If you’re telling a student the answer to a question it would never occur to them to ask, I can’t see how that’s a productive use of anyone’s time.
(This is not verbatim, but it’s the idea.)
If you’re telling a student the answer to a question it would never occur to them to ask, I can’t see how that’s a productive use of anyone’s time.
(This is not verbatim, but it’s the idea.)
There’s a new proposal at stackexchange.com for a Q&A site on Mathematics Learning, Studying, and Education.
Of course the entire mathtwitterblogosphere is a massive Q&A site on Mathematics Learning, Studying, and Education. But based on my experience of the incredible usefulness of the StackExchange sites Math StackExchange and MathOverflow, I think this site could become a great resource.
Possibly also a great forum for some much-needed productive dialogue between the K-12 and collegiate levels. For that to happen, though, it needs you. The bulk of the folks currently signed up for the beta of the new site are active on Math StackExchange and MathOverflow, which are dominated by college-and-up level math. The conversation is going to be so much richer with serious K-12 representation! Go sign up!
If you haven’t heard of the StackExchange sites before, they are a very thoughtfully constructed Q&A structure. It all started with StackOverflow, which was for working programmers to ask and answer practical coding questions. MathOverflow copied this idea for working research mathematicians. Math StackExchange is in principle for Q&A about math at any level, although as I mentioned, in practice it’s usually (though not always) about college and graduate level. Now there are also StackExchange sites on cooking, gaming, English language learning, and a million other things. The design of the software, and the culture of the sites, do an impressive job keeping the Q&A productive and on topic.
In the case of the math sites, the culture can also feel a little normatively intense (as in, there’s a “way we do things” that can be pretty strongly policed) and not always welcoming. Denizens of the sites will tell you that this is how they keep the conversation so productive and on-topic. But imho, it also stems from the deep ambivalence that the academic math world has about whether it wants to
(A) Share all its goodies and invite everyone into its kingdom, or
(B) Bull-guard the considerable stash of privilege that accrues from its high intellectual status.
(More on this in future posts.) The incredible usefulness of the sites makes it worth it; but also, this is part of why I want you guys to go populate the new Math Education site. You are clear in your hearts that math is for everybody. This is our chance to go talk with some folks who represent a culture that is working through that for itself. Meanwhile, we get to benefit from their perspectives, which have seen very different parts of the mathematical kingdom in their travels.
Disclaimer: I think lots and lots of individuals on Math.SE and MO think math is for everybody. I am not trying to stereotype the sites or mathematicians more generally. And I think it’s likely that the people from Math.SE and MO who gravitate to the new Math Learning/Education site are going to be skewed toward the folks who think that math is for everybody. What I am trying to do is to name some notes that I hear in the cultural soundscape of academic math as a whole, and Math StackExchange and MathOverflow in particular; but I’m not trying to identify those notes with any individuals.
I am at the Joint Mathematics Meetings this week. I had a conversation yesterday, with Cody L. Patterson, Yvonne Lai, and Aaron Hill, that was very exciting to me. Cody was proposing the development of what he called a “critical language of task design.”
This is an awesome idea.
But first, what does he mean?
He means giving (frankly, catchy) names to important attributes, types, and design principles, of mathematical tasks. I can best elucidate by example. Here are two words that Cody has coined in this connection, along with his definitions and illustrative examples.
Jamming – transitive verb. Posing a mathematical task in which the underlying concepts are essential, but the procedure cannot be used (e.g., due to insufficient information).
Example: you are teaching calculus. Your students have gotten good at differentiating polynomials using the power rule, but you have a sinking suspicion they have forgotten what the derivative is even really about. You give them a table like this
and then ask for a reasonable estimate of f'(4). You are jamming the power rule because you’re giving them a problem that aims at the concept underlying the derivative and that cannot be solved with the power rule.
Thwarting – transitive verb. Posing a mathematical task in which mindless execution of the procedure is possible but likely to lead to a wrong answer.
Example: you are teaching area of simple plane figures. Your students have gotten good at area of parallelogram = base * height but you feel like they’re just going through the motions. You give them this parallelogram:
Of course they all try to find the area by . You are thwarting the thoughtless use of base * height because it gets the wrong answer in this case.
Why am I so into this? These are just two words, naming things that all teachers have probably done in some form or another without their ever having been named. They describe only a very tiny fraction of good tasks. What’s the big deal?
It’s that these words are a tiny beginning. We’re talking about a whole language of task design. I’m imagining having a conversation with a fellow educator, and having access to hundreds of different pedagogically powerful ideas like these, neatly packaged in catchy usable words. “I see you’re thwarting the quadratic formula pretty hard here, so I’m wondering if you want to balance it out with some splitting / smooshing / etc.” (I have no idea what those would mean but you get the idea.)
I have no doubt that a thoughtful, extensive and shared vocabulary of this kind would elevate our profession. It would be a concrete vehicle for the transmission and development of our shared expertise in designing mathematical experiences.
This notion has some antecedents. First, there are the passes at articulating what makes a problem pedagogically valuable. On the math blogosphere, see discussions by Avery Pickford, Breedeen Murray, and Michael Pershan. (Edit 1/21: I knew Dan had one of these too.) I also would like to believe that there is a well-developed discussion on this topic in academic print journals, although I am unaware of it. (A google search turned up this methodologically odd but interesting-seeming article about biomed students. Is it the tip of the iceberg? Is anyone reading this acquainted with the relevant literature?)
Also, I know a few other actual words that fit into the category “specialized vocabulary to discuss math tasks and problems.” I forget where I first ran into the word problematic in this context – possibly in the work of Cathy Twomey-Fosnot and Math in the City – but that’s a great word. It means that the problem feels authentic and vital; the opposite of contrived. I also forget where I first heard the word grabby (synonymous with Pershan’s hooky, and not far from how Dan uses perplexing) to describe a math problem – maybe from the lips of Justin Lanier? But, once you know it it’s pretty indispensible. Jo Boaler, by way of Dan Meyer, has given us the equally indispensable pseudocontext. So, the ball is already rolling.
When Cody shared his ideas, Yvonne and I speculated that the folks responsible for the PCMI problem sets – Bowen Kerins and Darryl Yong, and their friends at the EDC – have some sort of internal shared vocabulary of problem design, since they are masters. They were giving a talk today, so I went, and asked this question. It wasn’t really the setting to get into it, but superficially it sounded like yes. For starters, the PCMI’s problem sets (if you are not familiar with them, click through the link above – you will not be sorry) all contain problems labeled important, neat and tough. “Important” means accessible, and also at the center of connections to many other problems. Darryl talked about the importance of making sure the “important” problems have a “low threshold, high ceiling” (a phrase I know I’ve heard before – anyone know where that comes from?). He said that Bowen talks about “arcs,” roughly meaning, mathematical themes that run through the problem sets, but I wanted to hear much more about that. Bowen, are you reading this? What else can you tell us?
Most of these words share with Cody’s coinages the quality of being catchy / natural-language-feeling. They are not jargony. In other words, they are inclusive rather than exclusive. It is possible for me to imagine that they could become a shared vocabulary of our whole profession.
So now what I really want to ultimately happen is for a whole bunch of people (Cody, Yvonne, Bowen, you, me…) to put in some serious work and to write a book called A Critical Language for Mathematical Problem Design, that catalogues, organizes and elucidates a large and supple vocabulary to describe the design of mathematical problems and tasks. To get this out of the completely-idle-fantasy stage, can we do a little brainstorming in the comments? Let’s get a proof of concept going. What other concepts for thinking about task design can you describe and (jargonlessly) name?
I’m casting the net wide here. Cody’s “jamming” and “thwarting” are verbs describing ways that problems can interrupt the rote application of methods. “Problematic” and “grabby” are ways of describing desirable features of problems, while “pseudocontext” is a way to describe negative features. Bowen and Darryl’s “important/neat/tough” are ways to conceptualize a problem’s role in a whole problem set / course of instruction. I’m looking for any word that you could use, in any way, when discussing the design of math tasks. Got anything for me?
In fairness, for all I know, somebody has written a book entitled A Critical Language for Mathematical Task Design. I doubt it, but just in case, feel free to get me a copy for my birthday.
I am taking a perhaps-undeserved dig here at a number of in-many-ways-wonderful curriculum and instructional design initiatives that have a lot of rich and deep thought about pedagogy behind them but have really jargony names, such as Understanding by Design and Cognitively Guided Instruction. (To prove that an instructional design paradigm does not have to be jargony, consider Three-Acts.) I feel a bit ungenerous with this criticism, but I can’t completely shake the feeling that jargony names are a kind of exclusion: if you really wanted everybody to use your ideas, you would have given them a name you could imagine everybody saying.
Back in the spring, I resolved to make a practice of having students summarize each others’ thoughts whenever I have classroom opportunities. This summer, I got the opportunity to give this technique a sustained go, when I taught at SPMPS (which was completely awesome btw). And:
It is an effing game-changer.
This summer, when I or a student put forth an idea, I regularly followed it with, “who can summarize what so-and-so said?” Or (even better), “so-and-so, can you summarize what so-and-so just said?” Following the models of Lucy West and Deborah Ball, I carefully distinguished summary from evaluation. “Not whether you buy it, just the idea itself.” When dipsticking the room on an idea, I would also make this distinction. “Raise your hand if you feel that you understand what was just said; not that you buy it, just that you understand what they’re trying to say.” Then, “leave your hand up if you also buy it.”
These moves completely transformed the way whole-class conversation felt to me:
* Students were perceptibly more engaged with each others’ ideas.
* The ideas felt more like community products.
* Students who were shy to venture an idea in the first place nonetheless played key roles as translators of others’ ideas.
Furthermore, for the first time I felt I had a reliable way past the impasse that happens when somebody is saying something rich and other people are not fully engaged. More generally, past the impasse that happens when somebody says something awesome and there are others for whom it doesn’t quite land. (Whether they were engaged or not.)
A snippet of remembered classroom dialogue to illustrate:
Me: The question before us is, do the primes end, or do they go on forever? At this point, does anybody think they know?
(Aside: This was after a day of work on the subject. Most kids didn’t see the whole picture at this point, but one did:)
[J raises his hand.]
J: They don’t end. If they ended, you’d have a list. You could multiply everything on the list and add 1 and you would get a big number N. Either N is prime or it’s composite. If it is prime, you can add it to the list. If it is composite, it has at least one prime factor. Its factor can’t be on the list because all the numbers on the list when divided [into] N have a remainder of 1. So you can add its factor to the list. You can keep doing this forever so they don’t end.
Me: Raise your hand to summarize J’s thought.
(Aside: although J has just basically given a complete version of Euclid’s proof of the infinitude of the primes, and although I am ecstatic about this, I can’t admit any of this because the burden of thought needs to stay with the kids. J is just about done with the question, but this is just the right thing, said once: the class as a whole is nowhere near done. This is one of the situations in which asking for summaries is so perfect.)
[Several kids raise their hands. I call on T.]
T: J is saying that the primes don’t end. He says this because if you have a list of all the primes, you can multiply them and add one, giving you a big number N. If N is prime, you can add it to the list. If N is not prime, and its prime factors are not on the list, you can add them.
Me: J, is that what you were trying to say?
(Notice that a key point in J’s argument, that the factors of N cannot already be on the list, was not dealt with by T, and J did not catch this when asked if T had summarized his point. This is totally typical. Most kids in the room have not seen why this point is important. Some kids have probably not seen why J’s argument even relates to the question of whether the primes end. All this has to be given more engaged airtime.)
Me: raise your hand if you feel that you understand the idea that J put forth that T is summarizing.
[About 2/3 of the room raises hands. I raise mine too.]
Me: Leave your hand up if you also find the idea convincing and you now believe the primes don’t end.
[A few kids put their hands down. I put mine down too.]
N [to me]: Why did you put your hand down?
Me [to class]: Who else wants to know?
[At least half the class raises hands.]
Me [to T]: Here’s what’s bugging me. You said that if N is not prime and its prime factors are not on the list, I can add them. But what if N is not prime and its prime factors are already on the list?
T [thinks for a minute]: I don’t know, I’ll have to think more about that.
[J’s hand shoots up]
Me [to T]: Do you want to see what J has to say about that or do you want to think more about it first?
[T calls on J to speak]
J: Can’t happen. All the numbers on the list were multiplied together and added 1 to get N. So when N is divided by 2, 3, 5, and so on, it has a remainder of 1. So N’s factor can’t be 2, 3, 5, and so on.
T: Oh, yeah, he’s right.
Me: Can you summarize his whole thought?
[T explains the whole thing start to finish.]
Me: Do you buy it?
Me: Who else wants to summarize the idea that J put forth and T summarized?
Unexpectedly, this technique speaks to a question I was mulling over a year and a half ago, about how to encourage question-asking. How can the design of the classroom experience structurally (as opposed to culturally) encourage people to ask questions and seek clarification when they need it? The answer I half-proposed back then was to choose certain moments in the lesson and make student questions the desired product in those moments. (“Okay everyone, pair up and generate a question about the definition we just put up” or whatever.) At the time I didn’t feel like this really addressed the need I was articulating because it had to be planned. Kate rightly pressed me on this because actually it’s awesome to do that. But I was hungering for something more ongoingly part of the texture of class, not something to build into a lesson at specific points. And as it turns out, student summaries are just what I was looking for! The questions and requests for clarification are forced out by putting students on the spot to summarize.
A last thought. Learning this new trick has been for me a testament to teaching’s infinitude as a craft. Facilitating rich and thought-provoking classroom discussions was already something I’d given a lot of thought and conscious work to; perhaps more than to any other part of teaching, at least in recent years. I.e. this is an area where I already saw myself as pretty accomplished (and, hopefully with due modesty, I still stand by that). And yet I could still learn something so basic as “so-and-so, can you summarize what so-and-so said?” and have it make a huge difference. What an amazing enterprise to always be able to grow so much.
I was having a conversation a few weeks ago with a computer programmer and math enthusiast whom I’ll call Dorian. He was arguing very passionately that talking about a square root of was the wrong way to introduce complex numbers. He recounted this moment in his own schooling: 16 year old Dorian, told by his teacher “we introduce a new number whose square is …,” asking, “but I can prove that the square of any number is positive, what about that?!” His teacher wasn’t able to satisfy his objection and made him feel that it wasn’t valid. He left the experience feeling angry and frustrated and that his question had been treated as a failure to understand.
Dorian later learned that complex numbers can be visualized as a plane containing the real line; that addition of points in this plane is just vector addition; and that multiplication is done by multiplying the distances from the origin and adding the angles from the positive real axis (see here for a brief explanation if desired). Here was a concrete model for the complex numbers, with concrete geometrical interpretations of the operations and . And it was clear to him that in this model, there is a point, in fact two points, whose squares correspond to the point on the real axis. But philosophically, this fact is a consequence of the concrete geometrical description of the operations in the plane, rather than an ontologically dubious starting point for the whole project.
Dorian concluded that actually this model, via the geometry of addition and multiplication in the complex plane, is a pedagogically superior introduction to the complex numbers. His argument is that it presents no ontological quandary. Nobody will object to a plane. Nobody will object, at least on philosophical grounds, to these new definitions of and , as long as you can prove they have nice properties and coincide with the old definitions on the real line. You’re not saying anything so wildly speculative as “postulate a square root of …”
I am not writing this post to get into the question of whether Dorian is right about this. I see lots to say on both sides. What I am writing this to say is that there is a lesson in Dorian’s story much deeper than the question of how to introduce the complex numbers. That is not the real question here as far as I am concerned.
The real question is this: when you’ve picked your approach and gone with it, how will you deal with the students it doesn’t work for?
Now you can always obsess about how to introduce a topic, and I believe there is basically always value in thinking and talking about the pedagogical consequences of different ways of looking at things. And I think some models for ideas are legitimately better than others. But no model will speak to every student. This point is so important, and was so lost on me as a young teacher, and is lost on so many (especially young) teachers that I have spoken with, so excited that they are about the way they have thought of to present negative numbers or whatever, as though miraculously everyone in the room will get it this time, that I need to repeat it:
There is no model that is the right model for each and every student, each and every time.
No matter how awesome your idea for how to think about XYZ concept is, there will be somebody in your class who will have no idea what you are talking about. To me, the big question here is, what are you going to do about it?
More specifically, how are you going to treat their thinking?
Now, I like to think that nobody reading this blog would be so callous as to intentionally make a student feel stupid for asking an honest question. But there are far subtler ways to do it. The one I most want to warn you against is the sin I know I’m guilty of: being so wrapped up in the awesomeness of your presentation that the kid who doesn’t get it does not compute to you. You say whatever you say out loud but in your mind you’re like, “wait – you don’t understand? Huh?” Or, you’re like, “oh my goodness can’t you just see it as I do?”
Regardless of what you say out loud, having such a response in the back of your mind invalidates whatever obstacle the student is facing. I want to suggest an alternative:
Take the case that any earnest failure of a student to see your point of view is actually coming from a legitimate mathematical objection.
This is how you treat dissatisfaction with honor.
I don’t care what the kid’s IEP says. Mathematical convention does not require us to check somebody’s Wechsler results before they are allowed to raise an objection. If they don’t buy it, they don’t buy it. Now it’s your turn to understand their objection and answer it.
“I don’t get it.” “I don’t buy it.”
A student I’ll call Manny, whom I had in my 2003-4 AP Calculus class, came to me around March and said something like, “this entire class is based on a paradox.” He objected to my (retrospectively totally hand-wavy) discussion of limits. It never gets there, so how can you talk about what happens if it were to get there?
I tried to answer Manny’s objections; I spent some time with him on it; but he left the conversation unsatisfied. Retrospectively it is clear to me that this is because (a) I didn’t get what the problem was, and (b) to my shame I didn’t consider the possibility that there was really much to it. Then, less than a year later, I read The Calculus Gallery, whereupon I learned that actually Manny’s objection was more or less exactly Bishop Berkeley’s famous objection that in due time forced mathematicians to invent real analysis. For a sense of the importance of this development, let me mention that I have read, though I don’t recall where right now, that the development of real analysis was really the event that led to the birth of modern mathematical rigor.
So, yes, I am on record as having treated as essentially invalid an objection that actually led to the creation of modern rigor. Don’t let that be you.
If they don’t get it, take the case that there’s a legitimate mathematical objection behind that. Treat their “I don’t get it” as “I don’t buy it.” Now getting them to buy it is your job.
Paul Salomon’s “imbalance problems”. You know how I love a thought-provoking picture.
I recently saw some video from Deborah Ball’s Elementary Mathematics Laboratory. I actually didn’t know what she looked like so I didn’t find out till afterward that the teacher in the video was, y’know, THE Deborah Ball, but already from watching, I was thinking,
THAT IS A F*CKING MASTER. I F*CKING LOVE HER.
It put me in mind of a professional development workshop I attended 2 years ago which was run by Lucy West. Both Ball and West displayed a level of adeptness at getting students to engage with one another’s reasoning that blew me away.
One trick both of them used was to consistently ask students to summarize one another’s train of thought. This set up a classroom norm that you are expected to follow and be able to recapitulate the last thoughts that were said, no matter who they are coming from. Both Ball and West explicitly articulated this norm as well as implicitly backing it up by asking students (or in West’s case, teachers in a professional development setting) to do it all the time. In both cases, the effect was immediate and powerful: everybody was paying attention to everybody else.
The benefit wasn’t just from a management standpoint. There’s something both very democratic and very mathematically sound about this. In the first place, it says that everybody’s thoughts matter. In the second, it says that reasoning is the heart of what we’re doing here.
I resolve to start employing this technique whenever I have classroom opportunities. I know that it’ll come out choppy at first, but I’ve seen the payoff and it’s worth it.
A nuance of the technique is to distinguish summarizing from evaluating. In the Ball video, the first student to summarize what another student said also wanted to say why he thought it was wrong; Ball intercepted this and kept him focused on articulating the reasoning, saving the evaluation step until after the original train of thought had been clearly explicated. Which brings me to a second beautiful thing she did.
Here was the problem:
The first student to speak argued that the blue triangle represents half because there are two equal wholes in the little rectangle at the top right.
He is, of course, wrong.
On the other hand, he is also, of course, onto something.
It was with breathtaking deftness that Deborah Ball proceeded to facilitate a conversation that both
(a) clearly acknowledged the sound reasoning behind his answer
(b) clarified that he missed something key.
It went something like this. I’m reconstructing this from memory so of course it’s wrong in the details, but in overall outline this is what happened –
Ball: Who can summarize what [Kid A] said?
Kid B: He said it’s half, but he’s just looking at the, he’s just…
Ball: It’s not time to say what you think of his reasoning yet, first we have to understand what he said.
Kid B: Oh.
Kid C: He’s saying that the little rectangle has 2 equal parts and the blue is one of them.
Ball [to Kid A]: Is that what you’re saying?
Kid A: Yeah.
Ball: So, what was the whole you were looking at?
Kid A [points to the smaller rectangle in the upper right hand corner]
Ball: And what were the two parts?
Kid A [points to the blue triangle and its complement in the smaller rectangle]
Ball: And are they equal?
Kid A: Yes.
Ball [to the rest of the class]: So if this is the whole [pointing at the smaller rectangle Kid A highlighted], is he right that it’s 1/2?
Many students: Yes.
Ball: The question was asking something a little different from that. Who can say what the whole in the question was?
Kid D [comes to the board and outlines the large rectangle with her finger]
Kid A: Oh.
I loved this. This is how you do it! Right reasoning has been brought to the fore, wrong reasoning has been brought to the fore, nobody feels dumb, and the class stays focused on trying to understand, which is what matters anyway.
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
Multiplying by 10,
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:
Dividing by 10,
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)
I haven’t talked openly about race or racial difference on this blog before, but I actually think about it a lot. One of the most damning legacies of our racist history has been systematic libel against the minds of black and brown children (and adults for that matter). Meanwhile, in our culture, math is the ultimate signifier of intelligence. So the math classroom has heightened power, both to inflict injustice and to rectify it. Given this, plus the diversity of teachers and students, a comfortable cross-race conversation about racial matters is a must for the profession. In the spirit of contributing to that conversation, I offer
A Note to My Fellow White People
Guys, we have to chill out a little. It has to be possible for somebody to say to you, “that was ignorant,” or “that was racially offensive,” or even “that was racist,” without you flipping out, getting offended or defensive, or needing to be reassured you are not a horrible person. It’s not a good look, on any level: it’s not dignified, and it makes it impossible to have a productive conversation about race across racial lines.
I was at a cafe a couple months back trying to get some schoolwork done when I found myself distracted by a profoundly uncomfortable conversation at the next table. There was a white man in his early 50s and two black women, one close to his age and one closer to mine. They seemed to be sharing a familiar and friendly meal. Things started to go south when the man admitted to being afraid of a young black man on the street. The younger of the women said something to the effect of, “you might have work to do on that.”
Her tone was warm: she wasn’t being accusatory but rather seemed to be offering her words in the spirit of holding her friend to a high standard. But the man immediately became anxious, although his face and words were all smiles and jokes. His first response was that white people make him more uncomfortable than black people, as though he could re-establish his lost racial coolness with sufficiently loud declamations of prejudice against white people.
The women weren’t having it. “You’re being ignorant against white people now.” I interpreted their response as saying, “you can’t get off the hook with this diversionary tactic.” But he kept trying. His anxiety was as audible to me as a fire alarm, even when he wasn’t talking. I tried to concentrate on my math but I couldn’t get anything done.
Things stayed in this state, a tense, anxious impasse overlaid by a thin layer of too-eager conviviality and jokes, for about 20 minutes, till they got up to leave, no noticeable progress having been made in the conversation. At this point the man, in that same overly-eager joking tone, almost-but-not-quite-explicitly asked for reassurance that everybody was still his friend. They gave him the reassurance. On their way out, the younger woman leaned over to my table and apologized for her “ignorant friend.”
I’m not telling you this story to put the man down or call him ignorant. I don’t remember the context of the conversation and I don’t have my own opinion about it. Also, I think in all likelihood he’s a completely nice and decent person, and so are the women.
The point of the story is the man’s intense anxiety at being put on the spot racially, and the way that anxiety dominated both the conversation and its goals (so that what started as an attempt to raise consciousness was aborted, and turned into a reassurance fest), and the social and public space (so that the younger woman felt the need to apologize to a neighboring table).
Now I don’t fail to have empathy for him. If you are a white person with a modicum of sense and decency, you know that you are the beneficiary of an unjust history. (Shout out to Louis CK.) Just knowing that you’re benefiting is already a little uncomfortable to begin with. Feeling like you might be participating in that injustice can make the discomfort acute. I’ve been there many times.
But, guys, we’ve got to get it together! It is necessary to learn how to be with that discomfort and still function. First of all, the story I just told you is about a grown-a** man! Trying to prove how un-racist you are, and then needing to be coddled and preened so that you know the trouble is past, is unbefitting of the dignity of an adult. So is any other response aimed at removing the source of your discomfort rather than tolerating it – throwing a fit, acting defensive or offended, etc. Shouldn’t we aspire to some grace here?
Secondly, it makes it impossible for the conversation to advance! If we want to avoid participating in injustice we have to be willing to tolerate the possibility that we already are participating. Otherwise how will we learn what to avoid? In the anecdote I’ve recounted here, the man’s anxiety shut down the ability of the conversation to make any progress. He was blessed with friends who were willing to hold him to a higher standard and he was too busy freaking out to get the benefit of that! The bottom line question is, would you rather spend your time and energy proving how un-racist you are, or would you actually like to learn how to make the world better?
All of this puts me in mind of a much more public incident. In 2009, Attorney General Eric Holder gave a speech at the Dept. of Justice Black History Month program in which he said that Americans are afraid to talk about race and called upon us to do better. Multiple commentators immediately jumped down his throat.
Thereby proving his point.
The Attorney General made an effort to hold the nation to a higher standard. At the time, we didn’t react with grace or manifest any interest in growing.
How about now?
IMO the best thing white teachers, or any teachers who find themselves teaching classes of black/brown students can do is to constantly hold their students to the same high standards they would hold their own biological children to. Giving these kids a high standard education is one of the few ways to equip these kids to deal with racism.