Reading this

article about the exploration-oriented math program I worked on this summer made me think again about how many people have a negative, knee-jerk reaction to the idea of changes in middle school math education. It's sort of weird because at the same time, a lot of people (often the same people) believe that the current system is a failure and that kids are not learning math.

The following observations on the topic are informed simply by my own experiences and opinions. I am going to be thinking of an exploration-based approach as discussed in the article as the alternative to the current approach that privileges memorization and repeated drills.

(1) Miscategorization

I think when people hear about any kind of innovative math teaching approach, they immediately think of "New Math" and assume that since it was a failure, this other approach will also be a failure, even though the correspondence between the two programs may be limited. In my experience, they often don't even wait to hear any details about what the innovative approach entails before shuddering with horror at "New Math" memories.

(2) Face Validity

In a lot of contexts, I've noticed that people put more emphasis on face validity (does it look on the surface as though it will work) than they should when deciding whether something is likely to be effective. For example, people untrained in survey research may look at a questionnaire and say "But we want to find out X; why are you asking Q, Y, and Z?" not realizing that asking X directly may not be the best way to get the information. There are all kinds of things that look like they shouldn't work, but do when you actually test them, and vice versa. For instance, many people would say that the DARE anti-drug program has high face validity because it addresses the major aspects of keeping kids off drugs, but in

reality, it doesn't work.

I believe that most people, if asked whether it would be possible to teach children how to speak a language fluently without explicit instruction in grammar, vocabulary, usage, etc., but rather through exposure to repeated examples of (or complete immersion in) the language with no attempt at explanation of what was going on, would say it doesn't seem plausible. Yet basically every child does this when they learn their mother tongue, and immersion programs are considered very

effective for teaching children additional languages.

People's mental models of how the brain works are generally not informed by any actual knowledge of cognitive science, so "obvious" or "commonsensical" ideas about learning are not grounded in theory and don't necessarily mean squat.

Just today, watching an episode of

*Sherlock Holmes*, I watched a horse trotting down the street, with a view from the back. A horse's legs are unbelievably

skinny! It is not plausible that a horse should be able to walk any distance on such scrawny-looking little legs, let alone be an excellent runner that can carry human weight at great speed and/or for long distances. And yet...

This doesn't mean that we should all accept that anything can be true and draw no distinctions. It's reasonable to attempt to judge the plausibility of assertions that we hear. But it seems important to recognize whether we actually have any basis for those judgments and what those underlying assumptions or facts consist of. And it's always a good idea to ask whether there is actual

*evidence* on the issue. Sometimes there's not, and we have to decide whether to tentatively favor one take on the situation or withhold judgment.

(3) The "Solid Foundation" Myth

Many people maintain that novel math approaches that downplay the priority of e.g. memorization of the multiplication tables do not allow children to get a "solid foundation" in the basics of arithmetic before moving on to other material, and without this foundation, it is impossible to ever learn higher math. I don't know whether this is true or not. But I don't think that it's necessarily true, despite the possible face validity of it.

One driver of the exploration-based approach discussed in the article is kids often have trouble with math starting in Algebra I. They have difficulty moving from the arithmetic of the earlier years to understanding math with all these variables. This approach introduces the fundamentals of algebra during 6th and 7th grade (without ever calling attention to that), with the idea that kids will not have to go through such a stark transition phase if they've been already doing it as they go on.

Robert put this pretty well: If you imagine that higher math is at the top of the mountain, and teachers are helping kids build a rock solid foundation from which to make their way up the mountain, it doesn't help much if the platform is only a few feet tall and the mountain itself is hundreds of feet tall. How do they get the rest of the way up? (This is assuming that the foundation really is solid and that all kids start from this point, some big ifs.)

I think of it like this: The traditional arithmetic approach really gives kids the idea that math is all about numbers and manipulating them in various specified ways. Ideally, kids develop fluency and speed in performing these manipulations which are, when you come down to it, pretty easy and straightforward. About the time they feel comfortable with that (or, in reality, often before), we pull out the algebra book and basically tell them that everything they think they know about math is wrong. They know that 6 * 9 = 54 because of the multiplication table, but not what's underlying it, so how do they make sense of 6 * X = 54?

(4) The Self-Confidence Paradox

I think many people also find it crazy that an exploration-based approach attempts to both challenge students to do harder problems starting at a younger age

*and *simultaneously improve kids' self-confidence toward math. I mean, it's obvious that mathematical self-esteem comes from feeling like you have complete mastery of the material and can answer all the problems, so isn't this attempt to ensure that even the smartest kids will encounter problems that are at the edge of or even beyond their abilities going to cripple them?

And I can attest that some of the problems are

*quite *difficult.

However, there is, to my mind, a certain logic in the idea of having kids encounter hard problems from early days because everyone reaches a point where math gets hard for them. (If this is not the case for you, it's because you quit too early. There's tons of math that nobody has figured out yet, I am sure.) If you have found math pretty easy up to this point, being unable to immediately grasp some concept or get the right answer to some problem in algebra, geometry, calculus, differential equations, topology, whatever is going to be kind of disconcerting. You may start to question whether you are "good" at math or not. You may start to lose your confidence in your ability to do math. (As I believe I've mentioned, Taylor series did this to me in Calculus 2 the first time, though I did encounter a question on my high school calculus final exam that prompted me to write "Only God knows the answer," a fact my teacher remembers to this day.)

You may not understand that working at a problem is just part of the normal process of doing math and that being unable to answer a given problem does not mean that you are stupid. Math ability is not a Superpower. It's something you can develop through effort. And getting comfortable with the idea that some problems are harder than others, and sometimes you'll encounter problems that you can't get the answers to, but that often after working on a problem, you can actually solve things that looked impossibly crazy hard at first glance...that's how you develop the confidence in your own ability to tackle hard problems. And how you learn to not get totally freaked out when something is beyond you.

(5) And What's the Ultimate Goal Anyway?

It is my opinion that, with the increasing amount of technology available for gathering and crunching numbers, knowing how to do math is more important than ever. But we need people who can figure out that given the data S, T, U, V, X, Y, and Z that X = S * T * U/(Y - Z) more than we need people who can calculate 0.283 * 73146 * 9.4 / (2.4 - 0.005) without using a calculator. Oh, and V? That variable wasn't relevant in answering the question at hand.

In summary:

- People may automatically think any new math is going to be "New Math" all over again.

- "Looking right" isn't much of a criterion for validity. Horses can run even with those wimpy-looking legs.

- It may be ultimately more effective to teach kids to

**climb** the mathematical mountain than help them build a "solid foundation" that doesn't get them very close to the top. Mountain climbing isn't easy, but has its rewards,

including "the ... exertion it requires, the satisfaction of overcoming difficulties by working with others, the thrill of reaching a summit, and the unobstructed view from a mountaintop."

- I am no longer afraid of Taylor series.

- A great many adults probably still believe that math is about numbers.

As for the exploration-based program in the article - does it work? This is, to coin a phrase, an empirical question. I would say there are reasons to be hopeful that it

*could *work better than the current system (needs a thorough evaluation), but whether it

*would* work better in practice

*...*that would be an implementation challenge, no question, even with substantial evidence in its favor.