A draft of the California math curriculum framework went online Wednesday for public comments and suggestions. While weighing in at 1,200 pages, the document is actually a readable grade-by-grade manual that puts meat on the bare-bones Common Core standards that the state adopted in 2010. It explains the rationale for key standards and puts them in context of what students will learn, while providing guidance on how they should be taught. Interspersed are numerous sample problems and illustrations that teachers can use in the classroom.
5th grade fractions
The Mathematics Curriculum Framework and Evaluation Criteria Committee made an effort to create a readable, comprehensible, as well as comprehensive, guide to the Common Core standards.
A look at a small but important piece, how to introduce fractions to third graders (pages 25-27 of the third grade chapter) provides a good example of how the committee strove to put each standard in a larger context and explain its relevance and its coherence.
“In grade three students develop an understanding of fractions as numbers, building on the idea of partitioning a whole into equal parts. An important goal is for students to see unit fractions as the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of the whole numbers. …
Then it explains why this is important:
Student proficiency with fractions is essential for success in more advanced mathematics such as percentage, ratios, and proportions and in algebra at later grades. …
After explaining what was done in grades 1 and 2 to prepare for whole-unit fractions, it says,
This is the first time students represent fractions using a number line and other visual fraction models to develop an understanding of fractional parts, sizes, and how to compare fractions. This work will continue through grade six, preparing the way for work with the complete rational number system in grades six and seven.
It lists and explains the standards:
In grade three students understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. (3.NF.1▲).
Students understand a fraction as a number on the number line and they represent fractions and compare fractions using visual fraction models (such as a circle, rectangle or a line segment) (3.NF.2-3▲).
The framework then gives a sample problem for students to place fractions on a number line, leading up to 1/2, 1/3/, 1/4 and 1/5. A commentary suggests ways to help students in anticipation that some will struggle.
Students learn to use the number line to represent whole numbers in second grade, but this is the first time they will use the number line to represent fractions.
Some students may have difficultly creating equal sized intervals, especially for thirds and fifths. Providing strategies, such as folding a separate piece of paper, might be helpful.
Although the Common Core Standards cover only transitional kindergarten through grade eight, the framework does provide suggestions for high school, where districts will have a choice between traditional subject courses (geometry, Algebra II and pre-Calculus) and integrated math courses, which blend traditional disciplines. Common Core is well suited for the latter. (Update: To quell rumors that it would not support integrated math courses, the University of California has issued a statement affirming its support for both the traditional and the integrated pathways.) And the document includes a chapter on options for accelerating math for those students ready to take Algebra I in eighth grade or, as an alternative for those taking Algebra I in ninth grade, combining three years of high school math into two years in order to enable students to take Calculus.
Assembled in only six months by a 19-member committee of math teachers, district administrators and college professors under the guidance of Tom Adams, director of the state Curriculum Frameworks & Instructional Resources Division, the framework borrows liberally from recommendations and ideas of other states, including New York, (through its EngageNY project), Massachusetts, Arizona and other state recipients of Race to the Top grants that had more than a year’s jump on California. That’s a key advantage of shared standards – learning from others, said Sue Stickel, the deputy superintendent of the Sacramento County Office of Education and chair of the math framework committee. “We do have a dialogue between states where before, with California’s own standards, we had a conversation with ourselves. This provides huge resources for everyone.”
For districts that have been putting off getting serious about Common Core, the framework could provide a jumping off point for discussion – and not a moment too soon. Unless the Legislature chooses to push back the start date – one option it will consider this year – the new Common Core assessments in math and English language arts will be given in the spring of 2015, two years from now and less than 18 months after the formal adoption of the math framework by the State Board of Education this November. The English language arts framework won’t be adopted until May 2014.
Teachers, math experts and the public will have the next 60 days to comment on the document before it advances to the Instructional Quality Commission, chaired by former State Superintendent of Public Instruction Bill Honig, which will make the final changes to the framework, based on public suggestions, before forwarding the document to the State Board this summer. Stickel said that regional county offices of education plan forums on the framework, and there may be other public sessions. The state Department of Education has provided an online survey for comments as well.
Cautions on moving too fast to Algebra
The most anticipated chapter of the framework pertains to the contentious issue over whether Algebra I should be taught in 8th grade, as has been encouraged by the state for the past decade, or in 9th grade, the standard sequence under Common Core. The new State Board of Education position, adopted this year, is that Common Core 8th grade math will be the default curriculum; students should take Algebra in 8th grade only if they’ve demonstrated they’re ready for it. And there are other options for accelerating math in high school, such as taking double courses in one year or a course in summer school, leading to AP Calculus or AP Statistics. But the State Board left it to the Framework Committee to provide more specific guidance on how to offer “compact” accelerated courses and who should take them.
Common Core’s 8th grade math is more rigorous than the current non-Algebra 8th grade alternative to Algebra I. Common Core 8th grade math includes linear equations and about a quarter of the Algebra I course, along with elements of geometry. But half of it is pre-Algebra, leading advocates of 8th grade Algebra to warn about a backsliding in the percentage of students – about two-thirds – who currently enroll in Algebra in 7th or 8th grade.
“This was a hard chapter for the Committee to write, for members felt it (8th grade Common Core) was already challenging,” Stickel said. As the committee observed, “Common Core Standards for grades 6-8 are comprehensive, rigorous and non-redundant. … Therefore, careful consideration needs to be made before placing a student into higher mathematics coursework in middle grades. Acceleration may get students to advanced coursework, but this might create gaps in students’ mathematical background.” It refers to the “great challenge” for students and teachers for handling more standards that are more rigorous in a compressed time frame.
In Appendix A of the standards, the writers of Common Core laid out a potential course combining 7th and 8th grade standards in 7th grade, creating the opportunity for Algebra I in 8th grade. The committee referred to this and warned that accelerated courses should cover all of Common Core standards, not skip or skim over any of them. And students should show a fluency in math and a conceptual understanding, as measured by a portfolio of their work, before assigning them to an accelerated path.
The language and the caveats in the chapter struck Doug McRae, a retired testing publisher and an advocate of Algebra in grade 8, as “negative” in tone. “The draft seems to convey a message to the reader that algebra by grade 8 should be reserved for an isolated few advanced students. That message is not consistent with the experience California has from the past 15 years of our ‘algebra by grade 8’ initiative,” he wrote in an email.
Not the traditional cycle
The primary purpose of the framework has been to provide guidance for publishers to create textbooks and materials. But the traditional cycle – standards first, followed by the framework, then textbook adoption, then teacher training and professional development using approved materials culminating with assessments – doesn’t apply to Common Core. A compressed time frame, with tests in 2014-15, won’t allow the districts to wait that long, and, under budget flexibility, districts have latitude to buy materials of their choice anyway. Low-cost and free digital materials and curriculum plans developed in other states have freed up teachers and districts to check out a wealth of options. The state Department of Education has already instituted the process of textbook approval, even without formal approval of the math framework.
Stickel said that one advantage of posting the framework online is that it can be updated with links to other useful sources as they become available.
EdSource’s trusted, in-depth reporting has never mattered more.
With the coronavirus affecting every aspect of California’s education, demand for EdSource’s reporting has increased tremendously.We can meet this demand, with help from readers like you.From now through December 31, NewsMatch will match your one-time gift or your new monthly donation for 12 months. Your contribution ensures that EdSource’s content continues to be available for free – without a paywall or ads. Make your donation today to DOUBLE your impact.