Most of us tend to teach mathematics in the same manner as it was taught to us. I think of that as our “default setting”. We are comfortable with it.; it “feels right” to us. Unfortunately, it is often not the best way to teach math (which is why most of the North American population does not understand mathematics!).
To teach otherwise, to use strategies and approaches that we did not experience in or school years, requires real effort to change. It makes us uncomfortable; it does not “feel right”.
I believe that when we teach mathematics meaningfully, we need to have students doing more than just following our instructions. When we show them how to do a particular computation (e.g., 27 x 46), demonstrating each step of the computation that leads to the answer, their subsequent work (i.e., the 50 problems to do on the page) only shows to us whether or not the students could follow all of the necessary sub-steps in order to arrive at the final answer. Such work does not show any understanding of multiplication, nor does it show that the students understand why the sub-steps produced the answer.
I contend (again!) that “understanding” lives in mathematical processes. The National Council of Teachers of Mathematics (NCTM) lists 5 math processes, namely these:
• Problem solving
• Reasoning and Proof
If we regularly incorporate these processes into our mathematics teaching, students cannot help but build mathematical understanding!!
I will add one caveat: you cannot add the processes for a week, examine the results, and say, “this doesn’t work!” The truth is, we must help students build skills in these areas. If they have not been talking and/or writing about their math thinking already, such communication will take time to build. If students have not been problem solving (in the truest sense of the word) then they will need to learn some strategies and approaches to help them solve problems. A similar case can be made for making connections, reasoning and proving, and representing.
But building competency in the processes is worth the time that it takes! When students are doing the hard thinking in math (and not just following rules that are meaningless to them) you will find you and your class enter a new place of teaching and learning!
Will you step out of your comfort zone so the students can go “where the magic happens”?