It would be easy to trash Musk's argument--by, for example, pointing out that there isn't a lot of K-12 math in building (or launching, or tracking, or maneuvering) a satellite--but instead I'll be a little more respectful.Speaking at the ISSR&D Conference in Washington D.C. Wednesday, the CEO of SpaceX and Tesla (TSLA) was asked about the education system. Musk explained that he believes schools aren't doing enough to help children grasp why they're learning each subject."You just sort of get dumped into math. Why are you learning that? It seems like, 'Why am I being asked to do these strange problems?'" Musk said. "Our brain has evolved to discard information that it thinks has irrelevance."Musk suggested learning be focused around solving a specific problem, such as building a satellite or taking apart an engine. Then students will encounter and master subjects such as math and physics on the path to solving their problem. Understanding how to use a wrench or screwdriver will have a clear purpose.

Musk's idea isn't new. What he's suggesting is called "problem-based learning", an old (Dewey promoted it a hundred years ago) pedagogical style which I describe as "inefficient at best". Barry Garelick of the Traditional Math blog wrote a brief post about Musk's PBL suggestion, and then wrote a follow-up post highlighting some of the comments from that post. One of those comments hit the nail on the head--in problem-based learning, so much of the time is spent on the "problem" that the kernel of math that's supposed to be gleaned from the "problem" is lost in the shuffle. Very little math gets learning in a class period, and that which does get learned is mostly a by-product. "Exactly!" scream the proponents of problem-based learning. But no.

Remember when manipulatives were the big thing in math? Many (many!) moons ago I found one that I really liked--Hands-On Equations. Used to teach students how to solve algebraic equations, it involved dice, pawns, and the idea of "legal moves" (e.g., it's "legal" to add a pawn to both sides) to provide a physical representation of algebraic operations. Gradually,

*through 26 lessons*, the program transitions students from solving problems with the manipulatives to solving them using standard algebra. Sounds great! My students loved it, I loved it, everyone had fun, the kids were engaged--anyone walking in to my class would think that this,

*this*was a place where learning was taking place. You could have checked every box on an evaluator's clipboard.

Cut to the end of those 26 lessons, though, and students did no better on a test than had previous classes who did not use Hands-on Equations. No better at all. Despite the program's built-in transition from manipulative to algebra. Students saw that transition as just part of the program, part of the game. They didn't make the leap from the "game" to the math. They learned the game well, they didn't learn the math. They spent a lot of time learning a little math.

And

*that*is what's wrong with Musk's idea. He made the classic rookie mistake; I won't be hard on him because it's such a common mistake. But people who are really smart, or very talented in a certain area, can see "connections" between the many things they know. That excites them, it's so cool! If they can share those connections, everyone else will be excited about the topic, too, and will learn! In the post-Sputnik days of "new math", the smart people got together and decided that if everyone learned basic set theory and different bases, our country's "math deficit" would be instantly erased! Today the silver bullet is matrices.

What they get wrong, though, is the confusion between cause and effect. Being excited and understanding the material and seeing connections doesn't

*cause*learning, it's the

*result*of learning. There is no way set theory and bases are going to help someone who doesn't

*already*understand math, and the same goes for matrices. You have to teach fundamentals. No one starts playing piano with a Bach concerto; they start with notes, and chords, and Chopsticks. So it is with math.

Now I hope that some won't (intentionally) misunderstand what I'm saying. I'm not saying that math should be taught as an abstraction; on the contrary, it's the language of science and the universe, and that's part of the reason we learn it at all. There's no way I would advocate divorcing math from the sciences, from engineering, from games. Math is learned best when it is taught with applications and examples. But the examples are there to highlight the math, not to subsume it.

Additionally, high school math takes us up to what was learned and developed in the 1600's (calculus). That's why "Train A" and "Train B" problems exist; there's no real-world need to solve such problems, they just subtract a little abstraction to make the problem easier to understand. Seriously, outside of some statistics (i.e., social science problems), what real-world problems are ordinary K-12 students going to solve using the math we teach them? Darned few! But we can help them understand real-world things, often with the help of physics, especially where driving, a real-world activity if ever there was one, is involved--doubling speed quadruples energy, speed going around a curve, how long it takes to stop if you lock up the brakes, how police determine your speed from the skid mark your car left on the road, etc.

So to close, I give Musk credit for having his heart in the right place. He's just a little off in the time scale--problem-based learning can only occur after the elementary learning has already taken place.

## 8 comments:

When my daughter did geometry, we used to curse the mania for 'real-world problems.' That book was stuffed with incredibly dull real-world problems about designing sprinkler systems and whatnot. We much preferred her usual Saxon books, which featured problems that weren't terribly realistic, but were at least amusing. (A favorite involved the ratio of Bolsheviks to tsarists in a crowd.)

I don't have that much sympathy for Musk because I think by and large he's a scam artist. I went through an extended dialogue with a Tesla fan over why Tesla is overhyped. First of all, even with the cost lowered, Tesla has not broken even in any year it has manufactured. More than that, the first generation Teslas will start to lose battery life in the next year or so. At that point the dynamic of how much recycling costs will be and how much environmental impact they will cause. While the electric car may make sense for those loft dwelling urbanists, for the majority of Americans that is still decades away. When the Tesla fan then claimed that the cars were the most popular ones on the road, I pointed out that their production is dwarfed by Toyota, Honda and even Subaru. Now Musk is pushing the hyperloop- which incidentally failed in trials last week, along with a list of other expensive futuristic systems that will cost billions to research and maybe never work. If someone wants to read meaningful speculation on the future, go read Freakonomics or go to their website.

PS. I also find it laughable that Swedish darling Volvo is going to stop designing fossil fuel cars after this year. Talking to friends at a party today, all of whom work for Lockheed or General Dynamics, they think Volvo investors will back off of that claim quickly.

I have to disagree. Space age math came to my school.when I was in fifth grade. What a revelation. Math changed from memorizing to playing and understanding. Finally, meat in the curriculum. Bases were the best thing that year..working with bases let us see how grouping and regrouping worked. Multiplication was no longer about memorizing. That year set me up for math study in college. I had no understanding previously,.as I was a trained algorithm follower and a good memorizer...daily multiplication tests in 4th grade made me good as I didn't want to be paddled.

Quick anecdote: my fourth grade teacher taught us to do mental math. The instruction was to visualize a chalkboard in your head. Literally see the numbers on the board, the mentally write down the answer using the memorized facts and algorithms. When I was in college, one of my engineering profs pointed out the advantages of mental math in making sales pitches or working with colleagues or folllowing lectures. We students did not have a clue on how he was so fast. Chisenbop was the rage at the time, but that clearly wasn't it. Turned out he had had space age math, and was using part to whole knowledge and grouping efficiently.

I taught my dc with space age math. Having the whys of arithmatic made life easy and they needed much less time to learn than the memorize crowd. Multiplication took ten weeks, not three years to get to the point of knowing all the facts. My gc won't be going to public school...waste of time.j

Love it! As usual you are spot on. :)

Mrs. Barton, one of those rare people I've referred to as a "superteacher", taught multiplication to *all* her 3rd grade students. Read more about her at

http://rightontheleftcoast.blogspot.com/2007/11/superteacher.html

We have done a tremendous amount of research with Level I of Hands-On Equations. For example, in one study involving 111 average 4th graders, the percentage of students answering an equation such as 4x+3=3x+6 increased from 8% on the pre-test to 79% on the retention test; likewise, the percentage of students answering an equation such as 2(2x+1)=2x+6 correctly increased from under 8% on the pre-test to 59% on the retention test. The retention test was given 3 weeks after completing the 7th lesson, which provided a pictorial solution approach. Furthermore, no manipulatives were used on the retention test. Hence, our evidence is very solid that students as early as the 4th grade experience a high level of success with Hands-On Equations, and they can transfer that learning to a pictorial notation.

Although Hands-On Equations provides a lesson at the end of the program, Lesson 26, on how to make the transition to the traditional written notation, it would be a mistake to think that that one lesson would suffice to accomplish that task. Rather the purpose of the program is to provide students with a foundational understanding of the algebraic principles involved in solving algebraic linear equations. That learning is verified by asking the students to verbalize what they are doing as they are solving the equations, e.g., "I am maintaining the balance by adding two pawns to each side."

Hence, Darren is very mistaken in making this statement, "They learned the game well, they didn't learn the math. They spent a lot of time learning a little math." I am positive that they learned a tremendous amount of math.

He said he taught this program eons ago. However, if he has access to these students I would encourage him to ask them what impact Hands-On Equations had on their subsequent mathematical work. He might get comments such as the ones below received from Kevin Bral.

"Dear, Dr. Borenson: You have no idea how using Hands-On Equations strengthened my mathematical abilities and the abilities of my two sisters.

"When I was taught the Hands-On Equations method of doing algebra as a 6th grader, I had no way to know that it was helping to develop the way I perceive math today. I just knew that it was making sense to me, and that I was having FUN!.... All students should be exposed to your method and given the opportunity to cultivate a love and aptitude for math, as we have. - Kevin Bral"

I am currently designing a history of algebra course (for the non-math major student), and it's even further back in time than you think. Many of those "train A/train B" problems exist in Babylonian arithmetic as cart problems, and growth problems. One of the most direct parallels is the simultaneous equation problems from some of the cuneiform tablets. Clearly rigged, completely impractical (See chapter 1 of J. Sesiano's _An Introduction to the History of Algebra, AMS, ISBN 987-0-8218-4473-1).

We need to teach the methodology first - algorithm and algebra go together - then the applications.

Dr. Borenson:

I don't claim that Hands-On Equations helped no one. I claim--and let's remember, that I was a motivated adoptee of your program--that I didn't get the results that I hoped for and you touted.

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