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Counting Fifty Dollar Bills

This yielded a few really interesting mistakes, all pointing me to the same conclusion: my 3rd Graders need to learn how large numbers work. Though, as always, the student thinking itself was interesting.

Sorry that this kid used the “highlighter.” But they got 6000 for those 30 fifty dollar bills.

Their count: 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 2000, 3000, 4000, 5000, 6000. Thousands are stupid, we should just eliminate them entirely and count “11 hundred, 12 hundred, etc.” It would solve a lot of problems! We’ll have to wait until I’m King of Math to do that one.

This same student made another mistake while working on this problem. They asked me if 1000 was made of three 50s. I asked, “wait what?” They clarified that since two 50s make 100, maybe three 50s make 1000?

Fascinating!

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The Mistake

Students don’t know how to read equations, and when they see two numbers they habitually add them together. Any blank is there as the result of an operation. The equals sign just means “make sure that you do this operation.”

My goal is to help students connect equations to the notion of equivalence — something that students in my experience already come into my classes with a decent understanding of, whether from experience or school.

I show this image, and talk about how we know that the pairs of apple buckets have the same number of apples (you can move one apple from one bucket to the other).

Then, we move to puzzles. I tried to leave different buckets “missing,” because I know that these are really four different types of problems. In particular, students are the most confused when the third bucket is missing (since they just tend to sum the first two numbers and put that in the third bucket).

Then, I want to nudge students towards connecting the arrow symbol to the equals sign and buckets to boxes of missing numbers:

This might also be a good time for this activity:

For an extra challenge, I ask students to only use the digits 0-9 each once.

Commentary

Every year I see this mistake in 3rd Grade. I’ve tangled with it over and over again, and I’ve also tangled with the research on the equals sign:

Does Understanding the Equal Sign Matter?

Equations and Equivalence in 3rd Grade

Ultimately, I don’t think the problem is in interpreting the equals sign exactly. It’s more about reading an equation, which is hard.

That said, I don’t want to ram directly into their rigid understanding of equals signs and equations, so I use arrows and buckets to help describe equivalence. Then, I just casually slide into using the equals signs and abstract equations in a similar way. That’s my current approach to making a change.

I used to have a big conversation in class about what the equals sign meant, but ultimately I became dissatisfied with that and moved to this approach.

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The Mistake

This is one of those things that sneaks up on me. I never think it’s going to be as hard for students as it is.

Even though I don’t have a picture of it, some students also say that arc AB is 100 degrees. In other words, we know that something is double the central angle and that something is equal to it, but we don’t know what.

Why?

Every mistake points to something that students don’t yet understand as well as they could. In this case, what is it? It’s such a simple relationship.

For a lot of areas of math, we don’t have strong intuitions about the topic until we get our hands messy with it. That’s not really the case with a lot of geometry, where we can see the diagram itself and often have visceral reactions to it.

What does the above diagram suggest, to the untrained eye? Maybe it’s difficult to see that the inscribed angle must be smaller than the central angle it shares an arc with?

What I Did

This was one of our online days, so I asked students to find the error and respond to these prompts in the chat.

I asked the first question because I wanted to draw attention to the most important thing to notice — that the central angle must be larger than the inscribed angle. I figured, that would make it easier to remember that it’s exactly half the central angle. Then, I figured it would be good to draw attention to the arc and clarify that the inscribed angle is half of that as well.

During the discussion I remembered my favorite way of thinking about this: like a rubber band or a slingshot that is pulled back. If you pull it back really far, doesn’t the angle on your finger get tighter? That’s what’s going on here too.

After calling on a few students to explain their responses, I asked them to use these ideas to try a similar problem.

After that, I paired students up (in breakout rooms) and asked them to try this amazing “misconceptions” activity that I found on Jo Morgan’s Resourceaholic collection.

Commentary

I guess the human eye isn’t really so sensitive to differences in angles! Makes me wonder whether I should incorporate some estimation activities into this unit. Would it be useful to estimate the sizes of the angles before learning any of the theorems, just to focus on the visual skill of seeing the inscribed angle as smaller? That sounds like it could help.

I don’t know much about research on geometric misconceptions. How does one train their eye to see things differently? The way I usually approach it in my teaching is to start by teaching the true relationship, then to clearly identify errors when they come up. Maybe I should identify those errors more systematically, earlier in the unit? Now that I have these little “mistake analysis” slides, I could use them soon after studying the inscribed angle relationship.

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So many skills involved: 0.2(d – 6) = 0.3d + 5 – 3 + 0.1d

This is a very challenging set of student work to look at as a teacher. It’s tempting to just mark this up as “confused about how to deal with complex equations,” but just look at all the moving parts! There are so many things going on here, and so many things that all of these kids know how to do.

For example, each of these kids distributes — not entirely correctly, but they do distribute.

In particular, not every kid handles the sign correctly — a bunch of kids are during 0.2(d – 6) into 0.2d + something. That’s interesting, and a discrete algebraic move that we could focus on later.

And then there is the multiplication of 0.2 and 6. Unclear whether that’s a mistake these kids would make under calmer situations.

Every kid knows they’re trying to get all the variable expressions on one side, though some kids subtract or move the 0.4d rather than subtract it from each side. (It makes me wonder if they would’ve made the same mistake if the 0.2d was on the right side, as then it would be subtracting a smaller thing from a larger thing. Kids often have a harder time seeing subtraction if it involves creating a negative quantity.)

So much complexity here! I’m sure there’s more that I haven’t noticed.

Thanks Deb!

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Solving equations, trying to distribute, lots of stuff going on!

What does this student already know? In other words, what don’t you have to teach again?

What’s going on with the distribution?

You’ve got time for a 5 minute activity to help this student. What activity do you pick?

(Thanks Shauna!)

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8 + 32 = 40 <--- wait, what's the mistake??

Greetings from Singapore!

This is from my son’s Primary 1 workbook. He had written 38 instead of 32 in the top row and 2 instead of 8 in the second row.

The submitter thinks that this teacher’s feedback was probably not so effective, and I’m inclined (from a distance, obviously) to agree:

The teacher’s marking indicates that my son should fix his mistake in the parentheses provided. Yet seeing the “mistake” made me pause. I suppose that if the focus is on simply getting the numbers transferred correctly, then it’s a mistake. Yet if the focus is on finding the sum, and stating the relationship between the two numbers, then perhaps it shouldn’t be considered a mistake. This is interesting to me not so much because it reveals what’s going on in the kid’s head as it does how often teachers have narrow ideas of what’s correct. As for what’s going on in the kid’s head, he could’ve simply been rushing. He correctly transferred the numbers in the other problems on the worksheet. This “mistake” could’ve been used to spark a discussion about why the correct answer was obtained and when it’s appropriate to shift quantities around and when it isn’t.

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What makes an equation tricky to solve?

Here are four equations, each using similar numbers. Which one would be the most difficult for a student to solve?

Conventional wisdom is that this question is too complex to really grapple with; it depends on the kids, the class, what they’re studying, etc.

I think the conventional wisdom has a point, but I know that I’ve learned a lot about the differences between these four problems, and there ought to be a way to teach what I’ve learned to others. Of course, only other people can be the judge of whether I know something worth teaching! Here’s a rough attempt to get at some of the pedagogical structure that underlies these problems.

Equations that can trigger arithmetic

Part of the problem with trying to make generalizations about how kids will tend to struggle or succeed in solving equations is that there are MANY ways to successfully solve an equation. I can’t say, “when kids see bla bla bla they do bla bla bla,” because just as many kids will as won’t do that, and it largely depends on the approaches to instruction that the teacher takes with the class.

So I want to play with the idea that it can be helpful to think of an equation as having triggers that can activate various strategies that a kid might potentially use for an equation.

Here’s at attempt to explain what I mean in the context of the four equations above. There are many different ways that a child might think about solving x – 12 = 9, especially after receiving instruction on how to solve these types of equations from a teacher. Here are just a few:

• An “undoing” metaphor which involves adding 12 to the right side
• A “balanced moves” metaphor which would involve adding 12 to both sides
• Just to get weird: a kid could multiply both sides by -1, add x to both sides, then perform 12 – (-9).

I’m sure there are many more potential strategies out there. What I think is special about x – 12 = 9, though, is that it is especially well-suited for one way of thinking, one strategy, in particular:

• Just treat it like an arithmetic problem — what minus 12 is 9?

The idea I’m playing with is that the equation contains a trigger for using an arithmetic strategy, i.e. a strategy that depends on no new algebraic metaphors, procedures or ideas, just whatever knowledge of the operations and arithmetic a student comes into this work with.

I want to be clear about two things. First, I’m not claiming that all students are going to approach this equation using arithmetic. Instead, I’m saying that these equations are especially well-suited for arithmetic, and a student who uses arithmetic sometimes to solve equations would very often use it in x – 12 = 9.

Second, I think that arithmetic is generally something that students have had a lot of experience in, compared to algebra. By the time an 8th Grader gets to an equation like x – 12 = 9, thinking about this equation in terms of arithmetic is likely to be productive and yield correct and helpful thoughts about the equation.

So, here’s a big generalization that I want to throw out there: equations that can trigger arithmetic are going to tend to be easier for kids.

Not all arithmetic is equal

One of my favorite things in math education thought is CGI. CGI stands for something, but I don’t really care about “Cognitively Guided Instruction.” What I love is the way it so convincingly analyzes the different strategies that young children will tend to use for various arithmetic problems. Here is an excerpt from a wiki article I wrote on CGI:

Their earliest work identified a taxonomy of addition and subtraction word problem types. Different problem within this taxonomy are differently difficult for children even when the “arithmetic” remains constant. For example, “Tom had 11 apples, and then lost 3 of them, how many does he have left?” might be easier for a student to represent than “Tom has 11 apples and 3 of them are spoiled, how many are not spoiled?” as the former problem has an action that a student can act out using counters, paper or some other representation.

Different arithmetic word problems trigger different strategies, even when they use the same numbers. And they have a systematic way of thinking about it!

(I hope it’s clear that they’re my paradigm for the sort of systematic approach I’m trying for here…)

Here is my understanding of the way arithmetic works. Kids form meaning of the operations in terms of contexts and word problems that are naturally associated with actions. So, subtraction gets associated with taking stuff away , and that meaning is there even when the contexts aren’t. Attempts to give kids alternate ways of seeing the operations are hard precisely because the operations are built so snugly on these paradigmatic actions. This determines which strategies get triggered by various subtraction problems, even without contexts, and even when they contain the exact same numbers.
And here’s a particular example, connected to the equations above: x – 12 = 9 and 21 – x = 9 are not the same type of problem. The first has the start as an unknown (if you think about it as an arithmetic problem, aka with the paradigmatic meaning of something takeaway 12 is 9); the second has the change as the unknown (aka there was 21 and then some got taken away and now there’s 9).
Neither equation is particularly easy to think through using arithmetic, but there’s a strategy available to young students for 21 – x = 9 that isn’t available for x – 12 = 9. To solve 21 – x = 9, you can go down from 21 until you get to 9, i.e. 21 minus 10 is 11, minus 11 is 10, minus 12 is 9, got it! 12!
You can’t really do that for x – 12 = 9 because you don’t know where to start. You’re left with trial and error. Of course, trial and error was an available strategy for 12 – x = 9 too, so there are fewer approaches available for x – 12 = 9.
In other words, if you think about it in terms of arithmetic, x – 12 = 9 is probably harder for students than 12 – x = 9.

Equations that don’t trigger arithmetic

But of course most equations can’t be solved easily by thinking of their paradigmatic meaning from arithmetic. Often you can de-trigger arithmetic in an equation by messing with the numbers, and intimidating any kid’s sense that arithmetic might be productive:

x – 12.05 = 9/2

You could think about this as arithmetic, but the arithmetic would not be a friendly way to go about this since the numbers are not so amenable to the strategies and experiences of arithmetic. That’s a precise sense in which this type of equation is more difficult — arithmetic would be unavailable to students who might solve it, and they’d have to rely on newer, more algebraic approaches.

You can also de-trigger arithmetic by tossing in more variables, but here some subtlety is needed:

21 – 10x = 9

It seems likely that students would not tend to use arithmetic naturally on this because it’s so much more complex looking than a typical arithmetic problem. At the same time, kids can learn to use their arithmetic on this type of problem, and for a lot of kids this can be productive as they’re getting the hang of algebra.

If a kid is comfortable with arithmetic, they could think of this as 21 – BLA = 9. BLA must be 12, so 10x must be 12, so x must be a tenth of 12, or 12/10.

Some people call this move “covering up” or “chunking” the equation, but the point is that it extends the power of arithmetic to apply to a greater class of equations.

Precisely because 10x – 12 = 9 is tougher on the arithmetic, it’s less likely for “covering up” to be helpful on this problem. So this a precise sense in which 10x – 12 = 9 can be tougher than 12 – 10x = 9 — there’s a strategy for one that the other doesn’t have.

That said, there are other interpretive issues that could arise for 12 – 10x = 9 if you use a more algebraic technique that wouldn’t arise for 10x – 12 = 9. In particular, the metaphor of undoing or unwinding (i.e. inverse operations) fits more naturally with 10x – 12 = 9 than 12 – 10x = 9. So that’s another precise sense in which one of these equations is trickier than the other.

In practice, I’d predict that 12 – 10x = 9 is easier for students very early in instruction who would be more likely to be triggered to use arithmetic, whereas later in instruction when kids have learned algebraic techniques 10x – 12 = 9 would be easier, because it fits better with the unwinding/undoing way of seeing an equation.

So, in short, not only are equations that have arithmetic triggers maybe easier, but equations that can be covered-up to allow arithmetic to hook-up with a student’s thinking will also be easier.

(Another subtle distinction: 12 – 2x = 9 allows for easier arithmetic than 12 – 10x = 9 because you can use your arithmetic to solve 2x = 12 but it’s tougher to use arithmetic for 10x = 12.)

Texts and teachers don’t make these types of distinctions

There are a few other dimensions that I think equations differ in important ways. Here are a couple that might make for good other case studies, if this becomes a series of posts:

• Equations that trigger problems with managing zero (like 2x – 5 = 7 – 3x)
• Operations that fit naturally with a “balancing” metaphor (adding/dividing, maybe multiplying) and operations that don’t fit as naturally (subtracting)
• Distributive property equations that are amenable to “covering up” vs equations that aren’t

This is rarely (never?) the way equations are presented to kids in texts or by teachers, though. I think the dominant approach is to classify problems by surface-level complexity and to vary everything else, in the hopes that kids will get exposure and practice with all these different types of problems. My point is that there is no organized, principled, systematic way that we have of thinking about the various different problem types of equations, so we just typically put them all in a blender and then present them to students all at once. And then we spot-check the difficulties for months and months.

And my theory is that if we have a theory of the problem types and micro-skills that give equations their underlying pedagogical structure, we might be able to design better resources and teaching for kids in this skill.

And my other theory is that if we’re attuned to this sort of thing, even if we don’t have a system, meaning if we’re looking out for these sorts of problem types and micro-skills, we can generate better feedback and responses to the stuff we see kids have trouble with in class and on assessments.

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Dear Diary: What types of equations are hard to solve?

I’ve been begging people lately to send me their solving linear equations mistakes. As long as people indulge me, this will be a series of posts devoted to thinking aloud about what sorts of linear equations are hard to solve, and what it is that makes them hard, and what exactly it means to know how to handle every equation that’s out there. The goal I have in mind is a way to be able to teach a (relatively) new teacher what sorts of equations they can expect their students to struggle with and (ideally) how to systematically teach this stuff so that kids see all the cases. (Besides for the obvious, e.g. an equation with lots of stuff in its that requires a ton of steps).

I don’t have settled thoughts about this, but I do need a place to work this stuff out. These posts are unfinished and sketchy. It’s basically my math mistakes diary. You should only read this if that sounds like fun to read…

…and here we go!

***

I think there are a lot of things that make this hard. I don’t know about Michelle’s students, but fraction multiplication is touch-and-go with my students, so there’s that factor.

One of the most important little “micro-skills” relating to equations that I teach my students is about scaling equations up so as to eliminate fractions. I think that would be useful here.

There’s an interesting choice about whether to do that scaling before or after distributing. That’s another interesting little choice that a student would have to make. I could imagine designing a little activity centered on a bit of strategy: when does it make sense to scale up? I think the message to kids could be that if you scale up before distributing, you might get to avoid some fraction arithmetic.

Maybe the activity could look like this. You start with that equation and then you present two options.

“Amanda took that equation and did this: 14m – 7 – 3m/5 = 24/5 – 18m/3”

“Billy’s first move looked like this: 35(2m – 1) – 3m = 6(4 – 3m)” [include annotations to make it clear what Billy did!]

And then you offer some prompts to draw attention to the choices you get to make?

Micro-skills here: knowing whether to distribute first, or whether to wait; scaling up an equation to avoid fractions

***

Awesome!

Just be explicit about the mistake…we scale both sides by -6, but the left side ends up only getting scaled by 6. The fractions seem sort of incidental to this one. Maybe this is related to the thing where multiplying by a negative feels like it should make the stuff negative, rather than giving everything its opposite sign?

To what extent is this an equations mistake vs a negatives mistake?

Lots going on here. The idea that we’re scaling both sides vs. just the fractions is tricky it seems. Clearly fractions make an equation tricky to solve as it presents a number of conceptual issues. To what extent is it helpful for a teacher to understand anything about how kids handle equations with fractions besides “with difficulty”? It seems like all the interesting choices to make when handling equations with fractions are about finding ways to transform that equation so that we can avoid those fractions.

There’s an interesting choice to make about whether you try to multiply both sides of an equation by 2 or by -2. What makes that interesting is again that multiplying by a negative introduces conceptual issues that aren’t introduced by x2. I feel like I’m getting close to saying that, as a matter of coaching, it makes sense to encourage kids to scale equations by positive numbers whenever possible. (Which, I think, is always.)

Micro-skills involved: What do you choose to scale each side of the equation by? When do you choose to scale each side of the equation? (Do you distribute first, or scale first?)

***

This was one of mine that I wrote about elsewhere:

Yesterday I gave students a no-grades quiz in algebra. A student who, I had been told at the start of the year, frequently struggles in math, has been having a lot of success lately. She knew exactly how to handle both of the systems of equations that were on this short quiz, but she got stumped at one of the resulting equations:

$-1.7x = 4.3x + 3.6$

I didn’t know what to say when she got stuck, exactly, but I was fairly confident that this was an example of a micro-skill that she was missing.

She and I agreed that she’d like me to write a little example on the side of her page, so I wrote this:

$-2x = 5x + 7$

[I drew some arrows going down from each side labeled “+2x.”]

$0 = 7x + 7$

My student read the example and then exclaimed (in a way I can only describe as “joyous”), Oh wait, you can make 0 there?!

Micro-skills: Knowing that you can make 0.

There seem to be a bunch of cases where you introduce and manage zeroes in solving equations. Another one of mine that I have rattling around in the archives:

Check out the bottom right. This pair of students got frozen at y = 2y + 5 because they subtracted y from each side and then weren’t sure if they were allowed to end up with 0. (Two years ago I talked to these kids about this, so I’m not guessing from their work.) This seems identical to the situation that this year’s student.

The moral of the story to me is that making zeroes is something that students need “permission” to do. Not like permission from me, the mathematical authority. I just mean that it seems to be something that beginners need help realizing is a kosher move.

As long as this picture is here, another little decision that kids need to make (see top-right) is whether to add or subtract something when trying to use a balancing move.

Microskill: “making a zero” to decide whether to add/subtract something from each side of equation

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Dear Math Mistakes: Calculating Inheritance

Dear Math Mistakes,

This piece of student work [below] puzzled us. What’s your take on what’s going on in the first one?

Sincerely,

Baffled at Ben-Gurion

***

For our readers who don’t speak Hebrew, here is my translation of the problems:

A) The oldest brother received 72,600 shekels, which was 33% of the inheritance. What was the total amount of the inheritance?

B) The second brother received 37% of the inheritance. How much did he receive?

C) What percentage of the inheritance did the third brother receive? How much did the third brother inherit?

***

Dear Baffled,

Fascinating stuff!

From the second two problems, this student seems to have a solid procedure for finding a given percentage of an original amount. If asked to find 20% of 5000, this student would compute (20 * 5000)/100.
But what do you do when you don’t know the original amount? This is what the first problem calls for, since the total inheritance is the unknown. One way that a particularly sophisticated algebra student might approach this would be to solve the following equation: (33 x ?)/100 = 72600.
It seems to me that this student is trying very hard to head towards something like this sophisticated approach. They’re looking at (20 x 5000)/100 and trying to figure out how to invert the procedure, to solve for the original amount. Neither attempt really lands at anything accurate or workable, but you can see an attempt to make sense out of inverting a formula.
In that first attempt — I’m assuming it’s a first attempt — the student tries to put the brother’s inheritance at the bottom of the fraction and to insert the other two brothers’ percentage at the top of the fraction. Again, nothing truly sensical here, but you see three attempts at inversion — 67 instead of 33, multiplication by 100 instead of division by 100, and putting the inheritance in the bottom of the fraction.
In the second attempt the student tries to scale 72600 by (100 + 33) to uncover the original total inheritance. I see this as attempting something like scaling up the 72600, which is not a bad idea at all, but this is still being embedded in a larger attempt at inverting the “forwards” procedure.
(It also seems to me possible that this student tried to solve (33 x ?)/100 = 72600 for “?” and somehow mushed together the 33 and 100, but then I wouldn’t know where the 67 came from.)
What to make of all this? All that really matters for the student is what they can learn next, and engineering this is always the hard, context-dependent part of looking at student work. Based only on knowing this, though, I’d say that they could be ready to learn to solve (33 x ?)/100 = 72600 for “?.” There are other less algebraic ways to go about this, but it seems to me like this student could be ready for the algebraic approach.
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@JustinAion’s fascinating square root mistake

Nobody asked me, but here’s my thought.

When people learn something new, they cling to a paradigmatic case. This paradigmatic image matters more for how they use this new idea than whatever rules or logic they might otherwise adhere to. For example, when kids are first learning about triangles, they don’t identify new triangles on the basis of properties. They look at their paradigmatic image of triangle and compare this new shape to it. This is why kids misidentify so many shapes, at first. Below, 10 might be declared a triangle, though 5 would be rejected.

Now, it would be sort of besides the point to lament young children’s tendency to identify shapes in this way. This is just what the beginning of the learning curve looks like. It helps teachers to be familiar with this tendency — we can directly address it in friendly ways — but it’s totally normal. In particular, it’s not really an artifact of instruction.

(I suppose if kids make it to a weirdly old age without being able to logically identify triangles, yeah, that would be an artifact of instruction.)

I think the situation with square roots that Justin points out is pretty similar to this. When kids solve equations by taking the square roots of both sides, a lot a lot of these cases involve a square root solution. It seems totally normal for kids to start seeing this as a paradigmatic case, and to think that all solutions to such equations involve square roots. Totally normal, not something to stress too much about.

In fact, I saw this mistake in my 8th Grade class last week. A kid was using the Pythagorean Theorem, and had put little square roots over the side lengths. No stress: told him that this wasn’t necessary; reminded him of the conceptual meanings that made this move incorrect; reminded myself to include more chances for him to practice this idea; set him off to try the next problem, but without the extra square roots.

I think this is just how learning happens.

I’ve sometimes read or talked to teachers who wished kids didn’t make these sorts of mistakes. And I guess it would be nice if kids could just reach an age where they operated as logical, analytical and meaning-oriented students at the start of their learning curves. I understand why we teachers feel a bit of nervousness when kids aren’t being guided by meaning.

But ‘being guided by meaning’ is another way of saying ‘being guided by logic,’ and this is not my understanding of how beginners hold on to new ideas.

It’ll take time, practice, corrections, maybe a big ol’ worksheet, but if a kid made it that far in solving these equations, they’ll make it the rest of the way. Keep it up!