Memo #5 Generalizations

Generalizations are a key component to the acquisition of algebraic thinking. Driscoll suggests that it is important to not only generalize arithmetic, but also functions and relations between numbers. For example, observing that powers of 2s (ie 1, 2, 4, 8, 16, 32...), 3s (1, 3, 9, 27, 81...) and so on all grow larger, but numbers like 1/2 to positive powers grow smaller and smaller, approaching 0, helps students grasp the concept of the function that is "repeated multiplication." In a way, if we think about fostering algebraic thinking, rather than procedural fluency that is warned about in Kilpatrick and Swafford, then we will students building rules for themselves, and understanding the functions they are applying, or tasks they are following through with.

One of the large hurdles for students is systems of equations, a topic that Driscoll touches on in Chapter four. As in many mathematical situations, sometimes all of the computation is not necessary, and the skillful student will see patterns and develop strategies to take shortcuts while still arriving at the same answer. The hardest part about all of this is inspiring students to pick up on the knowledge for themselves, since problem solving and algebraic thinking is not the easiest to teach.

I would suggest that instead of modeling problems, teaching procedural strategies, and then letting them go to fend for themselves on the homework, that instead we foster a general curiosity and a genuine interest in math by allowing them to come to terms with new ideas on their own. By this I mean that students come into the classroom with so much information already, and (I would like to think) an innate desire to learn. So instead of talking at them for 45 minutes, or an hour, or even two, we should rather entrust them with problems and puzzles that will instead lead to learning. Too often I see students who become frustrated and give up on problems that are not that different from what they know, but they don't see themselves as competent problems solvers. The truth is that they ARE capable of extending their knowledge, like Driscoll suggests. We just need to give our students plenty of time and support.