In early September, math teachers from across Kent County, Md., gathered for their first professional learning community meetings of the school year. Kris Hemstetter, a math and English/language arts coach for the 2,200-student district, was working with a small group of 3rd through 5th grade teachers and urged them to keep in mind “where our kids are, and where they’re going.”
Ms. Hemstetter was referring to a shift in the Common Core State Standards for mathematics known as “coherence,” the idea that math concepts should be linked both within and across grade levels. Unlike many previous state standards, the common core puts a heavy emphasis on coherent instruction, and consequently, many districts are finding ways to familiarize teachers with standards other than the ones they teach.
For the second year in a row, the elementary teachers in Ms. Hemstetter’s district are meeting monthly in cross-grade sessions after school, rather than in the grade-level silos previously used, to go over math concepts and instructional methods. For Ms. Hemstetter, the goal is to get teachers talking about how skills build on each other.
Many experts say organizing these kinds of cross-grade-level professional learning experiences is one of the best ways to encourage coherent instruction. While teachers can study, which outline student-learning sequences for particular math topics, those texts can be tough to penetrate without support. Intergrade professional learning, though logistically difficult, gives educators a chance to teach each other about the complicated learning progressions inherent in the standards, they say.
Math naturally lends itself to fluid instruction, according to Phil Daro, a senior fellow at America’s Choice, a Washington-based school improvement consulting group owned by Pearson Inc., and a lead writer of the common standards for math. But state academic standards have tended not to make fluidity a focus until now.
“Traditionally in the U.S., we taught one arithmetic for whole numbers and taught a second and different arithmetic for fractions, and then we taught a third arithmetic for expressions with letters in them,” he said. “Whereas mathematically, you could do it all with one arithmetic.” Other countries, including Singapore and Japan, have done a better job of teaching students to apply the same principles to many different types of problems, Mr. Daro said.
Coherence is named in the common standards, along with “rigor” and “focus,” as one of the three major mathematical “shifts” from previous state standards. “Mathematics is not a list of disconnected topics, tricks, or mnemonics; it is a coherent body of knowledge made up of interconnected concepts,” the standards document says. “Therefore, the standards are designed around coherent progressions from grade to grade.”
The standards ask students to “apply and extend previous understandings” when learning new skills—an indirect reference to cross-grade coherence. For instance, students are expected to use what they learned about multiplication in 3rd grade to multiply fractions by whole numbers in 5th grade, and then again when dividing fractions by fractions in 6th grade.
“As a teacher, what it means is you really should be familiar with the whole program, not just the part of the program where you add on new knowledge,” said Mr. Daro.
While many math teachers have long sought to ensure their instruction is consistent with what’s taught in other grades, the common core has made that objective more explicit, while seeking to make the connections more readily apparent.
The common standards for math were developed out of a series of documents, based on research about students’ cognitive development, that detail how instruction on a particular topic should advance over time. The standards’ writers took those “progressions documents” and divided the skills into grade levels at which they should be taught. The progressions documents can now be, a website hosted by the New York City-based nonprofit group Student Achievement Partners, which was founded by the lead standards writers.
The documents are, in a word, dense. But they do offer insight into how skills build on one another, and most experts on the standards suggest teachers consult them.
“What I’ve been recommending is that people form book clubs or study groups and read those progressions together,” said Mr. Daro. “They’re not written to be read while sipping a cup of tea. They’re written to be studied, and you don’t study alone.”
Three to five elementary teachers could meet once a week for a month to read a single progression, he said. “Will that be enough for them to understand everything in the progression? Probably not. Will that advance their mathematical thinking? Surely it will.”
The progressions documents take a lot of “unpacking,” said Ms. Hemstetter. Even so, she tells teachers to put them by their desks, along with the standards. “It’s not an easy read sometimes, but every time you read a portion of that progression document, it’s going to make more sense to you,” she said.
At a professional development conference held by the Maryland education department in August, a small group of teachers gathered at a session on fractions division. As the teachers discussed whether students should convert improper fractions to mixed numbers, which the common standards do not require, high school teacher Kevin Wajek chimed in from the back of the room. “As soon as you cross into high school, that’s the answer I want,” he said in reference to improper fractions. “Unless it’s a recipe, do not use mixed numbers again.” The middle school teachers turned his way to consider the plea.
The moment exemplified the way cross-grade learning can contribute to coherent instruction in math. Mr. Wajek, the math department chairman at Severna Park High School, in Maryland, said in an interview that coherence goes both ways—high school teachers need to know the beginnings of the learning progressions as well. “I should understand where that conceptual understanding came from, so that I can say, ‘Remember when ...’ ” and point to what students learned in previous grades, he said.
Windy Hill Middle School, in the 15,900-student Calvert County, Md., district, began organizing professional learning communities across grade levels three years ago, when the school first introduced the common core.
“We may discuss a topic we’ll talk about in 7th grade, and tie it into a topic in 6th grade, and where they’re going with it in 8th grade,” said Dawn Caine, a math teacher at the school. The middle school teachers have also met with 5th grade teachers from the nearby elementary schools at times, she said.
As of yet, though, that kind of cross-grade integration isn’t common practice in middle schools, according to Steven Leinwand, a principal research analyst at the Washington-based American Institutes for Research who specializes in math education. “Most middle school teachers are meeting as [single-] grade teams—one math, one English, one social studies, and one science teacher,” he said, calling that a major problem.
Cross-grade-level meetings have been helpful for Chris Austin, a 4th and 5th grade math teacher at Rock Hall Elementary in Rock Hall, Md., who attends Ms. Hemstetter’s professional learning community. Working with teachers below her grade level has given her a firmer grasp on the progressions. “I’m not as familiar with the 3rd grade document, so she can remind me to take it down a notch,” she said, pointing to a colleague who teaches 3rd grade. Knowing the progressions below her grade also allows her to fill in gaps for her students, Ms. Austin explained.
“We didn’t have progression documents before the common core to see why does this make sense,” said Ms. Hemstetter. Now, in the professional learning communities, “we can unpack the progression document, look at student work, and analyze 2nd through 5th grade to see why these progressions make sense,” she said. “It’s about the conversations.”
Coverage of efforts to implement college- and career-ready standards for all students is supported in part by a grant from the Bill & Melinda Gates Foundation, at
A version of this article appeared in the November 12, 2014 edition of Education Week as Teachers Band Together to Build Math ‘Coherence’