Mathematics

Math 255: Mathematics for Elementary and Middle School Teachers

Fall, 1999

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

Instructor Dr. Larry Krajewski
             MC 526
             e-mail: lkrajewski@centurytel.net

             Phone: 796-3658 [office]
                    782-1648 [home] (call before 10 p.m. please)
             Hours: 3M, 10:30-11:30W, 10F & by appointment

Prerequisites: Grade of C or better in Math 155

Text Mathematics for Elementary Teachers, by Tom Bassarear, Houghton Mifflin, 1997.

Final Exam 14 December (Tuesday), 7:40-9:40 a.m. (1:10 class)
             15 December, 9:50-11:50 a.m. (2:10 class)

---

Description

This course is designed to introduce the preservice K-9 teacher with ideas, techniques and approaches to teaching mathematics. Manipulatives, children's literature, problem solving, diagnosis and remediation, estimation, mental mathematics, equity issues, and the uses of the calculator are interwoven throughout the topics presented. 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. Each course is designed to contribute to the development of one or more of the knowledge bases in professional education.

This course contributes to the development of the knowledge bases: Knowledge of the Learner, Curriculum Design, Planning and Evaluation, and Instructional and Classroom Management.

---

Resources
You may qualify for free tutoring in the Learning Center.

Methodology
Lecture, class discussion, small group work, student presentations.

Todd Wehr Library The following books are on reserve:

      Solutions Manual for text

     Selected Bibliography for Gender Equity in Mathematics and

     Technology Resources Published in 1990-1996 ,Women & Mathematics Education

---

Goals

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:

---

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.

---

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

       A. Addition and Subtraction

       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

Six summaries of articles on the following topics (include a copy of the article in your summary; article must be at least two pages long in original citation.) Format is in the packet.

Problem Solving <due September 10>

Addition <due September 24>

Subtraction <due October 8>

Multiplication <due October 29>

Division <due November 12>

Number Theory <due December 3>

[Some good sources are Arithmetic Teacher, Teaching Children Mathematics, Mathematics Teaching in the Middle School, and School Science and Mathematics.]

A problem notebook with assigned problems from the text and class.

Two math activities, one for grades K-5, the other for grades 6-8. <due December 3>

A problem-solving portfolio representing your work during the semester.

Learning journal from class

exams

---

Evaluation

Points

Class journal 5

Problem notebook 10

Readings 3

Portfolio 30

Math activities 2

tests 50

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. 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.

---

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.

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)

---

BIBLIOGRAPHY