Learning To Teach Algebra

I was asked to talk about an interesting algebra problem that I've done. Instead, I want to talk about an interesting problem/issue that comes up when teaching algebra that I've encountered that I think everyone (teachers and students) need to be aware of.

Students learn whatever their teacher decides to show them. This can be an issue as sometimes students learn a specific method rather than the concept the method is covering. I remember thinking that math was so great because it wasn't up for interpretation. There was only one way to solve a problem and it produced a single answer. It wasn't until much later that I realized that it isn't as simple as that.

Let's consider multiplying two polynomials. If you are multiplying a constant by a binomial then you distribute the constant to both of the terms in the binomial individually and then add the two products together.

3(x+1)=(3*x)+(3*1) 

=3x+3 

Easy enough. What about multiplying a binomial instead of a constant though? When I learned it, my teacher showed us FOIL. Multiply the first terms of each binomial, outer terms, inner, and last and then add all the products together.

(3x+1)(2x+5)=(3x*2x)+(3x*5)+(1*2x)+(1*5)

=6x^2+15x+2x+5

=6x^2+17x+5

A few more steps, but not too bad I guess. Now what if we were to multiply a binomial by a different polynomial? Well the distributive is a trick for constants and FOIL is a trick for binomials, so now we have to learn ANOTHER trick for something with three terms. This is ridiculous that we have to learn a whole new topic just to add a term to a polynomial. Luckily, the scope of that class didn't cover multiplying anything more than two binomials, so I didn't have to learn any more tricks. Whew!

Then the next year came around and I had a new teacher. This teacher brings us through multiplying polynomials as well. They teach the distributive property and I'm feeling good. I learned that the previous year and I'm awesome at it. Then the teacher throws in a problem multiplying two binomials together, but we hadn't covered FOILing yet. I assume that the teacher just assumed that we all remembered the trick from the year before and FOIL like a pro without any review. Then the unthinkable happens. We were asked to multiply a binomial by a three-term polynomial before the teacher introduced us to the new trick. Most of the students were confused. The teacher, however, was also confused. He didn't understand why we didn't get it. We learned the distributive property, what more do we need? It took an entire day's class of us telling him that we learned the distributive property as a trick for monomials and him telling us that the distributive property works for all polynomials and FOIL is just a fun trick for two binomials but it is still just distribution. We were floored.

I thought it was so silly that my first teacher ever taught us to FOIL. Why wouldn't he just teach us the distributive property fully? Today, I talk to students younger than me and they have no idea what FOILing is. That's good I guess. It's better that they learned distributive property and are able to generalize rather than a trick that only works in specific cases. I realize now, however, that the FOIL method isn't trying to be something different from the distributive property, it's just a clever way of organizing all of your terms. If only there was some way of organizing terms for polynomial multiplication beyond two binomials...

Insert the Box Method here. What is the box method, you may be asking? I'm asking the same thing. Apparently, after FOIL was popular and even after teachers just taught basic distributive property, people were teaching the Box Method. Don't worry, it's not just a one-case trick nor is it trying to overwrite the distributive property. It's just a way to make distributing polynomials with a lot of terms easier and more organized. Basically, it's everything I ever wanted in a neat little box. Which of the following seems easier:

 or

The Box Method looks easier, but it's the same method and the same number of steps. It's exactly the same work but organized.

This issue can be expanded into factoring equations as well. I learned a Guess-and-Check method of factoring quadratic equations into two linear binomials and then never did anything else. Guess-and-Check isn't the best method though. You can't always easily find the roots. You know what method does work every time for finding roots of a quadratic equation? That's right. The Box Method. It goes both ways.

Have you ever needed to divide polynomials? Perhaps you learned to use long division like I did. You know what method is being taught now that's at least a little easier? You guessed it. The Box Method. It's almost like magic. It makes things so much easier, especially if it is used in all of the ways we've discussed. That way students can connect multiplying polynomials to dividing them to factoring one into multiple.

The point here is not to say that the Box Method is the greatest thing to ever happen to math. The point is that math isn't static; it's constantly evolving. Yesterday FOIL was the ah-ha method that people were raving over as being so much better than what came before it, today it's the box method, and tomorrow it will be something else. As teachers, we have to understand that what's really important are the base concepts (like the distributive property) and everything else is only there to help once we have that down. We can't get stuck teaching one-case tricks and hoping that the students see the base concept and how it is generalized themselves.