kitchen table math, the sequel: Susan J's brilliant lesson on per cent

Tuesday, September 4, 2007

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.

15 comments:

Jo Anne C said...

* moving the decimal point two places to the right is the same thing as dividing by 100

* moving the decimal point two places to the right is just an easy way of dividing by 100, a shortcut

Shouldn't this read moving the decimal place to the *left* is dividing.

Moving the decimal 2 places to the right would be multiplying by 100.

SusanS said...

Saxon 6/5 covers this quite a bit,if I remember correctly.

SusanS said...

Saxon also brings in the metric system around the same time to keep reinforcing the "divide by 10" and "multiply by 10" (or 100s or 1000s) concept.

Then, it tops it off with a chapter on just moving the decimal as a shortcut.

SusanJ said...

Speaking as a former high school chemistry teacher, I'm not big on teaching the metric system as part of math rather than in a science class.

The reason that moving the decimal point works has to do with place value. If there's time in the math class, I'd prefer for students to get really comfortable with decimal numbers and, possibly, learn to use scientific notation and understand how it relates to place value.

Students should be just as comfortable with the tenth's place as the ten's place.

Catherine Johnson said...

Good grief - thanks! (direction)

le radical galoisien said...

I remember for the GCE O-level exams you have the "summary" component, where in addition to answering comprehension questions about a passage you have to summarise an essay of a few thousand words into 150 or 180 (based on a select criteria) ... do they still do that in Britain, or is it just us Singaporeans now?

Catherine Johnson said...

The reason that moving the decimal point works has to do with place value. If there's time in the math class, I'd prefer for students to get really comfortable with decimal numbers and, possibly, learn to use scientific notation and understand how it relates to place value.

Students should be just as comfortable with the tenth's place as the ten's place.


yes, and that is a TALL order

We just finished the 3rd grade (3A) Primary Mathematics lesson & workbook on place value.

It was terrific.

I'll have to post some examples -- it has sequences of skip-counting that teach & reinforce the concept.

122, 123, 124, ____ ____ ____

256, 266, 276, ____ ____ ____

388, 488, 588 _____ ____ ____

And so on.

SusanJ said...

I would not have thought of skip counting as important to understanding place value. I guess I can see how it might.

The MOST important thing about place value is to be able to relate it to exponential notation. Going to the left in the decimal system you have first the 10^0 place, then the 10^1 place, then the 10^2 place, etc. Going to the right after the decimal point you have the 10^-1 place, then the 10^-2 place, etc.

Once you relate the place values to exponential notation, you can understand all the rules about moving the decimal point and also how to use scientific notation. You can also see how other bases, such as base 8, work.

Catherine Johnson said...

I would not have thought of skip counting as important to understanding place value. I guess I can see how it might.

well, I have to defer to you (and all the other ktm members who have studied college math), but my sense is that skip-counting in a third grade book is extremely helpful.

I can't really say more than that; I found it helpful myself -- and C. had one of those "oh, I get it" looks on his face that I think mean what they seem to mean....

SusanS said...

It is very helpful to the slow processing crowd. The speed picks up and the patterns finally emerge.

It also clears out some of the brain cobwebs of the special ed set. At least it does with my special edder.

It's really one of those "break it down" tricks that math heads might not spot due to their own natural abilities.

SusanJ said...

OK, you've both had experience with this and if you say it helps, I believe you.

However, a student who is going on in math needs to understand the relationship of exponential notation to the place values at some point. (Seventh grade?)

Do you skip count situations such as as 286, 296, 306 where the digits in two different places change at once?

Mr. Person said...

The texts I have experience with introduce exponential notation (for place value) in sixth grade. Negative exponents in seventh-eighth grades.

Skip counting is excellent place-value instruction, and I too never considered it before. I would imagine you would use it simply to highlight the idea of place value, so you would stick to sequences of digits that don't cross over zero.

On another note, I've found that, in order to explain WHY the "of" almost always means multiplication, it's helpful to start with the basics/concrete, then move to the abstract, then plop in more difficult numbers:

5 (cups) OF 6 (marbles) --> 5 x 6

2 (groups) OF 10 (mice) --> 2 x 10

1 (bag) OF 400 (hammers)--> 1 x 400

A (units) OF B (things)--> A x B

1.5 (cups) OF 6 (marbles) --> 1.5 x 6

1/2 (bag) OF 10 (mice) --> 1/2 x 10

.
.
.

30% of $25 --> 30 (hundredths) OF 25 (dollars) --> 30/100 x 25

SusanJ said...

I definitely agree that a lesson on "of" could be useful [and a key point in the per cent article]. Translating English into math is big problem and I don't have time to do the research necessary to write about this. I'm sure many people have addressed it.

However, I think you may be using the word "of" in two different senses in your examples.

According to my dictionary one meaning of "of" is, "From the total or group comprising." I think this is the meaning when we say one-half of a pizza or 25% of 100 marbles. (So I probably should have written that "of" almost always means multiply when we are talking about fractions and per cents. I may even change this.)

Another meaning is, "Containing or carrying." So when you write a "cup of marbles," you are using it in this second sense. I think most of these examples require dimensional analysis, e.g.
5 cups (6 marbles/cup).

Mr. Person said...

Yeah, I know what you're saying. The whole-number examples work better with illustrations, and perhaps using only the "group" phrasing.

But you CAN think of the examples I listed using only one meaning:

So many units OF so many things.

The point is to show that students obviously know what operation to use when they see "2 groups of 10," and to relate this to something less obvious, like "2% of 10."

In other words, "2 groups of 10" and "2% of 10," or "2/100 groups of 10" are operationally the same.

SusanJ said...

The point is to show that students obviously know what operation to use when they see "2 groups of 10," ....

I agree with you on this. And also that at this point however we can convince them that they can trust their intuition and of course they know that 2% "of" means 2% "times" is good.