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Numbers play a crucial role in our daily lives, whether we are buying a car or understanding the world and financial news reports we encounter every day. Well taught, mathematics is also a thing of beauty, exciting in its logic, elegance and coherence. It builds the analytic spirit on which intelligent and precise thinking depend.
The assumption that students have to have special talents to be successful in mathematics has reigned for too long in this country. Everyone needs to have broad, deep and useful knowledge of mathematics, a fact other countries have long understood. Our global economic competitors expect and demand the acquisition of mathematical knowledge from all of their citizens. We need to keep pace. If they can do it, we can do it. By adoption of these standards, we open a new era in California and demystify mathematics.
These standards express what every student in California can and needs to learn in mathematics. They are comparable to the standards of other countries, including Japan and Singapore, two high-performing countries on the Third International Mathematics and Science Study (TIMSS). As such, these expectations for students are significantly higher than is currently the case, but we believe that with hard work and sound preparation in the early grades, all students will be able to achieve these standards. While the standards raise the bar dramatically, they in no way should be interpreted as a ceiling for any child. Students who can move more quickly through the material should be encouraged to do so; students who are able to exceed these standards should be given the opportunity to succeed at even higher levels of study.
The standards focus on essential content for all students and prepare them for the study of advanced mathematics, for science and technical careers and for post-secondary study in all content areas. All students are required to grapple with and develop an abstract, analytical way of thinking; learn to deal effectively and comfortably with variables and equations; and use mathematical notation effectively to model situations. Our goal for California students is to:
- develop fluency with basic computational skills;
- develop understanding of mathematical concepts;
- become mathematical problem-solvers who can recognize and solve routine problems readily and can find ways to reach a solution or goal where no routine path is apparent;
- communicate precisely about quantities, logical relationships and unknowns via the use of signs, symbols, models, graphs and mathematical terms;
- gather data, analyze evidence and build arguments using mathematical reasoning to support or refute hypotheses; and, make connections among mathematical ideas and between mathematics and other disciplines.
The standards identify what all students should know and be able to do at specific grade levels. Nevertheless, local flexibility is maintained with these standards. Topics may be introduced and taught at one or two grade levels prior to when mastery is expected. Decisions about how best to teach the standards are left to teachers, schools and districts. These standards do not specify how the curriculum should be delivered. Teachers may use "direct-instruction," "explicit-teaching," "knowledge-based," "discovery-learning," "investigatory," "inquiry-based," "traditional," "progressive" or other teaching methods to teach students the subject matter set forth in these standards. At the middle and high school levels, the standards can be used by schools with either an integrated program or the traditional course sequence of Algebra I, Geometry, Algebra II, etc.
These standards enroll students in a mathematical apprenticeship in which they practice skills, solve problems on a daily basis, apply mathematics to the real world, develop a capacity for abstract thinking, and ask questions involving numbers or equations. Students need to know basic formulas, understand what they mean, and know when they should be applied. Students ought to struggle with thorny problems after they have learned to perform the simpler calculations on which they are based. Students should think about why mathematics works in addition to how it works; mathematical thought as well as mathematical results must be stressed. Students need to recognize that the solution to any given problem can be determined by employing more than one strategy and frequently raises new questions of its own: Does the answer make sense? Are there other, more efficient, ways to arrive at the answer? Does the answer bring up more questions? Can I answer those? What other information do I need?
Facts, skills, procedural knowledge, conceptual understanding, problem solving, application, reasoning and the eventual communication of the entire process are the threads which form the tapestry of mathematics. Strength and splendor are achieved when the threads are woven tightly together and in proper balance.
The kindergarten through grade 7 mathematics content standards are organized, grade by grade, with each of five strands (Number Sense; Algebra and Functions; Measurement and Geometry; Statistics, Data Analysis and Probability; and Mathematical Reasoning) included at each grade level. Focus statements that increase in complexity from kindergarten through grade 7 are included at the beginning of each grade level to indicate how the discrete skills and concepts together form a cohesive whole.
The standards for grades 8 through 12 are organized differently than those for kindergarten through grade 7. Strands are not used for organizational purposes because, unlike in the earlier grades, in grades 8 through 12 the mathematics studied naturally falls under discipline headings: Algebra, Geometry, etc. Many schools teach this material in traditional courses, while others teach this material in an integrated fashion. In order to provide local educational agencies and teachers with flexibility, the grades 8 through 12 standards do not mandate a particular discipline to be initiated and completed in a single grade. Nevertheless, however it is taught, the core content of these subjects must be covered and all academic standards for achievement must be the same.
The Mathematical Reasoning standards are different from the other standards in that they do not represent a content domain. Mathematical reasoning cuts across all strands. In addition, as stated in the Board's 1996 Mathematics Program Advisory:
- Problem solving involves applying skills, understandings and experiences to resolve new or perplexing situations. Solving problems challenges students to apply their conceptual understanding in a new or complex situation, to exercise their basic skills, and to see mathematics as a way of finding answers to some of the problems that occur outside of a classroom. Students grow in their ability and persistence in problem solving by virtue of extensive experience in solving problems at a variety of levels of difficulty and at every level in their mathematical development.
Problem solving, therefore, is an essential part of mathematics and is subsumed in every strand and in each of the higher order disciplines in grades 8 through 12. Problem solving is not separate from content. Rather, students learn concepts and skills in order to apply them and to solve problems in and out of school. Because there are fundamental techniques used in the solutions of problems that occur in all areas of mathematics and at all levels, the basic elements of problem solving are consistent across grade levels.
It is necessary that the problems students solve address important mathematics. As students progress from grade to grade, problems should require that students: (1) use increasingly more advanced knowledge and understanding of mathematics, (2) deal with problems that are increasingly more complex (applications and purely mathematical investigations), (3) increasingly incorporate inductive and deductive reasoning and proof and (4) increasingly require students to make connections among mathematical ideas within a discipline and across domains. Each year students need to solve problems from all strands by drawing upon grade level appropriate skills and conceptual understanding, with most of the problems related to the mathematics students study that year.