Year 6 Students’ Methods
of Comparing the Size of Fractions
Peter Gould
NSW Department of Education and Training
Background
- Students use their knowledge of whole numbers to order
fractions.
- Students order fractions correctly for the wrong reasons.
- Thinking quantitatively about fractions relies upon
equal-partitioning and the invariance of the whole.
- Different concept images are invoked at different times, so
no conflict appears.
- Adding and subtracting fractions is based on the concept of
equality of fractions.
Aim of the study
- The aim of this study is to explore students’ fraction concept
images by examining their explanations as to which of a pair
of fractions is the larger.
- Which strategies do Year 6 students use to compare the size
of quantity fractions?
- Do Year 6 students consistently apply the same strategy
across a range of fraction comparison questions?
Method
- Sample of 100 students, 11 to 12 year
old, Year 6, primary middle-class
school, Sydney metropolitan area.
- The students were given a sheet of
paper. The questions were read out
by the teachers.
- Example Question: Which is bigger,
one-half or one-third? How do you
know?
- Pairs that had to be compared:
Q1: ( 12 , 13 ) Q2: ( 41 , 15 ) Q3: ( 51 , 16 ) Q4:
1 ) Q5: ( 1 , 1 ) Q6: ( 2 , 5 ) Q7:
( 16 , 12
6 3
3 6
9 , 12 )
( 10
13
- The answers were classified as
correct or incorrect and the
explanations were coded according
the method they described.
Results
Unitary Non-unitary
Strategy
(Q1) (Q7)
Area model
44
34
Number and size of parts
10
0
Bigger denominator is smaller fraction (BIS)
6
12
Parts of a common whole (e.g., 3, 6, 9, 100)
7
5
Additive reasoning (9+3)/(10+3) = 12/13
0
6
Conversion procedure (Common Denominator)
4
10
Conversion procedure (Decimal)
4
0
Conversion procedure (Percentage)
3
3
Number of parts
4
4
Size of complement to 1
0
4
Other (e.g. fact, changed question, unclear,
18
22
guessed, no explanation)
Table : Strategies used to Compare the Sizes of Unitary and Non-unitary
Fractions
- 48% of the students used the same method for comparing
unitary fractions and non-unitary fractions.
- Some students used a new strategy for the final two
questions.
Q1 Q2 Q3 Q4 Q5 Q6 Q7
96 96 92 90 95 71 53
Table : Percentage of Correct answering,
N = 100
- The percentage of students that
correctly answered question 6 and 7
is lower than for the rest of the
questions.
- Although 41% answered all seven
questions with the correct response,
their responses were often achieved
through spurious reasoning.
Figure : Which is bigger, one-third or
one-sixth?
Conclusion
From the number of correct answers on standard assesments it is not necessarily possible to infer the
students’ understanding of the equality of fractions.
Gould, P. (2005). Year 6 students’ methods of comparing the size of fractions. In P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce, & A. Roche (Hrsg.), Building connections: research, theory
and practice. Proceedings of the annual conference of the mathematics education research group of Australasia (S. 393–400). Sydney: MERGA.