# Mathematics

## Spring, 2004

Course:  Math 255, Mathematics for Elementary and Middle School Teachers, 4  credits

Instructor:  Dr. Larry Krajewski

MC 526

e-mailllkrajewski@viterbo.edu

Phone:  796-3658  [office]

782-1648 [home] (call before 10 p.m. please)

Hours:   10 MWF, 12F & by appointment

Prerequisites:  Grade of C or better in Math 155

TextMathematics for Elementary Teachers, 2nd ed., by Tom Bassarear, Houghton Mifflin, 2000.

(This text is used in Math 355 also)

Final Exam:                      May 6 (Thursday), 9:50 – 11:50 a.m.

### Description

This course is designed to introduce the preservice K-9 teacher with ideas, techniques and approaches to teaching mathematics. Emphasis is on problem solving and problem posing, use of manipulatives and children’s literature, and understanding children’s thinking. The math content areas are the arithmetic operations and number theory.

The Viterbo College Teacher Education Program has adopted a Teacher As Reflective Decision Maker Model and the Wisconsin Standards For Teacher Development and Licensure also known as INTASC (Interstate New Teacher Assessment and Support Consortium) Standards. Each course is designed to contribute to the development of one or more of the INTASC Standards

These standards can be found at

http://www.dpi.state.wi.us/dpi/standards/index.html

Wisconsin Standards for Teacher Development and Licensure (INTASC)

Standards 1,2,3,4,6,8,9

A, B, F

Wisconsin Content Guidelines

1,2,3,4,5,6,7,8,9,10

Viterbo Core Abilities

Thinking, Communication, Cultural Sensitivity

The following materials are on reserve in the Todd Wehr Library:

• Sample Portfolio by Jennie Schoonover
• Math Activity Packets from previous classes
• Sample solutions to investigations

### Goals (INTASC 1)

To help students:

1.             learn to value mathematics;

2.             learn to reason mathematically;

3.             learn to communicate mathematically;

4.             become confident in their mathematical ability; and

5.             become problem solvers and posers.

### Objectives

Upon successful completion of this course, the student will be able to:

• explore, conjecture, reason logically and use a variety of mathematics  methods effectively to solve nonroutine problems. (INTASC 1)
• establish classroom environments so that his or her students can explore, conjecture, reason logically and use a variety of mathematics methods (INTASC 2, 3, 4, 6)
• effectively to solve nonroutine problems and develop a lifelong appreciation  of math in his or her life; (INTASC 1)
• become familiar with educational research on effective teaching of  mathematics. (INTASC 9)

### Student Responsibilities

One cannot benefit from or contribute to a class discussion or activity unless one is physically present (this a necessary condition, not a sufficient one). Attendance is required. Call me (796-3658) if you will  not be in class. A valid excuse is necessary to miss class. Unexcused absences may lower your grade for the course.

Assigned readings of the texts and handouts need to be done if meaningful discussion can occur.

As teachers you should appreciate the importance of class participation. Your active participation makes the course go. Math is not a spectator sport. Assigned problems and textbook exercises are ways for you to develop problem solving skills and reflect on your learning. Do the problems when they are assigned.

### Content

I. Foundations for Learning Mathematics

A. Getting Comfortable with Mathematics

B. Problem Solving and Reasoning

C. Patterns

D. Communication

E. Connections

F. Putting It All Together

II. Fundamental concepts

A. Sets

B. Functions

C. Numeration

III. The Four Operations of Arithmetic

B. Multiplication and Division

C. Mental Arithmetic and Estimation

IV. Number Theory

A. Divisibility and Related Concepts

B. Primes and Composites

C. Greatest Common Factor and Least Common Multiple

V. Extending the Number System - Integers <if time permits>

A. Operations With Integers

### Requirements

• Two math activities, one for K-5, the other for grades 6-8. (see handout)
• A problem-solving portfolio representing your work during the semester. (see handout)
• Learning journal from class (see handout)
• Three exams
• Four summaries of articles in professional journals on the following topics (see handout):

Problem Solving , Addition or Subtraction of Whole Numbers, Multiplication or Division of

Whole Numbers, Number Theory.

[Some good sources are Arithmetic Teacher, Teaching Children Mathematics, Mathematics

Teaching in the Middle School, AIMS, and School Science and Mathematics.]

### Evaluation

tests                                                                                         600                                        A: 90% - 100%

Portfolio (including 3 investigations)                                 200                                        B: 80% - 89%

Learning journal                                                                       80                                        C: 70% - 79%

Readings                                                                                  100                                       D: 60% - 69%

Math activities                                                                         20

total                                       1000

### A Note to You

This is a course for prospective elementary and middle school teachers about how children learn mathematics and how to create positive, developmentally- appropriate mathematics instruction. Since we have all gone to elementary school, we have learned and "know" the mathematics which will be addressed in this course; in fact, we may know it so well that we have forgotten what it was like to ever not know it. Or, because of inadequate past instruction, we may feel we "know" certain mathematical topics but have never really understood them, or we may even dislike the subject. If this is the case, consider this course an opportunity to break the cycle of negative, disempowering mathematics teaching. Mathematics can be an intellectual adventure, a powerful tool, and a creative experience for children.  As a teacher, you can make it so.

Some of you may have had mathematics courses that were based on the transmission, or absorption, view of teaching and learning. In this view, students passively "absorb" mathematical structures invented by others and recorded in texts or known by authoritative adults. Teaching consists of transmitting sets of established facts, skills, and concepts to students. I do not accept this view. I am a constructivist. Constructivists believe that knowledge is actively created or invented by the person, not passively received from the environment. No one true reality exists, only individual interpretations of the world. These interpretations are shaped by experience and social interactions. Thus, learning mathematics should be thought of as a process, of adapting to and organizing one's quantitative world, not discovering preexisting ideas imposed by others. It is one way to make sense of the world.

Consequently, I have three goals when I teach. The first is to help you develop mathematical structures that are more complex, abstract, and powerful than the ones you currently possess so that you will be capable of solving a wide variety of meaningful problems. The second is to help you become autonomous and self- motivated in your mathematical activities. You will not "get" mathematics from me but from your own explorations, thinking, reflecting, and participation in discussions. As independent students you will see your responsibility is to make sense of, and communicate about, mathematics. Charles Schultz, creator of "Peanuts", compared people to multispeed bikes and noted that "most of us have gears we do not use." Hopefully you will see mathematics as an open-ended, creative activity and not a rigid collection of recipes. And the last is to help you become a skeptical student who looks for evidence, example, counterexample and proof, not simply because school exercises demand it, but because of an internalized compulsion to know and to understand.

I want to help you learn to do something different from and better than what you have experienced as pupils in previous mathematics classes A mathematics methods class is about mathematics, about children as learners of mathematics, about how mathematics can be learned and taught, and about how classrooms can be environments for learning mathematics. It's a class where the students learn about learning mathematics while they themselves are learning mathematics.

As a teacher I have come to realize that when I teach mathematics I teach not only the underlying mathematical structures but I am also teaching my students how to develop their cognition, how to see the world through a set of quantitative lenses which I believe provide a powerful way of making sense of the world, how to reflect on those lenses to create more and more powerful lenses and how to appreciate the role these lenses play in the development of their understanding.

So I ask your help in establishing a mathematical community where one uses logic and mathematical evidence as verification rather than the teacher, where mathematical reasoning replaces the memorization of procedures, and where conjecturing, inventing, and problem solving are encouraged and supported.

You may find this experience frustrating at times. Persevere!  Eventually I hope you will own personally the mathematical ideas you once knew unthinkingly or only peripherally (and sometimes anxiously). I want you to become competent and confident using mathematical ideas and techniques. I want you to be ready to learn how to get other persons actively involved in problem solving. To nurture a mathematical idea in the mind of a child might be easier if it first thrived in the mind of the child's teacher.

In training a child to activity of thought, above all things we must beware of what I will call "inert ideas" - that is to say, ideas that are merely received into the mind without being utilized, or tested, or thrown into fresh combinations . . . Education with inert ideas is not only useless: it is, above all things, harmful. Except at rare intervals of intellectual ferment, education in the past has been radically infected with inert ideas . . . Let us now ask how in our system of education we are to guard against this mental dryrot. We enunciate two educational commandments, "Do not teach too many subjects," and again, "What you teach, teach thoroughly." . . . Let the main ideas which are introduced into a child's education be few and important, and let them be thrown into every combination possible. The child should make them his own, and should understand their application here and now in the circumstances of his actual life. From the very beginning of his education, the child should experience the joy of discovery. (Alfred North Whitehead, The Aims of Education)

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Americans with Disabilities Act

If you are a person with a disability and require any auxiliary or other accommodations for this class,

please see me and Wayne Wojciechowski, the Americans With Disabilities Act Coordinator (MC 320  <796- 3085>) within ten days to discuss your accommodation needs.

It is somewhat surprising and discouraging how little attention has been paid to the intimate nature of teaching and school learning in the debates on education that have raged over the past decade. These debates have been so focused on performance and standards that they have mostly overlooked the means by which teachers and pupils alike go about their business in real-life classrooms  - how teachers teach and how pupils learn.

Jerome Bruner