Math 658: The Psychology of Learning Mathematics,
Spring 2008
Theresa J. Grant
(269) 387–3842
terry.grant@wmich.edu
Course Description:
This course focuses on developing an understanding of
what we know about how people think about mathematics and how an understanding
of mathematics develops. The
readings will provide an overview of various theoretical approaches used to
better understand the teaching and learning of mathematics, with a focus on the
K–12 level. The reading and
writing assignments in this course will allow for insight into the existing
evidence accumulated on teaching and learning mathematics and inspire thoughts
for future directions in research.
Course Responsibilities:
The semester grade will be based on your class
participation, including taking on the role of facilitator, and your writing
assignments: weekly
one–pagers and 3 more extensive papers.
Class participation
A weekly assignment for the course is the reading and
contemplation of the assigned readings.
Since the format of the course is that of a discussion seminar, its
success depends on your preparation and careful reading of the assignments, and
then your participation in class discussions. Everyone should enter the classroom ready to discuss the required
material in depth.
Everyone will be asked to co–facilitate
class discussions several times during the semester. To prepare for these sessions you will:
In addition, each person will prepare two
presentations, on the two major written assignments described below. You will prepare handouts for
these presentations that include an annotated reference list. In some cases, you will choose readings
to be assigned.
Weekly Assignments
1) Learning Theorists,
due Feb. 21st [Choice of Learning
Theorist due Jan. 18th]
Each of you will choose a particular scholar
from the field of cognitive science or mathematics education to investigate in
depth. You will then submit a
paper, 7 to 10 pages in length that includes the following:
(a)
Historical background on the scholar—time period in which
the scholar lived and important educational/societal issues of the time that
impacted the scholar;
(b)
Philosophical and/or theoretical beliefs about learning in general
and in mathematics, specifically, and implications on teaching (again in
general and in mathematics). What
support does the scholar offer for those beliefs?
You
should focus your discussion on the credibility of the ideas; whether the
scholar provides evidence to support claims; whether the theory is useful; and
whether the theory Òrings trueÓ based on your own personal experience. You should read original writings by
the scholar as well as those written about the scholar by others in an attempt
to get to know the scholar as well as you are able.
Beyond
submission of the paper, you will also present this information to the class
with a handout that contains a detailed list of references on the scholar and a
summary of what you have read.
This handout should indicate which articles you recommend, and for what
purposes. In preparing for the
presentation, think about your audience – how can you build off of, and
connect to, what they have been reading and discussing in class this semester.
Some
possible learning theorists to study are:
Johann Pestalozzi &
Warren Colburn (together), Jerome Bruner, William Brownell,
H. Van Engen, Jean Piaget, Edward L. Thorndike, Richard Skemp, Alan Schoenfeld,
Robert Davis, Robbie Case, and Paul Cobb.
2) Topical Research on
Learning, due March 27 [Topic due Feb. 28th]
There are a multitude of research studies on the
way students learn mathematics in the context of particular mathematical
topics. It would be impossible to
cover all of these studies individually, so each of you will be required to
choose one topic to investigate in depth.
You will then submit a Òreview of researchÓ paper, 10–15 pages in
length that includes the following:
(a)
summaries of research findings on the topic;
(b)
critique of whether the data are believable and the research
designs appropriate and effective;
(c)
discussion of what the research findings suggest about the way
children learn mathematics and implications for teaching and whether these
ideas are discussed explicitly or implicitly in the articles;
(d)
suggestions for further research on the learning of the topic.
Topics
to choose from include algebra, functions, geometry, proportional reasoning,
rational numbers, data analysis, and whole number computation.
Beyond
submission of the paper, you will also present this information to the class
with a handout that contains a detailed list of references and a summary of
what you have read. This handout should indicate which articles you recommend,
and for what purposes. You should
also choose one or two articles for your classmates to read to prepare for your
presentation. Finally, as you
prepare for your presentation, think about your audience – how can you
build off of, and connect to, what they have been reading and discussing in
class this semester.
3) Synthesis of Learning
Theories and Implications for Teaching, due April 19th.
The
objective of this assignment is to encourage reconsideration and reflection on
the major themes of the course in light of the most current movement in
situated cognition. This
assignment will be in the format of a ÒpracticeÓ preliminary exam
(take–home) and will provide a series of questions to which you will
respond. Further details will be
provided later.
Tentative
Schedule
National
Council of Teachers of Mathematics.
(2000). Teaching and
Learning Principles, Principles and Standards for School Mathematics,
pp. 16–21.
Maher,
Carolyn and Davis, Robert.
(1990). Building
Representations of ChildrenÕs Meaning in Davis, R., Maher, C. & Noddings,
N. (Eds.) Constructivist Views on the Teaching and Learning of Mathematics,
JRME Monograph No. 4, pp. 79–90.
Jan. 17 Historical
Perspectives on Mathematics Learning:
Arithmetic
Thorndike,
E. L. (1920/1970). The psychology of drill in arithmetic: The strength of
bonds. (Selected portion reprinted
in J. K. Bidwell and R. G. Clason (Eds.)
Readings in the history of mathematics education,
pp.465–474). Washington, DC: National Council of Teachers of Mathematics.
McLellan,
J. A., & Dewey, J. (1895/1970). The psychology of number. (Selected portion reprinted in J. K.
Bidwell & R. G. Clason (eds.) Readings
in the history of mathematics education, pp.154–162). Washington, D.
C.: NCTM.
Brownell,
W. A. (1935/1970). Psychological considerations in the learning and the
teaching of arithmetic. In W. D. Reeve (Ed.), The teaching of arithmetic
(Tenth Yearbook of the National Council of Teachers of Mathematics) (pp.
1–31). New York: Columbia University, Teachers College, Bureau of
Publications.
Reprinted in J. K. Bidwell & R. G. Clason (eds.) Readings in the history of mathematics education,
pp.504–530. Washington, D. C.: NCTM
Van Engen, H. (1949). An analysis of meaning in arithmetic. Elementary School Journal, 49, 321–329, 395–400.
Additional
Readings
Resnick, L.B. &
Ford, W.W. (1981) Psychology of Mathematics for Instruction. Hillsdale, NJ: Lawrence Erlbaum
Associates, Inc. [Chapters 1
and 2].
Jan. 24 Historical
Perspectives on Mathematics Learning, continued
Ausubel, D. P. (1968). Educational psychology: A cognitive view. (pp. vi, 83–89, 107–115, 136–138). New York: Holt, Rinehart, and Winston, Inc.
Skinner,
B. F. (1969). The science of learning and the art of teaching. In The technology of teaching (pp.
9–28). Englewood Cliffs, NJ: Prentice–Hall, Inc.
GagnŽ,
R. M. (1965). Learning hierarchies.
In Conditions of learning (pp. 237–257). New York: Holt,
Rinehart, and Winston, Inc.
Additional
Readings
Resnick, L.B. &
Ford, W.W. (1981) Psychology of Mathematics for Instruction. Hillsdale, NJ: Lawrence Erlbaum
Associates, Inc. [Chapters 5
and 6].
Jan. 31 Piaget & Vygotsky
Siegler, Robert (1998). ChildrenÕs Thinking, (Ch 1, 1–16), Ch 2 (21–29; 49–60),
Englewood Cliffs, NJ:
Prentice-Hall.
Piaget,
J. (1964). Development and learning. Journal for Research in Science Teaching,
2, 176–186.
Vygotsky, L. S. (1978). Mind and Society:
The Development of Higher Psychological Processes. M. Cole, V.
John–Steiner, S. Scribner, & E. Souberman (Eds.). Reprints of original writings. Harvard University Press. [pp. 15 – 16, 11 – 14, 79
– 91.]
Davydov,
V. (1995). The influence of L. S. Vygotsky on education theory, research, and
practice. Educational Researcher, 24(3), 12–21.
Feb. 7 Information Processing
Siegler, Robert (1998). ChildrenÕs Thinking, Ch 3 (63–74; 93–95). Englewood Cliffs, NJ: Prentice-Hall.
Case,
R. and Sandieson, R. (1988). A developmental approach to the
identification and teaching of central conceptual structures in mathematics and
science in the middle grades. (pp. 236-259). In Hiebert, J. and Behr, M. (Eds.) Number
Concepts and Operations in the Middle Grades. Reston, VA:
NCTM.
Riley, M. S., Greeno, J. G., &
Heller, J. I. (1983). Development of children's problem-solving ability in
arithmetic. In H. P. Ginsburg (Ed.), The development of mathematical
thinking, (pp. 153–196). New York: Academic Press.
Antell, S. E., & Keating, D. P.
(1983). Perception of numerical invariance in neonates. Child Development, 54,
695–701.
Feb. 14 Early
Arithmetic & Out-of-school knowledge
Saxe, G.B. (1988). Candy selling and math learning. Educational Researcher, 17(6),
14–21.
Carpenter,
T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction
of concepts in grades one through three. Journal for Research in Mathematics
Education, 15(3), 179–202.
Lave, J., Murtaugh, M., & de la
Rocha, O. (1984). The dialectical
construction of arithmetic in grocery shopping. In B. Rogoff & J. Lave (Eds.), Everyday
cognition: Its development in
social context (pp. 67–94).
Cambridge, MA: Harvard
University.
Gelman, R. (1982). Basic numerical
abilities. In R. J. Sternberg (Ed.), Advances in psychology of human
intelligence, (Vol. 1, pp. 181–205). Hillsdale, NJ: Erlbaum.
Hiebert,
J.. (Ed.) (1986). Conceptual and procedural
knowledge: The case of mathematics.
Hillsdale, NJ: Erlbaum. Chapter 1 (pp. 1–27), Chapter 5 (pp.
113–132), and Chapter 7 (pp. 181–198).
Brown,
J. S., & Van Lehn, K. (1982). Towards a generative theory of
"bugs". In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition
and subtraction: A cognitive
perspective, (pp. 117–135). Hillsdale, NJ: Erlbaum.
Erlwanger, S.H. (1973/2004). BennyÕs conception of rules and answers
in IPI Mathematics. Journal of
ChildrenÕs Mathematical Behavior, 1(2), 7-26.
Reprinted in: T.P. Carpenter, J.A.
Dossey, & J.L. Koehler (eds.) Classics in Mathematics Education Research,
pp.49–58. Reston, VA: NCTM.
Schoenfeld, A.H. (1988). When good teaching leads to bad
results: The disasters of Òwell
taughtÓ mathematics courses. Educational
Psychologist, 23, 145–166.
Hiebert,
J. (1992). Reflection and
communication: Cognitive
considerations in school mathematics reform. International Journal of Educational Research, 17(5),
439–456.
Hiebert,
J., & Wearne, D. (1993).
Instructional tasks, classroom discourse, and studentsÕ learning in
second-grade arithmetic classrooms.
American Educational Research Journal, 30(2), 393–425.
Carpenter,
T. P., Fennema, E., Peterson, P.L., Chiang, C., & Loef, M. (1989/2004).
ÒUsing knowledge of children's mathematical thinking in classroom
teaching: An experimental study.Ó American
Educational Research Journal 26: 499-531.
Reprinted in: T.P.
Carpenter, J.A. Dossey, & J.L. Koehler (eds.) Classics in Mathematics
Education Research, pp.135–151. Reston, VA: NCTM.
Collins, A., Brown,
J. S., & Newman, S. E. (1989).
Cognitive apprenticeship:
Teaching the Crafts of Reading, Writing and Mathematics. In L. B. Resnick (ed.) Knowing,
Learning and Instruction: Essays
in Honor of Robert Glaser, 453 – 484. Hillsdale, NJ:
Lawrence Erlbaum Associates.
Cazden,
C. B. (1988) The structure of
lessons and Variations in Lesson Structure. In C. B. Cazden, Classroom Discourse: The Language of Teaching and Learning
(pp. 29–52, & 53–79).
Portsmouth, NH: Heinemann.
Brown,
J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture
of learning. Educational Researcher, 18, 32–42.
Palinscar
Reply.
Donmoyer,
R. (1996). Introduction: This
Issue: A focus on learning. Educational
Researcher, 25(4), 4.
Anderson,
J. R., Reder, L. M., & Simon, H. A. (1996). Situated learning and
education. Educational Researcher, 25(4), 5–11.
Hiebert,
J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., Oliver,
A., & Wearne, D. (1996). Problem solving as a basis for reform in
curriculum and instruction: The
case of mathematics. Educational Researcher, 25(4), 12–21.
Schoenfeld,
A.H. (1983/2004). Beyond the
purely cognitive: Belief systems,
social cognitions, and metacognitions as driving forces in intellectual
performance. Cognitive Science,
7, 329–363.
Reprinted in: T.P. Carpenter, J.A.
Dossey, & J.L. Koehler (eds.) Classics in Mathematics Education Research,
pp.111–133. Reston, VA: NCTM.
Prawat,
R. S. (1997). Problematizing DeweyÕs views of problem
solving: A reply to Hiebert et al.
Educational Researcher, 26(2), 19–21.
Smith,
J. P. (1997). Problems with problematizing
mathematics: A reply to Hiebert et
al. Educational Researcher, 26(2), 22–24.
Davis,
R. B., Maher, C. A., & N. Noddings, N. (Eds.) (1990). Constructivist
views on the teaching and learning of mathematics, Monograph No. 4. Reston, VA: National Council of
Teachers of Mathematics.
[Chapters 1–3, pp. 1–47]
Steffe,
L.P., & Kiernan, T. (1994/2004).
Radical Constructivism and Mathematics Education. Journal for
Research in Mathematics Education, 25, 711–733.
Reprinted in: T.P. Carpenter, J.A.
Dossey, & J.L. Koehler (eds.) Classics in Mathematics Education Research,
pp.69–82. Reston, VA: NCTM.
Cobb,
P., & Yackel, E., (1996/2004).
Constructivist, emergent, and sociocultural perspectives in the context
of developmental research. Educational Psychologist, 31, 175–190.
Reprinted in: T.P. Carpenter, J.A.
Dossey, & J.L. Koehler (eds.) Classics in Mathematics Education Research,
pp.209–226. Reston, VA: NCTM.
Additional
Readings:
Phillips,
D. C. (1995). The good, the bad, and the ugly: The many faces of constructivism. Educational Researcher,
24(7), 5–12.
von
Glaserfeld, E. (1996). Footnotes to "The many faces of constructivism" [Response to The good, the bad, and the
ugly: The many faces of
constructivism]. Educational Researcher, 25(6), 19.
OR
Second
Handbook of Research on Mathematics Teaching and Learning
AND
Topical Research
Study Presentations (Readings to be assigned
by presenters)
Classics in Mathematics Education
Research
Carpenter,
T.P., Dossey, J.A., & Koehler, J.L. (2004). Classics in Mathematics Education Research,
pp.209–226. Reston, VA: NCTM.
Teaching and Learning Mathematics for
Understanding
Hiebert,
J. & Carpenter, T. P.
(1992). Learning and
Teaching with Understanding. In D.
A. Grouws, ed, Handbook of
research on mathematics teaching and learning, p. 65–100. New York, NY: Macmillian Publishing
Company. [Chapter 4]
Eisenhart,
M., Borko, H., Underhill, R., Brown, C., Jones, D., & Agard, P.
(1993). Conceptual knowledge falls
through the cracks: Complexities of learning to teach mathematics for
understanding. Journal for
Research in Mathematics Education, 24(1), 8–40.
Steffe,
L. P., Nesher, P., Cobb, P., Goldin, G. Q. & Greer, B. (1996) Theories
of Mathematical Learning.
Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
Carpenter,
T. P. (1980). Research in Cognitive Development. In R. J. Shumway (Ed.), Research
in mathematics education (pp. 146–206). Reston, VA: The National
Council of Teachers of Mathematics.
Schoenfeld,
A.H. (1987). Cognitive science in
mathematics education: An
overview. In A.H. Schoenfeld
(Ed.), Cognitive Science and Mathematics Education (pp 1–32). Hillsdale, NJ: Erlbaum.
Kieran, C.A. (1994). Doing and seeing things differently: A 25–Year retrospective of mathematics education research on learning. Journal for Research in Mathematics Education, 25(6), 583–607.
Skemp.
R. R. (1987). The psychology of
learning mathematics.
Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
Carpenter,
T. P., Hiebert, J., & Moser, J. M. (1983). The effect of instruction on
children's solutions of addition and subtraction word problems. Educational
Studies in Mathematics, 14, 55–72.
Fennema,
E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S.
B. (1996). A longitudinal study of learning to use children's thinking in
mathematics. Journal for Research in Mathematics Education, 27(4),
403–434.
Wearne,
D., & Hiebert, J. (1988). A
cognitive approach to meaningful mathematics instruction: Testing a local theory using decimal
numbers. Journal for Research
in Mathematics Education, 19, 371–384.
Constructivism
Simon,
M. A. (1995). Reconstructing mathematics pedagogy
from a constructivist perspective. Journal For Research in Mathematics Education, 26(2),
114–145.
Steffe, L. P. & Kieren, T. (1994). Radical constructivism and mathematics
education. Journal for Research in Mathematics Education, 25(6),
711–733.
Steffe,
L. P. & DÕAmbrosio, B. S.
(1995). Toward a working
model of constructivist teaching: A reaction to Simon. Journal For Research in Mathematics
Education, 26(2), 146–159.
Simon,
M. A. (1995). Elaborating models of mathematics
teaching: A response to Steffe and DÕAmbrosio. Journal For Research in Mathematics Education, 26(2),
160–164.
Kamii,
C. (1985). The importance of social interaction, Young children reinvent
arithmetic, (pp. 26–70). New York: Teachers College Press.
Cobb,
P. (1994). Where is the mind?
Constructivist and sociocultural perspectives on mathematical
development. American Educational Research Journal, 23(7), 13–20.
Discourse
Cobb,
P., Wood, T., Yackel, E., & McNeal, L. (1992). Characteristics of classroom mathematics traditions: An interactional analysis. American Educational Research
Journal, 29(3), 573–604.
Forman,
E. (1996). Learning mathematics as
participation in classroom practice:
Implications of sociocultural theory for educational reform. In L. P. Steffe, P. Nesher, P. Cobb, G.
A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp.
115–129). Mahwah, NJ: Lawrence Erlbaum Associates.
El'konin,
D. B. (1975). Primary schoolchildren's intellectual capabilities and the
content of instruction. In L. P. Steffe (Ed.), Soviet studies in the
psychology of learning and teaching mathematics, (Vol. 7, pp. 13–54).
Palo Alto, CA: Stanford University, School Mathematics Study Group.
GalÕperin,
P.Y., & Georgiev, L.S. (1969).
The formation of elementary mathematical notions. In J. Kilpatrick & I. Wirszup
(Eds.), Soviet Studies in the Psychology of Learning and Teaching
Mathematics (Vol. 1, pp. 189–216). Reston, VA: The
Council.
Lampert,
M. (1986). Knowing, doing, and
teaching multiplication. Cognition
and Instruction, 3, 305–342.
Cobb,
P. (1990). Multiple perspectives. In L. Steffe & T. Wood (Eds.), Transforming
children's mathematics education, (pp. 200–215). Hillsdale, NJ:
Erlbaum.
Kilpatrick,
J. (1985). Reflection and
recursion. Educational Studies in Mathematics, 16, 1–26.
Problem Solving, Overviews:
Silver,
E. (Ed.) (1985). Teaching and Learning Mathematical
Problem Solving: Multiple Research
Perspectives. Hillsdale,
NJ: Lawrence Erlbaum Associates.
Lester,
F. (1994.) Musings about mathematical problem–solving
research: 1970–1994. Journal for Research in Mathematics
Education, 25 (6), pp. 660–675.
(Both
sources list research studies in references.)
Hiebert,
J., & Carpenter, T. P. (1982). Piagetian tasks as readiness measures in
math instruction: A critical
review. Educational Studies in Mathematics, 13, 329–345.
Carpenter,
T.P., Ansell, E., Franke, M.L., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: A study of kindergarten childrenÕs
problem–solving processes. Journal
for Research in Mathematics Education, 24(5), 428–441.
The Middle Grades & Beyond
Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.), Number Concepts and Operations in the Middle Grades (pp.211–228). Reston, VA: The Council.
Foltz,
C., Overton, W. F., & Ricco, R. B.
(1995). Proof construction:
Adolescent development from inductive to deductive problem–solving
strategies. Journal of
Experimental Child Psychology, 59 , 179–195.
Hart,
K. (1988). Ratio and proportion. In J. Hiebert & M. Behr (Eds.), Number
concepts and operations in the middle grades, (pp. 198–219). Reston,
VA: National Council of Teachers of Mathematics.
Mack,
N. K. (1993). Learning rational numbers with
understanding: The case of informal knowledge. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.) Rational
numbers: An integration of research (pp. 85–105). Hillsdale, NJ: Erlbaum.
Wearne,
D., & Hiebert, J. (1988). A cognitive approach to meaningful mathematics
instruction: Testing a local
theory using decimal numbers. Journal for Research in Mathematics Education,
19, 371–384.
Hiebert,
J. & Wearne, D. (1986). Procedures over concepts: The
acquisition of decimal and number knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge:
The case of mathematics (pp. 199–223). Hillsdale, NJ: Erlbaum.
Herscovics,
N. (1989). Cognitive obstacles
encountered in the learning of algebra.
In S. Wagner & C. Kieran (Eds.), Research issues in the learning
and teaching of algebra (pp. 60–86). Reston, VA: The
Council.
Ohlsson,
S. (1988). Mathematical meaning and applicational meaning in the semantics of
fractions and related concepts. In J. Hiebert & M. Behr (Eds.), Number
concepts and operations in the middle grades, (pp. 53–92). Reston,
VA: National Council of Teachers of Mathematics.
The High School Level & Beyond
Matz,
M. (1980). Towards a computational theory of algebraic competence. Journal
of Mathematical Behavior, 3(1), 93–166.
Linchevski,
L. & Herscovics, N. (1996). Crossing
the Cognitive Gap between Arithmetic and Algebra: Operating on the Unknown in the Context of Equations. Educational Studies in Mathematics,
30(1), 39–65.
Tall,
D. & Thomas, M. (1991). Encouraging Versatile Thinking in
Algebra Using the Computer. Educational
Studies in Mathematics, 22(2), 125–147.
Resnick,
L. B., Cauzinille–Marmeche, E., & Mathieu, J. (1987). Understanding
algebra. In J. Sloboda & D. Rogers (Eds.), Cognitive processes in
mathematics, (pp. 169–225). Oxford: Clarendon Press.
Tall,
David. (1990). Inconsistencies in the Learning of
Calculus and Analysis. Focus on
Learning Problems in Mathematics, 12(3–4), 49–63.
Heid,
M. K. (1988). Resequencing skills
and concepts in applied calculus using the computer as a tool. Journal for Research in Mathematics
Education, 19(1), 3–25.
White,
P., & Mitchelmore, M. (1996).
Conceptual knowledge in introductory calculus. Journal for Research
in Mathematics Education, 27(1), 79–95.
Batanero,
C., Estepa, A., Godino, J., & Green, D. R. (1996). Intuitive strategies and preconceptions
about association in contingency tables. Journal for Research in Mathematics
Education, 27(2), 151–169.