Mathematics Curriculum Framework - November 2000
When we cannot use the compass of mathematics or the torch of experience … it is certain that we cannot take a single step forward.
The 2000 Mathematics Curriculum Framework is one of seven sets of standards created to advance educational reform in Massachusetts. It is the work of teachers and administrators in pre-kindergarten through grade 12, mathematics education professors, university mathematicians, and community members working with staff from the Department of Elementary and Secondary Education.
Organization of the document
The guiding principles articulate a set of beliefs about teaching, learning, and assessing mathematics in Massachusetts. Five strands organize the mathematical content: Number Sense and Operations; Patterns, Relations, and Algebra; Geometry; Measurement; and Data Analysis, Statistics, and Probability. At the beginning of each strand are the broad concepts from NCTM Standards 2000; they guide the clustering of the learning standards in that strand. In grades PreK-6, each of the broad concepts is identified by an icon (, , , ), and the icon at the end of each standard indicates the NCTM concept to which it is linked. Because the standards are grouped by broad concepts, the order in which they appear in each grade span is not necessarily the order in which they might be taught.
The learning standards are organized by two-year grade spans or for single-year courses. The grade spans are PreK-K, 1-2, 3-4, 5-6, 7-8, 9-10, and 11-12, and the courses are Algebra I, Geometry, Algebra II, and Precalculus. Each learning standard in each grade span or course has a unique identifier that consists of:
- the grade level (K, 2, 4, 6, 8, 10, or 12) or course (AI = Algebra I, G = Geometry,
AII = Algebra II, and PC = Precalculus),
- the strand (N = Number Sense and Operations, P = Patterns, Relations, and Algebra,
G = Geometry, M = Measurement, and D = Data Analysis, Statistics, and Probability), and
- the standard number.
For example, standard 4.G.6 is the sixth standard of the Geometry strand in the grades 3-4 grade span. Standard AI.P.12 is the twelfth standard of the Patterns, Relations, and Algebra strand in the Algebra I course. This numbering system allows teachers to also organize the standards by grade span. For example, sixth grade teachers preparing their curriculum can distinguish grades 5-6 standards in each strand by identifying all of the standards beginning with a 6.
The learning standards specify what students should know and be able to do as learners of mathematics at the end of each grade span or course. Students are held responsible for learning standards listed at earlier grades as well as their current grade. Following the standards are selected problems and activities that clarify the standards and help teachers use them. Those marked with a † are drawn from NCTM Standards 2000. Each grade span also includes exploratory concepts and skills that are not intended to be assessed at that grade level. The exploratory concepts and skills usually appear as learning standards at a higher grade level and are assessed then.
Throughout this document, the standards are written to allow time for study of additional material at every grade level and for advanced courses in middle and high school. At the secondary school level, the order of the standards is not meant to imply a necessary sequence of topics. Not only is the sequence of topics flexible, but schools may also make choices about course organization and sequence. Schools may offer a sequence of courses emphasizing a particular mathematical subject or a sequence of multistrand courses. For schools choosing to offer courses that emphasize one subject, the document provides sets of standards for Algebra I, Geometry, Algebra II, and Precalculus. Schools may choose to offer Geometry before or after Algebra II. As the Department found in a survey of mathematics department chairs, teachers, and curriculum coordinators throughout the state, if students are to be able to take a Calculus course in grade 12, they must usually take Algebra I in grade 8.1 Indeed, a recent report from the U.S. Department of Education articulated the need to:
- Provide all students the opportunity to take Algebra I or a similarly demanding course that includes fundamental algebraic concepts in the 8th grade to enable students to take more advanced mathematics and science courses in all four years of high school.
- Build the groundwork for success in algebra by providing a rigorous curriculum in grades K-7 that moves beyond arithmetic and prepares students for the transition to algebra.
- Ensure that all students, parents, teachers, and counselors understand the importance of students' early study of algebra as well as continued study of rigorous mathematics and science in high school.2
To provide for more local control and flexibility in secondary school mathematics curricula, two assessment options at the end of grade 8 are being planned. One option will be based on the multistrand standards for grades 7 and 8. The other option will be an end-of-course assessment for Algebra I based on the standards listed for Algebra I.
Although the Department presents standards for only one post-Algebra II course—Precalculus—there are other mathematics courses that school districts might offer concurrent with or subsequent to Precalculus. Among these options are probability and statistics and discrete mathematics. Schools should also provide interested students with enrichment options in mathematics such as advanced placement courses, independent research, internships, or study of special topics.
Development of the standards
The development of these standards is based upon two reform initiatives in Massachusetts, the Education Reform Act of 1993 and Partnerships Advancing the Learning of Mathematics and Science, the Statewide Systemic Initiative funded by the National Science Foundation since 1992. The initial set of standards was completed and approved by the Board of Education in 1995. Because the Education Reform Act required that all standards documents be reviewed and revised periodically, a mathematics revision panel was appointed in the summer of 1998. After reviewing the comments that the Department had received on the original set of mathematics standards, the revision panel examined these standards, assessed their appropriateness, and presented a draft to the Board of Education in September 1999.
The panel drew on the work of the National Council of Teachers of Mathematics (NCTM); reports of the Mathematical Association of America, the American Mathematical Society, and the American Association for the Advancement of Science; NCTM's Principles and Standards for School Mathematics October 1998 discussion draft; data from the Third International Mathematics and Science Study; the National Research Council's National Science Education Standards; and results from the initial administration of the Massachusetts Comprehensive Assessment System (MCAS).
Based on the comments that the Department received on this draft, further revisions were made. Drawing on the published version of NCTM Standards 2000, the final draft reflects the work of many PreK-12 mathematics educators, mathematicians, and others who helped clarify and refine the standards and ensure their developmental appropriateness.