Last week, we wrapped up our winter sessions with over 50 elementary school math teams. Part of these sessions are devoted to having teachers work together to solve problems. Having teachers “do the math” helps brings meaning to important topics in mathematics education. We gave the following problem, from Van de Walle:
In a particular small town, 2/3 of the men are married to 3/5 of the women. What fraction of the entire population are married?
This is a challenging problem, but only because traditional algorithms get in the way of sense-making methods. The gut reaction is to do something with common denominators. Time after time, with each group, primary and intermediate. Through questioning, the mistake can be recognized.
“In this context, what does the 15 over here represent?” [points to 10/15]
“The total number of men.”
“And over here?” [points to 9/15]
“The total number of wom–OOOOOh…”
Sometimes, it takes longer to reach an ‘OOOOOh’:
“What does the 10 represent?”
“The number of married men.”
“And the 9?”
“The number of married wom–OOOOOh…”
Once teachers realize that having 10 men married to 9 women is somewhat problematic, most model the problem using colour tiles. Two out of three men being married becomes four out of six and six out of nine. Three out of five women being married is equivalent to six out of ten. Six pairs of husbands and wives can be formed. We have 12 out of 19 people being married.
Others think logically to solve the problem. The number of husbands must equal the number of wives. The number of husbands and wives are represented by the numerators. Therefore, the numerators must be made equal. With all due respect to Dr. Math, it just makes sense.
The use of manipulatives to construct meaning continues to be a focus of teachers involved in the numeracy project, both for themselves and for their students. Long before I became involved in this project, my fellow Numeracy Helping Teachers (Marc Garneau, Selina Millar, Sandra Ball, and Shelagh Lim) worked tirelessly to set a climate in which teachers and students felt comfortable using a variety of manipulatives.
At these sessions, we present teachers with problems, not practice. It’s a pleasure to work with such an amazing group of educators so willing to explore, take risks, and persevere. But as much fun as these sessions with teachers have been, I’m looking forward to the real fun: problem-solving with their students.
A much more inclusive context!