It's hard math for middle school.

If C. and I had worked our way through Art of Problem Solving's Competition Math for Middle School, neither of us would have gotten stuck on the factoring problem we both got stuck on:

Competition Math for Middle School, a book for mathematically gifted middle schoolers, explicitly teaches the answer to this question:r^{2}is a multiple of 24 and 10. What is the smallest value?

441, 256, and 576 ... are all perfect squares. If a number is a perfect square, each of the prime factors will have an even exponent in its prime factorization.Leafing through the two books, I am struck by the the amount of explicit, procedural teaching

J. Batterson, p. 139

*directed to mathematically gifted students*. No one's asking them to figure these things out or to "problem solve." Instead, they're being directly told that all of the prime factors in a perfect square will have even exponents.

Ditto for the how many diagonals in an n-gon, a perennial favorite amongst SAT math writers these days, it seems.

**: My wording above --**

*correction**explicit, procedural teaching*-- implies that these books teach procedures instead of concepts. That's not at all the case and isn't what I meant to convey.

What I meant to convey was the fact that gifted middle school students aren't being asked to figure out concepts for themselves; they're being explicitly told how the problems work, and why.

## 18 comments:

Test prep is test prep. While middle school competitions SHOULD be about discovery, they are just turn the crank like all HIGH STAKES testing. Why are you surprised?

I use middle school math problems in high school they usually offer insight without too much procedural knowledge - great for weaker students.

The Art of Problem Solving contest prep stresses problem solving, not memorizing factoids. But middle school math competitions are often more about speed than about solving difficult problems, so learning some shortcuts for common problem types does help on the contests.

I agree, though, that most SAT math problems are middle-school level. My son had no problem with them when he took the SAT at the end of 6th grade.

I give a talk to parents on Mastering the SAT. I tell them that essentially, the test is a test of mastery of fractions, pre algebra, middle school geometry, and some algebra 1. While yes, technically, there are some higher alg 2 and geometry questions, the way to succeed on the test is absolute mastery of middle school math--not middle school math as it is taught, but the content that SHOULD be taught when we speak of those subjects.

On the negative side, competition math is often taught from the procedural side--to the detriment of the students. A book that teaches all of these tricks to gifted kids is setting them up for a very hard fall somewhere in life, even if not in math competitions.

Solid middle school math should teach the *why* as well as the procedure.

To wit: I had a teacher at a fractions institute who admitted to us that she had *never understood* why when solving a "joint work" problem, you do what you do. She had led competitions for kids, and just told them what you do "here's the trick", but not until Wu's material did she know why.

I took the AOPS Advanced MathCounts class this summer (I'm a coach and wanted to see what it would be like for my students and what new things I could learn from it.) I appreciated that the class taught the shortcuts but also focused on how the shortcuts worked and how you could adapt them when the problem was given a new twist. (For example, to find the number of factors a number has, first find its prime factorization, then add 1 to each of the exponents, then find the product of these numbers. They explained why this made sense, then assigned different variations of problems on this topic.)

I'm no expert on the SAT (I'm a middle school teacher), but there does seem to be a large overlap between hard middle school math and what is on the SAT. We sometimes use SAT practice problems in our MathCounts practices. Our district's merit scholars often participated in MathCounts in middle school. Perhaps that is because the type of kid who stays after school to do math is the type of kid who is also successful on the SAT, or perhaps it is because MathCounts helps to prepare them for the SAT.

I give a talk to parents on Mastering the SAT. I tell them that essentially, the test is a test of mastery of fractions, pre algebra, middle school geometry, and some algebra 1.It's more than that -- it's competition math. Now, maybe a good curriculum **is** competition math, but that's not what I think I see in Singapore Math.

SAT math is also deliberately tricky, by which I mean the test uses misdirection just as a magician uses misdirection.

The test uses our natural "cognitive biases" -- up to and including the perceptual biases that produce optical illusions -- to make students think a question (or a solution) is something it isn't.

The fact that you take the test under time and emotional pressure makes the use of misdirection & cognitive biases that much more effective.

For example, to find the number of factors a number has, first find its prime factorization, then add 1 to each of the exponents, then find the product of these numbers.I learned that for the first time ever in Chung!

(But I didn't learn why it works.)

The prime factorization trick has a simple explanation in combinatorics.

If the are k occurrences of prime factor p in a number, then a factor will have 0, 1, 2, ..., k occurrences of that factor. That is, there are k+1 possible choices.

Since each factor has a unique factorization, you are just counting the ways that you can independently choose the exponent for each of the primes. That is, you need to multiply the number of choices (exponent+1) for each of the primes.

A good contest prep course would not teach the trick, but would teach the general principles of prime factorization and of counting, and use the trick just to illustrate more general problem-solving methods. I believe that is the way that AoPS teaches.

For those "joint work" problems, I could never remember the grids and boxes to do the trick. But when I realized that I was just adding up "holes per hour", it was easy. Of course they disguise the numbers by giving you "hours per hole", so you need to invert it...

...and that reminds me :)Resistors in parallel are another kind of "joint work" problem. Remember when we wanted to add up "holes per hour" because both guys are working for the same number of hours and each digs so many holes? Now we have these resistors across the same voltage and we want to know how many amps get through.

But ohms = volts/amp and we want amps/volt so we can add them.

Flip them, add them, and flip them again for the answer.

I once saw a test question about two cars that had different fuel consumption (miles per gallon). The more efficient car cost more and we were given the price of gas and how many miles the owner drove per day. The question wanted to know how many days it would take to break-even.

It was very tempting to just find the difference in miles per gallon, and then figure how many days it would take to save the difference in car prices. But the cars were not using the same number of gallons, but driving the same number of miles! We wanted to subtract gallons per mile, so we had to flip the numbers.

In Europe they use liters per 100 km, so they wouldn't have that problem.

gasstation - thank you!

Hey Rocky - good to see you!

Just noticed this post and double-took when I saw the book cover! Glenn Ellison was a classmate of mine in high school. Extremely impressive. As a high school senior he was acing Real Analysis at Yale. I knew he was at MIT, but not that he'd written a book for gifted high schoolers. I would love to know what Glenn thinks of Reform Math!

oh my gosh!

I'd love to know, too.

Haven't been able to spend much time with the book so far, but it looks great.

Judging from the introduction, he's not too keen on the usual middle school math.

Thanks for mentioning my book.

On the SAT = middle school competition math the two are similar. The SATs require more algebra than Mathcounts (and the difference is even bigger when comparing to other middle school competitions.) But Mathcounts is more demanding in several other areas. For example, you need to know more number theory and combinatorics and to be able to do harder problems.

My goal was definitely both to teach procedures and to teach why formulas work. Doing very well at math contests requires this. You may have memorized a formula telling you know many factors 3600 has, but you'll never have memorized a formula telling you how many odd, perfect square factors it has. That said, in middle school I think you often teach "why" by example. Formal proofs or derivations seem not to work as well.

In general I am not a fan of reform math. I like the general idea of teaching multiple ways to do things and can see that there were nice mathematical ideas behind some of the designs, but think that in practice they have three big problems:

1. Kids computational skills end up nowhere near what they should be.

2. Problems tend to be too easy. I know this is done to help kids who don't have good computational skills learn the concepts, but I think sometimes it's only harder problems that make you really learn the concept.

3. Often the textbooks/teachers fail to realize the benefits that motivated the curriculum design. Greatest common factor/least common multiple problems are a great example. I think they're in the middle school curriculum because some mathematicians realized that you could motivate them as things you needed to know about to reduce/add fractions and that there's lots of interesting/useful math you can introduce by studying them: you can teach connections to prime factorization; and you can teach the Euclidean algorithm. But in practice, teachers often don't know these things and may teach students to find GCF(a,b) by looking at all numbers from 1 to a to see if they're factors of a, doing the same to find factors of b, and then looking for the biggest number that's common to both lists. If this is what you do, it becomes a really stupid and inane topic.

My kids attend schools that use reform math textbooks and what I cover in my book reflects what was on IMLEM/Mathcounts that they needed to know better. I don't know Singapore math well enough to know if there are other things that I'd feel I had to teach in middle school if kids had used that as their primary curriculum.

Hi Glenn - Thanks for weighing in!

And thanks for the insight into SAT math/middle school competition math.

btw, I don't at all intend to imply that your book (or the Art of Problem Solving book) fail to teach understanding -- not remotely

I've added a correction - re-read my post & realized the wording implies that these books are mere recipe books.

That's not the case at all.

That's nice of you.

I hadn't read your noting the 'explicit, procedural teaching' as a criticism. Teaching procedures definitely is part of what I try to do.

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