On pg. 40 of
"In the case of fractions, the developmental evolution of each concept is important. At first, one probably defines a/b by taking a candy bar, or a pizza, or something else, dividing it into b pieces, and taking a of them. ..."
"When one encounters 'improper fractions,' this meaning fails. One cannot use this meaning to speak of (say) 5/4" We must introduce the concept of unit. We divide each unit into b equal pieces, and take a of them. Now, with these new definitions, we can easily deal with 5/4, although we shall need two units in order to do it."
So one step in our process to "map" is the developmental evolution of each concept (okay its a BIG step, but someone must have done this before). The challenge is to understand where the child is and what questions and experiences you can provide to help them progress. It is unrealistic to expect elementary school teachers, who must teach all subjects, to know let alone be masters of the concepts and maps in all subjects. So, how can you do this if the teacher may not understand certain concepts or the map?
How can we help students move from their existing mental models to more powerful and more sophisticated models? (please post answers as comments ;)
Obviously it depends on their models, any mis-conceptions they may have, and other factors.
The following problem is from the interview excerpt starting on pg 45. The The student in the interview was described as "a generally bright resourceful 5th grade girl" who reported that "in her previous school there was very little instruction in arithmetic". :
"I want to show you what most people think is a really hard problem: 1/2 + 1/3"
She was stumped and did not know how to solve the problem. Later the interviewer then brought out cuisenaire rods and asked: "If we want to talk about "one half" and "one fourth" which rods do you supposed we want to call "one"?
Once she had the physical model she was able to answer "three fourths." She did this rather quickly it seems and without using the rods completely. She had simply identified which rod was "one half" and which was "one fourth" and then "solved the problem just by thinking about it. The concrete model and "enabled her to build up an appropriate representation in her mind".
Today I will start working on an Etoys version of the problem, perhaps with a pre and post assessment.