6.1. The problem of comparing fractions
A consistent finding in the study has been the confirmation of the difficulty
that most children have when asked to compare fractions. Jack3, for example,
is a 9 year-old boy who had scored very low on almost all of the questions in
his pre-test. It was, thus, expected that he would have difficulty in the Virtual
Playground with the swings task, which involved increasing the area of the
swings (currently a 3 x 4 area of twelve blocks) by comparing two fractions
(the fractions 1/3 and 1/4) and choosing the number that represents the larger
amount. When the task was presented to him by the bird he immediately
replied that he would increase the area by 1/3. However, when asked by the
observer how he came up with that result, in other words, how many blocks
he believed that 1/3 and how many 1/4 represented, he replied that 1/4 is four
blocks and 1/3 is five blocks. He then continued with his decision to add five
blocks to the swings area. When he completed the placement of the blocks
(inevitably creating a non-rectangular area), he clicked on the red button to
switch to “playground mode” and see if his decision was correct. When he
saw that it was not, he reflected on his construction and concluded that the
area “did not have the right shape”.
Lisa, a 10 year old girl who had been taught fractions in school and had
average scores on her pre-test (Figure 8), made some decisions based on what
“looked right”. These decisions were evident in two cases, in which she made
mistakes with her fractions. In the case of comparison between 1/3 and 1/4,
she decided to increase the swings area by 1/4. When asked why, she replied:
“because I counted them and they are twelve, so divided by three they will not
be enough... so... [I decided that it will be] four”.
8. Observer: So you decided to increase by 1/4...
9. Lisa: Yeah.
10. Observer: And how many blocks is that?
11. Lisa: Uhm... four.
Lisa made the common mistake (identified by []) of choosing 1/4 as the
fraction that results in the larger number. However, she correctly added four
3 Pseudonyms have been given to all children that participated in the studies and that are
mentioned here.
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