I believe that one of the most important ways that teachers engage with high-performing students is by defining success in mathematics in ways that truly matter. In the last blog I mentioned four qualities I emphasize with my students: knowledge, diligence, cleverness, and collaboration. I will mention here something that is not on the list, but which I used to spend too much time assessing: memorization. It’s not that a good memory is a bad thing; it’s just that memorization does not equate very well with real learning, and I do not want my assessments to suggest that it does. Of course I want my students to know that the cosine of zero is one, but not as a disembodied factoid. If I can convince them to work with trigonometric functions to solve real problems, I hope that they will internalize such details along the way. I now look back on some of my old assessments, which seemed to encourage success by memorization, and fear that they reduced mathematics to a trivial pursuit.
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For example, we might feel as mathematics teachers that the quadratic formula is well worth memorizing, but we should keep in mind that it is very easy to look up if someone has need of it after graduation. Moreover, the ability to recite it by heart does not make one quantitatively literate, let alone a capable mathematician. One of my elementary school teachers felt that it was important to memorize all the English prepositions in alphabetical order, and I am somewhat amazed that I can still recite them today. (They are probably occupying the brain cells that other people are successfully using to remember their passwords for Internet sites.) Although I still use prepositions on a regular basis, I can’t say that I have ever referred to that memorized list to find one. I have had students who can recite 100 digits of pi , and I suspect that it will serve them as well in their ongoing education as my list of prepositions has served me. The point is that the contents of our memory banks, be they mathematical or otherwise, are the by-products of education, not the goals. They should not be the way we define success in mathematics, and they should certainly not be the way we define failure.
We have all had students who seem to be incapable of memorizing the formulas we want them to know (perfect square trinomials, difference of squares, special triangle ratios, and so on), but who can memorize within a month the blocking and the lines, often with Elizabethan idioms, for their parts in a three-hour school play. Rather than envying the drama teacher, we should gratefully accept the evidence that our students are, indeed, capable of memorizing stuff, which liberates us to measure genuine understanding in our mathematics assessments.
Incidentally, if college English professors expected those students to remember their play lines months later and recite them as a qualification to enter their Shakespeare classes, those students would all wind up in Developmental English. But that’s another blog.
So, getting back to mathematics, a quiz with ten questions that all depend on memorization is not a great quiz. A student who gets them all right might not understand anything, and, perhaps more importantly, a student who gets them all wrong has not demonstrated an inability to do mathematics. A better quiz might ask for the formula first, then invite the student to solve a few problems applying the formula. The student could lose the memorization point up front and get full credit later for correctly solving the problems using the incorrect formula. Another option might be to start with some “matching” questions that could jog the students’ memories and thus increase the probability that they begin with the right formulas.
For my money, the most helpful statement about assessments is still the old classic: Assess what you value and value what you assess. Remember that you, the mathematics teacher, have the capability and the responsibility to define for your students what it means to succeed (and correspondingly to fail) in mathematics. Every assessment we give delivers a message; let us make sure that it is a message worth sending. Next month I will return to the four qualities I do attempt to assess (knowledge, diligence, cleverness, and collaboration). I will also offer, with no intention of being prescriptive, a few ideas about how they might be assessed.
Stay tuned for the next post in this series from math author Dan Kennedy as he delves deeper into additional ideas for engaging high-performing students:
- Engaging High Performing Students
- Memorization in Mathematics
- Assessing Diligence
- Assessing Knowledge and Cleverness
- Plain Vanilla Problems and More
- Making Students into Problem Solvers
- In Praise of Formative Assessment
- Searching for Gauss
- The Mathematical Pipeline