Equations and Equivalence in 3rd Grade

So I was stupidly mouthing off online to some incredibly serious researchers about equivalence and the equals sign and how it’s not that hard of a topic to teach when — OOPS! — my actual teaching got in the way.

I had done the right thing. In my 3rd Grade class I wanted to introduce “?” as a symbol for an unknown so I put up some equations on the board:

15 = ? x 5

3 + ? = 10

10 + 3 = 11 + ?

And I was neither shocked, nor did I blink, when a kid told me that the last equation didn’t make any sense. Ah, I thought, time to nip this in the bud.

I listened to the child and said I understood, but that I would like to share how it does make sense. I asked whether anyone knew what the equals sign meant, and one kid says “makes” and the next said “the same as.” Wonderful, I said, because that last equation is just saying the left side equals the same as the right side. So what number would make them the same? 2? Fantastic, let’s move on.

Then, the next day, I put a problem on the board:

5+ 10 = ___ + 5

And you know what comes next, right? Consensus around the room is that the blank is 15. “But didn’t we say yesterday that the equals sign means ‘the same as’?” I asked. A kid raised her hand and explained that it did mean that, but the answer should still be 15. Here’s how she wanted us to read the equation, as a run on:

(5 + 10 =  15 ) + 5

Two things were now clear to me. First, that my pride in having clearly and decisively taken care of this issue was misguided. I needed to do more and dig into this more deeply.

The second thing is that isn’t this interesting? You can have an entirely correct understanding of the equals sign and still make the same “classic” mistakes interpreting an actual equation.

I think this helps clear up some things that I was muddling in my head. When people talk about the need for kids to have a strong understanding of equivalence they really are talking about quite a few different things. Here are the two that came up above:

  • The particular meaning of the equals sign (and this is supposed to entail that an equation can be written left-to-right or right-to-left, i.e. it’s symmetric)
  • The conventional ways of writing equations (e.g. no run ons, can include multiple operations and terms on each side)

But then this is just the beginning, because frequently people talk about a bunch of other things when talking about ‘equivalence.’ Here are just a few:

  • You can do the same operations to each side (famously useful for solving equations)
  • You can manipulate like terms on one side of an equation to create a true equation (10 + 5 can be turned into 9 + 6 can be turned into 8 + 7; 8 x 7 can be turned into 4 x 14; 3(x + 4) can be turned into 3x + 12, etc.)

When a kid can’t solve 5 + 10 = ___ + 9 correctly or easily using “relational understanding,” this is frequently blamed on a kid’s understanding of the equals sign, equivalence or the particular ways of relating 5 + 10 to __ + 9. But now I’m seeing clearly that these are separate things, and some tend to be easier for kids than others.

So, this brings us to the follow-up lesson with my 3rd Graders.

I started as I usually do in this situation, by avoiding the equals sign. I find that a double arrow serves this purpose well, so I put up an arrow relationship on the board:

2 x 6 <–> 8 + 4

I pointed out that 2 x 6 makes 12 and so does 8 + 4. Could the kids come up with other things like this, I asked?

They did. I didn’t grab a picture, but I was grateful that all sorts of things came up. Kids were mixing operations nicely, like 12 – 2 <–> 5 + 5, in general it felt like this was not hard, kids knew exactly what I meant and could generate lots of ideas.

My next move was to pause and introduce the equals sign into this conversation. Would anyone mind if I replaced that double arrow with an equals sign? This is just what the equal sign means, anyway. No problem, that went fine also.

Kids were even introducing great examples like 1 x 2 = 2 x 1, or 12 = 12. Wonderful.

Then, I introduced the task of the day, in the style of Open Middle (R) (TM) (C):

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Yeah, I quickly handwrote it with a sharpie. It was that sort of day.

I carefully explained the constraints. 10 – 2 + 7 + 1 was a true equation, but wouldn’t work for this puzzle. Neither would 15 – 5 = 6 + 4. And then I gave the kids time to search for solutions, as many as they could find.

Bla bla, most kids were successful, others had trouble getting started but everyone eventually had some success. Here are some pictures of students who make me look good:

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Here is a picture of a student who struggled, but eventually found a solution:

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Here is a picture of the student from the class I was most concerned with. You can see the marks along his page as he tries to handle things like 12 – 9 as he tries subtracting different numbers from 12. I think there might have been some multiplying happening on the right side, not sure why. Anyway:

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The thing is that just the day before, this last student had almost broken down in frustration over his inability to make sense of these “unconventional” equations. So this makes me look kind of great — I did it! I taught him equivalence, in roughly a day. Tada.

But I don’t think that this is what’s going on. The notion of two different things being equal, that was not hard for him. In fact I don’t think that notion is difficult for very many students at all — kids know that different additions equal 10. And it was not especially difficult for this kid to merge that notion of equivalence with the equals sign. Like, no, he did not think that this was what the equals sign meant, but whatever, that was just on the basis of what he had thought before. It’s just a convention. I told him the equals sign meant something else, OK, sure. Not so bad either.

The part that was very difficult for this student, however, was subtracting stuff from 12.

Now this is what I think people are talking about when they talk about “relational understanding.” It’s true — I really wish this student knew that 10 + 2 <–> 9 + 3, and so when he saw 12 he could associate that with 10 + 2 and therefore quickly move to 9 + 3 and realize that 12 – 3 = 9. I mean, that’s what a lot of my 3rd Graders do, in not so many words. That is very useful.

So to wrap things up here are some questions and some provisional answers:

Q: Is it hard to teach or learn the concept of equivalence.

A: No.

Q: Is it hard to teach the equals sign and its meaning?

A: It’s harder, but this is all conventional. If you introduce a new symbol like “<–>” I don’t think kids trip up as much. They sometimes have to unlearn what they’ve inferred from prior experiences that were too limited (i.e. always putting the result on the right side). So you’re not doing kids any favors by doing that, it’s good to put the equations in a lot of different forms, pretty much as soon as kids see equations from the first time in K or 1st Grade. I mean why not?

Q: If kids don’t learn how equations conventionally work will that trip them up later in algebra?

A: Yes. But all of my kids find adding and subtracting itself to be more difficult than understanding these conventions. My sense is that you don’t need years to get used to how equations work. You need, like, an hour or two to introduce it.

Q: Does this stuff need to be taught early? Is algebra too late to learn how equations work?

A: I think kids should learn it early, but it’s not too late AT ALL if they don’t.

I have taught algebra classes in 8th and 9th Grade where students have been confused about how equations work. My memories are that this was annoying because I realized too late what was going on and had to backtrack. But based on teaching this to younger kids, I can’t imagine that it’s too late to teach it to older students.

I guess it could be possible that over the years it gets harder to shake students out of their more limited understanding of equations because they reinforce their theory about equations and the equals symbol. I don’t know.

I see no reason not to teach this early, but I think it’s important to keep in mind that in middle school we tell kids that sometimes subtracting a number makes it bigger and that negative exponents exist. Kids can learn new things in later years too.

Q: So what makes it so hard for young kids to handle equations like 5 + 10 = 6 + __?

A: It’s definitely true that kids who don’t understand how to read this sort of equation will be unable to engage at all. But the relational thinking itself is the hardest part to teach and learn, it seems to me.

Here is a thought experiment. What if you had a school or curriculum that only used equal signs and equations in the boring, limited way of “5 + 10 = ?” and “6 x ? = 12” throughout school, but at the same time taught relational thinking using <–> and other terminology in a deep and effective way? And then in 8th Grade they have a few lessons teaching the “new” way of making sense of the equals sign? Would that be a big deal? I don’t know, I don’t think so.

Q: There is evidence that suggests learning various of the above things helps kids succeed more in later algebra. Your thoughts?

A: I don’t know! It seems to me that if something makes a difference for later algebra, it has to be either the concept of equivalence, the conventions of equations, or relational thinking.

I think the concept of equivalence is something every kid knows. The conventions of equations aren’t that hard to learn, I think, but they really only do make sense if you connect equations to the concept of equivalence. The concept of equivalence explains why equations have certain conventions. So I get why those two go together. But could that be enough to help students with later algebra experiences? Maybe. Is it because algebra teachers aren’t teaching the conventions of equations in their classes? Would there still be an advantage from early equation experience if algebra teachers taught it?

In the end, it doesn’t matter much because young kids can learn it and so why bother not teaching it to them? Can’t hurt, only costs you an hour or two.

But the big other thing is relational thinking. Now there is no reason I think why relational thinking has to take place in the context of equations. You COULD use other symbols like double arrows or whatever. But math already has this symbol for equivalence, so you might as well teach relational thinking about addition/subtraction/multiplication/division in the context of equations. And that’s some really tricky, really important mathematics to learn. A kid being able to understand that 2 x 14 is equal to 4 x 7 is important stuff.

It’s important for so many reasons, for practically every reason that arithmetic is the foundation of algebra. I can’t list them now — but it goes beyond equations, is my point. Relational thinking (e.g. how various additions relate to each other) is huge and hugely important.

Would understanding the conventions of the equals sign and equations make a difference in the absence of experiences that help kids gain relational understanding? Do some kids start making connections on their own when they learn ways of writing equations? Does relational understanding instruction simply fail because kids don’t understand what the equations their teachers are using mean?

I don’t know.

Dear Aunt Sally: The exit tickets are all over the place!

Dear Aunt Sally,

My students had a hard time with my lesson on solving equations. How do I know? I gave a short “exit ticket” and the results were…mixed. There were three questions:

  • 3x + 5x = 56
  • 3x + 7 – 5x = 45
  • 6x – 5 + 11x + 17 = 63.

The main issue is combining like terms. (I’ve included some pictures of student work for you, Aunt Sally.)

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That said, some students in each class definitely did understand the material.

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One more issue. In most of my classes, students didn’t struggle with the first problem. But in each of my classes, some students did, and in one of my classes very few students got that first one correct either. 

What do I do to help the students who didn’t understand the material, and what do I do with the kids who did?

Ryan in Florida

***

Dear Ryan,

You have described one of the perpetual struggles of teaching. How do I balance the needs of the many versus those of the few? I suppose back in Neanderthal times the fellow charged with teaching youth to spear a mammoth came home to the cave feeling similarly. “Some of those kids get it,” he’d say. “But what about the one who used the flat end of the spear? He’s going to starve to death. He would really benefit from getting back to the basics. Maybe I’ll split the group in half?”

Thank goodness we aren’t Neanderthals. We are modern teachers! This means we have access to the wonders of modern technology. I of course mean the blackboard and copy machine.

My favorite way to follow-up on these sorts of quizzes is with examples, so I put together three such activities for you:

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And then some mixed-up practice, to take another step in the right direction.Screenshot 2019-09-08 at 7.46.00 AM

A good example activity, in my experience, can seem so simple so as to hide its design. And in fact the actual design of the student work was mostly straightforward. The choices to make are mostly ones I made long ago — to prefer simplicity, to subtly use arrows and lines, and to make the work as much like a student’s as possible while making the work as clear as possible to read.

(The mistake, in particular, closely follows the design of an Algebra by Example mistake.)

Most of the work here is in narrowing in on a specific type of problem that is worth including in the examples. (The simplicity of the format works well with the more complex task of narrowing in.)

Here is a rule I tell myself while looking at a student’s mistake: Every mistake points to a family of situations very similar to the mistake that the student doesn’t yet know how to handle. Find that family, and teach it!

In this case I don’t assume the mistakes on the second problem (3x + 7 – 5x = 45) point not to issues “combining like terms” in general. I assume instead that the mistake points to issues where the like terms are separated visually in the equation and where subtraction is involved. Those things are distinctive and make the equation more difficult to solve correctly — that is the family that I focus in on for the example.

How the examples are used is your decision, Ryan, but here is how I would do it. Begin class by showing just the example, and ask students to silently study it. (They can let you know when they’ve read it all with their thumbs.) When finished, you can ask them to answer the explanation questions on their own or with a partner. If you want to interject with an explanation, by all means, but often students are ready to dive straight into the practice problems.

There are three examples/non-examples I’ve made. Use as many as your class would benefit from.

Then, there are four practice problems. Those are important because they are mixed practice. If we see each of these examples as pointing to a micro-skill, a small family of problems within the broad category of “two-step equations,” then these problems are interleaved in the practice set. That can be useful! Your students will have to think back and remember what they did with the examples.

Teachers of equation-solving know that there are always more mistakes that need to be addressed. The difference between good and bad teaching of this topic, as I see it, is whether the teacher can get specific and point to precisely what is hard for their students. Subtracting, different visual formats, handling a variable with coefficient 1 (x as 1x), and so many other little things — there will always be more of these. Some teachers just repeat and repeat and repeat without getting any more specific. Instruction should get more specific as practice gets more general.

So, keep at it! Pretty soon your young charges will be scattering the landscape with Woolly Mammoth carcasses, so to speak.

-Aunt Sally

Quick direct instruction and interesting practice at math camp

Today is my first day of summer break, sort of. I spent the last six weeks working for a wonderful math camp where the teaching is so much fun.

My class is the closest that students come to school content during camp. It’s a fractions class that students get placed into based on an assessment. If the camp thinks that students could use more time working on fractions — that’s who I teach.

That said, the course content is tricky because I don’t want to simply repeat what they’ve seen in school. That would be boring for a lot of kids, and I’m aiming to approach familiar ideas in unfamiliar ways. I’m trying to work on skills, but from interesting perspectives.

Here’s a one-two-three sequence from my fractions course that I think worked particularly well.

First, I ask students to think about visuals. This was a focus of the previous lesson, but I want to make sure every students has it at the front of their minds.

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Source for image is fractiontalks.com

I’m trying to give everyone a chance to figure out what fraction a piece is by multiplying. (“There are four pieces, this is divided into fourths, that would make sixteen in total.”)

I’m teaching this both because it’s a useful bit of visual fluency, but also because I want to use this as part of my direct instruction.

Next is the direct instruction. I’m trying to teach students a mental shortcut: if you’re dividing a fraction by an integer (e.g. 1/2 divided by 10 is 1/20) you can multiply the denominator by the integer because that’s simply making the pieces 10 times smaller. I use visuals to explain this.

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This is a classroom poster with a version of my explanation. Below it is the next mental strategy I teach in the course. 

I immediately give students a few chances to try out this new technique on some mental math problems. (Below is my little cheat sheet — this is what I ask students, but I don’t give them this paper.)

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That’s the basics. But how are we going to practice it further? And how are we going to keep it interesting, and make sure students start using this technique in other contexts?

I then move to the third activity in this lesson, some mobile problems (designed by me on the EDC site). They’ve been carefully designed to give us a chance to use that mental shortcut we’ve just studied.

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A lot of the lessons in my fractions course seem to follow something like this pattern: reminder, quick explanation, interesting practice.

What exactly is it that worked about this? I think this pattern of quick direct instruction followed by interesting practice is a useful one. Of course not every topic is amenable to quick direct instruction (some skills need to be taught in larger chunks) but some are. And after some quick “are we on the same page” questions, it was nice to follow it up with interesting practice. And what made it interesting? I think that it looked different than the direct instruction, but there was still the chance to use it frequently.

This is a way of engineering challenging classroom experiences around stuff that you want to just explicitly teach. I think a lot of people think of these things as incompatible, but they clearly aren’t. At the same time, for a lot of my groups during the year I am trying to make things more accessible — I’m not trying to make it more challenging.

Or maybe I should be? Maybe this pattern of instruction would work just fine in my school-year work. One issue during the year is that I’m much more cautious about whether the practice is actually going to help with the skill. There is a risk to practicing in a different context than instruction. It’s always possible that kids won’t make the connections, that it will be either too hard or students won’t actually practice the thing you thought they would.

So, I’m not sure whether this is something I’ve learned about teaching camp or teaching school or teaching math. Time will tell, I guess.

Encore!

Another school year: done.

I love the dismantling that happens after the kids leave. For months this place has been made just so, and it takes just moments for the entire accumulation process to be set in reverse. Posters are first in the line of fire, but sooner or later everything is headed either to a drawer or the trash. The recycling bins were stuffed with homework this afternoon.

I’ve never been involved with anything theatrical (ok fine once) but I imagine it’s similar to what it feels like to strike the set the day after everything closes. Everything of significance has got to go. It’s all trash now, but just a few hours before the whole thing was whatever it is you call trash’s opposite. This stuff was indisposable by mutual assent. Now: nah.

And what that made me think of was the artifice of this whole enterprise. School is such a weird performance of the strangest kind of pretend — they call me ‘mister’! People sometimes point to the artificiality of schools as a critique: this isn’t what learning looks like in its natural state. And of course that’s exactly right, it’s not. It’s all fake. And that guy up on the stage — did you know he’s not really a wizard?

Teaching is weird, it’s fake. Teaching is not medicine. Medicine is someone is sick and you can help them, the most natural thing in the world. Driving a cab is not weird. People want to get to x, they pay you to take them there. Natural. Getting paid to cook people food is a reasonable transaction, it is not a weird. Lawyering, on the other hand: super weird. And as long as we’re on the topic: basketball player, musician, researcher, middle manager, writer, actor, teacher, weirdos all.

So it’s summer and the show is over and all but, come autumn, let’s do it again! I’ve got a school I love working at and I’m ready to play pretend for another year. Year nine is over.

Two classroom things

#1

Ask anybody: I’m not some sort of god teacher in the classroom. But the other day a teacher complimented me on my classroom management, in particular my ability to hold attention without using my voice. And I was like, yeah! That’s something I do very much on purpose.

I forget if I came up with this on my own or (more likely) picked it up from something I read, but I try very very hard not to interrupt the flow of the class with a redirection to a kid. None of that “I’ll wait” or “settle down” or “I need all eyes up here.” Not if I can help it, at least.

The thought is that the content of the class shalt not be interrupted, and I undermine this premise when I use my voice to interrupt. Plus, it’s just confusing to kids who are following along to hear this other thing, and then to dive back in. PLUS, it directs everyone’s attention to some kid and their shenanigans. PLUS PLUS, it creates a little pocket of energy in the room for a few moments and that can get filled with nonsense or accidental attention-grabbing things.

So I use looks, pauses, taps on the shoulder, etc. I know (I think) a lot of other teachers do this stuff, but I do it too and maybe it’s worth sharing.

#2

Very frequently there are kids that aren’t feeling confident and you can sort of feel them withdraw from the classroom. Very frequently there is an implicit contract these kids have with their teachers: I will be silent and confused, but you will not mind because I will not cause you an ounce of trouble.

Forget that! Cause me trouble.

So there’s this thing that sometimes happens in my classrooms. It’s a bit awkward, but I think it works out OK, most of the time. Which is that sometimes I’ll see a kid who is settling into that withdrawal, and so I’ll cold call on them.

Putting a kid on the spot in front of the class — especially one who is having a rough time in class — is a tricky proposition, and my purpose is not to embarrass a kid for not engaging. It’s just that they’ve got this contract, and they’re trying to figure out if I’m on board or not. But I’m not — I’m going to invite you in to whatever our class is doing, and I’m going to keep on doing that.

“Sarah, do you agree that 40 divided by 4 is 10? Why does that make sense, do you think?”

I’d be curious to hear the recording of that conversation. In that moment I felt myself trying very hard to balance two contradictory stances: a jokey, non-threatening, supportive teacher who is very persistently insisting that you share your thinking.

I also felt myself very much caught up in another contradiction, which was not signaling to Sarah (or the rest of the class) that I was “dumbing things down” for her, while also trying to find a mathematically interesting way for her to engage. This was also a high-wire act, even if it only lasted a few seconds. In these moments one of my moves is to just produce more and more on my own, but to be insistent that the student take on why.

“So from skip-counting we know that ten 4s makes 40. Why would that help us solve 40 divided by 4?”

Sarah ended up saying something nice, which was that 4 x 10 made 40 and that this gave you the answer to the division problem. (Did I mention yet that this is in my 3rd Grade class? I should’ve.)

Having survived this ordeal, I saw her withdrawing again, not looking at the board, not hearing other people, the sorts of things that raise all of my alarms as a teacher.

So I did something unexpected, which is for the next problem I cold-called on Sarah again. And we did that same thing, the same conversation. DEFINITELY a gamble — this is a risky teaching move. Looking at her face, I started getting very nervous…I thought tears were a distinct possibility. But what could I do? I had to keep going. So I turned back to that last problem.

“You had said 4 x 10 makes 40 to help with the last problem. So how can we think about 48 divided by 4?” 

She made it through that conversation. Thank god for her friend, Kya, who gave her a big high-five after that conversation wrapped up. Sarah put on a big smile.

This was a big risk, but at least this time I feel that it paid off. Sarah is pretty good at skip-counting, and she was able to use that to solve some problems on her own without any prompting from me. I was able to give her a high-five also, to compliment her on thinking to do that all on her own.

And though certainly it could have gone bad, the whole point of this to me was that, no, I do not accept the terms of that contract.

What can “I’m not good at this” mean?

Here’s a quick note from the field.

Class today was largely about some fraction arithmetic, and one student was having trouble with it. In class, a couple times, this student said “I’m just not good at this” or “This doesn’t make sense” and “I’m bad at calculating stuff.”

But, all through class, the student stuck with it — I mean that they tried problems on their own and asked questions. And there was definite progress. The student was becoming able to handle this type of question on their own, and starting to make sense of things.

At the end of class I told this student that I thought they were getting the hang of this. I saw a sort of relief pass over this student. Then the student told me:

  • They had a hard time with this topic
  • But this student sort of blames previous teachers
  • This student recently signed up for extra math that is more like our class’
  • The math of our class is the “only math they’re good at”

So what did this student mean when they said “I’m not good at this” etc. in class? In retrospect, it was a statement of fear and anxiety precisely because it was the exception to the rule, at least the rule in our class. This student feels generally competent in our class, so much so that they’ve signed up for similar, optional math next year, and was feeling nervous that this good thing was in danger.

This doesn’t contradict the standard story that people tell about kids saying that they’re “bad at math.” (Especially because this student’s statements weren’t that they were bad at math, in general. It was more specifically about arithmetic and algebra.) But I do think that it illustrates one way that these statements can be an attempt to express something subtler than what they’re usually credited for.

Three things seem important to me about this case:

  1. The student clearly didn’t think that “being bad at arithmetic” was a fixed quality, as this student was simultaneously asking for help. This goes against the “fixed mindset” interpretation, I think.
  2. The student clearly doesn’t, in general, think that they’re bad at all math. These expressions of frustration can be about anxiety that a student’s good thing is under threat. So while it’s never a good thing to hear a kid say that they’re bad at something, it can be a local, specific issue rather than a global one about their status in math class.
  3. The student didn’t think that it was socially OK to be bad at arithmetic or bad at math. This wasn’t a student being proud of being bad at math, and it wasn’t a sign of anything but their own frustration.

I’d be curious to hear other people’s stories about students who make “I’m bad at…” statement in class, to learn more about the different contexts in which kids say this. I’m sure, at the end, we’ll find that they always come at moments of frustration. But I suspect that if a kid is saying that they’re bad at math in class, there’s more to that story, and maybe it’s a sign that (paradoxically) some fundamental things are working for that student in class.