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This paper explores the potential of using modeling with spreadsheets and computer algebra systems for the discovery of new mathematical knowledge. It highlights the didactic significance of combining experiment and theory in exploring mathematical ideas. Topics covered include Fibonacci numbers, the Golden Ratio, parabolas, loci, Fibonacci-like polynomials, and permutations.
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From Modeling in Mathematics Education to the Discovery of New Mathematical Knowledge Sergei AbramovichSUNY Potsdam, USAGennady A. LeonovSt Petersburg State University, RUSSIA
AbstractThis 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.
Conference Board of the Mathematical Sciences. 2001.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).
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
Fibonacci numbers • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … • Fk+1 = Fk+ Fk-1, F1 = F2 = 1 • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … • 1, 2, 5, 13, 34, 89, … • fk+1 = 3fk - fk-1, f1 =1, f2 = 2 • PARAMETERIZATION OF FIBONACCI RECURSION
Two-parametric difference equationOscar Perron (1954) • THE GOLDEN RATIO
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:
PROPOSITION 1.(the duality of computational experiment and theory)
Hitting upon a cycle of period three {1, -2, 4, 1, -2, 4, 1, -2, 4, …}
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)
Traditionally difficult questions in mathematics research Do there exist cycles with prime number periods? How could those cycles be computed?
Transition to a non-linear equation Continued fractions emerge
Loci of cycles of any period reside inside the parabola a2 + 4b = 0(explorations with the Graphing Calculator [Pacific Tech])
Proposition 2. The number of parabolas of the form a2=msbwhere the cycles of period r in equationrealize, coincides with the number of roots of when n=(r-1)/2 or when n=(r-2)/2.
Proposition 2a. • Every Fibonacci-like polynomial of degree n has exactly n different roots, all of which are located in the interval (-4, 0).
Proposition 3. For any integer K > 0 there exists integer r > K so that Generalized Golden Ratios oscillate with period r.
Proposition 4 (Maple-based MIP). Corollary (Cassini’s identity):
Permutations with rises. Direction of the cycle on a segment The permutationhas 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 permutationdescribes the cycle.
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.
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), 749-766.Abramovich, S. & Leonov, G.A. (2009). Fibonacci-like polynomials: Computational experiments, proofs, and conjectures. International Journal of Pure and Applied Mathematics, 53(4), 489-496.
Classic example of developing new mathematical knowledge in the context of educationAleksandr Lyapunov (1857-1918)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.
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.