A list of Activities and Resources - all of which are on the CD, except those with an (H) which are only available in Handout form.
Sample Course Guides
Sample activities that promote an active learning environment
Sample ideas that connect the content of the course to the elementary curriculum
Comments on Deciding WHAT to teach
There are several different resources that mathematics faculty can use to decide which content areas to include in their courses for prospective teachers (see, for example, the suggested references included on the workshop overview sheet, especially the CBMS document). Most imply or explicitly list a collection of understandings that are desirable for teachers. Not surprisingly, these lists include topics in Number, Operations, Algebra, Functions, Geometry, Measurement, Data Analysis, Statistics, and Probability. Most also imply or implicitly state the importance of the quality of these understandings by describing the cohesiveness, depth, and flexibility of mathematical knowledge that is required to teach well. It is this quality of understanding that I want to address here.
I like how Liping Ma, in Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States (1999), distinguishes the quality of the understandings of teachers about mathematics through discussion of the "basic" and the "fundamental" points of view. When elementary mathematics is viewed as "basic" mathematics, then the mathematics can be described as a collection of procedures. When elementary mathematics is viewed as "fundamental", then the mathematics can be described as elementary (the beginning of the discipline), primary (containing the rudiments of advanced mathematics) and foundational (supporting future learning). Supporting the "fundamental" point of view, Ma argues that elementary teachers require a profound understanding of fundamental mathematics that has breadth, depth, and thoroughness.
How can mathematics faculty promote such an understanding, particularly one which has breadth, depth, and thoroughness? Recommendations 3 and 4 of the CBMS document on the Mathematical Preparation of Teachers give some general guidelines. I include them here for your reference:
Recommendation 3 Courses on Fundamental ideas of school mathematics should focus on a thorough development of basic mathematical ideas. All courses designed for prospective teachers should develop careful reasoning and mathematical "common sense" in analyzing conceptual relationships and in solving problems. Attention to the broad and flexible applicability of basic ideas and modes of reasoning is preferable to superficial coverage of many topics. Prospective teachers should learn mathematics in a coherent fashion that emphasizes the interconnections among theory, procedures, and applications. They should learn how basic mathematical ideas combine to form the framework on which specific mathematics lessons are buildt. For example, the ideas of number and function, along with algebraic and graphical representation of information, form the basis of most high school algebra and trigonometry.
Recommendation 4. Along with building mathematical knowledge, mathematics courses for prospective teachers should develop the habits of mind of a mathematical thinker and demonstrate flexible, interactive styles of teaching. Mathematics is not only about numbers and shapes, but also about patterns of all types. In searching for patterns, mathematical thinkers look for attributes like linearity, periodicity, continuity, randomness, and symmetry. They take actions like representing, experimenting, modeling, classifying, visualizing, computing, and proving. Teachers need to learn to ask good mathematical questions, as well as find solutions, and to look at problems from multiple points of view. Most of all, prospective teachers need to learn how to learn mathematics.
One model for meeting the spirit of these guidelines requires math faculty who teach prospective elementary teachers to deliberately include the following features in their courses: