Teaching Note: Using an Unrelated Problem
So much of a student's learning is unrelated to the things we think about when we think about teaching. Lesson plan, the flow of topics, the questions we ask students, how clearly things are explained, these are all important. But so much of a class also depends on each student's mental state, and I think much of a teacher's success comes down to how well they are able to control their students' mental states. Do you bring the energy of a room up when you enter it? Do you command attention when you speak? Does the way you talk help keep others awake? This may help you be a much better teacher.
However, there are other tricks, tricks of the human body and how we react to various stimuli, that we can also use. For example, a New York Times article recently discussed how Doug Lemov is coming up with a list of techniques teacher use, such as standing still when giving instructions, because it draws more attention to you. But I suspect that much more effective than this is Indeed, equally important as the actual content of a class is achieving intellectual engagement for the students, for example through(*) some degree of student participation: I certainly found that, whenever I asked or answered a question in class, I immediately became more awake and followed the lecture much more clearly. Perhaps it was just a little shot of adrenaline by body got from speaking in front of others; regardless, that biological response enabled me to be a better learner. [Additional clarification: So suppose that you think you've lost the class presenting some math. How do you get back the students who have lost it/are falling asleep/are no longer paying attention?]
All of which brings me to an idea that I've only just now put into words.
I was reading an excerpt from a book called The Calculus of Friendship (and the book itself, from the excerpt, seems much better than the title implies) in which they stated the closed form for the Fibonacci numbers:
F_n = [(1 + sqrt(5))^n - (1 - sqrt(5))^n]/(2^n * sqrt(5))
My first, split-second reaction to this formula was: huh? That's not it...
Then, of course a moment later, I realized that it is in fact identical to the formula I am used to:
F_n = [((1 + sqrt(5))/2)^n] - ((1 - sqrt(5))/2)^n]/sqrt(5).
(This formula makes more sense when you substitute phi = (1 + sqrt(5))/2 and phihat = (1 - sqrt(5))/2.)
I thought to myself afterwards about how I would help students follow a derivation of this closed form. Sometimes, students get lost in the derivation and the symbols become meaningless to them as you go forwards. Perhaps there's a way to re-engage them and to get them to consider the formula anew?
Well, one possibility would be to state one version of the formula, prove the other version, and say, "wait a minute, are these two the same?" Of course all the students will see that they are quite quickly and will thus re-engage with the formula, losing the mental block they had already put up against it when they got lost. Now if you launch into a review of what just happened, having just had a victory over the formula, they are more likely to stick with you lead your dear students once more unto the breach.
These kinds of careful coordination of classroom flow are something I haven't mastered yet, but now that I think consciously about them, perhaps I can structure my questions to students more carefully to keep them with me throughout.
(*) I originally wrote something I didn't mean, so I have changed the text and added a clarification to try to fix what I meant.
However, there are other tricks, tricks of the human body and how we react to various stimuli, that we can also use. For example, a New York Times article recently discussed how Doug Lemov is coming up with a list of techniques teacher use, such as standing still when giving instructions, because it draws more attention to you. But I suspect that much more effective than this is Indeed, equally important as the actual content of a class is achieving intellectual engagement for the students, for example through(*) some degree of student participation: I certainly found that, whenever I asked or answered a question in class, I immediately became more awake and followed the lecture much more clearly. Perhaps it was just a little shot of adrenaline by body got from speaking in front of others; regardless, that biological response enabled me to be a better learner. [Additional clarification: So suppose that you think you've lost the class presenting some math. How do you get back the students who have lost it/are falling asleep/are no longer paying attention?]
All of which brings me to an idea that I've only just now put into words.
I was reading an excerpt from a book called The Calculus of Friendship (and the book itself, from the excerpt, seems much better than the title implies) in which they stated the closed form for the Fibonacci numbers:
F_n = [(1 + sqrt(5))^n - (1 - sqrt(5))^n]/(2^n * sqrt(5))
My first, split-second reaction to this formula was: huh? That's not it...
Then, of course a moment later, I realized that it is in fact identical to the formula I am used to:
F_n = [((1 + sqrt(5))/2)^n] - ((1 - sqrt(5))/2)^n]/sqrt(5).
(This formula makes more sense when you substitute phi = (1 + sqrt(5))/2 and phihat = (1 - sqrt(5))/2.)
I thought to myself afterwards about how I would help students follow a derivation of this closed form. Sometimes, students get lost in the derivation and the symbols become meaningless to them as you go forwards. Perhaps there's a way to re-engage them and to get them to consider the formula anew?
Well, one possibility would be to state one version of the formula, prove the other version, and say, "wait a minute, are these two the same?" Of course all the students will see that they are quite quickly and will thus re-engage with the formula, losing the mental block they had already put up against it when they got lost. Now if you launch into a review of what just happened, having just had a victory over the formula, they are more likely to stick with you lead your dear students once more unto the breach.
These kinds of careful coordination of classroom flow are something I haven't mastered yet, but now that I think consciously about them, perhaps I can structure my questions to students more carefully to keep them with me throughout.
(*) I originally wrote something I didn't mean, so I have changed the text and added a clarification to try to fix what I meant.