Concept+of+Fraction

A page for notes on the paper by Robert B. Davis - [|The Development of the Concept of "Fraction" from Grade Two through Grade Twelve. Final Report. Part One, Part Two and Appendix.]

Distinction between a fraction as a part of something and a fraction as a specific size.

//"On any given day a large number of students will assert 'I knew what to do; I just couldn't thing of it during the exam!' Finding items in the human memory is no small task; on the contrary, we very often fail to locate items that do exist in our memories ... somewhere!"//

During an interview the student is asked to write down the answer after they say it. I noticed where B. would say one answer and write another. Perhaps this is a test for how well the concept is formed. Of course there could be other reasons as well hmmmm

On pg. 40 //"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 thing to "map" is the developmental evolution of each concept. 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. Even when there are "specialists" to teach mathematics, they may not have a deep understanding of the concepts or the map. (Hmmm need to find a way to say this that doesn't get some people defensive) 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 leave answers here? ;) Obviously it depends on their models, any mis-conceptions they may have, and other factors. One step at a time.

So let's say a student can add 3/7 + 1/7, but when presented with the problem 1/2 + 1/3 does not know how to get the answer.