One of students’ problems is that they don’t view fractions as numbers at all, but as “meaningless symbols that need to be manipulated in arbitrary ways to produce answers that satisfy a teacher,” write the panel of experts who developed the recommendations.
Another common problem is that students focus on numerators or denominators as separate numbers rather than thinking of the fraction as a single number. This is why students frequently mistake 3/8 as a larger number than 3/5. Other students confuse properties of whole numbers with properties of fractions. For example, many high school students believe that just as there is no whole number between 5 and 6, there is no number of any type between 5/7 and 6/7.
The guide addresses not only the need to improve students’ understanding of fractions, but also the need to improve teachers’ understanding of them.
“U.S. teachers’ understanding of fractions lags far behind that of teachers in nations that produce better student learning of fractions, such as Japan and China,” the panel writes.
From more than 3,000 studies on teaching fractions and number lines selected for review, the panelists identified 33 studies that met evidence standards with or without reservations; these 33 studies are the basis for many of the panel’s recommendations to teachers.
Here are the 5 major recommendations:
Recommendation 1. Introduce students to the concept of fractions by drawing on their informal understanding of sharing and proportionality.
Start with equal sharing; then progress to sharing in fractional parts.
- Use equal-sharing activities to introduce fractions. For example, 12 candies can be partitioned equally among 4 groups of children. Divide sets of objects first; then progress to dividing a single object. This shifts the problem from asking students how many things each person should get to how much of an object each person should get.
- Use equal-sharing activities to develop students’ understanding of ordering and equivalence of fractions. haring activities can help students see that as the number of people sharing objects increases, the size of each person’s share decreases. Students can compare the size of pieces of candy bars that result when splitting a candy bar equally among 3, 4, 5 or 6 children.
- Progress to more advanced understanding of proportional reasoning concepts. Begin with activities that involve similar proportions, and then move on to activities that involve ordering different pro – portions. For example, the story of Goldilocks and the Three Bears illustrates similar proportions because the big bear needs a big chair, the medium-sized bear needs a medium-sized chair, etc. Shoe sizes illustrate co-variation or how one quantity (foot) increases as another (shoe size) increases.
Comment from panel: “One-half is an important landmark in comparing proportions; children more often succeed on comparisons in which one proportion is more than half and the other is less than half, than on comparisons in which both proportions are more than half or both are less than half (e.g., comparing 1/3 to 3/5 is easier than comparing 2/3 to 4/5.).”
Recommendation 2. Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers.
Although the part-whole interpretation of fractions is important, too often instruction does not convey that fractions are numbers with magnitudes (values) that can be either ordered or considered equivalent.
- Use measurement activities and number lines to help students understand that fractions are numbers, with all the properties that numbers share.
- Have students locate and compare fractions on number lines.
- Use number lines to improve students’ understanding of fraction equivalence, fraction density (the concept that there are an infinite number of fractions between any two fractions), and negative fractions.
- Help students understand that fractions can be represented as common fractions, decimals, and percentages, and develop students’ ability to translate among these forms.
Comment from panel: “Number lines can be used to illustrate that equivalent fractions describe the same magnitude. For example, asking students to locate 2/5 and 4/10 on a single number line can help them understand the equivalence of these numbers…Although viewing equivalent fractions as the same point on a number line can be challenging for students, the panel believes that the ability to do so is critical for thorough understanding of fractions.”
Recommendation 3. Students are most proficient at applying computational procedures when they understand why those procedures make sense.
Teachers should take the time to provide such explanations and to emphasize how fraction computation transforms the fractions in meaningful ways.
- Use area models, number lines, and other visual representations to improve students’ understanding of computational procedures.
- Have students use estimation to predict or judge the reasonableness of answers to problems involving computation with fractions.
- Address common misconceptions regarding computational procedures with fractions.
- Present real-world contexts with plausible numbers for problems that involve computing with fractions.
Comment from panel: “Misconceptions about fractions often interfere with understanding computational procedures. The panel believes that it is critical to identify students who are operating with such misconceptions, to discuss the misconceptions with them, and to make clear to the students why the misconceptions lead to incorrect answers and why correct procedures lead to correct answers.”
Recommendation 4. Proportional reasoning is needed in everyday contexts, such as adjusting recipes to the number of diners or buying material for home improvement projects.
These contexts offer many opportunities to develop students’ conceptual understanding.
- Develop students’ understanding of proportional relations before working on ratio, rate and proportion problems and computational procedures that are conceptually difficult to understand (e.g., cross-multiplication)
- Build on students’ developing strategies for solving ratio, rate, and proportion problems. Encourage students to use visual representations to solve ratio, rate, and proportion problems.
- Have students use and discuss alternative strategies for solving ratio, rate, and proportion problems.
Comment from panel: “Evidence from many types of problem-solving studies, including ones involving ratio, rate, and proportion, indicates that students often learn a strategy to solve a problem in one context but cannot apply the same strategy in other contexts. Stated another way, students often do not recognize that problems with different cover stories are the same problem mathematically. To address this issue, teachers should point to connections among problems with different cover stories and illustrate how the same strategies can solve them.”
Recommendation 5. Professional development programs should place a high priority on improving teachers’ understanding of fractions and of how to teach them.
- Build teachers’ depth of understanding of fractions and computational procedures involving fractions.
- Prepare teachers to use varied pictorial and concrete representations of fractions and fraction operations.
- Develop teachers’ ability to assess students’ understandings and misunderstandings of fractions.
Comment from the panel: “Appropriate use of representations for teaching fractions, a key aspect of the panel’s recommendations, requires that teachers understand a range of representations and how to use them to illustrate particular points.”
“Developing Effective Fractions Instruction for Kindergarten Through 8th Grade,” Sept 2010, by Robert Siegler et al. National Center for Education Statistics. The report is available at: http://ies.ed.gov/ncee/wwc/