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Equations with Parameters: A Locus Approach
Article

## Sergei Abramovich, State University of New York at Potsdam, United States ; Anderson Norton, Indiana University, United States

JCMST Volume 25, Number 1, ISSN 0731-9258 Publisher: Association for the Advancement of Computing in Education (AACE), Waynesville, NC USA

## Abstract

This paper introduces technology-based teaching ideas that facilitate the development of qualitative reasoning techniques in the context of quadratic equations with parameters. It reflects on activities designed for prospective secondary mathematics teachers in accord with standards for teaching and recommendations for teachers in North America. The main educational implications of proposed didactics include the emphasis on the geometrization of algebra, emergence of residual mental power that can be used in the absence of technology, and development of skills in problem posing.

## Citation

Abramovich, S. & Norton, A. (2006). Equations with Parameters: A Locus Approach. Journal of Computers in Mathematics and Science Teaching, 25(1), 5-28. Waynesville, NC USA: Association for the Advancement of Computing in Education (AACE). Retrieved May 19, 2019 from https://www.learntechlib.org/primary/p/6076/.

© 2006 Association for the Advancement of Computing in Education (AACE)

### Keywords

## References

View References & Citations Map- Abramovich, S., & Brown, G. (1996). Integrating problem solving, technology and experience of mathematical discovery in teacher education. Journal of Computers in Mathematics and Science Teaching, 15(4), 287-302.
- Abramovich, S., & Brouwer, P. (2003). Revealing hidden mathematics curriculum to pre-teachers using technology: The case of partitions. International Journal of Mathematical Education in Science and Technology, 34(1), 81-94. Abramovich, S., & Brouwer, P. (2004). Hidden mathematics curriculum and technology in K-12 teacher education. In D. E. Mc Dougall & J. A. Ross (Eds.), Proceedings of the 26th Annual Meeting of North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 1191-1199). Toronto: Ontario Institute for Studies in Education. Akst, G. (1998). Enhancing algebraic reasoning with technology. In The Nature and Role of Algebra in the K-14 Curriculum (P. 81). Proceedings of a National Symposium, May 27 and 28, 1997. Washington, DC: National Academy Press.
- Avitzur, R., Gooding, A., Herrmann, E., Piovanelli, M., Robbins, G., Wales, C., & Zadrozny, J. (2002). Graphing calculator -3.2. [Computer software]. Berkeley, CA: Paciﬁ c Tech.
- Brown, S. I., & Walter, M. I. (1990). The art of problem posing. Hillsdale, NJ: Lawrence Erlbaum.
- Browning, C.A., & Klespis, M. L. (2000). A reaction to Garofalo, Drier, Harper, Timmerman, and Shockey. Contemporary Issues in Technology and Teacher Education. Retrieved August 25, 2005 from, http://www.citejournal.org/Bruner, J. S. (1973). Beyond the information given: Studies in the psychology of knowing. New York: Norton.
- Bruner, J. S. (1985). Vygotsky: A historical and conceptual perspective. In J. V. Wertsch (Ed.), Culture, communication and cognition: Vygotskian perspectives (pp. 21-34). Cambridge: Cambridge University Press.
- Burril, G. (1998). Synthesis of day one. In The Nature and Role of Algebra in the K-14 Curriculum (pp. 52-54). Proceedings of a National Symposium, May 27 and 28, 1997. Washington, DC: National Academy Press.
- Chamblee, G. E., & Slough, S. W. (2002). Implementing technology in secondary science and mathematics classrooms: Is the implementation process the same for both disciplines? Journal of Computers in Mathematics and Science Teaching, 21(1), 3-15.
- Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers. Washington, DC: Mathematical Association of America. Dugdale, S., Thompson, P. W., Harvey, W., Demana, F., Waits, B. K., Kieran, C., McConnell, J. W., & Christmas, P. (1995). Technology and algebra curriculum reform: Current issues, potential directions, and research questions. Journal of Computers in Mathematics and Science Teaching, 14 (3), 325-357.
- Dugdale, S., Wagner, L. J., & Kibbey, D. (1992). Visualizing polynomial functions: New insights from an old method into a new medium. Journal of Computers in Mathematics and Science Teaching, 11(2), 123-141.
- Ellerton, N. F., & Clarkson, P. C. (1996). Language factors in mathematics teaching and learning. In A. J. Bishop et al. (Eds.), International Handbook of Mathematics Education, (pp. 987-1033). Dordrecht: Kluwer Academic Publishers.
- Feurzeig, W., Katz, G., Lewis, P., & Steinbok, V. (2000a). Two-parameter universes. Part 1. Consider a rectangular point … . International Journal of Computers for Mathematics Learning, 5(2), 169-178.
- Feurzeig, W., Katz, G., Lewis, P., & Steinbok, V. (2000b). Two-parameter universes. Part 2. Picture a quadratic polynomial … . International Journal of Computers for Mathematics Learning, 5(3), 263-274.
- Fey, J. T. (1989). School algebra for the year 2000. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of Algebra (Vol. 4, pp. 199-213). Reston, VA: National Council of Teachers of Mathematics; Hillsdale, NJ: Lawrence Erlbaum.
- Goldenberg, E. P., & Walter, M. I. (2003). Problem posing as a tool for teaching mathematics. In H. L. Schoen & R. I. Charles (Eds.), Teaching mathematics through problem solving (pp. 69-84). Reston, VA: NCTM.
- Halmos, P. R. (1975). The teaching of problem solving. American Mathematical Monthly, 82, 466-470.
- Hoyles , C., and Sutherland, R. (1986). Peer interaction in a programming environment. In L. Burton & C. Hoyles (Eds.), Proceedings of the Tenth International Conference for the Psychology of Mathematics Education (pp.
- Jiang, Z., & McClintock, E. (2000). Multiple approaches to problem solving and the use of technology. Journal of Computers in Mathematics and Science Teaching, 19(1), 7-20.
- Kaput, J. J. (1992). Technology and mathematics education. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515-556). New York: Macmillan.
- Kaput, J. J. (1995). A research base supporting long term algebra reform? Plenary paper. In D. T. Owens, M. K. Reeds, & G. M. Millsaps (Eds.), Proceedings of the Seventeenth Annual Meeting of North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 71-94). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
- Kersaint, G., Horton, B., Stohl, H., & Garafalo, J. (2003). Technology beliefs and practices of mathematics education faculty. Journal of Technology and Teacher Education, 11(4), 549-577.
- Kieran, C. (1993). Functions, graphing, and technology: Integrating research on learning and instruction. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 189-237). Hillsdale, NJ: Lawrence Erlbaum.
- Kilpatrick, J. (1987a). Is teaching teachable? George Pólya’s views on the training of mathematics teachers. In F. R. Curcio (Ed.), Teaching and learning: A problem-solving focus. Reston, VA: National Council of Teachers of Mathematics.
- Kilpatrick, J. (1987b). Problem formulating: Where do good problems come from? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123-147). Hillsdale, NJ: Lawrence Erlbaum.
- Laborde, C. (1995). Designing tasks for learning geometry in a computer-based environment. In L. Burton & B. Jaworski (Eds.), Technology in mathematics teaching (pp. 35-67). Bromley: Chartwell-Bratt.
- McArthur, D., Stasz, C., & Hotta, J. Y. (1987). Learning problem-solving skills in algebra. Journal of Educational Technology Systems, 15, 303-324. Mistretta, R. M. (2002). Teaching mathematics with technology: The role of higher education in linking theory with practice. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L. Bryant, & K. Nooney (Eds.), Proceedings of the 24th Annual Meeting of North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 3., P. 1145-1152). Columbus, OH: Clearinghouse for Science, Mathematics, and Environmental education.
- National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: The Author.
- National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: The Author.
- Noss, R. (1986). What mathematics do children do with Logo? Journal of Computer Assisted Learning, 3, 2-12.
- Pea, R. (1987). Cognitive technologies for mathematics education. In A. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 89-122). Hillsdale, NJ: Lawrence Erlbaum.
- Saul, M. (1998). Algebra, technology, and a remark of I. M. Gelfand. In The nature and role of algebra in the K-14 curriculum (pp. 137-144). Proceedings
- Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19-28.
- Silver, E. A., Kilpatrick, J., & Schlesinger, B. (1990). Thinking through mathematics: Fostering inquiry and communication in mathematics classrooms. New York: College Entrance Examination Board.
- Steen, A. L. (2004). How mathematicians can contribute to K-12 education. Notices of the American Mathematical Society, 51(8), 869.
- Ursini, S., & Trigueros, M. (2004). How do high school students interpret parameters in algebra? In M. J. Hoines & A. B. Fuglestad (Eds.), Proceed-
- Yerushalmy, M., & Chazan, D. (2002). Flux in school algebra: Curricular change, graphing technology, and research on student learning and teacher knowledge. In L. D. English (Ed.), Handbook of International Research in Mathematics Education (pp. 725-755). Mahwah, NJ: Lawrence Erlbaum. Yerushalmy, M., Chazan, D., & Gordon, M. (1993). Posing problems: One aspect of bringing inquiry into classrooms. In J. L. Schwartz, M. Yerushalmy,
- Zaslavsky, O. (1995). Open-ended tasks as a trigger for mathematics teachers’ professional development. For the Learning of Mathematics, 15(3), pp. 1520.

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### Computer as a medium for overcoming misconceptions in solving inequalities

#### Sergei Abramovich, State University of New York at Potsdam, United States; Amos Ehrlich, Tel Aviv University, Israel

Journal of Computers in Mathematics and Science Teaching Vol. 26, No. 3 (July 2007) pp. 181–196

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