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# Writing analogy: calligraphy copybooks, applications and books

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## Abstract

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.
• 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 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

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.

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?