Welcome to Maths. Please leave your common sense at the door

I come across many students who feel that it is unacceptable to solve a Maths problem on first principles. They always go searching for a formula and, when none can be found, give up on the problem. In this blog post, I give two examples to illustrate this phenomenon.

What is five percent of 100?
Last term, I put a question to my year 7 class which involved calculating 5% of $250? We had not covered percentages yet and I wanted to see how they would approach this. Some students had little difficulty with the calculation although the majority decided they couldn't do percentages. Some tried to remembered a formula they had learnt earlier.
Mr Baroudi, do you multiply by 100 over 1 or do you divide by 100 over 1?

I told them not to worry about any formal methods for the time being. Instead, I asked them, "What does 'per cent' mean?" Everyone seemed to know it meant "out of 100".
I drew the diagram below, one step at a time, asking them, "How much will we take out of this $100 (or $50)"?
What is 5% of $250?

At this point, you could hear a knowing hum around in the classroom. Then, some students wanted to know whether that was an acceptable way of solving the problem. I reassured them that it was!

What is the point of linear algebra?
Linear equations and graphs are covered to death in our curriculum. One thing that is missing from the textbooks is what it actually means for the relationship between two quantities to be "linear". Our textbooks hide this understanding from the students. Here is an example taken from a year 9 textbook (I trust the publisher won't mind the lack of a citation):
Rachel is given 5 CDs for her birthday and decides to purchase 2 CDs per month for the next 2 years.
a Determine the linear rule that connects the number of CDs, N, and the time (in months) since her birthday, t.
The first step in finding the rule is to find the gradient and the textbook's method involves a formal write-up, the two points (0, 5) & (1, 7), and the formula: 
Surely, this is a linear relationship. The amount by which the number of CDs rises each month is the same. This is why it is linear! What's more, this amount is given in the statement of the problem: "2 CDs per month". Why apply a formula to re-discover that the gradient is 2?
To be even more absurd, the textbook writers use the 2 CD per month growth in order to find the second point, (1, 7) and still pretended that more work was needed to find the gradient.

Here is another example where the textbook suggests using the same formula:
For a period of 6 months after planting a sapling, a certain kind of sunflower exhibits linear growth. After 3 months the height was 36 cm, and after 4 months the height was 45 cm. a Find the rule, connecting the height of the sunflower, H (in cm) and the time t (in months).
The two points used in the formula were 1 month apart. It shouldn't be hard to work out that the amount of growth in a month (the gradient) is 9 cm (45-36). Why are they using a formula except to discourage students from thinking about the problem?

As a Maths teacher, people often ask me whether I give my own kids worksheets at home. My answer is this: "I don't. I simply make sure that my kids are not replacing their common sense with learnt formulas."


Five Triangles said…
None of these linear equation questions can be solved with formulas, but they require a bit more effort than common sense:

Very interesting problems. Thank you for sharing them.
Anonymous said…
Unfortunately,there seems to be no thought given to the purpose of education but lots of wasted energy thinking about what and how much curriculum needs to be passed along to the students. Of course being able to think would be better than being good at jumping through maths hoops. Is there any government in the world daring enough to scrap the curriculum and methods of teaching mathematics? I doubt it.
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Thank you,
Kathleen Fultz
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