Yikes! I’m sure your anxiety increased simply by reading the word “fraction.” It’s odd, but it seems that, whenever that word is mentioned, a cringe seems to crawl across an adult’s face, painfully remembering their first experience with fractions. It is, for sure, a unique concept, but I don’t think it has to be that painful.
Going to the University of Illinois drastically changed my outlook on math instruction. I realized that our kids do not have a strong enough number sense, nor do they have a strong enough knowledge of the concepts behind numbers and operations. Currently, education is undergoing a great deal of change with the Common Core, and math is included in that. Students are expected to know more than they ever were before, which is saying a lot, because many students do not understand fractions in the first place! However, I think what many students are lacking is this sense of numbers–how numbers break down, build back up, and how these rather abstract symbols represent concrete quantities.
In fact, one of the fundamental mathemtical practices from the Common Core cites that mathematics instruction should require students to “reason abstractly and quantitatively.” Here’s what they have to say:
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
Wow, that is quite a mouthful. What it really means is that they need to be able to navigate the transfer between symbols (in math, those symbols are called “numbers”) and what those numbers actually mean (context). When it comes to fractions, those numbers (or the “referents”) refer to parts and wholes. This can be extremely confusing: These parts can either be parts of a set or parts of one whole that is divided into its constituent parts. Both the “one whole” and the “set” are both referred to as “the whole.” Are you confused yet?
Our kids are, too.
When in my mathematics class at the U of I, my outlook on fractions was changed when I learned about discrete representations versus continuous representations (Fostering Children’s Mathematical Power, Baroody). While one fraction can represent either scenario, it helps kids to categorize their knowledge of what the number could mean, to help them make sense of it concretely (contextualize), and actually think through the problem, eventually transitioning to numbers. This can, then, pave the way for finding fractions of sets as well as equivalent fractions down the road. I’ve attached some videos I created that differentiate between continuous and discrete, as well as a lesson on finding the fraction of a set very concretely. I feel very strongly that students should be able to do this concretely before they are given the algorithm.
Watch, and enjoy. I hope you find it helpful!