Susan J's brilliant lesson on per cent

Susan's lesson is wonderful.

Read these two lines:

Remember that it doesn't make any difference whether you write a fraction with a slanted division line between the numerator and denominator or with a horizontal division line between the numerator and denominator:

This direction comes exactly where it should, arriving at the precise moment in which a student may be starting to feel confused by the horizontal notation 33% = 33/100.

The use of light violet highlighting is fantastic, too:

Don't forget that in word problems, is almost always means equals and of almost always means times.

This next line is a terrific example, IMO, of the proper way to teach a procedure:

You can probably divide 300 by 100 in your head. But it is useful to remember that an easy way to divide a number by 100 is to move the number's decimal point two places to the left. You should also remember that when a number isn't written with a decimal point, you put the decimal point just to the right of the number.

One of the problems with "traditional" math, which I'm sure plagues a lot of reteaching parents, too, is that teachers & parents fall back on purely procedural teaching whenever a student is struggling with a concept. You can see that your student's working memory is already maxed out just dealing with the new material, and you know there's no point adding even more information to the load. He's not going to absorb it, and he may lose focus on the concept he's trying to master.

I've thought about this a lot, because I do more "straight" procedural teaching than I would like.

I'm constantly looking for an "in" to offer an explanation or make a connection between the new material and something C. (presumably) already knows. But if I were actually writing a curriculum, explanations wouldn't be an "add-on," and I wouldn't be looking for an "in." The explanation would be a seamless part of the sequence of instruction and example. My starter examples would be simple enough to do two things:

• incorporate an explanation within the structure of the example
  

• leave enough working memory free to allow the student to read and understand a simple explanation accompanying the example
  

The Singapore Math books work this way. Rarely do they give a student "too much" at the same time. In Singapore Math, the examples are the explanation to a large degree. The written text is spare, even terse.

That's what Susan has pulled off. She has set up a simple per cent problem that gives her a fraction with a 300 in the numerator and a 100 in the denominator. Both of those numbers -- 300 and 100 -- are already "chunked"; the load on working memory is extremely low. The student can hold them in WM while reading Susan's reminder that moving the decimal point two places is another way of dividing by 100.

In short, having set up the correct teaching example, she can in two sentences express and distinguish between two ideas that trip up many a middle school student:^*

• moving the decimal point two places to the left is the same thing as dividing by 100
  

• moving the decimal point two places to the left is just an easy way of dividing by 100, a shortcut, no more & no less^**
  

This is agile writing.

I would describe my own efforts to teach per cent this summer as clunky.

First of all, I don't write my own problems. I use whatever problem happens to be on the page before me in whatever workbook I'm using.

As a result, C. ends up with a fraction along the lines of 286/100, which he can't divide mentally. When he forgets he can move the decimal point, I remind him; then, as he's laboriously writing out 2.86 (because his handwriting, along with everything else, has also not been learned to fluency), I say, "What are we doing when we move the decimal point?" At this point he's focused on getting the number right and only vaguely registers the move-the-decimal-point explanation, which he's sick of hearing in any case.

It doesn't work (not well, at any rate), because the explanation is an add-on.


writing is all about structure

That's the famous William Goldman slogan about screenplays: structure, structure, structure.

Structure is practically impossible to "see" when we read, and it is the single hardest trick of the trade to pull off.

The answer to Susan's question -- Are math books too verbose? -- is yes, for the same reason the answer to that question is yes with nearly any piece of writing. Remember the UK writing assignment:

[Judith] Koren describes how two British women she knows became effective essayists and speakers. “Each week, they’d had homework exercises like this: While preserving every essential point, reduce a 100-word essay to 50 words, then to 20, then to 10. Reduce 500 words to 50, 1,000 words to 100. Week after week, year after year...."
(appeared in American Enterprise Magazine)

The reason you can keep cutting a piece of writing after you've already cut it down to the bone, the reason it gets better with each cut, is that you are correcting and refining the structure.

In a piece of educational writing about math, the structure is refined and the verbiage trimmed by choosing the correct example or sequence of examples of the concept being taught.


Singapore Math for afterschooling

This reinforces my decision to use the Singapore Math books for formal afterschooling. They are superb.

I don't think you can find an unnecessary word anywhere in the series.^***

^* The article The Effects of Cumulative Practice on Mathematics Problem Solving by Kristin H. Mayfield & Philip N. Chase has a fascinating observation about "stimulus discrimination training" in math practice sets -- will post ASAP.

^** This is so important for students just learning math. Especially when I was relearning arithmetic, I found myself constantly confused over the question of whether a particular procedure was "real" or just a shortcut. (Can't explain better than that at the moment.)

^*** This may be true of the Saxon books, too. However, the Saxon books cover all the standards in all the states, or nearly so, which makes them too unwieldy for the time I have to work with.