Good Brawls and Honoring Kids’ Dissatisfaction Friday, Mar 8 2013 

I was just reading some old correspondence with a friend J who periodically writes me regarding a math question he and his son are pondering together. The exchange was pretty juicy, about how many ways can an even number be decomposed as a sum of primes. But actually, the juiciest thing we got into was this:

Is 1 a prime number?

It was kind of a fight! Since I and Wikipedia agreed on this point (it’s not prime), J acknowledged we must know something he didn’t. But regardless, he kind of wasn’t having it.

Point 1: This is awesome.

Nothing could be better mathematician training than a fight about math. Proofs are called “arguments” for a reason.

When I went to Bob and Ellen Kaplan’s math circle training in 2009, I was heading to do a practice math circle with some high schoolers and Bob asked me, “what question are you opening with?” I said, “does .9999…=1?” He smiled with knowing anticipation and said, “oooh, that one always starts a brawl.”

Well, it wasn’t quite the bloodbath Bob led me to expect, but the kids were totally divided. One kid knew the “proof” where you go

0.999...=x

Multiplying by 10,

9.999...=10x

Subtracting,

9 = 9x

so x=1

and the other kids had that same sort of feeling like, “he knows something we don’t know,” but they weren’t convinced, and with only a minimal amount coaxing, they weren’t shy about it. The resulting conversation was the stuff of real growth: everybody in the room was contending with, and thereby pushing, the limits of their understanding. Even the boy who “knew the right answer” began to realize he didn’t have the whole story, as he found himself struggling to be articulate in the face of his classmates’ doubt.

Now this could have gone a completely different way. It’s common for “0.999… = 1″ to be treated as a fact and the above as a proof. Similarly, since the Wikipedia entry on prime numbers says, “… a natural number greater than 1 that has no positive divisors…,” we could just leave it at that.

But in both situations, this would be to dishonor everyone’s dissatisfaction. It is so vital that we honor it. Everybody, school-aged through grown-up, is constantly walking away from math thinking “I don’t get it.” This is a useless perspective. Never let them say they don’t get it. What they should be thinking is that they don’t buy it.

And they shouldn’t! If it wasn’t already clear that I think the above “proof” that 0.999…=1 is bullsh*t, let me make it clear. I think that argument, presented as proof, is dishonest.

I mean, if you understand real analysis, I have no beef with it. But at the level where this conversation is usually happening, this is not a proof, are you kidding me?? THE LEFT SIDE IS AN INFINITE SERIES. That means to make this argument sound, you have to deal with everything that is involved with understanding infinite series! But you just kinda slipped that in the back door, and nobody said anything because they are not used to honoring their dissatisfaction. As I have pointed out in the past, if you ignore all the series convergence issues, the exact same argument proves that …999.0=-1:

...999.0=x

Dividing by 10,

...999.9=0.1x

Subtracting,

-.9 = .9x

so x=-1

If you smell a rat, good! My point is that that same rat is smelling up the other proof too. We need to have some respect for kids’ minds when they look funny at you when you tell them 0.999…=1. They should be looking at you funny!

Same thing with why 1 is not a prime. If a student feels like 1 should be prime, that deserves some frickin respect! Because they are behaving like a mathematician! Definitions don’t get dropped down from the sky; they take their form by mathematicians arguing about them. And they get tweaked as our understanding evolves. People were still arguing about whether 1 was prime as late as the 19th century. Today, no number theorist thinks 1 is prime; however, in the 20th century we discovered a connection between primes and valuations, which has led to the idea in algebraic number theory that in addition to the ordinary primes there is an “infinite” prime, corresponding to the ordinary absolute value just as each ordinary prime corresponds to a p-adic absolute value. Now for goodness sakes, I hope you don’t buy this! With study, I have gained some sense of the utility of the idea, but I’m not entirely sold myself.

To summarize, point 2: Change “I don’t get it” to “I don’t buy it”.

Now I think this change is a good idea for everyone learning mathematics, at any level but especially in school, and I think we should teach kids to change their thinking in this way regardless of what they’re working on. But there is something special to me about these two questions (is 0.999…=1? Is 1 prime?) that bring this idea to the foreground. They’re like custom-made to start a fight. If you raise these questions with students and you are intellectually honest with them and encourage them to be honest with you, you are guaranteed to find that many of them will not buy the “right answers.” What is special about these questions?

I think it’s that the “right answers” are determined by considerations that are coming from parts of math way beyond the level where the conversation is happening. As noted above, the “full story” on 0.999…=1, in fact, the full story on the left side even having meaning, involves real analysis. We tend to slip infinite decimals sideways into the grade-school/middle-school curriculum without comment, kind of like, “oh, you know, kids, 0.3333…. is just like 0.3 or 0.33 but with more 3’s!” Students are uncomfortable with this, but we just squoosh their discomfort by ignoring it and acting perfectly comfortable ourselves, and eventually they get used to the idea and forget that they were ever uncomfortable.

Meanwhile, the full story on whether 1 is prime involves the full story on what a prime is. As above, that’s a story that even at the level of PhD study I don’t feel I fully have yet. The more I learn the more convinced I am that it would be wrong to say 1 is prime; but the learning is the point. If you tell them “a prime is a number whose only divisors are 1 and itself,” well, then, 1 is prime! Changing the definition to “exactly 2 factors” can feel like a contrivance to kick out 1 unfairly. It’s not until you get into heavier stuff (e.g. if 1 is prime, then prime factorizations aren’t unique) that it begins to feel wrong to lump 1 in with the others.

I highlight this because it means that trying to wrap up these questions with pat answers, like the phony proof above that 0.999…=1, is dishonest. Serious questions are being swept under the rug. The flip side is that really honoring students’ dissatisfaction is a way into this heavier stuff! It’s a win-win. I would love to have a big catalogue of questions like these: 3- to 6-word questions you could pose at the K-8 level but you still feel like you’re learning something about in grad school. Got any more for me?

All this puts me in mind of a beautiful 15-minute digression I witnessed about 2 years ago in the middle of Jesse Johnson’s class regarding the question is zero even or odd? It wasn’t on the lesson plan, but when it came up, Jesse gave it the full floor, and let me tell you it was gorgeous. A lot of kids wanted the answer to be that 0 is neither even nor odd; but a handful of kids, led by a particularly intrepid, diminutive boy, grew convinced that it is even. Watching him struggle to form his thoughts into an articulate point for others, and watching them contend with those thoughts, was like watching brains grow bigger visibly in real time.

Honor your dissatisfaction. Honor their dissatisfaction. Math was made for an honest fight.

p.s. Obliquely relevant: Teach the Controversy (Dan Meyer)

Notes from the Learning Lab: How to Dull My Curiosity Friday, Dec 14 2012 

I know I say this kind of thing a lot but I’m sitting here studying for a final, and this truth is just glaring and throbbing at me:

If you want to dull my curiosity, tell me what the answer is supposed to be.

If you want to make my curiosity vanish completely, do that and then add in a little time pressure.

There is nothing as lethal to my sense of wonder as that alchemical combination of already knowing how things are going to turn out (without knowing why), and feeling the clock tick.

Wherein This Blog Serves Its Original Function Wednesday, Nov 21 2012 

The original inspiration for starting this blog was the following:

I read research articles and other writing on math education (and education more generally) when I can. I had been fantasizing (back in fall 2009) about keeping an annotated bibliography of articles I read, to defeat the feeling that I couldn’t remember what was in them a few months later. However, this is one of those virtuous side projects that I never seemed to get to. I had also met Kate Nowak and Jesse Johnson at a conference that summer, and due to Kate’s inspiration, Jesse had started blogging. The two ideas came together and clicked: I could keep my annotated bibliography as a blog, and then it would be more exciting and motivating.

That’s how I started, but while I’ve occasionally engaged in lengthy explication and analysis of a single piece of writing, this blog has never really been an annotated bibliography. EXCEPT FOR RIGHT THIS VERY SECOND. HA! Take THAT, Mr. Things-Never-Go-According-To-Plan Monster!

“Opportunities to Learn Reasoning and Proof in High School Mathematics Textbooks”, by Denisse R. Thompson, Sharon L. Senk, and Gwendolyn J. Johnson, published in the Journal for Research in Mathematics Education, Vol. 43 No. 3, May 2012, pp. 253-295

The authors looked at HS level textbooks from six series (Key Curriculum Press; Core Plus; UCSMP; and divisions of the major publishers Holt, Glencoe, and Prentice-Hall) and analyzed the lessons and problem sets from the point of view of “what are the opportunities to learn about proof?” To keep the project manageable they just looked at Alg. 1, Alg. 2 and Precalc books and focused on the lessons on exponents, logarithms and polynomials.

They cast the net wide, looking for any “proof-related reasoning,” not just actual proofs. For lessons, they were looking for any justification of stated results: either an actual proof, or a specific example that illustrated the method of the general argument, or an opportunity for students to fill in the argument. For exercise sets, they looked at problems that asked students to make or investigate a conjecture or evaluate an argument or find a mistake in an argument in addition to asking students to actually develop an argument.

In spite of this wide net, they found that:

* In the exposition, proof-related reasoning is common but lack of justification is equally common: across the textbook series, 40% of the mathematical assertions about the chosen topics were made without any form of justification;

* In the exercises, proof-related reasoning was exceedingly rare: across the textbook series, less than 6% of exercises involved any proof-related reasoning. Only 3% involved actually making or evaluating an argument.

* Core Plus had the greatest percentage of exercises with opportunities for students to develop an argument (7.5%), and also to engage in proof-related reasoning more generally (14.7%). Glencoe had the least (1.7% and 3.5% respectively). Key Curriculum Press had the greatest percentage of exercises with opportunities for students to make a conjecture (6.0%). Holt had the least (1.2%).

The authors conclude that mainstream curricular materials do not reflect the pride of place given to reasoning and proof in the education research literature and in curricular mandates.

“Expert and Novice Approaches to Reading Mathematical Proofs”, by Matthew Inglis and Lara Alcock, published in the Journal for Research in Mathematics Education, Vol. 43 No. 4, July 2012, pp. 358-390

The authors had groups of undergraduates and research mathematicians read several short, student-work-typed proofs of elementary theorems, and decide if the proofs were valid. They taped the participants’ eye movements to see where their attention was directed.

They found:

* The mathematicians did not have uniform agreement on the validity of the proofs. Some of the proofs had a clear mistake and then the mathematicians did agree, but others were more ambiguous. (The proofs that were used are in an appendix in the article so you can have a look for yourself if you have JSTOR or whatever.) The authors are interested in using this result to challenge the conventional wisdom that mathematicians have a strong shared standard for judging proofs. I am sympathetic to the project of recognizing the way that proof reading depends on context, but found this argument a little irritating. The proofs used by the authors look like student work: the sequence of ideas isn’t being communicated clearly. So it wasn’t the validity of a sequence of ideas that the participants evaluated, it was also the success of an imperfect attempt to communicate that sequence. Maybe this distinction is ultimately unsupportable, but I think it has to be acknowledged in order to give the idea that mathematicians have high levels of agreement about proofs its due. Nobody who espouses this really thinks that mathematicians are likely to agree on what counts as clear communication. Somehow the sequence of ideas has to be separated from the attempt to communicate it if this idea is to be legitimately tested.

* The undergraduates spent a higher percentage of the time looking at the formulas in the proofs and a lower percentage of time looking at the text, as compared with the mathematicians. The authors argue that this is not fully explained by the hypothesis that the students had more trouble processing the formulas, since the undergrads spent only slightly more time total on them. The mathematicians spent substantially more time on the text. The authors speculate that the students were not paying as much attention to the logic of the arguments, and that this pattern accounts for some of the notorious difficulty that students have in determining the validity of proofs.

* The mathematicians moved their focus back and forth between consecutive lines of the proofs more frequently than the undergrads did. The authors suggest that the mathematicians were doing this to try to infer the “implicit warrant” that justified the 2nd line from the 1st.

The authors are also interested in arguing that mathematicians’ introspective descriptions of their proof-validation behavior are not reliable. Their evidence is that previous research (Weber, 2008: “How mathematicians determine if an argument is a valid proof”, JRME 39, pp. 431-459) based on introspective descriptions of mathematicians found that mathematicians begin by reading quickly through a proof to get the overall structure, before going into the details; however, none of the mathematicians in the present study did this according to their eye data. One of them stated that she does this in her informal debrief after the study, but her eye data didn’t indicate that she did it here. Again I’m sympathetic to the project of shaking up conventional wisdom, and there is lots of research in other fields to suggest that experts are not generally expert at describing their expert behavior, and I think it’s great when we (mathematicians or anyone else) have it pointed out to us that we aren’t right about everything. But I don’t feel the authors have quite got the smoking gun they claim to have. As they acknowledge in the study, the proofs they used are all really short. These aren’t the proofs to test the quick-read-thru hypothesis on.

The authors conclude by suggesting that when attempting to teach students how to read proofs, it might be useful to explicitly teach them to mimic the major difference found between novices and experts in the study: in particular, the idea is to teach them to ask themselves if a “warrant” is required to get from one line to the next, to try to come up with one if it is, and then to evaluate it. This idea seems interesting to me, especially in any class where students are expected to read a text containing proofs. (The authors are also calling for research that tests the efficacy of this idea.)

The authors also suggest ways that proof-writing could be changed to make it easier for non-experts to determine validity. They suggest (a) reducing the amount of symbolism to prevent students being distracted by it, and (b) making the between-line warrants more explicit. These ideas strike me as ridiculous. Texts already differ dramatically with respect to (a) and (b), there is no systemic platform from which to influence proof-writing anyway, and in any case as the authors rightly note, there are also costs to both, so the sweet spot in terms of text / symbolism balance isn’t at all clear and neither is the implicit / explicit balance. Maybe I’m being mean.

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.

C and M: I was a rigid grader. I put the numbers in the computer and whatever came out, that was your grade. I used an old-fashioned grading system that punished missed work harshly and made it very hard to climb out of a hole. Knowing I hated grading, this rigidity was how I protected myself: I didn’t have to make judgement calls, I just put the numbers in the computer and didn’t think about it. I didn’t allow myself to imagine what receiving the grades felt like. Both of you were students who had some bad student habits but showed tremendous growth over the time we worked together, stretching yourself to contribute positively to both your learning and the class community. I gave a lot of F’s that in retrospect I regret. Yours are the two I regret most.

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.

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.

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