Here is how I used to go about learning when I was a student, before I knew better.

During Math class, which I enjoyed, I paid close attention and whenever I had an inkling of how to do the next step - for example, after the instructor had stated a lemma but before

To study material from lecture, I would quickly skim over my lecture notes to memorize the main definitions and theorems. Then, to work through assignments, I would do many exercises with ease, then stop when I encountered a difficulty and, after giving it a fair try, either manage or fail to solve the problem, call it good and turn in what I had - usually about 80 or 90 percent of the assignment. That meant that, as the assignment usually had problems of increasing difficulty and I stopped before the last problem, my studying reinforced my knowledge of material which I already understood fairly well, but never made me get insights that I had not already acquired in class. I do not recommend that way of studying.

One year I made a friend with whom I worked closely, and, from her, I learned how to study.

To study material from lecture, here is what she did: she would reconstruct the lecture. She had her lecture notes in her notebook next to her, but usually kept it closed. She would try to write down definitions and main theorems from memory, checking by briefly opening her notebook after writing. Moreover, she would also try to reconstruct the arguments by writing down the statements of the lemmas and outlining a proof sketch for each of them! Of course, that was time consuming: even a well-understood two-hour lecture that was fresh in our memory would take at least one hour and often close to two hours to redo. But if there was any subtlety that had gone by unnoticed during class, it really put the spotlight onto it. Then, to do the assignments, she would put in about twice as much work as I used to: indeed, she proudly claimed that she would never turn in an incomplete assignment. Since the last problems were much more difficult than the rest, she would spend the larger part of her time working on those challenging problems. At first that seemed stupid to me: why would one spend so much time reviewing lecture notes, when there was no grade associated to that activity, and we had understood the class fine anyway? Why would one spend more than half of one's study time on a tiny part of the assignment, for a tiny part of the grade? What an inefficient use of one's time! But then, as I watched her work, a realization dawned on me: the assignment was a way for her, not just to practice, but to gain better understanding of the material. She was learning

*in depth*.

It sounds like grades sent you a bad signal about how to spend your time. Do you think that students you teach at Brown fall into this trap less often because of the alternate grading system there?

ReplyDeleteNo, I don't really think that the "alternate grading system" at Brown makes a difference; but I hope that the UTA program makes a difference. Being a UTA makes them think from the teacher's perspective: what is the point of this part of the lecture or assignment? What should the students get from this or that? Why have some hard problems? How to choose exercises to stimulate and challenge them? It's all part of the larger context of thinking of what learning is about, and it shifts the focus away from grades.

Deleteon how to attend lectures: should professors encourage this type of attendance by pausing every 20 minutes or so to let students complete the details of a proof on their own or work on an exercise? It seems like this would be a great way to keep students awake and engaged.

ReplyDeleteSounds good. I took one course in which that was done in a formal way: 40 minutes or so of lecture, then a 10 minute break for the instructor while students where working on an exercise directly related to what had been covered immediately before the break, and repeat.

DeleteIt was done regularly by our teacher in math sup and spe, by sending someone to the blackboard to help with simple proofs from time to time (two or three per two hour lecture). They were rigorous in who was going to take turns so no one went too often or too seldom, and the inevitability of someone's turn took away a lot of the anxiety. Besides, since we were all learning, there was no stigma for not being able to complete the proof - only for not knowing the previous material. That kept everyone engaged. And everyone could try and figure out if the person at the blackboard was a little slow. A few times, the teachers would send the student back with a sigh (and a grin on the student's face) and finish the proof themselves. That was the normal way of teaching back then. I still have my notes/books from all those classes and I cherish them. My dream is that I can someday walk through them with my kid(s).

ReplyDelete