions; inadequate proof frameworks; not unpacking the conclusion; and extraneous statements (e.g., writing a definition that can be found outside of a proof into it).
In the following section, we consider both a sample correct, and a corresponding sample incorrect student proof attempt, of the same four theorems. We are numbering the lines with bold square brackets for the purpose of referencing them when we comment on them.
Sample Correct Proof 1. The first theorem we consider is: Theorem. Let S be a semigroup with an identity element e. If, for all s in S, ss = e, then S is commutative. Our sample correct proof is given below.
Proof:
[1] Let S be a semigroup with identity e. [2] Suppose for all s ϵ S, ss = e.
[3] Let a, b be elements in S.
[4] Now abab = e, so (abab)b = eb = b. [5] But (abab)b = aba(bb) = (aba)e = aba.
[6] Thus aba = b, so, (aba)a = ba, and (aba)a = ab(aa) = abe = ab. [7] Thus ba = ab.
[8] Therefore, S is commutative. QED.
We imagine that an idealized student prover would first write the hypotheses [1] and [2], leave a space for the body of the proof, and then write the conclusion [8], thereby completing the first-level proof framework. Next our idealized prover would unpack the conclusion [8], perhaps using scratchwork, and if necessary, consult the definition of commutative, which is in the course notes. By doing so, our idealized prover would know that he/she has to introduce two arbitrary elements of the semigroup, say a and b [3]. Then the prover could write line [7], thereby completing the second-level proof framework. What is required next is some “exploring”, that is, some manipulations, that prover cannot know will be useful, until lines [4], [5], and [6] can be written.
Sample Incorrect Student Proof Attempt 1. Everything is reproduced below as written by the student, including the student’s scratchwork, except for the line numbers.
Proof:
[1] Let S be a semigroup with an identity element, e. [2] Let s ϵ 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.
SCRATCHWORK:
We assume the student wrote the first-level proof framework at the start, lines [1], [2] and [7]. Line [2], as written, only hpothesizes a single element s so that ss = e. Line [2] should have been “Suppose for all s ϵ S, ss = e.” With this change, the first-level framework would have been correct. Also, we cannot be sure line [7] was written before the rest of the proof. If the student did not write all of lines [1], [2] and [7] first, this would constitute a beneficial action not taken.
In addition, despite being aware of the definition of Abelian written in the scratchwork, the student did not write the second-level framework by introducing arbitrary a and b at the top, followed by “Then ab = ba” right above the conclusion. Had the student written the correct second sentence in line [2] and taken these two actions, the situation would have been appropriate for exploring and manipulating an object such as abab. We think that such exploration calls for some self-efficacy, but can lead to a correct proof.
Line [3] violates the mathematical norm of not including in the proof definitions that can easily be found outside the proof. Also, this does not move the proof forward. The next three lines [4], [5], and [6] are not wrong, but also do not move the proof forward because to prove commutativity, one needs two arbitrary elements. These actions not only do not move the proof forward, but might have been detrimental. Through non-conscious priming, they might have wrongly convinced this student that he/she had accomplished something and prematurely brought work on the proof to an end.
Sample Correct Proof 2. Next we consider the following: 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. Our sample correct proof is given below.
Proof:
[1] Let S and T be semigroups and f:S→T be a homomorphism. [2] Let G be a subset of S and G be a group with identity e.
Part 1. [3] Note that G is a subsemigroup of S so, by Theorem 20.4, f(G) is a semigroup.
Part 2. [4] Let y ϵ f(G). [5] Then there is x ϵ G so that f(x) = y. [6] Now f(e) ϵ f(G) and f(e)y = f(e) f(x) = f(ex) = f(x) = y. [7] Similarly, y f(e) = y. [8] Thus, f(e) is an identity for f(G).
Part 3. [9] Let q in f(G). [10] Then there is p ϵ G so that f(p) = q. [11] Now because G is a group, there is p'ϵG so that pp' = p'p = e. [12] Thus q f(p') = f(p) f(p') = f(pp') = f(e), and [13] f(p')q = f(p') f(p) = f(p' p) = f(e). [14] Thus, each q ϵ f(G) has an inverse, f(p'), in f(G).
[15] Therefore, is a f(G) group. QED.
consider three parts, namely, [3] f(G) is a subsemigroup of T, [4] f(e) is an identity for f(G), and [14] each q ϵ f(G) has an inverse in f(G).
Next our idealized prover can begin on Part 1. But this is almost immediate because of a previous theorem in the course notes that says that the homomorphic image of a semigroup is a semigroup. Hence, line [3].
Our idealized prover can then go on to prove Part 2. For this, he/she would need to use the meaning of the definition of an identity and consider an arbitrary element of f(G) [4], and have to conjecture that the identity of f(G) is f(e), the image of the identity e of G. This would lead to using the meaning of x f(G) and line [5]. Then using the meaning of homomorphism would give line [6], showing f(e) is a left identity for G. Then line [7] would follow by similarity and line [8] would conclude a proof of Part 2.
Lastly, our prover would work on Part 3. For this, he/she would use the meaning of inverse element and consider an arbitrary element q in f(G). He/she would then call on the meaning of q f(G) to notice [10] that q can be written as f(p) for some p in G and, using the meaning of homomorphism, show that the image of the inverse of p is the inverse of q by lines [11], [12], and [13]. Line [14] asserts the conclusion of Part 3, and the proof is complete, according to the meaning of the definition of group.
We observe that the above idealized student prover often unpacked the meaning of a definition, by using what we have called its operable interpretation, and then altering the names of the variables to fit the theorem at hand. We also note that actions were rarely warranted, as is customary, as if they were completely transparent. However, for some beginning students, perhaps even many, we have found that using definitions in this way is not at all transparent.
Sample Incorrect Student Proof Attempt 2. We next consider a sample student proof attempt of the same theorem.
Proof:
[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.
The student has written the first-level framework correctly, lines [1], [2] and [7], that is, assuming the last line [7] was written immediately after writing the first two lines. To complete the framework, the student should have next considered f(G) and noted that there are three parts to prove, as indicated in the sample proof above. These are beneficial actions the student did not take.
This student’s work may suggest that he/she had some intuitive grasp of the concepts involved, and it may be tempting to give partial credit to the student. But from the point of view of having a student learn to construct proofs, doing so may send the “wrong message”.
Further, this student’s work is reminiscent of Carrisa, who was attempting prove or disprovethe statement: Let be a 1-1 homomorphism from (G,) to (H,*). If G is an abelian group, then H is an abelian group. Carrisa concentrated on elements of G, the wrong place to start, and mistakenly said the statement was true. She ignored, or did not see, the fact that is not known to be onto (Melhuish, 2014, pp. 3-4). Had she written a complete framework for a proof of H being a group or H being Abelian, she might have seen that she had inadequate information to finish a proof.
Sample Correct Proof 3. Next we consider the following: Theorem. If A, B, and C, are sets and C\B C\A, then C∩A C∩B. A sample correct proof follows.
Proof:
[1] Let A, B, and C be sets. [2] Suppose C\B C\A. [3] Let x C∩A. [4] Suppose x B. [5] Then x C\B, [6] so x C\A. [7] Thus x A. [8] This is a contradiction.
[9] So x B and [10] thus x C∩B. [11] Therefore, C∩A C∩B. QED.
An idealized student prover would first construct the first-level proof framework [1], [2] and [11], then “unpack” the conclusion, that is, use the operable interpretation of set inclusion to construct the second-level framework, [3] and [10]. Because the hypothesis refers to negative information about B, that is, C\B C\A, our prover might think of doing a subproof by contradiction, and hence, suppose [4], x B. At this point, our prover would explore where this leads. Then use the operable interpretation of x C∩A to get x C. This together with the operable interpretation of set difference gives [5], x C\B. Using the hypothesis that C\B C\A and the operable interpretation of set inclusion and modus ponens, gives [6], x C\A. Then the operable interpretation of set difference gives [7], x A, which is the contradiction pointed out in line [8]. Thus, it is legitimate to write [9], x B. This finishes the proof, as lines [10] and [11] were already written.
The above proof is not particularly unusual or difficult to read. This includes the passage consisting of [5], [6], [7], and [8], even though no explicit warrants are provided. However, the explanation of it in the above commentary seems more difficult to understand. We suggest that this is because many readers have automated some of the actions involved. This illustrates the value of automation mentioned in the fourth paragraph of the section titled, “Theoretical Perspective”.
Sample Incorrect Student Proof Attempt 3. Next we consider a student proof attempt of the same theorem.
Proof:
This student did not write the entire first-level framework, but started in the right place with the hypotheses. lines [1] and [2]. The student did not attempt a proof of x B by contradiction despite being in a situation where he/she could not prove that x B directly. Instead, the student seemingly began a direct proof in line [3], taking an element of C∩A and unpacking what that meant [4]. After that, the student seemingly tried to use the hypothesis [5] and the fact that x is in C. It is not clear how the deduction, [6], follows from what precedes it. It is possible that the student lost his/her tain of thought, and thought wrongly that he/she knew x C\B. Also, there is no indication the student was starting a proof by contradiction. This leaves no reasonable way to conclude [7].
It was not helpful that the student did not write a full proof framework. If he had, and noticed the he did not know how to continue with a direct proof, he might have seen how to start a proof by contradiction. We also note that the student drew two Venn diagrams, one in which both C A and B A; we conjecture this was also not helpful.
Sample Correct Proof 4. Finally we consider a sample correct proof and an incorect student proof attempt of the following: Theorem. Let X, Y, C, and D be sets and f:X→Y be a function. If C D and D Y, then f -1(C) f -1
(D). Proof:
[1] Let X, Y, C, and D be sets and f:X→Y be a function. [2] Suppose C D and D Y.
[3] Let x X. Suppose x f -1(C), [4] so that f(x) C. [5] Then f(x) D, [6] which means x f -1
(D). [7] Therefore f -1(C) f -1(D). QED.
Our idealized prover would first write the first-level proof framework [1], [2], and [7]. By unpacking the conclusion [7], our prover would know that he/she needed to start with an element of f -1(C), which is line [3], and show x f -1(D), which is [6], completing the seond-level framework. The operable interpretation of the definition of f -1(C) and yields [4]. Using the fact that C D our prover would get line [5]. Then applying the operable definition of f -1(D),gives [6], and the theorem is proved.
Sample Incorrect Student Proof Attempt 4. Next we consider a student proof attempt of the same theorem, along with the student’s scratchwork. In his/her work there was a large blank space between lines [6] and [7].
Proof:
[1] Let X, Y, C, and D be sets and f:X→Y be a function.
[2] Suppose C D Y. [3] Suppose y C, [4] then y D and y Y. [5] Since f is a function, there is an x X so that (x,y) f.
[6] Suppose x f -1
(C), then
[7] Then x f -1 (D).
Scratchwork:
Function: Then f(D) = {y | there is an element d D so that f(d) = y}
The student has written lines [1], [2], [7], and [8] just as in the correct proof. So the student has written most of a proof framework, and to complete it, he should have written line [6] immediately after line [2]. The student then introduces [3] y C and legitimately concludes [4] y D, along with the extraneous fact that y Y. He/She then, but irrelevantly, states in line [5] that there is an element x X so that (x,y) f. This assumes incorrectly, but irrelevantly that f is onto. Indeed, lines [3], [4], and [5] are not helpful. Then in line [6], it seems that the student begins again with the correct assumption, which had it been done earlier, would have produces a proof framework. Apparently he/she could not figure out how to get to line [7], as indicate by the blank space. Although this proof was part of an in-class exam, the students were allowed access to all of their course notes. A beneficial action the student did not take would have been to write the operable interpretation of [8] into the scratchwork. Instead, it contains the definition of f(D).
Summarizing, the student who wrote the above “proof” took a number of detrimental actions that should not have taken been and did not take a number of beneficial actions which that should have been taken.
Teaching Implications and Future Research
Having isolated and illustrated a few proving difficulties that our students, and probably many others, very often have, we can suggest some teaching interventions that might alleviate these difficulties. What form these interventions might take and how one might gauge their effectiveness, is a matter for future research. Because what might be done, and how to do it and gauge its effectiveness are closely intertwined we discuss them together.
Perhaps a good place to start explicit teaching is with proof frameworks, described in detail in the theoretical perspective. As noted above in the sample proofs, a number of difficulties seem to be traceable to not writing part or all of proof frameworks. Also, the writing of a proof framework can be decomposed into parts that can be taught separately. Perhaps an intervention might begin by thoroughly teaching students how to write one kind of common proof framework. After that the others could probably be learned quickly. We have found that students tend to resist writing full proof frameworks. We think this is because it involves writing in a way that is not “from the top down”. In most of their past experience, texts were read and written from the top down. There should be enough practice for students, not only to understand what they are doing, but also to form a habit of consistently writing proof frameworks. That is, they should overcome their, possibly nonconscious, reluctance. Also, it would be good if the entire process of writing proof frameworks became automated. To accomplish this in a reasonable amount of time, it is probably better to ask students to practice constructing only a proof framework, not the entire proof, for each practice problem.
To gauge whether such an intervention has succeeded, one might interview students towards the end of the course, asking them to construct a few relatively easy proofs, and observe them to see if they wrote proof frameworks. One might also analyze examination proofs for difficulties that might be traceable to not having written a proof framework.
be many definitions. It would be useful if students knew their operable interpretation in an automated way. It would also be useful for students to eventually develop the ability to autonomously produce operable interpretations of formal definitions for themselves. To arrange both of these, one might consider occasional brief small group discussions developing operable interpretations for definitions about to be used in a course. At the end of the group discussions, a teacher might certify which interpretations would be accepted for the course. One might also consider very brief short-answer quizzes on collections of operable interpretations. Another kind of quiz might consist of a few fragments of proofs that students could extend a little using operable definitions. Again, to gauge the usefulness of such activity, one might want to draw on quizzes, interviews, and an analysis of some examination questions.
In the next iteration of the course, we hope to implement the above interventions and investigate their effectiveness.
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