National assessments consistently show that large numbers of students struggle with fractions from grade 3 onward, effectively reducing their academic and career opportunities. Fortunately, a growing body of academic research is starting to uncover strategies that actually work.

The following summarizes a review of studies on the difficulties with fractions by two leading researchers in the study of learning fractions in: Putting Fractions Together. Braithwaite, D. W., & Siegler, R. S. (2020, March 19). *Journal of Educational Psychology*.

**Whole number bias**

Numerous studies show that many students tend to think of fractions as two separate whole numbers rather than as a single number. This is called whole number bias and it leads to fundamental errors in comparing fractions, understanding fraction equivalence, and fraction arithmetic. It’s what causes students to claim that 2/9 > 1/2 because 2 > 1 and 9 > 2, that 9/18 is larger than 1/2, or that 3/5 +1/4 = 4/9. These and other difficulties with understanding individual fractions are why 50% of U.S. eighth graders were unable to put 5/9, 2/7, and 1/2 in order from least to greatest on a major national assessment.

**The centrality of magnitude**

Vital to understanding fractions and fraction arithmetic is the ability to represent or reason about *fraction magnitude* (size). The previous examples of errors caused by whole number bias, such as 2/9 > 1/2 or 9/18 is larger than 1/2, are all also errors in understanding fraction magnitude. And an inaccurate understanding of individual fraction size hinders performance on fraction arithmetic by robbing the student of the ability to recognize unreasonable answers. For example, in one recent study, U.S. middle school students asked to estimate sums of pairs of fractions on a number line were no more accurate than if they had simply marked the midpoint on the line for each answer. Even worse, a majority of answers were smaller than the student’s own estimates of one or both of the numbers being summed. These results illustrate that when students don’t understand fraction magnitude, they are unable to do even the most basic reasoning about addition, the simplest of arithmetic operations.

**The Integrated Theory of Numerical Development**

The Integrated Theory of Numerical Development holds that “numerical development involves increasingly precise representation of the magnitudes of increasing ranges and types of numbers, including whole numbers, fractions, and decimals.” One of the most important predictions of the theory is that understanding numerical magnitudes is closely related to understanding arithmetic. This prediction has been supported by multiple studies showing a strong correlations between understanding magnitude and arithmetic for both whole numbers and fractions. It has been further supported by experimental studies showing that interventions with a strong focus on fraction magnitude improve fraction arithmetic performance compared to traditional methods, even as less instructional time is focused on the learning and practicing of arithmetic.

**ExploreLearning Frax—a better way to learn fractions**

Frax delivers the latest research-proven instructional strategies in an adaptive game-based learning format to create a better way to learn fractions.

A few of the key factors in Frax that make a difference:

**In Frax, fractions are numbers first**. Each has a specific magnitude (size) and position on the number line alongside whole numbers and other fractions. Students work extensively with length models and number lines to interpret, represent, compare, order, and estimate fractions. In doing so they overcome whole number bias and develop a strong understanding of fraction magnitude.**Frax demystifies fraction arithmetic**. When students understand fractions as numbers they also better understand the arithmetic. They learn how to make sense of fractions operations and can draw connections to their work with whole numbers (e.g. the sum of two fractions must be larger than each individual fraction and therefore the sum of 1/2 + 1/3 can't be 2/5).**Frax is adaptive and individualized**so that students of all ability levels have early and ongoing success. In addition, the Frax online learning system consistently rewards students for both their effort and progress. Students come to understand that if they are willing to put in the work, they really can succeed in learning fractions.**Frax is game-based**and challenges students to perform a variety of tasks that build their fractions skills in a wide range of engaging scenarios. The math games are supported by brief, just-in-time instruction, allowing students to learn largely by doing rather than by watching and listening.