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Fractions are an important building block in students’ mathematical foundations. Understanding how they work and why is crucial for success in Algebra and more advanced math courses. But fractions are also notoriously difficult to master.
Studies have shown that a significant portion of students—about one third—don’t make much progress in their understanding of the topic between 4th grade, when operations with fractions are typically introduced, and 6th grade, when students are expected to be fluent in fractional arithmetic.
Fractions are so challenging in part because they don’t operate in the same way that whole numbers do, said Nancy C. Jordan, a professor of learning sciences at the University of Delaware. Numbers of the same magnitude can look very different: Take 2/4 and 8/16, for example. And sometimes, when the numbers in a fraction grow bigger, the magnitude actually gets smaller—1/4 is bigger than 1/8, for instance.
On Tuesday, Jordan presented her work on a fraction sense intervention for struggling 6th graders at an Institute of Education Sciences Math Summit, an online conference hosted by the U.S. Department of Education’s research wing.
Jordan’s work, which is funded by an IES grant, will scale up a program that she and her colleagues found improved students’ understanding of fraction concepts and measurement, as well as their ability to apply that understanding to solve problems.
The intervention “aims to make explicit mathematical connections,” said Jordan, demonstrating how fraction magnitudes are represented across different contexts.
Parts of a whole vs. values on a number line
Traditional fraction instruction emphasizes fractions as part of a whole, said Jordan. Think about an eight-slice pizza with two slices missing to represent 6/8, for example, or a group of four circles with three colored in to represent 3/4.
But teaching fractions this way, rather than representing them as numbers with their own magnitudes, can lead to misunderstandings, Jordan said. She shared examples of student work from pre-tests in her research. In one question, students were asked to shade in 3/4 of eight circles. To get the question correct, students would need to shade in 6 circles.
But when students got the question wrong, many shaded in three circles, because they thought of 3/4 as three parts—rather than a value between the numerals 0 and 1.
In Jordan’s intervention, teachers use a number line to represent fractions. This allows teachers to show fraction equivalence on the number line—to demonstrate, for example, that 3/4 is the same distance between 0 to 1 as 6/8. Teachers also link the number line to other fraction representations: fraction bars, a collection of items, or liquid in a measurement cup.
Teachers then help students connect these representations to numbers and equations, and students get regular practice distinguishing between and performing different operations.
Teaching fractions with a number line isn’t a new practice. It was emphasized in the Common Core State Standards introduced in 2010, which at the time represented a major shift in how fractions were taught in schools. The underlying idea behind this change is that number lines help students put fractions into context—demonstrating their relationship to integers.
But Jordan said that presenting fractions as part of a whole is still a common teaching method—as are procedural shortcuts that can leave students with little conceptual understanding of why operations work the way they do. She gave the example of the “butterfly” method of adding and subtracting fractions, which relies a multiplication trick to find a common denominator.
Jordan said her intervention demonstrates that even if students have misconceptions, it’s possible to help them develop deeper understanding in older elementary grades: “Even students who are struggling mightily with basic fractions after three years of instruction can learn to make sense of fractions.”
