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Calculating the cost of everyday purchases can be done in multiple ways. (Daniel Munoz/Reuters)
Calculating the cost of everyday purchases can be done in multiple ways. (Daniel Munoz/Reuters)

Math Education

Why doing math in multiple ways helps kids Add to ...

You buy 3 bags of apples with 24 apples in each bag. The total cost of the 3 bags is $12.24. You sell the apples for $0.75 each. How much money do you gain per apple?

One way of going about solving this problem would be to multiply 24 (apples) by 3 (bags) to get 72 apples, divide $12.24 by 72 to get a per-apple price of 17 cents and then subtract that from the 75 cents you’re charging to arrive at your cool 58 cent profit per apple.

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Setting aside for a moment the morality behind a 341 per cent mark-up on a piece of fruit, solving this math problem appears to be a fairly straightforward, multi-step operation, right? Not so fast. This is the type of math question that a great many Ontario Grade 6 students had trouble with according to the most recent EQAO provincial tests. This question draws on cognitive skills outlined in the area of Ontario’s math curriculum called Thinking, which is one of the areas our students struggled with most. And this spring, as every year, another batch of elementary students will have to grapple with this question.

When most people hear about EQAO results, they are typically seeing their local school’s or board’s, or maybe the province’s, results being reported in the media. These results tend to be the high-level report on the percentage of students who met the provincial standard of achievement.

Schools and boards get much more detailed and personalized results than that, however. In fact, they see how each of their students answered each one of the test questions. EQAO also provides them with reports that roll up their students’ achievement by the curriculum strands and skills that were assessed. That’s how we know, for example, that Ontario’s Grade 6 students performed best on math questions drawing on their skills in the category of Knowledge and Understanding and less well in the skill areas of Thinking and Application. That’s important information for those who are developing instructional programs for our students.

There’s been a lot of public discussion lately about the need for a “back to basics” approach to math instruction over “discovery learning.” The math question above is a good example to show why there’s actually room and need for both approaches. There are several ways that the 58 cent answer could have been arrived at, and all of them perfectly valid.

While a return to drilling multiplication tables may have made the 24 apples x 3 bags part of the problem solving a snap (though it will be the rare reader of this article who performed the $12.24 divided by 72 operation in their head), an overemphasis on this approach, to the exclusion of the higher-order skill area of application, would not serve the student well for the real world either. After all, it’s one thing to know how to multiply; it’s another thing to understand that multiplication is needed to solve this problem and which numbers should be multiplied.

This difference is borne out in the EQAO results, which show that students are actually doing relatively well understanding the basic mathematical concepts they’ve been taught but are having greater difficulty making use of them. As in most things, the best way forward is with a balanced approach where enough attention is given to both the fundamentals and the processes that help develop higher-order thinking and application skills. With those skills firmly in hand, our students will be well positioned for success as they move forward into each next step of their learning. Now how do you like them apples?

Bruce Rodrigues is CEO of the Education Quality and Accountability Office and a former school board Director of Education and secondary school math teacher.

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