Fractions are a foundational piece for tackling mathematics at all levels of schooling. Students need to understand how two numbers interact with each other as numerators and denominators, as ratios, and as proportions before they move on to more advanced math.
But fractions are also notoriously hard to teach. In the early grades, students may come in with an innate sense of fractions because they’ve learned to split a candy bar into halves or fourths. Teachers can get stuck moving students from this organic “parts-of-a-whole” understanding to thinking about fractions as unique numbers, with their own set of operational rules.
To move the needle on fractions, teachers need time to build their students’ conceptual understanding, said David Dai, an 8th and 9th grade math instructor at the Barton Academy for Advanced World Studies in Mobile, Ala., and a board member of the National Council for Teachers of Mathematics.
“We try [to] teach rules and procedures and algorithms for calculations, ... as opposed to developing a conceptual understanding for kids. Their foundation early on gets weaker,” Dai said. “Students only see fractions as computational pieces that they have to memorize rules and formulas for, where it should really be, ‘How do I make sense of [the] thing I’m looking at?’”
Dai spoke with Education Week about strategies teachers can use to clarify students’ misconceptions about fractions to make manipulating them easier. This interview has been edited for length and clarity.
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What’s the first thing teachers should do to clarify students’ misconceptions about fractions?
Show students fractions in different contexts. Too often, teachers give students singular experiences and expect them to generalize and abstract those ideas [to all types of fractions]. Give them opportunities to make sense of the part-to-whole relationships in multiple contexts. But then also think about how to present the relationship between different parts.
For instance, different contexts could mean splitting a pizza, or splitting up a classroom into boys and girls, or even splitting a food item into calories from different sources, like sugar.
Seeing the same concepts in multiple situations will [help students] develop a deeper understanding for those concepts. And then we can build on that idea for more procedural or formulaic processes later with fraction calculations.
How should teachers balance teaching fractions with visual examples and the number line?
If the fraction is less than one, like one-fourth of a whole, it makes sense to kids. It’s like looking at a pizza that’s cut into eight slices. ... If I were to organize one out of eight, and then two out of eight, and three out of eight [slices], students, generally, can sort those values in order fairly easily because the denominator is the same. They understand all of these pieces are the same size. If we’re taking that particular part-to whole relationship and transitioning over to a number line, and sorting them numerically, that makes a lot of sense for students.
Where things get a little tricky is when [the] pieces are not necessarily the same size. If we’re looking at a fraction that’s three-fourths, and we’re trying to compare that to three-eighths, most students are going to say that three-eighths is larger because your denominator eight is larger than the other fraction that has a denominator of four. This is a common misconception because students don’t really understand the size of each fraction piece.
I would encourage teachers to think about using a visual, like pizza slices, and the number line together. If we have examples that are concrete, like portioning pizzas or candy bars, which makes sense to kids, why not also have a number line [as a whole], and then split that like we’d split candy bars. Then we can show one-half and one-fourth and so on, on a number line.
How can teachers move students from these basic clarifications to more complex operations?
Context matters a lot for our early learners. Before we even start to [work on] abstract ideas of combining fractions, it’s important for kids to look combining these fractions in contexts that are familiar.
Let’s stick with candy. If they have three-fourths of a candy bar and another two-sixths of the same candy bar, they’re just trying to figure out how much of this one type of candy they have. They will grapple with this idea of combining fractions with two different denominators a lot better when the context makes sense. There’s a purpose and drive to this [activity] because they’re trying to make sure all the pieces are the same to distribute amongst the class. Kids inherently believe in this idea of fairness.
If we can build on that inherent sense of what is equal and fair, we can [help students] make sure that everyone has the same parts for us to compare and combine with an operation [like addition].
When it comes to multiplying or dividing fractions, students associate these operations on fractions as following the same rules as whole numbers. Too often, we rush into [teaching them] procedural tricks or mnemonic devices. Kids feel rushed into checking off boxes, and that’s where we start to lose them.
Instead, teachers can focus on connecting the language they use to explain an abstract operation, like multiplication, to an operation. [For example], I have three-fourths of something, and I’m taking half of that—are you going to have more or less than what you began with?
Generally, when students think about multiplication, things get larger. But now if I’m taking [away] one-half of something, the language that we use suggests I’m going to have less than what I began with. If students can process this in terms of the language and pair that with the operation, then they could say, “Oh! The resulting number I get should be less than what I started with!”
