With educators on the lookout for instructional materials that fit the content and vision of the common-core standards, a new set of “publishers’ criteria” aims to influence decisions by both the developers and purchasers of such offerings for high school mathematics.
Crafted by the lead writers of the math common core, the 20-page document issued last week seeks to “sharpen the alignment question” and make “more clearly visible” whether materials faithfully reflect both the letter and spirit of the math standards adopted by 45 states and the District of Columbia.
Also, revised K-8 criteria were issued, with changes to a version first put out last July based on feedback from the field. For one, an earlier call to limit the length of textbooks—to no more than 200 pages at the elementary level and 500 at the secondary—was removed.
Both sets of criteria attempt to drive home three core dimensions of the math standards: focus, coherence, and rigor.
The high school document drew a generally favorable review from W. Gary Martin, a professor of mathematics education at Auburn University who has helped develop tools for assessing the alignment of instructional materials to the new math standards.
New “publishers’ criteria” for the common core in math offer guidelines for high school instructional materials that align with the standards.
Focus: “In any single course, students using the materials as designed spend the majority of their time developing knowledge and skills that are widely applicable as prerequisites for postsecondary education.”
Coherence: “Coherence is about making math make sense. Mathematics is not a list of disconnected tricks or mnemonics. ...A special character of the mile-wide, inch-deep problem in high school is that there are often too many separately memorized techniques, with no overall structure to tie them all together.”
Rigor: “Educators will need to pursue, with equal intensity, three aspects of rigor: (1) conceptual understanding, (2) procedural skill and fluency, and (3) applications. The word ‘rigor’ isn’t a code word for just one of these three; rather, it means equal intensity in all three.”
Problems with purpose: “Each problem or exercise has a purpose—whether to teach new knowledge, bring misconceptions to the surface, build skill or fluency, engage the student in one or several mathematical practices, or simply present the student with a fun puzzle.”
Beware of the ‘microstandards': “A drive to break the standards down into ‘microstandards’ risks making the checklist mentality even worse than it is today. ... If the standards are like a tree, then microstandards are like twigs. You can’t build a tree out of twigs, but you can use twigs as kindling to burn down a tree.”
Math practices: “Over the course of any given year of instruction, each mathematical practice standard is meaningfully present in the form of activities or problems that stimulate students to develop the habits of mind described in the practice standards. These practices are well-grounded in the content standards.”
Matching the content: “Any discrepancies in high school content enhance the required learning and are clearly aimed at helping students meet the standards as written, rather than setting up competing requirements or effectively rewriting the standards.”
Variety of tasks: “There is variety in what students are asked to produce,” including answers and solutions, arguments, explanations, diagrams, and mathematical models.
Reasoning: “Materials provide sufficient opportunities for students to reason mathematically and express reasoning through classroom discussion, written work, and independent thinking. Reasoning is not confined to optional or avoidable sections of the materials.”
SOURCE: High School and Publishers’ Criteria for the Common Core State Standards for Mathematics
The criteria, he said, “provide a vivid picture of how effective instructional materials could support the [math standards]. If those adopting and developing materials really took these criteria to heart, the face of high school mathematics could significantly change in very positive ways.”
‘A Key Resource’
The publishers’ criteria are endorsed by several prominent organizations that provided feedback, including national groups representing governors, chief state school officers, state boards of education, and large urban districts, as well as Achieve, the Washington-based nonprofit that managed the process for developing the Common Core State Standards.
In fact, one of the endorsing organizations, the Council of the Great City Schools, signaled last year that more than 30 of its member districts would use the criteria in math (and a companion set for English/language arts) to guide their decisions in selecting materials. Also, California recently used the K-8 math criteria as part of its work to develop guidance for districts on selecting math materials.
“The states who do play a role in [reviewing] instructional materials are really looking at these as a key resource,” said Carrie Heath Phillips, a program director at the Council of Chief State School Officers.
On the issue of focus—addressing fewer math topics in greater depth—the high school document signals that materials should have a clear eye on readying students for postsecondary education.
“In any single course, students using the materials as designed spend the majority of their time developing knowledge and skills that are widely applicable as prerequisites for postsecondary education,” said the document, which will be revised next fall.
A table spells out domains and specific standards that deserve special attention in five areas: number and quantity, algebra, functions, geometry, and statistics and probability. For example, the table spotlights all three specific standards in a section on reasoning quantitatively and using units to solve problems. In algebra, it highlights every domain in the standards as containing widely applicable prerequisites, but identifies as “especially important” the first domain, focused on “seeing structure in expressions.”
“Materials must give especially careful treatment to the domains, clusters, and standards in Table 1, including their interconnections and their applications,” the criteria say.
Jason Zimba, a co-author of the criteria and one of the lead writers of the common-core math standards, said attention to postsecondary education is key and reflects the common-core focus on preparing students for college and careers.
“Students deserve to spend the majority of time on what will prepare them for the actual work of postsecondary education,” said Mr. Zimba, a founding principal of Student Achievement Partners, a New York City-based nonprofit working with states and districts on common-core implementation.
But Mr. Martin from Auburn said that was one dimension of the criteria that gave him pause. “They almost seem to be redefining the standards,” he said, "[suggesting] what we really meant was [this].”
J. Michael Shaughnessy, the immediate past president of the National Council of Teachers of Mathematics, said he found the items identified in the chart appropriate, “but I might include some other ones, too.”
Rigor and Coherence
When it comes to rigor, the publishers’ criteria document articulates a three-pronged definition centered on conceptual understanding of key math concepts, procedural skill and fluency, and application of mathematics.
“Curricula have not always been balanced in their approach to these three aspects,” it says.
On “coherence,” the document says high school coursework often emphasizes too many memorized techniques with “no overall structure to tie them all together. Taking advantage of coherence can reduce clutter in the curriculum.”
The criteria acknowledge it’s sometimes helpful to isolate part of a standard, but say an emphasis on reducing standards to smaller pieces, or “microstandards,” threatens their focus and coherence.
That message resonates with Denise M. Walston, the director of mathematics for the Council of the Great City Schools. “In so many instances, you’ve seen math taught as a set of discrete little facts,” she said, “so you lose the unity.”
The criteria, she said, are helpful to districts not simply in selecting new materials, but also in evaluating existing materials and figuring out where adjustments are needed.
Page Limit Scrapped
With the revised K-8 criteria, the decision to scrap limits on the length of textbooks came in response to feedback from curriculum experts that this kind of cap was not helpful, Mr. Zimba said.
The change was welcomed by Dennis J. Slattery, the editorial director for K-12 math at Pearson, who said that while “the intention is a good one” to focus on fewer math concepts, simply counting pages would be misguided.
“I don’t think they wanted teachers or schools to be evaluating on that type of criteria,” he said. Overall, Mr. Slattery said he was pleased with both sets of criteria and that his publishing house is taking them to heart.
“We’re paying attention to it because our customers are,” he said.
Another change replaced language in the first version saying “approximately three-quarters” of students’ time should be spent on the “major work” of the standards. The document now offers a range, saying materials should devote “at least 65 percent and up to approximately 85 percent of the class time to the major work of the grades, with grades K-2 nearer the upper end of that range, i.e., 85 percent.”
