I recently wrote an article for Edutopia about factoring polynomials using areas and why FOIL is absolute crap:

For one, I’m not a fan of FOIL (first-outside-inside-last) for a plethora of reasons. While I think it’s handy to have an acronym that reminds students of a procedure, it only works in a very special case. In this case, FOIL works only for multiplying a binomial by another binomial. Does FOIL lead students toward understanding multiplication of all types of polynomials, or understanding why the distributive property works even with variables? I’m not so sure.

Please do let me know what you think about it! And please do encourage me to write more about math. I could use it in the coming school year!

### 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

WRITE MORE ABOUT MATH PLEASE

When I want to highlight an example of something we require all math students to do that’s a 100% waste of time, I point to factoring polynomials. And I really don’t care what method you use. There isn’t a single scientist or engineer in the country who ever factors polynomials by hand (they have all heard of wolframalpha and other online resources). The only defense math educators can put forward is that somehow this tedious operation “teaches kids to think.” What it does is takes up a month of math education that might actually be used for learning things like creative problem solving, estimation techniques, statistics, or math modeling. So debating which method is superior for something that is a deadweight waste of time seems to miss the macro point about the core issue with U.S. math education — kids spend five years doing, for the most part, tedious calculation by hand. And the reason? They all make tidy little problems for standardized tests like the SAT. But there’s a reason only 15% of U.S. adults use any math beyond decimals, percentages, and fractions. We teach low-level operations in a way that almost no students care about, or retain — let alone know how to use in any meaningful way.

Author

First, Ted, let me say how glad I am that you’re so passionate about this.

Secondly, I’ll also say that, if the argument is that we have wolfram alpha or other online resources, then, in fact, math is dead. We can find derivatives, values of x, optimal areas, transformations, or any number of topics online with relative ease. In the 21st century, however, I would argue that it isn’t so much about the content, but approach. If we’re teaching approaches, then polynomials as conduits to areas and volumes are super-critical to the optimization work of engineers. I should know as I graduated from an engineering school.

I would also argue, as you almost did, that pedagogy is as critical, if not more so, than the content. Thus, skills like problem solving, estimation, statistics, and the like can be addressed through content that is diverse as linear equations as polynomials. Otherwise, your argument just tells me either that math is dead or that only the math you’re teaching matters, which doesn’t ring true to me.

Also, I would argue that more than 15% of Americans know their math, but it’s a matter of application, but I agree with your statement in this respect: teaching polynomials better would increase the percentage of students who know more than decimals and the like.

Best.

Math is so far from dead if we allow people to use on-line resources. It actually gives us a fighting chance to teach kids math in a way that is so much better than what we do today. I, too, have an advanced degree in engineering (PhD from Stanford in Math Modeling within the Engineering School), ran a math-intensive chip company doing digital-signal-processing, and have spent twenty-five years in venture backing the top scientists and engineers in the country. And, as far as I can tell, the only people around who think we’re making good use of kids’ time in school by drilling them endlessly on tedious computation they’ll never ever do (even if they are a top scientist or engineer) are math teachers and standardized test designers. If kids actually had a real understanding of the underlying concepts and how they’d actually use them, the current approach might be less outrageous. But they don’t. So we throw up this barrier — high school math — that is one of the main criteria for getting into college that has absolutely nothing to do with what almost any adult does later in life.

The best math experience I’ve seen in a classroom came when I observed a social studies class. The teacher had kids work in groups, use any online resources available, and take on the following challenge. Predict the world’s population in the year 2100. And present your results, explain your assumptions and approach, describe how you calculated your result, and be ready to debate other groups on the relative merits of approaches, as well as implications for the world you’ll grow up in. This challenge is so great and actually motivates kids to understand sources of data, the concepts behind segmenting data into like clusters, curve-fitting and extrapolation, and — most importantly — that math can be really creative and that there actually isn’t just one right answer to a meaningful math challenge (I can’t tell you how many math teachers have said to me, “Well, math is different since there’s only one right answer.” My response is always, “That’s because you’re teaching kids low-level calculating instead of useful, powerful, and engaging math.”

Now if kids had plenty of time to work on things like predicting the world’s population, we might not be so completely ashamed of ourselves for putting them through high school math as it currently exists. But they don’t. They’ll spend nine months in AP Calculus learning how to do derivatives and integrals by hand (mindless tasks you can dictate to a smart phone), yet give you a blank stare when you ask them to explain what an integral or derivative is, and how it might be useful.

I am passionate about math, but mostly outraged. I think what we do is completely inexcusable, and so many young kids conclude a) math is boring and/or b) they’re not good at math. But the real issue isn’t the student, but the fact that we insist on teaching kids low-level tasks that make for great SAT questions, but that they never, ever use. And the answer isn’t to teach the same content better. It’s to elevate the teaching of math to things that are interesting and that have value and applicability later in life. Otherwise, we might as well teach our kids five years of Sudoku.

I had never heard of FOIL until my recent interest in the state of school math. All those years ago we learned how to multiply out or expand brackets, by applying the distributive law (but they didn’t call it that) and as you have done, using rectangles.

Mnemonics are for people with small brain and good memories.

Regarding factorizing polynomials someone told me “There’s a formula for cubics and quartics, go and look them up”. The implication was clearly “Don’t expect any help from here, you are on your own!”.

Your example cubic 2x^3 – 11x^2 + 12x + 9 is as with all nice cubics best treated as a puzzle, as you have done.

With the coefficient 2 for x^3 the factors have to be of the form

(2x +or- A)(x +or- B)(x +or- C), and of course ABC = 9

So the choices for A, B and C are from 1 1 9 and 1 3 3 in any order.

I looked at the first and thought : (2x + 1) gives a root x = -1/2, then Oh Dear, that means some 1/8ths in the value. But no, there’s a 2X, so 1/2 or -1/2 could be a root. And so on. It is vital that the students get the connection bewteen roots and factors.

So the root is -1/2 and the factor is (2x + 1)

Then I write (2x + 1)(x^2 + px + q) and multiply out if necessary, to compare coefficients.

(they might have to solve the pair of simultaneous equations – good for the soul)

With a fourth degree polynomial I would reach for the nearest computer algebra program! Just as any practical person would.

Plotting a rough graph is often a good idea, if they have a clue about the various shapes of cubic graphs.

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And I’d like to see more math in your mix.

I’m 100% with you. My “I ban FOIL” post remains popular. I assume your first crack at it does it well. This clearly bears repeating. Thanks.

The area model for multiplication (from the article you linked), is neat, but I see you use it to lure kids in. After that, multiple distribution?

Jonathan

Author

Exactly! At least to give them a good grounding for those who can’t just see it numerically. FOIL is cute at best, but it can’t be expanded. Someone mentioned that, if we were to work in the fourth dimension, then we couldn’t visualize it, but at least the area model gives folks a grounding for both multiplication / division of polynomials. From there, I think it’s a lot easier.

Of course, you were a big influence in my transition away from FOIL, so there is that too, Jon.

It is a handy organization tool, and maybe a good stepping stone for helping people discover the larger concept of distribution and then factoring. It’s a scaffold, not an end all be all. Problematic when that’s where the development stops. No reason to hate though.

One problem I see about algebra generally is that kids feel that they are not in control of the situation. Suppose I forgot how to expand (a + b)(c + d). Then if I have control I can temporarily replace the (a + b) by anything simpler, say w, and get a simpler problem, expand w(c + d). If I cannot deal with this I put some numbers in. If I can I get wc +wd. This can be restored to (a + b)c + (a + b)d, again simpler (might have to remember that pq = qp), and so eventually obtain ac + bc + ad + bd.

This approach can easily be applied to (p + q + r)(x + y + z) by putting w for p + q + r , or about 6 alternative replacements. Not only does this give control, it also matches up with the way complicated expressions are, or can be, composed in computer programs.

This turns a tedious activity into something more mathematical.

Still tedious, but effective. I introduce multiple distribution in just this way. In fact, I use

w. I blur it. Expand it. Then reveal that it is really (a + b).I agree! I agree so wholeheartedly that I wrote a book about getting rid of tricks in the mathematics classroom. We are always adding to it and I’d love your input on other areas of math where you feel it’s equally important to Nix the Tricks.