With educators on the lookout for instructional materials that fit the content and vision of the common-core standards, a new set of 鈥減ublishers鈥� criteria鈥� 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 seeks to 鈥渟harpen the alignment question鈥� and make 鈥渕ore clearly visible鈥� whether materials faithfully reflect both the letter and spirit of the math standards adopted by 45 states and the District of Columbia.
Also, 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鈥攖o no more than 200 pages at the elementary level and 500 at the secondary鈥攚as 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 鈥減ublishers鈥� criteria鈥� for the common core in math offer guidelines for high school instructional materials that align with the standards.
Focus: 鈥淚n 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.鈥�
Coherence: 鈥淐oherence 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.鈥�
Rigor: 鈥淓ducators will need to pursue, with equal intensity, three aspects of rigor: (1) conceptual understanding, (2) procedural skill and fluency, and (3) applications. The word 鈥榬igor鈥� isn鈥檛 a code word for just one of these three; rather, it means equal intensity in all three.鈥�
Problems with purpose: 鈥淓ach problem or exercise has a purpose鈥攚hether 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.鈥�
Beware of the 鈥榤icrostandards': 鈥淎 drive to break the standards down into 鈥榤icrostandards鈥� 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鈥檛 build a tree out of twigs, but you can use twigs as kindling to burn down a tree.鈥�
Math practices: 鈥淥ver 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.鈥�
Matching the content: 鈥淎ny 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.鈥�
Variety of tasks: 鈥淭here is variety in what students are asked to produce,鈥� including answers and solutions, arguments, explanations, diagrams, and mathematical models.
Reasoning: 鈥淢aterials 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.鈥�
SOURCE: High School and Publishers鈥� Criteria for the Common Core State Standards for Mathematics
The criteria, he said, 鈥減rovide 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.鈥�
鈥楢 Key Resource鈥�
The publishers鈥� 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, that more than 30 of its member districts would use the criteria in math (and a ) 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.
鈥淭he states who do play a role in [reviewing] instructional materials are really looking at these as a key resource,鈥� said Carrie Heath Phillips, a program director at the Council of Chief State School Officers.
On the issue of focus鈥攁ddressing fewer math topics in greater depth鈥攖he high school document signals that materials should have a clear eye on readying students for postsecondary education.
鈥淚n 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,鈥� 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 鈥渆specially important鈥� the first domain, focused on 鈥渟eeing structure in expressions.鈥�
鈥淢aterials must give especially careful treatment to the domains, clusters, and standards in Table 1, including their interconnections and their applications,鈥� 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.
鈥淪tudents deserve to spend the majority of time on what will prepare them for the actual work of postsecondary education,鈥� said Mr. Zimba, a founding principal of , 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. 鈥淭hey almost seem to be redefining the standards,鈥� he said, "[suggesting] what we really meant was [this].鈥�
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, 鈥渂ut I might include some other ones, too.鈥�
Rigor and Coherence
When it comes to rigor, the publishers鈥� criteria document articulates a three-pronged definition centered on conceptual understanding of key math concepts, procedural skill and fluency, and application of mathematics.
鈥淐urricula have not always been balanced in their approach to these three aspects,鈥� it says.
On 鈥渃oherence,鈥� the document says high school coursework often emphasizes too many memorized techniques with 鈥渘o overall structure to tie them all together. Taking advantage of coherence can reduce clutter in the curriculum.鈥�
The criteria acknowledge it鈥檚 sometimes helpful to isolate part of a standard, but say an emphasis on reducing standards to smaller pieces, or 鈥渕icrostandards,鈥� threatens their focus and coherence.
That message resonates with Denise M. Walston, the director of mathematics for the Council of the Great City Schools. 鈥淚n so many instances, you鈥檝e seen math taught as a set of discrete little facts,鈥� she said, 鈥渟o you lose the unity.鈥�
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 鈥渢he intention is a good one鈥� to focus on fewer math concepts, simply counting pages would be misguided.
鈥淚 don鈥檛 think they wanted teachers or schools to be evaluating on that type of criteria,鈥� he said. Overall, Mr. Slattery said he was pleased with both sets of criteria and that his publishing house is taking them to heart.
鈥淲e鈥檙e paying attention to it because our customers are,鈥� he said.
Another change replaced language in the first version saying 鈥渁pproximately three-quarters鈥� of students鈥� time should be spent on the 鈥渕ajor work鈥� of the standards. The document now offers a range, saying materials should devote 鈥渁t 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.鈥�
