The analogy

Recently, I discovered a analogy about mathematical activities: what kind of writing task do you do?

• You just complete the calligraphy copybooks: follow the marked line with pencil. So you are not able to write free content nop form.

  

• You could write a formal application to Government for example [2]. With this kind of document, you are restricted with a lot of format constraints but you could freely write the content.

  

• And finally, you could write a book [3]. You are not restricted to form or content.

  

Following 5 Practices for Orchestrating Productive Task-Based Discussions in Science [1] these categories rise up the demanding of knowledge. And I think that students really “write a book” if they do Project-based learning.

An example of this analogy

I give you an example of this analogy for practicing fractions as operator. I want students to calculate \(\frac{3}{4}\) of \(16\).

Calligraphy copybooks

• Activity: “Calculate \(\frac{3}{4}\) of \(16\)”
• Students possible response: “\(\frac{3}{4} \text{ of } 16 = \frac{3 \cdot 16}{4} = 12\)”

Aplication

• Activity: “Two farmers want to divide a 4 x 4 plot but the first should have three times surface than the second.

Divide this plot to verify this requeriment"

• Students response:
  • understand what is “three times”
  • calculate somehow¹ that one farmer has 12 squares and other 4 squares.
  • draw
  

Book

• Activity: “Two farmers want to divide a 4 x 4 plot but the first should have three times surface than the second. What’s the best way to do it? Consider costs like fencing, buying seeds, irrigation, etc. and crop benefits.”²
• Students responses: ?

Update: I change the book analogy from this:

Can you find three different ways to divide this plot verifying this requeriment?

What is the division which has the minimum cost? (each fencing side has a cost of $10)?

Can you compare yours with your neighbours’?

Can you find out what is the minimum cost division among all possible divisions?"

to above.

References