Mathematical pedagogy II

One time I overheard a mother say about her daughter, "My girl is great at math except she has a hard time with word problems." I thought to myself was: "Then your girl isn't good at math!" Only in school is "math" a game of manipulating disconnected numbers. Math is word problems (although they become increasingly abstract). As many people do, this mother was confusing arithmetic for math.

Arithmetic is not the same things as math; the distinction is very important. Mathematics -- the exploration of pattern by means of proof -- has not substantially changed in the last 2500 years. But arithmetic -- the computing of numerical values -- has changed beyond all recognition in the last 50. The fastest human arithmetical savant might be capable of adding a few numbers per second. To say computers eclipse that isn't even close. Even a cheap cell phone can do 100 million arithmetical operations per second; an inexpensive computer can do 3 billion per second. Going from 10 to 10 billion in 50 years isn't "change", it's "conquest". It is absolutely absurd to ask a human being to do an arithmetical task. When your free cell phone, cheap watch, or low-end TV can out-perform you by a factor of a billion, it's time to stop competing.

An analogy. For thousands of years people have fetched well water by gracefully carrying large jugs on their heads. One day a pipeline is built and water can be delivered directly into everyone's house. The pipelines didn't improve the porting of water, they eliminated it. Imagine a teacher saying to a class: "We must learn the art of porting water on our heads! Carrying water on your head is good for your balance, strength, and poise!" When inevitably the children don't want to practice because they can just get water out of a tap, the teacher says: "The children of today are all loafers! Pipes are making everyone stupid and lazy."

Teaching arithmetic in 2009 is like making pupils port water on theirs heads. Arithmetic is a conquered technology. But for one exception -- estimation, to which I will return -- arithmetic simply isn't done by hand anymore. Game over.

Yet "math" in grades 2-6 is still dedicated to arithmetic. (We'd be better off teaching them porting water on their heads, at least that would promote exercise!) As I suggested in an earlier blog, I'm afraid that the principal accomplishment of existing numerical pedagogy is to make a significant portion of the population hate numbers. To illustrate: despite the years dedicated to the subject in school, you can't find any cashier nor any restaurant waitstaff who can sum a bill sans computer, much less compute tax, despite the obvious service advantages. Nobody can or wants to do arithmetic anymore; it's time we admit this and move on. Yet there seems to be a strong reluctance to change the curriculum because of some society-wide nostalgia for arithmetic.

In my opinion, eliminating arithmetic from grade school curriculum is a fantastic opportunity -- now we can dedicate those precious grade school years not to boring arithmetic but to fun and useful math!

In an age of ubiquitous computers lack of even rudimentary computer skills abound. The required skills are more than functional knowledge of email and web-browser -- what's missing is the knowledge of how to frame questions so that they can answered by a computer. The first tool that should be learned is the spreadsheet. If 50 years ago it was necessary that all people should be able to add and subtract, today it is necessary that all people should be able to use a spreadsheet.

Here's what I would do if it were my own children I was teaching. As would anyone, I would start with concepts of magnitude. How big is 7? Make a pile of 7 blocks. Is 9 a bigger than 3? A lot bigger or a little? Next, in continuity with existing curriculum, I'd introduce the concepts of addition and subtraction by adding or removing to a pile. But after that I'd break with tradition. From here forward I'd have my kid sitting in front of a computer spreadsheet program like Excel. We would explore questions learning how to program the spreadsheet to get an answer. We start with questions like "If I had had 3 blocks and then I was then given 6 by Ann and 4 from Bob, how many total would I have?" For this problem, we'd put "Mine" in a column and 3 next to it, then we'd put Anne and Bob's contributions in the next rows and finally type in the excitingly magical phrase "=sum(B1:B3)" and, presto, there's the answer. I'd advance through problems like this always with the of practice solving word problems by converting them into spreadsheets programs. There would never be a big sheet of arithmetic that the pupils was supposed to mechanically work though as is currently the case in grades 1-4. All problems would be word problems. All answers would be spreadsheets.

As we advanced from calculation with spreadsheets into algebra, we'd return to a somewhat traditional curriculum of symbol manipulation but using the computer to sanity check our conclusions and to solve associated word problems. But there would be a few changes I'd make.

First, as before, I'd never present a page full of meaningless algebraic equations to "solve for x". That kind of practice is too mechanical for my tastes. That said, I do think that algebraic manipulation is useful and has to be practiced, but I'd keep it grounded with word problems in every exercise. Second, I'd work to make sure the students can use algebra as a language -- that they can translate questions to and from that language. A game I've played with students before is what I call "reverse word problems". Given an equation like: "5 = x + 3" I'd say: "I have 5 oranges now. I had 3 and my friend gave me some number I can't remember, we'll call it x. How many did he give me?". Once students get the hang of this game, I have been amazed how quickly they can connect these meaningless abstract symbols to their lives. Once when playing this game a girl told me a used-car-lot-based story that involved a sale price on a used car. I loved it and asked her how she came up with it and she told me that her father sells used cars; this was a girl that just 30 minutes earlier had told me that she hated math because she didn't see the point of it!

A common response to proposals for introducing calculators into math curriculum is that the students who rely on calculators don't have any intuition for numbers and are prone to draw ridiculous conclusions when they mistype something on the calculator. For example they are capable of writing "10+5 = 50" because they pressed multiply instead of add and have no intuition for that being absurd. This is a very valid concern. I would address this in two ways. First, I would never use a calculator in a classroom, only spreadsheets. Calculators show only the current operand and never show the operation so the user can not go back and check that things were entered correctly. Calculators are therefore so error-prone that I never use them except for extraordinary circumstances. Second, and more importantly, I'd teach number estimation as completely separate exercise from computation. Estimation is wildly undervalued and should be taught regardless of whether or not you believe in eliminating the arithmetical curriculum. (As an interesting aside, I've noted that all my nerd friends over 60 can estimate extremely well because they were all slide rule masters which required them to keep track of magnitude in their heads while using the slide rule for the mantissa.)

I would teach number estimation with just two key ideas. First is the idea of an order-of-magnitude. Strip off all the significant digits and just deal with the magnitude. Multiply and divide are then just slapping on and off zeros. I'd do all this exercise with classic Fermi problems such as estimating the volume of the earth and so forth. All of these exercises would be played (with supervision) with Google. Google the diameter of the earth. Simplify this to an order of magnitude, cube it, etc. Then write down your order-of-magnitude estimate and then look up the real answer with Google. The second tool I teach is doubling and halving. Want 8 times of something? Double it three times. Want 9 times? Double it three times and add some more. Want to divide by 5? Half it twice and take some off.

In summary, if teaching my own children math, I'd abandon arithmetic, emphasize spreadsheets, avoid calculators, always use word problems, ground algebra with reverse word problems, and teach estimation as it's own computer-less skill with Fermi problems and Google.