Geometry Labs + Which One Doesn’t Belong

I love Henri Picciotto’s Geometry Labs text. I was preparing my geometry class for his inscribed angles activity, and saw this:

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Thanks to the Which One Doesn’t Belong people (and Christopher’s lovely book), I’m no longer able to look at sets of four things. It’s ruined me. I’m always deciding which of them is the odd one out.

Since there are subtle differences between the inscribed angle cases, I decided to cover up the words and ask my students which of the four diagrams was the weird one.

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This drew attention to the location of the centers, the location of radii, and the presence of isosceles/scalene triangles. (I know it’s May, but any chance to get kids to practice using this vocabulary is time well spent.)

This week in 4th Grade I’ve also been using Geometry Labs‘s chapter on tilings. (Sort of a random topic, but random topics are fun. Plus, I need to figure out where we stand on multiplication/division before one last push in our last weeks together.)

There I was, trying to figure out how to attune kids to the subtle classification differences between these two square tilings…

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…and while, admittedly, I clearly had “Which One Doesn’t Belong” on my mind, it seemed a pretty good fit for my need here too. I took out some pattern blocks and snapped a picture:

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There were lots of interesting aspects of this discussion, though my favorite had to do with whether the top-left and bottom-right tilings were different. I forget if we’ve talked about congruence yet in this class, but there were a lot of good ideas about whether tilting the tessellation made it a different tiling.

Not much else to share here, but I guess I’d say that I do this a lot. I don’t rewrite texts or worksheets or whatever very often. More often I add little activities before or after, to make sure kids can understand the activity, or to react to their thinking. That’s good for me (because I don’t have time to remake everything) and good for kids too (I write crappy curriculum).

What is it that I do?

I read a lot of teacher blogs these days.

(Incidentally, I turned MTBoS_Blogbot into an RSS feed, which was my reason for begging Lusto to make it in the first place.)

Anyway, I read a lot of teacher blogs. I see your beautiful activities, clever games and meaningful conversations. I wish I had an ounce of the teacherly creativity that Sarah Carter has, but really I don’t. It’s not what I do.

So, what exactly is it that I do?

In 8th Grade we’re going to study exponential functions. Class began with a lovely Desmos activity. They worked with randomly assigned partners.

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After thinking through these questions, I thought kids could begin learning about equations for exponential functions, and towards this it would be helpful to contrast linear table/equations with exponential ones.

In years past, I would have aimed to elicit these ideas out of a conversation. I’ve lost faith in this move, though. While it’s nice to get kids to share ideas, their explanations are often muddy and don’t do much for kids who don’t already see the point. (Just because a kid can say something doesn’t mean that they should.) This, at least, is what I suspect.

Better, I’ve come to believe, to follow-up an activity like this one with briefly and directly presenting students with the new idea. I worry more about visual presentation than I used to. Here is what I planned to write on the board, from my planning notebook:

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I put this on the board, so that it would be ready after the kids finished the Desmos activity: what could the equations of each of these relationships be? boom, here they are:

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Spot the differences between this and my plan! They were all on purpose.

During planning I hadn’t fully thought through what I was going to ask kids to do with this visual. At first, I stumbled. I gave an explanation along with the visual, but I got vibes that kids weren’t really thinking carefully about the equations yet. So I asked them to talk to their partners for a minute to make sure they both understood where the exponential equation came from.

You can tell when a question like that falls flat. There wasn’t that pleasant hum of hard-thinking in the classroom, and the conversations I overheard were superficial.

Remembering the way Algebra by Example (via CLT) uses example/problem pairs, I quickly put a new question on the board. I posted an exponentially growing table and asked students to find an equation that could fit this relationship.

There we were! This question got that nice hum of thinking going.

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The equation wasn’t there, originally, duh.

While eavesdropping on kids, I heard that L had a correct equation. I thought it would be good to ask L to present her response, as she isn’t one of the “regular customers.”

Her explanation, I thought, gave a great glimpse of how learning works. She shared her equation but immediately doubted it — she wasn’t sure if it worked for (0,5). After some encouragement from classmates she realized that it would work. Turns out that her thought process went like this: 10, 100, 1000, that’s powers of 10 and this looks a lot like that. But how can I get those 5s to show up in there too…ah! The example involved multiplication so this one can too.

(Of course, she didn’t say this in so many words. After class I complimented her on the explanation and she put herself down: I don’t know how to explain things. I told her that learning new stuff is like that — your mind outpaces your mouth — but I thought I had understood her, and confirmed that I got her process.)

With the example properly studied, I went on to another activity. Following my text, the next twist was to bring up compound interest. I worried, though, that my students would hardly understand the compound interest scenario well enough to learn something from attacking a particular problem.

While thinking about this during planning, I thought about Brian’s numberless word problems. (My understanding of numberless problems is, in turn, influenced by my understanding of goal-free problems in CLT.)

I took the example problem from my text ($600 investment, 7% interest/year, how much money do you have in 10 years?), erased the numbers and put the variables on the board.

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Then, I asked kids (again with the partners) to come up with some numbers, and a question. If you come up with a question, try to answer it. (A kid asking But I can’t think of a question is why this activity was worth it. And with some more thought, they could.)

I collected their work from this numberless interest problem, and I have it in front of me now. I see some interesting things I didn’t catch during class. Like the kid who asked ‘How much $ does someone lose from interest after 5 years?’ (And why would an 8th Grader know what interest is, anyway?) Or the kids who thought a 10% interest rate would take $100 to $180 over 8 years.

No indications from this work that anyone uses multiplication by 1.10 or 1.08 or whatever to find interest. Not surprising, but I had forgotten that this would be a big deal for this group.

For a moment I’m tempted to give my class feedback on their work…but then I remember that I can also just design a short whole-group learning activity instead, so why bother with the written feedback at all.

I’m not exactly sure what ideas in the student work would be good to pick up on. I should probably advance their ability to use decimals to talk about percent increase, but then again there was also that kid who wasn’t sure what interest was.

My mind goes to mental math. I could create a string of problems that use the new, exponential structure with decimals:

  • 600 x 1.5
  • 600 x 1.5 x 1.5
  • 1000 x 1.5^3
  • 50% interest on a $200 investment

That’s awfully sloppy, but it’s just a first draft.

Or maybe the way to go is a Connecting Representations activity that asks kids to match exponential expressions with interest word problems.

I’m not sure, but all this is definitely a good example of what I do. It’s what I’m learning how to do better in teaching, at the moment. It’s not fancy or flashy, and no one’s lining up to give me 20k for it, but it’s definitely representative of where I am now.

I’m not sure at all how to generalize or describe what it is this is an example of, though. Is it the particular flow of the 45-minute session that I’m learning to manage? Or is it the particular activity structures that I happen to have gathered in my repertoire?

None of those are satisfying answers. Maybe, instead, this is just an example of me basically doing what I think I should be doing. My reading is piling up, and I’m getting some coherent ideas about how learning and teaching can work. This lesson is a good example of how those principles more-or-less look in action. It might not be right (and it sure isn’t at the upper limits of what math class can be) but I’ve got a decent reason for most of the decisions I made in this session.

I think what I have to share, then, is how what I’m reading connects to how I’m teaching. This episode is an example of that.

Michael Disagrees With Tweets

In what may or may not begin a new series on this blog, I will now (politely and lovingly, I hope) disagree with a tweet.

On the internet, nobody knows if you can manage a classroom or not. Maybe twitter can solve this. Currently, you get a blue “verified account” check next to your name if you did something cool to deserve it, like being rich or popular. Maybe we could have something like that in education. (I’m a verified red apple educator!)

Until then, there’s no way to tell online who can or can’t run a classroom.

I suppose it’s true that someone who has never run a classroom probably can’t, and these people shouldn’t try to tell you about managing behavior. But take Tom. I don’t know Tom. I have no idea what sort of a teacher he was when he was in the classroom. How would Tom’s standard apply to Tom? How can I know if Tom can run a classroom or not?

This is always how it is with teaching. We don’t have access to each other’s classrooms, so we can only rely on each other’s descriptions of teaching. That’s true for everybody, teachers and non-teachers alike.

This matters a lot more to ex-teachers than to teachers, I think. The relationship between teachers and non-teachers is complicated. You might think that teachers are just suspicious of non-teachers, and that’s true, but we also care the most about what some non-teachers say. Someone on twitter once pointed out to me that classroom teachers are generally suspicious of non-teachers but very trusting of a few chosen non-teacher experts who have credibility. This struck me as totally true.

As a consequence of all this, some non-teachers find it helpful to try to hold on to the status of in-the-know teacher even though they have left the classroom.

To which I say, it’s not worth it. Don’t bother. The kindness that teachers offer other teachers isn’t because of a presumption that this other teacher gets it, or that they have useful information to offer that non-teachers don’t. Rather, I’d say, it’s just that: kindness. I would posit that it’s not that teachers are more trusting of others in the classroom, just that we try to be nice to each other, because the job is hard and knowledge is tentative and we all know how little status we each have. Once you leave the classroom your status has just bumped up in the education world, and that extra-kindness can no longer protect you from the skepticism of other teachers.

Which is fine, because you can still influence teachers in the one way you ever could: by describing what it is that you think will work.

Mental Math Gone Wrong?

Maybe this was a good idea, maybe not.

I was trying to figure out how to start class. My 8th Graders have been studying the Pythagorean Theorem. I knew I wanted to start with some mental math* but wasn’t sure how to start.

This desire to often begin class with some mental math is, at this point, sort of an instinct. On the one hand you need instincts when you’re planning class, because otherwise everything takes forever as you get sucked into a recursive vortex of decision-making. But is it a good instinct? I don’t know how to think about that.

The way I teach the Pythagorean Theorem, being able to mentally chunk a tilted square into triangles and squares (rather than trying to count each square or triangle) is an important part of the skill. It helps kids quickly see the area of squares, freeing up their attention to focus on the relationship between the squares built on the sides of triangles.

Yesterday, we explicitly talked about the Pythagorean Theorem in terms of the area of squares built on a right triangle’s sides. The plan for class was for kids to get better at using it in all sorts of different problems.

So, I decided to build a string of squares built on the hypotenuses of right triangles, and ask kids to find the square-areas in sequence, building up to a generalization. We start: What’s this square’s area? Put a thumb up (please don’t wave a hand in someone else’s face) when you’ve decided. What is the area? How do you know? OK here’s your next tilted square, etc.

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Sloppy picture so you know it’s the real deal

Here’s where my teaching got sort of mushy. The really important skill isn’t finding the area of tilted squares. What kids really are going to want to know, later on, is the Pythagorean relationship between right triangle sides and areas.

So here’s the question: did this string of problems draw attention to the important math?

Turns out, it didn’t. Kids made the generalization in the last step (as far as I could tell from eavesdropping on their conversations) entirely on the basis of the earlier examples. And those areas were found by chunking up the area. In other words, this was arithmetic-generalization. They didn’t use the Pythagorean relationship.

What were my options, when I realized this? I was happy that kids were able to mentally dissect these tilted-squares, but was a bit disappointed that they didn’t start noticing Pythagorus here. I lost a chance to help them try out using that relationship. Since the rest of the class was designed to help them practice this theorem, it became important for me to prompt their memory of it at the start.

What can you do, right? Impossible to predict kids perfectly. Except that I could have prompted the Pythagorean relationship after the first example didn’t go the way I expected it to. I could have said — after I made sure that students were not going to — that this tilted square’s area could be found using Pythagorus, and then I’m sure I would have gotten more kids to play out this relationship in their minds for the rest of the string.

That’s not what happened, though, so I weakly finished the string with my own personal observation that, hey, we could’ve used PT here. The kids shrugged. OK. I pulled out a quick problem that did prompt kids to use the Pythagorean Theorem, but by then I’m not sure I had everybody on board. We finish, and kids are getting jittery. We’ve used up* whatever whole-group learning time we were going to get at the start of class, so I started problem-solving time.

That’s definitely how I see things right now, at least. Again, I don’t know if this instinct is a good one.

Class went OK after that. But I’m still trying to figure out whether I did this right. Should I have designed the initial string differently? Should I have reacted differently?

(And, the sort of meta-question I have is what exactly it would mean for me — or any teacher — to know how to do this better. Where does that knowledge come from? Can it be shared?)

Two Ideas

Everybody should do more mental math — mental algebra, geometry, calculus, topology, whatever — at pretty much every level of math.

Whenever you’re tempted to write comments on a student’s paper, just pick out one common issue from the class’ work and start class responding to that, somehow.

 

Misery?

Stephen King taught high school English for two years:

I wasn’t having much success with my own writing, either. Horror, science fiction, and crime stories in the men’s magazines were being replaced by increasingly graphic tales of sex. That was part of the trouble, but not all of it. The bigger deal was that, for the first time in my life, writing was hard. The problem was the teaching. I liked me coworkers and loved the kids — even the Beavis and Butt-Head types in Living with English could be interesting — but by most Friday afternoons I felt as if I’d spent the week with jumper cables clamped to my brain. If I ever came close to despairing about my future as a writer, it was then.

“Jumper cables clamped to my brain.” I totally believe and experience this. It’s an obvious fact of my life…but is it true? Why would it be?

My work isn’t as intellectually involved as e.g. being a grad student, researcher, journalist, etc. Teachers don’t have to regularly learn new facts or disciplines. We don’t make our living as readers, writers or thinkers.

We don’t even work especially long hours. Yes, yes, endless grading. But even taking grading into account, it’s unclear to me how many extra hours we actually put in. I know, personally, that I tend to way over-estimate my out-of-work hours. I tend to count all sorts of quasi-work into the bucket, like all that time that I’m thinking about looking at student work but instead I’m writing a blog post on a Sunday night.

Grading and planning are like (to get back to King) little evil vampire children that rap on the window while we’re catching a break after a long day. Let us in, they say, you need to.

(I just finished reading Salem’s Lot which stars Matt Burke, veteran teacher, which is how I ended up down this road.)

The Bureau of Labor Services surveyed teachers and instead of asking how many hours they worked in a week, asked them how many they worked yesterday. (This includes out of school work.) The stunning results: responses amount to just under a forty-hour work week. We even, on average in this survey, work less on the weekends than a comparison groups that contains health care professionals, business and financial operations professionals, architects and engineers, community and social services professionals, managers.

I’m inclined to believe the more modest hour-estimates of the BLS, as they fit what I see in myself and colleagues at the different places I’ve taught. (I’d also say that, in the places I’ve taught, there are outlier teachers who just go nuts with work. If the BLS stuff doesn’t fit your picture, you might be such a teacher.)

But I’m also inclined to think that King’s brain wouldn’t be depleted if he were a journalist or a researcher or a bond-trader who worked till ten every night.

(Speaking of work hours and exhaustion: I read King’s Under the Dome while working fifteen-hour work days as a delivery truck driver in my summer after graduation. I’ve never been as desperate for a book as I was while working that job. That book took over my life while I was working — my wife [then girlfriend] still teases me about it. The job involved driving around campus, picking up and dropping off recycled furniture, which sort of wore me out. I was physically exhausted but mentally starving and I’d collapse in bed with that book for some of the most satisfying hours of reading I’ve experienced in my life.)

What could there be about teaching that makes it mentally exhausting? Or is this just standard working-adult exhaustion?

I can’t think of anything, which makes me wonder if Steve and I are making this up.

One thing I know about teaching, though, is that you rarely know if you’ve taught well. And that fits with what I know about writing — that it’s lonely work, done in a quiet space over long-periods of time. Unlike teaching, there are certain stone-cold ways of knowing that you’ve done good work — the acclaim of readers — but the lead-up to that moment (if it ever arrives) is the ultimate marshmallow test.

Maybe this is it: teaching exhausts us in a way that kills our willingness to write.

Beyond “Beyond Explaining”

There’s a good reason why educators often talk about the need to move beyond explanations. People who don’t know much about teaching think all the action in teaching is about the clarity of the explanation. (That, and getting kids to listen to your ultra-clear explanation.)

There’s much more to the job than that, of course. Michael Fenton puts this nicely in a recent post:

In my first few years in the classroom, I held the notion that the best way to improve as a teacher was to hone my explaining skills. I figured that if I could explain things more clearly, then my students would learn more. […]

The best way to grow as a teacher is to develop my capacity to listen, to hear, to understand. […] This doesn’t mean that I’ll stop working on those other skills. But it does mean I have a new passion for learning about listening—really listening—to students and their thinking.

I think this focus on listening is wonderful, and Michael did say that he’s going to keep working on his other skills, which is a nuanced take. But what about the title of the post, “Beyond Explaining, Beyond Engaging”? Philosopher Eric Schwitzgebel has a helpful distinction between a piece’s headline view and its nuanced view:

Here’s what I think the typical reader — including the typical academic reader — recalls from their reading, two weeks later: one sentence. […] As an author, you are responsible for both the headline view and the nuanced view. Likewise, as a critic, I believe it’s fair to target the headline view as long as one also acknowledges the nuance beneath.

So let’s take on that headline: should we go “beyond explaining”? If we’re trying to improve our teaching, it could be that getting better at listening has a higher payoff than getting better at explaining. But my experience has been that there isn’t any strict hierarchy of payoffs in teaching. Teaching evolves in funny ways. Last year I taught an 8th Grade class that pushed on my classroom management. This year I’ve spent a good deal of time learning how to tutor students with learning disabilities. I’d hate to say that explaining is some sort of basic teaching skill, the sort of thing novices focus on but more experienced teachers don’t need. Teaching is weirder, more cyclical, more web-like than that.

Maybe, though, we should move beyond explaining because it’s easy — or because pretty much everybody knows how to do it well after their first two years on the job.

That might be true, for all I know. If I doubt it, it’s only because it’s only over the past year that I’ve really started to understand some of the things that make a good explanation hum and lead to great student thinking, instead of slack-eyed drooling from the back rows of the classroom.

Besides, a lot of what I’ve learned about explaining comes from outside math education writers and speakers. Which started me thinking that maybe this knowledge (if it’s even true) isn’t as well known to math teachers as it could be.

Here’s what I think I know about giving good explanations to kids:

1. Study Complete Mathematical Thoughts; Don’t “Roll Them Out”

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This isn’t really an explanation, but it’s a complete example. Source: Algebra by Example

The first students I taught were subject to an especially painful type of instruction. I don’t know exactly how to describe it. Maybe an analogy would help. Imagine a magician (Ze Great Pershan-o) who is waaaay too detail-oriented: OK AND NOW CHECK IT OUT: I PUT MY HAND IN THE HAT! AND NEXT, I CLENCH MY FIST! HERE COMES THAT HAND SLOWLY COMING OUT OF THE HAT! ETC!

This is basically also how my explanations worked. HERE IS AN EQUATION! (WRITES AN EQUATION.) WE WANT TO SOLVE FOR X! (WRITES ‘X = ?’) HOW SHOULD WE DO THIS, CLASS? WELL, WE COULD DIVIDE BY 2. BUT THAT WOULDN’T HELP US VERY MUCH. LET’S INSTEAD SUBTRACT 2, ETC!

The phony enthusiasm was a problem. Another problem was that I was feeding the math one mini-idea at a time rather than presenting them the complete mathematical thought. I’ve come to think that when we do this — when we roll out the explanation, line by line — we lose a lot of kids.

Do you know that thing you do when you’re trying to understand something hard in a math textbook? How you put one finger at the top of the explanation, and then go line-by-line to make sure you understand each piece? But then you go back and try to make sense of the whole? That sort of self-explanation is where the learning can come from in an explanation, I think, and if we roll the explanation out, we’re making it harder for kids to look at how the pieces fit together.

So, when we’re ready to explain an idea to students, we ought to be offering them a complete mathematical thought. No need to dice it down to the atomic level, like Dumb Houdini or whatever.

Practically, this means that if I intend to “show the steps” in an explanation, I make a real effort to reveal them all at once. I project or photocopy artifacts like the ones that Algebra by Example create. If I’m working one-on-one, I’ll scribble a full example down on the page, rather than coaching a kid through that example step-by-step.

This idea, of course, isn’t my own. I came to understand this from reading about cognitive science, and especially cognitive load theory. There’s more to math than explaining, but there are ways of explaining things that preserve the math and others that make it harder for students to make meaning. Fully worked-out examples can help kids make meaning from our explanations, I think.

2. Use Arrows to Emphasize Process, Change and Action

This was an aha moment for me. I came to think people across education were saying similar things about how it’s easier for people to think about actions, rather than properties. You can hear this idea bouncing around research on how kids solve word problems, how our minds especially remember narratives, and what constitutes good writing.

Nobody has told me that these ideas are related. I’m a bit worried that I’m connecting totally unrelated people and ideas. Still, here’s the idea:

An interesting result of Cognitively Guided Instruction is that numerically equivalent word problems are often handled very differently by children. Consider these two problems:

Problem 1: Jill has 8 jellies, but a raccoon eats 5 of them. How many does she have left?

Problem 2: Jill has 8 jellies. A raccoon has 5 jellies. How many more jellies does Jill have?

Numerically equivalent problems — 8 minus 5 — but the first problem contains an action that is easy to represent. It would occur to a lot of kids that they could solve the first problem by dealing out 8 counters (or whatever) and then removing 5 off them from their pile. That’s directly modeling the action of Problem 1.

Problem 2 doesn’t contain an easy-to-notice action, so direct modeling would be less likely to occur to children. Carpenter and pals found that, in fact, kids didn’t use direct modeling strategies for Problem 2, and as a result Problem 2 was a bit trickier for kids to handle.

(This is 100% true, in my experience.)

Word problems with actions, essentially, contain stories that are easy for us to represent and understand. And stories are the sort of thing that our minds most easily grasp and symbolically represent. As Dan Willingham notes:

Research from the last 30 years shows that stories are indeed special. Stories are easy to comprehend and easy to remember, and that’s true not just because people pay close attention to stories; there is something inherent in the story format that makes them easy to understand and remember. Teachers can consider using the basic elements of story structure to organize lessons and introduce complicated material, even if they don’t plan to tell a story in class.

Incidentally, Tom Newkirk makes a similar observation about what constitutes good writing in his book Minds Made for Stories:

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Later, Newkirk argues that part of what makes good non-fiction writers good is they find subjects and actions to metaphorically represent abstract structures. They turn “evolution is a process whereby genes are randomly mutated” into “mutagens are constantly attacking our genetic material, altering it in ways that have the power to change the direction of an entire species” or whatever. Action, action, action.

(I’m also pretty sure this connects to Anna Sfard’s work on the way we tend to turn mathematical processes into mathematical objects but I’m not sure I have all the pieces put together yet.)

Back to Planet Classroom: What does this mean for my teaching? Practically, a lot of annotated arrows.

Instead of an example of simplifying expressions that looks like this

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…I try to recreate it with annotated arrows, to emphasize the actions involved:

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Our minds privilege stories, which means that they privilege change over inaction. If our explanations can include more doing things to things, this can help kids see what we mean a bit more easily.

For a lot of fun use of arrows to represent the actions in otherwise static representations, check out David Wees’ new tumblr for mathematical annotations. A prime example:

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Arrows — especially annotated arrows — can help transform examples (i.e. what correct work might look like) into explanations that help someone understand the examples.

3. Describe Mathematical Pictures, Though This is Harder Than You Might Expect

The What Works Clearinghouse (WWC) is this big federal initiative to try to sort through the evidence for various educational claims and give clear recommendations. The thing is, there is a ton of dissatisfaction with their standards for recommendations. Some people think their standards of evidence are weirdly strict. Others say they privilege large experimental or quasi-experimental studies over other forms of evidence.

Anyway, they have this report on helping students who struggle with math, and I like it. Their fifth recommendation is all about visuals:

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Why are visuals important? At least partly because words don’t distract you from pictures — you can pay attention to both at once. (Unlike reading a slide and hearing it explained to you, where the words interfere with each other. You might find yourself doubly distracted in that situation.)

This relates to dual-coding theory, a theory from cognitive science that deserves to be better know in math education. Like worked-out examples or the privileged role of narrative, it’s a legitimately useful bit of cognitive science to know.

If you’re looking to teach a strategy, describing a (complete!) mathematical picture (with arrows!) can help.

“So to figure out 7 minus 2, I started with 2 and counted up until I got to 7 and that was 5, so I figured out that 7 – 2 is 5.”

The thing, though, is that it’s very easy to mess this up. A “mathematical picture” is not “a bunch of written numbers or words.” That’s not describing a picture with words — that’s just using spoken words to describe written words. I don’t think that helps as much, according to dual-coding. Words can distract you from words.

I’m not saying that board work in the above tweet is bad at all. My point is just that these equations are going to compete for attention with any spoken explanation in a way that (dual-coding says) a picture wouldn’t. (Though check out those annotated arrows!)

A problem: when I think about it, there are a lot of mathematical topics that I can’t think of a good picture for. And even for ones where I can (e.g. the connection between area and multiplication) those visual representations aren’t obviously connected to their numerical ones to kids. Those connections need to be carefully taught. Ideally, they’re built into a curriculum.

I mean, it’s obvious to me that you can carve up a rectangle into 4 quarters and this represents multiplying (x + 3)(x +7), but anyone who’s tried knows that this isn’t obvious to high school math students.

So while it’s great to aim for “describing a mathematical picture” as an ideal for explanation, we’re limited by the mathematical pictures that kids understand and that we know.

I love learning new pictures for mathematical ideas. I feel like this year I really realized the power of the visual representation of the Pythagorean Theorem to help my kids understand its meaning and use:

But there are a lot of topics where I don’t know good visuals to go with the numbers, equations or words. I’d love if we could find more of them.

What Beyond Explanations Shouldn’t Mean

I could be wrong, but I don’t see a lot of people writing or thinking about how to give good mathematical explanations. This is despite the fact that the vast majority of teachers I talk to say they give explanations often, even if they are a bit embarrassed by this. (They shouldn’t be, I think.)

And the vast majority of thinkers and writing about teaching would tell me that they aren’t anti-explanation, just against a mono-focus on explaining as the core of teaching.

So why doesn’t anyone write about giving good mathematical explanations? Three possibilities I can think of.

  1. Unlike me, pretty much already knows how to give good mathematical explanations.
  2. People don’t think that improving our explanations is worth the ink. It’s a low pay-off instructional improvement.
  3. Anti-explanation ideals make it trickier to talk about improving our explanations.

I’m pretty sure it’s not Possibility 1. I think Possibility 2 sounds good, and Possibility 3 is a solid maybe and is anyway related to 2.

Either way, now you know what I’ve recently figured out about explaining stuff to kids:

  • Represent complete mathematical thoughts
  • Use annotated arrows to emphasize action and change in those representations
  • If possible, describe mathematical pictures

So, what’s next? Just last week I tried out a new representation of solving trig problems with my geometry students. It flopped:

But then I made a little tweak, and it went better. Which got me thinking: we’ve got this whole internet thing. Why aren’t people sharing more of these images? Is it less fun for us to share pictures of our own work? Does it seem self-promoting in a way that sharing other things (e.g. activities) doesn’t?

I’m not sure. But I think that this work is valuable, and is worth sharing. Explanations are nitty gritty, but it’s important nitty gritty.

Book Review: Schoolteacher

I.

Schoolteacher is a book that is referenced by pretty much every other scholarly book on education I’ve ever read. It’s a — the! — sociological study of k-12 teaching.

It’s also a book that I failed to read some three times before recently pushing myself through to the finish line.

It’s not that the book isn’t insightful. It is very insightful, the most revealing explanation of forces I’ve felt as a teacher that I’ve ever encountered. (More on that in a second.) It’s also not that the book is boring, or that it’s overly technical, and the dynamics that Dan Lortie describes aren’t even so complicated.

So I don’t really have a theory as to why I found it such a slog to get through my first three times.

What made the difference this last time was simple: I just skipped the boring parts. That took me to the second half of the book. That’s the part where Lortie starts including huge blocks of quotes from the teachers he interviewed, and these sections were much more fun to read:

Well, years ago I hit children. Of course I wouldn’t, you can’t do that now, but I have done that which I felt badly about. Recently I have taken work I didn’t like, I’ve just crumpled it up and threw it in the wastebasket. And afterward I have thought what a terrible thing to do because maybe that was the child’s best effort.

Yeah! Now we’re cooking.

So I read the second half that has all the interviews and then slogged back through that first half, though having a good deal of knowledge of Lortie’s argumentative strategy and goals.

Incidentally, reading this book for me felt eerily like reading Racism Without Racists, another sociology tract that I found impossibly boring until I skipped to the juicy quotes.

Sociologists, if you’re listening: don’t hide your quotes!

II.

Lortie begins with the history of the teaching profession in America. Except he’s not actually interested in telling this story, he just wants to point out that a lot of things haven’t changed since the beginning of teaching in America.

So, teaching begins with a single teacher in a schoolhouse; teaching still happens with a single teacher in a closed room. In the beginning, it wasn’t that hard to become a teacher; it’s still not that hard. Schools used to be managed locally, by citizens; they still are. Things more-or-less stay the same in teaching.

This line of argument has been ruined by would-be-reformers who say ridiculous things about “21st century learning” or “the out-dated Prussian factory model of teaching.” But Lortie has a good point. How much has teaching changed?

Put it this way: if a 19th century teacher had a time machine and traveled to 2017, how long would it take them to get the hang of modern teaching? The biggest issue we’d have would probably be cutting-out the beatings and extreme (for us) emotional cruelties and shaming.

Actually, Lortie has something interesting to say about this too:

During the latter half of the nineteenth century and the early decades of the twentieth, laws and school custom changed; increasingly sharp limits were placed on the teacher’s use of physical punishment. There is a paradox in this transformation of values and practices: the teacher’s use of physical coercion was limited at about the same time compulsory education became the rule.

The presumption that students attended school voluntarily became void just when teachers were forced maintain their authority through persuasion and other leadership qualities. Discipline took on a different coloration under such conditions: teachers had to learn how to ‘motivate’ students regardless of whether they or their parents wished them to be in school.

Granting the premise that teaching has changed less than your typical profession (e.g. farming), why hasn’t it?

Lortie calls this conservatism, and he argues that resistance to change built in to the profession at pretty much every level. Students go to school and decide to become teachers. If they hated school, though, they don’t become teachers. These school-loving dorks partially choose teaching because of the service element, love of schools, and flexible hours. (“Teachers are sensitive to criticism about this.”) Is this a recipe for radical employees who are eager to rewrite the rules of school? No, Lortie argues, this is a recipe for a profession of people who basically like teaching the way they were taught.

I wasn’t so convinced by this. Sure, teachers go into teaching because they like the job as it is. But isn’t this true of literally every other chosen profession? If you go into medicine presumably you tend to be ok with the way medicine is presently delivered to patients. Maybe the professions that should experience the most radical employees are the ones that aren’t chosen, then?

The other plank of conservatism is the lack of strong teacher preparation. We sort of slide into teaching, mostly, even if we go through traditional teacher training. That’s because teacher training doesn’t fundamentally alter our prior ideas about teaching. Lortie calls it [THIS IS THE PHRASE THAT EVERYONE CITES PERPETUALLY FROM SCHOOLTEACHER] the “apprenticeship of observation.”

This might be a good point to mention that the book was published in 1975. Lortie keeps on mentioning that school unions are just starting to collectively bargain and the Women’s Liberation Movement is shaking things up. If you don’t believe something that Lortie says, you can always dismiss it on the grounds that it’s outdated.

The inherent conservatism of teaching would be much more worrisome to me if I had the perspective of a reformer. I don’t, though. I’m much more interested in the lived experiences of teachers. So it was the second half of the book that interested me more.

III.

Before reading Schoolteacher I was familiar with the idea that teaching is a “special but shadowed” profession. (“Special but shadowed” is another catchy Lortie phrase.) To me, “special but shadowed” meant that in the eyes of outsiders teaching is both valuable and undesirable work. Lawyers have reputations for slime, traders for greed, but teachers are saintly in the eyes of others. But we’re also seen as refugees from more competitive jobs where we couldn’t cut it, and the money we make (good but not great) reflects this status.

(Interestingly, Lortie actually finds that a lot of his teachers are refugees from more competitive professions: “about a third of Five Towns [the district he interviewed] teachers reported that they had wanted to go into another line of work but were unable to do so because of external constraints.”)

It’s other people who see teaching as less-than. But after reading this book, what I came to think was that teachers see ourselves as doing special but shadowed work as well.

You can see this in the things teachers told Lortie in their interviews. Over and over again in these interviews, teachers reveal themselves to have work ideals that they aren’t able to live up to with any regularity.

Math teachers these days talk a lot about reaching every single student. Teachers in Lortie’s study shared these universalistic ideals too. To see how teacher expectations align with these ideals, Lortie asks a clever but indirect question — Please recall some occasion when you felt especially proud of something you achieved as a teacher. Please tell me about it.

If our expectations aligned with our ideals, a good day of teaching would be a day when the whole class understood something. But teachers, Lortie found, invariably mention successes with individual students rather than whole-group:

The most provocative difference between responses to the pride question and responses to other questions lies in the scope of outcomes claimed by the teacher. In speaking about their ideals, respondents emphasized reaching all students; some teachers, in fact, made such universality the focus of their answers. But the occasions associated with pride, in all but one instance, involved a single student or a small number of students. Pride, in short, is generated by ‘elitist’ outcomes, which are overtly rejected.

When Lortie asked teachers directly for their ultimate aims (I know it’s not easy to state clearly, but would you try to explain to me what you try most to achieve as a teacher? What are you really trying to do?), teachers talk about learning and achievement. But when he probed teacher expectations in indirect ways — Describe an outstanding teacher — teachers responded as if the best you could as a teacher is to nail the interpersonal element:

The elaborations elicited by direct questions concentrated on the ultimate outcomes of instruction, on learning changes in students. But when we ask teacher to describe outcomes achieved by outstanding colleagues, they emphasize results of a proximate and relational nature.

So on and so on. Name an ideal of the profession. How about “fulfilling each child’s potential?”

School systems often advertise their goals as including, for example, ‘the full realization of every child’s potential.’ It is clear that the aims of classroom teachers are less exalted.

I had always known that schools are placed in impossible positions by citizens and society. Schools are tasked with fixing democracy, creating citizens, Americanizing immigrants, training the workforce for everything and nothing, solving inequality, battling racism, being the ultimate solution to segregation and a million other problems that adults find intractable in larger society.

What I hadn’t considered was what this looks like for teachers. I had sort of imagined that teachers resist all of this, or at least that we understood what we can hope to achieve in our work.

What Lortie is saying is that teachers aren’t rejecting this rhetoric. We incorporate it in the form of our ideals and values. But we aren’t able to live up to them, because the form of the work makes living up to these values impossible.

IV.

You might wonder why teachers don’t just reject the crap rhetoric. Reach every child. Create a love for learning. Help children become curious people. If the lived reality of teaching is that — at best — it’s impossible to know if you’ve ever achieved any of these goals, shouldn’t the profession re-calibrate its expectations?

I think for Lortie the answer is about the way teachers are dependent on schools as institutions for our work. There are no independent classroom teachers, like doctors in private practice. Every teacher depends on being employed by a school, and every teacher has to suck it up because status-wise we’re controlled by administrators who are controlled by superintendents who are controlled by parents who are part of the same crazy democratic society that came up with all these unreasonable expectations in the first place. It’s a nice story, if one that I don’t know how to confirm.

In any event, though, here’s what we end up with: teachers are in professional tension. We have sky-high ideals and we also recognize that they’re rarely met.

My favorite chapter in the book describes this as “endemic uncertainty.” We never know if we’re doing a good job. The ideals are rarely fully achievable, so how do we know if we’re doing a good job? “Uncertainty is the lot of those who teach,” he says, and it’s a very good line.

How hard is it for teachers to figure out if we’re doing a good job? So hard that they had to change their questions about assessing outcomes because teachers kept on losing it and walking out of the interviews. Even with their revised question, prompting teachers to think about whether we’re achieving anything leads to some of the most depressing meditations possible on teaching:

I feel very inadequate and hopeless at times.

I do wonder, at the end of every single year, how much good have I done? And it’s hard to see.

You can go on for an eternity with nothing. They seem to be regressing.

This is all especially troubling for teachers because, for a lot of us, it’s the emotional rewards of teaching that mean the most to us.

A finding of Lortie’s is that teachers are focused on the present. The future and the past provide less consolation to us than they might be in other professions. We can’t look forward to that big raise, and we can’t try for lasting achievements. A lot of us won’t last a career in the classroom. As for the past, because of how easy it is to become a teacher and how uncertain we are of ever having achieved anything, we’re also not inclined to feel especially good about what we’ve already done. That puts a lot of pressure on the now. And right now, all we have is the kids in our room. When it’s going great we feel good. When it’s going badly we feel like crap. The past doesn’t matter, neither does the future. We’re present-oriented, and that means we’re chasing the rewards of the present.

But because of the endemic uncertainties of teaching, those rewards are rare.

V.

This was the new, big idea I got while reading Lortie: teachers tend to be divided on ourselves. Our ideals and our expectations are majorly out of whack. And, if you believe Andy Hargreaves, this causes teachers to experience a tremendous amount of guilt.

This idea is one that I think helps me understand a bunch of things that lately I’ve been finding puzzling about teaching.

WARNING: This is just me making stuff up.

Teaching Puzzle #1: Teachers often talk like educational progressives, but teach like traditionalists. This is weird, when you think about it. Why talk about what “we discovered” in class if what actually happened was you asked a bunch of leading questions and one kid answered them?

This makes sense, though, because teacher ideals are different than teacher expectations. We should expect teachers to talk in the language of our ideals, especially when we’re around administrators, coaches or parents that represent those ideals.

Teaching Puzzle #2: Traditionalists about teaching make a big deal out of their differences with progressives, even though their classroom practices seem pretty indistinguishable from progressive practices.

But this makes sense too. Traditionalists reject the ideals of teaching and the rhetoric that results from those ideals. For example, you sometimes get people debating whether every child can do math at a high level is true or not. People who think that it’s really important to reject this are trying to free themselves of the oppressive, guilt-inducing ideals of the profession.

These seem like the two options for teachers who try to talk about teaching: reject the ideals or reject experience.

VI.

Another thing that people take from Lortie is his observation that, unlike other professions, teachers haven’t developed a big body of professional knowledge about how to teach. Teachers all pretty much agree that you learn to teach on the job, and that seems to be pretty much the best way to do it.

Everyone has an explanation for why this is. Lortie’s is complex. Teachers are focused on the present (presentism); they’re expecting to leave after a few years, or they’re expecting to take breaks from the job; they’re more-or-less happy with the way things are (conservatism); they don’t see other teachers teach very often (individualism); they don’t even know what success looks like.

Lortie doesn’t say this (or at least not strongly) but I think the ideals/expectation gap is a big factor. It’s the gap that produces that potent cocktail of guilt and doubt. Our belief in the ideals of teaching creates a huge difference between what we think success in teaching is supposed to look like and what we experience daily. What teachers really have is a set of tools we’ve developed for making painful compromises on our ideals.

When professors/teacher educators/consultants/coaches talk about teaching, it’s always about how to reach those ideals. (See: ambitious teaching.) So we get tips for improving our teaching, and even if they work teachers don’t believe that they work because we still haven’t met the ideals. We’re never succeeding, so we never know that we know anything.

Traditionalists say, great, time to throw out the ideals.

But who wants to throw out their ideals? Not the majority of teachers. The majority of teachers really do think that the goal of school should be to help every child discover a love of learning. (Hey, I believe that too.) So the traditionalist option is simply not available for the majority of teachers.

Traditionalists say, OK, let’s change the culture and make it an option. (They don’t say it like that, but that’s what I’d imagine some would say.)

The cost of producing technical knowledge: either your ideals or your reality.

VII.

There’s a missing choice in all this, I think. What if great teaching is about finding great ways to navigate the space between your ideals and reality?

Call it elevated nitty-gritty talk about teaching.

It’s not possible to reach every student in your class; this is an unreachable ideal. But it is an ideal — reaching more students is better. What’s are ways to move in that direction?

Everybody talks as if the goal of teaching is maximizing knowledge; it’s not, and teachers know this. We care about relationships in ways both selfish and not. We want kids to be happy in school. The ideals of schooling point towards the future — but teachers are focused on the present. This puts us in tension, but it’s a tension that we ought to have as we both want to prepare students for their futures while also helping them through school in humane and kind ways. This tension isn’t incidental to the work of teaching. It is the work of teaching. The point of teaching is balance. What are especially good ways to find this balance?

We want kids to learn stuff, but we also want them to have a love of learning. We don’t want learning to be painful. If the knowledge-maximizing teaching method turned out to be hooking up a small charge to a kid’s ears and lightly shocking them, I’d rather teach inefficiently, thank you very much. Teachers mostly don’t want learning to be painful. What are ways of teaching that balance the tension between difficulty and pleasure in learning?

(If you’re saying that you can have both, congratulations, you are speaking on behalf of the ideals of teaching.)

I think teachers could develop technical knowledge that looked like elevated nitty-gritty knowledge.

You have to assume, in the history of teaching, that pretty much everything has been tried. I’m sure there’s a good reason why this couldn’t possibly work.

Part of the reason why it might not work is because of the pressures on teachers to fully accept the ideals of schooling. What parent wants to hear that teachers consign some kids to failure (not what we mean) and what administrator wants to defend that sort of talk? And those who hope to influence teachers — reformers, teacher educators, consultants and researchers — fiercly defend the ideals of the profession at every conference and in every article. We might turn Lortie’s conservatism on its head here — these are precisely people who elected to leave teaching to promote the ideals.

But what about in teacher spaces online? Freed somewhat from the pressures of the ideals, might we find a little bit of space between the nitty gritty of the classroom and unrealistic dreams?

VIII.

I have five books on my shelf that are all about why it’s so hard for reformers to change teaching*. They all draw heavily on Lortie, and it seems to me that this is what Schoolteacher is ultimately about.

* Tinkering Toward Utopia; Someone Has To Fail; From the Ivorty Tower to the Schoolhouse; Changing Teachers, Changing Times; Inside Teaching. I haven’t seriously read all of these.

I don’t know if I agree with everything in Schoolteacher, and I’m definitely sure I didn’t follow everything Lortie said. The book can get a bit listy at times, especially in those opening chapters.

Anyone who’s interested in changing teaching at any level ought to take a look at Schoolteacher. It has a lot of ideas, and a lot of pretty convincing quotes to support the latter chapters. I wasn’t always sure I could follow his arguments carefully enough to be sure they were true, but that didn’t bother me much. Lortie was trying to open up brand new space for others to explore, and based on my bookshelf and my read, it seems he succeeded.

How to Start Something In a #MTBoS

Why You Should Listen To Me

You shouldn’t.

OK Now Here’s The Post

Is it helpful or annoying to publicly muse on a community’s ability to get things done? Keep reading to find out!

The other day I was chatting on twitter, and out of the conversation came an idea to start a local math circle.

The other other day I was chatting on twitter, and we had the idea to start an organization that would help conference organizers find not-just-white-men to speak at their meetings.

What happens to these ideas? What do you need to do for them to actually happen?

The MTBoS has created and organized a lot of impressive things. Of these, Twitter Math Camp is objectively (objectively!) the most impressive. There are, of course, many others. I’m biased by my own involvement, but I think the Global Math Department is pretty impressive as well.

Every once in a while, these days, I hear about a new curricular website that somebody has put together. The single-purpose sites are collaborative, but not in the way that Twitter Math Camp is. Estimation180 or Open Middle or Visual Patterns or Math Mistakes all take ideas and materials from the community. There’s one or two people at the core, and then a community at the periphery.

To run an annual conference requires coordination of an entirely different sort. A significant number of people need to get their act together, together.

It seems to me that much of the ambitious coordination happening in the MTBoS right now is at the corporate level. A lot of the relational capital (so what if I made up a term) is being drawn into large organizational structures like Desmos, Illustrative Mathematics, NCTM. This is great — I’m a fan of each of those organizations.

My guess, though, is that this has slowed the pace of community organization at other levels. It takes a lot of people who trust each other and can get stuff done to make something happen. It seems to me that those people are getting busier and busier with their other highly-collaborative projects.

We’ve reached the part of the post where I speculate on what it takes to get a collaborative project off the ground in the MTBoS right now.

Step One: Decide that you actually care about this enough to make sure the ball doesn’t drop and the conversation doesn’t end. When conversation happens through asynchronous chat there’s always the chance that you’ll get ghosted. Every project that I’ve been part of lives or dies depending on whether there’s someone like this, someone who never says “sorry I’ve been crazy busy lately.” If this idea is going to happen, it might as well be you.

Step Two: Find one other person who you think is pretty close to your degree of commitment. Of course, you’ll never know, and people always are more committed at the start than they actually will turn out to be. This is normal, in my experience.

Step Three: Have a video chat or phone conversation ASAP. A phone conversation is best. The reason is because (a) having a phone conversation with a stranger is uncomfortable, and doing something mildly uncomfortable helps build trust and (b) you’re giving them your actual phone number and that (as ridiculous as it is to say) counts as an trust-building act in our near-dystopia too.

Step Four: I’m out of advice. Try to meet IRL if you can. Try to get more people involved, if you can. But the benefits of adding more people to your team, at first, are low. The main thing you need a collaborator for is to get past the part of your head that’s saying this idea is stupid and you’re an ego-maniac for thinking it was worth doing in the first place.

Two people is enough to start.

Step Later: Once things get going and you have a bunch of people involved, frequently test your group’s ability to keep working without you. Plan to step down, or to work on a side project for a year or two while someone else takes the lead. It’s sort of this sad thing where an organization takes a single high-energy person to get it going, but unless that person reduces their energy input the organization will never produce their own energy and become energy self-sufficient. (Yep, took that metaphor too far.)

I don’t know if this is true, of course, but it’s what I’ve seen so far.

It occurs to me that maybe I’m taking twitter too seriously again. True, I see a lot of ideas that never come to fruition. It’s true that the ephemeral nature of the Stream makes organization hard. But maybe that’s precisely what allows so many ideas to pop up. People would share far fewer ideas if they actually were intending to commit to them, and that’s fine.

But I think there’s room for more organization in the MTBoS than I currently see. And while I don’t think we should force anyone to do anything with their online experience — if people are happy with the way things are, that’s fine too — I suspect that people would take pleasure from forming the deeper relationships that come from doing something good together.

We have so much energy for talking about social justice, and yet there are no groups (that I know of) committed to promoting social justice in MTBoS. I think about this a lot. I don’t think it’s hypocrisy. I think we’ve communally lost the knack for organizing ourselves.

We aren’t aided by our tools. The technology we’re working with has been designed to get us to keep us using that technology. We need to fight that tendency if we want to form lasting groups online.

Back to writing a homework for tomorrow. Good night!

3rd Graders Tell You Whether or Not Fractions are Numbers

I’ve been thinking lately about what it means to be a number, for kids. My guess is that, for kids, something is a number if you can perform arithmetic operations with it. So, for example, when fractions are first introduced they aren’t seen as numbers, since you can’t count by them, add, subtract, multiply or divide them. Being a number involves living the life of a number, so to speak, and that life involves counting and arithmetic, mostly.

(Related: maybe measuring should be in that mix.)

If this idea is true, it would go counter to the conventional math education wisdom. When math education speakers or thinkers want to help kids realize that fractions are numbers, they usually recommend number line activities. The thinking seems to be that if we can show that fractions mingle with whole numbers, this will lend them legitimacy.

I think that kids find this unconvincing, though. I think being a number has more to do with whether the actions you can perform with numbers can also be performed with this new thing.

Or maybe it’s a 50/50 proposition? And both are important? Anyway, I wanted to look into this.

I asked my 3rd Graders, who have been studying fractions, whether they thought fractions were numbers. I asked them to write a few sentences explaining their thought. (I had to work to get those sentences. A bunch of them were very unhappy about writing.)

So, are fractions numbers?

Here are some of my favorite responses:

  • Yes, because to make a fraction you need numbers.
  • Fractions are a certain category of numbers because without numbers fractions would just be lines.
  • I do not think this is a number (1/2) when the 2 and the 1 are numbers, but not together, for 1 + 2 = 3 number, but this is not a number it is a half.
  • Yes a fraction is a number because a fraction is a number of pieces of something.
  • Fractions are numbers because I don’t know to explain it.
  • Fractions are numbers because you could add up to a number like 1/2 + 1/2 = 1 and a whole is a different way of saying 1.
  • NO a fraction is not a number a fraction is only part of a number.
  • Yes, because a fraction is a part of a whole number.
  • I think fractions are numbers because all math is numbers.
  • NO I think fractions are half of numbers. 1/2 is a fraction. 1 is a number.
  • A fraction: a problem of numbers in groups. Therefore not really a kind of number, but closely related.
  • Are fractions numbers?! Yes because 1/4 = 6 ’cause fractions are numbers but the bigger the number the less it is.

I have no idea what that last one means either.

So, how did my hypothesis hold up?

One kid used the fact that you can add fractions to make a whole number to decide that fractions are numbers. Though it wasn’t the mere fact of addition that helped this kid — it was the way that fractions could relate to whole numbers.

Reading these responses leads me to revise my hypothesis. Kids decide that fractions are numbers when fractions/whole numbers co-mingle in equations and problems.

The number line still, I think, is not necessarily relevant unless you count along the number line. Meaning, it’s not about the number line, it’s about the number line as a representation of counting that involves both whole numbers and fractions.

After asking my students to write their sentences, I led the class in some counting that I marked up on the number line: 0, 1/4, 2/4, 3/4, 1, 5/4, 6/4, 7/4, 2…

This, along with other arithmetic operations that involve both fractions and whole numbers, will help my kids increasingly see fractions as numbers.

UPDATE: I talked to my 3rd Graders about this again today. I just reminded them of the question, and then the room sort of exploded with arguments. Here are the arguments I heard.

When two parents have a kid that kid is still a person. So that’s like when you take a 1 and cut it up into thirds it’s still a number.

You can make numbers out of fractions like 1/2 + 1/2 so those have to be numbers too, because otherwise how could they make a number?

I like that these two arguments are sort of the flip sides of each other. I told my class that to me one of these sounded like “fractions are parents of numbers” and the other “fractions are children of numbers,” which a couple kids found hilarious.

One girl said that it really depended on the fraction. People talk about having half of something all the time, so half is a fraction. Other things — like 7/13 — that wouldn’t be a fraction because nobody talks about those. Later, I tried to put pressure on this position: what if I made a recipe that called for 7/13 of something? She said it really depended on what everybody did, not just me.

Her take reminds me of debates between prescriptivists and descriptivists about language.