index

MATH 300-07 -Mathematics Pedagogy
Fall, 2004
University of Connecticut
Neag School of Education
College of Liberal Arts & Sciences

Time: Monday 4:00-5:30, MSB 118
Meeting Dates : August 8, September 8,15,22,29, October 13,20,27, November 10, 17
Professors Tom DeFranco, Jean McGivney-Burelle, Chuck Vinsonhaler
Text: A Handbook for Mathematics Teaching Assistants, by Thomas W. Rishel

Chuck Vinsonhaler
Office: MSB 316
Phone: 486-3944
E-mail: vinsonhaler@math.uconn.edu
Web Page: http://www.math.uconn.edu/~vinsonhaler/
Office Hours: MWF 10, and by appointment

Thomas C. DeFranco, Ph.D. defranco@uconnvm.uconn.edu
486-3815

Jean McGivney-Burelle, Ph.D. mcgivney@uconnvm.uconn.edu
486-0288

Erin's tips for new TA's

Course Description
This course will provide students a foundation on theories of teaching and learning mathematics along with an understanding of current research trends influencing mathematics education. Special emphasis will be given to the NCTM Standards, the MAA Recommendations on College Teaching and current instructional techniques and strategies and ways to implement them in the college mathematics curriculum.

Objectives
a) Students will gain an understanding of how the mathematics curriculum has developed and evolved from an historical and psychological perspective and ways in which various learning theories/theorists have impacted and influenced the teaching of mathematics.

b) Students will be exposed to models of teaching and examine the role of domain knowledge, epistemology, pedagogy, curriculum, and assessment in teaching.

c) Students will explore ways to enhance the teaching and learning of mathematics by understanding and applying learning theories impacting the field of mathematics education

d) Students will examine innovative ways to teach mathematics at the college level as outlined by NCTM and the MAA .

Evaluation

Reflective Journal                   35%
Critique and Response            35%
Class Participation                  30%

Assignments

Journal Entries
Journals should provide an opportunity for students to reflect on and analyze their experiences during this course. You will be required to submit two (2) journal entries. Each entry (3-4 pages) should illustrate your ability to reflect on and analyze the assigned readings or the activities encountered in class or both. In addition, your writing should reflect ideas that you believe to be significant in your own teaching, assessing or learning of mathematical problem solving. Each journal should be typed and you will be graded on your ability to reflect on the situation--NOT on your ability to describe the situation.

Critique and Response
Working in pairs each student will be required to make a classroom observation and write a critique of their partners teaching. The critique should be constructive and based on the principles of teaching discussed in class. The partner being critiqued should then reflect on the review and write a response to the critique. The response should address each point raised in the critique and ways you would change your teaching based on the critique. The critique should be typed and handed in as part of your final grade. Be prepared to discuss your response to the critique of your own teaching.

Class Participation
Be prepared to lead a discussion of one of the assigned readings. In addition each student should be prepared to discuss and comment on the readings assigned for each class.

Date                    Topics
8/25 Participant Introductions
         Principles Of Good Practice For Teaching
         Activity-Teacher Knowledge Base

9/18 Learning Theory
Activity- A Private Universe
Activity- historical Perspectives on the Psychology of Mathematics for Instruction

10/2 Domain Knowledge
      Procedural and Conceptual Knowledge
     Pedagogy
      Large Group / Small Group Instruction
      Cooperative Learning
     Curriculum (MAA)
     Activity-Create a Course Syllabus
      Activity – Commuter Problem

10/16 Assessment and Student Presentations
     Altemative Assessment
     Scoring Rubric
     Writing Assignments
     Activity-Grading Exams, Grading Group Projects

10/23 Reflection and Student Presentations
Activity-Student Presentations
Revisit POGPFT

ADDITIONAL MATERIAL

What is most important for new TA's to learn? (Generated by a group activity)
1. Communication skills
2. Adequate Preparation
3. How to be clear about goals and expectations
4. Balancing teaching and your own studies
5. How to give and receive feedback, learn from mistakes.

Results of TA's classroom observation of each other.

Things done well - items most frequently mentioned.
1. Good eye contact, clear speaking
2. Good use of board
3. Quick review to begin class
4. Well organized, good examples
5. Experiments with small groups and other non-lecture methods

Things needing work - items most frequently mentioned
1. More eye contact, speak louder and more clearly
2. Respond more directly and encouragingly to questions
3. Involve students in discussion
4. Write all important information on the board, carefully

The Basics: PRACTICE
C. Vinsonhaler
Fall, 2000

Prepare
Reflect
Active Learning
Communicate
Talk to good Teachers
Individuals
Communicate
Experiment

Prepare. This is one thing EVERYONE can do, and is arguably the most important component of good teaching practice.

Decide what is important and emphasize it
Think about where students will get hung up (Reflect!)
Know how to do homework problems
Think about props, examples, activities that will bring the material to life (Communicate!)

Reflect. One of the sayings attributed to Mother Theresa is the following: “I have so much to do today, I’m going to spend two hours in prayer instead of one.” This is not to say that prayer is the only thing that can save your teaching. But you will benefit by taking time to think about what you are trying to do and what the best way to do it is.

Before class you can incorporate reflection into your preparation, as above.
Right after class is an excellent time to ask yourself how you could have improved what went on.

Active Learning. As a successful math student, you probably enjoy listening to a well-done lecture on an interesting bit of mathematics. But you probably know that you don’t “learn” the material in that way. To be able to use the mathematics and call it your friend, you have to engage it by working through the proofs yourself, and solving problems involving the results. The same is true of your students. No matter how brilliant your lecturing style, the students won’t learn unless you coerce (force?) them to engage the material.

You must assign and grade homework, or at least give quizzes on the homework.
You should encourage student participation during classroom time in any way you can.
Calling on a student by name is much more effective than a generic Are there any questions? “Pat, what bothers you about this equation.”

Communicate. As someone who has devoted considerable time and energy to the study of problem solving, I’m confident in asserting that communication is the world’s biggest problem.

Talk loudly
Write large, write often, write neatly. Leave important material on a board at the side.
STOP – Single Thought One Person. Look one individual in the eye as you make a point, then move on. Too much eye contact can make a listener nervous.
Repeat yourself. Emphasize the important points by making them again, in as many different ways as possible.

Talk to good Teachers. One can read countless (countable?) books and papers about teaching. One can get bored doing this, although some learning does occur. A more interactive way to engage the problem of good teaching is to talk to good teachers. If you do this, you will find that:

Good teachers care about their students;
Good teachers are happy to discuss teaching, happy to have you visit their classes;
Good teachers are good teachers – you will learn good teaching from them.

Individuals. Ralph Waldo Emerson said “The secret to education is respecting the pupil.”

Learn your students’ names: (1) make notes on your class list; (2) be in class 5 minutes early and talk to them
Automathography. During the first week, have your students write a 1-2 page letter telling you about their math backgrounds, some personal details that distinguish them as individuals, and their goals and expectations for the course.

Communicate. Remember that I said communication is the world’s biggest problem? Remember that I said to repeat yourself? There are so many aspects to good communication in the classroom that a second visit to this topic is mandatory.

Goals, expectations, clear assignments
Feedback – grade homework and tests promptly.
Encourage all answers – we learn more from our mistakes than our successes.

Experiment. You are going to make mistakes, so you might as well learn something and have some fun doing it. Try new ideas once in a while. Tell the class when you do – they can provide helpful feedback. If you read or hear about a pedagogical device that interests you, such as using small groups in the classroom, give it a whirl. At worst, you will “waste” fifteen minutes. At best, you might turn your class from a moribund mass of note-takers into a galvanized group of active learners.

Another view of teaching mathematics can be found at:
www.math.uiuc.edu/~reznick/ciu.html

References

[1] Case, B. A., (Ed.) You're the Professor, What Next? Ideas and Resources for Preparing College Teachers, The Committee on Preparation for College Teaching, MAA Notes No. 35, 1994.

[2] Davis, R.B., C.A. Maher, & N.Noddings (eds.), (1990). Constructivist views on the teaching and learning of mathematics. Reston, VA:NCTM.

[3] Eble, K. E. The Craft of Teaching: A Guide to Mastering the Professor's Art, Second Edition. San Francisco: Jossey-Bass, 1988.

[4] Grouws, D.A. (ed.).(1992). Handbook of research on mathematics teaching and learning. Macmillan Publishing Co., NY

[5] Kaput, J. J. and Dubinsky, E. (Eds.) Research Issues in Undergraduate Mathematics Learning: Preliminary Analyses and Results. MAA, Notes, No. 33, 1994.

[6] Krantz, S. G. How To Teach Mathematics: A personal perspective. AMS, 1993.

[7] National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA.

[8] National Council of Teachers of Mathematics (1991). Professional Standards for t the Teaching of Mathematics Reston, VA.

[9] National Council of Teachers of Mathematics (1995). Assessment Standards for School Mathematics. Reston, VA.

[10 ] National Research Council. Everybody counts: A Report to the Nation on the Future of Mathematics Education Washington, DC: National Research Council, 1989.

[11] National Research Council. Moving Beyond Myths: Revitalizing Undergraduate Mathematics. Washington, DC: National Research Council, 1991.

[12] Polya, G. How To Solve It. Princeton University Press

[13] Resnick, L.B. & Ford, W.W. (1981). The Psychology of Mathematics for 1
Instruction. Lawrence Erlbaum Associates. Hillsdale, NJ.

[14] Schoenfeld, A. (Ed.). A Source Book For College Mathematics Teaching, MAA,
1990.

[15] Schoenfeld, A.H. (1985). Mathematical problem solving. Academic Press, Inc.

[16] Sterrett, A. (Ed.) Using IVriting To Teach Mathematics. MAA, Notes, No. 16,
1992.

Articles

[17] Bailey, M.A. (1923). The Thorndike Philosophy of Teaching the Processes and Principles of Arithmetic, Tlze Mathematics Teacher, The National Council of Teachers of Mathematics, NY, 16(3) 129-140.

[18] Becker, J. R. and Pence, B. J. The Teaclung and Learning of College mathematics: Current Status and Future Directions. Research Issues in Undergraduate Mathematics reaming: Preliminary Analyses and Results, J. J. Kaput and E. Dubinsky (Eds.), MAA Notes No. 33, 1994.

[19] Britton, G. L. Journals and Essay Examinations in Undergraduate Mathematics. Using Writing To Teach Mathematics, Andrew Sterrett, (Ed.), MAA, Notes No. 16, 1992. pp.104-106.

[20] Broder, G.M. (1986). Constructivism: A theory of knowledge. Journal of chemical education, 63(10), 873-878.

[21] Brown, J.S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1),3242.

[22] Brownell, W.A. (1935). Psychological Considerations in the Learning and the Teaching of Arithmetic The Teaching of Arithmetic, The National Council of Teachers of Mathematics, Tenth Yearbook, Teachers College, Columbia Unive: (pp. 1-31).

[23] Bruner, J.S. (1966). Notes on a theory of instruction. Toward a theory of instruction, (pp. 39-72).

[24] Chickering, A. W. and Gamson, Z. F. Seven principles for good practice in undergraduate education. AAHE Bulletin, Mar., 1987, pp. 3-7.

[25] Confrey, J. (1990). What constructivism implies for teaching. In R.B. Davis, C.A. Maher, & N.Noddings (eds.), Constructivist views on the teaching and learning of mathematics. Reston, VA:NCTM, 107-122.

[26] Crocker, D. A. Constructivism and Mathematics Education. You're the Professor, What Next? Ideas and Resources for Preparing College Teachers, The Conanittee on Preparation for College Teaching, Bettye Anne Case, (Ed.), MAA Notes No. 35, 1994. pp. B-III-64-67.

[27] Dubinsky, E. Pedagogical Change in Undergraduate Mathematics Education. You're the Professor, What Next? Ideas and Resources for Preparing College Teachers, The Committee on Preparation for College Teaching, Bettye Anne Case, (Ed.), MAA Notes No. 35, 1994. pp. 114-119.

[28] Foley, G. D. Let Us Teach Exploration. You're the Professor, What Next? Ideas and Resources for Preparing College Teachers, The Conunittee on Preparation for College Teaching, Bettye Anne Case, (Ed.), MAA Notes No. 35, 1994. pp. B-III-61-63.

[29] Gagne, R.M. (1962). The acquistion of knowledge. Psychological Review, 69(4), 355-365.

[30] Gass, F. Working in Small Groups. You're the Professor, What Next? Ideas and Resources for Preparing College Teachers, The Committee on Preparation for College Teaching, Bettye Anne Case, (Ed.), MAA Notes No. 35, 1994. pp. B-III-80.

[31] Goldin, G. (1990). Epistemology, constructivism, and discovery learning in mathematics. In R. B. Davis, C.A. Maher, & N.Noddings (eds.) , Constructivist views on the teaching and learning of mathematics. Reston, VA:NCTM.

[32] Greeno, J.G. (1989). A perspective on thinking. American Psychologist, 44(2), 134 141.

[33] Hiebert, J. & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.) Conceptual and procedural knowledge: The case of mathematics. (pp. 1-27) Hillsdale, NJ:LawrenceErlbaum Associates.
[34] Hiebert, J., and Carpenter, T.P., (1992) Learning and teaching with understanding. In D.A. Grouws (Ed.) Handbook of research on mathematics teaching and learning. (pp.65-97) Macmillan Publishing Co., NY

[35] Kaput, J.J. (1992). Technology and mathematics. In D.A. Grouws (Ed.) Handbook of research on mathematics teaching and learning. (pp. 515-556) Macmillan Publishing Co., NY

[36] Kenney, E. A., A Reply to Quesnons from Mathematics Colleagues on Writing Across t/ze Curriculum. Using Writing To Teach Mathematics, Andrew Sterrett, (Ed.), MAA, Notes No. 16, 1992. pp.17-21.

[37] Kilpatrick, J. (1985). Reflection and recursion. Educatzonal Studies in Mathematics 16(1), 1-26.

[38] Kilpatrick, J. (1987). What constructivism might be in mathematics education. In J.C. Bergeron, N. Herscovics, & C. Kieran (eds.), Psychology of mathematics education. Montreal: Proceedings of the Eleventh International Conference, 3-27.

[39] Leder, G.C. (1992). Mathematics and gender:Changing perspectives. In D.A. Grouws (Ed.) Handbook of research on mathematics teaching and learning. (pp.597-622) Macmillan Publishing Co., NY

[40] Mcleod, M.B. (1992).Research on affect in mathematics education:A reconceptualization. In D.A. Grouws (Ed.) Handbook of research on mathe, teaching and learning. (pp.575-596) Macmillan Publishing Co., NY

[41] MOSAIC, (1992). The science of learning math and science, 23(2), 37-43.

[42] Noddings,N. (1990). Constructivism in mathematics education. In R.B. Davis C.A. Maher, & N.Noddings (eds.), Constructivist views orz the teaching and learning of malhematics. Reston, VA:NCTM, 7-18.

[43] Price, J. J., Learning Mathematics Through Writing: Some Guidelines. College Mathematics Journal. Nov. 1989.

[44] Rishel, T. W. Writing in the Math Classroom; Math in the Writing Classo or, How I Spent My Summer Vacation. Using Writing To Teach Mathematics, Andrew Sterrett, (Ed.), MAA, Notes No. 16, 1992. pp.30-33.

[45] Schoenfeld, A.H. (1992). Learning to think mathematically:Problem solving, metacognition, and sense making in mathematics. In D.A. Grouws (Ed.) Handbook of research on mathematics teaching and learning. (pp.334-370) Macmillan Publishing Co., NY

[46] Selden, A. and Selden, J. Constructivism in Mathematics Education: A View of How People Learrz. You're the Professor, What Next? Ideas and Resources for Preparing College Teachers, T1ze Comminee on Preparation for College Teaching, Bettye Anne Case, (Ed.), MAA Notes No. 35, 1994. pp. B-III-63-64.

[47] Shulman, L (1987). Knowledge and Teaching: Foundations of the New Reform. Harvard Educational Review. Vol. 57, No. 1, 1-22.

[48] Sipka, T. Wriang in Matize~natics: A Pletizora of Possibilities. Using Writing To Teach Mathematics, Andrew Sterrett, (Ed.), MAA, Notes No. 16, 1992. pp.11 - 14.

[49] Talman, L. A. Weekly Journal Entries-An Effective Too for Teaching Mathenzatics. Using Writing To Teach Mathematics, Andrew Sterrett, (Ed.), MAA, Notes No. 16, 1992. pp.107- 110.

[50] Thurston, W. P. Mathematical Education. You're the Professor, What Next? Ideas and Resources for Preparing College Teachers, The Committee on Preparation for College Teaching, Bettye Anne Case, (Ed.), MAA Notes No. 35, 1994. pp. B-III-2-8.

[51] von Glasersfeld, E. (1991). Introduction. In E. von Glasersfeld (ed.), Radical constructivism in mathematics education. Dordrecht:Kluwer, xiii-xx.

[52] von Glasersfeld, E. (1990). An Exposition of Constructivism: Why Some Like It Radical. In R.B . Davis, C.A . Maher, & N. Noddings (eds.) , Constructivist views on the teaching and learning of mathematics. Reston, VA:NCTM, 19-29.

[53] Wahlberg, M. Lecturing at the "Bored ". American Mathematics Monthly. JuneJuly, 1997, pp. 551-556.

[54] Weissglass, J. Small group learning. American Mathematics Monthly. Aug.-Sept., 1993, pp. 662-668.