How did I miss that Justin Lanier started blogging (finally!) last August?
His blog is called I Choose Math. Keep your eye on this one, he’s the real deal.
Here’s another one. It should be quick.
When a student says, “Is it like this?” or the equivalent, I used to err on the side of “yes.” I.e. even if I wasn’t sure exactly what they were saying, but I thought it sounded like it might make sense. I think this was somewhat a function of the fact that I adopted a generally encouraging posture (this is my personality but also a deliberate choice), but it itself was just sort of my reflexive response from within this posture (not a deliberate choice).
It never felt quite right, so over time I trained myself instead to say things like, “I can’t understand what you’re saying but I think you might be onto something, but I’m not sure.” I never had concrete evidence that my original response was doing something unhelpful though.
Now I do. In a recent conversation with one of my teachers, several times I said, “Let me explain back to you what I think you’re saying, and you tell me if it’s right…” And he said, “yes yes yes it’s like…” But I didn’t recognize my attempted explanation in what he seemed to be saying yes to. So, it’s official: this is TOTALLY UNHELPFUL. I’m disoriented; that’s why I asked the question. Unless I come away from your answer feeling sure that you understood me, your “yes” only serves to make me more disoriented.
Take-home lesson. Never say “yes” unless you are sure you have understood fully what the student is saying, and agree with it. As I’ve often discussed before, sometimes a “yes” is inappropriate even then; for example if there’s a danger that the student is trying to foist onto you the work of judging for her or himself. But if you have any doubt, then the “yes” is definitely inappropriate: the encouragement is fake, and the student is left being equally unsure as before, and now also having exhausted the resource of checking with you. Retrospectively the only student who even feels good hearing the “yes” in this situation is the one who is playing a Clever Hans game, and in this case it does him or her the disservice of encouraging the game.
One of the many reasons I put myself in a math PhD program is that it is an intense full-time laboratory in which for me to examine my own learning process, and my experience as a participant in math classrooms from the student side. I hope to record many lessons from this laboratory on this blog. Here is one.
As a teacher I have always strongly encouraged people to pipe up when they’re confused, whether working in groups or (especially) at the level of whole-class discussion. To encourage this, I do things like:
* I leave lots of wait time.
* I respond to questions (especially those expressing confusion) with enthusiasm when they are asked, and after they are discussed I point out concrete, specific ways in which the questions advanced the conversation.
* I give (very deeply felt) pep talks about the value of these questions.
* Sometimes I directly solicit questions from people whose faces make it seem like they have one.
I am behind all of these practices. However, in every class that I have taught, whether for students or teachers, including all those of an extended enough length so that the practices would have time to shape the culture, it has always seemed to me that participants are often not asking their questions. This has puzzled me a bit. I’ve generally responded by trying harder: leaving longer wait-time, making more of a point to highlight the value of questions when they happen, giving more strident and frequent pep talks. This hasn’t resolved the matter.
Now I am not about to pronounce a new solution. But I have what for me is a very new insight. I imagine some readers of this blog will read it and be like, “Ben, I could have told you that.” I’m sure you could have, but this wouldn’t have helped me: retrospectively, students have told me it many, many times. But I didn’t get it till I felt it. This is the value of putting yourself in their position.
What I’ve realized since beginning graduate school is that I had an incomplete understanding of why students don’t ask questions. I believed that the only reason not to ask a question is the fear of looking dumb. My approach has been entirely aimed at ameliorating this fear and replacing it with the sense that questions are honored and their contribution is valued.
Now one of the great advantages of going to grad school as an adult, rather than going fresh out of college, is that I have very, very little fear of looking dumb. (In the immortal words of my friend Kiku Polk, you get your “f*ck you” at 30.) To all my early-20′s people: your 20′s will be wonderful but if you make sure you keep growing, your 30′s will be better.
And one of the great advantages of going to grad school after over a decade as a teacher, is that I have a strong commitment to asking my questions, stemming from the value that I know they have both for myself and the class.
Perhaps as a consequence, I found that in all four of my classes last semester, I asked more questions than anyone else in the room.
Be that as it may, I frequently didn’t ask my questions.
As it turns out (and now, okay, maybe this is Captain Obvious talking, but a propos of all of the above, somehow I’ve been overlooking it for a decade), not wanting to hold up class is its own reason not to ask questions! Maybe it’s a basic piece of our social programming. If things are going one way in a room of 20 or 30 people, it feels sort of painful to contemplate forcing them in another direction on your account. Especially if you’ve already done it once or twice, but even if not. And more so the further your question seems to be from what the people around you (esp. the teacher) look like they want to talk about. All this is intensified if you’re not sure your question is going to come out perfectly articulate – not (necessarily or only) because of how this will make you look, but because you know that your interruption is going to take up more mental and social space if it has to start with a whole period of everybody just getting clear on what you’re even asking.
There is an added layer that it is often perceptible that the teacher desires for everyone to understand and appreciate what was just said as clearly as she or he understands and appreciates it. Last night I was in a lecture in which I was hyperaware of not always asking my questions, and part of the dynamic in that case was actually the professor’s enthusiasm about what he was saying! I did ask a number of questions, but one reason I didn’t ask more is that I sort of felt like I was crashing his party! My warm feelings toward this professor actually heightened this effect: messing up someone else’s flow is worse when it’s someone you like.
As I mentioned above, students have been trying to tell me this for years. I never got it, because on some level I always believed that the real problem was that they were afraid to look dumb. I remember a conversation with a particular student who was my advisee as well as my math student. When I pressed her on asking more questions in class, she said something to the effect of, “you know, you’re doing your thing up there, and I don’t want to get in the way.” I literally remember the voice in my head reinterpreting this as a lack of belief in herself. Now I think that that was part of it as well; but my response was all aimed at that, and so didn’t address the whole issue.
Now my process of figuring out how to operationalize this new insight in terms of teaching practice has only just begun, and one reason I am writing about this here is to invite you into this process. I am certainly NOT telling you to withhold your enthusiasm on the grounds that it might make kids not want to interrupt you with questions. Furthermore, evidently when I describe experiences from my graduate classes, I am describing a situation in which the measures you and I have been taking for years to encourage question-asking are mostly absent. I doubt most of my professors have even heard of wait time. Nonetheless, I am sure that this new point of view is fruitful in terms of actual practice. Below are my preliminary thoughts. Please comment.
If I want to really encourage question asking, what I have been doing (aimed at building a culture of question-asking) is necessary, but insufficient. It is also necessary to think about lesson structure with an eye to: how do I design the flow of this lesson so that (at least during significant parts where questions are likely to arise in students’ minds) asking their questions does not feel like an interruption? One model, which is valuable in other ways as well, is to have students’ questions be the desired product of a certain segment of class. For example, when the lesson arrives at a key idea, definition, or conclusion, ask students to turn to their neighbors and discuss the key idea and try to produce a question about it. Then have the pairs or groups report their questions. This way, the questions cannot be interruptions because they are explicitly the very thing that is supposed to be going on right then.
I like this idea but it has limited scope because it requires the point in the lesson at which the questions arise to be planned, and of course this can never contain all the questions I would want to have asked. Another thing to think about is the matter of momentum. I think my discussion of enthusiasm above really revolves around momentum. Enthusiasm generates momentum, but momentum is actually the thing that it hurts to get in the way of. Therefore I submit a second idea: the question of managing my/your own and the class’s momentum. Having forward momentum is obviously a big part of class being engaging, but perhaps it also suppresses spontaneous questions? Or under certain conditions it does?
(In a way this reminds me of the tension – one I am much more confident is an essential one of our profession – between storytelling and avoidance of theft – I discussed a particular case of this tension in the fourth paragraph here. Momentum is aligned with storytelling: a good story generates momentum. Avoiding theft is aligned with inviting questions.)
A last thought is that in a class of 20 or 30, having the class engage every question that pops into any student’s head at any time is obviously not a desirable situation. You might think I thought it was desirable based on the above. But the question is how to empower students to ask questions when we want them. I know that I for one have often known I wanted some questions so I could be responsive to them, and they weren’t forthcoming. The question is about how to change this. Part of the answer is about the culture, valuing the questions, encouraging the risks, and making everyone feel safe; but it’s the other part – how to structurally support the questions – that’s the new inquiry for me. As I said above, please comment.
Having learning as a full-time job is really, really delicious. But tonight when I stopped mathing and engaged the edublogosphere it felt like a relief to read about classrooms, populated by humans. (To my fellow humans: I love the differentiable manifolds but I love you more.) Thanks Jesse Johnson, Dan Goldner, and Kate No Wackness for your continuing dedication to learning (your kids’, yours and ours).
This is my favorite thing ever.
The particular thought that was driving me crazy at least from 7pm to 10pm, not that you care, was: if is any surjective continuous function between topological spaces that maps open sets to open sets, then I can prove that the inverse image of a compact set is compact. I studied a converse if and happen to be smooth manifolds and happens to be an injective immersion. But these are very very strong assumptions. How much can they be weakened?
Kate, this is A’s name for you. (An homage to your no bullsh*t approach.)
We interrupt our regularly scheduled long patch of radio silence to share with you an arresting, mathematically rich visual:
Thanks to Josh Kershenbaum for the tip.
To clarify something: this image was originally created as an answer to a question, but I didn’t create a WCYDWT tag for that angle on it. To me, the image lands as (a) very beautiful; relatedly, (b) haunting, hard to get out of my head; therefore, (c) incipiently highly narrative – it is asking us to surround it with story, whether the original story conceived by the maker of the image or some other; and, lastly, (d) unavoidably mathematical. I don’t have a clear sense of the next move, but I do think this image opens a rich vein of something for a math teacher to use. The question “What can you do with this?” isn’t rhetorical: what’s your next move?
(See exchange with Dan Meyer below.)
I just recently learned of an organization in NYC called the Grassroots Education Movement, which last Thursday premiered a documentary film with the awesome title The Inconvenient Truth Behind Waiting for Superman. They will apparently send you a copy for free; I just ordered mine.
Meanwhile, the city of New York continues to besiege its own public schools with budget cuts, looming layoffs, and a multi-year hiring freeze. (Having spent the year training 12 new teachers, let me not even get started on the hiring freeze.) Another thing that happened on Thursday was that East Side Community High School, a wonderful school on the Lower East Side where I used to teach and where the math teaching is strong enough that we placed four student teachers there this quarter, had its first fund-raiser. Like, big event, speakers, performances by students, pay to get in, as though it were a non-profit, carrying out its own civic mission and in need of private funding to do it, rather than a public school, charged with a civic mission by the state, which no longer sees fit to pay for it.
I missed both the documentary premiere and ESCHS’s fund-raiser because I was teaching the final class of a 3-session minicourse at Math for America on the fundamental theorem of arithmetic. Let me do a little reflecting on the execution:
At the end of the 2nd session, I gave participants about a half-hour to try to figure out something quite difficult. I attempted to scaffold this with some unobtrusive PCMI-style tricks in a previous problem set: sequences of problems with the same answer for a mathematically significant reason. It turned out not to be enough. There was high engagement the whole time, but no one seemed to be headed in my intended direction after that half-hour. On the other hand, that half-hour had made the group into a legit mathematical research community. What was afoot was a live process of trying things out, questioning, pressing on others’ logic, and generally behaving like research mathematicians. I was left with a dilemma. I had one session remaining. I wanted to protect that process, meaning I did not want to steal from them any of the deliciousness (or pain – also delicious) of the process they were in the middle of by offering them too much direction. But at the same time I felt I needed to guarantee that we would reach resolution. (Storytelling purposes.)
The solution I went with: I had them pick up in the final session where they left off, but I brought in a sequence of hints on little cut-up slips of paper. I tried to call them “idea-starters” as opposed to “hints” to emphasize that the game was you thinking on your own, and this is just to get you moving if you’re stuck, rather than I have a particular idea and I want you to figure out what it is, but I don’t think I was consistent with this, and I think they pretty much all called them “hints,” and I don’t think it really mattered. They were in an order from least-obtrusive to most-directive. None of them were very directive. Most importantly, I told the participants that if they wanted to get one, they needed to decide this as a table. (There were 6 tables with 3-4 folks each.)
How this went: a) it preserved the sense of mathematical community. I do not think there was much of a cost to participant ownership of what they found out. b) People were actually pretty hesitant to use the “idea-starters.” Most of them went untouched. This would probably be different with a different audience. (High schoolers instead of teachers?) c) The “idea-starters” worked great, but very slowly. I planned to spend 45 min letting them work in this arrangement, but after 45 min, most of the groups were still deep in the middle of something. After over an hour, I asked two groups to present what they had, however incomplete, for the sake of a change of pace and the opportunity for cross-pollination of ideas between the tables. I had actually meant to do a lot more of this but had forgot to mention it at the beginning. I let everybody work for another 10-15 min while these groups laid out their presentations. By the time they presented, I realized that there wasn’t enough time left for everyone to really get back to work afterward, but in any case their ideas had gotten more fully developed in that 10 min. so they actually had pretty much figured out everything I had wanted them to. I presented the final link in the logical chain, just to fill in the picture, in the last 5 minutes. It was pretty satisfying to me to watch the presentations, except that it happened so late in the session. This for two reasons. One was that I would have ideally liked to have time to encourage the participants to interrogate the presenters more, but there wasn’t time for that. The other was that I had intended to spend the last half-hour with the participants consolidating their understanding of the argument by applying it to a new situation in which they didn’t know the outcome and it would tell them; but we didn’t have time for that either. I really feel a loss about that.
If I were to repeat it I think I would interrupt much earlier to have people present partial work. The cross-pollination of ideas might or might not accelerate the figuring-out process. Either way I think the change of pace would have been good for concentration. Also, I could have put some of the questions I used as “idea-starters” into the Session 2 problem sets, trying to move some of the combustion I got in session 3 into session 2. But these would both be experiments as well. I hope I get a chance to try them.
This is a shout out.
Dan Goldner came on my screen last summer during Riley’s soft skills conference. My impression was that he was a bright-eyed bushy-tailed newbie (just off the student teaching year) with a surprising amount of classroom insight considering this.
At the end of the summer I met him IRL in a miscellaneously awesome context and have been following his blog Work in Pencil since then. In the fall of his first full-time year in the classroom, he didn’t produce a lot of content. (No surprise.) 2 posts in the early fall, 2 right before Christmas, then nothing for months. (I guess I’m not one to talk.)
But now he’s back, and I feel that my initial impression last summer, while quite positive, was actually an underestimate. I didn’t anticipate all this maturity. I want to let you read for yourself, so I’m going to minimize the endorsement verbiage. I will say this: disconcerting forthrightness and vulnerability combined with some real writing craft. Keep your eye on this one.
One of the student teachers I supervise is planning a lesson introducing the sine and cosine angle sum formulas. I wanted to give him some advice on how to make the lesson better – in particular, along the axes of motivation and justification – and realized that, never having taught precalculus, I barely had any! Especially re: justification. I basically understand these formulas as corollaries of the geometry of multiplication of complex numbers. I have seen elementary proofs, but I remember them as feeling complicated and not that illuminating.
So: how do you teach the trig angle sum formulas? And in particular:
* How do you make them seem needed? (I offered my young acolyte the idea of asking the kids to find sin 30, sin 45, sin 60, sin 75 and sin 90 – with the intention of having them be slightly bothered by the fact that they can do all but sin 75.)
* Do you state the formulas or do you set something up to have the kids conjecture them? If the latter, how do you do it? How does it fly?
* How do you justify them? Do you do a rigorous derivation? Do you do something to make them seem intuitively reasonable? What do you do and how does it fly?
* Do you do them before or after complex numbers, and do you connect the two? If so, how do you do it and how does it fly?
Any thoughts would be much appreciated.
Thanks to John Abreu, who sent me the following in an email -
Please find attached a Word document with the proofs of the trig angle sum formulas. After opening the document you’ll see a sequence of 14 figures, the conclusions are obtained comparing the two of them in yellow. Also, I left the document in “crude” format so it’ll be easier for you to decide the format before posting.
I must say that the proofs/method is not mine, but I can’t remember where I learned them.
with an attachment containing the following figures (click to enlarge / for slideshow) -
As far as I can tell, the proof is valid for any pair of angles with an acute sum.
Let be two complex numbers on the unit circle, at angles from the positive real axis. Then and , so by sheer algebra, . On the other hand, the awesome thing about multiplication of complex numbers is that the angles add – the product will be at an angle of from the positive real axis; thus it is equal to . This is QED for both formulas if you believe me about the awesome thing. Of course it usually gets proven the other way – first the trig formulas, then use this to prove angles add when you multiply. But I think of the fact about multiplication of complex numbers as more essential and fundamental, and the sum formulas as byproducts.