1. ©2001 CBMS Math Preparation of Teachers Teachers need to study the mathematics of a cluster of grade levels, both to be ready for the various ways in which grades are grouped into elementary, middle, and high schools in different school districts, and to understand the larger mathematical learning context in which the mathematics taught in a specific grade fits.

2. ©2001 CBMS Math Preparation of Teachers Teachers need to acquire a "deep understanding'' of school mathematics concepts and procedures. Teachers must develop "mathematical knowledge for teaching.'' This knowledge allows teachers to assess their students' work, recognizing both the sources of student errors and their students' understanding of the mathematics being taught.

3. ©2001 CBMS Math Preparation of Teachers Is elementary mathematics so simple that teaching it requires knowing only the "math facts" and a handful of algorithms? Quite to the contrary. This early content is rich in important ideas. It is during their elementary years that young children begin to lay down those habits of reasoning upon which later achievement in mathematics will crucially depend. When the goal of instruction is to help children attain both computational proficiency and conceptual understanding, teaching elementary school mathematics can be intellectually challenging.

4. ©2001 CBMS Math Preparation of Teachers Number and Operations To be prepared to teach arithmetic for understanding, elementary teachers, themselves, need to understand: A large repertoire of interpretations of addition, subtraction, multiplication and division, and of ways they can be applied. Place value: how place value permits efficient representation of whole numbers and finite decimals; that the value of each place is ten times larger than the value of the next place to the right; implications of this for ordering numbers, estimation, and approximation; the relative magnitude of numbers.

5. ©2001 CBMS Math Preparation of Teachers Number and Operations To be prepared to teach arithmetic for understanding, elementary teachers, themselves, need to understand: Multidigit calculations, including standard algorithms, "mental math," and non-standard methods commonly created by students: the reasoning behind the procedures, how the base-10 structure of number is used in these calculations. Concepts of integers and rationals: what integers and rationals (represented as fractions and decimals) are; a sense of their relative size; how operations on whole numbers extend to integers and rational numbers; and the behavior of units under the operations.

6. ©2001 CBMS Math Preparation of Teachers Algebra and Functions Although the study of algebra and functions generally begins at the upper- middle- or high-school levels, some core concepts and practices are accessible much earlier. If teachers are to cultivate the development of these ideas in their elementary classrooms, they, themselves, must understand those concepts and practices, including: Representing and justifying general arithmetic claims, using a variety of representations, algebraic notation among them; understanding different forms of argument and learning to devise deductive arguments. The power of algebraic notation: developing skill in using algebraic notation to represent calculation, express identities, and solve problems.

7. ©2001 CBMS Math Preparation of Teachers Algebra and Functions Although the study of algebra and functions generally begins at the upper- middle- or high-school levels, some core concepts and practices are accessible much earlier. If teachers are to cultivate the development of these ideas in their elementary classrooms, they, themselves, must understand those concepts and practices, including: Field axioms: recognizing commutativity, associativity, distributivity, identities, and inverses as properties of operations on a given domain; seeing computation algorithms as applications of particular axioms; appreciating that a small set of rules governs all of arithmetic. Functions: being able to read and create graphs of functions, formulas (in closed and recursive forms), and tables; studying the characteristics of particular classes of functions on integers.

8. ©2001 CBMS Math Preparation of Teachers Geometry and Measurement For many years, the geometry curriculum for the elementary grades consisted of recognizing and naming basic two-dimensional shapes, measuring length with standard and non-standard units, and learning the formulas for the area and perimeter of a rectangle (and possibly a few other shapes). Because many students arrive in high-school geometry courses unprepared for its content, topics in geometry have recently been accorded a more prominent role in the curriculum of the lower grades. To most elementary teachers, their own encounter with high-school geometry notwithstanding, much of this material is new. In order to teach it to young children, they must develop competence in the following areas:

9. ©2001 CBMS Math Preparation of Teachers Geometry and Measurement Visualization skills: becoming familiar with projections, cross-sections, and decompositions of common two- and three-dimensional shapes; representing three-dimensional objects in two dimensions and constructing three-dimensional objects from two-dimensional representations. Basic shapes, their properties, and relationships among them: developing an understanding of angles, transformations (reflections, rotations, and translations), congruence and similarity. Communicating geometric ideas: learning technical vocabulary and understanding the role of mathematical definition.

10. ©2001 CBMS Math Preparation of Teachers Geometry and Measurement The process of measurement: understanding the idea of a unit and the need to select a unit appropriate to the attribute being measured, knowing the standard (English and metric) systems of units, understanding that measurements are approximate and that different units affect precision, being able to compare units and convert measurements from one unit to another. Length, area, and volume: seeing rectangles as arrays of squares, rectangular solids as arrays of cubes; recognizing the behavior of measure (length, area, and volume) under uniform dilations; devising area formulas for basic shapes; understanding the independence of perimeter and area, of surface area and volume.

11. ©2001 CBMS Math Preparation of Teachers Data Analysis, Statistics, and Probability Statistics is the study of data, and despite daily exposure to data in the media, most elementary teachers have little or no experience in this vitally important field. Thus, in addition to work on particular technical questions, they need to develop a sense of what the field is about. Prospective teachers need experience in: Designing data investigations: understanding the kinds of question that can be addressed by data, creating data sets, moving back and forth between the question (the purpose of the study) and its design. Describing data: understanding shape, spread, and center; using different forms of representation; comparing two sets of data.

12. ©2001 CBMS Math Preparation of Teachers Data Analysis, Statistics, and Probability Statistics is the study of data, and despite daily exposure to data in the media, most elementary teachers have little or no experience in this vitally important field. Thus, in addition to work on particular technical questions, they need to develop a sense of what the field is about. Prospective teachers need experience in: Drawing conclusions: choosing among representations and summary statistics to communicate conclusions, understanding variability, understanding some of the difficulties that arise in sampling and inference. Probability: making judgments under conditions of uncertainty, measuring likelihood, becoming familiar with the idea of randomness.