Trapezoidal representations for an=5 2n
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Transcript Trapezoidal representations for an=5 2n
Discovering New Knowledge
in the Context of Education:
Examples from Mathematics.
Sergei Abramovich
SUNY Potsdam
Abstract
This presentation will reflect on a number of mathematics education courses taught
by the author to prospective K-12 teachers. It will highlight the potential of
technology-enhanced educational contexts in discovering new mathematical
knowledge by revisiting familiar concepts and models within the framework of “hidden
mathematics curriculum.” Situated addition, unit fractions, and Fibonacci numbers will
motivate the presentation leading to a mathematical frontier.
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.
Technological tools allow for the
development of entries into this space
for prospective teachers of mathematics
Example 1.
“Find ways to add consecutive numbers in
order to reach sums between 1 and 15.”
Van de Walle, J. A. 2001. Elementary and
Middle School Mathematics (4th edition) , p. 66.
1+2=3; 1+2+3=6; 1+2+3+4=10; 1+2+3+4+5=15;
2+3=5; 2+3+4=9; 2+3+4+5=14;
3+4=7; 3+4+5=12;
5+6=11;
6+7=13;
4+5=9; 4+5+6=15;
7+8=15.
Trapezoidal representations of integers
Polya, G. 1965. Mathematical Discovery, v.2, pp. 166,
182.
T(n) - the number of trapezoidal representations of n
T(n) equals the number of odd divisors of n.
15: {1, 3, 5, 15}
15=1+2+3+4+5; 15=4+5+6; 15=7+8; 15=15
Trapezoidal representations
for
an=32n
Trapezoidal representations
for an=52n
If N is an odd prime, then for all
integers m≥ log2(N-1)-1 the number
of rows in the trapezoidal
representation of 2mN equals to N.
Examples: N=3, m≥1;N=5, m≥2.
Abramovich, S. (2008, to appear). Hidden mathematics
curriculum of teacher education: An example. PRIMUS
(Problems, Resources, and Issues in Mathematics Undergraduate Studies).
Spreadsheet modeling
Example 2.
How to show one-fourth?
One student’s representation
Representation of 1/n
Possible learning environments (PLE)
Steffe, L.P. 1991. The constructivist teaching
experiment. In E. von Glasersfeld (ed.),
Radical Constructivism in Mathematics
Education.
Abramovich, S., Fujii, T. & Wilson, J. 1995.
Multiple-application medium for the study of
polygonal numbers. Journal of Computers in
Mathematics and Science Teaching, 14(4).
Measurement as a motivation for the
development of inequalities
1
2
3
4
4
1
2
3
4
From measurement to formal
demonstration
1
2
3
n
...
n
1
2
3
n
1
3
6
n(n 1)/ 2
...
n(n 1)/ 2
1
2
3
n
Equality as a turning point
2
1
4
9
n
2
...
n
1
2
3
n
Surprise!
From > through = to <
1
5
12
...
1
2
3
n(3n 1) /2
n(3n 1) /2
n
And the sign < remains
forever:
n
P(m,i)
i P(m,n), m 5
i 1
P(m,n) - polygonal number of side m
and rank n
How good is the approximation?
400
350
300
250
red : P(m,1 00)
100
blue:
P(m,i)
i
i 1
200
150
100
50
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Abramovich, S. and P. Brouwer. (2007). How to
show one-fourth? Uncovering hidden context
through reciprocal learning. International Journal
of Mathematical Education in Science and
Technology, 38(6), 779-795.
Example 3.
FIBONACCI NUMBERS
REVISITED
f k1 afk bfk1 , k = 1, 2, 3, É ; f0 = f1 = 1
a, b Ğreal numbers
When a=b=1
1, 1, 2, 3, 5, 8, 13, É
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:
fk1
lim
k f
k
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
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
fk1 af k bf k 1
gk f k / f k1
b
gk 1 a , g1 1
gk
Continued fractions emerge
Factorable equations of loci
Loci of cycles of any period reside
inside the parabola a2 + 4b = 0
Fibonacci polynomials
n
Pn(x) d(k,i)x
ni
i 0
d(k, i)=d(k-1, i)+d(k-2, i-1)
d(k, 0)=1, d(0, 1)=1, d(1, 1)=2, d(0, i)=d(1, i)=0, i≥2.
Spreadsheet modeling of d(k, i)
Spreadsheet graphing of Fibonacci
Polynomials
Proposition 2. The number of parabolas of the form
a2=msb where the cycles of period r in equation
b
gk 1 a , g1 1
gk
realize, coincides with the number of roots of
n
i
Pn(x) C2ni
x ni
i 0
when n=(r-1)/2 or
n
i
Pn(x) C2ni1
x ni1
when n=(r-2)/2.
i 0
Proposition 3.
For any integer K > 0 there
exists integer r > K so that
Generalized Golden Ratios
oscillate with period r.
Abramovich, S. & Leonov, G.A. (2008, to appear).
Fibonacci numbers revisited: Technologymotivated inquiry into a two-parametric difference
equation. International Journal of Mathematical
Education in Science and Technology.
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 as Lyapunov was preparing a new
course on probability theory
Each day try to teach something that you did not know the
day before.
Concluding remarks
The potential of technology-enhanced
educational contexts in discovering new
knowledge.
The duality of experiment and theory in
exploring mathematical ideas.
Appropriate topics for the capstone sequence.