A comparison of proof comprehension, proof construction,
proof validation and proof evaluation
Annie Selden John Selden
New Mexico State University
This paper considers how proof comprehension, proof construction, proof validation, and proof evaluation have been described in the literature. It goes on to discuss relations between and amongst these four concepts—some from the literature, some conjectural. Lastly, it considers some related teaching implications and research.
Introduction
In the mathematics education research literature on proof and proving, there are four related concepts: proof comprehension, proof construction, proof validation, and proof evaluation. There has been little research on how these four concepts are related. We first briefly describe these four concepts, then we consider how they are related. That is, how are they the same? How are they different? Finally, we discuss some related teaching implications and research.
The four concepts as described in the literature
Proof comprehension means understanding a textbook or lecture proof. Mejia-Ramos, Fuller, Weber, Rhoads, and Samkoff (2012) have provided an assessment model for proof comprehension, and thereby described proof comprehension in pragmatic terms. Their model includes both local comprehension and holistic comprehension. Local comprehension
includes: Writing the theorem statement in your own words. Knowing the definitions of key terms. Knowing the logical status of the statements in the proof. Knowing the kind of proof framework (e.g., direct, contrapositive, contradiction, induction). Knowing how/why each statement follows from previous statements (e.g., making implicit warrants explicit ). Holistic comprehension includes: Being able to summarize the main, or key, ideas of the proof. Identifying subproofs and how they relate to the overall structure of the proof. Instantiating difficult parts of the proof with an example to aid comprehension. Providing a summary of the proof. Using the ideas from the proof in another proof.
Proof construction (i.e., proving) means attempting to construct correct proofs at the level expected of university mathematics students (depending upon the year of their program of study). What is needed for successful proof construction? To date, more is known in the research literature about difficulties that often prevent students from proving a theorem (e.g., Selden & Selden, 2008; Weber, 2001) than about interventions that would help students’ proving.
Proof evaluation has been described by Pfeiffer (2011) as determining whether a proof is correct and “also how good it is regarding a wider range of features such as clarity, context,
sufficiency without excess, insight, convincingness or enhancement of understanding.” (p. 5).
However, in order to distinguish proof evaluation from proof validation, we will put aside the portion referring to validation and concentrate on features of proofs including clarity, context, convincingness, beauty, elegance, and depth (e.g., Inglis & Aberdein, 2015). We would also like to separate proof evaluation from the use of adjectives that we have found with student validations, where terms like “wacky” and “confusing” were used when evaluating other
students’ proof attempts (Selden & Selden, 2015).
The paucity of research on the interrelationships
To date, there does not seem to have been much research attempting to relate the four concepts. Here is what we have found: Pfeiffer (2011) conjectured that practice in proof evaluation, as she defined it, could help undergraduates appreciate the role of proofs and also help them in constructing proofs for themselves. She obtained some positive evidence, but her conjecture needs further investigation. Selden and Selden (2015) obtained some evidence that
improving undergraduates’ proof construction abilities would not necessarily enhance their proof validation abilities and suggested that proof validation needs to be explicitly taught.
Relationships between and amongst these four concepts
Proof comprehension
Mejia-Ramos, et al. (2012), in their assessment model, considered both local comprehension/understanding and holistic understanding of a proof. By local comprehension, they meant knowing the definitions of key terms, knowing the logical status of the statements in the proof, knowing the proof framework, and knowing how/why each statement followed from previous statements. Such local comprehension is also needed for proof validation as described by Selden and Selden (2003); see below.
By holistic comprehension, Mejia-Ramos, et al. (2012) meant being able to summarize the main ideas of the proof, identifying the modules [subproofs] and how they relate to the
proof’s structure, being able to transfer the ideas of the proof to other proving tasks, and
Can students be taught these strategies? Samkoff and Weber (2015) attempted to teach these five strategies, using reciprocal teaching, and found a qualified “yes”. Instantiating a theorem statement with an example helped students understand its proof. Students were also able to
identify proof methods, especially if they looked at the proof’s assumptions and conclusions.
However, students did not instantiate a line of a proof with a specific example. In addition,
Samkoff and Weber found that simply asking students to “know the definitions of the terms in
the theorem” was not enough. Moreover, simply asking students how to prove a theorm before reading its proof lead to superficial responses (e.g., “use epsilons”).
Furthermore, it seems that how one reads a proof depends on what one wants to “get out of it” (Rav, 1999). Indeed, Mejia-Ramos and Weber (2014) found that mathematicians commonly read published proofs to gain insight, not to check their correctness, and additionally, that mathematicians consider refereeing a proof to be a substantially different activity.
Proof construction
We limit our consideration to situations in which undergraduates are asked to prove theorems, not to conjecture them, as this is the more common situation in U.S. undergraduate mathematics education. What is needed for successful proof construction? It is not clear that this has been discussed much in the mathematics education research literature. However, the kinds of difficulties that can stop students from proving a theorem have been researched. These include: Difficulties interpreting and using mathematical definitions and theorems. Difficulties interpreting the logical structure of a theorem statement one wishes to prove. Difficulties using existential and universal quantifiers. Difficulties handling symbolic notation. Knowing, but not bringing, appropriate information to mind. Knowing which (previous) theorems are important (e.g., Selden & Selden, 2008: Weber, 2001).
One overlap of proof construction with both proof comprehension and proof validation seems to be in knowing and using definitions and theorems appropriately. For proof construction, one needs to bring definitions and theorems to mind at an appropriate time so one can use them. However, in proof comprehension and proof validation, definitions and theorems have already been invoked, so one does not have to think of them, rather one only has to decide if they have been used appropriately. In general, it would seem that creating a new proof oneself, would be harder than merely comprehending what has already been done by someone else or checking its correctness, provided it is not a “garbled” student proof attempt.
Proof validation
While proof validation has been described briefly as the reading of, and reflection on, a proof attempt to determine its correctness, much is involved. Selden and Selden (2003) elaborated on what it might take to validate a proof attempt, suggesting that doing so is more complex than simply reading from the top-down:
Validation can include asking and answering questions, assenting to claims, constructing subproofs, remembering or finding and interpreting other theorems and definitions, complying with instructions (e.g., to consider or name something), and conscious (but probably nonverbal) feelings of rightness or wrongness.
calculations, expansions of definitions, or constructions of subproofs. Towards the end of a validation, in an effort to capture the essence of the argument in a single train-of-thought, contractions of the argument might be undertaken. (p. 5).
If one compares this statement on proof validation with the Mejia-Ramos, et al. (2012) assessment model for proof comprehension, there seem to be several possible common features: Knowing the definitions of key terms. Checking the logical status of statements. Knowing which proof framework was used. Constructing subproofs. Perhaps summarizing the proof. But, the relation to considering examples is not so clear. However, in this regard, Weber (2008) found that his eight mathematicians used example-based reasoning in proof validation, that is, they often checked the truth of an implied warrant through use of a carefully chosen example. It may be that many mathematicians, through experience, have developed implicit knowledge of which examples are likely to be useful.
One big difference between proof comprehension and proof validation might be that in most proof comprehension situations one can reasonably assume a proof is correct, especially if it appears in a lecture or textbook. Indeed, one’s skepticism about the validity of a proof may depend greatly upon its source--whether from a textbook, a journal, a colleague, or a student. On this issue, Samkoff and Weber (2015) concluded, “It would not be surprising if strategies for [proof] validation differed from those of [proof] comprehension. ”
Proof evaluation
As described above, proof evaluation seems more like making value judgments about a finished proof or a published proof text. When a student’s proof attempt is being examined by another student, such judgments can be about not understanding what is written, rather than about its beauty, clarity, elegance, or depth. In the recent Selden and Selden (2015) validation study, students said they found parts of the proof attempts “confusing”, “convoluted”, or “a
mess”. One student found the notation “wacky”. Other student validators said too much or too little information was given in a proof. Thus, for students, it seems that “making sense” of (i.e., understanding/comprehending) a proof attempt (as written) is a prerequisite for proof validation to begin.
In an internet study, Inglis and Aberdein (2014) asked 255 mathematicians to consider whether a proof of their own choosing was “elegant”, “insightful”, “explanatory”, “polished”, and so forth. The mathematicians were provided 80 such adjectives. The authors concluded
that mathematicians’ adjective choices could be classified along four dimensions: aesthetics,
intricacy, utility, and precision. Additionally, we conjecture that evaluations such as those made by these mathematicians would require a certain familiarity with, and competence with, proof comprehension and proof construction. We feel one would need to have seen (i.e., comprehended) and constructed many proofs in order to make value judgments on characteristics such as elegance, insightfulness, and depth.
“proofs” course (Selden, McKee, & Selden, 2010), we first validate students’ proof attempts,
then go over them again to comment on their style (i.e., their adherence to the genre of proof).
In sum
There are more questions here than answers. One can not only ask, how are proof comprehension, proof construction, proof validation, and proof evaluation realted, but also how does one teach them? Which should be taught first or should they be taught in some combination? What is the effect of doing so?
It would seem that students’ proof comprehension would benefit from their attempts at proof construction and vice versa--suggesting these two concepts/skills should be taught together. Indeed, reading comprehension researchers (e.g., McGee & Richgels, 1990) state that reading and writing taught together result in better learning. In addition, before submitting a proof, whether for homework or a journal, one needs to validate it for oneself to ensure its correctness. Finally, it would seem that one should have a good grasp of the first three--proof comprehension, proof construction, and proof validation--before attempting to evaluate proofs as beautiful, elegant, insightful, obscure, and so forth.
Related teaching implications and research
What do mathematicains consider when preparing pedagogical proofs? What do students “get out of” proofs presented in lectures or textbooks? How can one teach proof comprehension? There has been some research on each of the above. While clearly informative, this research has not specifically considered the relationship of proof comprehension to proof construction, proof validation, or proof evaluation. For example, Lai and Weber (2014) found that mathematicians said that they considered both the intended audience and medium, whether lecture or textbook, in their proof presentations. However, they also found that although mathematicians valued pedagogical proofs featuring diagrams and emphasizing main ideas, they did not always incorporate these into the pedagogical proofs they constructed or revised.
Researchers are interested in proof comprehension because mathematics undergraduates, at least at the upper-divison level in the U.S., spend a lot of time watching and listening to proofs being demonstrated in lectures and are also assigned proofs to read in their textbooks. The question is: What do, and what should, students “got out of this”? To begin to answer this question, Fukawa-Connelly, Lew, Mejia-Ramos, and Weber (2014) examined what students “got out of” one real analysis professor’s proof of the theorem that if a sequence has the property that the distance between any two consecutive terms xn and xn-1 is less than rn,
where 0<r<1, then it converges. The professor’s lecture was much more detailed than what he wrote on the blackboard, but most students only copied down what was on the blackboard, and did not pay attention to the professor’s added oral remarks. As a result, the students did not comprehend much of what the professor intended to convey. Apparently, the students, unlike the professor, did not see the professor’s oral explanations as important.
self-explanation training tended to generate higher quality self-explanations and performed better on a comprehension test constructed according to the assessment principles of Mejia-Ramos, et al. (2012). The students also increased their cognitive engagement. Experiment 3, with 107 students in a lecture situation, showed that 15 minues of reading a self-study intevention booklet, describing self-explanation, also improved students’ proof comprehension, and this improvement persisted over time, suggesting proof comprhension can be taught effectively.
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