Mathematics has a logical structure. Every block in this structure is connected to every other by gap-free reasoning - all resting upon a foundation of a few agreed-upon definitions and fundamental propositions. Despite its abstract nature, mathematics beautifully describes and helps us to scientifically understand the physical universe. At the same time, mathematics is a necessary practical tool of everyday life. Without it, we would be unable to conduct business and design the buildings and machinery of the modern world.
A culture which gives high respect to mathematics and whose people are mathematically literate has a much needed precision and exactitude in its debate and discussion. Studying mathematics is a discipline that develops precise habits of mind. In mathematics, principles of reasoning must be strictly followed, ambiguities must be sorted out, and terms are not allowed to slip from one meaning to another without warning. We all need, when appropriate, to be capable of the exactness and logic for which mathematics trains the mind.
The beauty and utility of mathematics are part of the birthright and legacy of every one of our children. It is the privilege and responsibility of all of us - mathematicians, teachers, schools, and parents - to ensure that all children get the highest quality mathematics education we can give them.
The highest quality mathematics education means algebra and geometry for all students. One of the most important parts of Sheila Tobias's influential Overcoming Math Anxiety (1978) was the opening chapter, "The Primacy of Mathematics," in which she explains that young men and women who are not going on to college need algebra and geometry just as much as the college-bound. Research shows that algebra and geometry are not only preparation for college, but a gateway to good jobs for all young men and women.
These Standards set forth attainable goals. Researchers have found that they can predict that students who master fractions, decimals, and percentages and their inter-conversion will be successful in algebra. These Standards prepare all student by the end of seventh grade to take algebra and geometry. These Standards then cover in eighth and ninth grades the complete subjects of algebra and geometry, not just parts of them, so that all students - college-bound or not - will have the advantages that come with mathematical literacy.
This highest quality education must include the three components of balanced instruction in mathematics: basic facts and skills; conceptual understanding; and problem-solving ability. These are so intimately intertwined that it is difficult to keep them separated, and we should not often attempt to do so. Concepts are embedded in skills and make the skills meaningful and lasting. Conceptual understanding helps guide successful problem solutions. Interesting problems may lend themselves to reinforcement of basic skills and suggest generalizations to whole classes of similar problems.
When we say that a student has mastered a skill, we must mean that a student both can carry out the skill and can also understand why the way-of-getting-the-answer works and when and how it may be usefully applied.
This Mathematics Standards document may not say all of this explicitly for each separate standard, but this Standards document assumes the necessity of students learning, teachers teaching, and testers testing all three - facts and skills, conceptual understanding, and problem-solving.
Mathematics is a cumulative, hierarchical subject; learning new skills and concepts often depends on mastery of previous ones. If students are to begin school each year ready to learn, they must enter each grade in possession of all the prerequisite skills, including problem solving skills, knowledge, and conceptual understanding necessary to allow them to tackle new and more challenging work each year.
Such a progression does not depend on innate ability possessed by only a few students, rather, it is achievable by all, as shown by many high performing countries, but it requires hard work from everyone - students, teachers, and parents. We must therefore all be held to the highest possible standard; less than our best is unacceptable.
The legislature of the state of California recognized that our children are able to learn at much higher levels than those currently demanded of them, and therefore authorized the creation of Standards to set forth and establish rigorous learning goals for K-12 students.
These Standards are not intended to describe what children currently achieve. Rather, they establish tall yardsticks against which children, schools, and school districts, and the state at large may measure their gradual improvement. These Standards are not intended as floor competencies for school promotion or graduation nor to determine any student's course grades. They provide for all of us, instead, a tool which all members of the education community may use to diagnose the strengths and weaknesses of each student and of each mathematics program, and so facilitate steady gains.
No one who participated in writing these Standards expects that all students will master these objectives immediately. It would be astounding if next year every eighth grader in California mastered algebra. But these Standards do provide a bright beacon indicating where we wish to go and a measure of our progress in getting there.
These standards are clear and specific, so that the grade-by-grade competencies required for each child are clearly spelled out, and they are understandable by both parents and teachers. They are measurable and objective, as they must be to provide the basis for statewide assessments of our children.
These standards reflect the knowledge and skills necessary for California's workforce to be competitive in the information-based economy of the next century, and they are comparable in rigor and academic content to the content and performance standards used in the school systems of America's global competitors. It is now up to students, teachers, and schools to work hard to measure up to them.
This Mathematics Standards document is organized into Strands. These are categories that help teachers, test-writers, and writers of instructional material keep track of the many topics in K-12 mathematics. The six Strands in this document are based on the Strands in the California Education Round Table High School Mathematics Graduation Standards and the Strands in the Mathematics Framework for California Public Schools. Problem-solving --which like conceptual understanding is not in reality a separate mathematical topic, but rather a matter that pertains to all topics in mathematics--does not have a separate Strand, but is embedded in all the Strands and standards in this document.
Priorities in Kindergarten through Grade Six
Many teachers in the elementary grades (K-6) are not math specialists. This Mathematics Standards document assigns priorities to math topics in grades K-6 in order to assist those teachers and, at the same time, to assist test-writers. The numbers in the Priorities column are a way of indicating the relative amount of time needed for a topic and its importance for assessment. Higher numbers indicate greater importance.
Grade-by-Grade Emphasis
(The description of grade-by-grade emphasis pertains to the content of this Mathematics Standards document, but it closely parallels and is adapted from the Mathematics Framework for California Public Schools issued in September 1997.)
KINDERGARTEN
Kindergarten focuses on a basic understanding of numbers and comparisons. Students show that they have an introductory understanding of numbers, can represent them in various ways and can determine whether the results of simple arithmetic operations are reasonable.
Students are to compare different groups of objects, noting whether one group has the same number, more, or fewer objects than another group . In addition, as spatial aspects of mathematics are developed, students are to recognize various elementary geometric shapes and patterns.
GRADE 1
First grade focuses on number sense including place value and introductory addition and subtraction. Students show they have extended their use of numbers through 100 by counting, reading, writing, ordering, and comparing them. Students can count forward and backward, along with counting by twos, fives, and tens. Students demonstrate an understanding of place value at the level of two-digit numbers. Students show they can do elementary addition and subtraction problems. Students can solve basic addition and subtraction problems found in sentences with numbers, and in this context, can use and understand symbols such as blanks or boxes. Students can count up the value of a collection of coins.
In Grade 1, students demonstrate an ability to measure and compare objects according to measurement attributes. Students also identify, describe, and compare geometric shapes.
GRADE 2
Second grade focuses is on number sense and measurement. Second-grade students are able to work with larger numbers in many different contexts. Students understand the concepts of addition and subtraction and can apply these operations in a variety of situations. Students have memorized their addition and subtraction facts and have developed the roots of their understanding of the concepts of multiplication and division
Students can measure using standard units of measure in this grade for quantifying length, weight, volume, time and temperature. Second-grade students demonstrate expanded understanding of the properties of geometric figures including symmetry. Also, students can collect, organize, display, and interpret numerical data.
GRADE 3
Third grade focuses on number sense and geometry. As students increase their use of numbers to 10,000, they show increased addition and subtraction abilities. The third grade places an emphasis on the whole-number operations of multiplication and division. Students will know the multiplication facts and expand multiplication and division to work with larger numbers. Students also use and interpret fractions and use decimals in the context of money in this grade.
Student work in geometry this year shows further learning about common three-dimensional geometric objects (cubs, spheres, cones, etc.). They can also identify and describe congruence and similarity for two-dimensional figures and find the area of rectangles and squares.
GRADE 4
Fourth grade students are nearing completion of their learning about and mastery of whole number operations, with an emphasis on multiplication and division. Students show an understanding of common fractions and decimals and initial understanding of the use of ratios and proportions.
In measurement work, fourth grade students use measurement units in quantification and in conversions within a measurement system. With the introduction of the function strand, students understand the use of ordered pairs of numbers and their representation as points in a coordinate system.
GRADE 5
Fifth grade focuses on number sense and new concepts in geometry. They identify prime factors of whole numbers. Student can add and compare fractions with unlike denominators. Conversions are also made between fractions and their decimal and percentage equivalents. Students are also able to solve two-step problems in this grade.
Fifth grade students show new learning in geometry, including knowing ways to find the areas of parallelograms and triangles and the volume of a rectangular solid. They also measure angles in degrees in circles and common angles.
GRADE 6
Sixth grade focuses on number sense with the introduction of negative numbers as both integers and rational numbers. Students can use the four basic operations of arithmetic with rational numbers and understand exponents as indicating repeated multiplication. The introduction of negative numbers is paired with student ability to use all four quadrants of the coordinate grid.
Students also begin their algebra skills in earnest this year as they demonstrate abilities to work with variables and equations. This includes formulating conversions between measurement systems. They extend their geometric knowledge and can identify and use more properties of geometric figures and about find the circumference and area of a circle.
GRADE 7
Seventh grade focuses on algebra readiness. The mastery of rational numbers is an essential part of this focus. Students can perform operations on fractions and simplify them by utilizing prime factorization. Students show they can move from number to variable, from specifics to generalizations, and from description to a verified solution. Students should be able to present a verbal description as an expression, equation, or inequality; operate and simplify rational numbers. Students use proportions to express one quantity as a proportion of another; apply geometric properties and relationships; describe and graph patterns or functions; and interpret, analyze, and make generalizations or conclusions from a set of data.
GRADE 8
Eighth grade focuses on algebra. Symbolic reasoning and calculations with symbols are central in algebra. In particular, assessment should cover:
GRADE 9
Ninth grade focuses on geometry. In particular, assessment should cover:
GRADE 10
Tenth grade returns the focus to algebra. Content areas include the solution of systems of quadratic equations, advanced work with rational and radical expressions and advanced factoring, logarithmic and exponential functions, sequences and series, and imaginary and complex numbers.
GRADES 11/12
Because the State of California only requires two years of mathematics to graduate from high school, standards "for all students" end at the conclusion of tenth grade. The standards listed in this document for Grades 11/12 are only one year's worth of work and are equivalent to a course in mathematical analysis. This is considered an elective course.
The standards list the content of a full-year mathematical analysis course that blends together all of the concepts and skills that must be mastered prior to enrollment in a calculus course.
After they complete the work listed in these Standards for Grades 11/12, students would then have the option of taking several other electives, as outlined in the Draft 1997 Mathematics Framework:
Students who are college-bound and intend to take quantitative and scientific courses in college should take calculus in high school.
Students who do not intend to take quantitative and scientific courses in college could also take math analysis in Grade 12 following a year of electives or possibly even a year of no math. For this reason, the math analysis standards are identified as Grades11/12 in the Standards.
This design provides for a high level of mathematics achievement for all students while providing a variety of options in the last two high school years to meet the varied needs of individual students.