27

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?

anomaly
  • 535
  • 5
  • 11
  • 5
    Welcome to ME.SE. I feel like your question could do with some tidying up, but I can't pin down what specifically would be best, so I'll leave that to someone else. What I will say is that your experience sounds like mine. – Jessica B Apr 25 '16 at 16:54
  • 8
    Since this is not literally an answer... : I myself strongly prefer examples at every stage. A sufficient stock of examples should indeed suffice to compare to anything within some specified range... almost by definition, I think. And interpolation of a sufficient stock of examples comes very close to completely determining what theorem could be true of them. – paul garrett Apr 25 '16 at 18:04
  • 5
    I think there are really two questions here; one about examples, the other (more disguised) about how to learn students how to solve exercises. I think that both should be asked, but separately. The first one should be concluded by a more focused question than "any suggestion?": here it feels like you have not yet fully found your question. Concerning the second question, let me insist that the first step to formulate it is to realize that student should be taught how to solve exercise, rather than expected to come up with a good methodology by themselves. – Benoît Kloeckner Apr 25 '16 at 18:16
  • 2
    "One of the most frustrating things about my experiences teaching" -anything , to anybody. I totally empathize with you, to me examples just muddle things and create a thousand questions of the kind "but what if premises change?" ... But this just says that your mindset is probably much more receptive to abstractions than the mindset of the majority of people. – Alecos Papadopoulos Apr 25 '16 at 19:32
  • 2
    +1 I think this is a great question, not one I would have formulated, but something that perplexes me and drives me crazy, too. When I put up a definition it's like most of my students are willfully blind to it. I wonder if they aren't functionally illiterate. I've started immediately quizzing them on a series of T/F examples that may or may not match the definition, but it's of minimal help to their retention. Particularly aggravating when unambiguous well-defined terms are the entire power of the discipline. – Daniel R. Collins Apr 25 '16 at 20:07
  • 3
    @BenoîtKloeckner: I disagree about the last point. There isn't a static methodology that one can apply here; it's just a matter of understanding the material well enough to solve problems. For example, one question on a homework assignment was to prove that the eigenvalues of a hermitian matrix are real. There's no computation involved; it's just a matter of applying the definition and writing out the results. Yet about a third of the class tried to solve it by writing down an explicit 2x2 or 3x3 matrix, churning through a long computation, and then baldly asserting it holds in general. – anomaly Apr 25 '16 at 20:08
  • ... Although I did read someone on a blog once write "You never appreciate the importance of a definition until you have to use it in anger (i.e., prove something with it)". – Daniel R. Collins Apr 25 '16 at 20:09
  • 2
    @DanielR.Collins: I know, right? It's like they ignore the definition when you present it, then try to infer it from later exercises. I can't find it at the moment, but there was a great paper posted here about the precise definitions of continuity and differentiability. Students effectively made up their own definitions of both: continuous functions are ones without gaps or jumps in their graphs, and differentiable functions are ones without sharp points (e.g., $0$ for $|x|$). Very frustrating. – anomaly Apr 25 '16 at 20:42
  • 2
    @anomaly: you misread "methodology" for "set of methods" or "small list of tricks to apply". I meant that students need to be explained what it means to apply a theorem, what it means to prove a result, what it means to provide a counter-example, what it means to prove a statement... I don't think we should have a detailed argument here, but I stand by my point that your last paragraph could be made into a good, suitable question. – Benoît Kloeckner Apr 25 '16 at 20:43
  • 1
    @BenoîtKloeckner: This class was beyond the point when students would be expected to have basic mathematical literacy, and they had an entire massive textbook full of examples of proper proofs and definitions. The problem was that they weren't interested in it; they wanted to read a set of worked problems and tweak the numbers around to solve similar problems. But this is getting into something that should be a full post or at least a side thread. – anomaly Apr 25 '16 at 20:47
  • 3
    @AlecosPapadopoulos: But this is mathematics: abstractions are the entire point. :) Going into a math class with neither comfort with nor expectations of abstractions is like going into a French class and demanding to not have to learn any verbs. – anomaly Apr 25 '16 at 20:48
  • I posted a fleshed out question inspired from the second paragraph: http://matheducators.stackexchange.com/questions/10913/how-to-help-student-get-to-grips-with-exercises-definitions-theorems – Benoît Kloeckner Apr 25 '16 at 20:59
  • 1
    @anomaly: my fourth year math majors student have this kind of issues. I stand by my point: they were never really taught this, or got away with this wrong way of dealing with math up to know, or both. If they are in an advanced class without this knowledge or will, we are at least partly to blame. – Benoît Kloeckner Apr 25 '16 at 21:01
  • 1
    I agree this needs to be taught/explained explicitly; I commit to doing so; I find that it makes little to no difference. (Albeit: My community-college students are mostly at a much lower level than yours.) – Daniel R. Collins Apr 25 '16 at 22:45
  • 5
    I may make a proper answer eventually. For now don't forget that in most other human endeavours you do learn through a series of examples and non-examples (eg language aquisition, cooking). Students are simply doing what comes naturally. – DavidButlerUofA Apr 25 '16 at 23:59
  • 4
    I like this question. It is a continual frustration how little attention is paid to the theory which is presented. – James S. Cook Apr 26 '16 at 00:36
  • 2
    One complicating factor may be the (sometimes stark) difference between the definition, and working definitions. For example, I really wish the implication in the standard definition of an antisymmetric relation were replaced with its contrapositive, which often makes for a much better working definition. The actual definition is essentially useless if you're handed a set of ordered pairs and wish to know if the relation is antisymmetric. Of course, the definition of an $n$th root is quite a different matter, that my students are currently ignoring at their peril. – pjs36 Apr 26 '16 at 05:09
  • 3
    @pjs: Nth roots are the most exasperatingly impossible topic for students in my algebra classes, and simultaneously the most highly-correlated with success on the final exam every semester. – Daniel R. Collins Apr 26 '16 at 23:13
  • 3
    I wonder whether some of the problems described in the latter half of your question relate to how students read (in particular) or take in (more generally) mathematics. The notion of trying to solve problems without knowing all of the definitions resonates with me; I describe this issue in MESE 2164 on helping students to read textbooks (for which my response may have some use/relevance here). – Benjamin Dickman Apr 27 '16 at 01:35

3 Answers3

11

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.

paul garrett
  • 14,597
  • 2
  • 33
  • 71
  • 2
    +1 Because I've been so eager for anyone to try answering this. The "petty-despot... uncontestable rule-sets" part is good. If anything were to be edited, I might suggest the "trying to prank them" paragraph, because I really didn't follow what you were saying in that portion. – Daniel R. Collins Apr 26 '16 at 23:11
  • One of the 'pranks' that I like in the exercises, but would never put in a test, is asking for the derivative of pi squared. Or e^7. That may not be the sort of pranking Paul was referring to, though. I think it does test something conceptually important. (Does the student recognize when something is a constant?) But it's too much a 'gotcha!' for me to be willing to put it on a test. – Sue VanHattum May 01 '16 at 23:02
4

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

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.

JTP - Apologise to Monica
  • 9,770
  • 43
  • 70
4

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.

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

Wrzlprmft
  • 2,548
  • 16
  • 33