This is the third part of a multi-part blog post on the concept of area.

Pop quiz!

What’s the formula for the area of a triangle?

## 2012-06-26

## 2012-06-15

### The concept of area – Part 2

This is the second part of a multi-part blog post on the concept of area.

NAEP’s stark revelation of the extent to which students fail to understand the concept of area and lack basic problem solving skills warrants not a simple tweaking of current methodology, but a complete rethinking of how the topic of area is presented.

Existing mathematics curricula compartmentalize math skills into discrete chunks, the main benefits accruing to educators: it’s easier to teach and easier to write tests. The math bite approach to imparting skills is worse than the much-maligned notion of teaching to the test; it institutionalizes a testing regime, at the expense of developing students’ thinking skills. Even if they learn every standard, the parts remain disjointed without students gaining the real math wisdom that comes from synthesizing parts into a whole.

In earlier posts, we have tried to convey this notion of synthesis. We discussed our general philosophy: how applying concepts in useful ways abstracts the concept. We continue to advocate the concept → formal introduction → applications → abstraction sequence. Once concepts are synthesized, i.e., reach the abstract stage of understanding, actual thinking emerges and those concepts are ready to be applied in new, previously unencountered situations—and isn’t that the real point of education?

NAEP’s stark revelation of the extent to which students fail to understand the concept of area and lack basic problem solving skills warrants not a simple tweaking of current methodology, but a complete rethinking of how the topic of area is presented.

Existing mathematics curricula compartmentalize math skills into discrete chunks, the main benefits accruing to educators: it’s easier to teach and easier to write tests. The math bite approach to imparting skills is worse than the much-maligned notion of teaching to the test; it institutionalizes a testing regime, at the expense of developing students’ thinking skills. Even if they learn every standard, the parts remain disjointed without students gaining the real math wisdom that comes from synthesizing parts into a whole.

In earlier posts, we have tried to convey this notion of synthesis. We discussed our general philosophy: how applying concepts in useful ways abstracts the concept. We continue to advocate the concept → formal introduction → applications → abstraction sequence. Once concepts are synthesized, i.e., reach the abstract stage of understanding, actual thinking emerges and those concepts are ready to be applied in new, previously unencountered situations—and isn’t that the real point of education?

## 2012-06-05

### The concept of area – Part 1

This is Part 1 of a multi-part blog post on the concept of area.

In 2005, the National Assessment of Educational Progress, otherwise known as ``The Nation's Report Card'', presented the following question:

This
question was given to fourth graders, 47% of whom answered correctly.
This result would have been mediocre if tiling had been taught by fourth
grade and good if tiling had not been taught (i.e., students
were able to figure it out on their own)—but we don’t know which.

In 2005, the National Assessment of Educational Progress, otherwise known as ``The Nation's Report Card'', presented the following question:

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