Bringing Addition to a Multiplication Party

I just realized that two things I had thought to be quite different might, actually, be really similar.

First, a series of mistakes my 4th Graders make when they use addition thinking for multiplication problems:

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Second, my 4th Graders’ thought that multiplication by a negative would make a number positive, but smaller:

Yesterday a couple of 4th Graders asked, “Wait can you multiply by a negative?”

Any guesses as to what prompted this question?

Kids had been working on a multiplication puzzle and (accidentally) gotten themselves into a position where they needed to solve ___ x 20 = 10. If positive numbers make multiplication bigger, then shouldn’t negative multiplication make things smaller?

What is this mistake? Why should multiplication by a negative make a number smaller, but positive?

Here’s what I’m realizing: it comes from the thought that positive/negatives have opposite effects in multiplication/division. Which isn’t true, but it is true that positives/negatives have opposite effects in addition/subtraction.

The relevant opposites when it comes to multiplying aren’t positives/negatives, but instead numbers greater/less than 1. To draw the contrast really clearly, when it comes to adding the relevant opposites are numbers greater/less than 0.

This is not some out-there and abstract idea, though. When kids work with negative numbers they regularly reveal an understanding that positives and negatives should have opposite effects, as with 3 – (-5):

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We talk a lot about opposite operations, but do we talk enough about opposite numbers? We talk a lot about negatives as opposites to positives, but do we talk enough about numbers less than 1 as opposites to numbers greater than 1? How much of learning is trying to figure out the limits of thinking like addition?

 

What’s Going On in Pershan’s Classes This Week

A few weeks ago, I brought up negative numbers in my 4th Grade class. I framed this in terms of “giving kids permission” to bring up negatives in class.

Yesterday a couple of 4th Graders asked, “Wait can you multiply by a negative?”

Any guesses as to what prompted this question?

Kids had been working on a multiplication puzzle and (accidentally) gotten themselves into a position where they needed to solve ___ x 20 = 10. If positive numbers make multiplication bigger, then shouldn’t negative multiplication make things smaller?

This is very smart and interesting, and that’s all I have to say about that.


 

On a totally different note, there’s a dead pigeon right outside the window of my 9th Grade class. It’s becoming a distraction. (“Someone should really call animal control.”)

I was trying to talk about congruence and stuff and kids started getting fascinated and freaked out by the dead bird. So I climbed up on the window sill (there’s no draw string) to pull the shade down. And then the whole curtain contraption got ripped off the wall and missed a few kids’ head by an inch.

I get reminded about this daily. And I still haven’t called anyone about the bird.


 

I made a commitment this year to work on the relational side of teaching. To an extent, I am, but I’m realizing that it’s been entirely haphazard. Is there a way to be more systematic about this?

Another question: working on curriculum and learning can be serious intellectual work. Can working on relationships be serious intellectual work? How?


 

I was going to work on fractions in 4th Grade yesterday, but the multiplication practice that my kids dug into swallowed them up for the whole period. They loved it! (It was from these books.)

Multiplication practice can be fun for kids.

Then again, I’ve been trying to get one of my Geometry classes to work on flowchart proofs and that is a slog. I’d love for them to get great at this, but it looks like we’re going to move on without having everyone on board.

Sometimes skills practice sucks, and sometimes it doesn’t.

Chatting with @benjaminjriley about “discovery” talk.

Benjamin and I were having one of these enormous and confusing twitter conversations about this (probably not-quite-accurate) Galileo quote, and what educators mean when we talk of discovery more broadly.

First, Benjamin’s take, as I understand it.

Ben thinks that this quote is badly wrong. It is possible to teach people things! And there is plenty of knowledge that you simply cannot just discover all on your own. Even worse: discovering knowledge within yourself? Show me precisely where it is, within yourself, that your knowledge of my middle name is. The answer is that it’s not within yourself, obviously.

This quote is incredibly wrong, Ben says. And this makes it problematic for professional educators to share it, because it completely undermines the importance of the profession. If teachers don’t teach, then who needs teachers? Kids don’t need school if they can’t be taught. If it’s all about discovery — even worse, discovery about yourself — then you can’t really be taught. Teachers and schools, then, are useless.

Here was my question to Ben, though: if this is so incredibly and badly wrong, and if it so terribly misunderstands the work of teaching…then why does it appeal to professional educators? Shouldn’t professional educators be the first to understand how badly this rhetoric misrepresents the work?

I didn’t get a chance to hear Ben’s full answer on Twitter — he said it had to with the culture of teaching — but either way, his take is that professional educators have fallen into a characterization of their own work that is badly, badly wrong.


 

Now, my take.

I get suspicious of arguments like Ben’s. Are teachers badly misunderstanding their own work? I guess it’s possible, but why would we do something so self-destructive? And how come this idea, which is so obviously and clearly wrong, appeals to so many teachers?

I’m not saying that this is a knock-down argument against Ben’s take. But it makes me start to think — is there another way to interpret something like the quote Tracy shared?

I think that there is, and it has to do with the various meanings that “teach” and “discover” have.

What does it mean to teach? It could mean a few things:

  1. “I taught it to him” <–> “I explained it to him”
  2. “I taught it to him” <–> “I directly caused him to understand it”
  3. “I taught it to him” <–> “I set the conditions of his understanding”
  4. “I taught it to him, but he didn’t understand it” <–> “I attempted to cause learning, but failed”

(See here for more on the various meanings of “teaching.”)

What does it mean to discover?

  1. “I discovered it” <–> “I was the first to know it”
  2. “I discovered it for myself” <–> “I was the cause of my learning of it”
  3. “I discovered it for myself” <–> “I discovered the meaning for myself”

 

The quote that Tracy shared is

“We cannot teach people anything. We can only help them discover it within themselves.”

My most generous read of it is

“We cannot directly cause anyone to understand something. We can only help them discover the meaning for themselves.”

Which is absolutely true.

In fact, maybe it’s so obviously true that the statement becomes banal. Why would teachers make a big deal out of this statement? I think it’s because so many people outside education think that teaching involves directly causing kids to know stuff. Why don’t you just teach it? is something every educator has heard.

People who haven’t taught don’t typically understand the things that can get in the way of directly causing kids to know — their prior knowledge, their personalities, their attention, their peers, their priorities.


 

If my read is correct, it also is a pedagogy-neutral observation. Kids need to discover meanings. When I was in college, I often made sense of a teacher’s lecture — that was me discovering ideas for myself, while sitting in a lecture.

Ben and my conversation is one that has been had over and over about constructivism. For example, here is a passage from a really fantastic piece called “A Reformulation of Telling”:

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When teachers say that “you can’t teach” it’s just saying this — that all meaning has to come from the student, at the end of the day. Though the entire point of teaching is to help kids make this meaning — that’s the work!

I see no reason why this understanding of teaching is incompatible with public esteem. It’s serious work, and the public could understand that this is serious work.


 

It seems to me that Ben is not convinced by my rereading. If I understand him correctly, he thinks that when teachers share these quotes online or talk this way, they really mean that you can’t cause kids to learn stuff and that kids need to discover knowledge on their own.

I disagree. I think that this reading is both possible and likely, given that it doesn’t involve teachers badly mischaracterizing the nature of our own work. I’ve literally never met a teacher or seen teaching that was centered on “not teaching” as Ben reads it, and in fact that’s impossible to do while meeting external learning standards. Why would teachers engage in talk that directly contradicts what they’re doing in classrooms? It doesn’t make sense to me.

It would be great to do a bunch of interviews with teachers to try to gauge their meaning, if they assent to “discovery” talk. Has anyone looked into this? It might be a helpful thing for someone to study.


 

Here is where I’ll agree with Ben, though: there is a cost to teacher talk that emphasizes discovery and “not teaching.” It’s a line that is just waiting to be misheard, and whether that’s fair or not is irrelevant.

This sort of talk is easy to misunderstand, because of the various meanings of “teach” and “discover.” (I might as well have written a long post explaining to those who engage in discovery talk how that quote could be misunderstood in the other direction!)

Teachers know how important it is to know the landscape of potential misconceptions when they’re teaching math. If you care about helping people understand what the work of teaching is really about, we also need to take people’s misconceptions about our work into consideration.

Talking about teaching and learning is hard. It’s really really hard. It’s partly hard because the work is so invisible, and because the physical manifestations of that work (explaining, sitting, writing) are poor stand-ins for it. There is no agreed-on language, and many of the fiercest debates in education are best seen as desperate calls for clarity, each person silently begging the other to learn how they mean. That is what I think is happening here.

[It’s-My-Blog-So-I-Can-Write-About-Politics-If-I-Want-To Edition]

One of my favorite pieces on learning begins with a story:

Once upon a time, an astute and beneficent leader in a remote country anticipated increasing aggressions from a territory-hungry neighbor nation. Recognizing that the neighbor had more military might, the leader concluded that his people would have to out-think, rather than overpower the enemy. Undistinguished in its military armament and leadership, the country did have on remarkable resource: the reigning world chess master, undefeated for over twenty years. “Aha,” the leader said to himself, “we will recruit this keen intellect, honed so long on the whetstone of chess, teach him some politics and military theory and then outmaneuver the enemy with the help of his genius.”

Would this be a good idea? Could the chess master apply his skills to the battlefield — or would he have difficulty applying the insights of chess to human warfare? More generally, are cognitive skills context bound?

I think an entirely sensible answer to these questions is, maybe, who knows. It depends on the person. Some skills, for some people, become so much a part of who they are that they cease to be skills at all. There is no clear place where the skill stops and the person begins.

I can imagine that chess master developing an instinct towards caution in her gameplay. And then maybe, because she’s thoughtful and sharp, she starts seeing her life in terms of chess. Not right away — it takes many years. But she starts to understand her life partly in terms of chess, and she starts noticing opportunities in her life when that same chess-born instinct towards caution applies. And now there’s a two-way street between chess and her life, and the things she learns in one arena sooner rather than later show up in the other.

For me, teaching is structured empathy. I spend all my time trying to understand how other people think, in the hopes of changing how they think. This isn’t to take a stand on whether teachers should primarily be talkers or listeners. I sure hope that even the talkers out there spend a lot of time worrying about their students’ thinking. (For that matter, I’ve always found that talking is an important part of listening.)

I mostly worry about how other people think about math. I know a thing or two about how people think about numbers or shapes, but I couldn’t tell you how people think about molecules, slavery or elections. I’d have to spend a lot more time listening (and talking) in order to know anything about that.

At the same time, like that chess master, it seems as the habit of listening is no longer easy to separate from the rest of my experiences. In this sense, my teacher identity has taken over its host.

Yes, this piece is a plea for listening.

In school today I saw a lot of sad students. (Mostly happy ones.) The adults did a fantastic job supporting students today, giving them room for their fears and sorrow. If you’re living in NYC in 2016 you don’t experience a lot of elemental forces that leave you feeling powerless, but an election is (of course) entirely out of your control, and that’s a feeling that can leave you reeling if you haven’t developed a taste for powerlessness.

Take a day or two, and listen to the kids. Hear their fears and stories, understand how they think. But sooner rather than later we need to listen more widely, to voices that are literally far away from us. (Thanks, urban/rural polarization.)

You can’t persuade people if you can’t listen. You can’t continue the fight against bigotry if you can’t listen. You can’t fight racism if you can’t listen. The only way to challenge your neighbors is to understand them, with the tenaciousness and curiosity you’d bring to a kid with a bizarre strategy for solving an equation.

I can’t think of a much better use for a classroom than helping kids understand that you’re supposed to understand how the other 50% of the country feels.

In my attempts to understand why we elected DJT, demographics are conspiring against me. My values, education, incentives and experiences all stack the deck against understanding what the hell is going on out there.

The piece that has helped me the most to understand what just happened goes like this:

The rural folk with the Trump signs in their yards say their way of life is dying, and you smirk and say what they really mean is that blacks and gays are finally getting equal rights and they hate it. But I’m telling you, they say their way of life is dying because their way of life is dying.

We’re supposed to understand each other. I can’t figure out whether that’s trite or not (it is) but sometimes we teachers are trite.

Forgetting and Learning

On Friday morning, I asked my 4th Graders to think about this question in their heads:

Which is greater?

30 x 15 or 25 x 20

I was very curious as to their responses. It had been a few weeks since we had worked on an incredibly similar problem — Which field has a greater area, a 45 by 20 or a 40 by 25? — and I was curious if students would make a connection to a problem we had spent a full class period studying. (I guessed they wouldn’t.)

When working on the “Greater Area” problem, nearly all my students had thought that the two fields had equal area. They reasoned that 45 by 20 and 40 by 25 fields would have equal area since a “5” could be taken from the 45 and moved to the 20. It was hard to convince students that the areas were not equal.

Following up on that lesson, our class had focused on mental strategies for multiplication. Our main focus was on using multiples of ten to help with computation, a strategy my students called “Stuff.” (My fault for asking them what we should call it.)

Would kids approach my new “Which is Greater” problem differently after all these experiences?

After giving students time to think mentally, I handed out index cards and asked them to write down their mental-train-of-thought on the card. I told them that I didn’t want them to rethink it in writing — I wanted a recording of their mental approaches. I knew that these were difficult instructions to follow, but I wanted to preserve their original lines of thought as much as possible.

I was surprised — and a bit sad — to notice that several students thought that 30 x 15 and 25 x 20 were going to be equal products. Though the idea is incorrect, S displays it beautifully:

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N, I think, is trying to get at the same idea as S, though she has an entirely different observation to make:

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(I’m trying to think of an algebraic way to express N’s idea. I think she’s saying that a + b will always be equal to (a + 10) + (b – 10).)

It seemed these students didn’t remember what we had spent a full class period working on just two weeks before.

One student made a nice connection to the “Greater Area” problem. W roughly sketched each product as a rectangle, and told me that it looked to him like the 20 x 25 rectangle was bigger. In our previous work, I had asked my class to think about whether it was the thinner or the “squarer” field that held more area. It seemed to me that W was drawing on his memory that, perimeter being equal, “squarer” fit more area.

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Though my students weren’t making connections to the old problem, it made me happy to see that many of my students used multiples of ten to help their computations. Nobody had done that when we did this problem a few weeks before. 

One student skip-counted instead of using multiples of ten, though. That made me a bit nervous about her progress. (Maybe I shouldn’t worry. After all, she didn’t use skip-counting or anything to tackle the 40 x 25 version of this problem a few weeks prior.)

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I was interested in the variety of “first moves” involving multiples of ten that students made along the way towards a final answer:

20 x 20 = 400 and 30 x 10 = 300

15 x 10 = 150 and 25 x 10 = 250

30 x 10 = 300 and 25 x 10 = 250

2 x 25 = 50 and 3 x 15 = 45

30 x 10 = 300 and 20 x 10 = 200

Using multiples of ten is a procedure that students can decide to do in a number of different ways.

There were two other approaches that I found especially interesting. The first was Ella’s. She started with 10 x 20, but wrote that 10 x 20 = 201. I have no idea where this came from, though I understand why she added 5 at the end (to get to 25, I assume).

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Ella’s work shows how tricky it is to keep track the differences betwee multiplication and addition. Adding 5 back at the end is a typical addition strategy, and even though she has a lot of strong work on her notecard, that’s the sort of mistake that I’d like to teach her to avoid.

The other was D’s. It seems that D attempted to use the standard algorithm, but she concludes that 25 x 20 and 30 x 15 are equal on its basis:

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The truth is that she didn’t actually make any sort of error with the execution of the standard algorithm. But where did she get 500 from? (Not by adding 150 and 300, I don’t think.) My best guess is that, after performing the standard algorithm, some other line of thought occurred to her that made her think that the two products were equal, even though the rest of her mathematical work suggests the opposite.

The main difference between the students who got correct vs. incorrect answers was whether they drew on the structure of addition or the structure of multiplication in their thinking. Those who drew on addition ended up deciding that 5 can be moved between factors in 30 x 15 to produce an equivalent product, 25 x 20. This is true for addition: 30 + 15 = 25 + 20. Students who drew on multiplication drew on group-structure, multiples of 10 and doublings in their work. And, as E shows, some students used both structures at once, leading to mixed results. (E realized that the products were not equal, but she miscalculated them.)

This problem is coming after students have already been prompted fairly heavily to think about multiplication using multiples of ten. It was great to see that my students are using it, and the variety of ways in which they did their work makes me think that they are making the strategy their own.

At the same time, it’s fairly remarkable to me how many students didn’t make the connection to a very similar problem they worked on just a few weeks ago. I often forget just how different my memory is to that of my students’. For me, that class was a landmark. For them, it was probably just another class. After working on the “Greater Area” problem a few weeks ago, they walked right out of my class and into someone else’s lesson.

For kids, there are always many things happening at school. A teacher can’t count on just one lesson sticking — we need to connect several different experiences to have a shot of building some strong knowledge.

A Worked Example Workaround?

When you decide to explain some math to a kid, how should you explain it? Step-by-step, or all at once?

There’s an issue with step-by-step explanations: kids have to remember what you’ve already said in order to understand what comes next. This means that there’s often a lot to hold in their head!

There’s an issue with fully worked-out examples: by not developing math slowly, in full view of the student, you make it seem as if the solution was dropped out of the sky. It can present a false picture of math: as constituted entirely of an encyclopedia of procedures that mathematicians memorize, look-up, and employ on canned problems.

I was thinking, today, about whether there is a way to get the best of both worlds. My mind wandered towards my 8th Grade class. We’re studying slope, so I launched class by putting two right triangles under the document camera. Which is steeper?

Students debated, thought some more, offered good and better approaches. I kept a record of their ideas, which I scanned after class:

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I’m wondering, what if I started tomorrow’s class by projecting this image back on the board? I’d say, Here’s what we figured out yesterday. I’d like to give you two new triangles to look at today. I’m going to keep yesterday’s work up, in case it’s helpful.

Would that give a distorted view of mathematics and its development? Would that give students the benefits of worked-out examples?

6 + -2 as “Adding a Minusing 2”

For the last month or so since I read this article, I’ve been unable to shake the idea of “mathematical worlds”:

Rather than describing the challenges of integer learning in terms of a transition from positive to negative numbers, we have arrived at a different perspective: We view students as inhabiting distinct mathematical worlds consisting of particular types of numbers…Proficient students and adults may also inhabit multiple mathematical worlds from moment to moment.

This metaphor completely fits my experience with children. There is often doubt about what “world” we’re operating in. Are fractions allowed? Are we using right now, or does x^2 = -9 have no solutions?

(The other relevant metaphor is of a game, as in “I didn’t realize fractions were allowed.” Is there a connection between the metaphors or world and of game?)

One thing this metaphor helped me realize is that my students might bring up negatives in my class more often if I simply gave them permission to. How do you give permission? I decided to bring them up right away with my 4th Graders. (I culled the questions from here.)

I handed out a piece of paper to each student. I asked them to work completely on their own. The first question was 9 – 10 = __. Here were the responses:

  • 1 (7 kids)
  • -1 (4 kid)
  • ? (1 kid)

Next I asked, 3 – 5 = ___, and received similar responses. So far, kids were pretty confident.

The next question led to protests, though: 6 + __ = 4. Some protests, some talk of -2. (Wait, are we allowed to use negatives?) I collected their papers and, since kids seemed to have what to say about this, I asked volunteers to share their thinking.

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Isn’t it interesting how the first two questions have positive answers but the third has a negative response? I think this student swapped worlds between #2 and #3.

First went E, who explained that since 6 – 2 = 4, this had to be -2. I wasn’t sure exactly what she meant by this, but then H objected to this whole thing in an incredibly helpful way:

H: “Wait, I don’t think that makes sense. How can we say that we’re doing 6…plus minus 2?”

Me: I have no idea what you mean so I’m going to offer a belabored explanation of how we’re not adding “minus 2,” rather we’re adding “negative 2.”

H: “I mean like I don’t see how you can add minus like that.”

Me: Wait, I think I understand what you’re trying to say. You’re saying it’s like if we said 3 times divide 2, like, what would that mean?

H: Yeah, that doesn’t make any sense.

But then a bunch of other kids jumped in. They were totally able to make sense of “6 plus minus 2.” They said you were adding a minusing of 2. This is subtly different than anything I’d ever before understood kids to say. They weren’t treating “-2” as a point on the number line or even as a single number at all. They were making sense of -2, in this context, as an action. The action of subtracting 2.

All the while this was making me nervous. Middle and high school students have trouble distinguishing between the three slightly different ways in which we use the minus sign in mathematics. (See: “High School Students’ Conceptions of the Minus Sign”)

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Should I have said something to keep kids from thinking of “6 + -2” as “6 plus a minusing of 2”?

The more I thought about it, though, the more confident I became that this was an incredibly productive way to start thinking about adding a negative. Kids who think about 6 + -2 as “adding a minus 2” are already along the road towards thinking of the process of subtracting as an object (a subtracting). There is more work for such a thinker to do — they have to come to think of this subtracting of 2 as the sort of thing that you can operate on. (What does it mean to add 3 to a minusing of 2, as in -2 + 3?) But there is a clear conceptual path for these students to take.

(I’m thinking here of Anna Sfard’s idea of interiorization, condensation, and reification.)

This all is a brand new idea to me, a way of making sense of negative arithmetic that I’d never understood before. I continue to be blown away by the variety of ways that kids make sense of arithmetic with negatives — each problem situation seems to allow for slightly different conceptions. Really making sense of negative arithmetic might involve negotiating many different ways of making sense of many different types of problems. It’s only once kids have a few different mental models that they can start uniting their thoughts into one big idea of negative numbers.

We’re going to keep this up in 4th Grade. Actually, this all happened yesterday, and today kids ended up digging into the number line, its mirror-like symmetry and the “realm of the negatives” (which J connected to Minecraft and Portal, god save the children.)

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Our notes from the discussion today.

 

Kids seem to be having fun exploring this realm, and they’re doing good math. Why not give kids permission to enter this world?