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?

One thing that might come up for b – ax = c (a la Bridget’s tweet: http://donsteward.blogspot.com/2015/03/cover-up-method.html?m=1) is (a form of unwinding?) called the cover-up method

(Which reminds me, do you know about this? I learned about it at a conference several years ago and always wonder what it might turn out to be useful for: http://www.algebraicthinking.org/wiki/encyclopedia-algebraic-thinking)

The cover-up method would look at b – ax = c and “cover up” the ax to encourage the asking of “b minus what is c” and then reveal the ax to encourage the asking of otherwhat times a is what. Having listened to a lot of kids talk about algebra, I’d say that a certain percentage of kids will develop this for themselves and another percentage can be talked into it by peers or teachers, like most math strategies. It isn’t a crazy thing invented by math teachers for a particular type of problem, though, it’s actually a pretty powerful algebra move.

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What’s driving me nuts is that (a) you and Bridget are absolutely correct and (b) this feels to me like it has strong similarities to guess and check but (c) I’m struggling to figure out a way to articulate them.

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Isn’t misunderstanding negatives the main barrier to unwinding b-ax=c? Also balancing leads to b-c=ax pretty quickly. I know these are harder for students, but it just seems like not recognizing important characteristics because of minus vs negative.

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I think I disagree on this, goldenoj, but this might have to do with the particular way that I’m using “balance reasoning.”

Balance reasoning says something like, “If you take the same thing away from both sides of a balanced equation, it’s still going to stay balanced.” I’ll agree, then, that you can take c away from both sides, getting b – ax – c = 0.

But the idea that you can add ax to both sides is a non-trivial extension of this, one that doesn’t really come into the balance metaphor. You’re making a zero on the left side. That’s something new.

(Though, I’m struggling somewhat to explain why this doesn’t fit into the balance metaphor. I think it doesn’t.)

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I have to hand to John that I was thinking something similar. Why not just use an understanding of negatives to rearrange an equation of the form b-ax=c to the equivalent -ax + b = c? Would a student think to apply an understanding of integers to rewrite the equation? I’m not so sure. Based on my experience, many students continue to view subtraction as purely subtraction even after exposure to the idea that subtraction is the same as adding a negative number. Then again, the whole idea of balance/mobiles tends to break down in this case because we are left with the question of what opposite multiples (negative a) of a weight (x) look like on a diagram,

Part of the struggle too is knowing how students will comprehend and begin to investigate the equations. In the case of an equation like b-ax=c, I think the guess and check or cover up method that Max suggested would be the go to for many students working work this kind of equation for the first time. Guess and check is a comfortable strategy for most students (regardless of the math concept in question), while the cover up method reaches back to the intuitive understanding of equations that many students develop (i.e. something plus 18 is 30, so that something has to be 12).

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Totally agreed with you that the balance reasoning breaks down for b – ax = c. I think your idea of using a “negate both sides” strategy is workable, but I’m not sure if (a) students would eventually develop this on their own (b) whether students would like using it once exposed to it and (c) whether students could make it part of their repertoires.

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Michael-

I am really enjoying your discussion of algebra. I have used bar models with great success in developing algebraic thinking. You inspired me to post an explanation of how I use them (which I think will be useful to my students’ parents also). I address the b – ax = c question. I didn’t really talk about ax + b = cx + d, but bar models can handle that very well also. What I really like about bar models is that it puts so much more of the thinking back in the hands (minds) of the students. They can progress and make discoveries much more independently than any other method I have tried.

http://localmaximum.weebly.com/blog/using-bar-models-to-give-students-a-handle-on-algebra

Let me know what you think.

Doug

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This is interesting stuff, Doug. I like your use of bar models, and I haven’t really given that close consideration yet, but I need to. Thank you for pointing me towards your work.

I want to think something through, though. From your post:

When it comes to solving b – ax = c, this seems key. We transform b – ax = c into b + (-ax) = c. This is how you manage to represent b – ax in your bar model.

The thing is, once we are OK making the move from b – ax to b + (-ax), things get a lot easier, no matter the reasoning. While I think your bar models do a great job with equations, in general, I don’t think they offer any particular advantage when dealing with b – ax = c. Rather, once b – ax = b + (-ax) is on the table, kids can use the bar model or equality properties or some sort of unwinding or balance reasoning to get -ax = (c-b), and then equality properties to divide both sides by – a.

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Yes, I think you are right. I am often surprised at how something like b – ax or 2/3x can seem almost an afterthought one year, and be a tenacious sticking point the next. What the heck did we do or discuss the previous year that made it so easy? Those math talks I didn’t take notes on, or a few students who got it early and helped everyone else…

I have been revisiting your analysis of the various strategies (which has been really helpful in refining my thinking – thank you), and I think there is a subtle difference between bar models and balance thinking/backtracking. I do use balance models after we have been working with bar models for a while, so our sequence is to focus first on equations where an expression is set equal to a constant, and we work through negatives, fractional coefficients, etc. Then I pull out the balance model when we get to equations with variables on both sides.

I think the the lesson from bar models is broad and singular: how to transform equations, or to see or write equivalent equations from given equations. When solving our goal is to simplify equations, until we get the simplest relationship: x = ___. At each step, we are learning something new, and it fits the formal way of writing out solutions quite well: if 4x + 3 = 11, then that means that 4x = 8 and that means that x = 2. But later in high school, we are not always using algebra to solve equations. We sometimes transform an equation to a more complex one.

Balance thinking is similar, but backtracking, constructing a sequence of steps from interpreting and inverting what is written on the page, has a different feel to me. It requires a lot of experience and reasoning to justify the “rules” for backtracking, and then remembering the rules as you work. And it seems to me that it is one-way, almost entirely focused on finding the missing value. Bar models may have these advantages then: 1) it is a unified approach, 2) it generalizes well to transforming equations as well as solving, and 3) when students look at bar models the steps they need to take to simplify (or transform) the equation are more obvious, making it easier to get started with them.

The question that remains for me is how closely the students really see an equation as encoding the same info as a bar model. They seem me to me to be equivalent symbolic representations, but I need to investigate more fully what my students think. In the end, they do seem to have more success solving equations algebraically.

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