The other day, a math teacher sent me a lovely compliment about Newton’s Nemesis and asked if I knew much about how students with dyscalculia or other math learning disabilities perceived fractions. I really haven’t done much reading on the topic, so off to the library! I did a little reading this week and discovered a couple of intriguing studies, but am by no means, an expert.
In 2016, Katherine Lewis published some interesting findings from her dissertation on students with Math Learning Disabilities (MLD) and their knowledge of fractions. She completed the work at Berkeley under the direction of Alan Schoenfeld, a leader in mathematics education reform that I’ve admired since graduate school. Dr. Lewis conducted interviews of two young women with MLD as they compared fractions.
Children typically have fairly good number sense about 1/2, even in early grades, so they usually can determine if a fraction is more or less than ½. Children are also fairly successful in comparing fractions if the two fractions have the same denominator, such as 2/9 and 8/9. Since they know that 8 is more than 2, they can easily identify 8/9 as the larger number. But MLD students often find these comparisons difficult, and Dr. Lewis wanted to know why.
She interviewed two women with MLD, aged 18 and 19, who had persistent errors comparing the fractions such as 2/5 & 3/5 and 2/4 & 2/3.
When they compared fractions such as 2/5 and 3/5, they could create area or array models for their explanation. However, they would interpret the value of each fraction as its fractional complement. In other words, they would interpret a model of ¾ as the number ¼.
When comparing fractions such as 2/4 & 2/3, both women offered that 2/4 was equal to half, even though the interviewer didn’t ask them to make that equivalence. However, they both incorrectly thought 2/3 was equal to or less than ½. This error came about because the MLD students compared only the numerators (both are numerators are 2 so they must be the same) or only the denominators (there are more pieces if divided by 4, so that must be larger).
During the interviews, Dr. Lewis also noticed that the students perceived ½ as an action, such as halving something. They didn’t necessarily perceive ½ as 1 part out of 2 parts. She also noticed that one of the subjects sometimes wrote 5/10 as 5 1/10.
While these are only two students and we can’t generalize these findings, their perceptions of fractions are fascinating. How can we start where those students are and build a stronger foundation?
Interestingly in another study, Bugden & Anasari (2016) found perceptions of area can impact the intuitive recognition of numbers for students with dyscalculia. They tested 15 children with dyscalculia, aged 9-14, with a comparison group, and asked them to quickly compare two sets of dots that flashed on a computer screen. Which set was larger? The children with dyscalculia could recognize the larger set of dots with the same accuracy as children without dyscalculia, unless the area (size) of the dots was manipulated. If the larger number of dots had a smaller area, children with dyscalculia struggled to identify the larger number as larger!
So what if area models or arrays are problematic if they aren’t drawn precisely?
At the end of Dr. Lewis’ paper, she suggests that we need additional research to determine whether using length or weight models might be a better for MLD students than using area or array models. Her subjects struggled to compare and see the magnitude of the fractions with those models. When you combine this with the Bugden & Anasari 2016 study, she may have a really good suggestion for more research!
If we could leverage measurement models of division using length models that are halved or quartered then added together, perhaps we could make a better connection for our MLD students.
Lewis, K. E. (2016). Beyond Error Patterns. Learning Disability Quarterly, 39(4), 199–212. DOI: 10.1177/0731948716658063
Bugden, S., & Ansari, D. (2016). Probing the nature of deficits in the ‘approximate number system’ in children with persistent developmental dyscalculia. Developmental Science, 19(5), 817-833. DOI: 10.1111/desc.12324