Lessons Learned from Two Bad Common Core Math Standards

by John R. Walkup, Ph.D.

The Common Core State Standards receive a lot of heat, especially from parents. I won't step into that fray.  But I want to highlight two Common Core math standards that illustrate what happens when the design of content standards does not meet college/career-preparation.

Common Core State Standard, Math, Grade 5

The skill in summing fractions arises often in real life. Large chunks of our population cannot do it. As such, the importance of the following standards equals anything else in math education.

CCSS.MATH.CONTENT.5.NF.A.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.)

CCSS.MATH.CONTENT.5.NF.A.2

Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Hess Cognitive Rigor Matrix

On the surface, these standards appear reasonable. Some would even call them "rigorous."* They're wrong.

For the time being, let us concentrate on the first standard, 5.NF.A.1.

In terms of Cognitive Rigor, this standard states nothing more than "Add and subtract fractions with unlike denominators (including mixed numbers)."

Nothing about this standard compels teachers to create rigorous culminating activities. As a result, most student work will align to the lowest level of Depth of Knowledge (DOK-1), the straightforward application of learned steps.

Adding and subtracting also align to the apply-level of Bloom's Taxonomy. As such, this standard encourages teachers to assign activities aligned to the [3/1] cell of the Cognitive Rigor Matrix.

The [3/1] cell is over-sampled in everyday math instruction. Building on data collected in our statewide study of Oklahoma schools, I warn math educators in "Cognitive Rigor Defines 'Plug and Chug' Math" about assigning too much content targeting this cell.

Note a few other features of this standard:

• The added phrases beginning with “by replacing…” simply tell students how to perform the computation. 
• The parenthetical statement notifies the teacher of a formula that students can use to generate correct responses. 

None of these considerations raises the Depth of Knowledge level of the culminating activities. Worse, the standard does nothing to reinforce the concept of what happens when we sum unlike fractions.

In essence, this standard encourages "plug-n-chug" exercises. Such drudgery kills student enthusiasm and saps conceptual understanding, even if students can compute correct answers.

Academic vocabulary

Both Common Core standards undermine academic vocabulary by neglecting to mention an important math concept, the multiplicative identity. In more general terms, we can call this property the identity.

To compute 2/3 + 5/4, students would need to

• apply the identity (4/4) to the first term producing (4/4) X (2/3) = 8/12 and 
• apply the identity (3/3) to the second term producing (3/3) X (5/4) = 15/12.  

The fractions 2/3 and 8/12 are equivalent fractions, as are 5/4 and 15/12. The fact that the identity can generate equivalent fractions, that is, transform a number without changing its value, is an important concept.

The only mention of the identity appears in the glossary of the Common Core State Standards for mathematics. As stated:

Multiplicative identity property of 1: a × 1 = 1 × a = a

Will elementary school teachers make the connection that the identity is represented by such terms as 3/3 and 4/4? I doubt it.

Should we introduce this word in fourth or fifth grade? I think so.**

Conceptual understanding

No one disagrees that concepts are important in math education. While math lessons sometimes focus on conceptual understanding of students' answers, they tend to neglect conceptual understanding of the process used to generate these answers.

One of the best methods for reinforcing conceptual understanding is through visuals.

The Common Core presents visual representations of equivalent fractions as a fourth-grade standard. Unfortunately, the two fifth-grade standards above do nothing to encourage students to visually represent fraction summation.

There are many ways to visualize the identity acting on a fraction to produce an equivalent fraction. If we represent a fraction as the pieces of a pie, the identity cuts the pie into more slices.

We can picture the first fraction mentioned in the standard, 2/3, as a pie cut into three equal slices for which we choose two slices to eat. The identity slices the pie into 12 equal slices; we now choose eight slices to eat.

So, the identity changes the way we represent the fraction, but not its value. If we were told we could eat 2/3 of a pizza, we would end up just as full if we ate 8/12 of the pizza.

Therefore, 2/3 and 8/12 are equivalent fractions.

The process of summing equivalent fractions can then be visualized quite readily, as shown in the following figures using the addition 2/3 + 3/4 as an example.

For the last step, we just count the total number of slices (17).

The above visual representation represents only one of many ways to reinforce conceptual understanding. Regardless of the technique, the stated standards do nothing to ensure that students understand the process of fraction summation.

Formulas in standards

This standard breaks one of my golden rules for standards: Do not include formulas.

The formula stated in this standard is especially egregious. Yes, the standard urges teachers to teach students to transform "unlike" fractions into equivalent "like" fractions. The formula suggests skipping all that bother and generating the answer in one easy step.

By focusing on the answer and not the process, students can demonstrate 100% proficiency on Grade 5 assessments; a year later, they may fail every question. ("Was b on the bottom? Or was it c?")

Formulas also promote procedural knowledge at the expense of conceptual knowledge. When students use the formula to compute correct answers, do they know why it works? In short, formulas signal students that their teacher rewards correct answers over understanding, a rotten thing to do in math education.

So why did standards writers include the formula? I don't care. Get rid of it.

Least common multiple

Why haven't I mentioned using the least common multiple to simplify fraction summation. Clearly, including the formula in the standard negates use of this concept to sum fractions.

For example, to compute 3/8 + 1/6, we would prefer to find equivalent fractions with 24, not 48, as the denominator.

In Common Core states, students cannot apply their knowledge of the least common multiple in Grade 5 to sum unlike fractions. Why? Because least common multiples are not taught until Grade 6.

Once the Common Core introduces sixth-grade students to the least common multiple, the standards never ask students to revisit fraction summation with this concept in mind. I find this strange, because one of the most important uses of the least common multiple is to simplify the summation of unlike fractions.

(By the way, in earlier versions of this post I mistakenly used the term greatest common factor in place of least common multiple. If I make this mistake, students will too. Teachers should pay close attention to confusion between these two concepts.)

What about the second Common Core standard?

Some readers might object to my criticisms, noting that the second standard focuses on solving real-world problems where students use fractional models in their solution.

The standard says no such thing.

Standard 5.NF.A.2 does ask students to solve word problems, but not real-world problems. In other words, this standard asks students to do nothing more than read a (contrived) word problem involving fractions, translate the prose into a mathematical equation, then solve it.

The use of visual models is suggested as a mere option. If teachers use the formula in Standard 5.NF.A.1 to teach fraction summation, they can ignore this choice. Instead, they can ask students to use "equations to represent the problem" by assigning countless exercises like the following:

Jimmy ate 5/16 of his pizza. Sue ate 3/4 of her pizza. How much pizza did they eat?

The processing of information is a signature of DOK-2 questions, but the mental processing involved in solving this word problem is scant. In the end, Standard 5.NF.A.2 targets low-level DOK-1 activities just as much as Standard 5.NF.A.1. Students will read the problem, write down 5/16 + 3/4, and use the formula to generate the answer.

This is rigorous?

On the plus side, the tail end of the standard asks students to estimate the value of fractions, a good skill to learn. I would have switched the order by teaching students to estimate the summation before learning to calculate it exactly. But I'm good with it.

Update

Education Week apparently has gone on the pro-CCSS bandwagon, even claiming that the Common Core has shifted math instruction by emphasizing the use of number lines to teach fractions.

Not quite.

The teaching of fractions by placing their value on a number line is hardly a Common Core invention, nor is it a new-found emphasis. In the scores of lessons on fractions that I have observed prior to the Common Core era, number line placement was one of the more prevalent instructional strategies.

Oklahoma's older PASS standards list the use of a number line to teach fractions in grades 3 and 4. California's older math standards asked teachers to use number lines to teach fractions in grades 4, 5 and 6. The Common Core, on the other hand, only compels such an approach in grade 3. (However, the CCSS does express more detail about how one should teach the standard.)

I consider the abstraction of the number line more developmentally appropriate for grade 4. Although we can liken the number line to a physical object, with fractions marking off certain lengths from the end. the number line is in reality a mathematical abstraction. This is especially true if one considers negative values and the fact that the number line extends to infinity in both directions.

The fact that California decided to wait until grade 4 to teach placement of fractions on the number line is telling since California's older math standards usually had students grappling with topics at a younger age than the CCSS.

* Loosely speaking, I am using the word "rigor" as a synonym for "cognitive challenge." I have no interest in debating whether the word "rigor" is used properly in education. 

** Introducing a new word like identity into instruction requires full-blown vocabulary development, especially since the word has other meanings. Students would not only need to learn its definition, but also how to read it, say it, spell it, use it in sentences, and so on.

---

Seeking training at your school or district centered on Cognitive Rigor or Depth of Knowledge?  Call me at (559) 903-4014 or email me at jwalkup@standardsco.com. 

We will discuss ways in which I can help your teachers boost student engagement and deep thinking in their classrooms. I offer workshops, follow-up classroom observation/coaching, and curriculum analysis to anywhere in the country (and even internationally).

Follow me on Twitter at @jwalkup.