Favorite Posts by Other People in 2015

You want to know my methodology, right? Well, first I went back in twitter time and looked for pieces that I shared. Of those, I picked a favorite for each month, or two if it was a good month. I tried to vary up the people whose posts are in this list, so there aren’t any repeat appearances.

January:  Grace Chen, “Teacher Moves for Cultural Competence”

Bonus: Shanker Institute, “Update on Teacher Turnover”

February: Andrew Gael, “There’s More Than One Way To Skin a Task”

Bonus: Adam Lefstein, “Against Boldness in Teaching”

March: Fawn Nguyen, “Reversing the Question”

Bonus: Dan Willingham, “Computational Competence Doesn’t Guarantee Conceptual Understanding in Math”

April: Raymond Johnson, “A Few Thoughts from NCTM 2015”

Bonus: NCES Blog, Free and Reduced Lunch, a Proxy for Poverty?

Aside: Lots of good blogging about hints from AnnaHenriMike, Annie, Umussahar, Joe, Dylan and others too. Also, Max and I talked about complex numbers. I also talked to Justin’s undergrads about math mistakes.

May: Anna Weltman, “There Are No Kids Here”

Dan Goldner, “Anticipatory Set”

Bonus: Fallace and Fantozzi, “Was There Really a Social Efficiency Doctrine?”

June: Nada, I took June off from blogs/twitter. (New year’s resolution: less time on twitter?)

July: Dylan Kane, “Standards-Based Grading: Skepticisms”

Bonus: Eric Schwitzgebel, “Cheeseburger Ethics”

August: Carl Oliver, “What Would a Teacher’s ‘Off-Season Workout’ Look Like?”

Bonus: Joe Posnanski, “The Age of Tiger”

September: Daniel Schneider, “ELL Math – 3 Weeks In”

Bonus: Larry Cuban, “Research Influence on Classroom Practice”

October: Kent Haines, “Open Number Sentences: Is this _____ actually useful?”

Bonus: Melinda D. Anderson, “The Misplaced Fear of Religion in Classrooms”

November: Chris Lusto, “From Fallujah to Pennsylvania, My Life as a Marine and a Math Teacher”

Tina Cardone, “Expressions With Absolute Value”

Bonus: 

https://twitter.com/rohmansoldier/status/668885496286834688

December: Tracy Zager, “Counting Circle Variant: Tens and Ones”

Joe Schwartz, “22? 30? 50? 100?”

Bonus: Alan Jacobs, “My Year in Tech”

Bonus: Stephen Burt, “Overrated Writers”

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The Problems of Writing

There are times when writing about teaching feels like a pretty lonely affair.  It’s sort of a double-whammy of loneliness. At first, we turn to writing because there’s a something about the work that we can’t quite talk about with other teachers at our schools. That’s how I started, at least. I write, hoping to find someone who can give me that something I don’t get at school.

There’s another level of loneliness, though, that can sneak in to the teacher-writer’s life. It happens once the teacher starts worrying about the craft of writing. The teacher now – almost by accident! – has picked up a new art, a second set of tools, and might now be itching to figure out how best to use them. Not all teacher-writers share this itchiness, but there aren’t many places to turn if you do.

Magdalene Lampert – teacher and researcher – set out to do something about that first bit of loneliness. She video-taped every second of her school year and wrote a book about her decisions. Why?

“As the nature of teaching practice is made more explicit, it should be easier to teach well and to learn what good teachers know how to do. It should be easier for teachers to work together on improving what they do.”

Lampert’s aiming to make it easier for us to talk about teaching. It’s not my place to judge whether she succeeds. Instead, I want to take a closer look at her writing, as writing. Every teacher experiences isolation in the classroom. How does she aim to attack it?

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Her book (Teaching Problems and the Problems of Teaching) is an account of the major areas of her preparation and teaching. She tells us what she was planning to do, what she did, and why she did it.

The image I have of Lampert is her, alone in an office, meditating on the day’s work:

“Anthony interrupted the flow of the teaching I was trying to do but he gave me the opportunity to use what he had done as an occasion to teach something I wanted to teach. He had not raised his hand to ask for the floor. My work here was to manage the tension between encouraging Eddie and concurring with Anthony’s correction, as well as to acknowledge the correction while maintaining civility in students’ interactions with one another. I had to teach not only what a generalization would be, but also how to disagree with another student’s assertion.”

Do you hear all those “I”s, “me”s and “my”s? This is not a story about what went on in her classroom. This is a story about what went on in her head in her classroom.  This is the author, thinking about her thoughts.

Now, this works well for the reader (me) because Lampert’s head is an interesting place to spend some time. And Lampert is the absolute, unquestioned authority on what goes on in Lampert’s brain. She can tell us what she was trying, and I don’t doubt her.

Actually, I can’t doubt Lampert. All the action is happening in her head. As a result, her authorial voice is the voice of an expert, someone who is talking about what she knows best. The reader is almost eavesdropping.

Lampert’s style perfectly matches her purpose. There might be no better way to expose the complexity of the work of teaching. But it’s not the only way to write about craft.

One of my favorite writers is George Saunders, and he wrote a lovely piece about story-telling, “The Perfect Gerbil.” Saunders takes a very different stance than Lampert’s. Saunders is clearly addressing you, the reader, as a knowledgeable equal. Sure, he has something to say about writing. But just as he is a writer, you are a writer too. (In contrast: Lampert is a teacher, and that is that.)

Saunders’ piece takes us through his reading of a story by Donald Barthelme. This choice – to focus on the writing of someone else – is an important one. He could have walked us through the process of creating one of his own stories: Here’s what I was thinking when I wrote this paragraph, and so on. Focusing on his own work, though, would be placing himself in a position of authority relative to, me, the reader. There wouldn’t be room for my thinking.

The choice to think through someone else’s craft, then, is a show of goodwill to the reader. It’s an act of parity, of putting the reader and the writer in roughly the same place. It’s an awfully generous thing to do.

Writing about Barthelme, Saunders says:

“…he has given us a little something extra: a laugh, yes, but more important an acknowledgement that the writer is right there with us: he knows where ware, and who we are, and is involved in an intimate and respectful game with us. I think of this as the motorcycle-sidecar model of reading: writer and reader right next to one another, leaning as they corner, the pleasure coming from the mutuality and simultaneity of the experience.”

And this is precisely what Saunders is pulling off in his own piece.

This model of writing – the one that fosters mutuality and simultaneity – is certainly not the only good one. Lampert’s style is good too, for her purposes. The best parts of Teaching Problems feel to me like I’m trying (often failing) to keep up with a brilliant teacher. She manages to hold many things in her head at once. She’s driving the motorcycle all by herself, and I’m running after her, practically.

Lampert’s great, but here’s what I want to posit: her writing isn’t the best way to dispel the second sort of loneliness. By placing the reader in her mind, her area of greatest expertise, we aren’t given quite the invitation that Saunders offers us to engage in the work of writing. She isn’t saying to us, you know, it’s hard, but you could do this too.

What, you want Lampert to literally say ‘you could do this too’? No, absolutely not. Like I said, she has her own purposes. More to the point, even if she wanted to create parity between reader and writer, a simple statement wouldn’t be enough. Her stance as an expert is central to her writing. We are in her head, and we just don’t know enough to keep up with her. The reader can’t think about the things she’s thinking about. This is the missing invitation – we have no chance to think along with her. She does her work, and we do ours’.

However, this is not how writing about teaching has to be. Writing (thinking) about teaching can invite readers to do that very same work. Joe Schwartz recently wrote a piece – “22? 30? 50? 100?” – that exemplifies this to me. From the very first sentence (“Meet Alex.”) I feel Joe taking care to catch me up on the situation. Who is this kid? Who are you? What are you doing? (Struggling first-grader; devoted math coach; learning how this kid counts.) It’s only after establishing all this context that Joe drops the major intellectual puzzle, the one that drives the rest of the piece.

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It would have been so easy for Joe to write a piece that had none of these qualities. Today I worked with a kid who had some interesting issues with counting collections of objects, he might have begun, and the reader is already lagging behind. Knowing that I wanted to check up on his counting skills, he would continue, thinking something we could not possibly have thought, I dumped out some plastic dogs and asked him to count them, and I rest, because for a brief moment I’m all caught up.

In short, then, establishing context can be an act of generosity to the reader – he’s making room for us to think right along with him, rather than asking us to keep up. Is it surprising that some twenty-odd people wrote paragraph-length responses to his post, with ideas, analysis, their own suggestions and questions?

(And, by the way, Joe is also offering a master-class in hooking the reader. He’s dangled a huge piece of intellectual bait in front of starving teachers. Of course we bite!)

Teachers are given so little respect as thinkers and writers about teaching, that it can be tempting to write in a way that raises ourselves up from the reader and emphasizes our expertise. I’d suggest that Saunders and Joe point towards another possibility. If we write in a way that empowers the teacher-reader, we can invite them to become a teacher-writer. None of this work needs to be quite so lonely.

Teacher Writers vs. Teacher Researchers

“I know that it’s not possible to write a competent interview without some juggling and eliding of quotes; don’t believe any writer who claims he never does it…What’s wrong, I believe, is to fabricate quotes or to surmise what someone might have said. Writing is a public trust. The non-fiction writer’s rare privilege is to have the whole wonderful world of people to write about. When you get people talking, handle what they say as you would handle a valuable gift.” (On Writing WellWilliam Zissner)

I wonder if the journalist’s tools might be more helpful to me than the educational researcher’s. Even though journalists share few of our interests, their predicament more closely resembles our own. Out for an interview, at first there are two people talking. And hours later it’s a writer with a notebook trying to faithfully capture that conversation.

Every time I sit down to write about a classroom experience I worry about my honesty. Am I making this up? I look to student work, sometimes I had managed to scribble down a quote or two during class. I bet I could get handier with my notebook. I need more scraps of the past.

And I also wonder whether the work of the teacher researcher is more about writing than research. We have no communally accepted standards of evidence. There is no experimental design worth talking about when you’re alone with your students. We teachers can’t afford to have systematic research methods, not when thirteen kids are yelling for help.

So, we pay attention as best we can and then we write. But do we pay enough attention to our writing? What sort of stories do we tell? What our the cliches of our genre? How do we improve the craft? What’s captures attention, what’s interesting?

Here’s what’s interesting: Tracy Zager, Christopher Danielson, Andy Gael, Joe Schwartz.

Dissent of the Day

I said [Three Act problems] are most valuable to me before learning skills, or rather as the motivation for learning skills. I don’t expect that students will just figure everything out on their own, though. Act one helps generate the need for the tools I can offer them here in act two.

-“Teaching With Three Act Tasks: Act Two,” Dan Meyer

I’ve been thinking about it, and I think I disagree with Dan’s take here. I think there are important differences between providing instruction during, before or after a tough mathematical experience, and that instruction during a problem is often bound to be lost in the flood of ideas that a mind is awash in.

Here’s where I’m coming from. Over the past few class periods, my 4th Graders have been working on a lovely little activity. We watched a short video showing Andrew papering his cabinet with sticky notes. How many sticky notes would it take to cover the entire thing?

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I showed this video, and was disappointed by the tepid response from my students. Then I asked my students to estimate the number of stickies it would take to cover the cabinet. More blahs. And then I clarified that we’re trying to figure out how many stickies would cover the entire cabinet, and my kids exploded with ideas and excitement: “Wait, can you give us time to figure this out?”

Really, really great stuff.

While walking around, I noticed some kids getting lost in their calculations. Lots of great ideas, but constantly losing the thread.

IMG_4026 Other kids, though, used diagrams to preserve their line of thought. These kids, even if they were less computationally sophisticated than other students in class, were finding relatively more success in the problem.

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When I noticed this, I realized that this sort of diagramming was an important mathematical idea that I should make explicit to everyone. When pairs called me over to help them make sense of their confusing calculations, I made the suggestion: here’s a diagram, here’s how you can use it, this could help with where you’re stuck.

No dice, so I decided to pause class and say it to everyone: hey all! I noticed that the tricky thing isn’t just the calculations, but trying to keep track of what you’ve figured out and what you still need to work on. Diagrams can help, here’s a diagram, here’s how you can use it, you might try this.

As I walked around some more, I poked around to see if pairs had adopted my suggestion. No dice, still.

Bell rings, kids hand in their work, that’s that for the day.

The next day, I start class by saying, “I noticed a lot of us got stuck on the problem yesterday. We’re going to keep on working today, but here’s something that might help: here’s a diagram, etc.”

Dice!

What happened? Hard to know, of course, but here’s what I’m thinking: the first time around, my kids had a million mental distractions. Some were wondering if their calculations were right. Others were just trying to get a grip on a plan of attack for the problem. Others were trying to remember where on their page they had written their current tally of the stickies on the front and back.

In other words, these kids had a lot to think about during this problem, and they weren’t really able to dedicate the brain space needed to understand a new and unfamiliar strategy.

This is also how I make sense of something I’ve noticed in my Algebra 1 class. I haven’t yet given these kids activities that explicitly address the “cover-up” method for solving equations, but I keep trying to bring it up when kids ask me for help with equations in class. The thing is, it never seems to stick.

It seems to me that if we think “just-in-time” instruction works particularly well, my kids should be able to hold onto this method a bit better than they currently do. After all, they have a clearly felt need for some new bit of math (they called me over, right?) and they are getting the instruction during their felt moment of need. Super-duper effective setting for instruction, right?

But then it doesn’t stick. And I think it’s for the same reason that my 4th Graders didn’t take up the “draw a picture” suggestion: they’re too mentally distracted to really focus on the new idea and properly learn it. After all, learning a new idea in all its proper generality can be a pretty heady bit of work. When my kids call me over for help with their equations, they’re potentially thinking about many other mathematical things — where am I in the problem? did I make a mistake by subtracting? what’s 4 divided by 6? — and often can’t focus on the strategy itself.

This, then, is a sort of dissent against the Three Act model of instruction. New mathematical ideas are not best introduced in the middle of a problem if they’re going to get the mental real estate they deserve. Students are often productively distracted by a difficult problem, and unable to focus on the strategy or tool at hand.

The thing that works better, in my experience, is following up a tough experience with a new idea or tool. This seems to me closer to ideal. The students get to spend of time struggling with a tough problem, which I think is valuable all on its own; they thoroughly understand the problem context, since they spent careful time on it; when I introduce a new idea after this experience, they are in a strong position to focus on this tricky new idea itself rather than the million other things it takes to comprehend this new tool.

As Dan Schwartz writes:

This report is based upon work supported by the National Science Foundation under REC Grant 0196238.

OK fine, but he also writes:

Instruction that allows students to generate imperfect solutions can be effective for future learning.

But instruction that comes in the heat of the moment is not looking towards the future — it’s coming during the chaotic present, a time when the student’s mind is being bombarded with many tricky ideas that are specific to a particular problem context. I don’t think that’s a great time to introduce a new idea, but tomorrow might be.

How the Pentagon Has Just One Line of Symmetry

The homework for 3rd Graders was to draw as many lines of symmetry as they could for a regular pentagon. I thought they knew their way around this stuff — they’d had a lot of success working with a cut-out hexagon the week before, and had been getting a bunch of practice —  but I was surprised by the huge variety of responses in their homework, so I decided to figure out what was going on.

I figured we could settle this by having a whole-group conversation. Probably, the kids who found less than five lines stopped looking too early. If I shared a variety of work, I bet that kids would quickly recognize that these other lines split the pentagon into two congruent pieces. (In 3rd Grade speak, this is criterion is usually expressed as “you could fold it over perfectly.”)

It didn’t go down like that. I noticed that some of the kids who saw 2 or 3 lines switched positions during the early part of the discussion, but the 1s and 5s weren’t budging.

F and I offered (what I thought was) a pretty convincing argument against the possibility of there being only one line of symmetry. After all, they argued, if you think that the vertical line of symmetry works, just rotate the pentagon a bit and you have a new “top” of the shape to split in half.

It was around now — especially when T shared — that I started to form a theory about what the “one line” camp was thinking. Their responses to the arguments made me think that they didn’t see the regular pentagon as actually having this rotational symmetry. They saw it as irregular, in some way, so that these other lines wouldn’t actually fold over perfectly.

“Why don’t we cut them out?” R asked. Perfect, I said. I was in for a surprise!

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With the cut out pentagons, you could just see that the rotations returned the pentagon to its original shape, and made room for a new “vertical” line of symmetry. And, for sure, this helped some kids.

But Bo and Annie [pseudonyms] called me over, because they were caught in argument and couldn’t make progress.

Annie: “I know that you can rotate it, but there’s still only one line.”

What?

Annie: “I’m saying that you can draw that line if you rotate it, but it’s the same line?”

Do you mean that it wouldn’t be a new line?

Annie: “No, it would be a different line on the page, but I’m saying that it would still be the same line.”

Hold on, I think I’ve got you. Are you saying, look, you could draw a new line if you rotate it, but that’s not actually a different line, since it’s still basically that same, vertical line that you started with?

Annie: “Yes!”

Huh. So if you have all these different lines, you wouldn’t say that there are five lines of symmetry or anything?

Annie: “No, because they’re all the same.”

At the moment that I understood her, I no longer knew what to say. I’m used to thinking of all sorts of mathematical similarity and difference — congruence, modular congruence, similarity, equivalence, symmetry — but I hadn’t ever thought of mathematical sameness quite in the way that Annie did.

She’s saying (what is she saying, exactly?) that these lines aren’t different because they’re intellectually similar. There is, for her, just one line of symmetry because there’s basically only one idea of a line of symmetry. Sure, you could repeat that process five times, but that’s not the point. The point is that you get a line of symmetry by connecting a point and its opposite side.

And maybe she’s also saying that when you rotate the shape, you’re really making a new shape. That the regular pentagon is a brand new regular pentagon when you turn it in the air, and you have a new opportunity to draw that vertical line and divide it into parts.

I wonder what she’d say if she compared her pentagon thoughts to her hexagon results. And earlier in the week she had drawn a ton of lines of symmetry in a circle.

Maybe this has something to do with the unloveliness of the pentagon? Maybe pentagons feel different to her than these other shapes?

 

Are Blogs Better?

I saw one post come through my RSS reader over the past 24 hours or so that fits the bill, and it’s a lovely one from Tracy Zager:

We’ve all been thinking about what might help students get more comfortable switching back and forth between counting by tens and counting by ones. Today, Becky and I were talking with Debbie Nichols, who teaches 1st and 2nd grade. Together, we landed on the idea of passing out 10s and 1s – connected sticks of ten cubes and single cubes, base 10 rods and units, etc. – and then having a counting circle.

At first, author and reader both aren’t quite sure what’s going on with the kids and what will help. Then they try something, and even then we’re not quite sure what to make of it. Then the kids say something surprising and new and then we’ve gotten somewhere new. And the author opens up more questions than she answers, leaving room for anyone else to jump in:

Debbie’s planning to do the same thing with dimes and pennies on a different day. And, of course, we could give older kids multiples of ten and/or multiples of one. After doing it just four times, we noticed an increasing smoothness for some of the kids. They were noticing that they’d either move over or down on the hundreds chart.

Every other post I reviewed fell into these categories:

  • self-promotion, announcements
  • sharing a math problem
  • play-by-play of a lesson (“we started with this warm up…then kids worked on this question…then I asked this…then for homework…”)
  • polemic

All these other types of posts are valuable in their own way, but these days I put a special premium on posts like Tracy’s. I want more of them, please!

Davis vs. Walbesser on Research in Math Education

Robert Davis:

“In a society which has modernized agriculture, medicine, industrial production, communication, transportation, and even warfare as ours has done, it is compelling to ask why we have experienced such difficulty in making more satisfactory improvements in education…one is immediately struck by how often education has turned to agriculture, physics, biology, or medicine for suggestions concerning the best over-all conceptualization; how rarely has education turned to architects, novelists, concert pianists, dramatists, actors, clinical psychologists, painters, or psychoanalysts…In the belief that we are being scientifically objective and concerning ourselves with observable aspects of reality, we have often lapsed into counting (or otherwise recording) highly explicit events which are, in fact, too narrowly defined, and too unclearly related to the ideas inside the student’s head.

Henry Walbesser’s response:

“The effect is to question even the possibility of obtaining reliably observable data on human behavior. I find myself in disagreement with such a position…I would claim that we in mathematics education are only now beginning to attempt to be scientifically objective. To dismiss this possibility as “too explicit, too narrowly defined, or too unclearly related” to the learning is a naive position for a scholar such as Dr. Davis…Davis is asking that we replace the analytical tool of reliable observable behavior with the mystical construct of cognitive schema. As for myself, I find such a request completely unacceptable…It appears that [Davis’s] suggestion for action is one which resembles alchemy in its tactics, not experimental research. I find such a suggestion ridiculous and such a course of action without value for researchers.”

Cited in From Amateur to Professional: The Emergence and Maturation of the US Mathematics Education Research Community. All this is in 1967, a year the authors mark as the beginning of the emergence of the math edu research as a  distinctive community.

“I was working on the wrong map to begin with.”

Some people I interviewed wanted me to say I was sorry–I am and I did. Some people wanted me to say that I remembered –I did and I did not. And some people wanted me to say it was all a mistake–it was and it was not. It felt less like journalism than archaeology, a job that required shovels and axes, hacking my way into dark, little-used passages and feeling my way around, finding other pieces that did not fit, and figuring out that I was working on the wrong map to begin with. It would prove to be an enlightening and sickening enterprise, a new frontier in the annals of self-involvement. I would show up at the doorsteps of people I had not seen in two decades and ask them to explain myself to me. This is what they told me.

From The Night of the Gun by David Carr.

Here is the definition of boring: unchanging, inert, here this is and here it always has been. The following is what I know: this is what I know.

I’m so sick of reading boring writing about teaching. This is an inherently popular position to take, but I think I can be specific about the criticism. What made Carr’s memoir so much fun to read was his framing the work as “archaeology,” as his attempt to find the map of his own drug-addled life.

There’s a distinction to be made here. I’m not saying that what made Carr’s book interesting was that he exposed his journalistic practice. If his project took shovels and axes, those shovels and axes were mostly in operation off-screen. This book is not a “how-to” for journalistic investigations.

Instead, what made this so much fun was the real sense at every stage that we might learn something new. His priority in the first few pages of the book are not to establish his privileged authority — precisely the opposite! He makes quick work of his authorial knowledge, establishing the unreliability of his memory in the first chapter. Once he does that, anything can happen. We can be surprised!

Even in my favorite books about teaching math, surprise is dead from page 1. They’re all the same, more or less: here’s what some people do, here’s why it’s wrong, here’s what we know, here’s how to do it. Bor-ing.

(Some books have this cliche thing they do in the first pages: “When I started teaching, I used to do Mad Minutes while I blasted Megadeth from a stereo. Then I learned about progressive-based discovquery learning.” This is entirely different than what I’m talking about.)

I don’t know why the literature has to be this way. I don’t know why teachers aren’t demanding better reading. The only explanation for all this boring writing I can come up with is a depressing one — that people don’t really read these things anyway, or at least they don’t buy them to read them. Maybe we buy books for those little activity ideas, or to look nice on a shelf, or maybe someone else buys our books for us.

It didn’t have to be this way. I’ve been flipping through Benchara Branford lately. I never know what to expect when he’s talking about his classroom. Magdalene Lampert is not a prose superstar, but her writing is alive in the way most professional writing I see is not. (Are blogs better? Some are, definitely.)

Fewer books that sound like powerpoint; more archaeology, please!

“The aim is much humbler.”

In attempting to help others by an account of one’s own failures and successes, one runs the risk of assuming an appearance of authority and even dogmatism. Especially is this likely to be the case where brevity is essential. If I am unlucky enough at times to appear in this objectionable guise, I sincerely assure the reader it is merely appearance. I have myself fallen into too many pitfalls in the path of the teacher to feel at all inclined to domatize; yet, as I believe that (with much effort!) I have climbed out again, I may hope to warn others of their existence, and, if some still struggle in the pits, I may perchance help them to get out. No attempt is made to manufacture an infallible specific for perfecting mathematical education: the aim is much humbler. I hope that this account will lead others to test the value of my suggestions. I should like, too, to see others relating their experience. With a large amount of evidence thus collected from teachers of all ages and kinds of experience, there would be reasonable hope of deducing therefrom a body of principles, bearing up the teaching of mathematics, which might really merit the title of educational science.

From: Benchara Branford, A Study of Mathematical Education (1908)

Teacher research. Why didn’t this happen?

 

Three Minus One

On the first day back from summer, the math teachers were eating breakfast together at school and talking about kids. It was my first year teaching eighth grade, so I slide my roster across the table to Mr. B, who taught a group of seventh graders that drove him nuts last year.

“Watch out for S,” Mr. B said. “He lacks the capacity to sit still and listen.”

I wouldn’t go that far, but I know what he means. And S was having a hard time keeping up with some of the material from class, so I’ve been meeting with him once a week, during one of my planning periods.

S has a homework assignment about negative numbers. The kids of B-period had come in with a huge mess of ideas about negative numbers. I’ve made some progress with a bunch of them over the past few months, but not really with S, whose thinking still shows the magnetic tug of ways of thinking that have so far resisted my efforts to expose. Maybe his ideas aren’t stable enough to build on?

During our last meeting, S was working on a homework assignment I had given his class. (Was I surprised that it was a bit hard for them? No.) Here was the work:

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S: I had a hard time with these.

Me: Awesome. I think it’s often helpful to be more specific when you’re asking for help.

S: Like, I just didn’t get these, exactly. Wasn’t sure how to think about them.

Me: OK good, but can you be more particular? Which question? Did you know what the question was asking?

S: It was just all sort of hard.

And so on. Eventually, he asks about the first question.

S: So, for this one, is it asking to like pick which one of these is x?

No, no, that’s not what it means. Here’s what it means.

S: Ah, OK, got it that makes a lot of sense. So if x is 1, then this is 3 -1. And that’s -2, right?

I drag out a blank pause. I’m hoping he sorts this out on his own, because I am without ideas if he can’t.

S: Is it -2? Is it 2?

Here I jump in. I worry that he’s holding the problem in his head and trying to think about it too, so I write it down for him on his page: 3 – 1.

S: Ah, OK, it’s -2.

I pause again. In that pause, I feel a sort of sadness. Though it’s hard to identify in the moment, later I come to think that it’s sadness for the sort of chaos that S is showing in his world at this moment. What does it feel like to call into doubt something as basic to thinking as three minus one? It reminds me of the way reality slips away in so much of Phillip K. Dick’s writing:

“He felt all at once like an ineffectual moth, fluttering at the windowpane of reality, dimly seeing it from outside.”

S continued to flutter around three minus one. If three minus one was up for reconsideration, what else wasn’t, in that moment? I wonder if S had something in common with Descartes’ skeptic:

“So serious are the doubts into which I have been thrown as a result of yesterday’s meditation that I can neither put them out of my mind nor see any way of resolving them. It feels as if I have fallen unexpectedly in a deep whirlpool which tumbles me around so that I can neither stand on the bottom nor swim up to the top.”

I remember, in this moment, wondering whether I should talk or listen as S worked his way around 3-1. Would the experience of falling back to the correct answer be helpful for him, a sort of anchor to help him escape whirlpools to come? Or would the experience of hearing my instruction be more grounding? Would it be less embarrassing? Was he embarrassed?

I don’t know any of this, and I don’t think I ever will.

S: Oh wait it’s 2.

I asked him what helped him decide, but he was inarticulate. I offered the suggestion that thinking about other subtraction that he knows might be helpful when in this sort of doubt, but I’m sure he did this while trying to find his way. We moved on — there was more to talk about, and he led the way forward.

It is not hard for me to imagine being unmoored in the way S was. What do I know, after all, about learning and teaching? What did I know in that moment? All my beliefs were subject to revision, I had no confidence about my perceptions of S’s reality. This might be the key for better understanding what it’s like to be unsure of something as fundamental as three minus one. Because if I can’t be sure of how a child I see daily was thinking about three minus one, then what do I really know at all?