Text

Course topics

Course objectives

Class work

Assignments

This section of MATH 5020 is part of the Writing Intensive Program

How your grade will be calculated

Materials needed

This document describes a general plan for the course. Changes may be necessary.

Text: Mathematics for Elementary Teachers , first edition, and the accompanying Class Activities manual by Sybilla Beckmann, published by Addison-Wesley. These can be purchased from the UGA bookstore. Please bring the activity manual to class.

Course topics: Numbers: (Chapters 2 and; 3 weeks) The natural numbers, the whole numbers, the rational numbers (fractions), and the real numbers (decimals). The decimal system and place value. Representing decimals with bundled objects. Representing decimals on a number line. Comparing sizes of decimals. Finding decimals in between decimals. Rounding decimals. The meaning of fractions. The importance of the whole associated with a fraction. Improper fractions. Equivalent fractions. Simplest form of a fraction. Fractions as numbers on number lines. Comparing sizes of fractions: by giving them common denominators, by converting to decimals, and by cross-multiplying. Using other reasoning to compare sizes of fractions. Solving fraction problems with the aid of pictures. Percent. Benchmark percentages and their common fraction equivalents. Solving percentage problems with the aid of pictures. Solving percentage problems numerically.

Addition and subtraction: (Chapter 4, sections ; 1.5 weeks) Adding and subtracting fractions. Explaining why we add and subtract fractions the way we do. The importance of the whole when adding and subtracting fractions, especially in story problems. Recognizing and writing story problems for fraction addition and subtraction. Recognizing story problems that are not solved by fraction addition or subtraction. Mixed numbers. Understanding when percentages should and should not be added. Calculating percent increase and decrease with the aid of pictures. Calculating percent increase and decrease numerically. Percent of versus percent increase or decrease.

Multiplication: (Chapters 5 and 6; 4 weeks) The meaning of multiplication. Ways of showing multiplicative structure: with groups, with arrays, and with tree diagrams. Using the meaning of multiplication to explain why various problems can be solved by multiplying. Explaining why multiplication by 10 is easy in the decimal system. Why the commutative and associative properties of multiplication and the distributive properties make sense and how to illustrate them with arrays, areas of rectangles, and volumes of boxes. Using properties of arithmetic in solving arithmetic problems mentally. Writing equations that correspond to a mental method of calculation (to demonstrate the connection between mental arithmetic and algebra). The distributive property and FOIL. Using multiplication to estimate how many. The partial products multiplication algorithm. Using pictures and the distributive property to explain why the standard and partial products procedures for multiplying whole numbers are valid. Explaining why non-standard strategies for multiplying can be correct or incorrect. The meaning of multiplication for fractions. Recognizing and writing story problems for fraction multiplication. Recognizing story problems that are not problems for fraction multiplication. Explaining why the procedure for multiplying fractions works. Powers. Scientific notation. Multiplication of decimals: explaining why the procedure for the placement of the decimal point is valid. Multiplication of negative numbers. Understanding that multiplication does not always ``make bigger.''

Division: (Chapter 7, not including ; 4 weeks) The meaning of division (two interpretations, with or without remainder). Understanding when the answer to a story problem solved by whole number division is best expressed as a decimal, as a mixed number, or as a whole number with a remainder. Why dividing by zero is undefined. The scaffold method of division. Explaining why the scaffold and standard longhand procedure for dividing whole numbers works. Explaining why some non-standard methods of division are valid. The relationship between fractions and division. Calculating decimal representations of fractions. Explaining the relationship among remainder, mixed number, and decimal answers to division problems. Division of fractions and decimals: The meaning of division for fractions. Recognizing and writing story problems for fraction division. Understanding the distinction between dividing by 1/2 and dividing in 1/2. Explaining why the ``invert and multiply'' procedure for dividing fractions is valid. Explaining why the procedure for placement of the decimal point in decimal division problems is valid. Understanding that division does not always ``make smaller.''

Number Theory: (Chapter , 2.5 weeks) definitions of factors and multiples and concrete problems that use and illustrate these concepts, definitions of greatest common factor and least common multiple and concrete problems that use and illustrate these concepts. If time: the Euclidean algorithm. Prime numbers. The Sieve of Eratosthenes for producing lists of prime numbers. The trial division method for determining if a number is prime. Factoring counting numbers into products of prime numbers. If time: The proof that there are infinitely many prime numbers. If time: Irrationality of the square root of 2.

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.

Class work: This class is part of your preparation as a professional. As a professional, you should engage in collegial discussions about professional practice and you should constantly seek to enhance and refine your professional knowledge. To receive a full participation score, your work in class must consistently exhibit several or all of the following:

• interest in mathematical ideas
• interest in different ways of approaching mathematical ideas
• careful listening to different ways of solving a problem
• careful evaluation of proposed methods of solution
• attempts to connect the course material to your experiences with children and teachers at schools
• attempts to connect the course material to your future teaching
• attempts to connect the course material to the Georgia QCCs and to NCTM's Principles and Standards for School Mathematics
• interest in learning with and from others

Types of assignments and grading criteria:

Expect to have an assignment to turn in at every class. I encourage you to work on homework assignments with your classmates. Of course, you must adhere to UGA's Academic Honesty Policy. Therefore, always write your homework up on your own, using your own words to express the ideas you have discussed with others. Do not allow anyone to copy your work. When you discuss assignments with others, all partners should "give and take" ideas.

Late homework will not be accepted. Please consult with me as soon as possible if you are unable to hand in an assignment due to an illness or emergency.

Writing Intensive Program: This section of MATH 5020 is part of the Writing Intensive Program. The Writing Intensive Program is designed to help courses teach the writing process within various disciplines. Although you have taken English courses on writing, and although these courses will help you with all your writing, mathematical writing has its own special features. In mathematics, we seek coherent, logical explanations, in which the desired conclusion is deduced from starting assumptions. Our graduate assistant, Peter Petrov, has been trained by the Writing Intensive Program to help you learn to write good mathematical explanations. By participating in the Writing Intensive Program we have also learned about ways to use writing to deepen your understanding of the course concepts.

How your grade will be calculated:

We will grade all your work on a 5 point scale, and we will assign points as follows:

# of points	description	characteristics
5.25 points	exemplary	work that could serve as a model for other students
5 points	very good	correct work that is careful and thorough
4 points	competent	good, solid work that is largely correct
3 points	basic	work that has merit but also has significant shortcomings
2 points	emerging	work that shows effort but is seriously flawed
0 points	no credit	no work submitted, or no serious effort shown

Grading criteria: We will determine your score on assignments and tests by the extent to which your work meets the following criteria:

• The work is factually correct, or nearly so, with only minor, inconsequential flaws.
• The work addresses the specific question or problem that was posed. It is focused, detailed, and precise. Key points are emphasized. There are no irrelevant or distracting points.
• The work could be used to teach a student: either a child or another college student, whichever is most appropriate.
• The work is clear, convincing, and logical. An explanation should be convincing to a skeptic and should not require the reader to make a leap of faith.
• Clear, complete sentences are used. Mathematical terms and symbols are used correctly. If applicable, supporting pictures, diagrams, and/or equations are used appropriately and as needed.
• The work is coherent.

Your grade will be based on tests, homework, and a comprehensive final exam. I expect to give 2 tests and several announced quizes during the semester. I will calculate your course score using the following percentages.

I expect to assign letter grades as follows.

for scores from	up to	letter grade
4.7	5 or above	A
4	4.7	B
3.5	4	C
2.5	3.5	D
below 2.5		F

Materials needed: Please have a calculator available. Please bring your activity manual to class.