This article is the second in a series that provides alternate and common-sense interpretations of the Common Core math standards.
In the first article, I said Common Core (CC) lends itself to interpretations along reform math ideology, an ineffective method of teaching math that focuses on long, drawn-out ways to solve problems. In the wake of that first article, I have been asked, “How so, when the standards are touted as being neutral and not dictating pedagogy”? It is so because publishers, test makers, and schools take neutrality as a signal that techniques that caused many of our nation’s current math problems in the first place—and which many thought CC would fix—are to continue. On top of so-called neutrality, throw in Common Core’s Standards for Mathematical Practice, which require students to “explain” and “understand,” and the perfect storm exists to institutionalize the problems. In exchange, we get some expectations of “procedural fluency” that will not be defined until we see the actual tests.
People have also asked: “If reform math techniques are being followed (and enforced) and the test-makers are essentially testing to the teach (my phrase, not the readers’) why then would anyone want to follow the alternative approaches you have been writing about here?”
It’s a valid question. Here is my answer. I’m not saying to abandon the standards. I’m showing how the standards can be taught in ways that make sense, and in ways that have for many years been taught with success. I advise that teachers rearrange the order of some CC standards to impart standard algorithms earlier, rather than later. The practices we are seeing written about on the Internet that require students to draw endless pictures for every problem are not dictated by Common Core, and even a lead writer of the standards (William McCallum) has said so.
I also point out that the approach I write about is not the only alternative—it is a suggestion that may lead teachers to other equally sensible approaches. I welcome hearing other approaches. But we’re still left with the question: Where does this all leave us with respect to standardized tests, which may require reform math approaches to problems, as well as “explanations” and demonstrations of “understanding”? The fear is that students will do poorly on such tests because they will not know how to write explanations that demonstrate the so-called understanding. But such thinking confuses cause and effect. Forcing students to think of multiple ways to solve a problem, for example, or to write an explanation for how they solved a problem or why something works does not in and of itself cause understanding. It is investment in the wrong thing at the wrong time. The “explanations” most often will have little mathematical value and are on a naïve level since students don’t know the subject matter well enough. The result is at best a demonstration of “rote understanding.”
Of interest here is that PISA, the international exam given every several years, is essentially constructed along the same reform math principles; it tests for students’ ability to apply prior knowledge in new situations. Interestingly, the nations that teach math in the traditional fashion seem to do quite well. Basic foundational skills enable more thinking than a conglomeration of rote understandings.
With that as introduction, this article will address the third-grade standards that pertain to fractions.
What Fractions Are and Their Representation
The following standards pertain to teaching what a fraction is and how to represent it—with the ultimate goal that students can represent the fraction on a number line:
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.
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
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.
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.
These standards are formally written, but essentially define a fraction as the division of a whole into equal parts. This can initially be represented by shapes such as circles or rectangles, traditionally illustrated by pies, cakes, and candy bars. This is a natural way to introduce the concept, since when children are first learning about fractions, they need to focus on things they can see. See Figure 1 below, taken from “Arithmetic We Need, Grade 3,” (Brownell, et al.) 1955.
Ultimately, the goal of the standards is to move students from a pizza-understanding of fractions to a number-line understanding. But the concept of “a fraction is actually a number” means nothing to many in the third grade; it is abstract. So it is important that students first gain an initial understanding of what fractions are through shapes, so they can see what the numerator and denominator of a fraction represent. Then, a natural way to progress to the number line and to think about fractions as numbers is to use rulers, since students are already familiar with them. An example taken from “Arithmetic We Need, Grade 3” (Brownell, et al., 1955) is shown in Figure 2 below.
In this example, students can see and understand readily that a fraction such as can be represented by dividing a segment into four parts and marking off lengths by and counting three of them, which realizes the intent of standard 3.NF.A.2.B. The ruler concept can then be extended into lines of various lengths to show that line segments can be divided into equal parts. Figure 3 below, taken from “Growth in Arithmetic; Fourth Grade” (Clark et. al.; 1952) shows how this concept can be introduced—in particular, problems 1 through 6. A more challenging problem that illustrates this standard is problem 14.
While aspects of the standards listed above are within reach of most third graders, others may not be. In particular, the following additional standards on equivalence of fractions may be difficult for some third-grade students:
Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
The standards above address what equivalent fractions are, how to use them to compare size of fractions, and how to represent them pictorially. This should be done in increments, combining shape representations (i.e., circles and rectangles) with number-line representations, so students are not suddenly shifted to a system they do not understand.
Questions 4 through 8 in Figure 3 offer an example of how to lead in to the concept. Figure 4, taken from “Growth in Arithmetic, Grade 4” (Clark, et al., 1952), provides more information, showing what equivalent fractions “look like,” how they are represented pictorially, and how whole numbers can be represented as equivalent fractions (e.g., ).
The use of shapes can be extended to use of a number line to represent equivalent fractions, and can also use number lines to compare sizes of fractions. Students should be given examples of number lines that they can then use to either make evaluations of fractions, or to supply missing fractions on them. Figure 5 below provides an example from a textbook prepared by the School Mathematics Study Group (SMSG; 1962), which illustrates how exercises with number lines can be used to comply with the above standards. Note that this textbook was part of the 1960s New Math effort. While some of the topics taught in that era were inappropriately formal and abstract for the lower grades, some material in the books published under the auspices of SMSG were quite effective and realize the intents of some of the Common Core standards.
If students are required to draw their own number lines to illustrate equivalent fractions per these standards, it is advisable to 1) keep the problem straightforward, and 2) have the student use bar diagrams such as those shown in Figure 6 below. Figure 6 illustrates how 3/6 = 4/8 by dividing two bars of the same length into six and eight equal parts. Such diagrams can be done by measuring segments with a ruler:
Students should be allowed to use both types of pictorial representations: shapes (circles, rectangles, etc.) and number lines. Some students may find shapes easier to understand than number lines. For those students, a back-and-forth approach may make the transition to the number line easier. It should also be recognized that there may be some third-grade students who cannot fully make the transition; so shapes may be the main mechanism by which they understand equivalent fractions.
For Next Time
I will continue to address the standards for fractions, for fourth grade and higher, in subsequent articles. In the meantime, as I have said previously, I very much value and encourage readers’ interpretations of and approaches for teaching these standards. In so doing, I would also welcome seeing effective and time-tested approaches to teaching methods, as well as procedures and concepts described in other textbooks from various eras.
Brownell, William A., G. T. Buswell, I. Saubel. “Arithmetic We Need, Grade 3”; Ginn and Company. 1955
Clark, John R., C.W. Junge, H.E. Moser. “Growth in Arithmetic, Grade Four”; World Book Company; 1952.
School Mathematics Study Group. “Mathematics for the Elementary School, Grade Four”; Stanford University; 1962.
Image by grace_kat.