"There is no one way"

Saturday, September 26, 2015

How To

I am often asked "what is the best way to teach equation solving?" (or graphing, or factoring, etc.) Conversely, teachers often want to share with me what they deem the best way to teach students how to do certain things. These conversations are based on a widely-held belief that our job as math teachers is to teach how to, and that good teaching is about providing students with easy-to-remember strategies. To guarantee this works, review consists of reminding them of those strategies right before the test.

I don't mean to criticize teachers who buy into this: it is definitely the dominant culture. Most teachers, students, parents, and administrators genuinely believe that math instruction is teaching how to. There are even some math professors who believe that this is the form that math instruction should take initially, to lay the foundation for later understanding. Alas, that perspective is flawed, as I will try to show by way of analyzing one example: solving linear equations.

Here is a common approach. This is how you solve "one-step equations". This is how you solve "two-step equations". For more complicated equations, first do this, then do that. Later, there may be discussions of whether it's more efficient to do steps in this or that sequence. And so on. Remembering all these guidelines and when to apply them is difficult when you don't fully grasp what they mean. And in any case, this approach may have been appropriate when the solving of linear equation was an important skill, but those days are long gone: nowadays, linear equations can be solved by pressing a key. In fact, many equations can be solved that way, not just linear ones. Comparing sequences of steps in solving strategies is a highly technical activity, which is really neither interesting nor important if devoid of context and understanding. The concepts remain important, these particular skills at an advanced level, less so.

Don't get me wrong: I do think that there's a place for solving linear equations "by hand", alongside with various electronic, trial and error, and mental calculation approaches. But none of these methods are superior to the others. Students should be exposed to all of them, and make a case-by-case decision about which one to use in a given situation. What is important is to understand the underlying math.

First of all, students need to know what solving an equation even is. Unfortunately, some students know how to solve a linear equation, but don't know what they've achieved, other than "getting x by itself", as they put it. One way to clarify what this is about is to precede equation-solving by asking of two expressions: which is greater? For example, 6(x + 1) is greater than 12x/2. 6(x+1) is less than 6x+12. However, 6(x+1) may be greater than 12, or not, depending on the value of x. In fact, both sides can be equal, which happens if x = 1. Looking at examples like these helps set the groundwork to answer for what value of x are the expressions equal?

Another way to clarify what we're doing when solving equations is trial and error. In complicated cases, it may require many tries, or we may only be able to get an approximation. For example, if we are solving 6(x + 1) = 8, we find that if x = 0 6(x+1) is less than 8, and x = 1 it is greater. Is it reasonable to assume the actual answer is in between? Let's look at the graphs of y = 6(x + 1) and y = 8. Yes, it is reasonable: both graphs are lines, and they meet as expected for  a value of x between 0 and 1. Zooming in allows us to find a closer approximation of the value x which gives the same output on both sides of our equation. An exact answer, of course, can be found by symbol manipulation.

All this is to clarify what our goal is when solving an equation. It is well worth spending some time on that, but of course, once the goal is clear, there's more math to learn: the math underlying the traditional symbol-manipulation solving process is worth understanding. Of course, it includes "doing the same thing to both sides". The Common Core language of equivalent expressions and equivalent equations helps to throw some light on what we're doing as we write successive versions of the equation. But underlying the manipulations are also some prerequisite understandings: the distributive law is key, but also ideas like working backwards, opposites and reciprocals, and more generally how to complicate and simplify expressions.

Once students understand those ideas, they can find their own way through solving linear equations, or at any rate will need a lot less practice. If, as they do this, they are not as efficient as you are, it is really not a big deal. If one understands the ideas, using technology is fast and may reduce the risk of so-called "careless errors", (Of course, it does not eliminate the risk!) Note that students' accuracy when carrying out lengthy calculations may or may not reflect their understanding. It should not be overemphasized in assessment.

For specific curriculum suggestions on how to prioritize meaning over mindless manipulation when teaching equation-solving, take a look at Algebra: Themes, Tools, Concepts (free on my Web site), especially chapters 3 and 6. (In that book, instead of solving linear equations early on, we had the wisdom to do a lot of preparatory meaning-rich work and only hit equation-solving hard in Chapter 6. See especially 3.1, 3.4, and 3.9.) For a hands-on approach to this topic, see also my Lab Gear books.

More generally, when introducing any topic, if you find yourself doing a lot of teaching how to, step back, discuss the underlying ideas with your colleagues, and adjust the classroom balance towards more math, more thinking, and less treating the students as if they are little programmable entities.

--Henri

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