## The Cruel Irony of Algebra

You would think it would have occurred to me sooner, but it wasn’t until a few years ago that it really hit me what I think is the biggest problem most students have with Algebra: they don’t actually think the letters represent numbers. Here’s the kind of question I’ve asked that illustrates what I mean:

Chuck says that (xy)(wz) is always equal to (xyw)(xyz). Is he right?

Over the years, I have found that if a student is unsure of whether or not they are supposed to “distribute the xy”, they often just guess. When asked why, they say that they were unsure of the rule they were supposed to use, so they just took their best shot. For many years, I tried to show them which rule to use in various situations and the principles involved, hoping that over time they would catch on to the logic of algebra.

But invariably, for many of my students, even a slightly changed question presented what seemed like a freshly baffling challenge. After all, does the question below really seem all that different (other than to a math teacher)?:

Chloe says that (xy)(w+z) is always equal to (xyw) + (xyz). Is she right?

So what to do? While it may seem like taking two steps backwards in the march towards abstraction and generalization, these days I ask my students how they could possibly figure out for themselves if the two expressions are always equal, and the ensuing discussions leads us to the question of what the heck those x’s and y’s and z’s and w’s represent—numbers! So why not try out these equations with numbers? The cruel irony of algebra is that what is intended to make generalization easier actually becomes so abstract for many kids that numbers are the last thing on their minds. They end up seeing algebra as a bunch of arbitrary rules that are hard to predict.

Of course, just because two expressions are equal with a given set of numbers doesn’t mean they always will be—and we discuss that eventually as well. But as an entry point into algebraic identities, and as a gut check to see if my students “get” what algebra is about, I find this works because students believe what they can test and see for themselves. Having them practice distributing multiplication over addition (and also practicing NOT distributing multiplication over multiplication), while it has its place, I find isn’t a good substitute for the intuition that develops by playing with the raw numbers.

Next, semi-related entry: The joys and sorrows of “flip and multiply”!

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## 10 Comments

I tried it with x = 1 and y = 1 and a whole bunch of different values for w and z, and it seems like they both work 🙂

I agree that it’s a good idea to tell them to try numbers here, and to remind them that they don’t need to memorize rules but can think about them.

I think I’d add a geometric, visual approach: adding is putting two objects side by side, while multiplying puts them perpendicularly and gets an area or volume or whatnot. If you have that point of view, Chuck’s is immediately wrong just because of units: two areas multiplied isn’t even the same kind of thing as two volumes multiplied! Chloe’s thing says that a box with base xy and height w + z has the same volume as two boxes with base xy and heights w and z. That seems pretty obviously true without even plugging in any numbers.

In general we need more visual approaches and more use of units in mathematics!

I remember when I was in 7th or 8th grade realizing I could “cheat” and use numbers in my algebra to double check my work. I was especially fond of using prime numbers like 7, 11, and 13, since I felt it was unlikely that any accidentally cancelling or other such messiness can occur.

I recommend it to my students every time they ask me if a solution to a given equation is right. I hope by now they realize that the variables represent numbers (actually, don’t they represent number sets?). You’ve got me wondering though…

I agree that this is a huge hurdle for first year algebra students. One thing I’d like to add is that it’s actually a form of a more basic conceptual misunderstanding.

Students who are successful in algebra out of the gate look at 7+6 and 13 and see that the two are just different ways of expressing the same idea: the number thirteen. Therefore, they’re interchangeable. And for these students it’s not a big leap to think of variables as representations of numbers.

Whereas the students that struggle with algebra think of 7+6 as a “recipe” for getting to 13, kind of like how flour, eggs, and baking soda aren’t a cake until you actually mix them together and bake them. So they tend to think of all math as recipes — steps to be followed — rather than a generalization of arithmetic.

What @MathLaoshi is saying also relates to the meaning of the equal sign; if they see 7+6 = and then think = means “compute the answer and put it to the right” then an identity like x(yz)= (xy)z really doesn’t make any sense…. so partly has to start with early childhood teachers describing the equal sign correctly, and using numerical examples, e.g. does 52 + 98 = 51 + 99 …. saw something like that in a video with kindergarteners, who weren’t expected to add the numbers, just look at the pattern of being bigger or smaller by one…. older students can use variables to translate….

Wow, @dborkovitz, that would be amazing if young kids thought like “does 52 + 98 = 51 + 99 …. saw something like that in a video with kindergarteners, who weren’t expected to add the numbers, just look at the pattern of being bigger or smaller by one”

I recently worked with middle schoolers who are pretty deficient in their algorithms and asked them about things like 15 x 80 vs 5 x 240. Even after drawing a 5 by 3 grid of boxes labeled 80, they still had trouble seeing it. I think this kind of activity is a big part what we should mean by “developing number sense”.

Part of that issue might also be operation sense, do they really know what multiplication means? And if they master arrays for multiplying whole numbers, will someone help them extend to understanding what multiplication means for integers and fractions?

Did you try anything 3D for the multiplication you were getting at? A 5 x 3 x 80 rectangular solid (maybe starting w/smaller numbers and some multilink cubes or just blocks) and then counting layers two different ways…. I could see 80 in each square adding a layer of abstraction that was more difficult for them.

I’m definitely with you on getting students to try it with numbers — it really does seem to help them exactly as you’re saying (seeing the symbols as representations of numbers). I think students at all levels find it pretty convincing when they see a proposed identity work with a few arbitrarily chosen numbers… and this honestly doesn’t seem like a bad reason for believing that it works.

The next step, though, is often the one that stumps me: often it seems like the same students that struggle with seeing the algebra as more than just “arbitrary rules that are hard to predict” are also the ones who have a hard time making sense of *why* a given rule is true.

What I love about the plugging in numbers thing is that, after my very broad suggestion that they try it with numbers, the students can do that reasoning quite independently. But when they’re trying to figure out why, I have a harder time making analogous suggestions… The best I’ve got is stuff like “try visualizing each side” or “what does (x*y)^3 mean by definition?”, but even after these, I find myself having to do more handholding than I’d like.

Also, students who “sort of” get the conceptual explanation one day have completely forgotten it by the next, and then, for efficiency’s sake, you’re back to saying, “because the rule says…” The notion of rule-following itself is something that I would like to get better at teaching, despite the fact that it’s less glamorous than some other topics. Though it’s not all there is to math, it certainly makes a difference in how much students are able to accomplish in math. Is rule-following just logic, or is there more/less to it?

Anand,

Is this the first time students in your school just have to learn and memorize rules? Why don’t you just make them memorize it and just challenge them to come up with a case where it doesn’t work? That way they won’t question you!

Anand, I don’t think it’s surprising that the students who see algebra as a list of arbitrary, hard-to-predict rules also have trouble understanding why those rules work. If they really understood why the rules work, they would not see them as arbitrary or hard to predict. I think what you’re noticing in high school starts way back in first grade. It’s not just algebra that seems arbitrary; even arithmetic does. Students struggle to remember that when subtracting 37-19, they can’t just switch the ones digits and do 9-7, but they don’t know why they can’t. They follow the recipe for long division (daddy, mommy, sister, brother) but since they have no idea why these steps result in a correct answer, any little change, like a zero as one of the middle digits in the quotient, will throw them off. Then they “learn” fraction operations and they keep asking, year after year, “Do we need common denominators when we multiply?” This is why I feel foundations are so important. When I work with students of all ages, I try never to introduce new algorithms, symbols, rules, etc. in a way that my students will think is arbitrary. Whenever possible, I let the students discover the new rule on their own, through solving a problem. For example, to introduce the distributive property in 6th grade, I could give my students a list of foods, with prices, that would be needed for a 6th grade picnic. Then I would ask, if the school decides to have a picnic for the 7th graders as well, and they use this shopping list for each grade, what will the total cost be? Some students will calculate the total for one grade and multiply by 2. Others will multiply each item by 2 and then add. The class will see that both methods had the same result. After some more problems like this, I’ll ask them what they’re noticing about these problems and eventually ask THEM to come up with an algebraic statement to describe what will always happen. Maybe then we’ll crack open the textbook and see the nice little box where the distributive property is written out for us, and I’ll ask why they think it’s called that. Later, we can do a different set of problems to illuminate why we do not distribute multiplication over multiplication, like, how many seconds are in 2 days? Everyone will probably multiply 60(60)(24)(2). I’ll propose that we multiply the 60 seconds per minute by 2, the 60 minutes per hour by 2, and the 24 hours per day by 2. When they see why this is the wrong method for this problem, and a few others, I’ll remind them about the picnic problem and ask them why the two types of problems are different. Hopefully, with a foundation like this, they will come to high school being able to make sense of the algebraic identities. So, I agree with you and Tony that students need to understand how algrebra is an abstraction of arithmetic (and again the foundation of HOW we introduce them to variables is critical), but that’s only going to work if the students understand the arithmetic! I’m guessing that your students who don’t see that all those variables are really just numbers also don’t see why the rules for the numbers themselves make sense.