Re: Mathematical concepts

Herman Rubin <hrubin@xxxxxxxxxxxxxxxxxxxx> wrote:
> Marc Olschok <sa796ol@xxxxxxxxxxxxxxxxxxxxxx> wrote:
> >Herman Rubin <hrubin@xxxxxxxxxxxxxxxxxxxx> wrote:
> >>[...]
> >> Now what are you going to do with students who say, "Don't
> >> teach me the `theory'! Just tell me what I need to do the
> >> problems on the test!" Unless we are willing to tell them
> >> that they are going to be responsible for the concepts, not
> >> the manipulations, we will get more and more of this.
>
> >As long as the fate of students (within the educational system) depends
> >more on their performance in tests than on their understanding of the
> >concepts, the above situation is not likely to change.
>
> This is why I call it the MISeducational system. But it
> might even be the case that the results on tests given
> later will be better if the concepts are learned rather
> than the manipulations. Is the point of an education to
> get points and forget, or to learn for the future?

The latter of course. But this is discouraged within the current
test-driven system. This fundamental flaw will remain, even when
the question "manipulation vs concepts" is resolved by suitable
designed tests.

I also agree with you, that learning concepts instead of manipulation
will give better results on (suitable!) tests given later.
But for this to show, it helps to actually give such tests later,
not during or immediately after the course.
For some course-sequences comprehensive exams about the material
of two semesters could also be an improvement, because these fit
better with the cumulative nature (like language) of learning mathematics.
The current timing of exams instead enforces the short-term attidude
with respect to learning.

>[...]
> >There will still be students who are interested in learning and
> >understanding the concepts _and_ "train" for such tests. But they
> >will most likely view these activities as seperate issues.
> >If pressed, most will prefer graduating as 'testing machines'
> >(hoping to catch up on the concepts later) over dropping out
> >and starting a career as a well educated member of the working poor.
>
> Those are unlikely to be successful if they have to train for
> the tests. Will they even know which are the concepts? This
> is not idle; the concepts can be difficult to see, even for
> such as me.

Well, I already mentioned the timing of such exams, which has some
influence here, since conceptual understanding needs some time to ripen.

Another issue is the type of test. Compared to oral examinations
or homework assignments with presentation, written exams are quite
poor instruments for measuring students understanding.

Just a quote from 9 years ago:

| For a written exam, the students have to train rather than learn.
| They are put under restrictions of time and resources, both of which
| usually do not occur in mathematical work.
| The question is not whether they can explain Mathematics or solve problems,
| but whether they are able to solve problems under these restrictions.
| In this setup, deep understanding has almost no effect: there is neither
| space nor time for questions, concerning understanding.
| In fact, a more conceptual approach to a problem, that might give a better
| solution afterwards, will often need more time than provided, so that
| in this setting understanding might get in the way.
| The strategy
| - look at plenty of test problems before the test and memorise solutions
| - recognize a problem during the test
| - dump the precompiled solution on the paper
| will lead to 'better' results, than learning and understanding.
| I do not claim that the skills, needed for written exams are completely
| useless. They are very important for ... written exams.
| However, they do not have much relation to Mathematics.

This whole posting is still available at
<http://groups-beta.google.com/group/sci.edu/msg/70825a21e0e79e85?dmode=source&hl=en>
and was a response to
<http://groups-beta.google.com/group/sci.edu/msg/91a837d5920d772d?hl=en>

You might also remember the thread. ( Not much progress since then :-)

Marc
.

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