A proposed instructional theory for integer addition and subtraction

Journal for Research in Mathematics Education

Volumn 43, Issue 4, 2012, Pages 428-464

  Stephan, Michelle ^a   Akyuz, Didem ^b  

^a UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE   (United States)

^b MIDDLE EAST TECHNICAL UNIVERSITY   (Turkey)

Author keywords

Conceptual knowledge; Instructional intervention; Integers; Middle grades, 5 8; Modeling

Indexed keywords

EID: 84864223189     PISSN: 00218251     EISSN: None     Source Type: Journal    
DOI: 10.5951/jresematheduc.43.4.0428     Document Type: Article

References (53)

1. Akyuz, D. (2010). Supporting a standards-based teaching and learning environment: A case study of an expert middle school mathematics teacher. (Doctoral Dissertation). University of Central Florida, Orlando, FL.
2. Battista, M. T. (1983). A complete model for operations on integers. Arithmetic Teacher, 30(9), 26-31.
3. Bauersfeld, H. (1992). Classroom cultures from a social constructivist perspective. Educational Studies in Mathematics, 23, 467-481. doi:10.1007/BF00571468
4. Blumer, H. (1969). Symbolic interactionism: Perspective and method. Englewood Cliffs, NJ: Prentice-Hall.
5. Bofferding, L. (2010). Addition and subtraction with negatives: Acknowledging the multiple meanings of the minus sign. In P. Brosnan, D. B. Erchick, & L. Flevares (Eds.), Proceedings of the 32nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 703-710). Columbus, OH: The Ohio State University.
6. Cobb, P. (2003). Investigating students' reasoning about linear measurement as a paradigm case of design research. In N. Pateman (Series Ed.), M. Stephan, J. Bowers, & P. Cobb (with K. Gravemeijer) (Vol. Eds.), Journal for Research in Mathematics Education Monograph Series: Vol. 12. Supporting students 'development of measuring conceptions: Analyzing students learning in social context (pp. 1-16). Reston, VA: National Council of Teachers of Mathematics.
7. Cobb, P., Confrey, J., di Sessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher,. 52(1), 9-13. doi:10.3102/0013189X032001009 (Pubitemid 36209336)
8. Cobb, P., Gravemeijer, K., Yackel, E., McClain, K., & Whitenack, J. (1997). Mathematizing and symbolizing: The emergence of chains of signification in one first-grade classroom. In D. Kirshner & J. A. Whitson (Eds.), Situated cognition: Social, semiotic, and psychological perspectives (pp. 151-233). Mahwah, NJ: Erlbaum.
9. Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31, 175-190. doi:10.1080/00461520.1996.9 653265
10. Davis, R. B. (1967). Explorations in mathematics: A text for teachers. Palo Alto, CA: Addison-Wesley.
11. De Bock, D., Deprez, J., Van Dooren, W., Roelens, M., & Verschaffel, L. (2011). Abstract or concrete examples in learning mathematics? A replication and elaboration of Kaminski, Sloutsky, and Heckler's study. Journal for Research in Mathematics Education, 42, 109-126. http://www.nctm.org/publications/jrme. aspx
12. De Lange, J. (1987). Mathematics, insight and meaning: Teaching learning and testing of mathematics for the life and social sciences. Utrecht, the Netherlands: Vakgroep Onderzoek Wiskundeonderwijs en Onderwijs Computercentrum, Rijksuniversiteit Utrecht.
13. Dirks, M. K. (1984). The integer abacus. Arithmetic Teacher, 31(1), 50-54.
14. Forman, E., Larreamendy-Joerns, J., Stein, M., & Brown, C. (1998). "You're going to want to find out which and prove it": Collective argumentation in a mathematics classroom. Learning and Instruction, 8, 527-548. doi:10.1016/S0959-4752 (98) 00033-4 (Pubitemid 128340269)
15. Fosnot, C. T., & Dolk, M. (2005). "Mathematics" or "mathematizing"? In C. T. Fosnot (Ed.), Constructivism: Theory, perspectives and practice (2nd ed., pp. 175-191). New York, NY: Teachers College Press.
16. Freudenthal, H. (1968). Why to teach mathematics so as to be useful. Educational Studies in Mathematics 1, 3-8. doi:10.1007/BF00426224
17. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, the Netherlands: D. Reidel.
18. Gallardo, A. (2002). The extension of the natural-number domain to the integers in the transition from arithmetic to algebra. Educational Studies in Mathematics, 49, 171-192. doi:10.1023/A:1016210906658
19. Gallardo, A., & Rojano, T. (1994). School algebra: Syntactic difficulties in the operativity with negative numbers. In D. Kirshner (Ed.), Proceedings of the Sixteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 159-165). Baton Rouge, LA: Louisiana State University.
20. Glaeser, G. (1981). Epistémologie des nombres relatifs. Recherches en Didactique des Mathématiques, 2, 303-346. http://rdm.penseesauvage. com/
21. Gravemeijer, K. (1994). Developing realistic mathematics education. Utrecht, the Netherlands: CD-Press.
22. Gravemeijer, K., Bowers, J., & Stephan, M. (2003). Continuing the design research cycle. In N. Pateman (Series Ed.), M. Stephan, J. Bowers, & P. Cobb (with K. Gravemeijer) (Vol. Eds.), Journal for Research in Mathematics Education Monograph Series: Vol. 12. Supporting students' development of measuring conceptions: Analyzing students' learning in social context (pp. 103-169). Reston, VA: National Council of Teachers of Mathematics.
23. Gravemeijer, K., & Stephan, M. (2002). Emergent models as an instructional design heuristic. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing modeling and tool use in mathematics education (pp. 145-169). Dordrecht, the Netherlands: Kluwer.
24. Hefendehl-Hebeker, L. (1991). Negative numbers: Obstacles in their evolution from intuitive to intellectual constructs. For the Learning of Mathematics, 11(1), 26-32.
25. Janvier, C. (1985). Comparison of models aimed at teaching signed integers. In L. Streefland (Ed.), Proceedings of the Ninth International Conference for the Psychology of Mathematics Education (pp. 135-140). Utrecht, the Netherlands: State University of Utrecht.
26. Kaminski, J. A., Sloutsky, V M., & Heckler, A. F. (2008). The advantage of abstract examples in learning math. Science, 320, 454-455. doi:10.1126/science.l 154659 (Pubitemid 351590651)
27. Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb and H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 229-270). Hillsdale, NJ: Erlbaum.
28. Lappan, G., Fey, J., Fitzgerald, W, Friel, S., & Phillips, E. (1998). Connected mathematics project. Palo Alto, CA: Dale Seymour.
29. Liebeck, P. (1990). Scores and forfeits: An intuitive model for integer arithmetic. Educational Studies in Mathematics, 21, 221-239.
30. Linchevski, L., & Williams, J. (1999). Using intuition from everyday life in "filling" the gap in children's extension of their number concept to include the negative numbers. Educational Studies in Mathematics, 39, 131-147. doi:10.1023/A:1003726317920
31. Lytle, P. A. (1994). Investigation of a model based on the neutralization of opposites to teach integer addition and subtraction. In J. P. da Ponte & J. F. Matos (Eds.), Proceedings of the 18th International Conference for the Psychology of Mathematics Education (Vol. III, pp. 192-199). Lisbon, Portugal: University of Lisbon.
32. Mukhopadhyay, S., Resnick, L. B., & Schauble, L. (1990). Social sense-making in mathematics: Children's ideas of negative numbers. In G. Booker, J. Cobb, & T. N. de Mendicuti (Eds.), Proceedings of the Nth Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 281-288). Oaxtapec, Mexico: PME.
33. Murray, J. C. (1985). Children's informal conceptions of integer arithmetic. In L. Streefland (Ed.), Proceedings of the Ninth International Conference for the Psychology of Mathematics Education (pp. 147-153). Utrecht, the Netherlands: The State University of Utrecht.
34. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
35. Peled, I., Mukhopadhyay, S., & Resnick, L. B. (1989). Formal and informal sources of mental models for negative numbers. In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.), Proceedings of the Thirteenth International Conference for the Psychology of Mathematics Education (Vol. III, pp. 106-110). Paris, France: PME.
36. Rasmussen, C, & Stephan, M. (2008). A methodology for documenting collective activity. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design research methods in education: Innovations in science, technology, engineering and mathematics learning and teaching (pp. 195-215). New York, NY: Routledge.
37. Rasmussen, C, Stephan, M., & Allen, K. (2004). Classroom mathematical practices and gesturing. Journal of Mathematical Behavior, 23, 301-323. doi:10.1016/j.jmathb.2004.06.003 (Pubitemid 39234010)
38. Schwarz, B. B., Kohn, A. S., & Resnick, L. B. (1993/1994). Positives about negatives: A case study of an intermediate model for signed numbers. The Journal of the Learning Sciences, 3, 37-92. doi: 10.1207/sl5327809jls0301-2
39. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145. doi: 10.2307/749205
40. Smith, J. P. (1995). The effects of a computer microworld on middle school students 'use and understanding of integers. (Unpublished doctoral dissertation). The Ohio State University, Columbus, Ohio. Retrieved from http://rave.ohiolink.edu/etdc/view?acc-num=osul248798217
41. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. A. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267-307). Hillsdale, NJ: Erlbaum.
42. Steffe, L. P., von Glasersfeld, E., Richards, J., & Cobb, P. (1983). Children's counting types: Philosophy, theory, and application. New York, NY: Praeger.
43. Stephan, M. L. (2003). Reconceptualizing linear measurement studies: The development of three monograph themes. In N. Pateman (Series Ed.), M. Stephan, J. Bowers, & P. Cobb (with K. Gravemeijer) (Vol. Eds.), Journal for Research in Mathematics Education Monograph Series: Vol. 12. Supporting students'development of measuring conceptions: Analyzing students' learning in social context (pp. 17-35). Reston, VA: National Council of Teachers of Mathematics.
44. Stephan. M. L. (2009). What are you worth? Mathematics Teaching in the Middle School, 15, 16-23. http://www.jstor. Org/stable/41182947
45. Stephan, M., Bowers, J., Cobb, P., & Gravemeijer K. (Eds.). (2003). Journal for Research in Mathematics Education Monograph Series: Vol. 12. Supporting students' development of measuring conceptions: Analyzing students learning in social context (N. Pateman, Series Ed.). Reston, VA: National Council of Teachers of Mathematics.
46. Stephan, M., Cobb, P., Gravemeijer, K., & Estes, B. (2001). The role of tools in supporting students' development of measuring conceptions. In A. A. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics, 2001 Yearbook of the National Council of Teachers of Mathematics (NCTM) (pp. 63-76). Reston, VA: NCTM.
47. Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior, 21, 459-490. doi:10.1016/S0732-3123 (02) 00145-l
48. Streefland, L. (1996). Negative numbers: Reflections of a learning researcher. Journal of Mathematical Behavior, 15, 57-79. doi:10.1016/S0732-3123 (96) 90040-l
49. Thompson, P. W., & Dreyfus, T. (1988). Integers as transformations. Journal for Research in Mathematics Education, 19, 115-133. doi: 10.2307/749406
50. Toulmin, S. E. (1958). The uses of argument. Cambridge, England: Cambridge University Press.
51. Vlassis, J. (2004). Making sense of the minus sign or becoming flexible in 'negativity'. Learning and Instruction, 14, 469-484. doi:10.1016/j. learninstruc.2004.06.012 (Pubitemid 40256626)
52. Vlassis, J. (2008). The role of mathematical symbols in the development of number conceptualization: The case of the minus sign. Philosophical Psychology, 21, 555-570. doi: 10.1080/09515080802285552
53. von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Bristol, PA: The Falmer Press.