John Benson at Angles of Reflection has a post–a very important one–that’s been sitting at the top of the blog for a long time. Every so often I re-read that and think about it. It offers a telling vignette of how students are taught a rich technique for addition (“near doubles”) but do not use the technique. The students still revert to standard procedures of addition to “get the answer.” Then they tack on the new technique, which means they aren’t really learning about the structure of addition.
This explains something that has troubled me for some time. I frequently hear from parents that their child is not challenged by the curriculum that is being offered in their school. When I have looked at the curriculum I find that it is often very rich and full of many challenges. Perhaps the reason for this disconnect is that when the student looks at the material, the only thing the student is thinking about is how can I get this done and get a right answer, while I am looking at how many different things a student can learn from the variety of approaches taken.
I think the key to cutting off answer-getting in students is to notice that what they want is a good grade for just having the answer, so the teacher strategy is to give a middling grade for just having the answer, and a better grade for employing new strategies.
So, in the new year, I resolve to open up my questions more and to allow students to develop the question more in class. I’m going to invite more conjectures from them.
Here’s a glimmer of an idea I ripped off from an AMC question:
Five central angles are drawn in a circle. Their degree measures are integers in an arithmetic sequence. What could their measures be?Could you have only one set of five angles? Could you have more than one set? If so, what’s the smallest degree measure that the first (lowest) angle could be? If you are stuck, change the question. Suppose that the five angles are equal and solve. Could you use anything from that to answer the original question?What if it is a geometric sequence?
This thinking is making me wish that the definitions of the terms geometric sequence and arithmetic sequence get taught earlier–guess I can do that in about ten minutes in my own class. There’s a lot of problems we can make slightly more challenging by replacing an assumption of a set of equally numbered values with an arithmetic (or geometric) sequence.
Another thought, is that I should be including questions about 3D analogs to much of the 2D geometry I teach. I picked this suggestion up from Judah Schwartz, who has had and continues to have many good ideas about what math education ought to be. For instance, many geometry classes teach how to make a circumcircle of any triangle… but Judah has a good challenge: Can you draw a sphere through the 4 vertices of any tetrahedron?