With educators on the lookout for instructional materials that fit with the content and vision of the common-core standards, a new set of "publishers' criteria" aim 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 today 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.
In addition, a revised set of K-8 criteria were released today, with a variety of changes to the version first put out last summer based on feedback from the field (including districts that started to use them). One notable deletion was the explicit call for elementary math textbooks to not exceed 200 pages in length (and for middle and high school texts not to exceed 500 pages). Another change was to include more precise guidance on how much time should be devoted to the "major work" of the standards, differentiating in particular the K-2 level with that for the middle grades.
Both sets of 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.
"These criteria were developed from the perspective that publishers and purchasers are equally responsible for fixing the materials market," the high school document says. "Publishers cannot deliver focus to buyers who only ever complain about what has been left out, yet never complain about what has crept in. More generally, publishers cannot invest in quality if the market doesn't demand it of them nor reward them for producing it."
One of the endorsing organizations, the Council of the Great City Schools, signaled last year that more than 30 of its member districtsincluding Chicago, Los Angeles, and New Yorkwould 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 new high school criteria (as well as the revised K-8) drive home three core dimensions of the math standards: focus, coherence, and rigor. On the issue of focusthe notion of addressing fewer math topics in greater depththe document sends a clear signal 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," it says.
A table spells out domains and specific standards that deserve special attention in five broad 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 document says.
Jason Zimba, a co-author of the criteria and one of the lead writers of the common-core math standards, said this attention to postsecondary education is key.
"The Common Core State Standards initiative began with this idea of preparing students for college and careers," he said. "The criteria are making sure that message doesn't get lost in the trees of a [curricular] crosswalk" of the standards. "Students deserve to spend the majority of time on what will prepare them for the actual work of postsecondary education," said Zimba, a founding principal of Student Achievement Partners, a New York City-based nonprofit working with states and districts on common-core implementation.
The other two co-authors of the criteria (and lead writers of the math standards) are William McCallum, a math professor at the University of Arizona, and Philip Daro, an education consultant who is working on a common-core project for the Pearson Foundation. McCallum and Daro also are advisers to Student Achievement Partners.
'Rigor Isn't a Code Word'
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
• Applications of mathematics.
"'Rigor' isn't a code word for just one of these three," the document says. "Rather, it means equal intensity in all three." It adds that "to date, curricula have not always been balanced in their approach to these three aspects of rigor."
In reading through the document, I noticed it contains plenty of adjectivesmany, some, sufficient, thoughtful, and so onwhich raises the question of whether that opens the door to all kinds of interpretation.
Zimba suggested it's a careful balancing act to provide clear guidance without going too far.
"The criteria themselves are not a procurement document," he said. "When states and districts do adoptions or evaluations or purchasing, they use tools of more direct practicality. And what the criteria are trying to do is give direction and grounding to those tools."
He said the criteria are most explicit when it comes to the issue of focus. "Focus is so crucial to get right," Zimba said. "Those criteria are generally pragmatic and objective and meant to be easy to apply."
Combating 'Clutter in the Curriculum'
On "coherence," the document explains that this concept is about "making math make sense." It explains: "Mathematics is not a list of disconnected tricks or mnemonics. It is an elegant subject in which powerful knowledge results from reasoning with a small number of principles."
The document says 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. Taking advantage of coherence can reduce clutter in the curriculum."
Even as the criteria acknowledge that it's sometimes helpful or necessary to isolate a particular part of a standard, the document warns that an overemphasis on reducing the standards to smaller pieces threatens their focus and coherence.
"A drive to break the standards down into 'microstandards' risks making the checklist mentality even worse than it is today," the criteria say. "Microstandards would also make it easier for microtasks and microlessons to drive out extended tasks and deep learning."
There's far more in the high school criteria than I can address here, so I'll briefly highlight some other guidance:
• Materials do not invent connections not explicit in the standards without first attending thoroughly to the connections that are required explicitly in the standards.
• 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.
• 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 but is inevitable when using the materials as designed.
• Materials attend thoroughly to those places in the content standards that explicitly set expectations for multistep problems. Such problems are not scarce in the materials. Some or many of these problems require students to devise a strategy autonomously.
• Each math problem or exercise has a purpose, whether to teach new knowledge, bring misconceptions to the surface, or engage the student in math practices.
• Materials that devote roughly equal time to each content standard do not allow teachers and students to focus where necessary, as the standards "are not written at a uniform grain size."
• There is variety in what students are asked to produce, including answers and solutions, arguments, explanations, diagrams, and mathematical models.
• The visual design of materials isn't "distracting or chaotic, or aimed at adult purchasers, but instead serves only to support young students in engaging thoughtfully with the subject."
Since the criteria are just being issued today, I don't have comments from the field yet. But I'll be back later to share some of that reaction.
200-Page Limit Scrapped
With the revised K-8 criteria, Zimba said the decision to scrap the suggestion that elementary math textbooks should not exceed 200 pages came in response to consistent feedback from experts that this kind of arbitrary limit was not helpful. (In a blog post I wrote last summer, I reported mixed reaction to the idea, though some math education experts seemed to find it appealing.)
"That was the last bullet point in the document and attracted an awful lot of comment," Zimba said. He said that while experts in curriculum design he consulted seemed to "understand and agree" with the need for focus, a strict number limit was deemed problematic.
"That bullet point was doing more harm than good," he said. "But I'm glad we started that conversation."
Another change was to replace language in the first draft indicating that "approximately three-quarters" of students' time should be spent on the "major work" of the standards. (For instance, at the K-5 level, the major work generally consists of arithmetic and the aspects of measurement that support it, Zimba explained.)
He said many readers seemed to skip the "approximately" caveat and assume it must be precisely 75 percent. Some experts suggested that three-quarters of the time would be too much time in the middle grades, and others worried that it would be too little in the early primary grades.
The document now offers a percentage 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."
Other changes to the K-8 criteria, a summary page explains, include providing additional concrete examples throughout the document and providing additional information about meeting the needs of special populations (such as English-language learners and students with disabilities). In addition, there is a new "quality indicator" dealing with the issue of "lesson structure," as the earlier version's silence on this topic had been interpreted as a recommendation that lessons have no structure beyond posing problems.