Download Trapezoidal representations for an=5 2n

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
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.
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 equation
Oscar Perron (1954)
f k 1  af k  bf k 1 ,
a  b1
f0  f1  1

k 0
{F } : 1, 1, 2, 3, 5, 8, ...
Fk 1 1  5
lim

k  F
2
k

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:
lim
k 
f k 1
fk
Convergence
PROPOSITION 1.
(the duality of computational experiment and theory)
CC
What is happening inside the
2
parabola a +4b=0?
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
fk1  afk  bfk1
gk  f k / f k1


b
gk 1  a  , g1  1
gk
Continued fractions emerge
Factorable equations of loci
(Maple explorations)
Pascal’s-like triangle
The joint use of Maple and theory
The joint use of Maple and theory
Loci of cycles of any period reside
inside the parabola a2 + 4b = 0
(explorations with the Graphing Calculator [Pacific Tech])
Fibonacci-like polynomials
xa /b
2
n
n
i0
i0
i
n i
i
n i
P2n1 (x)   C2n
x

0,
P
(x)

C
x
 2n i1  0,
i
2n
Pn (x)  x
mod( n,2)
Pn1 (x)  Pn 2 (x)
P0 (x)  1, P1 (x)  x  1
Spreadsheet modeling of Fibonacci-like polynomials
Spreadsheet graphing of Fibonacci
Polynomials
Proposition 2.
The number of parabolas of the form a2=msb
where the cycles of period r in equation
g k 1  a  b / g k , g1  1
realize, coincides with the number of roots of
n
i
Pn (x)  C2ni
x ni
when n=(r-1)/2 or
i 0
n

i
n i
Pn (x)   C2n
x
i1
i0
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).
P2n1 (x)P2n1 (x)  xP2n2 (x)  1
xP2n (x)P2n2 (x)  P
2
2n1
(x)  1
Corollary (Cassini’s identity):
Fn1 Fn1  (Fn )  (1)
2
n1
Permutations with rises.
{g1 (a), g2 (a),..., g p (a)}
Direction of the cycle on a segment
gi1 (a)  gi2 (a)  ...  gip (a)
The permutation [i1 , i2 ,..., i p ] 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 [i1 , i2 ,..., i p ]
describes 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). Fibonaccilike 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 education
Aleksandr 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.