New answers tagged test-design
6
Comment-answer, but too long for a comment:
I think you are thinking about this wrong.
Tests are some of the MOST valuable hours in a course. They are high stakes performances (like in music or sports). Preparation for them drives a lot of learning. Then the actual execution and subsequent feedback is often much more valuable training than routine ...
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This is not really a math problem, it's a social problem. Some schools, such as West Point and Cal Tech, have their own honor code systems for this sort of thing. From what I understand, they work very well. However, most schools do not have any such system. Social scientists and psychologists have studied what factors promote or prevent cheating. Cheating ...
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What one has to do to test for conceptual understanding is hard to state in terms of general principles (although Polya's books on Plausible Reasoning do a pretty good job of addressing the issue) and maybe is best addressed via examples.
Here is one example. Consider a cubic polynomial in one variable that is increasing as a function of its argument. ...
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The "deeper" or at least off-beaten path solution should be much quicker than a standard but boring approach to create a system of equations or calculate a derivative or something like that. Give two-three problems and limit time so that they can solve them in time if they apply some sort of a trick. You'll get a lot of angry students :)
Here is an middle-...
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As stated in other comments, try not to "invite" your students to merely apply rules they do know. In the specific example you mention, I would prefer a multiple choice question of the form:
A truck's distance in metres from you as a function of time $t$ (in seconds) is given by a smooth function $p:[0,+\infty)\to\mathbb{R}$. Knowing that $p(19)=12$, $p'(...
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Asking students to explain why something happens can be useful for assessing understanding, although it is often harder to grade and works best with many demonstrations before the exam. (Students need to know what your expectations for a thorough explanation are.) I have found that asking students to critique a process will sometimes help me assess their ...
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One way is to use simple, atypical mathematical objects, fringe cases, non-examples and counterexamples, etc.
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Agreeing with comments and other posts: If you want more conceptual answers, give them less details in the set-up.
Using your velocity problem, here are a couple of examples of making it more conceptual:
Suppose that a truck's distance from you in meters at a time $t$ seconds after the big bang is given by the function $p(t)$. What does $p'(19)$ tell you (...
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I agree with @BrendanW.Sullivan's comment. That is, when teaching an undergraduate course, like calculus, students need more than procedural knowledge. For a deeper understanding, and efforts to evaluate such, students should be asked on exams to answer a few "free form" questions, like the one Brendan suggested.
A good question to ask following any ...
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If your assumption on how the students may get the right answer, maybe rephrase the question in a way where doing that is the wrong answer. For instance, you could ask for acceleration rather than velocity.
Still, I don't think many students will get the right answer by following the logic you mentiond without having an understanding on what's behind!
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