MATH 5001/7001 Arithmetic and Problem Solving 3 hours

Syllabus as of December 2002

Oasis Title: ARITH & PROB SOLV.

A deep examination of topics in mathematics that are relevant for elementary school teaching. Problem solving. Number systems: whole numbers, integers, rational numbers (fractions) and real numbers (decimals) and the relationships between these systems. Understanding multiplication and division, including why standard computational algorithms work. Properties of arithmetic. Applications of elementary mathematics.

Course Objectives: To strengthen and deepen knowledge and understanding of arithmetic, how it is used to solve a wide variety of problems, and how it leads to algebra. In particular, to strengthen the understanding of and the ability to explain why various procedures from arithmetic work. To strengthen the ability to communicate clearly about mathematics, both orally and in writing. To promote the exploration and explanation of mathematical phenomena. To show that many problems can be solved in a variety of ways.

Topical Outline:

Problem solving: Polya's principles.

Numbers: The natural numbers, the whole numbers, the rational numbers (fractions), and the real numbers (decimals). The decimal system and place value. Percent. Comparing sizes of numbers (whole numbers, fractions, decimals). Rounding.

Addition and subtraction: Why the standard algorithms for adding and subtracting whole numbers work. Adding and subtracting fractions. Percent increase and decrease. The commutative and associative properties of addition and their use in mental arithmetic.

Multiplication: The meaning of multiplication. Ways of showing multiplicative structure: with groups, with arrays, and with tree diagrams. Why multiplication by 10 is easy. The distributive property, the commutative and associative properties of multiplication: their relationships to areas of rectangles and volumes of boxes, and their use in mental arithmetic. Why the standard procedure for multiplying whole numbers works. Why the procedure for multiplying fractions works. Powers. (Optional: scientific notation.) Multiplication of decimals. Multiplication of negative numbers.

Division: The meaning of division. Why dividing by zero is undefined. Why the standard longhand procedure for dividing whole numbers works. The relationship between fractions and division.

Text: Mathematics for Elementary Teachers volume I, preliminary edition, by Sybilla Beckmann, Addison-Wesley, 2003

Chapter 1 Problem Solving, Chapter 2 Numbers, Chapter 3 Addition and Subtraction, Chapter 4 Multiplication, Chapter 5 Division, through section 5.4 only.

Math 7001: for graduate credit, students must complete an additional course project. The project could consist of several essays, or a longer paper, in which the student discusses some aspect of the course material in depth, or in which the student relates the course material to their future teaching (e.g., with a collection of lesson plans or with a discussion of some lesson plans). However, other creative ideas could also be acceptable. For example, students might think of a creative way to tie their course project for math 7001 to something they will be doing for one of their other courses.