I love working with pre-service elementary education majors. I frequently teach their content courses, and I am teaching them this semester. I usually spend a decent amount of time in my elementary education courses having the students explain why the standard algorithms for the operations on integers and fractions give correct answers. That is, I work to get the students to understand how the algorithm relates to the definition of the operation. This is something that they always have trouble with (I have taught the course 5-6 times).
But I think that this semester may be significantly better. The reason why is that I am applying techniques I learned in Black’s Assessment for Learning: Putting it into Practice. Here is what we did in class yesterday:
- I had the students determine qualities that make an explanation “good.” I prodded them on a couple of these, but we came up with:
- The explanation is relevant; that is, the explanation answers the question at hand.
- The explanation is appropriate for the audience (i.e. the explanation uses knowledge common to both the explainer and the explainee).
- The answer is correct.
- The answer is complete; there are no gaps that the audience would need to understand the explanation.
- The answer is concise; it is long enough, but no longer.
- I gave them three explanations for why the standard addition algorithm is really the same as definition of addition (roughly, “combining and counting”). Here are the explanation: one was decent, another was solely an explanation of how (not why) the algorithm works, and a third was somewhere in between.
- I asked them how well each of the explanations did in each of our categories from 1.
Initially, the students all loved the “how but not why” explanation (the second one). But when we delved into relevance, several students started saying that it did not answer the question. I could almost literally see light bulbs going off over several of the students’ heads. I think that this will greatly help their justification of several multiplication and division algorithms; I will keep you posted.
In some sense, I am kicking myself for not doing this before. I have (in theory, at least) been a proponent of helping students develop their metacognitive skills. It seems like that is what I was doing yesterday: giving them tools to think about how they are thinking about explanations.