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In my upper-level math classes, students read the textbook and discuss it in class. Learning mathematical ideas through reading is important, but it’s probably just as important to critique the assumptions of the author, which can have both positive and negative effects. 1/6
Positive: Sometimes steps will be judiciously omitted. Having the reader fill these in can create greater engagement, promoting greater independence later on.
Negative: At the same time, infelicitous expressions (such as “this is obvious”) can belittle the reader’s efforts. 2/6
Negative: At the same time, infelicitous expressions (such as “this is obvious”) can belittle the reader’s efforts. 2/6
One time I assigned students to go through a certain textbook passage, find three places where something was labeled “clear,” “obvious,” or “trivial” and discuss what work would actually be required to justify the statement. I think I need to do more of that. 3/6
Even good mathematical writing usually has some gatekeeping elements. I hope that confronting these directly, with attention to the threat of inferiority they can convey, will allow us to move through them as a community, to get at the real content we’ve gathered to study. 4/6
One of the best comments I ever received on an assignment in grad school came from a TA, who wrote next to a proof, “Never say that something is easy.” Mathematicians often like to say things are easy, mostly to communicate who belongs to the in-group. 5/6
It’s no harder to write “This uses a trick you may have seen before” or “Try a few examples and see if the pattern becomes clear” than to dismiss the whole cognitive process with “This is obvious.” Audience-dependent, of course, but as teachers we should cast our nets wide. 6/6
