## Some Miscellaneous Awesomeness Wednesday, Jul 18 2012

Just some awesome stuff I feel like pointing out:

Vi Hart does it again. That young woman has created a new art form.

Terry Tao’s airport puzzle. If you have to get from one end of the airport to the other to catch a plane, but you really need to stop for a minute to tie your shoe, is it best to do it while you’re on the moving walkway or not? (I learned this problem from Tim Gowers’ blog.)

Paul Salomon quotes Vi Hart quoting Edmund Snow Carpenter, and the quote is absolutely worth me quoting yet again:

The trouble with knowing what to say and saying it clearly and fully, is that clear speaking is generally obsolete thinking. Clear statement is like an art object: it is the afterlife of the process which called it into being.

Dan Goldner is doing my job for me. The original purpose of this blog was to read writing about math education, and to summarize and discuss it. I don’t do this very much any more (although expect summaries of a couple articles from the current JRME in the next few weeks months), but I do have a long list of things I wanted to read and discuss here but figured I’d probably never get to. On this list was the 1938 NCTM Yearbook, The Nature of Proof, by Harold Fawcett. But I’m taking it off; Dan’s got it covered.

## 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.

## Another One to Keep Your Eye On: Anna Weltman Saturday, Jun 23 2012

Here’s another blog to keep an eye on:

Recipes for Pi, by Anna Weltman.

I know Anna IRL. In fact, both of us have seen the other one teach. Thus prior to discovering her blog I already knew her as mathematically thought-provoking, endlessly creative, and deeply tuned in to student experience, not to mention a total sweetheart.

So I was excited to learn that she had started blogging in February, and her writing hasn’t disappointed. It’s sporadic, but who am I to complain about that, and more importantly it’s characterized by that same deep thinking about math and student experience that marks her teaching. Check it out.

Aside: Anna teaches at St. Ann’s School, along with Justin Lanier, Paul Salomon, and Paul Lockhart.

## Take the Effing Tests Guys Friday, Jun 1 2012

In December, Kate had a great idea, which I seconded.

We just got backed by Diane Ravitch.

I’d say it’s time to take this one to the streets.

## What She Said Monday, May 21 2012

Three weeks ago Sue VanHattum and Kate Nowak recommended Bob and Ellen Kaplan’s Math Circle Training Institute. If you are looking for a PD opportunity this summer and you are interested in cultivating students thinking for themselves, I strongly second their recommendation.

This is a weeklong training on the campus of Notre Dame in South Bend, Indiana where you learn how to run a math circle in the spirit of the Kaplans. What that means is that you ask thought-provoking questions and you facilitate students discussing them. Heaven, right? The setup is that in the morning, the Kaplans run a math circle on you, and in the afternoon they bus in local kiddies for you to try out your thought-provoking questions on, and watch others do it, and give and receive feedback. At lunch and at night you hang out with like minded educators talking about math and education. The \$850 includes room and board for the whole week.

I did this training in the summer of 2009 and it was a key step on my path to being the educator I am now. In 2007-8 I had come to the realization that my most central, pressing goal as an educator was to empower students to find their own mathematical curiosity, and I started stretching my pedagogical boundaries to find out what it would look and feel like to teach with this as the only goal. But I felt like I was reinventing the wheel. Reading the Kaplans’ book Out of the Labyrinth, I felt like I had found my comrades. Going to the Summer Institute, I felt like I had met them.

For example, Sue and Alex, and my fairy blogfamily Kate and Jesse Johnson. See what I mean?

Tangential to the math PD but also a wonderful benefit was the opportunity to spend a week on the Notre Dame campus. As a Jew I did not go into the experience expecting to be so moved by the shrines and sanctuaries of this Catholic institution, but I was. After my first experience with a labyrinth (the meditative kind), Alex McFerron said to me, “the Catholics really ace those sacred spaces.” True that.

## 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.

## Justin Lanier Thursday, Apr 26 2012

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.

## Best Calculator *Ever* Wednesday, Apr 18 2012

Somebody please tell me this is for real:

The QAMA calculator.

## My Former Students Are Grown-*ss Folks Thursday, Mar 29 2012

When I started out, veteran teachers at my school said to me, “you won’t really understand what you’re contributing until your students grow up and come back as adults.” I didn’t really understand what they were saying because it sounded unfathomably distant in the future.

But I am beginning to find out.

It’s just starting to sink in that the kids I taught in 2001-2, who were then high school freshmen, are now about 1 year older than I was when I taught them. The ones I taught in 2002-3 are 1 year younger.

BANANAS.

Example: I just had an email exchange with one of my 2002-3 students, who is now involved in math education (!) working for the Young People’s Project. This is the second former student I know of to get involved in math education. *Proud.* I would go so far as to say, *kvelling.*

## Elementary Mathematics from an Advanced Standpoint Monday, Mar 12 2012

Another of the many reasons I’m in grad school. I benefit as a teacher from understanding the content I teach in way more depth than I teach it. (I think everybody does, but it’s easiest to talk about myself.)

This does a number of things for me. The simplest is that it makes the content more exciting to me. Something that previously seemed routine can become pregnant with significance if I know where it’s going, and there’s a corresponding twinkle that shows up in my eye the whole time my students are dealing with it. A second benefit is that it gives me both tools and inspiration to find more different ways of explaining things. A third is that it helps me see (and therefore develop lessons aimed at) connections between different ideas.

So, this post is a catalogue of some insights that I’ve had about K-12 math that I’ve been led to by PhD study. The title of the post is a reference to Felix Klein’s classic books of the same name. The catalogue is mostly for my own benefit, and I don’t have all that much time, so I’m going to try to suppress the impulse to fully explain some of the more esoteric vocabulary, but I never want to write something here that requires expert knowledge to avoid being useless, so I’ll try to be both clear and pithy. (Wish me luck.)

Elementary level: Multiplication is function composition.

I’m developing the opinion that it’s important for especially middle and high school teachers to have this language. The upshot is that in addition to the usual models of multiplication as (a) repeated addition and (b) arrays and area of rectangles (and, if you’re lucky, (c) double number lines), multiplication is also the net effect of doing two things in a row, such as stretching (and possibly reversing) a number line.

The big thing I want to say here is that understanding this is key to understanding multiplication of signed numbers. I would go so far as to wager that anybody who feels they know intuitively why $-\cdot -=+$ understands it on some level, consciously or not.

When somebody asks me why a negative times a negative is a positive, I have often had the inclination to answer with, “well, what’s the opposite of the opposite of something?” (I have seen many teachers use metaphors with the same upshot.) The problem is that if you understand multiplication only as repeated addition and as the the area of rectangles, I’ve changed the subject with this answer. It is a complete nonsequitur. It’s probably clear why it has to do with negatives but why does it have to do with multiplication?

On the other hand, if on any level you realize that one meaning of $2\times 3$ is “double then triple”, then it’s natural for $(-2)\times(-3)$ to mean “double and oppositize, then triple and oppositize.” But for this you had to be able to see multiplication as “do something then do something else.”

Algebra I and Algebra II: Substitution is calculation inside a coordinate ring.

I just realized this today, and that’s what inspired this blog post. So far, I’m not sure the benefit of this one to my teaching beyond the twinkle it will bring to my eye, though perhaps that will become clear later. It’s certainly helping me understand something about algebraic geometry. The basic idea is this: say you’re finding the intersections of some graphs like $y=3x+5$ and $2x+y=30$. You’re like, “alright, substitute using the fact that $y=3x+5$. $2x+(3x+5)=30$, so $5x+5=30$…” and you solve that to find $x=5$, for an intersection point of $(5,20)$. A way to look at what you’re doing when you make the substitution $y=3x+5$ is that you’re working in a special algebraic system determined by the line $y=3x+5$, in particular the (tautological) fact that on this line, $y$ is exactly three $x$ plus five. In this system, polynomials in $x$ or $y$ alone work the usual way, but polynomials in $x$ and $y$ both can often be simplified using the relation $y=3x+5$ connecting $x$ and $y$. This algebraic system is called “the coordinate ring of the line $y=3x+5$.”

I can’t tell if it will even seem that I’ve said anything at all here. The point, for me, is just a sublte shift in perspective. I imagine myself sitting on the line $y=3x+5$; then this line determines an algebraic system (the coordinate ring) which, as long as I’m on the line, is the right system; and when I substitute $3x+5$ for $y$, what I’m doing is using the rules of that system.

Calculus: The chain rule is the functoriality of the derivative.

“Functoriality” is a word from category theory which I will avoid defining. The point is really about the chain rule. The main ways the derivative is presented in a first-year calculus class are as speed, or rate of change, on the one hand (like, you’re always thinking of the independent variable as time, whatever it really is), and the slope of the tangent line of a graph, on the other. There is a third way to look at it, which I learned from differential geometry. If you look at a function as a mapping from the real line to itself, then the derivative describes the factor by which it stretches small intervals. For example, $f(x)=x^2$ has a derivative of $6$ at $x=3$. What this is saying is that very small intervals around $x=3$ get mapped to intervals that are about 6 times as long. (To illustrate: the interval $[3,3.01]$ gets mapped to $[9,9.0601]$, about 6 times as long.)

Seen in this way, the strange formula $[f(g(x))]'=f'(g(x))\cdot g'(x)$ for the chain rule becomes the only sensible way it could be. The function $f(g(x))$ is the net effect of doing $g$ to $x$ and then doing $f$ to the answer $g(x)$. If I want to know how much this function stretches intervals, well, when $g$ is applied to $x$ they are stretched by a factor of $g'(x)$. Then when $f$ is applied to $g(x)$ they are stretched by a factor of $f'(g(x))$. (Note it is clear why you evaluate $f'$ at $g(x)$: that is the number to which $f$ got applied.) So you stretched first by a factor of $g'(x)$ and then by a factor of $f'(g(x))$; net effect, $f'(g(x))\cdot g'(x)$, just like the formula says.

(As an aside, for the sake of being thematic, note the role here of the fact that the multiplication comes from the composition of the two stretches – multiplication is function composition. When I say “the derivative is functorial” what I really mean is that it turns composition of functions into composition of stretches.)

Calculus: The intermediate value theorem is $f(\text{connected})=\text{connected}$. The extreme value theorem is $f(\text{compact})=\text{compact}$.

This is a good example of what I was talking about at the beginning about the twinkle in my eye, and connections between ideas. When I used to teach AP calculus, the extreme value theorem and the intermediate value theorem were things I had trouble connecting to the rest of the curriculum. They were these miscellaneous, intuitively obvious factoids about continuous functions that were stuck into the course in awkward places. They both had the same clunky hypothesis, “if $f$ is a function that is continuous on a closed interval $[a,b]$…” I didn’t do much with them, because I didn’t care about them.

I started to see a bigger picture about three years ago, in a course for calculus teachers taught by the irrepressible Larry Zimmerman. He referred to that clunky hypothesis as something to the effect of “a lilting refrain calling like a siren song.” I was also left with the image of a golden thread weaving through the fabric of calculus but I’m not sure if he said that. The point is, he made a big deal about that hypothesis, making me notice how thematic it is.

Last year when I taught a course on algebra and analysis, having benefited from this education, I made these theorems important goals of the course. But something further clicked into place this fall, when I started to need to draw on point-set topology knowledge as I studied differential geometry. Two fundamental concepts in topology are compactness and connectedness. They have technical definitions for which you can follow the links. Intuitively, connectedness is what it sounds like (all one piece), and compactness means (very loosely) that a set “ends, and reaches everywhere it heads toward.” (A closed interval is compact. The whole real line is not compact because it doesn’t end. An open interval is not compact because it wants to include its endpoints but it doesn’t. A professor of mine described compactness as, “everything that should happen [in the set] does happen.”)

Two basic theorems of point-set topology are that under a continuous mapping, the image of any connected set is connected and the image of any compact set is compact. These theorems are very general: they are true in the setting of any map between any two topological spaces. (They could be multidimensional, curved or twisted, or even more exotic…) What I realized is that the intermediate value theorem is just the theorem about connectedness specialized to the real line, and the extreme value theorem is just the theorem about compactness. What is a compact, connected subset of $\mathbb{R}$? It is precisely a closed interval. Under a continuous function, the image must therefore be compact and connected. Therefore, it must attain a maximum and minimum, because if not, the image either “doesn’t end” or “doesn’t reach its ends,” either of which would make it noncompact. And, for any two values hit by the image, it must hit every value between them; any missing value would disconnect it. So, “if $f$ is a function that is continuous on a closed interval $[a,b]$…”