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Azevedo, F. S. (2000). Designing representations of terrain: A study in metarepresentational competence. Journal of Mathematical Behavior, 19(4), 443 480. Bamberger, J. (2006). Restructuring conceptual intuitions through invented notations: From path-making to map-making. In E. Teubal, J. Dockrell, & L. Tolchinsky (Eds.), Notational knowledge: Historical and developmental perspectives. Rotterdam/Taipei: Sense Publishers. diSessa, A. A. (1993). Toward an epistemology of physics. Cognition and Instruction, 10(2 3), 105 225. diSessa, A. A. (1995). Designing Newton s laws: Patterns of social and representational feedback in a learning task. In R.-J. Beun, M. Baker, & M. Reiner (Eds.), Dialogue and interaction: Modeling interaction in intelligent tutoring systems (pp. 105 122). Berlin: Springer-Verlag. diSessa, A. A. (2000). Changing minds: Computers, learning, and literacy. Cambridge, MA: MIT Press. diSessa, A. A. (2002). Students criteria for representational adequacy. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 105 129). Dortrecht: Kluwer. diSessa, A. A. (2004). Meta-representation: Native competence and targets for instruction. Cognition and Instruction, 22(3), 293 331. diSessa, A. A. (2007). Systemics of learning for a revised pedagogical agenda. In R. Lesh, E. Hamilton, & J. J. Kaput (Eds.), Foundations for the future in mathematics education. Mahwah, NJ: Lawrence Erlbaum. diSessa, A. A., Hammer, D., Sherin, B., & Kolpakowski, T. (1991). Inventing graphing: Meta-representational expertise in children. Journal of Mathematical Behavior, 10(2), 117 160. Freeman, N. H. (1995). The emergence of a framework theory for pictorial reasoning. In C. Lange-Kuttner & G. V. Thomas (Eds.), Drawing and looking: Theoretical approaches to pictorial representation in children. New York and London: Harvester Wheatsheaf. Gee, J. P. (1996). Social linguistics and literacies: Ideology in discourses (2nd ed.). London: Taylor & Francis. Goody, J. (1977). The domestication of the savage mind. Cambridge, UK: Cambridge University Press. Gravemeijer, K. P. E. (1994). Developing realistic mathematics education. Utrecht: CD- Press / Freudenthal Institute. Hughes, M. (1986). Children and number: Difficulties in learning mathematics. Oxford, UK and New York: Blackwell. Kress, G. (2003). Literacy in the new media age. London and New York: Routledge. Nemirovsky, R., & Tierney, C. (2001). Children creating ways to represent changing situations: On the development of homogeneous spaces. Educational Studies of Mathematics, 45, 67 102. Ong, W. J. (1982). Orality and literacy: The technologizing of the word. New York: Routledge.
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Scribner, S., & Cole, M. (1981). The psychology of literacy. Cambridge, MA: Harvard University Press. Sherin, B. (2001). A comparison of programming languages and algebraic notation as expressive languages for physics. International Journal for Computers and Mathematical Learning, 6, 1 61. Sherin, B., diSessa, A. A., & Hammer, D. M. (1993). Dynaturtle revisited: Learning physics through collaborative design of a computer model. Interactive Learning Environments, 3(2), 91 118. Street, B. V. (1995). Social literacies. London and New York: Longman.
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Appendix A: How Programming Representations Relate to Algebraic Ones
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I aim here to show how the programming representation for Newton s laws relates to the usual, algebraic representation. It is intended to help those less comfortable with the physics. In addition, I want to expose and talk more about how particular representation does, or does not, alter the conceptual landscape. In general, the footnotes in this appendix are for physics experts. Start with the conventional representation of Newton s laws: F = ma First, we drop the m (or, equivalently, set it to one). Mass (m) is just simply another thing to learn about. In fact, mass is not important in terms of the main conceptual problems students have with Newton s laws. Hence: F=a We can use the definition of acceleration as the rate of change of velocity, the ratio of the change of velocity to the change in time: F = v t
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In traditional physics, this ratio is taken in the limit, as t approaches zero. (If t is large, this formula give the average force.) But, in the programming world, time is not continuous; it jumps, one tick of the clock (one iteration of the program s loop ) at a time.9 The best we can do for small times is to take the smallest t possible, one tick.10 Furthermore, for convenience, we can choose units of time so that t is just 1 one tick of the clock.11 So we have: 245
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Andrea A. diSessa F = v Force is precisely the change in velocity (of the object to which the force is applied) for each tick of the clock. If force is an impulse, which occurs in brief (one-tick) bursts like the rocket engine proposed in our designing Newton s laws task, one gets a change in velocity equal to the magnitude of the force during a tick precisely when the rocket fires, and none otherwise. The programming form for a thrust is thus: change v v + F; that is, change v to the old velocity, v, + F This is precisely the kick part of the program spaceship program. If the force persists, this command would be executed each tick of the clock.
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