**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:

- Thoroughly
read the assigned articles, and do additional reading on the topic.
- Consider
the important issues raised by individual articles, and by the set of
articles as a whole.
- Prepare
a list of discussion questions to promote reflection on the assigned
readings, both individually and collectively.
- Prepare
a reference list for your colleagues that includes written summaries of
the readings you have done.
- Meet
with me before class (preferably on Monday) to discuss your plans for your
facilitation of the discussion.

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__

- Read
assigned articles.
- Create
individual entries for every assigned reading in a computerized database
(e.g., EndNote, ProCite).
Each entry must include keywords, summary of the article, and
further detail on select points.
If your software has the capability to download abstracts from
ERIC, and you think this would be useful, you may do so. ERIC abstracts are not a
substitute for your own notes –these different ÒsummariesÓ should
complement (rather than duplicate) each other. I may collect these from time to time (see
below).
- Come
up with at least one good question about the readings (either about a
particular reading, or a question that crosses the readings). This question must be e-mailed to
me by Wednesday at 1 p.m.. I
will then send a list of the questions I receive that afternoon to help us
all prepare for the next dayÕs class.
- Write a one–page
commentary (
__not__summary) on the set of readings taken**as a whole**(__not__individually). The focus should be:**What do you now**[Note: Some weeks I will require that your individual entries on each article be attached.]__know__about how children learn mathematics, based on the evidence/arguments offered by the authors?

**1) ****Learning Theorists,
due Feb. 21 ^{st} **[Choice of Learning
Theorist due Jan. 18

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. 28^{th}]

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 19 ^{th}.**

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.