EDUC 572.1
Psychology of Mathematical Thinking (3 units)
Instructor: Noriyuki Inoue, Ph.D.
Email: inoue@sandiego.edu
Office: AW 2-107
Telephone: 619-260-7669
Course Description
Students will learn diverse psychological theories and research on mathematical thinking and discuss the
educational implications from multiple perspectives. Focus is placed on contemporary debates on
mathematical learning in mathematics education that are originated in key psychological theories and
research on learning processes, developmental mechanisms, the role of affective factors, and personal,
social, and cultural constructions of meaning in mathematical learning.
Course Objectives/Candidate Outcomes
Upon completion of this course students will be able to demonstrate:
Academic Excellence & Critical Inquiry and Reflection
 Deep understanding of the major psychological theories and research agenda on mathematical thinking
commonly discussed in the community of mathematics educators and researchers such as learning
processes, cognitive strategies, personal constructions of meaning, epistemological beliefs, self-concept,
culture of the mathematics classroom, and the socio-historic understanding of mathematical learning.
 Knowledge of the important educational implications from the contemporary theories and research on
cognitive, affective, social, and cultural dimensions of mathematical thinking and learning.
 Knowledge of the strengths and limitations of applying psychological theories and research in
mathematics education.
 Knowledge of the strengths and limitations of the research and assessment methodologies commonly
used in the research on mathematical learning.
Community and Service
 Understanding of the socio-cultural roots of mathematical learning.
 Skill and disposition in order to be a constructive participant in the discussion of the psychological
processes involved in mathematical learning and development.
 Understanding of applications of electronic communication forums for purposes of professional growth,
research, and instruction.
Ethics, Values and Diversity
 Knowledge of the research targeted on under-represented students’ approach and attitudes to
mathematical learning.
 Knowledge of how to apply diverse psychological research on mathematical learning to support underrepresented students in mathematics classrooms.
 Disposition to support diverse students’ mathematical learning by considering social, cultural, and
historic factors in teaching and assessing their mathematical learning.
Textbooks/Readings
Reserved readings
These materials are available on the class WebCT (webct.sandiego.edu). Please use your USD email
ID/password for WebCT.
Ball, D. L., Hill, H.C, & Bass, H. (2005). Who knows mathematics well enough to teach third grade, and
how can we decide? American Educator.
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Baroody, A. J. (1989). Manipulatives don't come with guarantees. Arithmetic Teacher, 37.
Bishop, A.J. (1991). Mathematical enculturation: A cultural perspective on mathematics education. Boston,
MA: Kluwer Academic Publications. Chap. 1
Boaler, J. (2006). Urban success: A multidimensional mathematics approach with equitable outcomes. Phi
Delta Kappan. Jan.
Brown, J.S., Collins, A.& Duguid, P. (1989). Situated Cognition and the Culture of Learning. Educational
researcher. pp. 32 -42.
Clements, D. (1996). Concrete manipulatives. Arithmetic Teacher.
Cobb, P. & Yackel, E. (1998) A constructivist perspective on the culture of the mathematics classroom. In
F. Seeger, J. Voigt, & U. Waschescio (Eds.), The culture of the mathematics classroom. Cambridge
University Press.
Gerdes, P. (1997). On culture, geometrical thinking, and mathematics education. In A. B. Powell & M.
Frankenstein (Eds.), Ethnomathematics: Challenging eurocentrism in mathematics education. Albany, NY:
SUNY Press.
Ginsburg, H. P. (1996). Toby's Math. In R. Sternberg & T. Ben-Zeev, The Nature of Mathematical
Thinking. Mahwah, NJ: Lawrence Erlbaum Associates.
Ginsburg, H. P. (1989). Children's Arithmetic: How they learn it and How you teach it. Pro Ed. Chap. 5, 6
Ginsburg, H.P. & Asmussen, K.A. (1988). Hot Mathematics. In G. B. Saxe & M. Gearhart (Eds.),
Children's mathematics: new directions for child development, no. 41. San Francisco: Jossey-Bass.
Ginsburg, H.P., Inoue, N. & Seo, K.H. (1999) Young children doing mathematics: Observation of everyday
activities. In J. V. Copley (Ed.), Mathematics in the Early Years. Reston, VA: National Council of Teachers
of Mathematics.
Inoue (2006?). Qualitative evaluations of the effectiveness of mathematical explanations given by preservice teachers. Manuscript submitted to the Journal of Mathematics Teacher Education.
Inoue, N. (2005). The realistic reasons behind unrealistic solutions: The role of interpretive activity in word
problem solving. Learning and Instruction.
Kohn, A. (1999). The costs of overemphasizing achievement. School Administrator.
Kohn, A. (1994). Grading: The issue is not how but why. Educational Leadership.
Ma, L. (1999). Knowing and teaching elementary mathematics. Laurence Erlbaum Associates. Chap. 4 & 5
National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics.
Reston, VA: NCTM.
Papert, S. (1980). Mindstorms: Children, data processors, and powerful ideas. New York: Basic Books.
Piaget, J. (1972). Comments on mathematical education. In H.E. Gruber & J.J. Voneche (Eds.), Essential
Piaget: An interpretive reference and guide. Northvale, NJ: Jason Aronson, Inc.
Pollak, H. O. (1997). Solving problems in the real world. In L. Steen (Ed.), Why numbers count:
Quantitative literacy for tomorrow's America. New York: College Board.
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Rogoff, B. (1993). Children’s Guided Participation and Participatory Appropriation in Sociocultural
Activity. In R.H. Wozniak& K.W. Fischer (Eds.), Development in context: Acting and Thinking in Specific
Environemtns (pp.121-153). Hillsdale, NJ: Lawrence Erlbaum Associates.
Saxe, G. B. (1988). Candy selling and math learning. Educational Researcher, 14.
Schoenfeld, A. H. (1991). On mathematics as sense-making: An informal attack on the unfortunate divorce
of formal and informal mathematics. In J. F. Voss, D. N. Perkins, & J. W. Segal (Eds.), Informal reasoning
and education. Erlbaum.
Seo, K. (2003). What children's play tells us about teaching mathematics. Young Children.
Shulman, L. S. (1987). Knowledge and teaching: Foundations for the new reform. Harvard Educational
Review.
Siegler, R. S. (1987). The perils of averaging data over strategies: An example from children's addition.
Journal of Experimental Psychology, 116.
Stigler, J., Fernandez, C., & Yoshida, M. (1996). Traditions of school mathematics in Japanese and
American elementary classrooms. In L. P. Steffe, P. Nesher, P. Cobb, G.A. Goldin & B. Greer. (Eds.),
Theories of mathematical learning. Mahwah, NJ: Lawrence Erlbaum Associates.
Uttal, D. H. (1995). Beliefs, motivation, and achievement in mathematics: A cross-national perspective. In
M. Carr (Ed.), Motivation in Mathematics. Cresskill, NJ: Hampton Press, Inc.
Voigt, J. (1996). Negotiation of mathematical meaning in classroom processes: Social interaction and
learning mathematics. In L. P. Steffe, P. Nesher, P. Cobb, G.A. Goldin & B. Greer. (Eds.), Theories of
mathematical learning. Mahwah, NJ: Lawrence Erlbaum Associates.
Zvonkin, A. (1992). Mathematics for little ones. Journal of Mathematical Behavior, 11.
Additional readings may be handed out in class.
Course Requirements/Activities
Class participation/assignments: You are required to actively participate in class discussions, online
activities, project presentations and questions and answer sessions. Your preparation for the class and
active contribution to the class constitutes a essential part of the learning activity in the course.
Occasionally, you will be given assignments. (There will be no make up for in-class assignments. Please
make sure that you attend all the sessions.)
Theoretical presentation: You will sign-up for at least one of the readings with * in the reading list, and
present the research findings/implications/essential ideas discussed in the reading. Your presentation needs
to address key theoretical issues discussed in the article and open up an essential discussion that are
relevant to teaching. A set of guiding questions will be given to you one week prior to your presentation.
The entire presentation is expected to be about 15 minutes followed by 5-10 minutes of Q&A session. The
use of PowerPoint is required.
Clinical interview report: You will sign at for at least one of the interview report projects assigned by
your instructor and conduct a clinical interview of a student on the topic discussed in the class according to
the given protocol, and give a short PowerPoint presentation on the interview in class. Detailed guidelines
will be given in class.
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Mid-term exam: There will be a mid-term exam that is take-home, open-textbook, essay style. You are
expected to give clear and insightful answers to the exam questions based on our class discussions and
readings. Completing each exam would 3 to 5 hours, depending on your progress.
Final project: You will identify a topic of your interest related to the psychology of mathematical thinking
and develop a small pilot study on the topic. First, you will identify your own research question, locate at
least three relevant literature, critically examine what needs to be done to answer your question, design a
small study on the topic, and submit a 3-6 page proposal of your project. After receiving feedback from
your peers/instructor, you will actually conduct the study with at least n=4, analyze the result, and submit
the final paper. More detailed guidelines will be given in class.
(Extra point project: You are encouraged to propose an extra point project of your own. The proposal
must be approved by the instructor before you actually work on the project. Be creative in devising
interesting and substantial projects. The points you can add with the extra point project depend on the
quality of your project. This is not a mandatory assignment.)
Assessment Plan/Grading Criteria/Rubric
The final grade is calculated based on the following criteria:
Class participation/assignments: 10%
Theoretical presentation: 5%
Clinical interview report: 5%
Essay exam: 40% (1st exam 20%, 2nd exam 20%)
Final project: 40% (Proposal 5% , Final paper 35%)
The following table shows the correspondence between letter grades and 100 point scale scores.
Letter grade
A
AB+
B
BC+
C
CD+
D
DF
100 pt score
9490-93
87-89
83-86
80-82
76-79
73-75
69-72
66-68
63-65
60-62
0-59
Equivalent score
96
92
88
85
81
78
74
71
67
64
61
0
Course Outline
Dates
Topics
Reading
Week 1
Introduction, overview of the course
Mathematics and psychology
What is mathematics?
Different paradigms in mathematics education
What's behind the NCTM standards?
NCTM (2000)
Week 2
Developmental theories and research on mathematical thinking I
Piagetian theory and cognitive constructivism
Informal mathematics
Piaget (1972)
Ginsburg et al (1999)
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Meaning making in mathematical learning
Clinical interviews and informal assessments
Week 3
Developmental theories and research on mathematical thinking II Ginsburg (1996)*
Early mathematics
Seo (2003)*
Everyday mathematics
Zvonkin (1992)*
Mathematics in play
Week 4
Children's arithmetic
Symbolic representations
Number facts and basic operations
Clinical interview report I (Informal mathematics)
Ginsburg (1989) Ch.5*
Ginsburg (1989) Ch.6*
Sielger (1987)*
Week 5
Problem solving and sense-making
Role of manipulatives and technology
Geometric thinking
Clinical interview report II (Number facts)
Final project guideline is handed out
Papert (1980)*
Baroody, (1989)*
Schoelfeld (1991)*
Clements (1986)*
Week 6
Developmental theories and research on mathematical thinking III Brown et al (1989)*
Socio-cultural roots of mathematical cognition
Rogoff (1993)*
Socio-cultural constructivism
Saxe (1988)*
Socio-cultural norms of mathematics classrooms
Clinical interview report III (Manipulatives)
Week 7
Real world problem solving
Word problem solving
Motivation and mathematical learning #1
Clinical interview report IV (Socio-cultural norm)
Mid-term exam is handed out
Pollak (1997)*
Greer (2005)*
Kohn (1994) *
Inoue (2005)
Week 8
Motivation and mathematical learning #2
Students' epistemological beliefs about mathematics
Negotiations of meanings
Clinical interview report V (Real world math)
Mid-terms exam due
Ginsburg et al (1988)*
Kohn (1999)*
Voigt (1996)*
Week 9
Comparative studies on mathematical learning
Mathematical learning around the world
Clinical interview report VI (Motivation)
Final project proposal due
Stigler et al (1996)*
Uttal (1995) *
Gerdes (1988) *
Week 10
Teaching and mathematical thinking
Studies on pedagogical content knowledge
What does it really take to promote understanding?
Communications in mathematics classroom
Clinical interview report VII (Culture and mathematics)
Shulman (1987) *
Ball (2005) *
Ma (1999) *
Week 11
Integration
Clinical interview report VIII (Teachers)
Cobb (1998)*
Bishop (1988)*
Boaler (2006)*
Week 12
Library day
Week 13
Final project presentations I
Final Project due
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Inoue (2006)
Week 14
Final project presentations II
* These articles are for theoretical presentations. Please sign up for your presentation in class.
Other information
Cancellation of class
The class might be canceled due to unavoidable reasons. Also, your instructor may be late or cannot come
to class due to unforeseeable reasons. In case no information is given to you and the instructor does not
come to class for 20 minutes, please regard the class was cancelled. If a class was canceled, the assignment
due in the class will be due next time.
Disabilities
If you have a special need that may effect learning activities, please notify the instructor at the beginning of
the course. Special arrangements could be made depending on the need and condition.
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