Last week, I ended with a sample exam I might give a calculus course which included both Skills problems and Conceptual problems. Before presenting the final installment of this series on assessment, I thought I’d take a few moments to discuss the genesis of this exam format.

Again, the assumption here is that we are working in a more traditional system, where students must be assigned grades, and these grades must in large part be based on performance on exams.

Given IMSA’s statements about the advanced nature of their curriculum, I had concerns about the fairly traditional exams we gave in mathematics. In my mind, there was little to distinguish our exams from those given in any other rigorous calculus course.

The reason given to me by other mathematics faculty was that there just wasn’t time in a roughly hour-long exam to assess conceptual understanding. I wasn’t convinced, and I started thinking of an alternative. I did agree, however, that it wouldn’t work to have a conceptual question on an exam which might take half the exam period for a significant fraction of the students to complete.

What I finally settled upon was including a range of conceptual problems for which students only needed to provide a reasonable approach to solving. If you chose a conceptual problem which happened to be centered on a student’s weakness, you wouldn’t be able to assess a broader conceptual understanding. And if you insisted the problem be worked completely through, you encountered significant time constraints.

I’d like to share one last anecdote. I recall a parent visitation day one Saturday, which happened to be the day after I gave a calculus exam. Two of the parents approached me after the session and told me how much their son or daughter enjoyed my exam. This indicated to me that for the student who *can* perform the routine procedures easily, they *want* to be challenged to think outside the box, and indeed they *thrive* on such challenge. Shouldn’t we, as educators, find ways to stimulate *all* of our students, rather than be content with having students in the middle earn their B’s, making sure the struggling students earn their C’s, and relegating the very capable students to a sustained boredom?

And now for the last installment….

“Where does this bring us? Here are some key points as I see them.

- We should move away from assigning grades punitively.
- We should reconsider the “point'”system of evaluating student performance. Referring to the TIMSS (Third International Mathematics and Science Study): “In our study, teachers were asked what ‘main thing’ they wanted students to learn from the lesson. Sixty-one percent of U.S. teachers described
*skills*they wanted their students to learn. They wanted students to be able to perform a procedure, solve a particular kind of problem, and so on….On the same questionnaire, 73 percent of Japanese teachers said that the main thing they wanted their students to learn from the lesson was to think about things in a new way, such as to see new relationships between mathematical ideas.” (Stigler and Hiebert,*The Teaching Gap,*ISBN 0-684-85274-8, pp. 89-90.) A point system reflects the assessment of procedural knowledge. - “We can think of all assessment uses as falling into one of two general categories — assessments
*FOR*learning and assessments*OF*learning.” (From an internal document distributed to mathematics teachers at IMSA.) But why? The distinction is artificial. There are many other ways to compartmentalize assessments, such as timed/untimed, individual/group, skill/conceptual, procedural/relational, short-term/long-term, etc. The main argument for focusing on the “for/or” distinction is its relationship to student motivation — but we are given no context for it. I suggest that our typical IMSA student*is*highly motivated — certainly in relation to the average student in a typical high school classroom. - We should consider the assignment of letter grades in general. Right now, it would be impractical to suggest that we have formal written evaluations of each student in each class. But is it desirable? And if so, what resources are necessary to support such a system?
- We should discuss the assessment of problem-solving.

Will any of these suggestions help to illuminate the power of ideas? I’m not sure. With the current need to assign grades, and their current cultural meaning and importance — especially when it comes to applying to college — there will be the necessary compromises in the classroom. I realize that many suggestions are of the “move away” rather than the “move toward” type. But I suppose that if there is something I am moving toward, it’s giving students at all levels more of a BC Fast-Track experience regardless of the depth of content.

This means actively moving toward a classroom environment where earning good grades is subordinate to learning complex concepts. Of course the two are not mutually exclusive — but I’d rather have students earn good grades because they learned, rather than learn in order to get good grades.

Of course many issues brought up in these remarks have been left hanging or only tentatively developed. These brief comments are meant to suggest questions for discussion, not definitive answers.

I can’t resist ending with the following challenge from Maslow: “In order to be able to choose in accord with his own nature and develop it, the child must be permitted to retain the subjective experiences of delight and boredom, as *the* criteria of the correct choice for him. The alternative criterion is making the choice in terms of the wish of another person. The Self is lost when this happens. (Maslow, source cited earlier, p. 58.) Is it possible to create a mathematics curriculum which can survive this test of course selection?”

Thanks for staying with this series! No, there is no simple resolution to any of the issues described in this essay. But that doesn’t mean we shouldn’t be involved in a conversation about them….

One of the truisms of assessment is that learners figure out what we actually value pretty quickly. If we talk problem solving, but assess computational speed, they will decide what we really value.

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Yes, I agree. As teachers who value conceptual understanding and problem solving, we need to be willing to take the time needed for meaningful assessment of these aspects of learning mathematics. It does take more work, but I think it is necessary.

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