Does Understanding the Equal Sign Matter?

Six years ago I was so worried about students’ mathematical conceptions and misconceptions that I created a website to get better at identifying them. This project quickly began to question its own legitimacy. Are conceptions stable? What do we mean by “conception” or “misconception”? Are there other causes of student mistakes?

Lately, I’ve been coming to doubt a lot of the power of conceptions to explain mathematical thinking. But this is definitely not something I’m sure about, and a recent conversation with Kent and Avery made me even less sure of myself.

In wondering how I might test my ideas — or make more sense to Kent and Avery — I thought of a wonderfully clear piece of research by Eric Knuth and others about conceptions of equality. The researchers here argue that students don’t properly understand the equal sign, and that this poor conception is responsible for troubles they have with solving equations and algebra, in general.

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I wondered: what would happened if I read this piece through the lens of my new worries? Would I find my worries resolved? Or would they find renewed strength?

The researchers call the trouble-inducing notion of equality the “operational view.” This isn’t nearly as evocative as their other name they have for this, which they call seeing equals as a “do something signal.” This view is limiting. Much more useful for algebra is the “relational view,” the more sophisticated idea that what’s to the left and right of an equal sign must be of the same numerical value.

So, why does seeing the equal sign as a “do something signal” matter for algebra? This passage summarizes two possibilities from Kieran and Carpenter.

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It’s worth digging in here. Kieran says that the misconception surrounding equality makes it difficult for kids to make sense of algebraic expressions. Why? Because it seems to students that you must do something with a variable like a. After all, if we give meaning to expressions via equations and students see equations like a + 3 = 10, then a + 3 means “do something to a and 3 to get 10.”

If students had a relational view of equality, this would be better. How? Then these students would see a + 3 = 10 as saying that “a + 3” has the same value as “10.” They would treat “a + 3” as a complex algebraic object, not as a variable with an operation.

Undoubtedly, it’s true that students would benefit from seeing a + 3 as a composite object. But it seems to me a bit like cheating to say that this has to do with seeing the equals sign as the “do something signal.” The issue isn’t so much the equals sign as a signal, the issue is that the way kids are reading a + 3 makes the ‘do something’ interpretation natural. The reason why kids read the equality as “do something” is, I think, plausibly explained by their reading of “a + 3” as “a mystery number plus 3” instead of “the composite expression a + 3,” which would indicate the more sophisticated understanding

And what of Carpenter? He says that kids with a “do something” conception should have trouble with solving 5x + 32 = 97 by subtracting 32 from both sides. “What kind of meaning can students who exhibit misconceptions about the equals sign attribute to this equation?” he asks. “Virtually all manipulations on equations require understanding that the equal sign represents a relation.”

I’m stumped as to why he says this. If you think of equality as “do something,” then there’s a perfectly fine way to understanding subtracting 32 from both sides. The equation states, if you’re of a “do something” mind, that if you multiply some number by 5 and then (do something) add 32 to it then you get 97. You should subtract 32 from both sides, then, because if you didn’t add 32 then your answer would have been 32 less.

In other words, I see no reason why “backtracking” (or “undoing the steps”) wouldn’t be available to a “do something” student. What about seeing the equals sign as “do something” would get in the way of backtracking?

As with Kieran, there is something else going on. Students often have trouble seeing equations as objects as opposed to processes. In other words, a student who sees the entire equation as describing a process (“you get 97 when you multiply some number by 5 and add 32 to it”) it’s hard to take a step back and say, OK, how would this process have gone different if you didn’t add that 32? That step back is a meta moment; it’s when you’re able to start talking about the process, and that’s a big change for kids.

But is that change all about the equal sign? I don’t think that it is. We’re putting too much into the equal sign. The conception of equality is made to stand for much bigger and more complex shifts in understanding.

(Update, 12/20: It seems to me that I might have not properly described the “do something” signal. I think they’re saying that the equal sign literally tells you to do something to a and 3. That’s why it’s impossible to solve an equation like a + 3 = 10 — it’s nonsense, or it’s a signal to combine a, 3 and 10 in some way, like 13a or something. It’s true — if kids literally think that = is a command to do something then they can’t do algebra. They also can’t do ___ + 3 = 10 or 3 + ___ = 10. I’m saying there’s a nearby conception of = that allows kids to make sense of these equations. It’s the “pick a number and then do something” conception.)

And what about the research article itself? It aims to give evidence to support Carpenter’s contention. They ask students questions about the equal sign, and then ask those same students to solve equations. Does one’s interpretation of the equal sign impact one’s ability to solve equations?

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So, are these related questions?

Here is the figure showing the relationship between correctly solving an equation and relational/non-relational views on equals:

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Here is the figure that shows the relationship between use of an algebraic strategy for solving the equation and relational/non-relational takes on equality:

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In sum, if you think about the equal sign in a shallow way then you probably weren’t solving an equation using algebra in 8th Grade. And you were less likely to correctly solve equations across all middle school grades.

(Random aside: when writing about research I always prefer the past tense “you were less likely” rather than “you are less likely.” The latter supports the idea that a particular study provides a universal truth, the former allows you to focus on the current study.)

The authors of this study take this as evidence that conceptions of the equals sign impact your ability to solve equations. “These findings suggest that understanding the equal sign is a pivotal aspect of success in solving algebraic equations.”

What confuses me is how they know the arrow points in that direction. It seems just as plausible to me that students’ ability to solve equations in productive ways drives their understanding of equality. In other words, maybe how you think about equality and how you solve equations co-develop. Why should we think that how you think about the equal-sign is a gatekeeper?

The authors ask, “Does understanding the equal sign matter?” My answer is a strong yes, but I seem to interpret the question differently than the authors do. They are asking, “Is understanding the equal sign a prerequisite for learning to solve equations and do algebra?” They answer yes, but I’m unconvinced. But I certainly think that understanding the equal sign matters. It matters because as you learn algebra your understanding of the equal sign should change. It should change because the strategies that you’re learning to solve an equation often require you to think of equations in new ways, and maybe even there is no other way to gain this conception of equality than to learn algebra.

This last claim of mine — it feels like that might be going too far. But remember the terms of the discussion. The conception of equality it supposed to enable students to successfully solve equations and do algebra. Is it possible to offer students a sturdy enough conception of equality for algebra without doing algebra? I’m not sure at all.

One thing I’ve been telling Kent is that I prefer to focus on strategies rather than conceptions. The above gives a sense of where I’m coming from. The hope of Knuth seems to be that if you give kids a great way to understand equality as elementary students, they’re going to have a much easier time learning algebra. The conception will be strong enough to guide future work.

It doesn’t seem like that to me. It seems to me that a conception of equality that can support algebra develops alongside a student’s grasp of algebra. In other words, there are no shortcuts and no inoculation possible in the elementary years. If you’re interested in doing work with elementary students that would support their future algebra work, I think you have to do some algebra with them.

(Project LEAP — from many people related to this article! — is doing this research work for equations. Project Z is doing this for negative integer work. It’s a promising route, I think.)

In short: if you want to improve kids’ future conceptions, you can help them by helping them come to strategies they can use and understand. You can’t take a shortcut through conceptions.

(Random aside: is there a difference between a conception and a belief? It seems to me that conceptions are types of beliefs, and that beliefs are confusing and unstable and maybe not good things to focus on in explaining human behavior. I know Lani Horn has written about this.)

Here are some questions I’m left with:

  • Am I misreading the statistics?
  • Why did 7th Graders have a spike in relational understanding (43%), compared to 6th (32%) and 8th Graders (31%) in the study?
  • Is a strong conception of equality without algebra a resource for learning algebra? What’s the path — it feels like there should be one, but I can’t name it.

 

9. How do kids think about solving equations?

The last 8 posts in this series have been about working through different aspects of how kids might think about solving linear equations. In this post I want to try to synthesize what I now know about all this and ask some questions that I don’t yet know the answers to.

There are different types of linear equations that we want students to solve. What makes it worthwhile to distinguish the different types of equations is the different sorts of reasoning that is possible for different problems.

An aside: Some teachers balk at the idea that we should think of teaching in this case-by-case way. These teachers would rather equip students with general methods that work in any sort of equation rather than helping students develop strategies for different types of problems. I think that this sort of teaching strategy is often a mistake. Novices in many fields often have a hard time grasping general techniques, and what experts consider to be general strategies are often only comprehensible to students as a family of related techniques. In others words, general techniques are almost always something that we obtain by making sense of related, more specific techniques. I believe that this is true based on experience and the confirmation of expert/novice research, the failure of Polya’s strategies to improve problem solving in studies, and math education research like CGI that offers a description of how many students come to understand arithmetic. 

In the previous posts, I looked at problem types that differed along the structure of the equation:

  • ax + b = cx + d
  • ax + b = c
  • b – ax = c
  • a(bx + c) = d, etc.

It’s only worthwhile to distinguish problem types on the basis of the different thinking that kids can do with them. Ideally, we would want to look at kids’ work and talk to students in order to really nail down what thinking is possible, but in the previous posts we worked out something that seems reasonable. (The “we” that worked this out was me, a handful of really sharp commentators and a few sharp people on twitter. You know who you are. Respect.) The table below represents, I think, what we hashed out:

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(I obviously haven’t consulted with anyone about the correctness of this table. Let me know if you disagree. This blog is always about working out ideas, not about proselytizing for them.)

This table suggests a clear hierarchy of difficulty of solving equation problems. The equation structures that allow for the most strategies will tend to be the easiest for students, since it’s more likely that a way of approaching the problem that a student will try will bear fruit. The harder equations are the ones where the familiar strategies won’t work. Balancing won’t help you with b – ax = c. Neither will unwinding. Covering up might, but “covering up” might also be harder to learn because it isn’t as useful in the types of equations that students are most likely to practice at first — the ax + b = c and ax + b = cx + d types of problems.

Notice, also, that we can tell a story about the origins of a common misconception. Students will often try to move parts of an equation from one side of the equation to the other without keeping the sides balanced. This is a model that is useful to explain the sort of things that happen in ax – b = cx – d equations, where the subtraction makes it harder to represent the helpful algebraic move in terms of balancing weights on a mobile or any other physical quantities. Instead, it looks like we’re moving a number from one side to the other. This model sticks because, though it’s false, it can also be used to help students solve (incorrectly) so many other types of equations. It’s useful, even though it’s totally wrong, which is what makes it so tempting.

This all suggests to me that I have to give careful attention to the trickiest problem types — the ones that involve subtraction. What are the resources students might have to make sense of these types of problems?

  • We could help students develop equality properties in the “easier” problem types, and then ask them to apply that sort of reasoning to the tricker problems.
  • We could try to bend the balancing model so that it incorporates negative quantities. We might do this by treating ax – b as ax + (-b), and then removing a negative quantity from both sides of the balanced equation.
  • We could try to bend the unwinding model so that it too applies more broadly. This would also involve the manipulation of expressions, along the lines of b – ax = -ax + b.

My gut says that we can’t just choose one of these routes. Different approaches might make sense for different students, depending on how they’re solving the simpler equations. And though we eventually want all students to solve equations using equality properties, that might be a rocky road for some students. They might be only comfortable with unwinding when the rest of the class is working on the trickier equations, and they’ll need help using their own methods in these new cases.

That said, the goal is still to get everyone to use equality properties.

I think this summarizes what I’ve done. Now, what haven’t we done?

We haven’t yet created a curricular approach to developing these strategies. I’m interested in using mental equation solving activities to help students practice these strategies, along the lines of the number talks that I’ve seen work in my elementary classes with arithmetic. I’d love to have a collection of “equation strings” that target these strategies.

I thought about how the different structure an equation has can prompt different thinking. But that’s only one dimension along which an equation can differ. Equations can also differ in the numbers that are used in the place of a, b, c, and d in the table above. Some of the models are ruined by fractions, decimals or negative numbers. Others aren’t. I haven’t done any of that analysis yet, mostly because I wasn’t sure where to start. Maybe someone else can do that work, or maybe I just have to wait for when I can talk to algebra students once classes start.

Another dimension for these equations is the complexity of the expression. a(bx + c) = d wasn’t part of the above analysis. Neither was ax + bx + cx = d. There are infinite possibilities, and I wasn’t sure how to wrap my head around it without the cases multiplying and multiplying so that the whole thing got out of hand. I’d love for someone to pick this up and show me how it’s done.

On twitter, Kristin Gray really made it clear to me that it’s insanely hard to identify what approach a student has taken just based on their work. This made me quite nervous — maybe balancing reasoning is an illusion, and kids don’t really use that sort of a model when they’re working on equations. Maybe balancing reasoning is really indistinguishable from using equality properties. And how could we know if a student is unwinding or using balancing? Like I said, all of these are valid concerns that make me quite nervous. I need to get back into the classroom and talking with children so that I can hear how they talk about their thinking. I also need to see larger collections of student work beyond what I have lying around my apartment or mathmistakes.org.

Finally, now that I have all this student thinking down I think I could plan ahead for giving feedback that helps students move within the table above. What do I say if a student is using balancing, but I want them to be able to see the equality properties perspective? How do you get a kid from using the “move to the other side” model to the balancing model? How can I help kids who are only using arithmetic learn to unwind an equation? I haven’t done any of this work, but I think that I (or someone else) could.

I don’t know if I’m going to continue working on solving equations right now, but if I do I have some clear steps forward:

  • Write some mental equation solving activities ala equation strings.
  • Look at student work and listen to kids describe their thinking about ax + b = c and ax + b = cx + d equations to confirm that balancing/unwinding/equality properties really are distinctive ways of thinking about solving an equation.
  • Look at student work and listen to kids describe their thinking about equations with negatives, decimals, fractions, or more complex expressions to understand how they think about them.
  • Make notes about the feedback that would help kids move from one strategy to the other within all this.

That’s it, for now.

8. “We’re taking it away from one side and putting it on the other.”

In the video above, the student describes adding 4 to both sides as “taking the 4 away from one side and putting it on the other.” This is difference then “balance reasoning” or using the subtraction property of equations.

The problem is that this “Give and Take” model for what we can do with equations runs into problems about 50% of the time. It only works when the signs work out, and when it doesn’t it leads to errors:

In the above, the student (I think) took the 3x away from one side of the equation and put it on the left side of the equation. This is perfectly in line with how the student in the video describes his move.

Where does this “Give and Take” model come from? The video above gives a clue, since the student offers the explanation to make sense of adding 4 to both sides. The story that this student developed for himself was that we’re getting rid of the 4 and putting it on the other side. This is naturally appealing, since the right side of the equation ends up with a 0 and the left side ends up with 4 more.

The “give and take” model is a way of seeing equation-moves. It helps make sense of why we would add quantities to both sides of the equation, but it makes wrong predictions about what would work for situations that don’t call for adding something to both sides, like 2x + 3 = 3x – 4.

What can we do about the give/take model? Can it be avoided?

Maybe not. Students need some way to make sense of what we’re doing in 2x + 6 = x – 4. Balance reasoning is no help with the adding 4 to both sides part of this. (Why? Because we almost never have to add stuff to both sides in balance problems, and because negative quantities don’t make a ton of sense in the balance context.)  This equation isn’t primed for backtracking, since there are variables on both sides and backtracking doesn’t handle these situations well. Perhaps we want to tell kids to first deal with the variables, using balance reasoning to get x + 6 = -4 and then using backtracking, covering up or guess and check to handle this equation. Fair enough, but this is fragile. Students should be able to make sense of the entirely appropriate move of adding 4 to both sides for  2x + 6 = x – 4 somehow. For a lot of kids, the best way to think of this case is taking stuff away from one side and slapping it on to the other.

It seems unlikely that there’s some other powerful context and type of thinking that we’ve missed. At this point, the devil is in the details. I don’t know if there’s another way to make sense of why we would add 4 to both sides in this case other than the properties of equality, or as an abstraction from balance thinking (i.e. “the mobile will always stay balanced if we add 4 to both sides.”)

I guess we just need to be careful about giving kids too many of these types of equations without justifying the move in terms of balance or equality properties.

7. How kids might think about different types of equations

I think what is really becoming clear to me is that the real work I have to do is in clarifying how kids can make sense of equations that go beyond the simplest cases.

The simplest cases are ax + b = c and ax + b = cx + d, where all the numbers are integers. They’re relatively simple for students because they can use balance reasoning or backtracking/unwinding to tackle them.

Things get more complex with equations of the type c x (ax + b) = d where the numbers are integers. I would expect these to be not-that-much harder for kids, since they can use unwinding and covering up to crack these. Still, these equations afford a new strategy, that of using the distributive property to transform this into an equation of ax + b = c type. I would bet that this is the best way to get manipulating expressions on the table, as a strategy.

b – ax = c is a lot harder, even with a, b, c as integers. It’s nearly impossible to directly use either unwinding or balancing on this equation. (Though if b > c then students can at least use the “cover up” method.) To use either balancing or unwinding, some arithmetic manipulation likely needs to be involved, treating b – ax = c as the same as b + (-ax) = c. This is useful for kids, but it’s a slippery move that many will struggle with. Another algebraic move could be transforming b – ax = c into b = c + ax. This involves having a strong grasp of the addition property of equations, something that students might develop from balance reasoning. Still, this is tough.

I don’t know whether it’s worth it to identify b- ax = c +/- dx as a distinctive type of problem. The same problems with balancing and unwinding apply, and covering up seems unlikely to help much. It seems likely to me that kids will be tempted to use balance reasoning, because of the surface resemblance of this type of equation to ax + b = cx + d. I would expect that to be the major source of errors. If we’re creative with the balance model, we can try to find a way to represent this and to use balance reasoning on this. That seems tricky to me, and I’m feeling like that might not be worth it. The other move, as before, is to use some algebraic manipulation to turn b – ax into b + (-ax). But, again, this is hard.

So far, so good. But where things get really complicated, I think, is when negative numbers and fractions get involved in any of these problem types. (As I write this, my thought is “well duh.”)

How kids think about equations with fractions, decimals and negatives in them is something I’m entirely unclear about. I think this is a place where I need to focus.

I think there’s a case to be made that all this work with integer equations should go before we spend much time working on equations with fractions/decimals/negatives involved. These trickier numbers might distract from the reasoning that we’re working on developing. This might be a situation where it makes sense to be as fluent as possible with something like balance reasoning in its natural setting before stretching the metaphor to include things like 1.4x + 9 = 0.7x + 8.

(Though, come to think about it, that’s actually pretty natural in a balance setting because weight can take on any positive value. A better example would have been -0.2x + 5 = 2x + 3. To me, this mistake in my writing underscores how little I know about non-integer equations and how kids can think about them.)

Next steps: non-integer equations, and thinking about feedback that could help students progress when working with different integer equations.

6. Equation Strings

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From http://numberstrings.com/2015/01/30/true-or-false-can-a-number-string-develop-relational-thinking/

In my elementary school classes, I find it very helpful to ask students to do mental math in addition to written math. I’ll often put a series of related problems on the board and ask kids to figure out the answers in their heads, and then we’ll share strategies. Some people call these number talks, other people talk about number strings. Whatever you call them, these are helpful for my arithmetic teaching for three reasons:

  • Working in our heads makes certain brute-force strategies harder, so we get a focus on more efficient strategies. An example: it’s harder to do 7 x 9 = 9 + 9 + 9 + 9 + 9 + 9 + 9 in your head, so it’s an opportunity for kids to practice picking a more efficient strategy for a tricky situation, like 7 x 9 = 7 x 10 – 7.
  • Students are likely to come up with different strategies for these problems, and this gives students a chance to contrast other strategies with their own. I think this helps kids better understand all these strategies.
  • Finally, stringing a few related problems together can allow us to be explicit about the connections between simpler arithmetic problems (e.g. 10 x 7) and more complex arithmetic problems (e.g. 9 x 7).

I think there’s a good analogy between learning to fluently and efficiently multiply and learning to fluently and efficiently solve equations. I also think I might find it useful to introduce “equation talks” or “equation strings” this year.

Now that I’ve clarified for myself what strategies kids might use for solving equations, I feel equipped to think about what equation talks might be target those strategies.

For targeting “balance thinking”:

  • x + 1 = 2x
  • 2x + 1 = 3x
  • 4x = 1 + 3x
  • 5x + 2 = 3x

For targeting “backtracking” or “unwinding”:

  • 2x = 4
  • 2x + 2 = 4
  • 2(x + 1) + 2 = 4
  • [2(x + 1) + 2]/2 = 4

For targeting “covering up” in b – ax = c problems:

  • 5 – x = 3
  • 5 – 2x = 3
  • 10 – 2x = 3
  • 100 – 2x = 3

For targeting balance reasoning with decimals:

  • x + 3 = 2x
  • 0.5x + 3 = 2.5x
  • 3 + 1.4x = 5.4x
  • 1.1x + 6 = x

And, so on. I think this will be helpful.

(Friendly wager: I bet we’ll see a book called “Algebra Talks” or something like that published in the next five years. Takers??)

5. Looking at research on how kids think about solving ax + b = c

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This is from Does Understanding the Equal Sign Matter? Evidence from Solving Equations  (Knuth, Stephens, McNeil, Alibali, 2006). The article is about an aspect of solving equations that I haven’t focused on yet — the meaning of the equals symbol — but their research is useful for me in some other way.

They were interested in the connection between conceptual understanding of the equals symbol and the ability of students to solve equations. In doing this work, though, they asked a bunch of students to solve equations and coded their strategies. I’ve been interested in getting clearer on the ways kids think about solving equations, so this is a good bit of evidence to serve as a check on what I’ve been blabbering about.

The researchers identified 6 different sorts of responses to 4m + 10 = 70 and 3m + 7 = 25:

  • Answer only
  • No response/don’t know
  • Guess and test
  • Unwind
  • Algebra
  • Other

It’s good to know that students are using unwind (i.e. backtracking) in solving equations, and that the researchers could distinguish this from algebraic moves. (The difference is this: “in using an unwind strategy students start with the constant value from one side of the equation and then perform arithmetic operations on that value.” They coded something as “algebra” if it operated on both sides of an equation, like “subtract 10 from both sides.”)

What about the absence of strategies such as “balance reasoning”? It’s not surprising that this strategy was absent, since the “ax + b = c” equation type doesn’t really give much of an opportunity for that sort of thinking. It’s much more likely for balance reasoning to come up in equations of the type “ax + b = cx + d.”

Still, I wonder how could these researchers would have distinguished between balance reasoning and algebra had they studied these types of equations? I suppose that some students could make the balance reasoning explicit by drawing some sort of figure. Besides for that, though, I don’t know. If you take away an x from each side, that’s going to look like algebra.

I wonder if it would come through in students explanations of why you can take away an x from each side. This makes me think that balance reasoning (as opposed to use of algebraic principles) will be harder to identify in students’ thinking, but might be identifiable in student explanations.

In sum, reading the Knuth paper gave me confidence that unwinding/backtracking really is a strategy that students use and develop as part of their equation solving repertoires. It’s not some magical way of thinking invented by math teachers that kids only use if you tell them to.

This also helped me identify two different problem types for solving equations:

  • c x [ax + b] = d, where unwinding/backtracking is a potential strategy, in addition to using equality properties and guess-and-check, but balance reasoning isn’t likely to show up.
  • ax + b = cx + d, where balance reasoning is a potential strategy, in addition to using equality properties and guess-and-check, but unwinding/backtracking is unlikely to help.

As long as I’m here, I might as well add a third problem type:

  • b – ax = c, where neither backtracking nor balance reasoning are much help.

Are there any strategies, beyond algebra, that are helpful for students as they try to solve b – ax = c? I feel like we’re close to getting all the big picture ways of thinking about solving equations on the table. There are so many types of equations, though, and there are lots of mini-moves that kids need to learn.

I feel like digging into what kids are likely to develop out of guess-and-check with b – ax = c would be interesting. I wonder if I can find any papers that detail student work for those types of equations. I wonder if anything on mathmistakes.org can be helpful. I wonder how close I can get by just spitballing based on experience?

4. How do kids think about unsimplified equations?

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Balance problems provoke students to use a special sort of thinking — what I’ve been unhelpfully calling “balance reasoning” — and this sort of thinking can be helpful for solving equations. This has potential to be helpful for students in their path on the way to using equality properties to quickly solve equations.

But I really, really wish that I didn’t have to rely on balance problems for this.

That’s not because I have anything against balance problems. They’re lovely! Students should get a chance to play with them. But I’ve taught too many sad high school students who shiver when an equation shows up in class. I want these kids to be happier in those classes, to be able to see more lovely math. I really, really would like my students to be able to solve equations, and relying on balance problems to do some of this work makes me nervous.

I get nervous about relying on balance problems because it’s really a subtle thing to help students make connections between two different scenarios. It’s possible, of course, but it’s hard. This might be a melodramatic way to put things (sorry) but using these problems is a risky move.

(And the same can be said for using “pick a number” problems.)

Is there any way I can gain the benefits of balance problems without shifting contexts from equations to mobiles?

The benefit of balance problems is “balance reasoning,” a particular way-of-seeing an equation. Can this way-of-seeing be developed by students while sticking to the solving equations context? If so, what experiences would they need to have in order to develop this way-of-seeing?

I’m not sure. I’ll put out a guess, but what I (or somebody else) would really need to do is watch kids work on different sorts of linear equations and listen to them talk about how they solved them.

Here’s my guess: most students will eventually develop balance reasoning after lots of experience with unsimplified equations. By unsimplified, I mean equations like these:

  • 2 + x = x + x
  • a + a + a + a – 12 = a + a + a + 8
  • 5 – b – b = 3 – b

Unsimplified equations are (more or less) the equations you’ll get from translating balance problems into equations.

I bet that kids who work on these equations using guess-and-check will eventually be able to understand (either by discovery, instruction from friends, discussion, direct instruction from me) how to use balance reasoning as a short-cut for their inefficient methods. And then I bet we can name it, formalize it, and extend it to less obvious situations (like 5 – b – b = 3 – b, which is tough to represent in a balance problem).

(I wonder whether these students would draw on a balance metaphor, or whether it would just be a regularity that they notice? If students do notice this pattern, how would they make sense of it? Which is another way of asking, how would they remember it?)

It’s important to note that unsimplified equations are not quite the same thing as the sorts of equations we eventually want kids to be able to solve. Not everyone who can solve an unsimplified equation will be able to solve the simplified version of that equation. Kids might not see that 4a – 12 = 3a + 8 is the same as its unsimplified version. This is the same dilemma that we had with balance problems in the first place (transfer of skills from one type of problem to another). But I’m less worried about the jump from unsimplified to simplified equations than I am about the jump from balance problems to equations. We’re all still talking about equations, it’s a smaller leap than the leap from a context that might look drastically different from an equation to a kid.

If all this is right (and who knows if it is!) then students can develop balance reasoning in an equations context, rather than a different scenario. And there are advantages to that — the leap from equation to equation is less tricky for learning than the leap from balance scenario to equation.

There still might be reasons to study balance problems in class, or even to use balance problems to develop balance reasoning.

  • It often happens in my classes that kids get fed up or frustrated after a week of playing with any one scenario. They need something that isn’t just different, but it has to feel different to them. Balance problems certainly feel different than equations.
  • The surface level difference between balance problems and equations is also helpful when dealing with students who have bad associations with equations, variables and numbers. Balance problems could be a sneaky way to get balance reasoning on the table for a group of kids, before we arm them with it for equations.
  • Because balance problems look different, on the surface, they could make great application problems for students who understand equations in some other context. Or maybe it’s helpful to start a unit with a few balance problems, to establish a metaphor that we can later connect to the unsimplified equations?
  • Maybe we think that the learning of balance reasoning will be more effective if it takes place in the balance problem context, for some reason.

I’m influenced here by the way students can get better at arithmetic in elementary years. There’s no substitute for work with actual numbers in number contexts. Students start multiplying with some basic fact knowledge, but they extend this knowledge by finding shortcuts and regularities (successive doubling, doubling and halving, finding friendly numbers, breaking apart by place value). This makes me wonder whether it’s a mistake to jump to balance problems to develop balance reasoning for solving equations. (Though balance contexts might serve an important purpose for all the reasons mentioned above.)

If any of the above is right, then the next step of this project needs to outline problem types for solving linear equations. (Analyzing problem types was the first published part of the Cognitively Guided Instruction project.) Here are some problem types I know of so far:

  • Simplified vs. unsimplified equations
  • Equations with addition vs. subtraction
  • One-variable equations vs. multiple-variable equations

The importance of these problem types is that different linear equations allow for different sorts of reasoning. I want to get systematic about what each of those types of linear equation are, and the thinking that they each invite.

Then, I really really really need to start looking at some student work to see if the picture I’m developing resembles reality at all.

3. How do kids think about “pick a number” problems?

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Source: http://algebra.mrmeyer.com/

I want to help students move from guess-and-check to using equality properties in the equation work. There are some problems that might be helpful in this work — like balance problems — but there are limitations to their usefulness. Balance problems allow for a sort of “balance thinking” that has close analogies in the world of solving equations — when you take things off of both sides things stay balanced (subtraction property) and when you divide each side equally each divided part is balanced (division property).

It’s hard, though, to represent negative numbers or fractions in balance problems, and there’s little reason to use anything resembling either the addition or multiplication property in that work. So there are limits to the usefulness of balance problems.

What other problems might be helpful? Another promising problem type is the “pick a number” problem, as seen at the top of the post. (There has got to be a better name for this type of problem. They’re not number tricks. “Closed number tricks”?)

I’d like to get precise about what makes pick a number problems promising for learning to solve equations. There are two things that I’m looking for to gauge potential usefulness of a new scenario or problem type: resemblance and novelty.

  1. Resemblance: My students will able to use guess-and-check to solve equations, but I want them to be able to use equality properties. Can guess-and-check and equality properties both be used for this problem type?
  2. Novelty: To be useful, some different way of thinking about this problem type, beyond guess-and-check and equality properties, needs to be available to students. Otherwise, the problem is just functionally equivalent to my equations.  Is there a new strategy students can use to think about this problem type?

“Pick a number” problems allow for guess-and-check, and you could also translate the steps into an equation and solve it conventionally. So there is resemblance.

The novelty is “backtracking” thinking. This is the thinking that, step by step, reverses the steps of the recipe until the original “pick a number” is revealed.

So, kids are likely (I bet) to think about backtracking problems in one of three ways:

  1. Guess and check
  2. Backtracking
  3. Equality properties

Suppose that kids got very, very good at these “pick a number” problems. What benefits could that have for their solving equation skills? Could backtracking help them develop equality property strategies?

I see two possible teaching strategies along these lines:

  • Ask students to translate equations into steps, and then have students use backwards thinking in its natural setting.
  • Develop generalizations in the “pick a number” problem that can be extended to solving equations.

The first option seems like a bad idea, since there are all sorts of equations that cannot be usefully translated into steps. Even basic equations like 4 – x = 10 don’t really work well with backtracking reasoning. (Pick a number. Start with 4. Subtract your number. You get 10.) You’d have to help students understand a specific class of equations that work well with backtracking, and that sounds like a headache.

I like the second option more. We could form generalizations from backtracking that are easily applicable to equations. “To find your number, you need to undo the operations that are done to it” is as true for “pick a number” problems as it is for equations. Of course, we would need to clearly show how that reasoning applies. But work in these problems could help students use the inverse of operations to reveal the starting number. Balance problems only gave you the subtraction and the division properties, but “pick a number” problems also give you multiplication, addition, squares and square roots properties of equality.

The big limitation is that variables have to be on one side for “pick a number” problems to work. These problems are unlikely to develop tools that are helpful for anything that has a variable on each side, because backtracking only works if we’re imagining the equation to have one, single picked number. How can you reverse the steps in “I multiply my number by 7, I add 18, I get 10 times my starting number”?

That’s the first big limitation of this problem type.

The second big limitation is the same one that balance problems had: this is a new scenario. Any skills that we want students to take away from “pick a number” or balance problems to equations will not come for free. We will need to spend time helping students see how the skills they use in one context are applicable in another. There’s an art to this, but maybe it’s not worth it.

I want to think about this more in the fourth post in this series.

2. How do kids think about balance problems?

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When I teach students to solve linear equations, I’m aiming to help move them from guess-and-check to using the properties of equality.  I’m in search of the ways of thinking that could help students move towards the more sophisticated techniques.

Balance problems (i.e. mobile problems) seem promising. They seem promising because they can be solved using guess-and-check, just as linear equations can. They can also be solved using the properties of equality, if you translate the situation into a linear equation.

In short, the thinking that students can do with balance problems is close enough to the thinking used to solve equations to be useful. If there happens to be a special way of thinking about balance problems, that way of thinking might help students develop facility with properties of equality. Much in the same way that visual pattern problems can help shepherd students from recursive to functional thinking, balance problems might help students move from guess-and-check to equality-property-thinking.

And, in fact, there is a special way of thinking that balance problems allow for. I don’t know what to call it. Maybe I’ll call it “balance reasoning.” That seems almost tautological, but whatever, balance reasoning! Balance reasoning is available for balance problems.

This is the reasoning that goes like, “If the left side has four circles and the right side has one circle and two triangles, then it’ll have to stay balanced if we take one circle off of each side. So it must be that three circles on the left side and two triangles on the right side are still balanced.”

In sum, students can use three types of reasoning for balance problems:

  1. Guess-and-check
  2. Balance Reasoning
  3. Properties of Equality

Noticing patterns in what we do with balance reasoning can help us develop properties of equality within balance problems. With enough experience with the sort of “remove the same things from each side” thinking we could formulate a technique: “If you can, take off the same number of things from each side.” This is a close analogue to the addition property of an equation.

With enough experience with other balance problems, we could get something like the division property of equality: “If you have all squares (or whatever) on one side and all triangles (or whatever) on the other, split up one side and give it to the other.”

Yay, kids can now solve equations! Right?

Well, not so fast. All of this is taking place in an unusual, foreign context. Balance problems are not equations. How do we make the move back into solving equations? I think we need to design activities that make explicit the connections between balance problems and equations. We translate back and forth between them. We formulate new rules for solving equations that explicitly mirror the rules that we know and love from balance problems.

So, now kids can solve equations! Right?

No, because balance problems aren’t natural settings for a lot of equations. Anything with negatives or fractions or subtraction is awkward to use with balance problems. Sure, we can extend balances to work with all sorts of signed variables or subtraction (like -2x – 3 = 0) but it’s awkward. It’s mathematically fascinating and valuable for its own sake, but unlikely to be much help when a kid is staring at an equation and feeling the urge to guess-and-check.

In short, balance problems can be solved using three modes of reasoning. One of those modes has promise for helping students think using properties of equality, but not all the properties of equality.

The goal of using balance problems in this context would be to develop properties-of-equality-thinking. Our feedback should be oriented towards that. Balance problems are useful, but only in a very particular way.

In the next post, I’ll think a bit about developing facility with other properties of equality.

1. How do kids think about solving equations?

I want to get ready for the school year. I’ll be teaching Algebra 1, and that’s where solving equations happens. These skills are seriously useful, so I want to do right by my students. I want to teach this topic well.

As far as I can tell, the most useful way to get better at teaching something is to better understand how kids will think about that something. So: how do kids think about solving equations?*

Actually, there are a few different ways of wording this question I might focus on.

  1. How do kids think about solving equation problems?
  2. How can kids think about solving equation problems?
  3. How will kids think about solving equation problems?

These three questions have different connotations to me. Do suggests the present reality. If I surveyed 1,000 students, what sorts of thinking would they in trying to solve an equation. Can implies future possibility. This might include useful, but unnatural models. Will implies future inevitability. An answer to “How will kids think?” might include common misconceptions or models that students tend to fall into.

I think I’m interested in answering all three questions. I want to know the ways that my students might think about this problem when they walk into my class, I want to know which models might be helpful for them to obtain, and I want to know what ways of thinking they are likely to fall into.

With all that said, I sit down with a pen and paper and try to get clearer on the potential thinking. A big problem quickly arises. There are a lot of ways that students do, can or will think about linear equations.

Here was the list that I started making for how to solve 2x + 3 = 10.

  1. Mental Guess and Check – “2 times 4 plus 3 is 11… 2 times 2 plus 3 is 7…”
  2. Guess and Check on paper-  Same, but on paper so it’s easier to notice how close we’re getting.
  3. Organized Guess and Check – “2(0) + 3 = 3.,  2(1) + 3 = 5,  2(2) + 3 = 7 …”
  4. Make a Table – Which is a way of organizing an organized guess and check
  5. Use a graph – Graph y = 2x+3 and look for where y is 10.
  6. Use a scale model – (Either with manipulatives or with paper.) Draw 2 circles and 3 boxes on one side of the scale, 10 boxes on the other. Proceed…
  7. Use a number line – Start at the origin. Draw an arrow to the right, label it “x.” Do that again. Then indicate a move to the right of three units. The whole thing lands at 10 on the number line. Proceed…
  8. Work Backwards – “We’re adding 3 at the end to make 10, so everything else makes 7. So what times 2 makes 7…”
  9. Use Properties of Equality – “Subtract 3 from both sides. Divide by 2.”

There are certainly more ways that students might think about solving equations that I haven’t listed here.

This listing could go on and on, but I don’t think this list-making will help me very much. What I need isn’t an encyclopedic knowledge of how kids think about equations. What I need is a framework for thinking about teaching solving equations. I need to know what my goals are, the value of different curricular choices, the feedback that would be helpful to give, the challenges I should anticipate.

So, I go back to my notebook. I know that I prioritize one of these ways of thinking above all others. The most useful, powerful way to solve equations is to use the properties of equality.

I think the most likely situation is that my students will come into my class being comfortable with using some version of guess and check to solve an equation.

I’ve got the beginning (guess and check) pinned down. I’ve got the ending (properties of equality) pinned down. Anything else is a toss-up. But this gives me some clarity. I’m essentially looking for mental models that can help shepherd my students from guess-and-check to using the properties of equality.

In the next piece in this series, I’ll look at candidates for that middle model.