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When I talk to other research mathematicians, there is pretty uniform agreement that we want our (university) students to understand the maths we teach them rather than just memorise processes. As far as I can tell, we have a reasonably consistent idea of what that means (although much less agreement of what to expect of students in a given context).

However, in the context of curriculum discussions, I've found that non-mathematicians are unwilling to accept 'understanding' as a suitable word to describe the desired outcome. I've also read a couple of times that students don't necessarily share our view of the term, although I haven't had time to check out the references.

So my question is:

Are there alternative phrases that can be used to describe what we mean by mathematical 'understanding' (at undergraduate level) that are better at communicating the meaning to non-mathematicians?

I have come across the phrases 'relational understanding' and 'instrumental understanding', which I think are helpful, but would only be meaningful to the minority who have come across them before.

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    $\begingroup$ Is it sufficient to say something like, we want our students to gain procedural fluency and then develop a conceptual understanding? That is the sort of colloquial language that I use if the other party is less familiar with terms like 'concept image', 'APOS', etc. $\endgroup$ – Benjamin Dickman Dec 5 '15 at 11:23
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    $\begingroup$ "Conceptual understanding" seems to be in the ballpark (although I would swap the order compared to Benjamin above). Maybe "be able to communicate and explain mathematical ideas". Or "be able to confirm and check that mathematical claims are true". $\endgroup$ – Daniel R. Collins Dec 5 '15 at 17:57
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    $\begingroup$ How about grok? :-) $\endgroup$ – Joseph O'Rourke Dec 6 '15 at 0:43
  • $\begingroup$ Is this distinct from mathematical maturity? Because that's the first phrase that came to mind. I'm not sure I can define it, but to borrow the classic phrase, you know it when you see it. $\endgroup$ – pjs36 Dec 6 '15 at 20:24
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    $\begingroup$ @pjs36 I think of mathematical maturity as a general frame of mind, whereas understanding refers more to specific topics. Even a student who is already mathematically mature still needs to study in order to understand various parts of mathematics. (I've occasionally oversimplified by describing mathematical maturity as consisting of (1) the ability to distinguish a mathematical proof from a page of garbage and (2) a preference for the former.) $\endgroup$ – Andreas Blass Dec 6 '15 at 22:53
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When desired learning outcomes are listed in a curriculum, who are they listed for? It depends on the situation, but often the target audience includes not only those who teach the course, but also administrators and prospective students. While your colleagues who are ordered to teach a course might understand what 'understanding' means, the other groups of people might not, and they are not to be ignored. I think understanding is a good goal but you need to explain it to others.

You need to make 'understanding' more concrete by asking yourself and your colleagues how it manifests itself (how it can be measured). The word as I understand it means several things and cannot — and should not — be captured in a single term. And even mathematicians may disagree on what the term actually means. Maybe something along the following lines would communicate your goal better than "The students should learn mathematical understanding":

  • The students should be able to explain ideas behind proofs instead of being only able to repeat the proofs when prompted.
  • The students should be able to use ideas from proofs they have seen to prove results they have never encountered before.
  • The students should have a collection of examples readily available and be able to use them to quickly check if a given mathematical statement is plausible.
  • The students should recognize false and meaningless mathematical statements. They should be able to judge whether a proof is valid or not.
  • The students should communicate their reasoning clearly and following the conventions of the field.

I believe this list would be more understandable to non-mathematicians than just the word 'understanding'.

This was just my list. You can make yours. (I can append my list if there are good suggestions.)

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  • $\begingroup$ Depending on the class, I'd add in "the student should be able to apply a theorem or idea to a situation in context" such as recognizing where it is important to use an integral in a pressure problem. This is more applicable to lower-division courses, however. $\endgroup$ – Opal E Dec 7 '15 at 20:53
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You mentioned: I have come across the phrases 'relational understanding' and 'instrumental understanding'...

The original article, by Richard Skemp, is well worth reading. I think you'll find your own favorite way of explaining this to non-mathematicians once you've read Skemp's piece, and perhaps a few of the many blog posts about it.

I recommend:

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  • $\begingroup$ I have read through all these links. While they are interesting, I don't find them directly applicable in answering the question. Also, that's the first time I've known someone refer to UQAM in English, despite working there for two years. $\endgroup$ – Jessica B Dec 17 '15 at 11:59

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