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From Modeling in Mathematics Education to the Discovery of New Mathematical Knowledge Sergei Abramovich SUNY Potsdam, USA Gennady A. Leonov St Petersburg State University, RUSSIA
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Abstract This paper highlights the potential of modeling with spreadsheets and computer algebra systems for the discovery of new mathematical knowledge. Reflecting on work done with prospective secondary teachers in a capstone course, the paper demonstrates the didactic significance of the joint use of experiment and theory in exploring mathematical ideas.
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Conference Board of the Mathematical Sciences. 2001
Conference Board of the Mathematical Sciences The Mathematical Education of Teachers. Washington, D. C.: MAA. Mathematics Curriculum and Instruction for Prospective Teachers. Recommendation 1. Prospective teachers need mathematics courses that develop deep understanding of mathematics they will teach (p.7).
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Hidden mathematics curriculum
A didactic space for the learning of mathematics where seemingly unrelated concepts emerge to become intrinsically connected by a common thread. Computational modeling techniques allow for the development of entries into this space for prospective teachers of mathematics
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Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … Fk+1 = Fk + Fk-1, F1 = F2 = 1 1, 2, 5, 13, 34, 89, … fk+1 = 3fk - fk-1, f1 =1, f2 = 2 PARAMETERIZATION OF FIBONACCI RECURSION
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Two-parametric difference equation Oscar Perron (1954)
THE GOLDEN RATIO
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Spreadsheet explorations
How do the ratios fk+1/fk behave as k increases? Do these ratios converge to a certain number for all values of a and b? How does this number depend on a and b? Generalized Golden Ratio:
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Convergence
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PROPOSITION 1. (the duality of computational experiment and theory)
CC
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What is happening inside the parabola a2+4b=0?
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Hitting upon a cycle of period three
{1, -2, 4, 1, -2, 4, 1, -2, 4, …}
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Computational Experiment
a2+b=0 - cycles of period three formed by fk+1/fk (e.g., a=2, b=-4) a2+2b=0 - cycles of period four formed by fk+1/fk (e.g., a=2, b=-2) a2+3b=0 - cycles of period six formed by fk+1/fk (e.g., a=3, b=-3)
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Traditionally difficult questions in mathematics research
Do there exist cycles with prime number periods? How could those cycles be computed?
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Transition to a non-linear equation
Continued fractions emerge
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Factorable equations of loci (Maple explorations)
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Pascal’s-like triangle
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The joint use of Maple and theory
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The joint use of Maple and theory
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Loci of cycles of any period reside inside the parabola a2 + 4b = 0 (explorations with the Graphing Calculator [Pacific Tech])
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Fibonacci-like polynomials
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Spreadsheet modeling of Fibonacci-like polynomials
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Spreadsheet graphing of Fibonacci Polynomials
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Proposition 2. The number of parabolas of the form a2=msb where the cycles of period r in equation realize, coincides with the number of roots of when n=(r-1)/2 or when n=(r-2)/2.
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Proposition 2a. Every Fibonacci-like polynomial of degree n has exactly n different roots, all of which are located in the interval (-4, 0).
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Proposition 3. For any integer K > 0 there exists integer r > K so that Generalized Golden Ratios oscillate with period r.
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Proposition 4 (Maple-based MIP).
Corollary (Cassini’s identity):
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Permutations with rises.
Direction of the cycle on a segment The permutation has exactly n rises on {1, 2, 3, …, p} if there exists exactly n – 1 values of j such that ij < ij+1 . Example: [1, 2, 3, …, n] – permutation with n rises The permutation describes the cycle.
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Proposition 5. In a p-cycle determined by the largest in absolute value root of Pp-2(x) there are always one permutation with two rises, one permutation with p rises, and p-2 permutations with p-1 rises.
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Abramovich, S. & Leonov, G. A. (2008)
Abramovich, S. & Leonov, G.A. (2008). Fibonacci numbers revisited: Technology-motivated inquiry into a two-parametric difference equation. International Journal of Mathematical Education in Science and Technology, 39(6), Abramovich, S. & Leonov, G.A. (2009). Fibonacci-like polynomials: Computational experiments, proofs, and conjectures. International Journal of Pure and Applied Mathematics, 53(4),
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Classic example of developing new mathematical knowledge in the context of education Aleksandr Lyapunov ( ) Central Limit Theorem - the unofficial sovereign of probability theory – was formulated and proved (1901) in the most general form (allowing random variables to exhibit different distributions) as Lyapunov was preparing a new course for students of University of St. Petersburg. Each day try to teach something that you did not know the day before.
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Concluding remarks The potential of modeling in mathematics education as a means of discovery new knowledge. The interplay of classic and modern ideas The duality of modeling experiment and theory in exploring mathematical concepts Appropriate topics for the capstone sequence.
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