The other day my daughters were assigned a perplexing math question for homework. It was a question straight out of their Grade 7 math textbook, which is the French (immersion) version of Math Makes Sense 7. Math Makes Sense is a Trillium-approved, "constructivist-lite" math textbook series published by Pearson Education Canada, and widely used across the province of Ontario. Here is the question:

Use a place value chart. Explain why you add one or more zeros to the end of a number that you multiply by 10, by 100, or by 1000. [translation mine]The girls thought about it for a while. They understood that adding the zeros had something to do with the fact that you move the decimal place to the right when you multiply by 10, 100, or 1000, but they got stuck on that word "why." Why do you move the decimal, thereby adding the zeros?

Now, this type of question is not uncommon in the Math Make Sense series. Proponents of what is variously called "discovery," "constructivist" or "reform" math would say it exemplifies the kind of challenging question that leads children into authentic mathematical "discovery." But does it?

The problem with this question, and others of its ilk that we have encountered over the years with this series (and with Nelson Mathematics —the "competition" to Math Makes Sense), is that the type of analytical reasoning needed to answer it adequately is not commonly taught in the contemporary math classroom. What the writers of the question are looking for is a kind of conceptual grasping, written in English. For instance, here is the answer provided in the back of the book:

For example: When I multiply a number by 10, it becomes 10 times bigger. In a place value chart, each digit of the number moves one position to the left. The digit 0 occupies the last position. [translation mine]For a series that prides itself on furnishing teachers and students alike with a conceptual approach to mathematics, this answer is quite curious. It substitutes one mechanical trick—adding zeros—for another: moving the decimal place. But both tricks are answers to a "how" question, and not to the "why" question posed.

The inconvenient fact of the matter is that it is nearly impossible to answer the question in a way that is mathematically precise using English alone. A mathematically correct answer requires a mixture of notation (with which kids at this level are mostly unfamiliar) and English. In fact, it requires a proof like this one:

A decimal number is written as \(a_k \ldots a_3 a_2 a_1 a_0\) (for some \(k\))

and represents the value \[\sum_{i\ge 0}^k a_i 10^i.\]

So \(10^d\) is represented by a 1 followed by \(d\) 0's.

Given a decimal number \(x\) represented by \(a_k \ldots a_3 a_2 a_1 a_0\),

what does the representation of \(10^dx\) look like?

\[10^d x = 10^d (\sum_{i \ge 0}^k a_i 10^i) = \sum_{i \ge 0}^k a_i

10^{i+d} = (\sum_{i \ge d}^{k+d} a_{i-d}10^i) + \sum_{0 \le i < d} 0 \cdot 10^i\]

So \(10^dx\) has the representation \(a_k \ldots a_3 a_2 a_1 a_0\)

followed by \(d\) 0's,

as we were required to show.*

Show me the Grade 7 student who can "discover" that.

(See also THIS MATH DEPRESSES ME)

*Proof courtesy of Prabhakar Ragde, professor of Computer Science at the University of Waterloo.

The book's answer is terrible. It just teaches the kids that the teacher's expectations are inscrutable, and that you should just BS your way through explanations.

ReplyDeleteIt seems to me that you might be able to give a real answer, without using formulas, if you went into a brief explanation of how base ten works. But that seems like a lot to expect of seventh graders.

Chris -- Yes, the answer in the book is terrible. Kids who write something like that, and get it marked right will think they understand the question and will be rewarded for their BS, and kids who don't answer because it's virtually unanswerable, will just think they're bad at math. (Actually, that pretty much sums up my problem with these textbooks.)

ReplyDeleteI think you're right, though, that one could conceivably give an answer by referencing the base 10 system, but neither the textbook, nor in-class lessons have given the kids enough background knowledge about base 10 to allow them to do so.

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ReplyDeleteAugh. I can't answer that question now! I want to give the same answer I wanted to give through most of my university math classes: "BECAUSE YOU JUST DO." No, I shouldn't really have a math degree.

ReplyDeleteDelphine (Grade 3) brought home her very first ever Math (or anything else) test today. She did well but lost .5 marks for saying that a number which increases by ten goes up by "one in the tens column". Which it does. The teacher wanted her to say it goes up by "ten in the tens column". Which would mean it goes up by 100, I guess. Anyway, I put a comment on the test - I figure if they want me to read and sign it they'd better be ready for me to second-guess their marking. AND HERE'S MY MATH DEGREE TO BACK THAT UP!

P.S.: Did you know Prabhakar before Twitter? You guys seem really tight.

P.P.S.: I just removed the last comment because it had spelling mistakes and I couldn't figure out how to edit it.

Wow, the way I think about that question, and the way I've taught it to my daughter, is to ask, "what's so special about 10 that it has this magical property that multiplying or dividing by 10 is just a matter of shifting place value?" The answer I try to lead them to is "because we're in a base 10 system."

ReplyDeleteIf we were using base 2, if we multiplied by 2 we could just shift place.

I would add that "if we multiply by 10 we just add a zero" will get your kids in big trouble if they try to apply it to numbers with digits after the decimal point. For example, what is 3.54 x 10?

That's why I stress "shifting place value" instead of "adding a zero".

Amy -- I suspect you'll be writing a lot of those kinds of notes. I don't know who finds the current math textbooks/programs more frustrating, parents who understand math or parents who don't.

ReplyDeleteFedUpMom -- Your point about multiplying a decimal like 3.54 by 10 demonstrates exactly why the question is so idiotic. I didn't mention this in the post, but on the page directly preceding the question, there's a little table showing how when you multiply by 10 you add one zero, by 100, 2 zeros, etc. Augh!

Wow, that's bad. I hope you've talked about this with your kids. If I were tutoring them, I would ask, "when is it correct to say that if you multiply by 10, you just add a zero?" and lead them to the answer "when you have a whole number, with nothing after the decimal point."

ReplyDeleteNotice that even in the case of a whole number, you can bollux up the "add a zero" rule just by putting in a decimal point. Like this:

8 x 10 = 80 (OK, just add a zero)

8.0 x 10 does not equal 8.00 (don't just add a zero!)