This post continues a series begun in 2014 on implementing the Common Core State Standards (CCSS).  The first installment introduced an analytical scheme investigating CCSS implementation along four dimensions:  curriculum, instruction, assessment, and accountability.  Three posts focused on curriculum.  This post turns to instruction.  Although the impact of CCSS on how teachers teach is discussed, the post is also concerned with the inverse relationship, how decisions that teachers make about instruction shape the implementation of CCSS.

A couple of points before we get started.  The previous posts on curriculum led readers from the upper levels of the educational system—federal and state policies—down to curricular decisions made “in the trenches”—in districts, schools, and classrooms.  Standards emanate from the top of the system and are produced by politicians, policymakers, and experts.  Curricular decisions are shared across education’s systemic levels.  Instruction, on the other hand, is dominated by practitioners.  The daily decisions that teachers make about how to teach under CCSS—and not the idealizations of instruction embraced by upper-level authorities—will ultimately determine what “CCSS instruction” really means.

I ended the last post on CCSS by describing how curriculum and instruction can be so closely intertwined that the boundary between them is blurred.  Sometimes stating a precise curricular objective dictates, or at least constrains, the range of instructional strategies that teachers may consider.  That post focused on English-Language Arts.  The current post focuses on mathematics in the elementary grades and describes examples of how CCSS will shape math instruction.  As a former elementary school teacher, I offer my own personal opinion on these effects.

Certain aspects of the Common Core, when implemented, are likely to have a positive impact on the instruction of mathematics. For example, Common Core stresses that students recognize fractions as numbers on a number line.  The emphasis begins in third grade:

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.2.A
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.2.B
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.

When I first read this section of the Common Core standards, I stood up and cheered.  Berkeley mathematician Hung-Hsi Wu has been working with teachers for years to get them to understand the importance of using number lines in teaching fractions.^[1] American textbooks rely heavily on part-whole representations to introduce fractions.  Typically, students see pizzas and apples and other objects—typically other foods or money—that are divided up into equal parts.  Such models are limited.  They work okay with simple addition and subtraction.  Common denominators present a bit of a challenge, but ½ pizza can be shown to be also 2/4, a half dollar equal to two quarters, and so on. 

With multiplication and division, all the little tricks students learned with whole number arithmetic suddenly go haywire.  Students are accustomed to the fact that multiplying two whole numbers yields a product that is larger than either number being multiplied: 4 X 5 = 20 and 20 is larger than both 4 and 5.^[2]  How in the world can ¼ X 1/5 = 1/20, a number much smaller than either 1/4or 1/5?  The part-whole representation has convinced many students that fractions are not numbers.  Instead, they are seen as strange expressions comprising two numbers with a small horizontal bar separating them. 

I taught sixth grade but occasionally visited my colleagues’ classes in the lower grades.  I recall one exchange with second or third graders that went something like this:

“Give me a number between seven and nine.”  Giggles. 

“Eight!” they shouted. 

“Give me a number between two and three.”  Giggles.

“There isn’t one!” they shouted. 

“Really?” I’d ask and draw a number line.  After spending some time placing whole numbers on the number line, I’d observe,  “There’s a lot of space between two and three.  Is it just empty?” 

Silence.  Puzzled little faces.  Then a quiet voice.  “Two and a half?”

You have no idea how many children do not make the transition to understanding fractions as numbers and because of stumbling at this crucial stage, spend the rest of their careers as students of mathematics convinced that fractions are an impenetrable mystery.   And  that’s not true of just students.  California adopted a test for teachers in the 1980s, the California Basic Educational Skills Test (CBEST).  Beginning in 1982, even teachers already in the classroom had to pass it.   I made a nice after-school and summer income tutoring colleagues who didn’t know fractions from Fermat’s Last Theorem.  To be fair, primary teachers, teaching kindergarten or grades 1-2, would not teach fractions as part of their math curriculum and probably hadn’t worked with a fraction in decades.  So they are no different than non-literary types who think Hamlet is just a play about a young guy who can’t make up his mind, has a weird relationship with his mother, and winds up dying at the end.

Division is the most difficult operation to grasp for those arrested at the part-whole stage of understanding fractions.  A problem that Liping Ma posed to teachers is now legendary.^[3]

She asked small groups of American and Chinese elementary teachers to divide 1 ¾ by ½ and to create a word problem that illustrates the calculation.  All 72 Chinese teachers gave the correct answer and 65 developed an appropriate word problem.  Only nine of the 23 American teachers solved the problem correctly.  A single American teacher was able to devise an appropriate word problem.  Granted, the American sample was not selected to be representative of American teachers as a whole, but the stark findings of the exercise did not shock anyone who has worked closely with elementary teachers in the U.S.  They are often weak at math.  Many of the teachers in Ma’s study had vague ideas of an “invert and multiply” rule but lacked a conceptual understanding of why it worked.

A linguistic convention exacerbates the difficulty.  Students may cling to the mistaken notion that “dividing in half” means “dividing by one-half.”  It does not.  Dividing in half means dividing by two.  The number line can help clear up such confusion.  Consider a basic, whole-number division problem for which third graders will already know the answer:  8 divided by 2 equals 4.   It is evident that a segment 8 units in length (measured from 0 to 8) is divided by a segment 2 units in length (measured from 0 to 2) exactly 4 times.  Modeling 12 divided by 2 and other basic facts with 2 as a divisor will convince students that whole number division works quite well on a number line. 

Now consider the number ½ as a divisor.  It will become clear to students that 8 divided by ½ equals 16, and they can illustrate that fact on a number line by showing how a segment ½ units in length divides a segment 8 units in length exactly 16 times; it divides a segment 12 units in length 24 times; and so on.  Students will be relieved to discover that on a number line division with fractions works the same as division with whole numbers.

Now, let’s return to Liping Ma’s problem: 1 ¾ divided by ½.   This problem would not be presented in third grade, but it might be in fifth or sixth grades.  Students who have been working with fractions on a number line for two or three years will have little trouble solving it.  They will see that the problem simply asks them to divide a line segment of 1 3/4 units by a segment of ½ units.  The answer is 3 ½ .  Some students might estimate that the solution is between 3 and 4 because 1 ¾ lies between 1 ½ and 2, which on the number line are the points at which the ½ unit segment, laid end on end, falls exactly three and four times.  Other students will have learned about reciprocals and that multiplication and division are inverse operations.  They will immediately grasp that dividing by ½ is the same as multiplying by 2—and since 1 ¾ x 2 = 3 ½, that is the answer.  Creating a word problem involving string or rope or some other linearly measured object is also surely within their grasp.

I applaud the CCSS for introducing number lines and fractions in third grade.  I believe it will instill in children an important idea: fractions are numbers.  That foundational understanding will aid them as they work with more abstract representations of fractions in later grades.   Fractions are a monumental barrier for kids who struggle with math, so the significance of this contribution should not be underestimated.

I mentioned above that instruction and curriculum are often intertwined.  I began this series of posts by defining curriculum as the “stuff” of learning—the content of what is taught in school, especially as embodied in the materials used in instruction.  Instruction refers to the “how” of teaching—how teachers organize, present, and explain those materials.  It’s each teacher’s repertoire of instructional strategies and techniques that differentiates one teacher from another even as they teach the same content.  Choosing to use a number line to teach fractions is obviously an instructional decision, but it also involves curriculum.  The number line is mathematical content, not just a teaching tool.

Guiding third grade teachers towards using a number line does not guarantee effective instruction.  In fact, it is reasonable to expect variation in how teachers will implement the CCSS standards listed above.  A small body of research exists to guide practice. One of the best resources for teachers to consult is a practice guide published by the What Works Clearinghouse: Developing Effective Fractions Instruction for Kindergarten Through Eighth Grade (see full disclosure below).^[4]  The guide recommends the use of number lines as its second recommendation, but it also states that the evidence supporting the effectiveness of number lines in teaching fractions is inferred from studies involving whole numbers and decimals.  We need much more research on how and when number lines should be used in teaching fractions.

Professor Wu states the following, “The shift of emphasis from models of a fraction in the initial stage to an almost exclusive model of a fraction as a point on the number line can be done gradually and gracefully beginning somewhere in grade four. This shift is implicit in the Common Core Standards.”^[5]  I agree, but the shift is also subtle.  CCSS standards include the use of other representations—fraction strips, fraction bars, rectangles (which are excellent for showing multiplication of two fractions) and other graphical means of modeling fractions.  Some teachers will manage the shift to number lines adroitly—and others will not.  As a consequence, the quality of implementation will vary from classroom to classroom based on the instructional decisions that teachers make.  

The current post has focused on what I believe to be a positive aspect of CCSS based on the implementation of the standards through instruction.  Future posts in the series—covering the “bad” and the “ugly”—will describe aspects of instruction on which I am less optimistic.