A comparison of proof comprehension, proof construction, proof validation and

proof evaluation

Annie Selden and John Selden

New Mexico State University

KHDM Conference

Didactics of Mathematics in Higher Education as a Scientific Discipline

Hannover, Germany, December, 2015

Introduction

We consider how these four concepts have been described in the literature and how they might be related.

•That is, how are they the same? How are they different?

•How should they be taught?

2

Proof Comprehension•Researchers are interested because mathematics undergraduates (at least at the upper-division level) spend a lot of time watching and listening to proofs being demonstrated in lectures.

•Also, they are assigned proofs to read in their textbooks.

•The question is: What do, and what should, they get out of this?

3

•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.

4

Local Comprehension

This includes:

•Knowing definitions of key terms

•Knowing the logical status of the statements in the proof and the proof framework (e.g., direct, contrapostive, contradiction, induction)

•Knowing how/why each statement follows from previous statements (e.g., the warrants)

5

Holistic Comprehension

This includes:

•Being able to summarize the main (i.e., key) ideas of a proof

• Identifying subproofs and how they relate to the structure of the proof

• Instantiating the difficult parts of a proof with examples (to aid comprehension)

6

And, perhaps somewhat more controversial (for initial comprehension):

•Being able to transfer the ideas of a proof to other proving tasks

7

Do Students Do Any of These?

Weber (2015) found 5 strategies good 4th year US university math students used. They were:

•Trying to prove a theorem before reading its proof

• Identifying the proof framework being used in the proof*

8

•Breaking the proof into parts or sub-proofs*

• Illustrating difficult assertions in the proof with an example*

•Comparing the method used in the proof with one‘s own approach

•Also, a majority of mathematicians in a follow-up study wanted their students to implement these.

*These are essentially in the above proof comprehension assessment model.

9

Can Students be Taught these Strategies?

Weber and Samkoff (in press) attempted to teach these strategies, using reciprocal teaching.

They found a qualified “yes”:• Simply asking students to “know the definitions of the

terms in the theorem” was not enough.

• Instantiating a theorem statement helped students understand its proof.

10

• Simply asking students how to prove a theorem before reading its proof led to superficial responses (e.g., “use epsilons”).

•Asking a student to break a proof into subproofs (in advance of reading it) difficult for students.

• Students were able to identify proof methods, especially if they looked at the a proof’s assumptions and conclusions.

• Students did not instantiate a line of a proof with a specific example.

11

Proof Construction

•What is needed for successful proof construction? (John gave some ideas on this.)

•More is known in the research literature about difficulties that often prevent students from proving a theorem.

12

Some Student Proving Difficulties

• Interpreting the logical structure of a theorem (without this one doesn’t know what to prove and can’t construct a proof framework

• Interpreting and using definitions and theorems

•Using existential and universal quantifiers

13

• Interpreting and using symbolic notation

•Knowing, but not bring to mind, appropriate information

•Knowing which (previous) theorems are important (Weber’s strategic knowledge)

14

Commonalities of Proof Comprehension and Proof Construction

Both require:

•Knowing and using definitions and theorems appropriately.

•However, in proof comprehension and proof validation (reading and checking for correctness), the definitions and theorems have already been invoked, so one only needs to check if they have been used correctly, not think of them when needed in a proof construction.

15

Proof Validation

Proof validation is the reading of, and reflection on, a proof attempt to determine its correctness.

It can include:

•Asking and answering questions

•Assenting to claims,

•Constructing subproofs

16

•Remembering or finding and interpreting other theorems and definitions

•Complying with instructions (e.g., to consider or name something)

•Conscious (but probably nonverbal) feelings of rightness or wrongness

17

•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. (Selden & Selden, 2003, p. 5)

18

Commonalities of Proof Comprehension and Proof Validation

Both require:

•Checking the logical status of statements

•Knowing which proof framework was used (and in the case of proof validation, whether it is appropriate)

•Constructing subproofs

19

•Perhaps, summarizing a proof

•Perhaps, checking a proof with example-based reasoning (Weber, 2008)

•Perhaps, the only difference seems to be that in proof comprehension one can reasonably assume that the proof is correct (because it appears in a lecture or textbook).

20

•But, Samkoff and Weber (in press) state, “It would not be surprising if strategies for [proof] validation differed from those of [proof] comprehension.”

•How one reads a proof depends on what one wants to get out of it. (Rav, 1999)

•Mathematicians use different strategies for [proof] comprehension and refereeing.

(Mejia-Ramos & Weber, 2014)

21

Proof Evaluation

•We, unlike Pfeiffer (2011), would like to separate evaluation from validation. That is, from use of adjectives like elegant, insightful, explanatory, in the case of (correct) proofs. (Inglis & Aberdein, 2014)

•And from use of adjectives that we have found student validators using like wacky, confusing, in the case of student proof attempts.

22

Teaching Implications

•How does one teach these four concepts?

•Which should be taught first or should they be taught in combination?

•What is the effect of doing so?

23

We have no answers, just some suggestions and a little evidence:

• It seems that students’ proof comprehension would benefit from attempts at proof construction and vice versa, suggesting these two concepts/skills should be taught together, even though they are different.

24

• Indeed, reading comprehension researchers state that reading and writing used together result in better learning. (e.g., McGee & Richgels, 1990)

• In addition, before submitting a proof, whether for homework or a journal, one needs to validate it for oneself to ensure its correctness. So proof construction and proof validation also go together.

25

•Caveat: We taught an inquiry-based proof comprehension course to NMSU undergraduates (as described by John), demonstrating (at each class meeting) validation of students’ proof attempts, but this was insufficient.

26

How do we know?

•We interviewed the students at the end of the course, using the same protocol as in our 2003 validation study.

•But, the NMSU students, who were at the end of the course, were not as good at validation as the students from our earlier validation study, who were at the beginning of a transition-to-proof course, suggesting specific teaching is needed.

27

•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, etc.

28

Thank you

Comments/Questions

29

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.

30

• 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.

• Samkoff, A, & Weber, K. (in press). Lessons learned from an instructional intervention on proof comprehension. Journal of Mathematical Behavior. Downloaded November 24 from www.researchgate.com.

31

• 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.

32

• 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. (2015). Effective proof reading strategies for comprehending mathematical proofs. International Journal for Research in Undergraduate Mathematics Education, 1(3), 289-314.

• Weber, K. (2008). How mathematicians determine whether an argument is a valid proof?, Journal for Research in Mathematics Education, 39(4), 431-459.

33