Given the shortage of highly trained mathematics teachers in California, schools and undergraduate institutions must actively encourage talented mathematics students to go into teaching careers. Undergraduate internships in K-12 classrooms, followed up with guided reflection and discussion, can be effective recruitment tools and enhance the value of credential year study.

K-6 teachers need a command of the mathematics described in this Framework for Algebra I, Geometry, and Algebra II. Further, prior to a credential program, K-6 teaching candidates should have a full year of mathematics content courses that cover at least the material described in Appendix B.

Junior high or middle school mathematics teachers need a command of mathematics beyond that of K-6 teachers. Prior to a credential program, junior high/middle school mathematics teachers should have a least 24 semester hours of courses that are part of an approved K-12 mathematics credential program.

High school mathematics teachers need the full background required for state secondary certification in mathematics before entering a credential program. College and university mathematics departments should design their programs so that their majors do not encounter unnecessary obstacles in meeting the state requirements.

In addition to mathematics content courses, preservice teachers at all levels need a broad understanding of pedagogical issues in mathematics. They need experiences/courses that enable them to

The shortage of qualified mathematics teachers affects primary and secondary schools in different ways. At the primary level there exist teachers who possess multiple subject credentials that do not meet the new demands of the curriculum in mathematics. Although these teachers may be highly effective in other areas and contribute to the overall program at their schools, they should be encouraged to turn to mathematically qualified colleagues for help in strengthening their own mathematics instruction.

At the junior high or middle school and at the high school levels, it is not uncommon for teachers to preside over mathematics classes that they are not certified to teach. Such teachers are granted emergency credentials through a process that is based in local school districts and is largely outside the credentialing process.

To deal with these very different problems, three steps are recommended:

Ongoing professional development should be expected of all teachers. Professional development for immediate classroom application should take place locally on a regular basis. Teachers should also be involved in long-term professional development of their mathematical knowledge and skills.

All too often new teachers ìwash outî in or just after their first year of teaching, thereby wasting much of the huge investment that went into their education. School administrators and colleagues must take steps to help new teachers succeed. Careful placement and active mentoring can help, as can activities listed below for all teachers. These can alleviate the isolation that can be a problem for all teachers but is most acute in the first year.

Teachers within a given school or locality should meet together regularly to discuss content and pedagogy in the mathematics curriculum. They should be encouraged to set up an ongoing collegial support system like those that have been identified with successful mathematics programs. The focus of such collegial meetings would be to share successful lessons and teaching approaches and to coach one another in ways to improve student understanding. School and district administrations should support such in-house efforts by making time and space available for them.

Formal local inservice programs, including School Improvement Program (SIP) days, should address the needs of teachers involved. Further, teachers should be active participants in planning and organizing meetings, short courses, and workshops offered by local districts, colleges, universities, and professional organizations. These programs need to be assessed independently to determine their effectiveness.

The goal of long-term professional development is sustained intellectual growth of mathematics teachers and enhancement of their mathematical knowledge and skill. Long-term professional development of this type should be expected and actively encouraged and rewarded both by school administrators and by state and national efforts. Institutions of higher education and other institutions with sufficient expertise should be enlisted in an effort to make opportunities for high-quality, long-term professional development available to all mathematics teachers. Long-term professional development programs in mathematics should be externally assessed to ensure that they achieve their goals of enhanced mathematical knowledge and capability. Teachers should be encouraged to share the benefits of their long-term professional development efforts as appropriate with their colleagues in local inservice programs such as School Improvement Program (SIP) days. Teachers should also be encouraged to belong to appropriate national and local professional organizations.

Since professional development is essential to implementing the high standards defined by this Framework, a variety of issues should be considered when designing such activities. The following is not a check-off list of what should be done, nor is it exhaustive. Indeed, any particular focused and in-depth program may only be able to deal with a few issues each year. Districts need to think carefully about their own priorities in deciding how to balance these considerations.

Source: These descriptions are updates from Recommendations on the Mathematical Preparation of Teachers, prepared by the Committee on the Undergraduate Program in Mathematics, Washington, DC, Mathematical Association of America, 1983, pp. 9-15. As stated in the document cited, ìthe course descriptions in these recommendations are not intended to be complete course descriptions but rather to identify topics that are particularly important for teachers.

The course descriptions assume as prerequisites three years of college preparatory mathematics, including two years of algebra and one year of geometry, and demonstrated mastery of the basic skills of the K-7 mathematics curriculum as outlined in this Framework. The courses described below should provide preservice teachers with the mathematical knowledge and background necessary to implement a curriculum that balances mathematical skills, development of conceptual understanding, and problem solving. To that end, the courses should exemplify this balance.

The objectives of these two courses should be to provide teachers with the ability to:

The courses should provide teachers with:

The course provides prospective teachers with part of the background needed for teaching the content of the elementary mathematics program as outlined in this Framework, including the development of the whole number system, measurement, and geometry.

While there may be considerable variation in the topics selected to achieve the objectives the following are particularly suitable for inclusion in such a course:

1. Number: Prenumeration concepts (attributes, classification, ordering, patterns). Properties of joining, separating, and comparing sets, set equivalence, set inclusion, and the use of sets in problem solving (Venn diagrams, exhaustive listings). Base ten numeration system; place value and its relation to grouping in operations, models for each of four basic operations; properties of the basic operations, common error patterns in student computation with basic algorithms.

Divisibility; prime and composite numbers; infinitude of primes, divisibility rules for whole numbers through 11 (excluding seven), lowest common multiple, greatest common divisor, and relative primeness.

2. Measurement: Use of standard and nonstandard (paper clips, erasers, body measures...) units in measuring length, perimeter, area, capacity, volume, mass, weight, angle, time, temperature. Selection of units of measure and methods of estimation.

3. Geometry: Basic two- and three dimensional objects: line, plane, square, rectangle, parallelogram, rhombus, trapezoid, triangles (scalene, right, isosceles, and equilateral), circles, rectangular solids, cubes, prisms, cones, pyramids, and sphere. Their properties: parallelism, perpendicularity, congruence (not only triangle), similarity, symmetry, length, area, and volume in basic metric and standard units, angles and angle measurement, the effect of scale on surface and volume. Use of geoboards, paper folding, and other models in illustrating geometric concepts. Basic concepts and properties of geometric transformations. Contexts in which geometric concepts arise, use of their properties in meaningful, nontrivial problem solving situations.

This course focuses on the development of the real number system and its subsystems, probability, statistics, and basic computer concepts. The prerequisite is Fundamental Mathematical Concepts I.

While, as in the case of Fundamental Mathematical Concepts I, there may be considerable variation in the topics selected to achieve the objectives stated above, the following topics are particularly suitable for inclusion in such a course.

1. Numbers: Extension of whole numbers to integers, models for integers, four basic operations with integers, properties of integers (order, absolute value,...). Extension of integers to the rational numbers, positive and negative, in any form, such as fraction, decimal, or percent and conversion between any of these representations. The four basic operations with rational numbers. Ratio and proportion. Properties of rational numbers. Extension of the rational numbers to irrational numbers such as pi, certain square roots, the irrationality of the square root 2; Pythagorean Theorem and a proof; properties of real numbers, use of square roots of positive numbers in meaningful contexts; relationship of real numbers to geometry. The arithmetic mean of a set of numbers. Integer and simple fraction exponents of real numbers. Scientific notation.

2. Algebra: Variables and functions, how they arise in modeling the world around us. Substantiating whether or not a given number satisfies a given equation. Solutions of linear equations and inequalities. Use of the Cartesian plane to graph shapes and linear functions. Use of formulas to express quantitative relationships. Qualitative understanding of how graphs of functions over time tell a story.

3. Probability: Relative frequency experiments; methods of counting (tree diagrams, exhaustive listings, permutations, combinations); sample spaces; joint events, independent events, dependent events, complementary events. Basic properties of probability.

4. Statistics: Organization and presentation of data (line, circle, and bar graphs; stem and leaf plots); role of scales and possible bias in graphs, analysis of the observed distribution (mean, median, mode, range, variance, standard deviation, percentile, quartile). Meaning of population, sample, and random sample. Sampling as it applies to consumers of sample research literature.

5. Technology: Impact of computers on contemporary school mathematics curricula; nature of an algorithm; elementary programming; writing simple programs correlated to the elementary school curriculum; role of computing in mathematics.