Quite the contrary, I want the student to understand a concept well enough
and be comfortable enough with seeing new questions so that when I do ask
them a question on an exam that is of a different form (although covering
the same content) then they will be comfortable with it. I cannot provide
the student with this level of comfort unless I provide the student with a
wide variety of questions to begin with. I didn't say that I wanted a
repository of 30 questions that are the same, instead I would (I'm dreaming
here) like a repository of 30 question which are all _different_ but
require understanding for the same concept. Then I could ask the students
to work through 5 of them on their own and I use a completely different
question for the exam (regard this as a simplistic example).
In my classes I often give a new "type" of question on exams and I never
get complaints from the students about not having seen it before. The
reason is that they have worked through (on their own) many different types
of questions (as opposed to the same types over and over). I would like to
see a repository of exercises that would help me provide the students with
this variety.
An example of such a repository of questions that do not give GC users an
advantage (as far as I can see):
Concept: The derivative of a function is a function whose
dependent variable gives the slopes of the tangent lines of the original
function.
Types of questions:
1. Give a graph of f'(x) and the y-intercept of f(x) and ask for a rough
sketch of f(x).
2. Give a graph of a function f(x) and ask for a rough sketch of f'(x) and
f''(x).
3. Give two graphs one of which is the derivative (or the second)
derivative of the other and ask for the student to identify which is which
4. Give graphs of various f(x) and various f'(x) and ask the student to
match them (someone mentioned this in a previous post; sorry I don't
remember the name).
5. Give a graph of f(x) and ask varous questions such as
a) on what interval is f'(x) positive?
b) on what interval is f'(x) increasing?
c) for which x values is f'(x)=0?
d) etc.
6. Give a graph of f'(x) and ask varous questions such as
a) on what interval is f''(x) positive?
b) on what interval is f(x) increasing?
c) for which x values is f''(x)=0?
d) etc.
7. Give a complete list of where f'(x) and f''(x) are
positive/negative/zero/undefined and some points on y=f(x) and ask for a
specific graph of f(x).
8. Given a brief list of where f'(x) and f''(x) are
positive/negative/zero/undefined and ask for various graphs that match
those characteristics (for example, have the student give as many
significantly different graphs as possible where f'(2)=0. You may think
students won't know where to stop on this question, but the most answers I
usually get to this question is 5 graphs but more often only two. It is a
good question to discuss in class and emphasizes local behaviour of graphs
and when two graphs are significantly different. You might be surprised by
the students' ideas on this question.)
9. I'm tired now. I may be able to think of a couple more but I would
like to hear what interesting ideas others have (of course, I'm in the
process of writing my final exam in Elements of Calculus and I'm always
open to new ideas).
I hope I have dispelled the notion that I only give students questions they
have seen before.
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Murphy Waggoner
Department of Mathematics
Simpson College
701 North C Street
Indianola, IA 50125
waggoner@storm.simpson.edu
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