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• Scaffolding Problem Types from “Plain Vanilla” to “Chocolate Amaretto”

In my previous blogs in this series, I have attempted to make the following points.

• While engaging our best students, we cannot abandon our duty to engage them all.
• A student can be high-performing in various ways.
• Our assessments define what we mean by “high-performing.”
• We should assess what we value and value what we assess.
• A good mathematics student is valued for diligence, knowledge, cleverness, and collaboration, which are elicited and assessed in different ways.
• Homework is an excellent way to encourage diligence and collaboration.
• To encourage and assess knowledge and cleverness, we must get past the idea that we need to prepare our students for every problem we will pose to them.
• If we want to challenge our students, we need to scale our grades.
• If we do scale grades, our assessments can actually encourage thinking (by requiring it), thereby assessing true knowledge and cleverness.

In my last blog, I noted that scaling grades enables us to move beyond the (educationally crippling) idea that we need to design tests that give our weakest students a fighting chance of getting 65% of the maximum score. While such a test might help us to pass a weak student, a strong student could get something like 98% of the maximum score and not be challenged (or even intrigued) along the way. By scaling grades, we get to protect our weaker students and challenge our best students at the same time. In fact, we get to challenge our weaker students as well, and that is not a bad thing. They might occasionally surprise us with their insights!

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It should go without saying that fair assessments should contain plenty of “plain vanilla” problems. These would be problems closely resembling homework problems, class examples, or problems from previous assessments. Since solving these problems would require no insights beyond what they have previously seen, diligent students could prepare for them by studying. If they have success, it could be due to memorization and mimicry rather than understanding, but that is okay; we value diligence. Also, it could be a sign of deeper understanding. Still, to assess real knowledge and cleverness (and to make our assessments more interesting) we need to ask some problems that go beyond plain vanilla. To extend the metaphor, we need to ask some “chocolate amaretto” questions.

There should be no need to be overly prescriptive here; a few examples should suffice. Once a teacher is liberated to ask flavorful problems without fear of betraying some students, everyone can enjoy making them up.

Let’s say the class has been doing factoring, having begun the year with the usual discussion of functions, and let’s say they are taking a test without calculators. (It’s a good idea to have some assessments with and some without, but that will be another blog.)

• A vanilla problem would be: Factor x2 – 9.
• Less vanilla problem might be: Find the range of the function P(x) = (x + 3)(x – 3).
• Chocolate amaretto problem might be: Suppose x2 – 9 is a factor of P(x). What is P(3)?

Notice that these problems are not particularly difficult, but an ability to go through the motions of factoring does not guarantee student success. A little thought is required in to recognize that the un-factored form of a quadratic function might convey more information about the range while finding P(3) requires an understanding of what it means to be a factor.

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Now let’s say we are dealing with linear equations. Our students have seen the midpoint formula and the distance formula in an earlier lesson, and they know all about slope.

• Vanilla problem: Find an equation of the line through (3, 4) with slope –2.
• Less vanilla: Find an equation of the line that is the perpendicular bisector of the segment with endpoints (2, –2) and (6, 10).
• Chocolate amaretto: A circle of radius 5 centered at the origin passes through the point (3, 4). Find an equation of the line tangent to the circle at that point.

I should add that a problem like the perpendicular bisector problem above actually becomes vanilla once the students have seen one just like it. Oddly enough, a problem like the circle problem retains its extra flavor after students have learned about derivatives in a calculus class since the obvious (vanilla) approach involving the derivative is significantly harder than the geometric approach of finding the negative reciprocal of the slope of the radius.

It is the non-vanilla problems that will, over time, turn your students into problem solvers. In fact, it will not take much experience with them to convince many teachers that they ought to be challenging their students with such problems every day, during class, in what the ed biz calls “formative” assessment. I have only been ignoring that important aspect of pedagogy up until now because of the need to get past the barrier of “grades” first. If we can now agree that we can pose problems on our tests for which we have not deliberately prepared our students (while keeping grades realistic through scaling), then the corollary will be that we can devote more class time to us watching the students do math rather than the other way around.