In my work as a mathematics consultant for the past six years, I have become convinced that a clear conceptual understanding of fractions is the biggest deficit that I run into with both teachers and students. Fractions are the gateway to proportional reasoning and algebraic thinking. Any student who doesn’t understand the structure of fractions and their operations will struggle in middle and high school mathematics.
This is not news to middle and high school teachers. When I was a high school math teacher, it frustrated me that my students didn’t understand fractions. However, at that time I didn’t have the tools myself to break down fraction ideas conceptually. I, too, had learned to manipulate fractions via algorithms and was good at memorizing procedures. I’ve spent years after my stint as a high school teacher working with elementary and middle math teachers. I’ve used many resources to help reinforce my foundational understanding of how fraction concepts unfold over the grades so that I could help teachers make sense of how to work with fractions. Now all teachers can access an excellent free resource online: the Progressions Document for Numbers and Operations: Fractions
The Progressions Documents for those of you that are not familiar with them are the original documents, each written by a group of talented educators, that were first step in developing the Common Core (CCSSM). The progressions documents were used to guide the three lead writers of the CCSSM in writing the individual standards. The Common Core approach to fractions in Grades 3 – 5 focuses on conceptual understanding of fractions and their operations. There is a clear connection between fraction concepts and later algebraic concepts. The Progressions Document for Numbers and Operations: Fractions should be required reading for all math teachers grade 3 – 12.
In their article Fraction proficiency and success in algebra: What does research say?, George Brown and Robert J. Quinn assert the following:
There are at least three critical achievements in the mathematical life of a student: mastering the idea of ten as a unit, understanding fractions, and grasping the concept of the unknown. Consequently, when attempting to learn algebra without the aid of understanding fractions, “it is no wonder that many students’ seeming mastery of fractions begins to fall apart” (Driscoll, 1982, p. 107) Australian Mathematics Teacher, v63 n3 p23-30 2007
Robert Siegler of Carnegie Mellon, in his article Early Predictors of High School Mathematics Achievement points out that understanding of fractions and long division by 10-year-old students is a predictor of later success in Algebra. He refers to research by Case and Okamoto (1996) who support the idea of introducing fractions to students as rational numbers on a number line at the beginning of their study of fractions: “During mathematics learning, the central conceptual structure for whole numbers, a mental number line, is eventually extended to rational numbers.”
Siegler is also the lead author on a 90 page Practice Guide “Developing Effective Fractions Instruction for Kindergarten Through 8th Grade” which includes the following Five Recommendations for Teaching Fractions:
Recommendation 1 is to build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts. Learning is often most effective when it builds on existing knowledge, and fractions are no exception. By the time children begin school, most have developed a basic understanding of sharing that allows them to divide a region or set of objects equally among two or more people. These sharing activities can be used to illustrate concepts such as halves, thirds, and fourths, as well as more general concepts relevant to fractions, such as that increasing the number of people among whom an object is divided results in a smaller fraction of the object for each person. Similarly, early understanding of proportions can help kindergartners compare, for example, how one-third of the areas of a square, rectangle, and circle differ.
Recommendation 2 is to ensure that students know that fractions are numbers that expand the number system beyond whole numbers, and to use number lines as a key representational tool to convey this and other fraction concepts from the early grades onward. Although it seems obvious to most adults that fractions are numbers, many students in middle school and beyond cannot identify which of two fractions is greater, indicating that they have cursory knowledge at best. Number lines are particularly advantageous for assessing knowledge of fractions and for teaching students about them. They provide a common tool for representing the sizes of common fractions, decimals, and percents; positive and negative fractions; fractions that are less than one and greater than one; and equivalent and nonequivalent fractions. Number lines also are a natural way of introducing students to the idea of fractions as measures of quantity, an important idea that needs to be given greater emphasis in many U.S. classrooms.
Recommendation 3 is to help students understand why procedures for computations with fractions make sense. Many U.S. students, and even teachers, cannot explain why common denominators are necessary to add and subtract fractions but not to multiply and divide them. Few can explain the “invert and multiply rule,” or why dividing by a fraction can result in a quotient larger than the number being divided. Students sometimes learn computational procedures by rote, but they also often quickly forget or become confused by these routines; this is what tends to happen with fractions algorithms. Forgetting and confusing algorithms occur less often when students understand how and why computational procedures yield correct answers.
Recommendation 4 involves focusing on problems involving ratios, rates, and proportions. These applications of fraction concepts often prove difficult for students. Illustrating how diagrams and other visual representations can be used to solve ratio, rate, and proportion problems and teaching students to use them are important for learning algebra. Also useful is providing instruction on how to translate statements in word problems into mathematical expressions involving ratio, rate, and proportion. These topics include ways in which students are likely to use fractions throughout their lives; it is important for them to understand the connection between these applied uses of fractions and the concepts and procedures involving fractions that they learn in the classroom.
Recommendation 5 urges teacher education and professional development programs to emphasize how to improve students’ understanding of fractions and to ensure that teachers have sufficient understanding of fractions to achieve this goal. Far too many teachers have difficulty explaining interpretations of fractions other than the part-whole interpretation, which is useful in some contexts but not others. Although many teachers can describe conventional algorithms for solving fractions problems, few can justify them, explain why they yield correct answers, or explain why some nonstandard procedures that students generate yield correct answers despite not looking like a conventional algorithm. Greater understanding of fractions, knowledge of students’ conceptions and misconceptions about fractions, and effective practices for teaching fractions are critically important for improving classroom instruction.
The CCSSM writers, using the approach recommended by the research of Case and Okamoto (1996), introduces fractions as units on the number line starting in Grade 3. This is a new approach that is at present not understood and fully embraced by elementary teachers who don’t have a deep understanding of fractions on the number line, or of fractions as single numbers with a single value (magnitude). Fractions are often understood only as a comparison of part to whole. While the part to whole interpretation of fractions is an important model to understand, it breaks downs when operations such as addition, subtraction, multiplication, and division are applied to two or more fractions. One of the misconceptions that students have is that operations (addition, subtraction, multiplication, and division) on fractions have the same effect as operations on whole numbers. Note: In the following examples, we will only take into account positive rational numbers, since introducing negative values adds another layer of complexity to operations, which is not introduced until Grade 6 and also presents many misconceptions for students.
When two whole numbers are added, the resulting sum is larger than either addend. Adding two positive fractions will also result in a sum that is larger than either addend. Multiplying two whole numbers will result in a larger whole number. However, multiplication does not have the same effect on fractions, which can be very confusing to students unless the misconception is recognized and explained with concrete examples.
When the two factors in a multiplication problem are fractions, each factor acts as a “scalar” on the other fraction. In the case of 1/2 x 3/4, the product will be less than 1/2 (since we are finding 3/4 of 1/2) and the product will be less than 3/4 (since we are finding 1/2 of 3/4).
The operation of division also presents a different story than division with whole numbers. When asked to divide a larger whole number by a smaller whole number, say 6 divided by 3, the answer is 2, which is less than 6. When asked to divide 6 by 1/2, the answer is 12, which is more than 6. This presents a lot of misunderstanding and rather than delve into the reasons that this happens (1/2 is smaller half as big as 1, so twice the number of 1/2s will fit into 6), most students simply memorize the “invert the second fraction and multiply” rule, but often forget whether that rule is for division or for multiplication, or which one they invert, or whether they have to get a common denominator when multiplying and dividing.
All of these misconceptions can be circumvented, but only if teachers anticipate the misunderstanding and provide guidance to students to understand how operations on numbers can affect the numbers differently, depending upon the value of the numbers being operated on. Are both numbers less than one? Are both numbers more than one? Is one number more and one and another less than one? And, of course, the introduction of integers in Grade 6 adds another layer of complexity.
Elementary students can grasp these ideas when they are introduced in a concrete way, using manipulatives, the number line and the area model. However, students are often given the rules and, since the rules are counter-intuitive to their understanding of how whole numbers work, they are often confused.
Unit Fractions: The Building Blocks of Fractions
CCSSM bridges fractions and algebra by introducing fractions as units. For example, 1/3 is introduced as the quantity formed by 1 part when a whole is partitioned into 3 equal parts. (3.NF.A.1)
We count things using units. When using 1 as our unit, we count 1, 2, 3, 4, 5…By the same token, when we use 1/3 as our unit, we count 1/3, 2/3, 3/3, 4/3, 5/3…When we use apple as our unit, we count 1 apple, 2 apples, 3 apples, 4 apples, 5 apples. When we use x as our unit, we count 1x, 2x, 3x, 4x, 5x…The introduction of fractions as units should help students understand why it is important to have a common denominator when adding and subtracting. The denominator expresses the size of the unit. The numerator tells how many units of the denominator exist.
Linking the understanding of units fractions to the number line, CCSSM first defines a fraction such as 3/5 as being the quantity formed by 3 parts, each with size 1/5. (3.NF.A.1). This idea is extended to the number line in 3.NF.A.2a. The whole on the number line is first defined as the interval between 0 and 1. Then, if the interval is divided into, say, 4 equal parts, each part has size 1/4. The fraction 1/4 is then defined as the endpoint of the part based at 0. Finally, 3.NF.A.2b completes the number line story by explaining that the placement of 3/4 on the number line is achieved by starting at 0 and marking off 3 intervals each of length 1/4 to make an interval of size 3/4 and that the endpoint of that interval represents the number 3/4 on the number line.
These ideas are complex, not only for students, but for teachers as well. The language in the standards (seen below) is not easy to comprehend. A significant amount of professional development for teachers needs to focus on how the concept of fraction unfolds between Grades 3 and 6.
- CCSS.Math.Content.3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
- CCSS.Math.Content.3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
- CCSS.Math.Content.3.NF.A.2a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
- CCSS.Math.Content.3.NF.A.2b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
Students who intuitively know that you can’t add apples and oranges often add 2/3 + 4/5 and get 6/8. If they understand 2/3 as 2 copies of 1/3 of a whole and 4/5 as 4 copies of 1/5 of a whole, it will be clear why they can’t be added in the form that they are presently in.
When adding fractions with unlike denominators in Grade 5, the CCSSM does not require that a lowest common denominator must be found, only that a common denominator must be found.
5.NF.1: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3+5/4=8/12+15/12=23/12. (In general, a/b+c/d=(ad+bc)/bd.)
The most efficient common denominator is found by multiplying the denominators of the addends. This is how fractions are multiplied in algebraic expressions, when the actual numbers are not known.
–> multiply a/b by 1 in the form of d/d –> a/b * d/d = ad/bd
–> multiply c/d by 1 in the form of b/b –> c/d * b/b = cb/bd
–> ad/bd + cb/bd = (ad + bc)/bd
The Grade 3 CCSSM standards and those in Grades 4 through 6 will hopefully result in students and elementary teachers having a conceptual understanding of fractions that will transfer to algebraic understanding in later years. However, due to the pipeline effect, it will be several years before many students reach high school having learned about fractions using this approach. Even then, the understanding will most likely be fuzzy. I highly recommend that middle and high school teachers reinforce the conceptual understanding of fractions outlined in the Progressions Document for Numbers and Operations: Fractions in their classes. It will pay off.