## Saturday, April 3, 2010

### Fraction Addition: Dienes' Theory of Mathematical Variability

In the final lesson with my first year diploma students (they will go on to read at least two more mathematics education courses as part of their pre-service training), we had a chance to review the test they took and key learning theories.

I used examples from fraction additions to discuss theory of variability (Zoltan Dienes). How is 2/5 + 2/5 different from 2/3 + 2/3? Both are symbolic representation of fraction additions. But mathematically they are not the same. Dienes refer to this as mathematical variability. In 2/3 + 2/3, students need to have the knowledge of mixed numbers and improper fractions. Such knowledge is not needed in doing 2/5 + 2/5. These two tasks possess mathematical variability.

In Singapore, these are taught in different grade levels. 2/5 + 2/5 is taught in Primary 2 while 2/3 + 2/3 are taught in Primary 4.

How about 1/8 + 1/8? This is taught in Primary 3. Why is addition of like fraction (2/5 + 2/5, 1/8 + 1/8 and 2/3 + 2/3) are considered not the same and are taught in different grade levels? According to Dienes, there is mathematical variability among the tasks. The Singapore approach seems to suggest the need for students to be systematically introduced to each variation.

In 1/8 + 1/8, students need to know equivalent fractions. Mathematically the three tasks are not the same and this is mathematical variability that Dienes wrote about.

Is 5/8 + 5/8 a variation of the three tasks given above?