Assessing Knowledge and Cleverness in the Math Classroom


It took me many years of teaching to figure out that my well-intended attempts to make mathematics easy for my students were stifling their creativity. I was preparing them to take tests that I would create, and the obvious expectation was that I would prepare them for those tests. In some sense, posing a problem for which I had not fully prepared them in advance would have been a violation of their trust in me as their teacher. The problem with this line of thinking is what it says about thinking, as I eventually summarized in the following Teacher’s Dilemma:

  • Thinking takes time.
  • If you can do something without thinking, you can do it very well.
  • If you can do something very well, you must have been well-prepared.
  • Therefore, if you prepare your students well, they will proceed through your tests without thinking.

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You must admit, there is some truth to this whimsical syllogism. Indeed, the more pressure we feel as teachers to prepare students for our own tests, the less thinking we will want them to do. That is why I have a more optimistic outlook on the Common Core initiative than many of my colleagues. While it is admittedly terrifying that we will not have control over the state/national assessments, at least that keeps the game somewhat honest. If they prove to be good, honest, assessments of actual learning, they may eventually bring thinking back into the equation. But I digress. There are many students who enjoy taking tests because they are good at it.

These are the students who will earn the “high performing” label. If they can only succeed because they know exactly what to expect on our tests, then they have a particular skill for mimicry that will probably serve them well later in life, but probably not a life that involves much mathematics. In the more likely event that they succeed because they are either knowledgeable or clever, then they have the right stuff: the very qualities that we value in a good student. That is why our assessments should not shy away from testing knowledge and cleverness, and that is why we have to expect our students to solve problems for which we have not meticulously prepared them. If we can convince them in advance that we value knowledge and cleverness, then it is not even a violation of their trust to pose some problems that we do not expect everyone to solve.

If they surprise themselves (and their teacher) by solving them correctly, much happiness will ensue. If they do not, then life goes on. Or does it? Ironically, the answer to that question depends not on learning, but on grades. If we want to challenge our students to think, if we want them to make real use of creativity, if we want to assess true knowledge and anything that could be construed as cleverness, then we must have a way of scaling our grades to conform to our expectations. If we want to challenge our clever and knowledgeable students (and we do), we must anticipate the effect that could have on the rest of the class.

A student who gets 95% of maximum credit on a challenging test has accomplished something truly significant and might well deserve a 99. An average student who gets 65% of max on the same test might actually have a very respectable score, something we might feel deserves a grade of, say, 84. I would propose that we let mathematics take it from here. Let’s say we give the 65% student an 84 and the 95% student a 99. The two points (65, 84) and (95, 99) determine a linear equation, y = (1/2)(x – 95) + 99 that can be used to scale every student’s grade. It is consistent, it is fair, and it keeps everyone in the game.

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Teachers who know a little statistics will realize that the slope of the line is actually cutting the standard deviation in half, which might be the very thing we need to do in order to fairly assess knowledge and cleverness in a class with widely varying abilities (or confidence, or maturity, or whatever). This grading strategy can be explained in advance to students (after all, it is a nice application of math in the real world) and can even be used as a motivational tool. You can point out how that second number in the ordered pair (65, 84) is completely dependent on how well the 65% student was meeting your expectations. In fact, I like to use the class mean in one of my two ordered pairs so I can pin the second number on how well the class as a whole is meeting my expectations. If the class mean is 65% and I think the class average ought to be 84 (based on my long years of experience with similar groups), then I use the pair (65, 84). If I change it to (65, 80) and keep the high student’s ordered pair the same, the effect is pretty dramatic on the scores at the low end of the line! So even the struggling students – especially the struggling students – need to meet or exceed my expectations in order to get the most out of my grade-scaling strategy.

While this fixation on the mathematics of grade-scaling might not make much of an impression on an average student, your high-performing students will find it refreshingly candid, and they might even buy into the theory that raising the class mean by helping everyone to exceed your expectations is a win-win situation for them and for their colleagues. You should never underestimate the effect that your best students can have on your worst students if you can get them to think as a team. Grades are, admittedly, a rather trivial basis for promoting altruistic behavior, but at least you do not have to begin by convincing students to care about them (especially those high-performing students, who probably became high-performing by caring about grades).

Once you allow grade scaling to free you from the artificial (and educationally crippling) requirement of making up assessments upon which your least capable student can score 65% of maximum, you can make your tests and quizzes more challenging, more interesting, and more worthy of everyone’s attention. You can pose clever questions that actually inspire thinking and make students proud of their insights, even if they fail to find complete solutions. Obviously, you will also want to include enough “plain vanilla” problems to reward diligence, but consider using them occasionally to set up a few “chocolate amaretto ripple” problems along the way. In next month’s blog we’ll take a closer look at the metaphorical difference between vanilla and chocolate amaretto ripple.

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About the Author

Dan Kennedy

Math Author, Professor of Mathematics at the Baylor School