I could have easily declared the following as a math teacher, but I’m being more demonstrative now:

No. More. FOIL.

Anyone who’s followed these posting in the last couple of years knows that I’m all for finding efficient ways of remembering how one works through different elements of math. I’m also for remembering processes so long as, later on, there’s a stronger element of true understanding there. Yet, what inevitably ends up happening is one of three scenarios:

1. They confuse “First, Outside, Inside, Last” i.e. trying to combine the two terms right next to each other when they’re not like terms.

2. They can’t factor because the mnemonic wasn’t taught to them for backwards compatibility.

3. They move on to trinomial multiplication and run out of letters.

I’m of the opinion that the geometric method just works whichever way around. It gives a visual representation to my students of how any polynomial can be multiplied or factored for that matter. For my ELLs particularly, making the transition from concrete to abstract is that much more important. Furthermore, I find FOIL, like so many other gimmicks, limited to their scope. They almost impose limits on what our children can and need to know for their future maths.

In the younger grades, I can somewhat understand trying to focus on a certain set of cases for studying math. When developing number sense, children need a certain set of axioms by which to ground their understanding of our math system. However, by the time they get to 8th grade, some of these gimmicks rear their ugly head when integers get involved. (PEMDAS and Keep-Keep-Change come to mind here). Thus, they’re so stuck in how the “last” teacher taught them that unlearning the previous methods become difficult.

With my students in 8th grade, I have an obligation to leave these students in good shape for high school. Most of my alumni can tell you that my teaching got them at least through 1st semester of freshman year, if not through all of it. If we think of our teaching (and our students) as part of a continuous learning process and not an assessment driven segment that someone down the assembly line may (or may not) pick up down the line. Limiting the amount of gimmicks (or developing fresh and profound ones, whatever that means) increase the likelihood that our students can delve into these topics, no matter what level of math they’re in.

Because I’d rather my students be the ones foiling and not getting FOILed.

**Mr. V, who got one thing he can tell you: you’ve gotta be free …**

**p.s. – JD provided the basis for this a year ago, but it’s definitely worth going over. **

### About Jose Vilson

José Luis Vilson is a math educator, blogger, speaker, and activist. For more of my writing, buy my book *This Is Not A Test: A New Narrative on Race, Class, and Education*, on sale now.

## Comments 12

Ay I so agree. Working with the middle school students that I tutor, they rely so much on those tricks that when it comes to actually doing the work or explaining it, they get all tripped up. So frustrating and in essence I end up having to reteach things.

I wish you taught all my students :)

Jose,

Very interestsing, especially since I taught Special Education and these mnemonics were a GOD-send for my students! I guess it depends on where the students are academically, when they arrive in your classroom. For me, I had (8th grade) students who were required to take Algebra (countywide initiative) but some could barely multiply without calculators. Rather than throw in the towel and take the advice of some colleagues (Don’t bother teaching them Algebra), I forged ahead anyway! That year was….er…filled with twists and turns, but I enjoyed it because it MADE me re-think how I learned those concepts in order to teach them to my students. More importantly, it made me realize how fortunate I was to not have a learning disability (although I am starting to question whether I have developed one later in life) in school and that gone are the days where teachers can assign work in a book with little to no instruction or modeling. There were many days when I relied on students to teach each other, since they understood their brains better than I did.

Great post!

But but but… I didn’t get long division in 4th grade (I’m 32). I didn’t get long division in 5th grade, and by then, math hmk was bringing me to tears. 6th grade I sorta got it but also got ahold of calculators, and was able to work my way through remainders. (FOIL, on the other hand, was a godsend that year.) In high school, I had to take the PSAT and the SAT w/o a calculator. Guess who did poorly the first two times because confronted with the same long division? Then I got a tutor, 2 years my junior, who taught me PEMDAS, and long division has never bothered me again. You can get rid of these pneumonics if you can replace them with something else, but remember, some of us otherwise good students who aren’t always so good in math can be helped by learning them. It never bothered me to be told, here’s a helpful pneumonic but it’s only going to take you so far…” but it sure did hinder learning so many steps in elementary school when I had nothing to organize the steps.

Monise:

I understand the use of mnemonics in special education, but it’s not a substitute. What would students who use FOIL as a crutch do with (x^3 + 2x) * (x – 2)? If you’ve taught them via distribution, you can use x=10 and show them that 1020*8 isn’t anything close to what they came up with once you plug in x=10 (probably x^2 – 4 or x^4 – 4).

Author

Wow, lots of loaded comments. Change IS hard to come by. Where to begin:

Thanks for the nod, Mamita. I tell you what, everything has to be in moderation. If people believe there’s a ‘trick’ to everything, then the more complicated stuff gets MORE complicated rather than establishing a firm base to work from.

And I guess that’s my point, Monise and pamela. As much as I loved FOIL and PEMDAS, I quickly found myself outclassed in calc and even other algebra. I had a hard time working through the next level because I relied so long on FOIL, I had to work hard to unlearn it so I could get binomial multiplication and factoring. Those inverse processes don’t often lend themselves to the “next level.”

For instance, what if we threw in Sherman’s example in this madness? Students might get lost and use “FOIL” as a crutch. Rather, can we focus on using a geometric model? There, I can throw in a series of variables and we’d just be finding a bunch of areas. And in reverse, I can also find a bunch of lengths given the areas involved.

This might be a good place to start.

Keep in mind that I said I was a Special Education teacher..in a self-contained classroom, with kids whose math computation levels were anywhere from 3-6th grade without calculators. I was very strong in Math in h.s. because I had teachers who ‘went hard,’ whether we got it or not…I was working on a college prep diploma track in high school and I did not have a disability, but I found a way to ‘get it.’ My students needed whatever i could give them to get them through Algebra because they were required to take an End-of-Course Test in Algebra, not Calc, Trig, or Finite Math. My point was that, in the situation, I had to do what i needed to do so that we would not spend months on the Order of Operations because, again, I had to cover Algebra (I am certified in Special Education) with 13-18 boys (with LD, EBD, ADD, ADHD) and no parapro..if they got frustrated, my entire lesson was for naught. I hope I did not imply that mnemonics were only used as substitutions for my students. When kids start behind, you have to meet them where they are to give them what they need.

Sherman, I see your point. I’m a teacher, but not math beyond 7th grade. And sure, if faced with something unfamiliar, I would resort to something in my toolbox rather than give up, so if someone had not pointed out the limitations of my tool, I could err.

I was so frustrated in math in school… as a teacher, it is the source of my patience. Surely as frustrated, sometimes helpless, and often pained I felt about math is how some of my students feel about either the entire subject or some aspect of what I’m teaching, I see it on their face and realize at that moment it is their long division, and I will try to explain it again, another way, whatever it takes. Math also showed me how we can learn in fits and starts…. one minute I’m solving for x and y just fine and I have a sense of the balance needed and then stoichiometry comes along and I am feeling Stupid with a capital S again. Math taught me to struggle when other things were coming easy, and now that nothing in teaching comes easy, it was probably a good lesson to grow up with. But you won’t catch me taking calculus any time soon.

I’d like to chime in with a second to Jose’s opinion: applied to primary and intermediate elementary schools. I’ve taught in K, 4th and 5th, and even at those young ages using devices instead of reasoning can catch up with kids’ understanding. I’ve seen even the standard algorithm used as a crutch to replace meaning– frequently students use this set up only to get an inaccurate answer, but they have NO idea how to go back and see what went wrong. I think starting with firm understandings drawn from mathematical reasoning is the best way to get young kids to see bigger, more complicated ideas down the road. Maybe we all in the primary sector should be having this discussion, so folks in middle and high school could work from better foundations of accurate, flexible, and efficient students.

Gwen, you have a point too. The temptation would certainly be less in the elementary grades to jump right to it instead of using it as a reminder or organizational tool if the standards were perhaps more developmentally appropriate for the students. They WANT long division and fractions, etc. to be covered sooner so they can get more stuff in later, but doesn’t mean it’s going to work when the students go to process it. Of course, the rules of RTTP and threatening teachers one more time, with the paycheck, will not going to help this situation much. (The aforementioned problem may not be true in all states, but in CA it’s a complication of “rigorous” standards.)

I don’t emphasize long multiplication (thanks for the shout out, btw), but somehow I think it is the most natural. Just have them multiply same as they would multiply 173 x 42. Most of them don’t love it, but long multiplication is an algorithm that most people get.

Long division, on the other hand… :)

Jose, I agree with your rant on gimmicks. They drive me bonkers. FOIL certainly is one of them. Ever seen X-factoring (for trinomials)? I consider “canceling” a dangerous gimmick as well. However, I’ve always perceived algebra tiles and area models much the same way.

Author

Yes, I have. And frankly, we all love “gimmicks” because they’re easily remembered. “Canceling” is a nice gimmick at times. Algebra tiles and the like are awful in a way because they’re only good for small operations. How about the huge ones? Doing it the old fashion way takes a fraction of the time. Right?