Writing analogy: calligraphy copybooks, applications and books

Xavier Bordoy

Table of Contents


"millorar una activitat" "analogia" "enriquir" "bitàcola"


Different types of activities and their differences

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.
A caligraphic copybook page. You can see a zoomed version
  • You could write a formal application to Government for example (Govern de les Illes Balears 2006). With this kind of document, you are restricted with a lot of format constraints but you could freely write the content.
Formal application example. You can see the original document
  • And finally, you could write a book. You are not restricted to form or content.
The book cover of Don Quijote de la Mancha. You can see the zoomed version

Following 5 Practices for Orchestrating Productive Task-Based Discussions in Science (Cartier et al. 2013) 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 34\frac{3}{4} of 1616.

Calligraphy copybooks

  • Activity: “Calculate 34\frac{3}{4} of 1616
  • Students possible response: “34 of 16=3164=12\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.


Cartier, Jennifer L., Margaret S. Smith, Mary Kay Stein, and Danielle K. Ross. 2013. Discussions in Science. National Council of Teachers of Mathematics. http://www.nctm.org/store/Products/5-Practices-for-Orchestrating-Task-Based-Discussions-in-Science/.

Govern de les Illes Balears. 2006. Llibre d’estil. amadip.esment. http://www.caib.es/conselleries/relinst/sgtrelinst/llibrestil/00index.html.

  1. Try and failure, 3416\frac{3}{4} \cdot 16, equations (3x+x=163x+x = 16), etc.↩︎

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