This article probes some transitions from naive realism to formal mathematical modeling, using a Direct Manipulation Environment (DME) for Newtonian Mechanics. It takes the form of a case study of two pre-university students as they work on a system of connected particles using the DME Interactive Physics (Knowledge Revolution). The article analyses the ways in which students developed an argument to account for the system's behavior. A discussion follows which examines the interplay of mathematical reasoning, modes of interaction with technology, and setting, in the process of formal mathematical modeling. Throughout the article, comparisons are made between the direct manipulation metaphor and interpretations of the microworld notion to aid analysis.
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References
1.
AckermannE. (1991). From decontextualized to situated knowledge: Revisiting Piaget's water-level experiment. In HarelI. & PapertS. (Eds.), Constructionism (pp. 269–294). New Jersey: Ablex Publishing Co.
2.
BerryJ., & GrahamT. (1991). Using concept questions in teaching mechanics. International Journal of Mathematics Education in Science and Technology, 22, 749–757.
3.
BlissJ. (1994). From mental models to modeling. In MellarH.BlissJ.BoohanR., & TompsettC. (Eds.), Learning with artificial worlds: Computer-based modelling in the curriculum (pp. 27–32). London: Falmer Press.
4.
BrousseauG., & OtteM. (1991). The fragility of knowledge. In BishopA.Melin-OlsenS., & Van DormolenK. (Eds.), Mathematical knowledge: Its growth through teaching (pp. 56–69). Dordrecht: Kluwer.
5.
ConfreyJ. (1990). A review of the research on students conceptions in mathematics, science and programming. Review of Educational Research, 16, 3–56.
6.
diSessaA. A. (1986). Notes of the future of programming: Breaking the utility barrier. In NormanD. A. & DraperS. W. (Eds.), User centered system design. New perspectives on human-computer interactions (pp. 125–152). Hillside, NY: Lawrence Erlbaum.
7.
diSessaA. A. (1989). A child's science of motion: Overview and first results. In LeronU. & KrumholtzN. (Eds.), Proceedings of the Fourth International Conference for Logo and Mathematics Education (pp. 207–227). Jerusalem.
8.
DoerrH. (1997). Experiment, simulation and analysis: An integrated instructional approach to the concept of force. International Journal of Science Education, 19(3), 265–282.
9.
EdwardsL. (1995). Microworlds as representations. In diSessaA.HoylesC., & NossR. (Eds.), Computers and exploratory learning. NATO ASI Series F. Computers and systems sciences (Vol. 146, pp. 127–154). Berlin: Springer-Verlag.
10.
GrahamT., & BerryJ. (1996). A hierarchical model of development of student understanding of momentum. International Journal of Science Education, 18(1), 75–89.
11.
HarveyB. (1985). Computer science: Logo style. Cambridge, MA: MIT Press.
12.
HealyL., & HoylesC. (1998). Justifying and proving in school mathematics: Summary of the survey results. End of project report. Institute of Education, University of London.
13.
HoylesC. (1993). Microworld/Schoolworld: The transformation of an innovation. In KeitelC. & RuthvenK. (Eds.), Learning from computers: Mathematics education and technology. NATO ASI Series F. Computers and systems sciences (Vol. 121, pp. 1–17). Berlin: Springer-Verlag.
14.
HoylesC., & SutherlandR. (1989). Logo mathematics in the classroom. London. Routledge.
15.
HutchinsE. L.HollanJ. D., & NormanD. A. (1986). Direct manipulation interfaces. In NormanD. A. & DraperS. W. (Eds.), User centered system design. New perspectives on human-computer interactions (pp. 87–124). Hillside, NY: Lawrence Erlbaum.
16.
Interactive Physics (1986–1999). Knowledge revolution. 66 Bovet Road, Suite 200, San Mateo, California.
17.
KaputJ., & RoschelleJ. (1999). The mathematics of change and variation from a millennial perspective: New content, new context. In HoylesC.MorganC., & WoodhouseG. (Eds.), Rethinking the mathematics curriculum (pp. 155–170). London: Falmer Press.
18.
KieranC. (1992). The teaching and learning of school algebra. In GrouwsD. (Ed.), Handbook of research on mathematics teaching and learning. New York: NCTM.
19.
KitchenA.SavageM., & WilliamsJ. (1997). The continuing relevance of mechanics in a-level mathematics. Teaching Mathematics and its Applications, 16(4), 165–170.
20.
LabordeC., & LabordeJ-M. (1995). What about a learning environment where Euclidean concepts are manipulated with a mouse? In diSessaA.HoylesC., & NossR. (Eds.), Computers and exploratory learning. NATO ASI Series F: Computer and systems science (Vol. 146, pp. 241–262). Berlin: Springer.
21.
McDermottL.ShafferP., & SomersM. (1994). Research as a guide for teaching introductory mechanics: An illustration in the context of the attwood machine. American Journal Physics, 62(1).
22.
NossR., & HoylesC. (1996). Windows on mathematical meaning. Dordrecht: Kluwer Press.
RoschelleJ. (1998). Beyond romantic and skeptic: A microanalysis of conceptual change in kinematics. International Journal of Science Education, 20(9), 1025–1042.
26.
SmartT., & BurnsS. (1988). Sixth form girls using computers to explore Newtonian Mechanics. In HoylesC. (Ed.), Girls and computers. General issues and case studies of Logo in the mathematics classroom (pp. 64–73). Institute of Education: Bedford Way papers 34.
27.
SmithJ.diSessaA., & RoschelleJ. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition. The Journal of the Learning Sciences, 3(2), 115–163.
28.
StevensonI. J. (1996). Constructing curvature: The iterative design of a computer-based microworld for non-euclidean geometry. Unpublished PhD Dissertation. University of London.
29.
StevensonI. J. (1999). A journey through geometry: Sketches and reflections on learning. For the Learning of Mathematics, 19(2), 42–47.
30.
StevensonI. J. (2000a). Modeling hyperbolic space: Designing a computational context for learning non-euclidean geometry. International Journal of Computers for Mathematics Learning. 5(2), 1–25.
31.
StevensonI. J. (2000b). Mechanics, modeling and pedagogy: Reflections on the design of contexts for new technologies. In RowlandT. & MorganC. (Eds.), Research in mathematics education Volume 2: Papers of the British Society for Research into Learning Mathematics (pp. 137–150). London: British Society for Research into Learning Mathematics.
32.
StevensonI. J. (2001). Disruption and disclosure: Learning to model spherical geometry. For the Learning of Mathematics, 21(1), 34–38.
