Exams
A quote from Marc Olschok at the Department of Mathematics, SUNY at Buffalo:
(I think this applies to many other subjects as well)
(1) _what_ is tested and _when_ is it tested ?
(2) what _types_ of tests are used ?
(1) One thing, administrators and designers of undergraduate programs
do not seem to grasp is that understanding needs some time to ripen.
Short term testing will put those in advantage, who concentrate on
their test-taking skills instead of deeper understanding.
I am sure that on an exam given 2 month after the course you would
do far better than someone who had only given attention to the
short term tests during the course.
A comprehensive (1yr) oral exam about a _subject_ can tell much more
about the mathematical ability of a student than a lot of tests
during a particular _course_ .
(2) Ultimately, a particular test will only test the skills needed to
take this particular test. A test will be useless unless these
‘test-taking skills’ are related to those skills that are supposed
to be tested.
It might be helpful to look at three popular ways of testing:
homework (with presentation), oral exam, written exam.
With weekly assigned homework problems, the students have to think
about problems for a while, they need to try and retry solutions and
finally they need to write it up. If they have to present their solution
they need to communicate as well (this also regulates excess copying).
The setup can be modified easily to include small groups, working on
problems. Indeed a few carefully chosen problems (about 4 or 5) are
enough.
At the same time, students learn to utilize resources like libraries
and fellow students.
Tenancy, patience, care and the ability to write and speak Mathematics
is tested and developed at the same time with this method.
These abilities are indeed necessary for doing Mathematics.
In an oral exam, the students can adjust to the level of the
examiner (no joke intended) as far as precision is concerned.
The students need an understanding, deep enough to actively explain
material. At the same time they have the chance of showing what they know
instead of revealing what they do not know.
Sufficient understanding of the material and the use of mathematical
language are necessary for such an exam; skills, they need later
on within Mathematics and other fields.
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.
Currently, the emphasis is on short term tests in the form of written exams.
Perhaps this is due to the image of ‘objectivity’ they enjoy.
At the same time, grades determined by these exams are valued high
by administrators.
Only a few students have courage enough to withstand this pressure and hold
on to learning in this environment.
For the majority, the result is graduation in test taking, rather
than Mathematics.
