# The California Mathematics Content Standards

**Introduction
to Grades Eight
Through Twelve**

The standards for grades eight through

twelve are organized differently from those

for kindergarten through grade seven. In

this section strands are not used for

organizational purposes as they are in the

elementary grades because the mathematics

studied in grades eight through twelve

falls naturally under discipline headings:

algebra, geometry, and so forth. Many

schools teach this material in traditional

courses; others teach it in an integrated

fashion. To allow local educational

agencies and teachers flexibility in teaching

the material, the standards for grades

eight through twelve do not mandate that

a particular discipline be initiated and

completed in a single grade. The core

content of these subjects must be covered;

students are expected to achieve the

standards however these subjects are

sequenced.

Standards are provided for Algebra I ,

geometry, Algebra II, trigonometry ,

mathematical analysis , linear algebra ,

probability and statistics, advanced

placement probability and statistics, and

calculus. Many of the more advanced

subjects are not taught in every middle

school or high school. Moreover, schools

and districts have different ways of

combining the subject matter in these

various disciplines. For example, many

schools combine some trigonometry ,

mathematical analysis , and linear algebra

to form a precalculus course . Some

districts prefer offering trigonometry

content with Algebra II.

Table 1, “Mathematics Disciplines, by

Grade Level,” reflects typical grade-level

groupings of these disciplines in both

integrated and traditional curricula. The

lightly shaded region reflects the minimum

requirement for mastery by all

students. The dark shaded region depicts

content that is typically considered elective

but that should also be mastered by

students who complete the other disciplines

in the lower grade levels and

continue the study of mathematics.

Many other combinations of these

advanced subjects into courses are possible.

What is described in this section are

standards for the academic content by

discipline; this document does not endorse

a particular choice of structure for courses

or a particular method of teaching the

mathematical content.

When students delve deeply into

mathematics, they gain not only conceptual

understanding of mathematical

principles but also knowledge of, and

experience with, pure reasoning. One of

the most important goals of mathematics

is to teach students logical reasoning . The

logical reasoning inherent in the study of

mathematics allows for applications to a

broad range of situations in which answers

to practical problems can be found with

accuracy.

By grade eight, students’ mathematical

sensitivity should be sharpened. Students

need to start perceiving logical subtleties

and appreciate the need for sound mathematical

arguments before making

conclusions. As students progress in the

study of mathematics, they learn to

distinguish between inductive and deductive

reasoning; understand the meaning of

logical implication; test general assertions;

**Table 1. Mathematics Disciplines, by Grade Level**

Grades | |||||

Disciplines | Eight | Nine | Ten | Eleven | Twelve |

Algebra I | |||||

Geometry | |||||

Algebra II | |||||

Probability and Statistics | |||||

Trigonometry | |||||

Linear Algebra | |||||

Mathematical Analysis | |||||

Advanced Placement Probability and Statistics |
|||||

Calculus |

realize that one counterexample is enough

to show that a general assertion is false;

understand conceptually that although a

general assertion is true in a few cases, it is

not true in all cases; distinguish between

something being proven and a mere

plausibility argument; and identify logical

errors in chains of reasoning.

Mathematical reasoning and conceptual

understanding are not separate from

content; they are intrinsic to the mathematical

discipline students master at

more advanced levels.

Prev | Next |