Interesting discussion at Andrew Gelman’s blog, touched off by Mark Palko’s (justified, in my view) condemnation of synthetic division. I don’t have enough experience of the high school curriculum to have an informed opinion, myself, but this seems a good place to say that I’ve been delighted with the math materials that CJ brings home from kindergarten. They do a lot of histograms; going around surveying other kids (sample topics: how many letters in your name? what is your favorite color of leaf?) and then making a bar graph of the result, sometimes constructing the bar itself as a stack of kids’ names. It’s very concrete; it weaves the math in with other parts of the curriculum; it makes it very apparent how to answer questions like “what color of leaf is most popular?” by inspection of the graph. I’m into it. Use the comment box to tell me what parts of K-12 math should be expunged, what math activities you like for kindergartners, etc.

## Where is the deadwood in the K-12 math curriculum?

**Tagged**curriculum, histograms, k-12, mmsd, synthetic division

Rewriting 1/sqrt(2) as sqrt(2)/2.

Unfortunately high school education is full of algorithms designed to explain important concepts but unfortunately are filter out the important parts in hopes of making things quicker and more systematic.

My favorite example of this is FOIL. It gives a quick way of computing say (2x+3)(3x+1) but as soon as students have to multiply (x^2+2x+3)(3x+1) they are sunk. Furthermore, most teachers never take the time to explain that the process you learn to multiply numbers in 3rd grade (line them up, keep track of place value and carrying) is exactly the same thing as the distributive property. In my experience, having students work this out in detail is a great way of tying together previous knowledge with future learning.

I’d be curious to know if people have tracked the introduction of some of these algorithms into the standard curriculum. Certainly “rinse & repeat” methods are faster and easier to teach but I’m convinced that it would be better to reduce the amount of stuff in the curriculum (like conic sections) and focus on how the concept behind the algorithm gives rise to the algorithm itself. Of course this is easier said then done.

Also, having kids count the number of ways to make change, say for a dollar, is a really fun and accessible activity. It is a great introduction to recursion and how to formulate an algorithm by noticing patterns.

Oh man, if I had a nickel for every time I tell my students that they don’t need to simplify their answers (because simplifying *doesn’t even make sense*)…

I think we should teach people how to program much earlier. Then I would replace a reasonable amount of learning to do pencil-and-paper calculations with programming a computer to implement those algorithms. Rather than learning synthetic division, you just learn how to program long division of polynomials.

In a perfect world this programming would be cumulative. Each student would start with a computer that could barely do any math, and end with a computer that could pass a calculus exam that had no word problems.

That is still infinitely better than students who insist on simplifying 1/sqrt(2) as 0.707.

I don’t see why we should get rid of synthetic division, FOIL, or any other problem-solving technique. I certainly don’t believe we should insist that students solve multiplication or division problems by only those techniques. But speaking as somebody who teaches first-year calculus regularly, the main problem is that students don’t know *any* technique for solving these kinds of problems, not that they follow algorithms too rigidly. I also completely agree with Allen: there is little value in “rationalizing denominators”.

Jordan and some may have noticed one of my rants on G+ recently tangentially related to this. But I think a lot of what counts toward “problem solving skills” in K-12 curriculum just aren’t really. (My opinion overlaps significantly with Lalit Jain’s above.)

I see this when I tutored in college, and now I also see this at Math.StackExchange. Students would ask about “how to solve this type of problems” with the expectation you hand them a tool specifically tailored so that they can just algorithmically plug in the relevant numbers and out pop an answer. I think of it as the “excel spreadsheet” style of “problem solving”.

There’s a load of jokes about how mathematicians like to reduce a problem to a previously solved case. But these students are not even taught (or trying to learn) how to reduce a problem to a previously solved case, except perhaps in the most rudimentary sense of direct substitution of numbers and variables.

To abuse the classic saying about teaching a man to fish versus given a man a fish: a lot of what I experienced in high school and what I see from teaching/talking to recent high school graduates indicate that K-12 mathematics (or perhaps just the 9-12 part), instead of teaching a man to fish, gives him an entire bag full of fishing poles, each one marked “trout”, “salmon”, “eel”, tench”, “sturgeon” etc.

Just to add something to this: another prime candidate is the quadratic formula: ax^2 + bx + c = 0 can be solved by x = (-b +/- sqrt{b^2 – 4ac})/2a. Students struggle to memorise this fact, and in high school I’ve seen kids type this formula into the memory banks of their graphing calculator so they can look it up and apply it on tests.

(a) What is much more useful than for students to memorise that formula is for them to see and how one can obtain this formula by completing the square. Once one learns the process it is a lot easier to remember the end results. (And should you forget it, you can just rederive it on the spot.)

(b) Why bother teaching how to solve the quadratic formula when we allow the use of graphing calculators on tests? Most of those calculators can perfectly well compute the roots of quadratics and cubics using known formulae, and numerically obtain approximates to higher order polynomials easily. And some even can do symbolic manipulations.

It is ridiculous to call FOIL a problem-solving technique. It is at best a mnemonic for a special case of a general problem-solving technique. Sure, by using the associative properties of addition you can expand (a+b+c)(d+e+f) = ((a+b)+c)((d+e)+f) and then apply FOIL three times; but the way many students cannot handle that simple product above while they can perfectly FOIL products of binomials, suggests to me that the reliance on FOIL actually reduces their problem solving skills.

The idea that one can use tools in slightly imaginative ways is one I would consider much more of a problem solving skill than algorithms like synthetic division or FOIL. Just to give another example: quite a few students, despite knowing the quadratic formula, cannot give a root of the polynomial equation

z^4 + 4 z^2 – 9 = 0

I am going to defend FOIL.

I have no experience teaching high school (although I see the results in calculus)… but I’ve always thought of FOIL as a good thing. There are essentially two steps to multiply out (x^2+2x+3)(3x+1): (1) come up with a systematic order, in your head, in which you want to write out the six products; (2) follow your recipe and do the six multiplications. In particular I don’t personally do any intermediate step such as (x^2 + 2x + 3)(3x) + (x^2 + 2x + 3)(1). I am curious, do any readers of this blog?

My intuition is that FOIL, *taught properly*, is a good thing: it is a good example of (1), it generalizes, and remembering an example is typically a good way of remembering how to solve a general kind of problem.

I think you’re right that distributivity is abstract, and people should also learn a concrete way of thinking about it. But the point is that we shouldn’t be teaching “FOIL” we should instead be teaching lexicographic ordering. The students just know the word FOIL and that doesn’t help them for bigger products (which is fine for them because algebra classes never ask them to do bigger products).

i don’t understand why synthetic division should not be taught in high school. then why teach derivatives now that even calculators can do them symbolically?

i just taught synthetic division today. this comes after we did transformation of functions: translation, expansion, even inversion. now i can combine synthetic division and function transformation to explain how Ruffini-Horner method of finding approximate roots of polynomials work.

It’s very funny, I had never heard of this FOIL thing, and I had to look it up online to see what it could possibly be…

I have no memory of the precise way I was taught such things in France (except that I remember we did _lots_ of exercises on expanding products, recognizing common factors, gathering like terms together), nor do I know how it is currently done, but it seems to me very fragile. Does anybody when this was first introduced?

P.S. “it” in “it seems to me very fragile” refers to the FOIL technique…

I graduated high school in Maryland public schools in 2002, and I never, ever learned synthetic division. Thank goodness, it makes no sense!

I’d like to expunge the cross multiply algorithm from the K-12 curriculum.

First, it seems absolutely nonsensical to multiply across an equal sign, and it’s often taught when students are still sorting through what equal signs and fractions and proportions mean. Teaching a cross multiply algorithm before students have had a solid SOLID foundation in algebra (i.e. before 9th grade mathematics) is a mistake. And even when they’ve had a course or two in algebra, they still confuse equal signs and fractions and when you can cross multiply. Second, cross multiply algorithm is a shortcut to clear denominators except that it’s often used in solving for a variable “x” that’s in the numerator, i.e. 3/4=x/12. In these cases, students would cross multiply and then divide through by 4, but really they should’ve just multiplied by 12 to begin with. The cross multiply algorithm not only confuses concepts but is not at all a shortcut in the instances it’s most commonly used.

Unfortunately, most students can’t complete the square.

I tried this, with the same idea when I was a TA for college algebra, because the quadratic formula is useless if they don’t know what it finds for them or when they use it (Many of them did not know when to use the quadratic formula and when they should be finding a vertex). I would walk through it with them over and over. I spent the better part of the semester on it. We would do it with explicit numbers; we would do it abstractly. They never got it. They eventually found an algorithm for completing the square for explicit numbers, i.e. by multiplying out m(x +k)^{2} + h, and finding values for m, k and h by comparing like terms. They never got that when we set that equal to 0 and solved for x, we’d get the quadratic formula.

I never learned the distributive property as FOIL. I just learned the distributive property. I learned that I had to make sure every term gets multiplied together. We should teach students to come up with a systematic order to keep track of the products. We do not need to give them a systematic order. We can at the very least expect our students to try and organize and keep track and juggle 6 things.

I also disagree that the systematic order should be done in our heads. I do not write out (x^2 + 2x + 3)(3x) + (x^2 + 2x + 3)(1), but for products with many more terms and negative signs, I absolutely do write it out. Similarly, for students who are just learning how to multiply such products, I see nothing wrong with them writing it out to keep track.

Personally, I’d prefer my students to write much more out on paper.

Dear Willie Wong,

> It is ridiculous to call FOIL a problem-solving technique.

How many thousands of college freshmen have you instructed? ANY technique which helps students solve mathematics problems is a problem-solving technique. We professional mathematicians can posture and preen over what is “real” mathematics in K-12. When you are in the trenches in a tutoring session (mandatory for all faculty at my university), whatever helps a student understand is worth using.

Before now, I had never heard of synthetic division and FOIL. I followed Jordan’s synthetic division link, and was absolutely baffled by what I saw. This is supposed to promote understanding? In K-12 during the1950s through the 1960s I had a mix of traditional and “new math” and I vaguely recall seeing this sort of weird algorithm-stripped-of-math-content. If I were now presented with that algorithm with no explanation, I would see that it had some kind of input and some kind of output but struggle to figure out what the darned thing is really for. You don’t have that problem with traditional equations.

A number of years ago there was a controversy over the math books used in Madison schools at the time. I’ve never seen them, but I believe the complaint was that they were extremely verbose and they de-emphasized the manipulation of equations. Never learning the economy and efficiency of equations is a poverty of tools.

Seconding Noah on the importance of lex-ordering, rather than FOIL.

It always astounds me that when people are asked to count some simple thing (how many words of length 4 are there in H and T? How many with 2 Hs and 2 Ts?) and they just start listing them at random, hoping that the point at which their creativity runs out is also the point at which they’ve found them all.

“Problem solving” is usually used to describe techniques for solving problems *that you don’t know how to solve*. Thus FOIL is as far as possible from a “problem solving” technique. FOIL is never going to help people outside of a math class, while “problem solving” techniques are often useful for all problems.

cl wrote:

“We can at the very least expect our students to … juggle 6 things.”

Love that idea — this must be the sort of school Allen Knutson went to. ;)

k: I think the argument is that synthetic division should be dropped in favor of polynomial long division. The latter (a) actually makes it obvious why you’re doing what you’re doing, and (b) is applicable to a much wider range of problems. Personally, I agree with these arguments. I learned synthetic division once, had no idea why I was learning it, and promptly forgot it in favor of long division.

A possibly relevant side observation: I’m rather surprised with how long it took me to realize that long division was essentially running the distributive property in reverse. Now that I’ve realized that, I find it much more intuitive, and a little faster, to do it as reverse-distribution.

Anecdote: our babysitter is student-teaching at a local high school right now, and she is somewhat anti-FOIL — she says her students know the distributive property and know FOIL, but don’t know (and perhaps have never been told?) that the two are related, and are really struggling with multiplying together sums of more than two things.

She also says that synthetic division is already mostly gone, as is long division; she supports the expunging of synthetic division but mourns that of long division.