I had no idea it was this bad. I have been hearing that parents were puzzled at math methods being taught to their children, but I figured it was just a mild inefficiency of method that they were not familiar with. We forget things, and when Jonathan and Ben were in more advanced maths I had to stare at things a while and look at the previous chapters (which I never did in high school) to figure it out. But they were in Christian schools which taught math in more old-fashioned ways. I recognised what was in front of me, but had forgotten it. I could get it back. (Though they usually got there first while we were staring at it together.)
Holly Math Nerd, who I have seen quoted before on the internet, has an essay I can only describe as chilling, Light Bulb Moments Are Not Accidents.
The clearest example came with a real-world problem: 6,990 ÷ 260. Framed concretely, this was a question about how many more paychecks it would take to pay off my car loan if I stopped making extra payments, with each paycheck covering half a payment.
Without prompting, she immediately saw that 260 × 2 = 520 meant 2 was the first step — and did the multiplication mentally. No boxes. No number lines. No written explanation of her “strategy.” No developmentally inappropriate requirement to do meta-analysis of her strategy in real time.
Just fluency, surfacing the instant the problem was allowed to be orderly.
This is the part that’s hard to explain to people who haven’t watched it happen: the so-called “conceptual” method didn’t deepen her understanding. It buried it.
It increased cognitive load, scattered attention, and replaced a stable procedure with constant decision-making.
The standard algorithm didn’t feel old-fashioned to her. It felt like relief.
Some of you are familiar with Richard Feynman's experience on the California State Curriculum Commission in 1964 New Textbooks For the "New" Mathematics. This is the same type of error allowed to continue unchecked for 60 years. It stems from the idea that the theory should be taught first, before there is any data to apply it to. Children's brains don't work that way. Heck, our brains don't work that way. Even in later years, when children have some abstract reasoning ability, you don't teach the idea of the periodic table and expect the student to figure it out, labeling it as they go. You put the periodic table in front of them and then start pointing out the patterns and connections. Once they get the general idea, then it doesn't necessarily matter much if neodymium drops out of their memory. But they aren't going to get the idea cold. If you want to teach maps, you start with places the child already knows, not the idea of a map.
9 comments:
Yeah but, ya know, teaching data is *boring* if you already understand the patterns, and if you don't understand the patterns it's even more painful.
Can someone get Tom Lehrer over here, please?
I always revert to the terrific Wire episode where the Baltimore street urchins are completely jazzed by the use of probability rules to maximize profits in the craps games they're already addicted to. Certainly I have a terrible time learning any abstract system without some experiences to tie it to, like figuring out how much paint I'll need to buy by calculating the area of some odd shape. It also reminds me of Bailey White kids in reading class. She discovered that they were maniacs for a maritime disaster story, but she thought her disaster books were too difficult, so she taped over some of the paragraphs with simplified ones. They tore up the covers to get to the real text in their hunger for more bone-chilling details of shipwrecks. What kids wants to learn to read as an abstract exercise? You read to get at the stories.
When I was in college as an undergraduate, I took a math course called "Math for the Liberal Arts." Aside from a few courses on things like calculating surface area in case you ever wanted to build a deck or something, almost the entire semester was devoted to gambling games. It was extremely instructive for exactly the reasons Texan99 suggests: everyone was already interested in the data, i.e. whether you made or lost money, so the theory then became a lot more interesting as well.
Great method. Probabilities, both precise and estimated, are good things to know.
You of course caught me when you mentioned maps in your last sentence. It just so happens that I was talking last evening with a high-school geography teacher who was lamenting the difficulty of getting her students to take an interest in soils. But the soils she discusses are all soils these students have never seen, and very likely never will see. And I don't just mean the soil itself but also its expression as vegetation and landforms. I suggested that she might have more success arousing interest if she focused on soils in this very county, and on the surface expressions that her students may have noticed or could notice once they were pointed out.
It is possible that the most intelligent students may benefit when abstract principles are taught first, but all the rest must be led to abstract principles by the long path of examples. But in the field I know best, I very strongly believe that an intelligent student armed with theory overestimates his understanding of the world.
@ JMSmith - ...because they have to unlearn so many things by jumping to conclusions before they can learn new things. A standard problem for the intelligent is to get out over their skis, because sometimes it works.
There are two parts to learning a model of the world -- the mechanics of the model, and its limits.
I have never been a huge fan of math books or papers that are nothing but proofs without examples. Please motivate your reasoning. Show your work. Not just the proofs; that's the answer.
I used to be so puzzled by geography textbooks with their dry, pointless lists of products and soils and so on. It seemed like gibberish. But my father was an avid gardener, so I was primed to care about what made soil fertile. Even more helpful might have been lessons centered on my grandfather's very productive North Carolina farm. Or something like Little House on the Prairie might have broken through to me. In the end, I was quite taken by a Heinlein juvenile called Farmer in the Sky, about a settlement on Ganymede where the new residents had to crush rock and fertilize it with very precious, very limited manure. Of course I had no concept where food or almost any other valuable product came from; they just appeared like magic.
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