Viterbo University - La Crosse, WI

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, assessment, , equity issues, and the uses of the calculator are interwoven throughout the topics presented. The math content are geometry and rational numbers.

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.

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
Selected Bibliography on Gender Equity in Mathematics
Various books on problem solving

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

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 effectively to solve nonroutine problems and develop a lifelong appreciation of math in their lives; (INTASC 2,3,4,6)
become familiar with educational research on effective teaching of mathematics. (INTASC 9)
design and use several forms of assessment such as portfolios, journals, open-ended problems, tests, and projects (INTASC 8)

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. Geometry

A. Spatial Reasoning

B. Van Hiele levels

C. Two-dimensional geometry

D. Three-dimensional geometry

E. Translations, reflections and rotations

F. Symmetry

II. Measurement

A. Length

B. Area

C. Volume

III. Rational Numbers

A. Models

B. Ordering

C. Renaming

D. Addition and subtraction

E. Multiplication and division

F. Decimals

Requirements

Two math activities, one on geometry and one on fractions
Four investigations
Learning journal from class
Three exams
A problem notebook with assigned problems from the text and class. You must work out the

solutions.

Six summaries of articles in professional journals on the following topics:

Geometry <due September 5>

Assessment <due September 26>

Technology <due October 10>

Measurement <due October 24>

Fractions <due November 14>

Equity <due December 5>

[Some good sources are Arithmetic Teacher, Teaching Children Mathematics, Mathematics Teaching in the Middle School, AIMS, and School Science and Mathematics.] (see handout)

Completion of a minimum of 12 hours of field experience working with an elementary student on mathematics
A journal of your sessions with your elementary student [NOTE: YOU MUST MEET WITH YOUR STUDENT AND FULFILL THIS REQUIREMENT IN ORDER TO PASS THIS COURSE.]

Evaluation

Points

Tests 300

Investigations 180

Student journal 60

Problem notebook & math activities 30

Learning journal 18

Readings 12

total 600

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)

BIBLIOGRAPHY

Baratta-Lorton, Mary, Mathematics Their Way Addison Wesley, 1976.

Biggs, Edith E. and James R. MacLean, Freedom to Learn, Addison Wesley, 1969.

Bresser, Rusty and Carn Holtzman, Developing Number Sense, Math Solutions, 1999.

Brown, Stephen I.and Marion I. Walter, The Art of Problem Posing, 2nd ed., LawrenceErlbaum, 1990.

Burns, Marilyn, About Teaching Mathematics, Math Solutions, 2000.

_____. Math By All Means: Place Value, Grade 2, Math Solutions, 1994.

_____ A Collection of Math Lessons, Grades 6 through 8, Math Solutions, 1990

_____ MATH - Facing an American Phobia, Math Solutions, 1998.

A Call for Change, Recommendations for the Mathematical Preparation of Teachers of Mathematics,

Mathematical Association of America, 1991.

Chapin, Suzanne H. and Art Johnson, Math Matters.Grades K-6. Understanding the Mathematics You

Teach, Math Solutions, 2000.

Connolly, Paul and Teresa Vilardi (editors), Writing to Learn Mathematics and Science, Teacher's College

Press, 1989.

Countryman, Joan, Writing to Learn Mathematics, Heinemann, 1992.

Crosswhite, F. Joe and Robert Reys, Organizing for Mathematics Instruction, 1972 yearbook, NCTM,

1972.

Dahlke, Richard and Roger Verhey, What Expert Teachers Say About Teaching Mathematics, Grades K-8,

Dale Seymour Publications, 1986.

Davidson, Neil (editor), Cooperative Learning in Mathematics: A Handbook for Teachers, Addison-

Wesley, 1990.

Davis, Robert B., Learning Mathematics, Ablex, 1984.

Developing Mathematical Reasoning in Grades K-12, 1999 Yearbook, NCTM, 1999.

Driscoll, Mark and Here Confrey (editors), Teaching Mathematics: Strategies that Work, Northeast

Regional Exchange, Inc. 1985.

Fennema, Elizabeth and Thomas Carpenter, Cognitively Guided Instruction, Program Implementation

Guide, WisconsinCenterfor Educational Research, 1990.

Grouws, Douglas A. et.al., (editors), Perspectives on Research for Effective Mathematics Teaching,

NCTM, 1988.

Hiebert, James et al, Making Sense - Teaching and Learning Mathematics with Understanding, Heinemann,

1997.

Johnson, David R., Every Minute Counts, Dale Seymour Publications, 1982.

_____________ . Making Minutes Count Even More, Dale Seymour Publications, 1986.

Kohn, Alfie, The Schools Our Children Deserve, Houghton Mifflin, 1999.

_____, What to Look for in a Classroom, Josey Bass, 1998.

Krulik, Stephen and Robert E. Reys, Problem Solving in School Mathematics, 1980 yearbook, NCTM, 1980.

Kuhs, Therese M., Measure for Measure - Using Portfolios in K-8 Mathematics, Heinemann, 1997.

Kulm, Gerald (editor), Assessing Higher Order Thinking in Mathematics, American Association for the

Advancement of Science, 1990.

Leutzinger, Larry, ed., Mathematics in the Middle, NCTM, 1998.

Ma, Liping, Knowing and Teaching Elementary Mathematics, LawrenceErlbaum, 1999.

Math Talk, The Mathematical Association (U.K.), Heinemann, 1987.

Mills, Heidi et al, Mathematics in the Making - Authoring Ideas in Primary Classrooms, Heinemann, 1996

O'Brien, Thomas C., Toward the 21st Century in Mathematics Education, Teacher Center Project, Southern

IllinoisUniversityat Edwardsville, 1982.

Ohanian, Susan, One Size Fits Few, Heinemann, 1999.

Ohanian, Susan, Day By Day Math - Activities for Grades 3 - 6, Math Solutions, 2000.

Paulos, John Allen, Innumeracy, Hill and Wang, 1988.

Polonsky, Lydiaet al, Math for the Very Young, John Wiley, 1995.

Polya, G., Mathematical Discovery Vols I and II, John Wiley, 1965.

Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, 2000.

Raphel, Annette, Math Homework that Counts - Grades 4 - 6, Math Solutions, 2000.

Resnick, Lauren and Leipold Klopfer (editors), Toward the Thinking Curriculum: Current Cognitive

Research, 1989 ASCD Yearbook, Association for Supervision and Curriculum Development,

1989.

Schoen, Harold L. and Marilyn J. Zweng, Estimation and Mental Computation, 1986 Yearbook, NCTM,

1986.

Schoenfeld, Alan H. Problem Solving in the Mathematics Curriculum, MAA Notes Numbered,

Mathematical Association of America, 1983.

_________________. Cognitive Science and Mathematics Education, LawrenceErlbaum Associates, 1987.

_________________. Mathematical Problem Solving, Academic Press, 1985.

Silver, Edward A., et.al., Thinking Through Mathematics, The College Board, 1990.

Skinner, Penny, It All Adds Up! Engaging 8-to-12-year-olds in Mathematical Investigations, Math

Solutions, 1999.

Steen, Lynn Arthur and Donald J. Albers (editors), Teaching Teachers, Teaching Students, Birkhauser,

1981.

Stenmark, Jean Kerr, et.al., Assessment Alternatives in Mathematics Project EQUALS, 1989.

___________. Family Math, Lawrence Hall of Science, 1986.

Sterrett, Andrew (editor), Using Writing to Teach Mathematics, MAA Notes Number 16, Mathematical Association of America.

Stigler, James W. and James Hiebert, The Teaching Gap, Free Press, 1999.

Suydam, Marilyn N. and Robert E. Reys, Developing Computational Skills, 1978 Yearbook, NCTM, 1978.

The Teaching and Learning of Algorithms in School Mathematics, 1998 Yearbook, NCTM, 1998.

Teaching and Learning Mathematics in the 1990s, 1990 Yearbook, NCTM, 1990.

Trafton, Paul R. and Albert P. Shulte, New Directions for Elementary School Mathematics, 1989 yearbook,

NCTM, 1989.

Tsuruda, Gary, Putting It Together, Heinemann, 1994.

Wisconsin Model Academic Standards in Mathematics, Wisconsin DPI, 1998.

Zaslavsky, Claudia, The Multicultural Math Classroom, Heinemann, 1996.

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