Writing analogy: calligraphy copybooks, applications and books

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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: “\(\frac{3}{4} \text{ of } 16 = \frac{3 \cdot 16}{4} = 12\)


  • 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


  • 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.


[1] Cartier, J.L., Smith, M.S., Stein, M.K., and Ross, D.K. 5 practices for orchestrating productive task-based discussions in science. National Council of Teachers of Mathematics, 2013.

[2] Govern de les Illes Balears. Llibre d’estil. amadip.esment, 2006.

  1. Try and failure, \(\frac{3}{4} \cdot 16\), equations (\(3x+x = 16\)), etc.

  2. Does the cost vary if we choose joined (arc-connex) regions than disjoint areas?