Math 658:  The Psychology of Learning Mathematics, Spring 2008

 

Theresa J. Grant

4427 Everett Tower

(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

 

 

Writing 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 “rings true” 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 “review of research” 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 “practice” preliminary exam (take–home) and will provide a series of questions to which you will respond.  Further details will be provided later.


Tentative Schedule

Jan. 10                 Current Vision of Mathematics Teaching and Learning

                                    National Council of Teachers of Mathematics.  (2000).  Teaching and Learning Principles, Principles and Standards for School Mathematics, pp. 16–21.

                                    Implications of Learning Theories on Teaching

                                    Maher, Carolyn and Davis, Robert.  (1990).  Building Representations of Children’s 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.

                                    Gagné, 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).  Children’s 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).  Children’s 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.

 

 

Feb. 21                 Results of Traditional Instruction/Misconceptions and “Bugs”

                                    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).  Benny’s conception of rules and answers in IPI Mathematics.  Journal of Children’s 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 “well taught” mathematics courses.  Educational Psychologist, 23, 145–166. 

 

 

Feb. 28                 Design and Evaluation of Alternative Instruction

                                    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 students’ 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). “Using knowledge of children's mathematical thinking in classroom teaching:  An experimental study.” 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. 

                                    Additional Readings

                                    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.

 

 

SPRING BREAK

 

 


March 13           Situated Cognition and Problem Solving

                                    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.

                       Additional Readings

                                    Prawat, R. S.  (1997).  Problematizing Dewey’s 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.

 

March 20           Constructivism and Social Aspects of Learning

                                    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.209226. 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.

 

 

March 27 – April 17: 

                           Multiple Perspectives on Mathematics Teaching & Learning

OR

                           Second Handbook of Research on Mathematics Teaching and Learning

 

 

AND

 

                                    Topical Research Study Presentations   (Readings to be assigned by presenters)

 

 


ADDITIONAL READINGS, to help you prepare for the prelim

 

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.

 

Overviews of Theories of Learning

                  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.

 

Implications for Teaching

                  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. & D’Ambrosio, 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 D’Ambrosio.  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.

 

Design of Alternative Instruction

                  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.

                  Gal’perin, 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.)

 

Elementary Grades

                  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 children’s 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.