# Mathematics Education

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## Book Description

Mathematics education is one of the most publicized and contested fields of endeavour in the area of education more generally. The entrails of international comparative mathematics achievement surveys are pored over by the media, politicians and educators alike; and, while for the last fifty years at least it has been assumed by most everyone in modern societies that mathematics should be a compulsory subject in all schools, parents and scholars alike argue furiously about whether traditional teaching and rote practising of mathematical skills is better or worse for pupils than conceptual teaching based on children’s own constructed ideas. University mathematics professors tend either to deplore the dropping of standards in their students, and thus the dropping of standards in teachers, or heartily embrace the new learning techniques made possible through careful use of the new technologies.

As academic thinking about and around mathematics education continues to flourish and develop, this new title in the Routledge series, Major Themes in Education, meets the need for an authoritative reference work to make sense of the subject’s vast literature and the continuing explosion in research output. Edited by Alan Bishop, a prominent scholar in the field, this Routledge Major Work is a four-volume collection of foundational and cutting-edge contributions that cover all of the major themes in mathematics education.

The first of the four volumes (‘Mathematics, Mathematics Education, and the Curriculum’) brings together key work on the goals of mathematics education, as well as vital material on the relationship of the curriculum with numeracy, assessment, technology, and the place of marginalized students. The second volume (‘Mathematics Teaching and Teachers’) gathers the most important thinking on topics such as pedagogical practices; mathematics teachers’ beliefs, attitudes, and values; professional development; teacher education; and teachers as researchers. The third volume covers the central theories of ‘Mathematics Learning and Learners’. The final volume in the collection (‘The Contexts of Mathematics Education’) gathers vital material from the rich body of literature that explores the social, cultural and political contexts in which mathematics education sits.

With comprehensive introductions to each volume, newly written by the editor, which place the collected material in its historical and intellectual context, this Routledge Major Work is an essential work of reference. It is destined to be valued by specialists in mathematics education and scholars working in related areas—as well as by educational policy-makers and professionals—as a vital one-stop research tool.

## Table of Contents

PROVISIONAL CONTENTS

Volume I: Mathematics, mathematics education, and the curriculum

Part 1: Mathematics and Mathematics Education

(a) Histories of Mathematics

1. Luis Radford, ‘On Psychology, Historical Epistemology and the Teaching of Mathematics: Towards a Socio-Cultural History of Mathematics’, *For the Learning of Mathematics*, 1997, 17, 1, 26–33.

2. G. G. Joseph, ‘Different Ways of Knowing: Contrasting Styles of Argument in Indian and Greek Mathematical Traditions’, in P. Ernest (ed.), *Mathematics, Education and Philosophy: An International Perspective* (Falmer Press, 1994), pp. 194–204.

3. N. Kleiner and N. Movshovitz-Hadar, ‘The Role of Paradoxes in the Evolution of Mathematics’, *The American Mathematical Monthly*, 1994, 101, 10, 963–74.

(b) Conceptions of Mathematics from an Educational Standpoint

4. Tommy Dreyfus and Theodore Eisenberg, ‘On the Aesthetics of Mathematical Thought’, *For the Learning of Mathematics*, 1986, 6, 1.

5. Efraim Fischbein, ‘Intuition and Proof’, *For the Learning of Mathematics*, 1982, 3, 2.

6. S. MacLane, ‘Mathematical Models: A Sketch for the Philosophy of Mathematics’, *American Mathematical Monthly*, 1981, 88, 7.

7. S. Restivo, ‘The Social Life of Mathematics’, in S. Restivo, J. P. Van Bendegem, and R. Fischer (eds.), *Math Worlds: Philosophical and Social Studies of Mathematics and Mathematics Education* (State University of New York Press, 1993).

(c) Culture, Mathematics, and Mathematics Education

8. Ubiratan D’Ambrosio, ‘Ethnomathematics and its Place in the History and Pedagogy of Mathematics’, *For the Learning of Mathematics*, 1985, 5, 1, 44–8.

9. B. Barton, ‘Making Sense of Ethnomathematics: Ethnomathematics is Making Sense’, *ESM*, 1996, 31, 210–33.

10. A. J. Bishop, ‘Western Mathematics: The Secret Weapon of Cultural Imperialism’, *Race & Class*, 1990, 32, 2, 51–65.

(d) Society, Technology, and Mathematics Education

11. Paulus Gerdes, ‘Conditions and Strategies for Emancipatory Mathematics Education in Underdeveloped Countries’, *For the Learning of Mathematics*, 1985, 5, 1.

12. C. Keitel, ‘Numeracy and Scientific and Technological Literacy’, in E. W. Jenkins (ed.), *Innovations in Science and Technology Education*, Vol. 6 (UNESCO, 1997), pp. 165–85.

13. W. Blum and M. Niss, ‘Applied Mathematical Problem Solving, Modelling, Applications, and Links to Other Subjects: State, Trends, and Issues in Mathematics Education’, *Educational Studies in Mathematics*, 1991, 22, 37–68.

Part 2: Education and the Mathematics Curriculum

(e) Goals of Mathematics Education

14. R. B. Davis, ‘The Culture of Mathematics and the Culture of Schools’, *Journal of Mathematical Behavior*, 1989, 8, 143–60.

15. T. A. Romberg and J. J. Kaput, ‘Mathematics Worth Teaching, Mathematics Worth Understanding’, in E. Fennema and T. A. Romberg (eds.), *Mathematics Classrooms that Promote Understanding* (Lawrence Erlbaum Associates, 1999), pp. 3–19.

16. P. Davis and R. Hersh, ‘The Ideal Mathematician’, *The Mathematical Experience* (Penguin, 1980), pp. 34–44.

(f) Mathematics Curricula in Schools

17. D. Robitaille and M. Dirks, ‘Models for the Mathematics Curriculum’, *For the Learning of Mathematics*, 1982, 2, 3, 3–21.

18. J. Confrey, ‘Conceptual Change Analysis: Implications for Mathematics and Curriculum’, *Curriculum Inquiry*, 1981, 11, 3, 243–57.

19. L. Streefland, ‘The Design of a Mathematics Course: A Theoretical Reflection’, *Educational Studies in Mathematics*, 1993, 25, 109–35.

(g) Mathematics Curricula at Tertiary and Vocational Levels

20. M. Harris, ‘Looking for the Maths in Work’, in Harris (ed.), *Schools, Mathematics and Work* (Falmer Press, 1991), pp. 132–44.

21. D. O. Tall, ‘Comments on the Difficulty and Validity of Various Approaches to the Calculus’, *For the Learning of Mathematics*, 1981, 2, 2, 16–21.

22. B. Cornu, ‘Limits’, in D. O.Tall (ed.), *Advanced Mathematical Thinking* (Kluwer, 1992), pp. 153–66.

(h) Assessment, Evaluation, and the Mathematics Curriculum

23. B. Cooper, ‘Authentic Testing in Mathematics? The Boundary Between Everyday and Mathematical Knowledge in National Curriculum Testing in English Schools’, *Assessment in Education*, 1994, 1, 143–66.

24. J. Kilpatrick, ‘The Chain and the Arrow: From the History of Mathematics Assessment’, in M. Niss (ed.), *Investigations in Assessment in Mathematics Education* (Kluwer, 1992), pp. 31–46.

Volume II: Mathematics teachers and teaching

Part 1: Mathematics Teachers

(a) Mathematics Teachers’ Knowledge

25. D. L. Ball, S. T. Lubienski, and D. S. Mewborn, ‘Research on Teaching Mathematics: The Unsolved Problem of Teachers’ Mathematical Knowledge’, in V. Richardson (ed.), *Handbook of Research on Teaching* (American Educational Research Association, 2001), pp. 433–56.

26. H. Freudenthal, ‘Should a Mathematics Teacher Know Something about the History of Mathematics?’, *For the Learning of Mathematics*, 1981, 2, 1, 30–3.

(b) Mathematics Teachers’ Beliefs, Attitudes, and Values

27. A. G. Thompson, ‘The Relationship of Teacher’s Conceptions of Mathematics Teaching to Instructional Practice’, *Educational Studies in Mathematics*, 1984, 15, 105–27.

28. C. Chin, Y.-C. Leu, and F.-L. Lin, ‘Pedagogical Values, Mathematics Teaching, and Teacher Education: Case Studies of Two Experienced Teachers’, in F.-L. Lin and Thomas J. Cooney (eds.), *Making Sense of Mathematics Teacher Education *(Kluwer Academic, 2001). pp. 247–69.

(c) Pre-Service Mathematics Teacher Education

29. T. J. Cooney, B. E. Shealy, and B. Arvold, ‘Conceptualizing Belief Structures of Preservice Secondary Mathematics Teachers’, *Journal for Research in Mathematics Education*, 1998, 29, 306–33.

30. J. Hiebert, A. K. Morris, and G. Glass, ‘Learning to Learn to Teach: An "Experiment" Model for Teaching and Teacher Preparation in Mathematics’, *Journal of Mathematics Teacher Education*, 2003, 6, 210–22.

(d) Mathematics Teachers’ Professional Development

31. D. Clarke, ‘Ten Key Principles from Research for the Professional Development of Mathematics Teachers’, in D. B. Aichele and A. F. Coxford (eds.), *Professional Development for Teachers of Mathematics* (National Council of Teachers of Mathematics, 1994), pp. 37–48.

32. C. Laborde, ‘The Use of New Technologies as a Vehicle for Restructuring Teachers’ Mathematics’, in F.-L. Lin and T. J. Cooney, *Making Sense of Mathematics Teacher Education* (Kluwer Academic Publishers, 2001), pp. 87–109.

Part 2: Teaching Mathematics

(e) Issues in Teaching Mathematical Topics

33. J. Boaler, ‘Learning from Teaching: Exploring the Relationship Between Reform Curriculum and Equity’, *Journal for Research in Mathematics Education*, 2002, 13, 4, 239–58.

34. Z. Markovits, B.-S. Eylon, and M. Bruckheimer, ‘Functions Today and Yesterday’, *For the Learning of Mathematics*, 1986, 6, 2, 18–24.

35. J. Gregg, ‘The Tensions and Contradictions of the School Mathematics Tradition’, *Journal for Research in Mathematics Education*, 1995, 26, 5, 442–66.

(f) Pedagogical Theories and Practices

36. P. Cobb et al., ‘Characteristics of Classroom Mathematics Traditions: An Interactional Analysis’, *American Educational Research Journal*, 1992, 28, 573–604.

37. S. Crespo, ‘Learning to Pose Mathematical Problems: Exploring Changes in Pre-Service Teachers’ Practices’, *Educational Studies in Mathematics*, 2003, 52, 243–70.

(g) Classroom Cultures and Interactions

38. H. Bauersfeld, ‘Hidden Dimensions in the Reality of a Mathematics Classroom’, *Educational Studies in Mathematics*, 1980, 11, 23–41.

39. I. M. Christiansen, ‘When Negotiation of Meaning is also Negotiation of Task: Analysis of the Communication in an Applied Mathematics High School Course’, *Educational Studies in Mathematics*, 1997, 34, 1, 1–25.

(h) Teaching and Assessing

40. D. J. Clarke, ‘The Interactive Monitoring of Children’s Learning of Mathematics’, *For the Learning of Mathematics*, 1987, 7, 1, 2–6.

41. A. G. Thompson and D. J. Briers, ‘Assessing Students’ Learning to Inform Teaching: The Message in the NCTM Evaluation Standards’, *Arithmetic Teacher*, 1989, 37, 4, 22–6.

42. P. J. Black and D. Wiliam, ‘Inside the Black Box: Raising Standards through Classroom Assessment’, *Phi Delta Kappan*, 1998, 80, 2, 139–48.

(i) Teachers as Researchers

43. S. Lerman, ‘The Role of Research in the Practice of Mathematics Education’, *For the Learning of Mathematics*, 1990, 10, 2, 25–8.

44. B. Jaworski, ‘Mathematics Teacher Research: Process, Practice and the Development of Teaching’, *Journal of Mathematics Teacher Education*, 1998, 1, 3–31.

45. B. Clarke, D. Clarke, and P. Sullivan, ‘The Mathematics Teacher and Curriculum Development’, in A. Bishop et al. (eds.), *International Handbook of Mathematics Education* (Kluwer Academic Publishers, 1996), pp. 1207–34).

Volume III: Mathematics Learners and Learning

Part 1: Mathematics Learners

(a) School Learners

46. S. H. Erlwanger, ‘Benny’s Conception of Rules and Answers in IPI Mathematics’, *Journal of Children’s Mathematical Behavior*, 1973, 1, 2, 7–26.

47. N. Gorgorio, N. Planas, and X. Vilella, ‘The Cultural Conflict in the Mathematics Classroom: Overcoming its "Invisibility"’, in A. Ahmed, H. Williams, and J. M. Kraemer (eds.), *Cultural Diversity in Mathematics (Education)* (Horwood Publishing, 2000), pp. 179–85.

(b) Adult Learners

48. D. Coben, ‘Mathematics or Common Sense? Researching "Invisible" Mathematics Through Adults’ Mathematics Life Histories’, in D. Coben, J. O’Donaghue, and G. E. FitzSimons (eds.), *Perspectives on Adults Learning Mathematics: Research and Practice* (Kluwer Academic Publishers, 2000), pp. 53–66.

49. C. Hoyles, R. Noss, and S. Pozzi, ‘Proportional Reasoning in Nursing Practice’, *Journal for Research in Mathematics Education*, 2001, 32, 1, 4–27.

(c) Disadvantaged and Marginalized Learners

50. T. N. Carraher, D. W. Carraher, and A. D. Schliemann, ‘Mathematics in the Streets and in Schools’, *British Journal of Developmental Psychology*, 1985, 3, 21–9.

51. S. Zeleke, ‘Learning Disabilities in Mathematics: A Review of the Issues and Children’s Performance across Mathematical Texts’, *Focus on Learning Problems in Mathematics*, 2004, 26, 4, 1–14.

(d) Gifted Learners

52. K. Tirri, ‘How Finland Meets the Needs of Gifted and Talented Pupils’, *High Ability Studies*, 1997, 8, 2, 213–22.

53. D. Buerk, ‘An Experience with Some Able Women who Avoid Mathematics’, *For the Learning of Mathematics*, 1982, 3, 2, 19–23.

(e) Gender Issues

54. M. Walshaw, ‘A Foucauldian Gaze on Gender Research: What Do You Do When Confronted with the Tunnel at the End of the Light?’, *Journal for Research in Mathematics Education*, 2001, 32, 5, 471–92.

55. G. C. Leder and H. J. Forgasz, ‘Single-Sex Classes in a Co-Educational High School: Highlighting Parents’ Perspectives’, *Mathematics Education Research Journal*, 1997, 9, 3, 274–91.

(f) Cultural Issues

56. A. Chronaki, ‘Researching the School Mathematics Culture of "Others"’, in P. Valero and R. Zevenbergen (eds.), *Researching the Socio-Political Dimensions of Mathematics Education: Issues of Power in Theory and Methodology* (Kluwer Academic Publishers), pp. 145–65.

57. G. De Abreu, ‘Understanding How Children Experience the Relationship Between Home and School Mathematics’, *Mind, Culture and Activity*, 1995, 2, 119–42.

Part 2: Learning Mathematics

(g) Issues in Learning Mathematical Topics

58. C. Kieran, ‘Concepts Associated with the Equality Symbol’, *Educational Studies in Mathematics*, 1981, 12, 317–26.

59. N. Movshovitz-Hadar, ‘The False Coin Problem, Mathematical Induction and Knowledge Fragility’, *Journal of Mathematical Behaviour*, 1993, 12, 253–68.

60. G. Vergnaud, ‘Multiplicative Conceptual Field: What and Why?’, in G. Harel and J. Confrey (eds.), *The Development of Multiplicative Reasoning in the Learning of Mathematics* (SUNY Press, 1994), pp. 41–59.

61. J. Adler, ‘A Language of Teaching Dilemmas: Unlocking the Complex Multilingual Secondary Mathematics Classroom’, *For the Learning of Mathematics*, 1998, 18, 1, 24–33.

(h) Theories of Learning Mathematics

62. S. Pirie and T. Kieren, ‘A Recursive Theory of Mathematical Understanding’, *For the Learning of Mathematics*, 1989, 9, 3, 7–11.

63. A. Sfard, ‘On Two Metaphors for Learning and the Dangers of Choosing Just One’, *Educational Researcher*, 1998, 27, 2, 4–13.

64. R. Skemp, ‘Relational and Instrumental Understanding’, *Mathematics Teaching*, 1976, 77, 20–6.

65. L. P. Steffe and T. E. Kieren, ‘Radical Constructivism and Mathematics Education’, *Journal for Research in Mathematics Education*, 1994, 25, 711–33.

(i) Language, Visualization, and Mathematics Learning

66. N. Presmeg, ‘Visualisation in High School Mathematics’, *For the Learning of Mathematics*, 1986, 6, 3, 42–6.

67. M. Setati et al., ‘Incomplete Journeys: Code-Switching and Other Language Practices in Mathematics, Science and English Language Classrooms in South Africa’, *Language and Education*, 2002, 16, 2, 128–49.

(j) Beliefs and Affective Aspects of Learning Mathematics

68. M. Lampert, ‘When the Problem is Not the Question and the Solution is not the Answer: Mathematical Knowing and Teaching’, *American Educational Research Journal*, 2001, 27, 29–63.

69. G. A. Goldin, ‘Affect, Meta-Affect, and Mathematical Belief Structures’, in G. C. Leder, E. Pehkonen, and G. Törner (eds.), *Beliefs: A Hidden Variable in Mathematics Education* (Kluwer, 2002), pp. 59–72.

70. D. B. McLeod, ‘Affective Issues in Mathematical Problem Solving: Some Theoretical Considerations’, *Journal for Research in Mathematics Education*, 1988, 19, 2, 134–41.

71. D. Moreira, ‘Facing Exclusion: The Student as Person’, in P. Gates and T. Cotton (eds.), *First International Mathematics Education and Society Conference 6th–11th September* (Nottingham University, Center for the Study of Mathematics Education, 1988), pp. 253–61.

Volume IV: The Contexts of Mathematics Education

Part 1: Societal and Cultural Contexts

(a) Parental and Community Aspects

72. R. Merttens, ‘Teaching Not Learning: Listening to Parents and Empowering Students’, *For the Learning of Mathematics*, 1995, 15, 3, 2–9.

73. M. Civil, ‘Culture and Mathematics: A Community Approach’, *Journal of Intercultural Studies*, 2002, 23, 2, 133–48.

(b) Numeracies and Mathematics Education

74. R. Noss, ‘New Numeracies for a Technological Culture’, *For the Learning of Mathematics*, 1998, 18, 2, 2–12.

75. R. Zevenbergen, ‘Technologizing Numeracy: Intergenerational Differences in Working Mathematically in New Times’, *Educational Studies in Mathematics*, 2004, 56, 97–117.

(c) Technologies and Mathematics Education

76. S. Schuck and G. Foley, ‘Viewing Mathematics in New Ways: Can Electronic Learning Communities Assist?’, *Mathematics Teacher Education and Development*, 1999, 1, 1, 23–37.

77. W. Dörfler, ‘Computer Use and Views of the Mind’, in C. Keitel and K. Ruthven (eds.), *Learning from Computers: Mathematics Education and Technology* (Springer-Verlag, 1993), pp. 159–86.

(d) International Comparisons of Mathematics Achievement

78. C. Keitel and J. Kilpatrick, ‘The Rationality and Irrationality of International Comparative Studies’, in G. Kaiser, E. Luna, and I. Huntley (eds.), *International Comparisons in Mathematics Education* (Falmer Press, 1999), pp. 241–56.

79. F. K. S. Leung, ‘The Mathematics Classroom in Beijing, Hong Kong and London’, *Educational Studies in Mathematics*, 1995, 29, 4, 297–325.

80. D. Zhang, S. Li, and R. Tang, ‘The "Two Basics": Mathematics Teaching and Learning in Mainland China’, in L. Fan et al., *How Chinese Learn Mathematics* (World Scientific, 2004), pp. 189–201.

Part 2: Research and Theoretical Contexts

(e) Developments in Research Approaches

81. H. Ginsburg, ‘The Clinical Interview in Psychological Research on Mathematical Thinking: Aims, Rationales, Techniques’, *For the Learning of Mathematics*, 1981, 1, 3, 4–11.

82. M. A. Eisenhart, ‘The Ethnographic Research Tradition and Mathematics Education Research’, *Journal for Research in Mathematics Education*, 1988, 19, 2, 99–114.

(f) Histories of Mathematics Education

83. G. M. A. Stanic, ‘The Growing Crisis in Mathematics Education in the Early Twentieth Century’, *Journal for Research in Mathematics Education*, 1986, 17, 3, 190–205.

84. A. G. Howson, ‘Seventy-Five Years of the International Commission on Mathematics Instruction’, *Educational Studies in Mathematics*, 1984, 15, 4, 75–93.

(g) Philosophies of Mathematics Education

85. P. Ernest, ‘The Dialogical Nature of Mathematics’, in Ernest (ed.), *Mathematics, Education and Philosophy* (Falmer Press, 1994), pp. 33–48.

86. E. Wittmann, ‘Mathematics Education as a "Design Science"’, *Educational Studies in Mathematics*, 1995, 29, 4, 355–74.

(h) Theories in Mathematics Education

87. P. Cobb, ‘Experiential, Cognitive and Anthropological Perspectives in Mathematics Education’, *For the Learning of Mathematics*, 1989, 9, 2, 32–42.

88. E. Fennema, H. Walberg, and C. Marrett, ‘Explaining Sex-Related Differences in Mathematics: Theoretical Models’, *Educational Studies in Mathematics*, 1985, 16, 3, 303–4.

89. G. Leder, ‘Sex-Related Differences in Mathematics: An Overview’, *Educational Studies in Mathematics*, 1985, 16, 3, 304–9.

90. E. Fennema and P. L. Peterson, ‘Autonomous Learning Behavior: A Possible Explanation of Sex-Related Differences in Mathematics’, *Educational Studies in Mathematics*, 1985, 16, 3, 309–11.

91. J. Eccles, ‘Model of Students’ Mathematics Enrollment Decisions’, *Educational Studies in Mathematics*, 1985, 16, 3, 311–14.

92. D. R. Maines, ‘Preliminary Notes on a Theory of Informal Barriers for Women in Mathematics’, *Educational Studies in Mathematics*, 1985, 16, 3, 314–20.

93. A. Sfard, ‘Reification as the Birth of Metaphor’, *For the Learning of Mathematics*, 1994, 14, 1, 44–55.

(i) International Cooperation in Mathematics Education Research

94. J. Cai, ‘Why do US and Chinese Students Think Differently in Mathematical Problem-Solving? Exploring the Impact of Early Algebra Learning and Teachers’ Beliefs’, *Journal of Mathematical Behavior*, 2004, 23, 2, 133–65.

95. B. Nebres, ‘International Benchmarking as a Way to Improve School Mathematics Achievement in the Era of Globalization’, in G. Kaiser, E. Luna, and I. Huntley (eds.), *International Comparisons in Mathematics Education* (Falmer Press, 1999), pp. 200–12.

(j) Globalization, Post-Colonialism, and Critical Perspectives

96. B. Atweh and P. Clarkson, ‘Internationalisation and Globalization of Mathematics Education: Towards an Agenda for Research/Action’, in B. Atweh, H. Forgasz, and B. Nebres (eds.), *Sociocultural Research on Mathematics Education* (Erlbaum, 2001), pp. 77–94.

97. R. Vithal and O. Skovsmose, ‘The End of Innocence: A Critique of "Ethnomathematics"’, *Educational Studies in Mathematics*, 1997, 34, 2, 131–57.