From Hewitt's article, I think his ideas and arguments have connections to the discussion/reading about instrumental and relational mathematics as well as Thinking Classrooms.
The idea of informing students about facts, especially if it's "necessary" knowledge, knowledge they could likely find out for themselves given time, is similar to the method of instrumental mathematics giving students a formula/method. Students simply accept, memorize and use it without knowing why it works. Even if the reasoning (proof) is given to them by the teacher, the students themselves are not the one doing the critical thinking and therefore they are still memorizing the process rather than exploring, learning, and understanding it themselves. Meanwhile, letting students use critical thinking, reasoning and recall of past knowledge to make their own discoveries is more meaningful. Making the discovery/achievement themselves makes the information they learned much more memorable than being spoon-fed the information. I've experienced forgetting the formula for the area of a circle many times because I simply learned what the formula is without knowing where it comes from. However, after figuring out the proof for the area of circle, things clicked and I actually remember it now.
After reading this article, I will definitely be more mindful of "arbitrary" vs "necessary" knowledge when making lesson and unit plans. The interactive activities in my lessons I plan can focus on the "necessary" and have students try to reason by themselves or collaboratively on what the pattern is. For example, for graph transformations, students can try figuring out the pattern for each change of values in the general graph transformation formula. A vertical whiteboard (in Thinking Classrooms) approach would be very good in achieving having students figure out the problems together. I definitely also would give students a puzzle to try at the start of lessons to reason through to build their comfort critically thinking, problem-solving and collaborating with others in class. I will also be more mindful to not give hints too early or often and instead take a step back to let students think.
Great thinking and writing about arbitrary vs. necessary in math teaching, Lisa! The connections with instrumental vs. relational learning are helpful. I think you're quite right too that a teacher's explanations and proofs may be taken up by most students as 'just something else to memorize'. It's sometimes hard for us as teachers not to give hints and explanations, but instead to let learners engage and reason for themselves -- but it is well worth holding back and giving students space to figure things out, with some minimal scaffolding!
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