The text was discussed the teacher's role in the classroom, being to differentiate arbitrary and necessary concepts. The author says that students need to be informed when a concept is arbitrary. On the other hand, teachers need to create tasks and activities that help students comprehend and develop a better understanding of necessary concepts and mathematical truths. The biggest takeaway from the article for me is that arbitrary concepts have to be memorized, while necessary concepts have to be understood by students. As a teacher, I think that it is important for students to be aware of this, as it will help them know how to study for tests and exams.
So how would this affect the way I teach my math classes and plan my lessons? I think that the type of activity provided to the class really depends on what type of concept they are learning. For example, arbitrary concepts can be studied and memorized through flash cards and definition activities. On the other hand, necessary concepts can be studied through homework problems, proofs, group problem solving, multi-step critical thinking puzzles, etc. This shows the importance of teachers strategizing activities around the types of concepts students are learning.
Thanks Nathaniel. I can see that you have a good sense of what Hewitt means by arbitrary and necessary, but you still seem to be taking a predominantly teacher-centred view of learning, where the assumption is that understanding comes from teacher explanation, rather than from student reasoning. Homework problems are most often drills of what was already told in class, rather than chances to reason for oneself -- although critical thinking puzzles and other kinds of problem solving offer more possibilities. I suggest you keep working on ways to give students chances to use their considerable reasoning power, so that they can make sense of the necessary understandings without always having the teacher just tell them things!
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