Writing analogy: calligraphy copybooks, applications and books
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 TaskBased Discussions in Science [1] these categories rise up the demanding of knowledge. And I think that students really “write a book” if they do Projectbased 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
[1] Cartier, J.L., Smith, M.S., Stein, M.K., and Ross, D.K. 5 practices for orchestrating productive taskbased discussions in science. National Council of Teachers of Mathematics, 2013.
[2] Govern de les Illes Balears. Llibre d’estil. amadip.esment, 2006.
[3] Wikipedia. El ingenioso hidalgo don quijote de la mancha. 2016. https://es.wikipedia.org/wiki/Don_Quijote_de_la_Mancha#/media/File:El_ingenioso_hidalgo_don_Quijote_de_la_Mancha.jpg.