# 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 (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.

- And finally, you could write a book (Wikipedia 2016). You are not restricted to form or content.

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 \(\frac{3}{4}\) 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 somehow
^{1}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

Cartier, Jennifer L., Margaret S. Smith, Mary Kay Stein, and Danielle K. Ross. 2013. *Practices for Orchestrating Productive Task-Based 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.

Wikipedia. 2016. “El Ingenioso Hidalgo Don Quijote de La Mancha.” https://es.wikipedia.org/wiki/Don_Quijote_de_la_Mancha#/media/File:El_ingenioso_hidalgo_don_Quijote_de_la_Mancha.jpg.