A study conducted a few years ago by a Carnegie Mellon University research team identified a major skill gap in U.S. students—inadequate understanding of fractions and division. The team found that fifth graders’ understanding of fractions and division predicted how high school students performed in algebra and in math overall. These findings reinforce what educators already know: There’s an urgent need to improve instruction of fractions and division.

Children are expected to learn fractions in primary school. But it is in upper elementary and middle school that their math foundations are really tested. The difficulty children have should not be surprising considering the complexity of the concepts being taught—for example, fractions with unequal denominators. What this suggests is that upper elementary and middle school teachers need to understand fractions very well. In addition, they must be able to teach them at a deeper level.

In this article, we’ll offer tips for making the teaching of difficult fractions easier and more comfortable for teachers and more effective and fun for students.

**The Readiness Factor**

Are students ready to be challenged by more complex fractions? Have they built the foundation of skills necessary to succeed? That’s a question teachers must consider before taking their instruction to a higher level. Because, without experience and a solid grounding in basic skills, children will not only find math increasingly difficult, but they’ll also get poor grades and possibly suffer from math anxiety.

What prior skills are needed to succeed at fractions and division? A strong curriculum teaches these foundational concepts in the early grades: Equal sharing, whole and parts, equivalent fractions, comparing fractions, estimating, operations and unit fractions. Of these skills, unit fractions is by the far most important because all fractions are built from this concept.

**Two Ways of Thinking About Division**

As with many terms in mathematics, it’s easy to assume we all understand what we mean when we refer to division. But that’s not the case. Many of us have not been taught that there are two ways of looking at division: Partitive and quotitive.

Partitive division refers to dividing a whole into several equal parts, such as sharing 2 bags of potato chips equally among 4 people. The idea behind partitive division is equal sharing or distributing equally into a specified number of parts. Quotitive division is different. Here we start out knowing the size of the parts, but asking how many of them exist—for example, how many $2 bags of potato chips can you buy with $10?

Why is it important students understand there are two ways of thinking about division? Both ways of thinking will give them the same answers and students can use them interchangeably. That’s helpful because sometimes they will be able to find the answer more quickly using one method over the other. In addition, providing students with different ways to approach a problem teaches them to think and reason. It leads to deeper understanding, sparks curiosity, and builds confidence. And, in a safe and supportive environment, it proves to students they can tackle difficult fractions and succeed.

For a deeper look at complex fractions, especially division, check out SDE’s webinar Success with Fractions—Next Steps: Simple Strategies for More Difficult Fractions (Gr. 3–6) by veteran classroom teacher and Singapore math expert Anni Stipek.