The mathematics curriculum taught in the Piedmont Unified School District is changing due to California’s state-wide adoption of the Common Core Mathematics Standards. This special edition of the PUSD Spotlight is devoted to the changes now underway in the District’s math program, and issues to be decided in the coming months concerning the next phase of Common Core Math implementation.
The District is holding a series of parent education programs, discussions, and Board of Education meetings to help parents understand the changes taking place in the classrooms now and over the next few years. Please visit the following link for the complete schedule of meetings, all of which are open to the public:
All are encouraged to attend to learn more about how curriculum and classroom instruction will shift over the next few years, and to provide input concerning implementation of Common Core Math.
For those who cannot attend any of these meetings, questions and comments about the implementation of Common Core Math may be addressed to your student’s site principal; Cheryl Wozniak, Director of Curriculum and Instruction at firstname.lastname@example.org; or Randall Booker, Assistant Superintendent of Educational Services at email@example.com.
We also want to encourage families to attend our upcoming Education Speaker Series: Why We Need to Change the Way We Teach Math
On Tuesday, January 27 from 7:30 – 9 pm in the Piedmont High School Student Center, 800 Magnolia Avenue, education reform advocate Dan Meyer, one of Tech & Learning‘s “30 Leaders of the Future” and an Apple Distinguished Educator featured on CNN and Good Morning America, will discuss Common Core Math and the importance and method of teaching students to formulate and solve problems rather than memorize and apply formulas. Tickets $10 at the door. Series members are free.
An Overview of Common Core Math and Pending Decisions
California’s State Board of Education (“SBE”) adopt standards for learning math in California public schools. On August 2nd, 2010, the SBE adopted the Common Core math standards. The SBE concluded that the new Common Core standards are a significant improvement over the previous math standards, which had been adopted by the SBE in 1997 (“the 1997 standards”). The SBE found that the 1997 standards were “a mile wide and an inch deep” – covering too many topics in too little depth.
Under Common Core, students at all levels will focus on fewer math concepts and principles, and will be expected to develop greater mastery and conceptual understanding of each. There will be greater emphasis on understanding the connections among different math concepts, within and between grade levels. The SBE determined that the Common Core standards are more rigorous than the 1997 standards, and provide a stronger foundation for more advanced math and for application of math concepts in the sciences.
Over the last two years, the Piedmont Unified School District started phasing in the Common Core standards, primarily in the elementary schools and middle school. Students who were enrolled in Intro to Algebra through Calculus AB during the 2013/14 school year (in general, eighth graders and up) are continuing under the 1997 standards and are not affected by the implementation of Common Core Math.
At the elementary level, Common Core Math focuses on concepts, skills and problem solving relating to addition and subtraction (kindergarten through 2nd grade) and multiplication and division of whole numbers and fractions (3rd through 5th grade). Elementary math is now more rigorous because of the emphasis on both conceptual understanding and application skills. This “rigor” is not about making math “harder” or introducing concepts at an earlier age. Instead, it is about requiring students to apply math concepts to solve problems in a variety of ways, and to explain their reasoning, to promote a deeper understanding of mathematics. Similarly, Piedmont Middle School offers three new math courses — CC6, CC7 and CC8.
In the next few months, PUSD will make key decisions concerning the next phase of Common Core implementation. These decisions, which concern middle and high school courses, are summarized below.
- Sequencing of Algebra and Geometry: Traditional or Integrated Courses
Although the State mandates the k-12 math standards, each school district has discretion to decide how to sequence algebra and geometry standards. Under the “traditional” approach, algebra and geometry concepts are taught as separate, successive courses over a period of three years — Algebra I, Geometry, Algebra II (typically in 8th, 9th and 10th grade). Under the “integrated” approach, these concepts are blended over the same three-year period into courses typically titled Integrated Math I, II, and III. The majority of other countries follow the integrated approach for algebra and geometry, so this is often referred to as the “international” approach.
Please note this decision concerns sequencing only — the content of the courses over the three year period is the same.
Also, please note that this decision concerns algebra and geometry only — elementary math, middle school math (CC6, CC7, CC8), and the class that follows algebra and geometry (Math Analysis, typically in 11th grade) are integrated courses.
- Opportunities to Accelerate: Compression Pathways
Each school district has discretion to decide how to support learners who require either additional challenge or additional support. In addition to a grade-level progression or “pathway” through the math standards, each district may offer alternatives: pathways that “compress” material so students may accelerate through the math standards; and pathways that “expand” courses to allow more time to master content.
The guiding consideration during the development of alternate pathways is how the District can best support all learners, whether they are able to accelerate through the curriculum, follow a grade-level progression, or require more time to master the math concepts. Other considerations include: the interrelationship between math and science courses and the need to ensure readiness for advanced science curriculum; the need for opportunities to compress or expand at multiple points and at points that correspond with cognitive development; consideration of student stress, particularly during junior year; and practical considerations concerning instruction and how the content standards can be coherently compressed or expanded.
The PUSD Board also already decided that, during the transition to Common Core Math and after implementation is completed, the District will:
- continue to offer pathways to Calculus AB and BC for qualified students; and
- offer at least one opportunity to “compress” math at both the middle school and high school level (students may compress at either or both points).
- Common Core Course Materials and Textbooks
Common Core-aligned text books are new and must be carefully evaluated. District math teachers are “piloting” curriculum from a variety of sources, developing some of their own, and assessing what works and what needs refinement. This pilot includes both traditional and integrated curriculum, so teachers are prepared for whatever the Board decides.
Background: Why did the District change to Common Core math?
Until recently, the mathematics curriculum currently taught in the Piedmont Unified School District was based on California State Standards adopted by the California State Board of Education (SBE) in 1997. The SBE has since reevaluated the 1997 standards and concluded that they cover too many topics in too little depth. The SBE determined that the 1997 standards do not adequately prepare students for the mastery of algebra, which is considered essential to a student’s success and continued interest and engagement in mathematics.
The SBE has now adopted the Common Core State Standards (CCSS) for mathematics curriculum. The CCSS specify math content and math practice standards to build a foundation for the mastery of algebra. This approach is based on what is considered both cognitively and developmentally appropriate at each grade level, to promote better comprehension and retention of math concepts. Common Core math is also based on the content and practice standards used to teach math in other developed countries where students typically outperform their United States’ counterparts. The SBE determined that the CCSS are more rigorous than the 1997 standards, and provide a stronger foundation for more advanced math and for application of math concepts in the sciences.
How does Common Core Math differ from what we have now?
Under Common Core Math, students at all levels will focus on fewer math concepts and principles, and will be expected to develop greater mastery and conceptual understanding of each. There will be greater emphasis on understanding the connections among different math concepts, within and between grade levels.
The CCSS establish content standards (such as addition and subtraction for grades k-2). In addition, the CCSS establish standards for the practice of math. The practice standards are designed to promote greater problem-solving and reasoning skills, and require students to explain and defend their thinking. These practice standards will be embedded at every level of math education.
How is math changing at the elementary level?
Common Core Math requires focus on fewer math concepts and the ability to apply those concepts in a variety of ways. At the elementary level, Common Core Math focuses on concepts, skills and problem solving relating to addition and subtraction (kindergarten through 2nd grade) and multiplication and division of whole numbers and fractions (3rd through 5th grade).
Similar to the changes in higher-level math, elementary school math will become more rigorous than it is now because of the emphasis on both conceptual understanding and application skills. This “rigor” is not about making math “harder” or introducing concepts at an earlier age. Instead, it is about requiring students to apply math concepts to solve problems in a variety of ways, and to explain their reasoning, to promote a deeper understanding of mathematics.
All elementary teachers are working together, across the three elementary sites and across grade levels, to modify curriculum for each grade level and link concepts across grade levels.
Do all public school districts in California have to implement Common Core Math?
Yes. The District is working closely with the Alameda County Office of Education and other school districts to identify “best practices” concerning professional development, lesson plans, selection of course materials, sequencing of courses at the middle and high school level, pathways for accelerated learning, pathways for support, and other aspects of implementation.
Does this District have any discretion over how to implement Common Core Math?
Yes. For example:
- Sequencing of Concepts: Traditional or Integrated Courses
Each district has discretion to follow either a “traditional” or “integrated” model for teaching algebra and geometry. Under the traditional approach, algebra and geometry concepts are taught as separate, successive courses over a period of three years (Algebra I, Geometry, Algebra II). Under the integrated (or “international”) approach, these concepts are blended over the same three-year period (Integrated Math I, II, and III).
- Pathways for Acceleration
Each school district has discretion to determine how to support learners who require either additional challenge or additional support. In addition to a grade-level progression or “pathway” through the math standards, each district may offer: pathways that “compress” material so to accelerate through the math standards; and pathways that “expand” courses to allow more time to master content.
- Curriculum and Course Materials
Although the State has mandated standards for the math concepts that students must learn in a particular course (for example, the ability to use a mix of ratios, fractions, proportions and percentages to solve a problem), the Common Core State Standards are not curriculum. Each school district may innovate and develop its own curriculum to help students learn the concepts and principles needed to meet the Common Core standards. Similarly, each district may choose its own course materials and texts.
Regardless of whether the District chooses a traditional or integrated sequence for algebra and geometry, it will take several years to pilot and adopt Common Core-aligned textbooks.
Last year, the PUSD Board of Education approved a plan for the multi-year implementation of Common Core (CC) Math, starting with new middle school math courses in the 2014/15 school year. Piedmont Middle School is now offering three new courses: CC6, CC7, and CC8. Students who were enrolled in Intro to Algebra through Calculus AB during the 2013/14 school year are continuing in the 1997 California Math Standards pathways and are not affected by the implementation of Common Core Math. That stated, their teachers are integrating CC instructional practices as part of their teaching, as these students will be taking the California Assessment of Student Performance and Progress (CAASPP) — the test that replaces the STAR tests and that is aligned with the Common Core standards.
The PUSD Board also decided that, during the transition to Common Core Math and after implementation is completed, the District will:
- continue to offer pathways to Calculus AB and BC for qualified students; and
- offer at least one opportunity to “compress” math at both the middle school and high school level (students may compress at either or both points).
Will students still have the opportunity to take AP Calculus?
Yes. The District is committed to maintain a pathway for qualified students to take AP Calculus. As some students need additional support, other students need to be challenged to maintain their interest and skill development in mathematics. Some students will choose to take college-level mathematics courses in high school, and the District is committed to provide a pathway to support this choice.
In the next few months, the Board will consider the next phase of Common Core math implementation. First, the Board will decide whether to adopt traditional or integrated courses to follow CC8. After the Board decides the course sequences, the Board will determine alternative pathways — how and when students may speed up or slow down their progression through the new curriculum standards.
A Brief Note About Course Names
Although the Common Core content and practice standards for middle school and high school mathematics are different from the 1997 standards, the course names may be the same or similar to current course names. This will almost certainly cause some confusion.
For example, the 1997 “Algebra I” course, which was taught primarily to 8th graders, bears little resemblance to the new “CC Algebra I,” which is designed for 9th graders. CC8, the new course designed primarily for 8th graders, will have some of the same content but be more rigorous than the 1997 “Algebra I” course.
How are the Traditional and Integrated Approaches Similar?
Traditional and integrated courses cover the same content and practice standards. Across the three courses, students in the traditional sequence will study the same content and mathematical practices as students in the integrated sequence. In other words, the two approaches have the same entry and exit point.
How are the Traditional and Integrated Approaches Different?
The difference between the traditional and integrated course sequences is how the state-mandated math standards are organized into courses. For example, in the traditional sequence, geometry is its own course. In the integrated sequence, geometry standards are included in all three courses.
Why Consider a Change from the Traditional to the Integrated Approach?
The state-wide adoption of Common Core Math requires change for all K-12 math courses. As the District plans for these changes, it has discretion to decide how to sequence algebra and geometry content. The District is taking this opportunity to carefully consider the pros and cons of both the traditional and international approaches to sequencing this content.
A primary advantage of the traditional approach is that it is familiar to teachers and parents. The traditional approach has served many PUSD students well; however, there are questions about how well it serves all learners.
A primary advantage of the integrated approach is that it allows for introducing the increasingly complex concepts in algebra and geometry over a period of several years, as students are more developmentally ready. This approach is intended to promote greater conceptual understanding and coherence of the connections among algebraic and geometric principles. A secondary advantage is that the international approach lends itself to compression (acceleration for advanced learners) more so than traditional courses.
Although PUSD currently uses the traditional approach to sequencing algebra and geometry standards, other math courses are integrated. Specifically, the math that both precedes and follows Algebra I, Geometry and Algebra II — elementary math and Math Analysis (or “Trigonometry”), respectively — are integrated. Also, the new CC8 is an integrated course, with elements of Algebra and Functions, Geometry, and Statistics.
It is important to note that, regardless of the choice between traditional and integrated course sequences, the changes in math education are significant. Given the state-wide adoption of the Common Core math content and practice standards, all math courses are changing.
Middle and High School Math: Alternatives to Grade-Level Pathways
In addition to choosing between traditional and integrated course sequences, the District has discretion to offer alternatives to the grade-level pathway for mathematics.
Prior to the implementation of Common Core Math, students had the opportunity to accelerate through the math curriculum by skipping 6th Grade Math. At the end of 5th grade, students could take a test to determine eligibility to skip 6th Grade Math in its entirety and advance to Pre-Algebra. Skipping 6th Grade Math in its entirety was considered an appropriate pathway to accelerate through the math curriculum, because there was some repetition in the content taught in 5th and 6th grade.
Also prior to the implementation of Common Core Math, students who needed additional support to master algebra had the opportunity to “expand” algebra into a two-year course. These students remained on a path to complete Math Analysis (also known as “Pre Calculus”) and be prepared for college-level math.
Under Common Core Math, the District may adopt alternate pathways to: “expand” courses for students who need additional time and support to master concepts; and “compress” courses to accelerate through the content.
The guiding consideration during the development of the pathways is how the District can best support all learners. Other considerations include: the need for opportunities to compress or expand at multiple points and at points that correspond with cognitive development; practical considerations concerning how to compress two courses into one year, for example, or three courses into two years, etc.; the need to ensure student readiness for advanced science curriculum; the need to ensure opportunities for AP Calculus in high school; and consideration of student stress, particularly during junior year.
How will the District Decide on Course Sequences and Pathways?
To inform the pending decisions, the Board created a Math Task Force (MTF) consisting of math teachers, parents, students, and administrators. The MTF is charged with considering and recommending options for math course sequences and math pathways in middle and high school. To date, the MTF met eight times since the start of the school year. The MTF will continue to meet monthly through February 2015, until it submits recommendations to the Board. Please refer to the November 2014 MTF Progress Report for more information about the work of the MTF.
The Math Task Force developed criteria for evaluating the traditional and integrated approaches and making a recommendation to the Board. For example, the MTF will consider whether one approach or the other better supports: the cognitive development of students; mastery of both the content and practice standards; opportunities for differentiation; and practical considerations about how to achieve coherent compression (speeding up) and expansion (slowing down) of the curriculum for students who do not follow the grade-level progression.
The MTF also discussed possible pathways for compression and identified preliminary considerations including the following:
- there may need to be transitional pathways in addition to “final” pathways;
- the pathways should provide flexibility to compress and expand at various points, including the opportunity to take summer classes; and
- questions must be addressed about how to assess a student’s readiness for compression.
In addition to input from the MTF, pending decisions about course sequences and pathways will be informed by: research into the relative effectiveness of traditional and integrated course sequences; practical experience and insight gained by the PUSD math faculty during the first semester of this school year; input from math experts at Alameda County Office of Education; and input from members of the Silicon Valley Math Initiative (SVMI). SVMI is a consortium of school districts working together to implement Common Core math.
What Does the Research Say?
There are few research studies investigating the relative effectiveness of these approaches. The District commissioned Cheryl Holzmeyer, Ph.D. to prepare an analysis of the research and available data comparing the traditional and international models. Please visit the following link for the complete report, which was issued in December 2014:
Common Core Mathematics Options: Assessing Traditional & Integrated Curricular Possibilities, A Report for the PUSD Mathematics Task Force, Cheryl Holzmeyer, Ph.D.
In summary, Holzmeyer found that studies comparing the effectiveness of traditional and integrated math courses tend to overstate the differences between the two approaches. More to the point, she cautions against inflating “the role of curricula alone in students’ overall learning experiences, attributing too much causality to curricula apart from the wider array of variables emphasized in the research literature from teacher professional development to school and community resources.”
Holzmeyer states that “some research suggests that an integrated math curriculum has the potential to facilitate students’ mathematical thinking more effectively than a traditional math curriculum especially in certain areas, such as conceptual understanding.” At the same time, she notes that both approaches have potential strengths and weaknesses, and that several studies and articles emphasize the importance of other factors such as instructional practice, teacher professional development, and class size.
What are Districts that are Similar to PUSD Doing?
There is no current count of which approach districts are choosing traditional and integrated course sequences. A questionnaire by the California County Superintendents Educational Services Association in Fall 2013 found slightly more districts choosing integrated (34 percent) than traditional (26 percent) but 40 percent had not yet decided. Nonetheless, these numbers are now out of date as more districts are finalizing their decisions.
A recent article describing how districts are split concerning traditional and integrated courses can be seen here: (http://edsource.org/2014/districts ) This article describes “strong, sometimes passionate disagreements” among parents and teachers concerning the choice between traditional and international math courses.
What are Other States Doing?
California was one of 46 states, along with Washington, D.C., that adopted the nationwide Common Core standards. (Three states–Indiana, South Carolina and Oklahoma–have since rescinded them, and two more states may follow.) Some states, including New York, mandate the traditional course sequence. Other states, including Utah and West Virginia, mandate the international sequence. Rather than mandate either approach, California allows school districts to choose what works best for them.
Are There Course Materials for Both Traditional and Integrated Courses?
Yes. Whether the District chooses traditional or integrated course sequences, there is published Common Core math curriculum for either approach.
Common Core-aligned materials for either the traditional or integrated approach are new and must be carefully evaluated to ensure they meet the needs of all students. District math teachers are “piloting” new curriculum from a variety of sources, and developing some of their own curriculum, and continually assessing what works and what needs refinement. This pilot includes both traditional and integrated curriculum, so the teachers are prepared for whatever the Board decides.
In addition, the Silicon Valley Math Initiative is supporting PUSD and over 100 districts in California with transitional Common Core course materials for both the traditional and integrated approach.
Will Teachers Be Prepared to Teach Both Traditional and Integrated Courses?
Yes. Regardless of whether the District chooses a traditional or integrated course sequence, math instructional practices are shifting. To support teachers in the transition to Common Core math, the District is committed to providing professional development for all math teachers, and this training will prepare teachers regardless of whether the Board decides on the traditional or integrated approach.
The Key to Successful Implementation of Common Core Math: Teachers
David Foster, the Executive Director of SVMI, emphasized that what matters most is the classroom implementation, not the course sequences. Although Foster recommends the integrated approach for a variety of reasons (please visit the following link to review Mr. Foster’s presentation to the Piedmont Community on December 1, 2014.
he has stated, “All that ever matters is teaching.”
Consultant Cheryl Holzmeyer, Ph.D. made a similar point in her analysis of the traditional and international models. Holzmeyer found that research comparing traditional and international math courses tend to overstate the differences, and she cautioned against underestimating key variables such as teacher professional development, instructional practice, and school and community resources.
In PUSD, teachers at all levels are collaborating on implementation of Common Core math, to promote coherence between grade levels and to take advantage of the full range of expertise among District staff.
During the 2013/14 and 2014/15 school years, professional development and common planning time for teachers is focusing on implementation of Common Core. This is designed to help teachers: Incorporate the new standards; develop curriculum and instructional practices to help students meet these standards; identify course materials; and prepare for new methods for assessing student learning.
The District is emphasizing collaboration among the math faculty and with math experts at the Alameda County Office of Education (ACOE) and Silicon Valley Math Initiative (SVMI), and use of teacher-leaders and peer coaches, and this model will continue throughout the implementation.
Since the summer of 2014, math teachers participated in nearly 500 hours of professional development training related to Common Core Math.
Time Frame for Implementation
The implementation of Common Core math will be imperfect, and this will be the case in every district. Over the next few years, the District will continue to focus professional development and collaboration time on mathematics, to support the faculty and continually evaluate and refine curriculum and instructional practice. PUSD hopes to roll out its full slate of Common Core courses at both the middle and high schools over the next three years
The primary consideration, throughout the implementation phase and beyond, will be support and increased opportunities for all learners.