Text

Course topics

Course objectives

How your grade will be calculated

Types of assignments and grading criteria

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

Class work and attendance policies

Materials needed

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

Text: Mathematics for Elementary Teachers , volume I, preliminary 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: (Chapter 2; 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 3, sections 3.1, 3.3 - 3.5; 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: (Chapter 4; 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 5, not including section 5.7; 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 6, sections 6.1 - 6.3; 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.

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

Please also see the more detailed grading criteria below.

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

Types of assignments and grading criteria:

You will work on several different types of assignments throughout the semester: don't hand in assignments, checked assignments, and scored assignments.

I encourage you to work on assignments with your classmates. Of course, you should adhere to UGA's Academic Honesty Policy, as described in http://www.uga.edu/ovpi/honesty/main.html. 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.

Don't hand in assignments: Most sections in the text include a number of exercises that have detailed solutions. The exercises will be assigned for you to solve without handing in your work. Read the solutions only after you have seriously attempted to solve the exercises: by grappling with the exercises you will learn much more than if you simply read the solution right away. The exercises will help prepare you to solve the problems, so do not skip them. The solutions to the exercises provide you with many examples of good mathematical explanations: use them as models for your own writing of mathematical explanations. In many cases, there is more than one way to solve an exercise. Therefore your solution need not be identical to the given solution in order to be correct. If in doubt, please check with me or with the teaching assistant.

Checked assignments will give you an opportunity to develop ideas and deepen your thinking without holding you to the polished level of performance that is expected on the scored assignments. Some checked assignments may ask you to solve a problem in the book and some may ask for exploratory writing. A checked assignment will receive a grade of check, of check-minus, or of 0 as follows.

A score of check, which counts as 5 points, will be given to work with the following characteristics:

• The work addresses the problem that was posed and makes significant progress.
• The work is neatly written and is understandable.

A score of check-minus, which counts as 3 points, will be given to work that represents a serious attempt but that fails to meet the standards set for a check.

A score of 0 will be given to work that was not handed in or that does not represent a serious attempt.

A checked assignment may be re-assigned as a scored assignment, or there may be a test question related to the checked assignment. Therefore, you should make your best effort on these assignments, and you should follow class discussions on them closely.

Scored assignments will usually ask you to write polished mathematical explanations of facts or phenomena in elementary mathematics. We will determine your score on such an assignment by the extent to which your work meets the following criteria:

• The work is factually correct, or nearly so, with only minor flaws (for example, a minor mistake in a calculation).
• The work addresses the specific question or problem that was posed. It is focused, detailed, and precise. There are no irrelevant or distracting points.
• The work is clear, convincing, and logical; it has the following characteristics:
  • The work could be used to teach another (college) student, possibly even one who is not in the class.
  • The work could be used to convince a skeptic.
  • The work does not require the reader to make a leap of faith.
  • Key points are emphasized.
  • If applicable, supporting pictures, diagrams, and/or equations are used appropriately and as needed.
  • The work is coherent.
  • Clear, complete sentences are used.

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, Graham Matthews, 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.

Class work: We will frequently work in small groups during class. When you work in a group, please make sure that everyone in your group has a chance to think about the question and has an opportunity to discuss and debate it. At times, this may mean that you should "hold back" a little, at other times you may need to ask your group to wait a moment for you to think about something. Although it can be tempting to listen to someone else's solution before thinking deeply about a problem, the process of grappling actually helps us learn and understand. Therefore please be sensitive to each other and allow everyone time to think.

Attendance is required. Unexcused absences will result in a lowering your course score by .25 for each unexcused absence beyond 1. After 3 or more unexcused absences, your grade may be lowered even further, or you may be dropped from the course. You are responsible for all information and announcements given in class, even if your absence is excused. Please make every effort to arrive in class on time. Late arrivals can be disruptive, and can cause you to miss important material or announcements. Two tardy arrivals will count as one absence.

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