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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 raises some teaching implication questions and suggests a few possible answers.
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.
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. Examples of their assessment items include: Write the given theorem statement in your own words. Identify the type of proof framework used in the proof. Make explicit an implicit warrant in the proof. Provide a summary of the 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). To date, many student difficulties have been noted (e.g., Selden & Selden, 2008).
Proof validation has been described as the reading of, and reflection on, proof attempts to determine their correctness. Some validation studies have been conducted with undergraduates and mathematicians (e.g., Inglis & Alcock, 2012; Selden & Selden, 2003; Weber, 2008). The
general finding is that undergraduates check “surface features” of proofs such as equations,
whereas mathematicians look for the logical structure and the correctness of implied warrants.
Proof evaluation was 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).
The paucity of research on the interrelationships
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conjecture needs further investigation. Selden and Selden (2015) obtained some evidence that just improving undergraduates’ proof construction abilities would not 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 and 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, they 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 instantiating the proof with examples. While the first two of these are probably also useful for proof validation, the final
two have more to do with generalization of a proof’s techniques, which are not needed just for
checking a proof’s correctness.
Weber (in press) found five strategies that good 4th year university matheamtics students use to foster comprehension when reading proofs. They are “(i) trying to prove a theorem before reading its proof, (ii) identifying the proof framework being used in the proof, (iii) breaking the proof into parts or sub-proofs, (iv) illustrating difficult assertions in the proof with an example, and (v) comparing the method used in the proof with one‘s own approach” (Abstract) and suggested there might be more.
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 undergraduate mathematics. 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 a student from proving a theorem has been researched. These include: difficulties interpreting and using mathematical definitions and theorems; difficulties interpreting the logical structure of a theorem statement; difficulties using existential and universal quantifiers; difficulties handling symbolic notation; knowing, but not bringing, appropriate information to mind; and knowing which theorems are important (Selden & Selden, 2008).
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proof oneself, would be harder than merely comprehending what has 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 their 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.
Proof validation can also include the production of a new text—a validator-constructed modification of the written argument—that might include additional 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) statement on proof comprehension, there seem to be several possible common features: checking the logical status of statements, knowing which proof framework was used, constructing subproofs, and perhaps summarizing the proof. But, the relation to considering examples is not so clear. However, 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.
Proof evaluation
As described above, proof evaluation seems more like making value judgments about a finished proof text. However, if it is a student’s proof attempt that is being examined by another student, these judgments can be about not understanding what is written. In the Selden and Selden (2015) validation study, students utter evaluative comments of the following sort. Students found parts of the proof attempts “confusing”, “convoluted”or “a
mess”. One found the notation “wacky”. Some student validators said too much or too little information was given. Thus, for students, it seems that understanding/comprehending a proof attempt (as written) is a prerequisite for proof validation to begin.
Inglis and Aberdein (2014) asked 255 mathematicians to consider whether a proof of their
choosing was “elegant”, “insightful”, “explanatory”, “polished”, etc. The mathematicians were provided 80 different adjectives like these. The authors concluded that mathematicians’ adjective choices could be classified along four dimensions: aesthetics, intricacy, utility, and precision. Evaluations such as those made by these mathematicians would require a certain familiarity with and competence with proof comprehension and proof construction. This is because one would need to have seen (i.e., comprehended) and constructed many proofs in order to make value judgments on characteristics such as elegance.
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Proofs are written in a certain genre (Selden & Selden, 2013) and advice is often given to both student and mathematician authors on how to write them (e.g., Tomforde, n.d.). In our
“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).
Teaching implications
There are more questions here than answers. How does one teach these concepts? Which should be taught first or should they be taught in combination? What is the effect of doing so?
It would seem that students’ proof comprehension would benefit from 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 used 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 constructon, and proof validation, before attempting to evaluate proofs as beautiful, elegant, insightful, obscure, etc.
References
Inglis, M., & Aberdein, A. (2015). Beauty is not simplicity: An analysis of mathematicians’ proof
appraisals, Philosophia Mathematica, 23(1), 87-109.
Inglis, M., & Alcock, L.(2012). Expert and novice approaches to reading mathematical proofs, Journal for Research in Mathematics Education, 43(4), 358-390.
McGee, L. M., & Richgels, D. J. (1990). Learning from text using reading and writing. In T. Shanahan (Ed.), Reading and writing together: New perspectives for the classroom (pp. 145-168). Norwood, MA: Christopher Gordon.
Mejia-Ramos, J. P., Fuller, E., Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educ. Studies in Mathematics, 79(1), 3-18.
Pfeiffer, K. (2011). Features and purposes of mathematical proofs in the view of novice students: Observations from proof validation and evaluation performances. (Doctoral dissertation) National University of Ireland, Galway.
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. (2015). Validations of proofs as a type of reading and sense-making. In K. Beswick, T. Muir, & J. Wells (Eds.), Proc. of the 39th Conf, of PME, Vol. 4 (pp. 145-152). Hobart.
Selden, A., & Selden, J. (2013). The genre of proof. In M. N. Fried & T. Dreyfus (Eds.), Mathematics and mathematics education: Searching for common ground (pp. 248-251). Springer: New York.
Selden, A., & Selden, J. (2008). Overcoming students’ difficulties in learning to understand and
construct proofs. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics education (pp. 95-110). MAA: Washington, DC.
Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Jour. for Research in Mathematics Education, 34(1), 4-36.
Tomforde, M. (n.d.). Mathematical writing: A brief guide. Downloaded Sept. 26, 2015 from Tomforde_MathWriting.pdf.
Weber, K. (in press). Effective proof reading strategies for comprehending mathematical proofs. International Journal for Research in Undergraduate Mathematics Education.
