A Constructivist Perspective on Teaching and Learning Mathematics

Phi Delta Kappan

Volumn 78, Issue MAR, 1996, Pages 492-499

  Schifter, Deborah ^a  

^a EDUCATION DEVELOPMENT CENTER   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0010202913     PISSN: 00317217     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Review

References (43)

1. Anne Hendry, "Math in the Social Studies Curriculum," in Deborah Schifter, ed., What's Happening in Math Class? Volume 1: Envisioning New Practices Through Teacher Narratives (New York: Teachers College Press, 1996), p. 10.
2. Ibid., p. 11.
3. Ibid., pp. 11-12.
4. Personal communication with Karen Schweitzer, 1993.
5. Paul Cobb, "The Tension Between Theories of Learning and Instruction in Mathematics Education," Educational Psychologist, vol. 23, 1988, pp. 97-103; Jere Confrey, "Learning to Listen: A Student's Understanding of the Powers of Ten," in Ernst von Glasersfeld, ed., Radical Constructivism in Mathematics Education (Dordrecht, The Netherlands: Kluwer Academic Publishers, 1991), pp. 111-38; Constance Kamii, Young Children Reinvent Arithmetic: Implications of Piaget's Theory (New York: Teachers College Press, 1985); and Ernst von Glasersfeld, "An Exposition of Constructivism: Why Some Like It Radical," in Robert B. Davis, Carolyn A. Maher, and Nel Noddings, eds., Constructivist Views on the Teaching and Learning of Mathematics: Journal for Research in Mathematics Education Monograph No. 4 (Reston, Va.: National Council of Teachers of Mathematics, 1990), pp. 19-30.
6. Paul Cobb, "The Tension Between Theories of Learning and Instruction in Mathematics Education," Educational Psychologist, vol. 23, 1988, pp. 97-103; Jere Confrey, "Learning to Listen: A Student's Understanding of the Powers of Ten," in Ernst von Glasersfeld, ed., Radical Constructivism in Mathematics Education (Dordrecht, The Netherlands: Kluwer Academic Publishers, 1991), pp. 111-38; Constance Kamii, Young Children Reinvent Arithmetic: Implications of Piaget's Theory (New York: Teachers College Press, 1985); and Ernst von Glasersfeld, "An Exposition of Constructivism: Why Some Like It Radical," in Robert B. Davis, Carolyn A. Maher, and Nel Noddings, eds., Constructivist Views on the Teaching and Learning of Mathematics: Journal for Research in Mathematics Education Monograph No. 4 (Reston, Va.: National Council of Teachers of Mathematics, 1990), pp. 19-30.
7. Paul Cobb, "The Tension Between Theories of Learning and Instruction in Mathematics Education," Educational Psychologist, vol. 23, 1988, pp. 97-103; Jere Confrey, "Learning to Listen: A Student's Understanding of the Powers of Ten," in Ernst von Glasersfeld, ed., Radical Constructivism in Mathematics Education (Dordrecht, The Netherlands: Kluwer Academic Publishers, 1991), pp. 111-38; Constance Kamii, Young Children Reinvent Arithmetic: Implications of Piaget's Theory (New York: Teachers College Press, 1985); and Ernst von Glasersfeld, "An Exposition of Constructivism: Why Some Like It Radical," in Robert B. Davis, Carolyn A. Maher, and Nel Noddings, eds., Constructivist Views on the Teaching and Learning of Mathematics: Journal for Research in Mathematics Education Monograph No. 4 (Reston, Va.: National Council of Teachers of Mathematics, 1990), pp. 19-30.
8. Paul Cobb, "The Tension Between Theories of Learning and Instruction in Mathematics Education," Educational Psychologist, vol. 23, 1988, pp. 97-103; Jere Confrey, "Learning to Listen: A Student's Understanding of the Powers of Ten," in Ernst von Glasersfeld, ed., Radical Constructivism in Mathematics Education (Dordrecht, The Netherlands: Kluwer Academic Publishers, 1991), pp. 111-38; Constance Kamii, Young Children Reinvent Arithmetic: Implications of Piaget's Theory (New York: Teachers College Press, 1985); and Ernst von Glasersfeld, "An Exposition of Constructivism: Why Some Like It Radical," in Robert B. Davis, Carolyn A. Maher, and Nel Noddings, eds., Constructivist Views on the Teaching and Learning of Mathematics: Journal for Research in Mathematics Education Monograph No. 4 (Reston, Va.: National Council of Teachers of Mathematics, 1990), pp. 19-30.
9. Phillip J. Davis and Reuben Hersh, The Mathematical Experience (Boston: Birkhäuser, 1980); Paul Ernest, The Philosophy of Mathematics Education: Studies in Mathematics Education (London: Falmer Press, 1991); Imre Lakatos, Proofs and Refutations (Cambridge: Cambridge University Press, 1976); Magdalene Lampert, "The Teacher's Role in Reinventing the Meaning of Mathematics Knowing in the Classroom," in Merlyn J. Behr, Caroline B. Lacampagne, and Margariete M. Wheeler, eds., Proceedings of the 10th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (DeKalb: Northern Illinois University, 1988), pp. 433-80; Bruno Latour, Science in Action (Cambridge, Mass.: Harvard University Press, 1987); and Thomas Tymoczko, New Directions in the Philosophy of Mathematics (Boston: Birkhäuser, 1986).
10. Phillip J. Davis and Reuben Hersh, The Mathematical Experience (Boston: Birkhäuser, 1980); Paul Ernest, The Philosophy of Mathematics Education: Studies in Mathematics Education (London: Falmer Press, 1991); Imre Lakatos, Proofs and Refutations (Cambridge: Cambridge University Press, 1976); Magdalene Lampert, "The Teacher's Role in Reinventing the Meaning of Mathematics Knowing in the Classroom," in Merlyn J. Behr, Caroline B. Lacampagne, and Margariete M. Wheeler, eds., Proceedings of the 10th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (DeKalb: Northern Illinois University, 1988), pp. 433-80; Bruno Latour, Science in Action (Cambridge, Mass.: Harvard University Press, 1987); and Thomas Tymoczko, New Directions in the Philosophy of Mathematics (Boston: Birkhäuser, 1986).
11. Phillip J. Davis and Reuben Hersh, The Mathematical Experience (Boston: Birkhäuser, 1980); Paul Ernest, The Philosophy of Mathematics Education: Studies in Mathematics Education (London: Falmer Press, 1991); Imre Lakatos, Proofs and Refutations (Cambridge: Cambridge University Press, 1976); Magdalene Lampert, "The Teacher's Role in Reinventing the Meaning of Mathematics Knowing in the Classroom," in Merlyn J. Behr, Caroline B. Lacampagne, and Margariete M. Wheeler, eds., Proceedings of the 10th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (DeKalb: Northern Illinois University, 1988), pp. 433-80; Bruno Latour, Science in Action (Cambridge, Mass.: Harvard University Press, 1987); and Thomas Tymoczko, New Directions in the Philosophy of Mathematics (Boston: Birkhäuser, 1986).
12. Phillip J. Davis and Reuben Hersh, The Mathematical Experience (Boston: Birkhäuser, 1980); Paul Ernest, The Philosophy of Mathematics Education: Studies in Mathematics Education (London: Falmer Press, 1991); Imre Lakatos, Proofs and Refutations (Cambridge: Cambridge University Press, 1976); Magdalene Lampert, "The Teacher's Role in Reinventing the Meaning of Mathematics Knowing in the Classroom," in Merlyn J. Behr, Caroline B. Lacampagne, and Margariete M. Wheeler, eds., Proceedings of the 10th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (DeKalb: Northern Illinois University, 1988), pp. 433-80; Bruno Latour, Science in Action (Cambridge, Mass.: Harvard University Press, 1987); and Thomas Tymoczko, New Directions in the Philosophy of Mathematics (Boston: Birkhäuser, 1986).
13. Phillip J. Davis and Reuben Hersh, The Mathematical Experience (Boston: Birkhäuser, 1980); Paul Ernest, The Philosophy of Mathematics Education: Studies in Mathematics Education (London: Falmer Press, 1991); Imre Lakatos, Proofs and Refutations (Cambridge: Cambridge University Press, 1976); Magdalene Lampert, "The Teacher's Role in Reinventing the Meaning of Mathematics Knowing in the Classroom," in Merlyn J. Behr, Caroline B. Lacampagne, and Margariete M. Wheeler, eds., Proceedings of the 10th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (DeKalb: Northern Illinois University, 1988), pp. 433-80; Bruno Latour, Science in Action (Cambridge, Mass.: Harvard University Press, 1987); and Thomas Tymoczko, New Directions in the Philosophy of Mathematics (Boston: Birkhäuser, 1986).
14. Phillip J. Davis and Reuben Hersh, The Mathematical Experience (Boston: Birkhäuser, 1980); Paul Ernest, The Philosophy of Mathematics Education: Studies in Mathematics Education (London: Falmer Press, 1991); Imre Lakatos, Proofs and Refutations (Cambridge: Cambridge University Press, 1976); Magdalene Lampert, "The Teacher's Role in Reinventing the Meaning of Mathematics Knowing in the Classroom," in Merlyn J. Behr, Caroline B. Lacampagne, and Margariete M. Wheeler, eds., Proceedings of the 10th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (DeKalb: Northern Illinois University, 1988), pp. 433-80; Bruno Latour, Science in Action (Cambridge, Mass.: Harvard University Press, 1987); and Thomas Tymoczko, New Directions in the Philosophy of Mathematics (Boston: Birkhäuser, 1986).
15. Cobb, op. cit.; Confrey, op. cit.; Davis. Maher, and Noddings, op. cit.: Deborah Loewenberg Ball, "With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics," Elementary School Journal, vol. 93, 1993, pp. 373-97; idem, "Halves, Pieces, and Twoths: Constructing Representational Contexts in Teaching Fractions," in Thomas P. Carpenter, Elizabeth Fennema, and Thomas Romberg, eds., Rational Numbers: An Integration of Research (Hillsdale, N.J.: Erlbaum, 1993), pp. 157-96; Eleanor Duckworth, 'The Having of Wonderful Ideas' and Other Essays on Teaching and Learning (New York: Teachers College Press, 1987); Catherine T. Fosnot, Enquiring Teachers, Enquiring Learners: A Constructivist Approach for Teaching (New York: Teachers College Press, 1989); and Deborah Schifter and Catherine T. Fosnot, Reconstructing Mathematics Education: Stories of Teachers Meeting the Challenge of Reform (New York: Teachers College Press, 1993).
16. Cobb, op. cit.; Confrey, op. cit.; Davis. Maher, and Noddings, op. cit.: Deborah Loewenberg Ball, "With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics," Elementary School Journal, vol. 93, 1993, pp. 373-97; idem, "Halves, Pieces, and Twoths: Constructing Representational Contexts in Teaching Fractions," in Thomas P. Carpenter, Elizabeth Fennema, and Thomas Romberg, eds., Rational Numbers: An Integration of Research (Hillsdale, N.J.: Erlbaum, 1993), pp. 157-96; Eleanor Duckworth, 'The Having of Wonderful Ideas' and Other Essays on Teaching and Learning (New York: Teachers College Press, 1987); Catherine T. Fosnot, Enquiring Teachers, Enquiring Learners: A Constructivist Approach for Teaching (New York: Teachers College Press, 1989); and Deborah Schifter and Catherine T. Fosnot, Reconstructing Mathematics Education: Stories of Teachers Meeting the Challenge of Reform (New York: Teachers College Press, 1993).
17. Cobb, op. cit.; Confrey, op. cit.; Davis. Maher, and Noddings, op. cit.: Deborah Loewenberg Ball, "With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics," Elementary School Journal, vol. 93, 1993, pp. 373-97; idem, "Halves, Pieces, and Twoths: Constructing Representational Contexts in Teaching Fractions," in Thomas P. Carpenter, Elizabeth Fennema, and Thomas Romberg, eds., Rational Numbers: An Integration of Research (Hillsdale, N.J.: Erlbaum, 1993), pp. 157-96; Eleanor Duckworth, 'The Having of Wonderful Ideas' and Other Essays on Teaching and Learning (New York: Teachers College Press, 1987); Catherine T. Fosnot, Enquiring Teachers, Enquiring Learners: A Constructivist Approach for Teaching (New York: Teachers College Press, 1989); and Deborah Schifter and Catherine T. Fosnot, Reconstructing Mathematics Education: Stories of Teachers Meeting the Challenge of Reform (New York: Teachers College Press, 1993).
18. Cobb, op. cit.; Confrey, op. cit.; Davis. Maher, and Noddings, op. cit.: Deborah Loewenberg Ball, "With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics," Elementary School Journal, vol. 93, 1993, pp. 373-97; idem, "Halves, Pieces, and Twoths: Constructing Representational Contexts in Teaching Fractions," in Thomas P. Carpenter, Elizabeth Fennema, and Thomas Romberg, eds., Rational Numbers: An Integration of Research (Hillsdale, N.J.: Erlbaum, 1993), pp. 157-96; Eleanor Duckworth, 'The Having of Wonderful Ideas' and Other Essays on Teaching and Learning (New York: Teachers College Press, 1987); Catherine T. Fosnot, Enquiring Teachers, Enquiring Learners: A Constructivist Approach for Teaching (New York: Teachers College Press, 1989); and Deborah Schifter and Catherine T. Fosnot, Reconstructing Mathematics Education: Stories of Teachers Meeting the Challenge of Reform (New York: Teachers College Press, 1993).
19. Cobb, op. cit.; Confrey, op. cit.; Davis. Maher, and Noddings, op. cit.: Deborah Loewenberg Ball, "With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics," Elementary School Journal, vol. 93, 1993, pp. 373-97; idem, "Halves, Pieces, and Twoths: Constructing Representational Contexts in Teaching Fractions," in Thomas P. Carpenter, Elizabeth Fennema, and Thomas Romberg, eds., Rational Numbers: An Integration of Research (Hillsdale, N.J.: Erlbaum, 1993), pp. 157-96; Eleanor Duckworth, 'The Having of Wonderful Ideas' and Other Essays on Teaching and Learning (New York: Teachers College Press, 1987); Catherine T. Fosnot, Enquiring Teachers, Enquiring Learners: A Constructivist Approach for Teaching (New York: Teachers College Press, 1989); and Deborah Schifter and Catherine T. Fosnot, Reconstructing Mathematics Education: Stories of Teachers Meeting the Challenge of Reform (New York: Teachers College Press, 1993).
20. Cobb, op. cit.; Confrey, op. cit.; Davis. Maher, and Noddings, op. cit.: Deborah Loewenberg Ball, "With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics," Elementary School Journal, vol. 93, 1993, pp. 373-97; idem, "Halves, Pieces, and Twoths: Constructing Representational Contexts in Teaching Fractions," in Thomas P. Carpenter, Elizabeth Fennema, and Thomas Romberg, eds., Rational Numbers: An Integration of Research (Hillsdale, N.J.: Erlbaum, 1993), pp. 157-96; Eleanor Duckworth, 'The Having of Wonderful Ideas' and Other Essays on Teaching and Learning (New York: Teachers College Press, 1987); Catherine T. Fosnot, Enquiring Teachers, Enquiring Learners: A Constructivist Approach for Teaching (New York: Teachers College Press, 1989); and Deborah Schifter and Catherine T. Fosnot, Reconstructing Mathematics Education: Stories of Teachers Meeting the Challenge of Reform (New York: Teachers College Press, 1993).
21. Cobb, op. cit.; Confrey, op. cit.; Davis. Maher, and Noddings, op. cit.: Deborah Loewenberg Ball, "With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics," Elementary School Journal, vol. 93, 1993, pp. 373-97; idem, "Halves, Pieces, and Twoths: Constructing Representational Contexts in Teaching Fractions," in Thomas P. Carpenter, Elizabeth Fennema, and Thomas Romberg, eds., Rational Numbers: An Integration of Research (Hillsdale, N.J.: Erlbaum, 1993), pp. 157-96; Eleanor Duckworth, 'The Having of Wonderful Ideas' and Other Essays on Teaching and Learning (New York: Teachers College Press, 1987); Catherine T. Fosnot, Enquiring Teachers, Enquiring Learners: A Constructivist Approach for Teaching (New York: Teachers College Press, 1989); and Deborah Schifter and Catherine T. Fosnot, Reconstructing Mathematics Education: Stories of Teachers Meeting the Challenge of Reform (New York: Teachers College Press, 1993).
22. Cobb, op. cit.; Confrey, op. cit.; Davis. Maher, and Noddings, op. cit.: Deborah Loewenberg Ball, "With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics," Elementary School Journal, vol. 93, 1993, pp. 373-97; idem, "Halves, Pieces, and Twoths: Constructing Representational Contexts in Teaching Fractions," in Thomas P. Carpenter, Elizabeth Fennema, and Thomas Romberg, eds., Rational Numbers: An Integration of Research (Hillsdale, N.J.: Erlbaum, 1993), pp. 157-96; Eleanor Duckworth, 'The Having of Wonderful Ideas' and Other Essays on Teaching and Learning (New York: Teachers College Press, 1987); Catherine T. Fosnot, Enquiring Teachers, Enquiring Learners: A Constructivist Approach for Teaching (New York: Teachers College Press, 1989); and Deborah Schifter and Catherine T. Fosnot, Reconstructing Mathematics Education: Stories of Teachers Meeting the Challenge of Reform (New York: Teachers College Press, 1993).
23. Karen Schweitzer, "The Search for the Perfect Resource," in Deborah Schifter, ed., What's Happening in Math Class? Volume 2: Reconstructing Professional Identities (New York: Teachers College Press, 1996), p. 48.
24. Ibid.
25. For further descriptions of such courses, see Deborah Schifter, "Mathematics Process as Mathematics Content: A Course for Teaching Mathematics," Journal of Mathematical Behavior, vol. 12, 1993, pp. 271-83; Schifter and Fosnot, op. cit.; Deborah Schifter and Martin A. Simon, "Assessing Teachers' Development of a Constructivist View of Mathematics Learning," Teaching and Teacher Education, vol. 8, 1992, pp. 187-97; and Martin A. Simon and Deborah E. Schifter, "Towards a Constructivist Perspective: An Intervention Study of Mathematics Teacher Development," Educational Studies in Mathematics, vol. 22, 1991, pp. 309-31.
26. For further descriptions of such courses, see Deborah Schifter, "Mathematics Process as Mathematics Content: A Course for Teaching Mathematics," Journal of Mathematical Behavior, vol. 12, 1993, pp. 271-83; Schifter and Fosnot, op. cit.; Deborah Schifter and Martin A. Simon, "Assessing Teachers' Development of a Constructivist View of Mathematics Learning," Teaching and Teacher Education, vol. 8, 1992, pp. 187-97; and Martin A. Simon and Deborah E. Schifter, "Towards a Constructivist Perspective: An Intervention Study of Mathematics Teacher Development," Educational Studies in Mathematics, vol. 22, 1991, pp. 309-31.
27. For further descriptions of such courses, see Deborah Schifter, "Mathematics Process as Mathematics Content: A Course for Teaching Mathematics," Journal of Mathematical Behavior, vol. 12, 1993, pp. 271-83; Schifter and Fosnot, op. cit.; Deborah Schifter and Martin A. Simon, "Assessing Teachers' Development of a Constructivist View of Mathematics Learning," Teaching and Teacher Education, vol. 8, 1992, pp. 187-97; and Martin A. Simon and Deborah E. Schifter, "Towards a Constructivist Perspective: An Intervention Study of Mathematics Teacher Development," Educational Studies in Mathematics, vol. 22, 1991, pp. 309-31.
28. For further descriptions of such courses, see Deborah Schifter, "Mathematics Process as Mathematics Content: A Course for Teaching Mathematics," Journal of Mathematical Behavior, vol. 12, 1993, pp. 271-83; Schifter and Fosnot, op. cit.; Deborah Schifter and Martin A. Simon, "Assessing Teachers' Development of a Constructivist View of Mathematics Learning," Teaching and Teacher Education, vol. 8, 1992, pp. 187-97; and Martin A. Simon and Deborah E. Schifter, "Towards a Constructivist Perspective: An Intervention Study of Mathematics Teacher Development," Educational Studies in Mathematics, vol. 22, 1991, pp. 309-31.
29. Personal communication with Karen Schweitzer, 1992.
30. Schweitzer, pp. 50-51.
31. Ibid., p. 55.
32. David K. Cohen et al., Effects of State-Level Reform of Elementary School Mathematics Curriculum on Classroom Practice (East Lansing: National Center for Research on Teacher Education and Center for the Learning and Teaching of Elementary Subjects, Michigan State University, Research Report 90-14, 1990); and Brace Joyce, ed., Changing School Culture Through Staff Development (Alexandria, Va.: Association for Supervision and Curriculum Development, 1990).
33. David K. Cohen et al., Effects of State-Level Reform of Elementary School Mathematics Curriculum on Classroom Practice (East Lansing: National Center for Research on Teacher Education and Center for the Learning and Teaching of Elementary Subjects, Michigan State University, Research Report 90-14, 1990); and Brace Joyce, ed., Changing School Culture Through Staff Development (Alexandria, Va.: Association for Supervision and Curriculum Development, 1990).
34. See Ball, "With an Eye"; idem, "Halves, Pieces"; Ruth M. Heaton, "Continuity and Connectedness in Teaching and Research: A Self-Study of Learning to Teach Mathematics for Understanding," paper presented at the University of Pennsylvania Ethnography in Education Research Forum, Philadelphia, 1991; Magdalene Lampert, "Arithmetic as Problem Solving," Arithmetic Teacher, March 1989, pp. 34-36; and idem, "The Teacher's Role." See also Nancie Atwell, "Writing and Reading from the Inside Out," in Jane Hansen, Thomas Newkirk, and Donald Graves, eds., Breaking Ground: Teachers Relate Reading and Writing in the Elementary School (Portsmouth, N.H.: Heinemann Educational Books, 1985), pp. 147-68; and George Hillocks, Jr., "Teaching, Reflecting, Researching," in Donald A. Daiker and Max Morenberg, eds., The Writing Teacher as Researcher (Portsmouth, N.H.: Heinemann, 1990), pp. 15-29.
35. See Ball, "With an Eye"; idem, "Halves, Pieces"; Ruth M. Heaton, "Continuity and Connectedness in Teaching and Research: A Self-Study of Learning to Teach Mathematics for Understanding," paper presented at the University of Pennsylvania Ethnography in Education Research Forum, Philadelphia, 1991; Magdalene Lampert, "Arithmetic as Problem Solving," Arithmetic Teacher, March 1989, pp. 34-36; and idem, "The Teacher's Role." See also Nancie Atwell, "Writing and Reading from the Inside Out," in Jane Hansen, Thomas Newkirk, and Donald Graves, eds., Breaking Ground: Teachers Relate Reading and Writing in the Elementary School (Portsmouth, N.H.: Heinemann Educational Books, 1985), pp. 147-68; and George Hillocks, Jr., "Teaching, Reflecting, Researching," in Donald A. Daiker and Max Morenberg, eds., The Writing Teacher as Researcher (Portsmouth, N.H.: Heinemann, 1990), pp. 15-29.
36. See Ball, "With an Eye"; idem, "Halves, Pieces"; Ruth M. Heaton, "Continuity and Connectedness in Teaching and Research: A Self-Study of Learning to Teach Mathematics for Understanding," paper presented at the University of Pennsylvania Ethnography in Education Research Forum, Philadelphia, 1991; Magdalene Lampert, "Arithmetic as Problem Solving," Arithmetic Teacher, March 1989, pp. 34-36; and idem, "The Teacher's Role." See also Nancie Atwell, "Writing and Reading from the Inside Out," in Jane Hansen, Thomas Newkirk, and Donald Graves, eds., Breaking Ground: Teachers Relate Reading and Writing in the Elementary School (Portsmouth, N.H.: Heinemann Educational Books, 1985), pp. 147-68; and George Hillocks, Jr., "Teaching, Reflecting, Researching," in Donald A. Daiker and Max Morenberg, eds., The Writing Teacher as Researcher (Portsmouth, N.H.: Heinemann, 1990), pp. 15-29.
37. See Ball, "With an Eye"; idem, "Halves, Pieces"; Ruth M. Heaton, "Continuity and Connectedness in Teaching and Research: A Self-Study of Learning to Teach Mathematics for Understanding," paper presented at the University of Pennsylvania Ethnography in Education Research Forum, Philadelphia, 1991; Magdalene Lampert, "Arithmetic as Problem Solving," Arithmetic Teacher, March 1989, pp. 34-36; and idem, "The Teacher's Role." See also Nancie Atwell, "Writing and Reading from the Inside Out," in Jane Hansen, Thomas Newkirk, and Donald Graves, eds., Breaking Ground: Teachers Relate Reading and Writing in the Elementary School (Portsmouth, N.H.: Heinemann Educational Books, 1985), pp. 147-68; and George Hillocks, Jr., "Teaching, Reflecting, Researching," in Donald A. Daiker and Max Morenberg, eds., The Writing Teacher as Researcher (Portsmouth, N.H.: Heinemann, 1990), pp. 15-29.
38. See Ball, "With an Eye"; idem, "Halves, Pieces"; Ruth M. Heaton, "Continuity and Connectedness in Teaching and Research: A Self-Study of Learning to Teach Mathematics for Understanding," paper presented at the University of Pennsylvania Ethnography in Education Research Forum, Philadelphia, 1991; Magdalene Lampert, "Arithmetic as Problem Solving," Arithmetic Teacher, March 1989, pp. 34-36; and idem, "The Teacher's Role." See also Nancie Atwell, "Writing and Reading from the Inside Out," in Jane Hansen, Thomas Newkirk, and Donald Graves, eds., Breaking Ground: Teachers Relate Reading and Writing in the Elementary School (Portsmouth, N.H.: Heinemann Educational Books, 1985), pp. 147-68; and George Hillocks, Jr., "Teaching, Reflecting, Researching," in Donald A. Daiker and Max Morenberg, eds., The Writing Teacher as Researcher (Portsmouth, N.H.: Heinemann, 1990), pp. 15-29.
39. See Ball, "With an Eye"; idem, "Halves, Pieces"; Ruth M. Heaton, "Continuity and Connectedness in Teaching and Research: A Self-Study of Learning to Teach Mathematics for Understanding," paper presented at the University of Pennsylvania Ethnography in Education Research Forum, Philadelphia, 1991; Magdalene Lampert, "Arithmetic as Problem Solving," Arithmetic Teacher, March 1989, pp. 34-36; and idem, "The Teacher's Role." See also Nancie Atwell, "Writing and Reading from the Inside Out," in Jane Hansen, Thomas Newkirk, and Donald Graves, eds., Breaking Ground: Teachers Relate Reading and Writing in the Elementary School (Portsmouth, N.H.: Heinemann Educational Books, 1985), pp. 147-68; and George Hillocks, Jr., "Teaching, Reflecting, Researching," in Donald A. Daiker and Max Morenberg, eds., The Writing Teacher as Researcher (Portsmouth, N.H.: Heinemann, 1990), pp. 15-29.
40. See Ball, "With an Eye"; idem, "Halves, Pieces"; Ruth M. Heaton, "Continuity and Connectedness in Teaching and Research: A Self-Study of Learning to Teach Mathematics for Understanding," paper presented at the University of Pennsylvania Ethnography in Education Research Forum, Philadelphia, 1991; Magdalene Lampert, "Arithmetic as Problem Solving," Arithmetic Teacher, March 1989, pp. 34-36; and idem, "The Teacher's Role." See also Nancie Atwell, "Writing and Reading from the Inside Out," in Jane Hansen, Thomas Newkirk, and Donald Graves, eds., Breaking Ground: Teachers Relate Reading and Writing in the Elementary School (Portsmouth, N.H.: Heinemann Educational Books, 1985), pp. 147-68; and George Hillocks, Jr., "Teaching, Reflecting, Researching," in Donald A. Daiker and Max Morenberg, eds., The Writing Teacher as Researcher (Portsmouth, N.H.: Heinemann, 1990), pp. 15-29.
41. Schweitzer, p. 55.
42. Ibid., p. 64.
43. Ibid., pp. 64-65.