# Some work I’m proud of

T was a third grader when all this went down.

At a previous session, I had asked T what she knew about multiplication, and she had told me, among other things, that $4\times 6$ is four sixes, and because that’s 24, she also knew that six fours would be 24. I asked why she said so and she didn’t know why. I asked her if she thought this would always be true for bigger numbers, or could it be possible that there were some big numbers like 30,001 and 5,775 for which 30,001 5775’s was different than 5,775 30,001’s. She wasn’t sure. I asked her if she thought it was a good question and she said she thought it was.

So this session I reminded her of this conversation. I forget the details of how we got going on it; I remember inviting her to wonder about the question and note that there is something surprising about the equality between four sixes and six fours. She could count up to 24 by 4’s and by 6’s and mostly you hit different numbers on the way up, so why do the answers match? And would it be true for any pair of numbers? But the place where I really remember the conversation is when we started to get into the nuts and bolts:

Me: Maybe to help study it we should try to visualize it. Can you draw a picture of four sixes?

T draws this –

Me: Cool! Okay I have a very interesting question for you. You know how many dots are here –

T: 24 –

Me: and you also told me that six 4’s is also 24, right?

T: yes.

Me: – so that means that there must be six 4’s in this picture! Can you find them?

T: I don’t understand.

Me [writing it down as well as saying it this time]: You drew a picture of four 6’s here, yes?

T: Yes.

Me: And that’s 24 dots, yes?

T: Yes.

Me: And you told me before that 24 is also six 4’s, yes?

T: Yes.

Me: So it must be that right here in this picture there are six 4’s!

[It clicks.] T: Yes!

Me: See if you can find them!

At this point, I go wash my hands. An essential tool that has developed in my tutoring practice is to give the student the social space to feel not-watched while they work on something requiring a little creativity or mental looseness, or just anything where the student needs to relax and sink into the problem or question. The feeling of being watched, even by a benevolent helper adult, is inhibitive for generating thoughts. Trips to wash hands or to the bathroom are a great excuse, and I can come back and watch for a minute before I make a decision about whether to alert the student to my return. I also often just look out the window and pretend to be lost in thought. Anyway, on this particular occasion, when I come back, T has drawn this:

T: I found them, but it’s not… It doesn’t…

I am interrupting because I have to make sure you notice how rad she’s being. The child has a sense of mathematical aesthetics! The partition into six 4’s is uglifying a pretty picture; breaking up the symmetry it had before. It’s a kind of a truth, but she isn’t satisfied with it. She senses that there is a more elegant and more revealing truth out there.

This sense of taste is the device that allows the lesson to move forward without me doing the work for her. Her displeasure with this picture is like a wall we can pivot off of to get somewhere awesome. Watch:

Me: I totally know what you mean. It’s there but it doesn’t feel quite natural. The picture doesn’t really want to show the six 4’s.

T: Yeah.

Me: You know what though. You had a lot of choice in how you drew the four 6’s at the beginning. You chose to do it this way, with the two rows of three plus two rows of three and like that. Maybe you could make some other choice of how to draw four 6’s that would also show the six 4’s more clearly? What do you think? You wanna try to find something like that?

T: Yes! [She is totally in.]

At this point I go to the bathroom. I hang out in the hall for a bit when I get back because she seems to still be drawing. Finally,

Me: Did you find out anything?

T: I drew it a lot of different ways, but none of them show me the six 4’s…

She’s got six or seven pictures. One of them is this –

Me: Hey wait I think I can see it in this one! (T: Really??) But I can’t tell because I think you might be missing one but I’m not sure because I can’t see if they are all the same.

T immediately starts redrawing the picture, putting one x in each column, carefully lined up horizontally, and then a second x in each column. As she starts to put a third x in the first column, like this,

she gasps. Then she slides her eyes sideways to me, and with a mischievous smile, adds this to her previous picture:

The pieces just fell into place from there. Again I don’t remember the details, but I do remember I asked her what would happen with much bigger numbers – might 30,001 5,775’s and 5,775 30,001’s come out different? And she was able to say no, and why not. Commutativity of multiplication QED, snitches!

# Hard Problems and Hints

I have a friend O with a very mathematically engaged son J, who semi-often corresponds with me about his and J’s mathematical experiences together. We had a recent exchange and what I was saying to him I found myself wanting to say to everybody. So, without further ado, here is his email and my reply (my take on Aunt Pythia) –

Dear Ben,

J’s class is learning about volume in math. They’ll be working with cubes, rectangular prisms and possibly cylinders, but that’s all. He asked his teacher if he could work on a “challenge” that has been on his mind, which is to find a formula for the volume of one of his favorite shapes, the dodecahedron. He build a few of these out of paper earlier in the year and really was/is fascinated with them. I think he began this quest to find the volume thinking that it would be pretty much impossible, but he has stuck with it for almost a week now. I am pleased to see that he’s not only sticking with it, but also that he has made a few pretty interesting observations along the way, including coming up with an approach to solving it that involves, as he put it, “breaking it up into equal pieces of some simpler shape and then putting them together.” After trying a few ways to break/slice up the dodecahedron and finding that none of them seemed to make matters simpler, he had an “ah ha” moment in the car and decided that the way to do would be to break it up into 12 “pentagonal pyramids” (that’s what he calls them) that fit together, meeting at the center of rotation of the whole shape. If we can find the volume of one of those things, we’re all set. A few days later, he told me that he realized that “not every pentagonal pyramid could combine to make a dodecahedron” so maybe there was something special about the ones that do, i.e., maybe there is a special relationship between the length of the side of the pentagon and the length of the edge of the pyramid that could be used to form a dodecahedron.

He is still sticking with it, and seems to be having a grand time, so I am definitely going to encourage him and puzzle through it with him if he wants.

But here’s my question for you…

I sneaked a peak on google to see what the formula actually is, and found (as you might know) that it’s pretty complicated. The formula for the volume of the pentagonal pyramid involves $\tan 54$ (or something horrible like that) and the formula for the volume of a dodecahedron involves $15 + 7\sqrt{5}$ or something evil like that. In short, I am doubtful that he will actually be able to solve this problem he’s puzzling through. What does a good teacher do in such a situation? You have a student who is really interested in this problem, but you know that it’s far more likely that he will hit a wall (or many walls) that he really doesn’t have the tools to work through. On the other hand, you really want him to find satisfaction in the process and not measure the joy or the value of the process by whether he ultimately solves it.

I certainly don’t care whether he solves it or not. But I want to help him get value out of hitting the wall. How do you strike a balance so that the challenge is the right level of frustrating? When is it good to “give a hint” (you’ve done that for me a few times in what felt like a good way… not too much, but just enough so that the task was possible).

In this case, he’s at least trying to answer a question that has an answer. I suppose you could find a student working on a problem that you know has NO known answer, or that has been proven to be unsolvable. Although there, at least, after the student throws up his hands after giving it a good go, you can comfort her by saying, “guess what… you’re in good company!” But here, I’d like to help give him some of the tools he might use to actually make some headway, without giving away the store.

I think he’s off to a really good start — learning a lot along the way – getting a lot of out the process, the approach. I can already tell that many of the “ah ha” moments have applicability in all sorts of problems, so that’s wonderful.

Best, O

Dear O,

Wow, okay first of all, I love that you asked me this and it makes me really appreciate your role in this journey J is on, in other words I wish every child had an adult present in their mathematical journey who recognizes the value in their self-driven exploration and is interested in being the guardian of the child’s understanding of that value.

Second: no matter what happens, you have access to the “guess what… you’re in good company” response, because the experience of hitting walls as you try to find your way through the maze of the truth is literally the experience of all research mathematicians, nearly all of the time. If by any chance J ends up being a research mathematician, he will spend literally 99% or more of his working life in this state.

In fact, I would want to tweak the message a bit; I find the “guess what… you’re in good company” a tad consolation-prize-y (as also expressed by the fact that you described it as a “comfort”). It implies that there was an underlying defeat whose pain this message is designed to ameliorate. I want to encourage you and J both to see this situation as one in which a defeat is not even possible, because the goal is to deepen understanding, and that is definitely happening, regardless of the outcome. The specific question (“what’s the volume of a dodecahedron?”) is a tool that’s being used to give the mind focus and drive in exploring the jungle of mathematical reality, but the real value is the journey, not the answer to the question. The question is just a tool to help the mind focus.

In fairness, questing for a goal such as finding the answer to a question and then not meeting the goal is always a little disappointing, and I’m not trying to act like that disappointment can be escaped through some sort of mental jiu-jitsu. What I am trying to say is that it is possible to experience this disappointment as superficial, because the goal-quest is an exciting and focusing activity that expresses your curiosity, but the goal is not the container of the quest’s value.

So, that’s what you tell the kid. Way before they hit any walls. More than that, that’s how you should see it, and encourage them to see it that way by modeling.

Third. A hard thing about being in J’s position in life (speaking from experience) is that the excitement generated in adults by his mathematical interests and corresponding “advancement” is exciting and heady, but can have the negative impact of encouraging him to see the value of what he’s doing in terms of it making him awesome rather than the exploration itself being the awesome thing, and this puts him in the position where it is possible for an unsuccessful mathematical expedition to be very ego-challenging. This is something that’s been behind a lot of the conversations we’ve had, but I want to highlight it here, to connect the dots in the concrete situation we’re discussing. To the extent that there are adults invested in J’s mathematical precociousness per se, and to the extent that J may experience an unsuccessful quest as a major defeat, these two things are connected.

Fourth, to respond to your request for concrete advice regarding when it is a good idea to give a hint. Well, there is an art to this, but here are some basic principles:

* Hints that are minimally obtrusive allow the learner to preserve their sense of ownership over the final result. The big dangers with a hint are (a) that you steal the opportunity to learn by removing a part of the task that would have been important to the learning experience, and (b) that you steal the experience of success because the learner doesn’t feel like they really did it. These dangers are related but distinct.

* How do you give a minimally obtrusive hint?

(a) Hints that direct the learner’s attention to a potentially fruitful avenue of thought are superior to hints that are designed to give the learner a new tool.

(b) Hints that are designed to facilitate movement in the direction of thought the learner already has going on are generally better than hints that attempt to steer the learner in a completely new direction.

* If the learner does need a new tool, this should be addressed explicitly. It’s kind of disingenuous to think of it as a “hint” – looking up “hint” in the dictionary just now, I’m seeing words like “indirect / suggestion / covert indication”. If the learner is missing a key tool, they need something direct. The best scenario is if they can actually ask for what they need:

Learner: If I only had a way to find the length of this side using this angle…
Teacher: oh yes, there’s a whole body of techniques for that, it’s called trigonometry.

This is rare but that’s okay because it’s not necessary. If the teacher sees that the learner is up against the lack of a certain tool, they can also elicit the need for it from the learner:

Teacher: It seems like you’re stuck because you know this angle but you don’t know this side.
Learner: Yeah.
Teacher: What if I told you there was a whole body of techniques for that?

Okay, those are my four cents. Keep me posted on this journey, it sounds like a really rich learning experience for J.

All the best, Ben

# Steven Strogatz on what math education often gets wrong

If you’re telling a student the answer to a question it would never occur to them to ask, I can’t see how that’s a productive use of anyone’s time.

(This is not verbatim, but it’s the idea.)

# Dispatches from the Learning Lab: Yup, Time Pressure Sucks

Continuing the series I began here and here, about snippets of new-feeling insight about the learning process coming from my new role on the student side of the desk…

This one is funny, because I knew it, I mean I knew it in my bones, from a decade working with students; but yet it’s totally different to learn it from the student side. I’m a little late to the blogosphere with this insight; I’ve been thinking about it since December, because it kind of freaked me out. Even though, like I keep saying, I already knew it.

Learning math under time pressure sucks. It sucks.

It sucks so much that I ACTUALLY STOPPED LIKING MATH for about 5 days in December.

I didn’t know this was possible, and I don’t think anyone who’s ever worked closely with me in a mathematical context (neither my students, colleagues, or teachers) will really believe it. But it’s true. It was utterly, completely unfun. There was too much of it and too little time. It was like stuffing a really delicious meal down your throat too quickly to chew, or running up the Grand Canyon so fast you puke. Beautiful ideas were everywhere around me and I was pushing them in, or pushing past them, so hard I couldn’t enjoy them; instead they turned my stomach, and I had the feeling that the ones I pushed past in a hurry were gone forever, and the ones I shoved in weren’t going to stay down.

I had some independent study projects to work on during winter break, and what was incredible was the way the day after my last final exam, math suddenly became delicious again. Engaging on my own time and on my own terms, that familiar sense of wonder was back instantly. All I had to do was not be required to understand any specific thing by any specific date, and I was a delighted, voracious learner again.

Now part of the significance of this story for me is just the personal challenge: most of the grad students I know are stressed out, and I entered grad school with the intention of not being like them in this respect. I was confident that, having handled adult responsibilities for a decade (including the motherf*cking classroom, thank you), I would be able to engage grad school without allowing it to stress me out too much. So the point of this part of the story is just, “okay Grad Program, I see you, I won’t take you for granted, you are capable of stressing me out if I let you.” And then regroup, figure out how to adjust my approach, and see how the new approach plays out in the spring semester.

But the part of the story I want to highlight is the opposite part, the policy implication. Look, I frickin love math. If you’ve ever read this blog before, you know this. I love it so much that most of my close friends sort of don’t feel that they understand me completely. So if piling on too much of it too quickly, with some big tests bearing down, gets me to dislike math, if only for 5 days, then the last decade of public education policy initiatives – i.e. more math, higher stakes – is nothing if not a recipe for EVERYONE TO HATE IT.

And, not learn it. Instead, disgorge it like a meal they didn’t know was delicious because it was shoved down their throat too fast.

In short. The idea of strict, ambitious, tested benchmarks in math to which all students are subject is crazy. It’s CRAZY. The more required math there is, and the stricter the timeline, the crazier. I mean, I already knew this ish was crazy, I’ve been saying this for years, but in light of my recent experience I’m beside myself. If you actually care about math, if you have ever had the profound pleasure of watching a child or an adult think for herself in a numerical, spatial or otherwise abstract or structural context, you know this but I have to say it: the test pressure is killing the thing you love. Its only function is to murder something beautiful.

If you teach, but especially if you are a school leader, and especially if you are involved in policy, I beg you: defend the space in which students can learn at their own pace. Fight for that space.

# Angle Sum Formulas: Request for Ideas

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.[1] 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.

Notes:

[1]Let $z_1,z_2$ be two complex numbers on the unit circle, at angles $\theta_1,\theta_2$ from the positive real axis. Then $z_1=\cos{\theta_1}+\imath\sin{\theta_1}$ and $z_2=\cos{\theta_2}+\imath\sin{\theta_2}$, so by sheer algebra, $z_1z_2=(\cos{\theta_1}\cos{\theta_2}-\sin{\theta_1}\sin{\theta_2})+\imath(\cos{\theta_1}\sin{\theta_2}+\sin{\theta_1}\cos{\theta_2})$. On the other hand, the awesome thing about multiplication of complex numbers is that the angles add – the product $z_1z_2$ will be at an angle of $\theta_1+\theta_2$ from the positive real axis; thus it is equal to $\cos{(\theta_1+\theta_2)} + \imath\sin{(\theta_1+\theta_2)}$. 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.

# Over the Course of an Instant…

As you may recall, I’m teaching analysis to this class of teachers, developing the $\epsilon$$\delta$ limit. Two weeks ago I bewildered everybody. Last week and this week, I set out to bewilder everyone even further.

Let me say what I’m going for here. The $\epsilon$$\delta$ limit is a notoriously difficult definition.1 How to scaffold my class to handle this difficulty? I am banking on the following strategy: make them need the definition. Make them unsatisfied with anything less. Continue poking holes in their current understanding, continue showing them inconsistencies between what they believe and the language they have to describe it, till they have no choice but to try to build something new. Then, let them try to build it. If they build the very thing I’m going for, rejoice. If they build something equally precise and powerful, rejoice. If they cannot build either (the most likely outcome, since the “right answer” took the world mathematical community 150 years to come up with), then it will still make powerful sense to them because it satisfactorily answers a question they were already engaged in trying to answer. That’s the plan anyway.

I will leave you with the two problem sets from the last class, and the readings and presentation from this one. I am very proud of the presentation. After that, I’ll write down one new thought for where to take this.

We engaged people’s attempts to define infinite decimals from the previous class, then abruptly shifted topics:

I let them work long enough so everyone got to do the first section of problems. My goals were:

1) Make participants recognize that they believe the speed of a moving object is something that exists in a particular moment of time.
2) Make them recognize that their naive definition of speed (distance / time) doesn’t actually handle this case.
3) Realize that we thus have a similar definitional problem as with repeating decimals.

We got this far. Then, with just 7 or so minutes left, I gave them another problem set:2

This problem set was designed to get somebody who has never studied calculus basically to take a simple derivative, to bring them into the conversation, and to refresh everyone else’s memory about the basic idea of derivatives. The last problem was on there just so that the calculus folks had a challenge available if they wanted it. Anyway, I had people finish the “Algebra Calisthenics” and “Speed” sections for homework.

This class, we began by engaging this homework, getting a feel for the standard calculus computation in which you identify the speed of an object in a moment as the value toward which average speeds seem to be headed as you look at smaller and smaller intervals. Then we began to press on what this really means.

I handed out a xerox of the scholium from the end of the first section of Book 1 of Newton’s Principia. (The last page of this pdf.) This is where Newton tries to explain what the hell he’s even talking about. I directed their attention to this telling sentence:

An in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities, not before they vanish, nor afterwards, but with which they vanish.

Then, I showed them the following presentation. Wanting to share this with you is the real reason for this blog post. I had a lot of fun making it.

What’sCalculusReallyDoing (as pdf)

What’sCalculusReallyDoing (as powerpoint)

Then I passed out a choice excerpt from the awesome criticism of early calculus by Bishop George Berkeley. (Specif, section XIV.)

I asked for the connection between the definitional problem we have here and the definitional problem we had 2 classes ago regarding infinite decimals. (“They both involve getting closer and closer to something but never getting there.”) Then I asked them to try to come up with definitions to address these problems.

This is such a non-sequitur but here’s my one additional thought. I’ve been thinking about how to push participants to recognize a definition as unsatisfying. Tonight, reading Judith Grabiner’s 1983 essay in the AMM about Cauchy and the origins of the $\epsilon$$\delta$ limit (here it is as a pdf), I had an idea that is totally new to me. Retrospectively I think it’s sort of obvious, but I totally never thought of it before:

To get people to recognize that a definition is mathematically inadequate, have them try to use the definition, for example to prove something! In my case, all of them think that 1/3 = 0.333… Great. So, if we have a candidate definition of the meaning of limits or convergence, can we use it to prove 1/3 = 0.333…? If not, maybe we need a better definition.

(I had this idea when I read Grabiner’s statement that thought Cauchy gave the definition of the limit purely verbally and a bit vaguely, he translated it into the more rigorous language of inequalities when he actually started using it to prove theorems.)

[1] This is for at least 2 distinct (though related reasons): first of all, it’s got three nested quantifiers. “For all $\epsilon>0$, there exists a $\delta>0$, such that for all $x$ satisfying …” That just makes it inherently confusing. Secondly, it does not in any way psychologically resemble the intuitive image it is intended to capture. This is the definition of the limit. When I think of limits I have these beautiful visual images of little points getting closer to something. When I try to identify a limit, I just imagine the thing that they’re getting closer to. That’s the whole story. When I try to get rigorous, I replace this beautiful and simple image with three nested quantifiers. Yuck.

[2] You will notice some interconnections in the sequence of problems. After a few good experiences with this last year and then hearing how much fun everyone had at PCMI, I am beginning to feel like these sequences of densely but subtly interconnected problems are really, really awesome. Constructing them is a deep art and I am a tiny apprentice. But you can get started humbly and still see payoff: it was certainly a cool moment today in class when we went over these problems and a number of folks who had done out Speed problems #1-3 “the long way” realized that they could have applied their answer to Algebra Calisthenics #2 to do these three problems in moments in their heads.