How to help student get to grips with exercises, definitions, theorems?

Question

This question is an attempt at implementing my suggestion that the second paragraph of that question could be turned into an important question. The discussion starting in comments there would be more interesting under the form of developed answers, in my opinion.

I think most higher educators in maths have felt the despair of seeing so many students

• not being able to check a definition on the most basic examples,
• being clueless on how to solve an exercise that turns out to contain terms they do not understand but that they did not thought of looking up,
• being unable to apply correctly a theorem except by copy-pasting the exact sentence that was used in a previous exercise (and which of course is ill-suited to the case at hand), etc.

If we assume that these problems are entirely theirs to deal with, then we only have to bear it and leave it be. More constructively, let us assume that it is part of our job to help them solve this issues (I do believe it is part of our jobs, and probably the main part at that).

What can we do about it?

A good answer should start with a diagnosis as to where this issues are coming from, and suggest a course of action or at least hints we can use with students. Suggestions that can be carried on individually are especially welcome, but suggestion that involve changes in curriculum or collective action from a whole pedagogical team can be interesting too (they just probably won't happen).

Answer 1

One hypothesis is that many students are functionally illiterate, and in fact cannot parse or understand the carefully-constructed sentences that we present to them. They might be succeeding in many other courses using an examples-induction approach, and only in the math discipline (and perhaps computer science) does the lacuna become unavoidable.

A second hypothesis that I've been working on is that many in the math discipline generally don't spend enough time emphasizing the importance of those definitions. I've been in multi-disciplinary teachers conferences where someone will present Bloom's Taxonomy and say, "We spend most of our time at the lowest level of definitions; the struggle is to get our students into the higher tiers of applications, et. al.", at which everyone at the table nods. To this I think: In the math discipline, our students are routinely jumping directly into applications without really understanding the foundational terms of discussion well enough.

Should instructors be spending more time testing on the definitions directly? Do most instructors lack such time due to other heavy curricular requirements? Should questions like that be on our final exams (almost uniquely lacking, compared to other disciplines)? Are math instructors so skilled at instantly remembering definitions that this passes under our radar much of the time?

The remediation that I've attempted is to commit to presenting a definition, and then immediately running a gauntlet of yes-or-no questions to the class about whether a series of examples meets the definition or not. (Using presentation software here helps for time purposes; when previously doing it by hand I frequently ran out of time.) For example: Define polynomials, then immediately ask: Is this a polynomial? (a) $5x^2 + 7x$, (b) $\frac{2}{x}$, (c) $\frac{1}{2}x$, (d) $8$. Then proceed to any theorems/proofs/applications after that point. Also: I have weekly online quizzes that test most conceptual issues arising directly from the definitions.

However, I have not found that to be making a great difference in end-results. Full disclosure: This is the first semester where I've been able to fully present things that way, and the class grades so far are the lowest that they've been in over 4+ years since I've been keeping current master records. That's likely due to the course population this spring semester, though.

Answer 2

One hypothesis I've seen (unfortunately, I can't find the paper at the moment) is that students sometimes don't understand that when a mathematician uses a word, it means just what they define it to mean--- neither more nor less. The definitions in mathematics are performative acts, not descriptions as even in other areas of science. When one defines, for example, continuity of real functions using the $\delta$-$\epsilon$ formulation, that is continuity; it's not some fuzzy, general idea of "not having any gaps in its graph" that the definition is merely groping for. In my own experience, some students attempt to learn or even define concepts by reading through some examples or worked problems, then coming up with a working definition themselves by looking for the similarities among them.

I've never found anything particularly effective. The best proposal I've seen is to aggressively show students counterintuitive examples. In the example of continuity above, we could consider (expanding the definition to more general metric spaces or topological spaces) maps to and from discrete spaces, the $p$-adic topology, the Zariski topology, etc. I don't like the idea, though, of getting away from inferring concepts from examples by showing more examples. Maybe the idea is to get further away from concrete, specific examples and remain in the abstract. It's not hard, for example, to use naive ideas or intuition to guess whether functions like $\sin x, 1/x$, $\operatorname{sign}(x)$, etc. are continuous. When you want to prove, for example, that the image of a connected set is connected, you're pretty much forced to actively use the definitions abstractly.

Thus maybe part of the hang-up is the discrepancy between insisting on formal definitions and abstractions, yet invoking them in situations where students don't feel they're necessary or desirable. For that matter, up to a point students don't necessarily distinguish (or have the tools to distinguish) between an argument, computation, etc. that's inherently invalid, versus one that's just arbitrarily disallowed by the teacher. I'm sure most people here have some memory in high school of using a clever, justifiable trick that the teacher disallowed because it wasn't what was then being tested, was outside the curriculum, or something similar.

Answer 3

One thing that I more and more realize is necessary but often left untold, is to explain some basic principles about how to approach an exercise they don't know at first how to solve. My experience makes me think they often don't guess a few things we assume are obvious, or we assume where told them in high school or previous classes. A few students get it by themselves, many don't and are thus driven toward mere reproducing of standard exercises, and consider it unimaginable to solve anything remotely different.

Here is a tentative list of hints they need to be told explicitly (at various stages of their studies):

• look up or recall or write down the definition of every term appearing in the statement which is not crystal clear to you,
• a mathematical object can be called the name defined in a mathematical definition if, and only if, it exactly and precisely matches the requirements written in the definition,
• to apply a theorem (or proposition), find which objects in the case at hand which you want to play the role of each object in the theorem to be applied, then check the assumption of the theorem on these, then you get the conclusion of the theorem,
• search your notes for results that uses the assumptions of the statement to be proven,
• search your notes for results that produce the sort of conclusion claimed by the statement to be proven,
• if the above ends don't meet, sum up the properties you have and the ones you can get from applying theorems, and the properties you would need to apply theorems yielding the desired conclusion, and iterate,
• if things get messy, one option is to try the statement on simple examples and work them out, then proceed to more complex examples until you get an idea,
• if you have to decide whether all objects of a kind satisfy a given property, answering no only needs to produce one counter-example; failing repeatedly to produce one might indicate the answer is yes, and understanding what fails in the tried examples might help you find out a proof.

Answer 4

1. Emphasize definitions in homework (eg, penalizing people who don't invoke the definition)
2. Repeat definitions in class (eg, whenever I work out an example, I pause and ask the class what the definition is, and then write it down)
3. Ask for definitions on exams, and let them know that you will ask for definitions on exams.

I hadn't thought too hard about it before, but if you wanted to further emphasize this, you could keep a running list of definitions that you expect them to know, and even encourage them to make a list of such things along with the definitions themselves as a homework/study aid.