By now, most math teachers have heard the FOIL debate and come to the same heart-felt conclusion (see here and also here), but I’m going to ask that you keep an open mind as I reintroduce our old friend.
Aaaannnd I need to start with a story:
About a year ago, I met a father/daughter duo who were too excited to tell me why FOIL was awful, and what I should do instead as a math teacher. I decided to let them go on without comment, even though I’m a little sassy, and I figured their opinion was the same as mine.
But anyway, the girl started by explaining that FOIL only works for binomials (hmmm) and when students need to multiply a binomial by a trinomial (or any other polynomial pair), FOIL just doesn’t work. Furthermore, she suggested that the better alternative would be just to explain how the distribution property can be applied to polynomial products in general. Just skip FOIL.
Her experiment was to teach two groups of students two different methods to multiply polynomials, one using only FOIL and one without.
Naturally I cringed at the prospect. That doesn’t even make sense. How could you expect a student to multiply trinomials with FOIL? I’m just going to assume I missed something.
And before I go on with this post, I want to state that I was hugely impressed that this high school student was interested in math education. I think the processes in her brain that led her to these conclusions are what we want to see in all of our students.
BUT, the more she talked about her project, I started to question my own opinion and her hypothesis. So I left our conversation with my mind racing (it often does unfortunately).
- First the experiment: I discounted the experiment because obviously we should be teaching students the meaning of and how to use distribution regardless of whether we introduce FOIL. It would be reckless to consider otherwise.
- Then I started thinking about how I would structure an experiment to test whether it’s better or worse to present FOIL to my students. I decided I didn’t really care about this kind of research, so I moved on!
- After about 30 seconds of that, I decided to think about whether I could get my students to come to the same conclusion as this girl. Surely evaluating these methods was a higher level thinking procedure.
- Then I started worrying that discounting the pneumonic device because some students get confused wasn’t necessarily fair either.
- I feel like FOIL is a word in our math lexicon whether we like it or not. I want my students to know what people are talking about when people speak math to them.
- And lastly, so what if I personally like saying “First, Outer, Inner, Last” in my head when I’m multiplying. It’s soothing. I AM NOT ASHAMED ANYMORE!
So I left that afternoon with a new perspective on FOIL/standard distribution. I want my students to learn both. I want them to decide whether FOIL is useful. I want them to be the ones to choose whether or not they’re interested in the product (ok that might have been a pun, but it was unintentional!).
This is more like what I’d do.
- Teach the students FOIL and how the pneumonic helps us multiply binomials and then teach the students how to multiply polynomials in general/give them a reading of a FOIL slamming post.
- Afterwards, let the students split into two groups and host a debate between the two sides. Students can meet as groups and work on anticipating the opposition’s points/developing counterpoints.
- They could look at or brainstorm alternate methods. Here’s a cool one that we’ve all seen before!
- Ultimately, and this is the most important step, let the students know that whatever they choose is fine. I’ll support them mathematically no matter what.
All that said, I’m still going to put a trinomial times a trinomial on the test.