I’ve been thinking about fractions a lot lately (well for about 5 years now actually) with some math teachers of junior grade students (ages 9 through 12) and research collaborators. Fractions are a major area of concern for teachers and students in North America. The National Assessment of Educational Progress in the US reported in 2005 that fractions are “exceedingly difficult for children to master” (p.5) Students also seem to have difficulty *retaining* fractions concepts (Groff, 1996). So there are some challenges to tackle. In breaking down this content area to learn more, this year we have focused on looking at the affordances of different representations of fractions. Questions like: Which representations of fractions do students rely on to solve problems, and why? Do some representations help students solve certain types of problems more easily?

One thing we have observed repeatedly in classrooms is that students rely on circle representations of fractions almost exclusively, even though they may not seem to be a great fit for the problem at hand. For example, one day when our teacher team was working in Kerry M’s classroom, the students were asked to draw a math model that would help them solve a distance problem. The problem went something like this: Suzie walks 2 km from her home and is two-thirds of the way to school while Bemel walks 3 km from his home and is half the way to school. What fraction model could you draw to help you represent the problem, and then solve it? There were a range of representations recorded in the class, but interestingly, many students used circle area models and partitioned the circles. Several students used linear models (number lines) to help them, which seemed quite a good fit with the problem about distance. Others used rectangular area models – often squares joined together in a long line similar to a linear model. When we asked students why they would use a circle area model to represent and solve the problem, one student explained: “It’s just easier to use circles. I was thinking about pies and how much of the pie was gone.”

As an observer, I was having a hard time imagining how the circles and pies fit with walking to school. But rather than get into those details here, what this response provoked me to think about is: why is there an over-reliance on circle representations for fractions, even when they aren’t well matched to the situation?

Is it intuitive or cultural? What are we doing to favour the circle area model over other models in our classrooms, in textbooks, in newspapers, in the manipulative materials made available to students? Is pizza the only context for fractions? It might work really well for fair shares, but that is only one component of fractions understanding that we need to understand. How does the circle fraction help a nurse or doctor figure out a critical one-quarter dosage of medication? Or to return to a food scenario, consider these area diagrams:

Three cakes are cut into two pieces. Which cake piece would you want to eat? Does it matter?

And when might we want to use discreet models (that show parts of a set)?

One of the things that we have learned through our research collaborative is that representations are a critical component of fractions understanding, and although we are gaining insights through our explorations with students, there is still a LOT to learn about how we represent fractions! For more reading, check out these articles:

Bruce, C. & Flynn, T. (2011). Which is greater: One half or two fourths? An examination of how two Grade 1 students negotiate meaning. *Canadian Journal for Studies in Science, Mathematics and Technology Education*, *11*(4), 309–327.

Bruce, C. & Ross, J. (2009). *Conditions for effective use of interactive on-line learning objects: The case of a fractions computer-based learning sequence*.** Electronic Journal of Mathematics and Technology*** *[online serial] *3*(1*).* Available http://www.radford.edu/ejmt

**And stay tuned for a new Digital Paper on Fractions!**

Underlying the development of fractional thinking is a number system that is different from the numbers that students have already had experience with. Fractions have different rules for naming, quantifying, ordering, adding, subtracting, multiplying, dividing, etc. Students will need to develop an understanding of these rules and be able to apply them when working with fractions. Using a variety of visual and numerical representations for fractions can support students to build up experiences with the different areas of fractions (fractional constructs).