Thursday, October 13, 2011

A Grade 7 Math Question

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.


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