## Sue’s Book Is Ready for Press and Needs Crowdfunding! Friday, Jun 20 2014

Hey y’all, I am incredibly excited about Sue’s book, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. If you have been around the math education blogosphere for more than a short time, you probably are too.

It needs crowdfunding to cover publication costs. I am about to help out and I invite you to do so too!

## Purging Thursday, Jul 5 2012

This is an impulsive and probably self-indulgent post.

When I moved to New York almost 6 years ago, I stowed two crates of hanging files in my grandmother’s closet. They are artifacts of my 2000-2005 teaching career in Boston. One crate, curricular materials; the other, student work. They had made it past one round of purging – this was the stuff I chose to bring with me to New York.

But they’ve been gathering dust since 2006 and I figured I owed it to my grandmother to get them out of her hair, so I picked them up on Tuesday. They’re sitting on my living room floor. I have absolutely no sensible place in my apartment for them. I am next to them, on the couch, a bag of paper recycling at my feet.

I didn’t budget time for these guys, and the time efficient move is to not even think about it; just dump it all.

I can’t bring myself to do this. That said, knowing how I get, if I start going through it paper by paper then (a) I will be here till next week and (b) at least half of it I will not be able to throw out.

Maybe I can make this blog post some kind of middle ground.

* Here’s the Jeopardy game I played with my Algebra I and Calculus classes they day before winter break! Optimization for $300: This is the maximum amount of money you can make selling cookies if you know that you could sell 100 cookies for$1 each, and that every time you raise the price $0.25, you lose 10 customers. Final Jeopardy (Algebra): $x$, given that $a=4, b=2, c=-1, d=37$, and $ax+b=cx+d$. Mr. Blum-Smith trivia for$200: Mr. Blum-Smith’s grandmother was kissed by this former US president. (Same grandmother whose apartment has been housing all this sh*t! Correct response: who is Bill Clinton?)

* Here are my various attempts at teaching about proof in Algebra I! My first year, I tried to teach a “proof unit.” It culminated with a “proof project,” where I had students attempt to prove one of six eclectic elementary theorems (e.g., sum of first $k$ odd numbers is $k^2$; any composite has a factor $>1$ but $\leq$ its square root; …). I remember being essentially unsatisfied. In the notes I made to myself after implementation (ed note: HOW CAN I THROW THESE OUT! F*CK!) I was starting to realize the whole thing was ill-conceived. I was smashing together the problem of actually figuring out what’s going on (interesting, unexpected, no guaranteed outcome) with the formal process of making it into an argument. I was setting the kids up. In my fourth year, I revisited the idea except with more coherence because the whole thing was based on creating a “number trick” (“think of a number; add 6; multiply by 2; … ; you got 42!”) and proving it worked. Still, the proof aspect of the unit was stilted and poorly motivated because the kids couldn’t see the need for the amount of formality I was insisting on.

* Here is a unit I wrote my student teaching year, about tessellations and symmetry, based on Escher. Here are pages of transparencies with Escher prints and other tessellations. Here are the 5 envelopes of tessellating polygons (triangles, rhombi, a nonconvex quadrilateral, some special pentagons…) I designed on the computer and lovingly cut out of paper. I never taught this unit again.

* Order of operations. I used to use this activity I stole from my own 7th and 8th grade math teacher, Steve Barkin, an institution of the Cambridge public schools. Take the year (I used to use the kids’ birth year, or just make it 1994 if I wanted it to be easier), and using the digits in that order, put any math symbols you want between them to get as many of the numbers from 1 to 100 as you can.

* Ah! And an inheritance from Steve I never actually made use of: a kind of integer number sense activity where you label the vertices of a graph with integers so that the numbers on adjacent vertices differ by 10, or else one of them is double the other. Like this! Fill in the blanks: $12\leftrightarrow ? \leftrightarrow ? \leftrightarrow 13$. Solution: $12\leftrightarrow 6\leftrightarrow 3\leftrightarrow 13$.

* CAN THEORY. This was the name of my linear-equations-in-a-single-variable unit, the core topic of my Algebra I class. I took the name and the idea from Maurice Page, then the math coordinator of the Cambridge Public Schools. The unit became what it was in my classroom in collaboration with my awesome colleagues Jess Flick (then Jess Jacob) and Mike Jenkins. The whole unit was based on physically modeling the equations with plastic cups and poker chips on a table; I put a piece of tape down the middle of the table and the rules were, all cups have to hold the same number of chips and both sides of the table have to have the same number of chips total. You figure out how many chips go in the cup. I beat that model to death every year. I tweaked the model in various ways to accommodate negative and fractional coefficients and solutions. That was the one topic I would have counted on nearly all my students still having mastery of the following year.

* Qualitative graphs! One of the years of my collaboration with Jess and Mike, we implemented an idea Mike brought to the table of a unit in Algebra I that was about interpreting qualitative features of cartesian graphs. The culminating project was, you picked a container (we had all kinds of shapes – beakers, vases, wine glasses, etc.), you filled it steadily with water and measured its height against the amount of water it contained, and you drew a graph of that. Before you did the experiment you predicted what the graph would look like. Afterward, you wrote an explanation of the features of the graph (changes in slope; concavity; inflection points) and discussed how they related to the shape of the container. My experience of the unit was that it was very difficult for kids, but it definitely felt like some proto-calculus skills.

That was the easy stuff. (I know; I’m being dramatic.) STUDENT WORK:

No, I can’t even open this up. GRRR. To every student I taught in 2000-2005: I am about to dump into a bag of paper recycling a whole lot of both your and my blood, sweat and tears. RRRRR okay. I have to immortalize a few memories. This will be spotty and haphazard, please forgive me. I am leaving most of you out in the below, but to all of you let me say that I hope you learned half as much from me as I did from you.

W: Best handwriting ever. Every homework assignment literally looked like the inscription on the One Ring. May you bring that level of love to everything you do.

D and M: The two black women in a calculus class I had allowed to be dominated by the personalities of cocky (mostly white) boys, you had the courage, and the respect for me and my potential for growth, to tell me what this felt like. I am grateful you did and sorry you had to. You are both rock stars and I regret that my class wasn’t a better environment for expressing that.

N and M: You stand out in my mind in your willingness to put in time and effort to understanding what you didn’t before. You put in after-school time to the degree it could have been a part-time job. That kind of commitment got you past hurdles higher than many adults I know have ever had to face. In my life I have come to understand that anybody can learn anything, and you guys helped teach me that.

W: As a math student you were an amazing combination of depth of thought and engagement, on the one hand, and desperate difficulty mastering computational techniques, expressing yourself in writing, or doing anything at all in a subinfinite amount of time, on the other. You asked some of the most thoughtful and interesting questions in class that I have ever heard. You practically never finished a test, even if you came after school for 3 hours to work on it. You were uniquely gentle and generous with myself and your classmates at all times. Rest in peace, W.

## Dispatches from the Learning Lab: Partial Understanding Monday, Apr 30 2012

So here’s another one that I suppose is kind of obvious, but nonetheless feels like big, important news to me:

It’s possible to only partly understand what somebody else is saying.

Let me be more specific. When you’re explaining something to me, it’s possible for me to get some idea from it in a clear way, to the point where my understanding registers on my face, but nonetheless the other 7 ideas you were describing I have no idea what you’re talking about.

<Example>

I am a 9th grader in your Algebra I class. You’re teaching me about linear functions. You are explaining to the class how to find the $y$-intercept of a linear function, in slope-intercept form, given that the slope is $4$ and the point $(6,11)$ lies on the line. You explain that the equation has the form $y=mx+b$ and that because we know the point $(6,11)$ is on the line, that this point satisfies the equation. Thus you write

$11=4\cdot 6+b$

on the board. At this point I recognize that we are trying to find $b$ and that we have an easy single-variable linear equation to solve. My face lights up and you take mental note of my engagement. Maybe you even ask for the $y$-intercept, and since I recognize that this must be $b$ I calculate $11-24 = -13$ and raise my hand.

Meanwhile, I have only the vaguest sense of the meaning of the phrase “$y$-intercept.” I have literally no understanding of why I should expect the equation to have the form $y=mx+b$. I have a nagging feeling of dissatisfaction ever since you substituted $(6,11)$ into the equation because I thought $x$ and $y$ were supposed to be the variables but now it looks like $b$ is the variable. Most importantly, I do not understand that the presence of the point on the line implies that its coordinates satisfy the equation of the line and conversely, because on a very basic level I don’t understand what the graph of the function is a picture of. This has been bothering me ever since we started the unit, when you had me plug in a bunch of $x$ values into some equations and obtain corresponding $y$ values, graph them, and then draw a solid line connecting the three or four points. Why am I drawing these lines? What are they pictures of?

Occasionally, I’ve asked a question aimed at getting clarity on some of these basic points. “How did you know to put the 6 and 11 into the equation?” But because I can’t be articulate about what I don’t understand, since I don’t understand it, and you can’t hear what I’m missing in my questions because the the theory is complete and whole in your mind, these attempts come to the same unsatisfying conclusion every time. You explain again; I frown; you explain a different way; I say, “I don’t understand.” You, I, and everyone else grow uncomfortable as the impasse continues. Eventually, you offer some thought that has something in it for me to latch onto, just as I latched onto solving for $b$ before. Just to dispel the tension and let you get on with your job, I say, “Ah! Yes, I understand.”

</Example>

This example is my attempt to translate a few experiences I’ve had this semester into the setting of high school. The behavior of the student in that last paragraph was typical of me in these situations, though it would be atypical from a high school student, drawing as it does on the resources of my adulthood and educator background to self-advocate, to tolerate awkwardness, even to be aware that my understanding was incomplete. Still, often enough I ended up copping out as the student does above, understanding one of the 8 things that were going on, and latching onto it just so I could allow myself, the teacher and the class to move on gracefully. Conversations with other students indicated that my sense of incomplete understanding was entirely typical, even if my self-advocacy was not.

The take-home lesson is two-fold. Point one is about the limitations of explaining as a method of teaching. Point two is about the limitations of trusting your students’ (verbal or implied) response to your (verbal or implied) question, Do you understand?

The basic answer (as you can tell from the example) is, No, I don’t.

Now I myself love explaining and have done a great deal of it as a teacher. I fancy myself an extremely clear and articulate explainer. But it couldn’t be more abundantly clear, from this side of the desk, how limited is the experience of being explained to. I mean, actually it’s a great, key, important way to learn, but only in small doses and when I’m ready for it, when the groundwork for what you have to say has been properly set.

I am somewhat chastened by this. I am thinking back self-consciously to times when I’ve explained my students’ ears off rather than, in the immortal words of Shawn Cornally, “lay off and let them fucking think for a second.” It’s like I was too taken with the clarity and beauty of the formulation I was offering, or in too much of a hurry to let them work through what they had to work through, or in all likelihood both, to see that more words weren’t going to do any good. Beyond this, I’m thinking back on the faith I’ve put in my ability to read students’ level of understanding from their faces. I maintain that I’m way better at this than my professors, but I don’t think I’ve had enough respect for how you can understand a small part of something and have that feel like a big enough deal to say, and mean, “Oh I get it.” Or to understand a tiny part of something and use that as cover for not understanding the rest.

## Dispatches from the Learning Lab: Why I Don’t Always Ask My Question Tuesday, Jan 24 2012

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.

What’s up?

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.

## 0.99999… Tuesday, Oct 5 2010

So I’m teaching this course this year. It’s for the math faculty of a high school. It’s called:

MA600 Algebra and Analysis with Connections to the K-12 Curriculum

I am unspeakably excited, and want to do the best job possible.

The class: 7 teachers, deeply committed to kids, serious, not real talkative, rightly protective about their time, which is in short supply, but eager to get sh*t done.

The content: Basically, all of mathematics, seen as a unified whole.

It’s met twice. The second class was last Thursday. I need to get my thoughts sorted out here. I’m expecting this to help me visualize the next moves more clearly, just by doing it, but I’d love your thoughts too.

I didn’t really know anything about the mathematical background of the group when I wrote the syllabus, so for the first class I gave them a getting-to-know-you problem set with a wide range of problems and just let them work the whole time. Magically the experience of watching folks work on the problems and then later looking at what they did on paper gave me just enough information to plan the direction of the class’ first unit. We’re beginning with analysis. My first goal: the $\epsilon$-$\delta$ definition of the limit. (I.e., the definition of the limit, for the snobby among you.) My second: the completeness axiom.

The plan: generate the need to define the limit by working with 2 everyday concepts that are actually limits. Namely, infinite decimals, and instantaneous speed. My hope is that by pressing on these concepts, we’ll see that in spite of our familiarity with them, we don’t actually understand them unless we have a precise way to talk about limits. Then, develop the definition out of the need to fully understand the familiar. Then, develop the completeness axiom out of the desire to make sure infinite decimals have a limit.

Here’s what we did:

I opened class with a problem set designed to get them thinking about the meaning of decimals in particular, and various other contexts for the idea of limits. I shamelessly bit the format from PCMI. The problems span a wide range of skills and I didn’t leave enough time to do them all, so people could attack problems appropriate to their skill level. This is now my favorite way to differentiate problem sets, a propos of a) using it in some NYMC workshops last year, and b) hearing about how wonderful it was for everyone at PCMI.

Then, since we are all just getting to know each other, I did a short presentation on the mindset I wanted us to be in:

(Scribd did not handle slide 6 very well, which is too bad because I was proud of that slide. This is my first PowerPoint presentation ever. Actually I did it in Keynote.)

Then, we got to business. I put this up:

I asked them to talk about it with their tables. (I had them in 2 pairs and a group of three, in three tables in a horseshoe shape in front of the board. I like this and think I’ll keep it. Easy transitions from pair/group to whole-class; tables feel separate enough so you don’t feel like your conversation with your partner is in front of everybody; but everyone’s close enough so we can all talk. On day 1 I put us all around one table, for a sense of collegiality and common purpose, but it was too close; you couldn’t discreetly check in with your neighbor, for example.)

There was a widespread sense of mathematical discomfort, and rightly so. Infinite decimals enter most people’s math educations with no attention to the fact that they actually violate everything you’ve learned about math up to that point. You don’t get the full story until analysis, but unless you really get intimate with and own that content, you probably don’t connect what you learn there to what your teacher introduced without comment somewhere between 3rd and 7th grade, as though it weren’t a mind-boggling idea. “When you expand 1/3 as a decimal, the 3’s just keep going.” Or, “3.14159… It never ends or repeats.” Um, excuse me? It NEVER ENDS?

So it’s no surprise everybody has an underdeveloped idea of infinite decimals, and therefore that objects like 0.99999… cause some dissonance. This is very productive dissonance. I’m hoping it carries us all the way to the completeness axiom; we’ll see.

One of the three tables produced the standard argument that if x = 0.9999…, then 10x = 9.9999…, so 9x = 10x – x = 9, so x = 1; but even this table found this conclusion unsatisfying. I asked them why. The table that had produced the argument said, “usually this method gives you a fraction.” I asked for an example. They produced one from the problem set:

x = 1.363636…
100x = 136.363636…
99x = 100x – x = 135
x = 135/99 = 15/11

I asked how many folks found this argument convincing. 7 out of 7. (Well, one raised hand was kind of hesitant.)

I asked the same question about the same argument with .9999…. 4 out of 7. Then I dropped this:

How many people found this one convincing? 0 out of 7.

Reasonable.

Right?

Then what’s the difference?

At first, they cast about a bit, but then one of them said, “1.363636… has a finite limit, but …9999.0 doesn’t.” Their ideas began to coalesce around this type of language. Another one said, “we can actually estimate 1.363636…, for example we know it’s between 1 and 2.”

From the point of view I am ultimately heading for, this is the rub. Infinite decimals suggest convergent series, and the standard way to give them meaning as real numbers is that they are equal to the limit of the convergent series they suggest. …9999.0 suggests a wildly divergent series, so it cannot become a real number in the same way. (To bring home that convergence is the heart of the matter: there is an alternative way to define distance between numbers, the 10-adic metric, according to which it is actually …9999.0 that has the convergent series, and in this alternative system the above proof is valid and it actually does equal -1.) What I’d like us to do is a) define limits precisely; b) use this to prove that when a series has a limit, you can do the above type of manipulations to find it; c) try to prove that the series suggested by an infinite decimal always has a limit; d) realize that we can’t prove this without articulating the completeness axiom; e) articulate the axiom; and f) prove from the axiom that any infinite decimal has a real number limit. (Somewhere along the line, produce an $\epsilon$-$\delta$ proof that 0.9999… = 1.) Now, how to orchestrate this…

For next time I told them to try to craft a definition of the meaning of an infinite decimal 0.abcd… I gave them a few minutes just before the end to discuss this with their groups. I’m expecting to learn a lot about their thinking from what they come up with, but I’m not counting on anyone to have a mathematically satisfying answer. I’ll be pleased if somebody does though.

As I think about next class, here’s what’s on my mind:

1) When we develop the $\epsilon$-$\delta$ limit, what I’m going for is for this definition to feel like a satisfying relief. I know how easy it is for this definition instead to feel like a horrible monstrosity designed to oppress analysis students. I think what I have to do is keep them thinking about the reasons why anything less than this definition is too vague, which means I need to keep coming up with objects and problems that throw monkey wrenches into whatever more naive definitions they go for. (Of course, if they come up with something equally precise as the $\epsilon$-$\delta$ limit but different, that would be amazing.) I feel like we’re off to a good start on this, but I want a fuller catalogue of head-scratchers (like …9999.0 = -1) to push the level of precision higher.

2) Relatedly, I sense a danger that the “real answers” will be unsatisfying because it’ll feel like “wait, I already said that.” For example, the participant who said that the difference between 1.3636… = 15/11 and …9999.0 = -1 is that “the first one has a finite limit”… I mean this is basically the answer. But it’s not based on a precise definition of limit yet, so it’s not what I want yet. I’m afraid of a “what was the big deal?” moment when we’ve got the real sh*t up there. I think the way to avoid this lies in that catalogue of head-scratchers I need to develop, so that nothing less than the real thing is satisfying. What do you think?

3) Where to go immediately next. Basically the question is: stick with decimals? Or change gears completely and press on the notion of instantaneous speed? Most (not all, I think) of the teachers have had a calculus course, but think at most 1 or 2 of them have internalized the philosophical lesson that instantaneous speed needs to be defined as a limit in order for us to even access it. I’m attracted to the idea of switching gears because I’m drawn to the connection between the disparate realms: two highly familiar, but totally different, objects – infinite decimals and speed in a moment – both getting pressed on to the point where you realize you never fully understood either one, and then you realize that the missing idea you need is the same thing in the two cases. (A precise way to talk about what number some varying quantity is “heading toward.”)

Actually as I write this out, it seems clear to me that switching gears is the way to go. I think it’ll give us a clearer understanding of what we’re missing with the decimals. Also, it’ll allow us to access all this rich historical stuff around the development of calculus. For example, maybe I’ll share with them some choice quotes from Bishop Berkeley’s The Analyst, to help articulate why the 18th century definition of the derivative was inadequate.

Anyway. Very excited about all this. Will definitely keep you posted.