acceding that temporary ugliness is an essential feature of doing mathematics; you can’t build cathedrals without putting up scaffolds. ​ secrets the erasers and wastebaskets of mathematicians could tell us, if only they could talk!

This situation crops up so often in mathematics that the acronym “TFAE” (for “The Following Are Equivalent”) has become a standard part of a mathematician’s education.

And close reading is intimately tied to the kind of “close thinking” that math requires, especially the more advanced, theoretical kind of math. You need to be able to pass back and forth flexibly between the micro and the macro. You need to see both the big picture and the minutiae, both the forest and the trees.

and throughout the process instilling ownership, confidence, and competence among the students. ​ Under my approach, a student taking an exam is handed a sheaf of all the reading-summaries she’s written, class by class. It’s an open-book exam where each student has written her own book, if she���s availed herself of the opportunity. If the student knows this ahead of time, there’s a big incentive for writing that book, which requires that she keep up with the reading during the weeks preceding the exam. ​ One thing my students tell me is that they often end up not consulting the summaries at all during an exam, but that they’re still glad they wrote them. My students will spend much of their time after graduation solving problems for which no solution-sheet exists and trying to convince others that their solutions will work. To do that well, they need to acquire communication skills, which I can help them cultivate now by having them present solutions to their peers and get feedback on those presentations.