Breaking students from the habit of relying on examples

Question

One of the most frustrating things about my experiences teaching math (at the university level, if that matters) is that students seem very reluctant to actually learn the material. Instead, they seem to want to be presented with a series of examples, then generalize from those examples, rather than the other way around. This works well up till calculus and then fails completely afterward, when math becomes more than simply a set of computational algorithms. I've never had any success getting students to think mathematically; any suggestions?

Also, like anyone who's taught mathematics, or even answered questions on stackexchange, I've been frustrated (and irrationally annoyed) by students who come to me with no ideas whatsoever about a problem, saying that they "don't know where to start." A bit of prodding often reveals that they don't understand one of the terms in the problem. I'm not sure how they were planning on solving it without that, but my pet theory is that they tried to compare that problem to their list of examples, found nothing, and gave up. (If they had found such an example, that might have allowed them to solve the problem without the missing definition, effectively treating the example as a black box.) Is there any credence to that idea, or is it something else entirely?

Answer 1

This question is very usefully provocative, as evidenced by the comments, and the pro-example versus [sic] pro-abstraction notions... and the apt comment(s) suggesting that, in particular, the genuine issue is not that students have not "learned the definition", but that their intellectual methodologies are inadequate to the task... or maybe some really are a bit lazy or work-avoidant (sounds better!) while not being actively opposed to the mathematics.

E.g., I was a bit baffled by the comment that implied that it is perhaps dangerously deficient to characterize things as: continuity is that the graph has no jumps, and a differentiable functions' graph has no corners... considering that this is exactly my own intuition, although/and I can conform to the ambient standards-of-proof as need be.

I would claim that the underlying problem is that students are typically rewarded more for compliance and conformity than for other forms of astuteness. Thus, they have often learned to ignore their own intuition, since the instructor/exam/homework is actively trying to prank them in that regard. "Definitions" are therefore not at all codifications of extensive prior experience of serious people, but just set-ups for pratfalls of the victims, ... EDIT: for example, prank questions might have very little mathematical content, but, rather, play upon delicacies of wording, or the tension/conflict/ambiguity between mathematical use and colloquial use. No way to reason these out. Rather, one must know what the examiner is thinking. Similarly, textbooks can create conventions/rules/definitions with little genuine mathematical content, but, nevertheless, with very precise boundaries, the latter lending themselves to questioning.

In particular, in all my experience with standard courses/textbooks/exams, it is not the case that definitions [sic] are helpful, much less clarify pre-existing ambiguities or troubles or confusions. Instead, for exam and such purposes, they are far too often (and, then, this "taints the well") used to prank students in a quasi-legalistic sense. An example of the worst sort of crap is any question about "how many axioms does a group satisfy?". EDIT: again, such a question has essentially no mathematical content, because any finite list can be put together into a single axiom by conjunction, of course. But students may be simply required to remember what a particular book says on a particular page (as opposed to intrinsic mathematical assertions). Similarly, notational conventions can be "tested".

The other issue, which is maybe not literally "laziness", but "passivity", is (I claim) partly a result of years of being beaten down by petty-despot "teachers" of mathematics, who portray the subject as consisting of ineffable, uncontestable rule-sets... It is surely fairly futile, but I think any approach that does not tell kids to "trust, but maybe refine/improve, their intuition", is doomed to disengagement.

The same appears to be true of grad students in mathematics at good universities, etc., btw.

So, yes, I'm resisting answering the question as asked... but eminently willing to edit, considering that I do think this is an important sort of question.

Answer 2

Instead, they seem to want to be presented with a series of examples, then generalize from those examples, rather than the other way around.

To be honest: This does not seem that wrong in general. Most of the generalisations offered by mathematics are motivated by examples, e.g., the concept of continuity is motivated by the fact that the vast majority of real-life relations is continuous (at least in very good approximation). Moreover, examples are a good way to discover proofs. That being said, the power and importance of mathematics comes from abstraction and generalisation, but if you cannot illustrate this power by connecting the example level with the abstract one, your students will not appreciate the latter. This holds in particular, if your students are not actually aspiring to become mathematicians, but study physics, computer science, or similar.

Hence:

• Explain to them the way mathematics works:

  • examples motivate definitions;

  • statements based on suitable definitions have wide, easy, and robust applicability (in a sense, this is why we bother with abstract mathematics in the first place), while appealing to example-based intuition is prone to errors;

  • examples are a good way to discover proofs, but they never are proofs themselves.

  For a practical example: Once we show that some structure complies with the vector-space axioms, the whole apparatus of linear algebra becomes available to us and due to the rigidity of mathematics, we do not have to spend any thought on whether we can actually apply it – example-based intuition cannot do this.¹

• Guide them by example (yes, yes, I know): Do not only present them polished proofs, but show them how to come up with a proof. Write down all the relevant definitions and known properties, and continue from there. Also try to give an example where you can translate some example-based intuition to a proper proof.

• For students of other disciplines like physics or computer science, debunk the myth that they only need mathematics as a computational tool. For example, explain to physics students that vector spaces are the basic language of quantum mechanics and they cannot properly understand it without understanding vector spaces.

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^{¹ Blatant self-advertising: For an even more practical example for this example, see this didactics paper of mine (preprint).}

Answer 3

$c^2=a^2+b^2-2ab\cos C\\$

The law of cosines. When I lay out the problem so that a, b, and c don't line up with the equation, half the students are suddenly lost.

Similarly, I offer a right triangle, but play an awful trick, I label the hypotenuse with the letter A, and the 2 legs are B and C. I watch as $a^2+b^2=c^2\\$ is followed regardless, with nonsensical results. It gets marginally better if I start with say, x, y, z, but not much.

Examples are great, until the student is stumped with the similar triangle question, confused, as they solved 4 problems each with a man and flagpole with shadows for each, but now you've presented a building with its shadow.

My best approach is to vary the examples so much that they don't fall into the traps I tried to illustrate. Find ways for the example to be just that, but help them open their mind to apply the example to other concrete examples.