Effectiveness of Requiring Students to Repeat Proofs Presented in Class

Question

In teaching mathematics, I've avoided asking students to replicate proofs I've demonstrated in class, believing this approach primarily tests memorization rather than understanding or critical thinking. However, I've observed a potential downside: students might pay less attention to the proofs when they know they won't have to reproduce them. While my goal is to cultivate an environment that encourages students to independently engage with and think through proofs, I'm reconsidering the strategy.

Is there pedagogical value in occasionally requiring students to repeat simpler proofs that were covered in class? How can this practice be balanced with the objective of promoting deep understanding and independent problem-solving skills?

Answer 1

I view a course as a story. The definitions introduce the characters, the theorems are the plot, and hopefully there is some sort of overarching message that plot is trying to convey.

If you were teaching Beowulf it would not be unreasonable to ask who Grendel is and how he dies. It is just a basic plot point: if you don't remember this you are not really following the story and have no hope of learning anything deeper. You might want them to develop their own interesting take on what the story is trying to convey but they cannot do this if they do not follow the plot.

It is the same if you cannot prove that every subgroup of a cyclic group is cyclic: you are just not really following the story. It is necessary but not sufficient for a student to know how to tell that part of the story if they are to have any hope of learning group theory.

Beginning students of mathematics have literally no idea what they are doing. The case is similar to people who have done very well in spelling competitions taking an English literature class and expecting to do well because of their spelling prowess. The experience is so completely unlike what they are expecting that they do not even know what to focus on. So a large part of your job will be to convey what it is that they are even trying to do. While it is a baby step, parroting back the basic proofs is a step towards comprehension.

When I was teaching these kinds of courses I would make sure to have a small collection of "core theorems". I expected the students to be able to reproduce the proofs. I made it clear that my intention was not for them to memorize the proofs. I asked them to find a personal narrative for the kind of discovery process someone would have gone through to discover the proof and to retrace this discovery path until it felt familiar. Some students, undoubtedly, did just memorize. However some of them really did learn something!

EDIT: I should also mention the research of Lara Alcock.

https://laraalcock.com/papers/

In particular I would like to point to self explanation training as a strategy for increasing proof comprehension. I wouldn't implement this kind of work as part of a summative assessment (a bit too subjective), but it is a valuable component of "helping students realize what they are even supposed to be doing".

Answer 2

1. As a first intuition, I would rather ask for a proof of a special case or a different case; something which asks for some level of understanding (of what to alter), not only copying and pasting. It can be essentially the same proof for you and still a significant challenge for the students, and one that requires engagement with the original proof.
2. However, just basic mathematical literacy (reading and writing) is challenging for many students, so copying a proof might turn out to be somewhat challenging to some of them. With this, you would be training them in copying, rather than thinking, so I would be wary of this as a repeated practice.
3. In a situation without notes available or in an oral situation, asking to repeat a proof is okay. Memorizing a proof usually happens by the way of understanding, though some people do brute-force memorize complicated things. However, I would still consider alternatives like presenting a proof attempt or a sketch of one and asking the student to find mistakes there and fix them, if possible, or at least explain what the problem is.

Answer 3

I used to often ask questions of the form "State and prove Theorem X" in proof-based courses. My hope was that, instead of memorizing verbatim, students would boil the proof down to a few key ideas, memorize those, and trust themselves to fill in the details on the fly during the test. And the very best students did this. But the majority memorized verbatim without seeming to understand much. In fact, I once asked for a proof where the textbook's version had a typo, and many students reproduced the typo on the exam!

Now, I prefer to look at specific proof techniques that I want them to know, and write questions that test their ability to do that technique in a (slightly) unfamiliar setting.

Answer 4

You will never be sure that the students really grasp a proof, if they can not express it or at least its essential part in there own words. Knowing proof really helps them to understand, what they do. So it's not just "memorizing" encourage them to talk about the essential idea of the proof, not just a formal memorized text.

Answer 5

If you're training musicians, you might have them do exercises in 2 different categories:

1. improvising from scratch in real time, on the spot, without prior rehearsal

2. rehearsing and analyzing difficult pieces by master composers

Category 1 is about developing creativity within your repertoire, completely mastering the techniques that you're already familiar/comfortable with.

Category 2 is about extending your repertoire to include new techniques that are initially very unfamiliar/uncomfortable. Initially, these techniques are advanced relative to your ability, so the only way to practice them properly is by imitating and analyzing them. But over time, as you build a baseline level of competence imitating/analyzing these techniques, they can gradually be moved into category 1 (and replaced with even more advanced techniques in category 2).

Basically every skill domain is the same way, including math.

• Category 1 contains general techniques like proof by induction.

• Category 2 contains key theorems like the fundamental theorem of calculus.

On an exam, it would be reasonable to ask students to

• solve previously-unseen problems in category 1, and

• imitate and analyze known solutions to previously-seen problems in category 2. ("Analyze" could mean, for example, identifying an error in a given proof.)

Lastly, I should emphasize that moving techniques from category 2 to category 1 requires both imitation and analysis, not just one or the other.

• If you don't imitate, then you won't be able to execute the mechanics of the technique. One who analyzes but doesn't imitate is called a critic.

• If you don't analyze, then you won't know how to tweak the mechanics of the technique to new situations, and you also won't know whether it's even appropriate to apply the technique in a new situation unless you're explicitly told to use the technique.

Imitating without analyzing produces a robot / ape who can't think critically; analyzing without imitating produces a critic who can't act on their own advice.

Answer 6

I vote yes, do it. If you want them to know the proofs (and you do), demonstrated recall will help assure that it happens.

I think your hesitancy to have students duplicate work, and do something new instead, is misplaced. Memorization of math proofs (or algorithms) is not either as evil or as simple some people think. Students still incorporate a lot of the knowledge of step by step workings, if they spit out a memorized proof.

It's not like you evaluate them based on word for word text recall (but for the key ideas logically presented). I mean do you think it is wrong (evil "memorization") if an AP Bio test has kids write essays summarizing a biological system?

It won't be exactly memorization, say in the sense of a meaningless random number sequence, to "memorize" a proof. Part of the memorization may be short term like that. But they will also get at least some long term schema in their memorization. And, really having a deeper stock of memorized proofs, makes working on fresh problems easier (have more tools in your kit).

This doesn't mean you can't have any fresh problems. But they should be slight variations. You can't expect students to make huge leaps. Not in general and especially not in exam situations.

Answer 7

I want to emphasize an alternative approach mentioned by Tommi:

I would rather ask for a proof of a special case or a different case; something which asks for some level of understanding (of what to alter), not only copying and pasting. It can be essentially the same proof for you and still a significant challenge for the students, and one that requires engagement with the original proof.

In my opinion this approach together with an open-book exam is the best strategy to encourage true engagement and retention of the proofs that you have covered. When the students can refer to the proofs covered in class, but have to adapt them to solve new problems, they will definitely have to pay attention to them to get to know them well!

Answer 8

Asking students to reproduce proofs can be very effective. The degree of effectiveness is largely determined by how you ask students to do it. As this answer describes, some reproduction tasks can lead students to focus only on the words and symbols of a proof, missing the underlying structure and ideas. Here's a practice that was effective at getting me to think about the latter when I was a beginning student.

In my first course with proofs, I had to memorize and present important proofs that had been shown in lecture. It wasn’t sufficient to simply memorize the words, because the instructor would always ask follow-up questions to probe for understanding. If they weren’t satisfied with my understanding of a certain proof, I would have to go back and study more, then present the proof again. Each presentation was essentially a small, low-stakes oral exam. (This was a large course with many students, so the logistics were more complex than what I've described here. There was a peer-to-peer aspect as well, which I think had its own benefits. )

This supported my learning in several ways. Studying proofs in order to present them helped me learn the structure and style of mathematical arguments. Preparing for potential questions helped me understand the proofs and concepts from lecture more deeply. And through the presentations, I practiced talking about mathematics and became more comfortable participating in mathematical conversations.

This can't replace students coming up with proofs on their own. But writing a proof is challenging, even when one is comfortable with mathematical language and understands related proofs. Beginning students haven't reached this threshold yet. Familiar proofs are a context in which students can practice the mechanics of proving without the added complexity of having to construct a novel argument. And even once students become more proficient with proof-writing, they still benefit from studying important proofs outside of class to solidify their understanding.

Answer 9

Allow me to add an answer from a student's perspective (many many years ago).

In high school my maths marks were pretty mediocre, not bad enough to bar me from college, but still. Proofs were always something I considered I just need to glue myself to a chair and memorize - and it wasn't a pleasant experience... nor very successful when under pressure. Getting to college to study computer science, I added some maths subjects - which I approached with the same mindset. In my second year of trudging along like this, a (well-loved) professor took pity on us and took one whole hour's class, not in the curriculum, to talk about the whole logic behind proofs; how things start with the basic, given building blocks (axioms), and given a set of operations (previous theorems), you'd follow a number of logic steps to get to the proof.

It was like an epiphany. I never again memorized a single proof. Exam prep might involve working them out, just for practice. Also, my maths marks (and enjoyment) considerably improved from there on. As a more useful side effect, the same "way of doing logic" also helped me in other subjects as well as general life. Even though I haven't done any pure math since then.

To answer your question: I believe asking proofs (of some or other variation) is useful, BUT as an educator you need to ensure that your students understand the reason for proofs and are able to work these basic tools of their (temporary?) trade without memorization. Not asking for proofs leaves you with an assumption of their knowledge (which is probably wrong in many cases - students are masters at minimizing effort for getting acceptable results). Asking proofs tells you they either (A) understand or (B) are good at memorizing (without deep understanding - which is not desirable). Equip them to understand, then challenge them to apply the understanding.