If you randomly split a class of primary children into two groups, what would happen if one group learned to add, subtract, multiply and divide with whole numbers and decimals while the other group learned to do these arithmetical operations with fractions? Past research suggests that the children working with fractions would have a more difficult time learning1 2. Why is that?
Fractions are less intuitive
In our previous article What Makes Learning Fractions so Hard? (Part 1), we learned that fractions have two different types of values, local and global. Whole numbers, on the other hand, have only one value (a global value) and this value corresponds directly to the number’s magnitude. For example, the number 2 represents two objects while the number 8 represents eight objects. Once you know what 2 and 8 mean, it is easy to comprehend that 8 is greater than 2. Also with decimals, 7.4 is greater than 7.3. This makes sense because 4 is greater than 3.
With fractions, things become much less intuitive and more difficult to conceptualise. Consider the fractions ½ and ¼. To a child, they see both fractions as having a 1 in the numerator which would make them equal to each other. However, in the denominators, one fraction has a 4 while the other has a 2. Children will often assume that ¼ is greater than ½ because 4 is greater than 2. This is, of course, not correct. While 4 is indeed greater than two, in this case the local value of 4 does not represent four objects but rather how many pieces of one whole object there is. It takes more practice for children to understand that the global value of ½ is 50% while the global value of ¼ is only 25%.
Whole Number Bias
One of the primary problem areas children fall into when learning fractions is the tendency to use whole number properties to interpret fractions. This is called the “whole number bias.” In a study we conducted with primary school children3, the results showed this bias occurs due to a lack of comprehension concerning what a fraction means and what the numbers inside a fraction represent. Most of the primary level pupils in our study had only a very limited understanding of fractions; they referred to pieces of pie or pizza, butdid not really understand which local value of the fractions correlated with the pieces of the pie or pizza and which correlated with the whole. What was even more difficult to grasp for the children in our study was the fractions’ represented abstract value (i.e. their global values).
What Can Be Done?
Since one of the key difficulties in learning fractions seems to stem from a lack of comprehension concerning how the local and global values relate to actual objects, we felt that a curriculum which emphasised the teaching of this type of comprehension had the potential to show positive results. We decided to create just such a curriculum and we tested it with children. In our next article, we will share the results.
References
1) Ball, D. L. (1993). Halves, Pieces, and Twoths: Constructing Representational Contexts in Teaching Fraction. In T. Carpenter, E. Fennema & T. Romberg, Rational numbers: An integration of research (pp. 157-196). Erlbaum.
2) Charalambous, C., & Pitta-Pantazi, D. (2007). Drawing on a Theoretical Model to Study Students’ Understanding of Fractions. Educational Studies in Mathematics, 64(3), 293-316. https://doi.org/10.1007/s10649-006-9036-2
3) Gabriel, F., Szűcs, D., & Content, A. (2013). The Development of the Mental Representation of Fraction Magnitude. PLOS ONE, 8(11), e80016. https://doi.org/10.1371/journal.pone.0080016