The math skills students develop in middle school are critical for success in the higher level math courses they will encounter in high school and beyond.1 In particular, a strong understanding of fractions, rates, and ratios lays the foundation for algebra I and serves as a milestone for grade 6, the last time students often receive focused instruction on these concepts.2
In Illinois, REL Midwest is partnering with education leaders and school districts to develop a Teaching Fractions Toolkit to target gaps in middle school students’ math achievement. This toolkit will support grade 6 teachers in implementing evidence-based practices for teaching fractions and will build on the What Works Clearinghouse (WWC) practice guide Developing Effective Fractions Instruction for Kindergarten Through 8th Grade.
Below, we highlight the five recommendations from the practice guide and note how they tie into the partnership’s planned work. For each recommendation, we also indicate the WWC level of evidence supporting it as well as the Every Student Succeeds Act (ESSA) evidence tier.
Recommendation 1. Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts (Minimal Evidence/Tier 4).
If you have ever watched a group of children share a package of cookies, you will soon realize that kids have a well-honed sense of sharing and proportionality! Fractions represent many aspects of a student’s life—sharing a bag of candies among classmates, dividing up a chocolate bar among friends, or cutting a ribbon of taffy into pieces of equal length. Teachers can engage students in these types of activities as a way to teach fractions. While doing so, teachers can encourage students to think about how many pieces are in the overall whole (bag, bar, and ribbon) and then point out how the parts of the whole are equal (serving size of candy, share of the chocolate bar, piece of ribbon). Learn more.
Recommendation 2. Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward (Moderate Evidence/Tier 3).
If we consider all numbers from 0 to 2, there are both whole numbers and fractions. Initially, a number line can help students see how a fraction relates to the whole number—for example, four quarter (1/4) pieces make up one whole. Teachers can stack number lines to help students order and compare magnitude of numbers. For example, by dividing one number line into fourths and another number line into sixths, students can see that 1/4 of something is greater than 1/6. Many middle school educators include manipulatives, such as fraction pieces, as part of their instruction, enabling students to physically handle fractional parts of something to conceptually understand what the fraction represents relative to a whole or other fractions. Learn more.
Recommendation 3. Help students understand why procedures for computations with fractions make sense (Moderate Evidence/Tier 3).
Computational procedures provide efficiency when working with fractions. However, without a conceptual understanding of fractions, students have difficulty determining which procedure to use, given the situation. Using visual representations, such as number lines and area models, helps students "see" the math. For example, teachers can use number lines and area models to help students conceptually understand how to add, subtract, multiply, and divide fractions. Operations with fractions result in solutions that are often counterintuitive to what students learned with whole numbers. For example, dividing 1/2 by 1/4 gives a solution of 2, a number that is larger than the two numbers you started with! Likewise, multiplying 1/2 by 1/4 produces 1/8, which is smaller than what you started with. Using stacked number lines, students can see why these solutions are correct.
Visual representations are particularly important when working with fractions, which require re-unitization to attain common denominators before applying an operation, such as when adding and subtracting fractions. Using a visual representation helps students estimate the solution. As stated in the previous recommendation, by using manipulatives as part of their instruction, teachers can enable students to physically move items to mimic what is going on operationally. Learn more.
Recommendation 4. Develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication as a procedure to solve such problems (Minimal Evidence/Tier 4).
The use of visual fraction representations does not end in elementary school. As middle school students begin to work with more complex numbers and situations involving ratios, rates, and proportions, visual representations help students understand the relationships between quantities. A bar diagram helps students visually interpret the difference in the rates of two cars traveling the same distance or determine unit rates when comparing grocery prices. A ratio table can aid in determining ingredient amounts when increasing or decreasing the number of servings for a recipe. As with other computational algorithms, cross-multiplication provides an efficient way to work with numbers in a proportional situation, but many students have difficulty setting up the problem if they do not conceptually understand how quantities relate to one another. Learn more.
Recommendation 5. Professional development programs should place a high priority on improving teachers’ understanding of fractions and of how to teach them (Minimal Evidence/Tier 4).
Many adults learned fractions procedurally rather than conceptually, and teachers are no exception. This leads to knowing algorithms but being unsure of where and when to use them. To support teachers in working with students, and education leaders and instructional coaches in working with teachers, it is important to provide space for educators to work with the same materials that students will use so that educators can gain a learner experience as well as an educator perspective. Middle school teachers need to understand the full spectrum of fraction development, from a basic understanding in the early grades, to ratios and proportional reasoning in middle school, to the use of fractions in the study of algebra, geometry, and statistics. Learn more.
The Teaching Fractions Toolkit partnership is producing a toolkit of professional development and supports for grade 6 math teachers on evidence-based practices for teaching fractions, rational number computation, and related concepts. In early 2023, the partnership will work with a smaller group of teachers in Illinois to test and refine the teacher and student supports included in the toolkit, before moving to a more widespread study of these resources in schools across Illinois.
Booth, J. L., & Newton, K. J. (2012). Fractions: Could they really be the gatekeeper’s doorman? Contemporary Educational Psychology, 37(4), 247–253. http://eric.ed.gov/?ID=EJ977998
Hoffer, T., Venkataraman, L., Hedberg, E. C., & Shagle, S. (2007). Final report on the national survey of algebra teachers for the National Math Panel. National Opinion Research Center at the University of Chicago.
National Council of Teachers of Mathematics. (2007). The learning of mathematics: 69th NCTM yearbook.
Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107(3), 909–918. https://eric.ed.gov/?id=EJ1071576