Two side-by-side articles in this past week’s Education Week show a disconnect between what the Common Core authors vs. actual classroom teachers think are the biggest challenges posed by the Common Core.
First, there’s an interview with William G. McCallum , the lead author of the Common Core Math Standards. McCallum cites coverage of fewer topics as the biggest change brought by the Common Core, and fractions, ratios, and proportional relationships as the biggest challenges to teachers.

For teachers, on the other hand, what seems to be most novel and challenging is the Math Standards’ emphasis on conceptual understanding and verbal communication. This is particularly true in the case of teachers of language-impaired students and students whose native language isn’t English. The latter are the focus of the other article.

When he began working the Common Core State Standards into his instruction three years ago, New York City middle school mathematics teacher Silvestre Arcos noticed that his English-language-learner students were showing less progress on unit assessments than his other students.

"It wasn't necessarily because they didn't have the numeracy skills," recalled Mr. Arcos, who is now a math instructional coach and the 7th grade lead teacher at KIPP Washington Heights Middle School, a charter school in New York. Rather, they were struggling with the linguistic demands of his new curriculum, which was oriented heavily toward word problems and explication of solutions.

To address the issue, Mr. Arcos began incorporating strategies that are typically the province of language arts teachers into his math lessons. Especially when working with his English-learners, he provided detailed instruction in close reading, sentence annotation, and writing fluency.Nor is Mr. Arcos alone:

Mr. Arcos' recognition that the new math standards may require greater attention to the needs of English-language learners is not uncommon among educators who work with such students. Particularly in the Standards for Mathematical Practice that preface and inform the grade-level objectives, the common core emphasizes the importance of explaining solutions and relationships, constructing arguments, and critiquing the reasoning of others. While such expectations are proving difficult for many students, educators say, they pose unique challenges for those not fully proficient in English.When I was a 6th grader in a school outside Paris, immersed among native French speakers, math class offered refuge from the linguistic challenges of my other classes. From the beginning, with minimal knowledge of French, I was able to follow what was going on on the chalkboard. And could figure out what to do on homework and tests. Nor do I feel like my math experience would have been any richer had the word problems involved more elaborate French sentence structures and vocabulary, or had I been required to explain my reasoning in French. In fact, the best elementary school math class I had was that 6th grade math class, with its solidly conceptual and engaging French math curriculum. When I returned to the U.S. for 7th grade, I was, in fact, ahead in math relative to my peers. Had my French math class gotten bogged down with “detailed instruction in close reading, sentence annotation, and writing fluency,” I would surely have instead ended up significantly behind.

So is “detailed instruction in close reading, sentence annotation, and writing fluency” in math class really what’s best for our students—whether or not they are non-native English speakers?

Not surprisingly, professors of mathematics

*education*, as opposed to professors of

*mathematics*, applaud this continued dilution of math with English:

In addition, the common core's emphasis on verbal expression and reasoning in math are widely seen as beneficial to English-learners. "The more language you use in the math classes, the more [ELL] students are going to learn, both in math and language," said Judit N. Moschkovich, a professor of mathematics education at the University of California, Santa Cruz.Especially because it becomes one more excuse to have students work in groups:

At the same time, a Teacher Notes panel provides specific activities teachers can use to help English-learners engage with the language of the lesson. One such exercise says: "Have students work with partners to discuss the graphic organizer and fill in the sentence frames [provided]. Then have them use the word bank [provided] to fill in the summary frame."As for Common Core Math Standards lead author William McCallum, he seems blind to the problems posed by the Standards’ perceived emphasis on verbal expression. When asked if there’s anything he might change about the Common Core, all he mentions are the geometry progression in the elementary school Standards and the level of focus in the high school Standards:

“I think the geometry progression could be evened out a bit in elementary school. I think in high school there could be more focus. High school was difficult because everybody has their pet topic, and it was difficult to resist those pressures.”As for challenges of particular Common Core-inspired problems or of conceptual understanding, McCallum blames these on mis-implementations or misinterpretations:

“What's interesting to me is that both the supporters and the critics of the common core, I think, are overemphasizing conceptual understanding—and understandably because everybody's always demanded procedural fluency, and the conceptual understanding is what's new. But that doesn't make the other requirement go away.”Well, what’s interesting to

*me*is that the lead writer of the Standards (a) thinks that conceptual understanding is something new in math education (b) has written something that he acknowledges is being mis-implemented and misinterpreted, and (c) has failed to do anything to stop this.

## 5 comments:

What I don't understand is why, instead of having students verbally show that they understand, can't they design problems that can't be solved unless the student has conceptual understanding? The Singapore math books are full of such problems--and interestingly, back when they were first developed, a huge number of the students using them were English language learners.

Anonymous: Depends what you mean by conceptual understanding. A problem that asks for how many 2/3 oz servings are in 1 3/4 oz of yogurt requires that a student recognize that the problem is solved by division as well as a procedural knowledge of how to do fractional division. If the student cannot explain why the "invert and multiply" rule works but shows he understands what the problem is asking and how to solve it, does that mean he/she lacks conceptual understanding?

I agree with Katharine that McCallum and others should be making more public statements about what CC math standards require and what they do not. After I wrote my first article offering alternative interpretations of CC math standards (in Heartlander), McCallum showed some interest and even blogged about it.

He even acknowledged that the standard algorithm for multidigit addition and subtraction can be taught in grades earlier than 4th--the grade in which that algorithm is mentioned in the CC standards: "By the way, the standard algorithms for addition and subtraction are “strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.” The standards do not forbid them being introduced in Grade 2—nor do they require that."

From a recent Atlantic article: "Tougher assessments tend to avoid multiple-choice questions, relying on open-ended prompts that challenge a student to gather evidence from a supplied text or use narratives to express a mathematical concept."

Your point being what?

Conceptual understanding is just that - conceptual. Too many educators mistake it for proper mathematical understanding. Conceptual is at the level of motivating why you are learning the material in the first place. It's at the pie chart level, and that simple kind of understanding of fractions provides no support for understanding rational fractions. Many teachers say that the toughest math class in high school is Algebra II because that's when those conceptual understandings fall apart. Understanding should be built on identities, not pies or bars.

Some students need more motivation than others, but it is neither necessary or sufficient for real understanding. That is built from doing and understanding nightly individual problem sets.

As for testing, how can standardized tests ever test conceptual understanding or problem solving in the sense of applying mathematical ability to new problems - ones not covered in all of the problem variations encountered in homework?

What are teachers, potted plants? Are they incapable of making those judgments even with seeing the kids every day? What is the purpose of yearly standardized tests? Are they used because we can't trust teachers or are they used just as a safety net? If they are used as a safety net, then it would be much simpler to just test the basics - results that can give specific things to correct. As it is now with NCLB, our schools get vague scores on "problem solving" and no other details on why the results aren't as good this year. Shouldn't they already know what the problems are?

Why on earth would anyone expect a problem solving yearly standardized test be part of a critical thinking feedback teaching loop?

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