# General-Purpose Tools

High school math can be filled with specific tools for one purpose only.  Use this box to solve a word problem about people painting houses, but this other box for rate-time-distance problems, and there are plenty more where those came from. Use FOIL to expand a binomial multiplied by another binomial, but don’t try it on a trinomial!

In reflecting on my own practice I realized I taught inequalities to my Algebra I students in one way, then to my Precalculus and AP Calculus students in a completely different way!  Why did I do this?? It makes no sense, and contributes to students’ feelings that mathematics is a giant toolbox you either know or don’t know how to use.

Here are some ideas about how to throw away some of the special-purpose tools…

• For word problems, encourage students to test answers to the word problem, not to guess the answer, but to define the process used to test an answer.  This process (called “guess-check-generalize”) works for all word problem “types”.  This means you don’t have to talk about word problem types, just give problems.  This method (testing numbers and looking for generality) is a useful skill in plenty of other places, and is basically what is described in Common Core’s practice #8, “Look for and express regularity in repeated reasoning”.
• Instead of FOIL, use an “expansion box”: you can introduce it when working with numbers.  Expansion boxes act like the area model for multiplication seen in elementary  and middle schools: a grid that works equally well when multiplying 23 by 17 … 2 1/2 by 3 1/3… (x-1) by (x^2+x+1)… (3+i) by (3-i).  Oh, and it also works for the product of two binomials.  Phil Daro, one of Common Core’s lead authors, specifically calls out FOIL as an artifact of high school mathematics that has no larger utility.
• For inequalities, teach the Precalculus / Calculus method earlier. The solution to -3x + 7 < 16 is closely tied to the solution to -3x + 7 = 16. Determine the solution(s) to the equation, then graph them on a number line to determine test zones.  Test somewhere in each zone to see if the zone makes the inequality true or false, then shade the ones that are true.

When teaching Algebra 1, my students learned to solve -3x + 7 < 16 by the “basic moves of inequalities” but those moves were very difficult to justify to my students. What I didn’t think about is that those basic moves don’t carry beyond linear inequalities. And I shouldn’t have been too surprised that they didn’t remember these rules in the long run: there was nothing to tie it all together except memorization. Similarly, I can’t tell you how many times I bashed the wall trying to get my students to solve |x-3| < 5. And even after breaking through that wall, the same wall was rebuilt weeks later as if nothing had happened.

By using general-purpose tools, there is less clutter in the curriculum and more “hooks” for students to connect topics and concepts together.  The methods we teach students should work for more than just today’s lesson — if it’s just for today, what’s the real purpose of teaching it to our students?

What’s your opinion? What other “special-purpose tools” are out there, and what are the alternatives?

Bowen is one of the lead authors of CME Project, a high school mathematics curriculum focused on mathematical habits of mind. Bowen leads professional development nationally, primarily on how math content can be taught with a focus on higher-level goals. Bowen is also a champion pinball player and once won \$1,000 for knowing the number of degrees in a right angle.

### One Response to General-Purpose Tools

1. Michael says:

I really like your ideas! Especially the guess and check idea for word problems. I am particularly against the use of FOIL. There is no method of teaching polynomial multiplication that ultimately ends up being easier to understand than the distributive property. If we just gave students enough time and practice to solidify the distributive property, there would be no problems multiplying binomials, multinomial or any polynomials for that matter. At some point students would pick up on the patterns. Say no to FOIL and hello to the distributive property!