There's old soviet animation "38 parrots". The plot of the movie was the animals were measuring a boa, and they concluded that the boa is longer in parrots than in monkeys. I always thought this is a cool joke.

My 2nd grade son has shown his classwork to me. They measured a rectangle by covering it by different shapes: squares, triangles etc, and then they computed area of the rectangle. For example, 20 squares were used to cover the rectangle, then the rectangle's area equals 20 rectangles. If 16 triangles match the shape of the rectangle, then the area equals 16 triangles.

Then they compared the areas. They deduced that area of the rectangle covered by squares is bigger than rectangle's area covered by triangles, because 20 is bigger than 16. That's right, same rectangle, different units, which area is bigger.

The idea of measuring is being able to compare different objects, for example, this house is 600 sq.ft, and it's probably small for my family; this one is 1500 sq.ft, and this one is probably ok. You could definitely measure your TV in inches, and then in centimeters, and conclude that your TV is twice bigger in centimeters than in inches, but it doesn't make any practical sense.

There's a math concept of measure, which is generalization of length, area and volume concepts. Generally speaking, measure is a function defined on a set, which maps its elements into real numbers, i.e π»:S⟼β. You are definitely free to build any kind of measure, and then measure your set in parrots, carrots or elephants. You can also compare the measures in some ways, because these are real-valued functions, having some good properties. I don't recall though any multiple measures juggling in university math program, I would say it is something more pertinent to theoretical research works.

I had a lengthy chat with my son's teacher, arguing that they definitely can compare amounts of the units used to cover the rectangle or they can compare the areas of different objects in same units, but they shouldn't compare areas in different units. I didn't succeed.

So is it something that 2nd grade kids are really expected to deal with or did the teachers misinterpret the Ontario math curriculum, which says:

My 2nd grade son has shown his classwork to me. They measured a rectangle by covering it by different shapes: squares, triangles etc, and then they computed area of the rectangle. For example, 20 squares were used to cover the rectangle, then the rectangle's area equals 20 rectangles. If 16 triangles match the shape of the rectangle, then the area equals 16 triangles.

Then they compared the areas. They deduced that area of the rectangle covered by squares is bigger than rectangle's area covered by triangles, because 20 is bigger than 16. That's right, same rectangle, different units, which area is bigger.

The idea of measuring is being able to compare different objects, for example, this house is 600 sq.ft, and it's probably small for my family; this one is 1500 sq.ft, and this one is probably ok. You could definitely measure your TV in inches, and then in centimeters, and conclude that your TV is twice bigger in centimeters than in inches, but it doesn't make any practical sense.

There's a math concept of measure, which is generalization of length, area and volume concepts. Generally speaking, measure is a function defined on a set, which maps its elements into real numbers, i.e π»:S⟼β. You are definitely free to build any kind of measure, and then measure your set in parrots, carrots or elephants. You can also compare the measures in some ways, because these are real-valued functions, having some good properties. I don't recall though any multiple measures juggling in university math program, I would say it is something more pertinent to theoretical research works.

I had a lengthy chat with my son's teacher, arguing that they definitely can compare amounts of the units used to cover the rectangle or they can compare the areas of different objects in same units, but they shouldn't compare areas in different units. I didn't succeed.

So is it something that 2nd grade kids are really expected to deal with or did the teachers misinterpret the Ontario math curriculum, which says:

estimate, measure, and record area, through investigation using a variety of non-standard units (e.g., determine the number of yellow pattern blocks it takes to cover an outlinedshape)

And that reminds me of the old joke -- or perhaps a Shel Silverstein poem? -- about the kid who traded his dollar for 2 quarters, his 2 quarters for 3 dimes, his 3 dimes for 4 nickels, and finally his 4 nickels for 5 pennies and proudly showed his father how much more money he had, since 5 is more than 1. (This obviously only works with a compatible currency, but there are probably equivalents everywhere.) Perhaps if you told the teacher that triangles are nickels and squares are quarters?

ReplyDeleteOn the other hand, to maintain a good relationship with the teacher perhaps it would be better to explain this privately to your child.

I wish I knew that story :)

ReplyDeleteTrue, I'm somewhat concerned though about the education quality. If you want to introduce the kids to non-sense/paradox/whatever else, then you have to be very careful and explain it properly.

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