discovering new knowledge in the context of education: examples from mathematics. sergei abramovich...

Discovering New Knowledge in the Context of Education: Examples

from Mathematics.

Sergei Abramovich

SUNY Potsdam

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.

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.

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).

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

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

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; 4+5=9; 4+5+6=15;

5+6=11; 6+7=13; 7+8=15.

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; 4+5=9; 4+5+6=15;

5+6=11; 6+7=13; 7+8=15.

Trapezoidal representations of integersTrapezoidal 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

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=32n

Trapezoidal representations for

an=52n

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

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?

Example 2.How to show one-fourth?

One student’s representationOne student’s representation

Representation of 1/nRepresentation of 1/n

Possible learning environments (PLE)

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).

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

Measurement as a motivation for the development of inequalities

€

11

+2

2+

33

+4

4> 4

From measurement to formal demonstration

From measurement to formal demonstration

€

11

+2

2+

33

+...+n

n> n

€

11

+3

2+

63

+...+n(n +1)/2

n> n(n +1)/2

Equality as a turning pointEquality as a turning point

€

1

1+

4

2+

9

3+ ...+

n2

n= n2

Surprise! From > through = to <

Surprise! From > through = to <

€

1

1+

5

2+

12

3+ ...+

n(3n −1) /2

n< n(3n −1) /2

And the sign < remains forever:

And the sign < remains forever:

€

P(m,i)

ii=1

n

∑ < P(m,n), m ≥ 5

P(m,n) - polygonal number of side m and rank n

How good is the approximation?How good is the approximation?

0

50

100

150

200

250

300

350

400

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

€

red : P(m,100)

blue :P(m,i)

ii =1

100

∑

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.

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

Example 3.FIBONACCI NUMBERS

REVISITED

€

f k +1 = af k + bf k−1 , k = 1, 2, 3, … ; f0 = f1 = 1 a, b – real numbers When a=b=1 1, 1, 2, 3, 5, 8, 13, …

Spreadsheet explorationsSpreadsheet explorationsHow 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:

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:

€

limk →+∞

fk+1

fk

ConvergenceConvergence

CCCC

PROPOSITION 1.(the duality of computational experiment and

theory)

PROPOSITION 1.(the duality of computational experiment and

theory)

What is happening inside the parabola a2+4b=0?

What is happening inside the parabola a2+4b=0?

Hitting upon a cycle of period three

Hitting upon a cycle of period three

Computational ExperimentComputational 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)

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

Traditionally difficult questions in mathematics

researchDo there exist

cycles with prime number periods?How could those

cycles be computed?

Do there exist cycles with prime number periods?How could those

cycles be computed?

Transition to a non-linear equation

Transition to a non-linear equation

€

gk +1 = a +b

gk

, g1 =1

€

gk = fk / fk−1

Continued fractions emerge

€

fk+1 = afk + bfk −1

Factorable equations of loci

Factorable equations of loci

Loci of cycles of any period reside inside the parabola a2 +

4b = 0

Loci of cycles of any period reside inside the parabola a2 +

4b = 0

Fibonacci polynomialsFibonacci polynomials

€

Pn(x) = d(k,i)xn−i

i =0

n

∑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 modeling of d(k, i)

Spreadsheet graphing of Fibonacci 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

realize, coincides with the number of roots of

when n=(r-1)/2 or

when n=(r-2)/2.

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.

€

gk +1 = a +b

gk

, g1 =1

€

Pn(x) = C2n−ii

i =0

n

∑ xn−i

€

Pn(x) = C2n−i+1i

i =0

n

∑ xn−i+1

Proposition 3.Proposition 3.

For any integer K > 0 there exists integer r > K so that Generalized Golden Ratios

oscillate with period r.

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:

Technology-motivated inquiry into a two-parametric difference equation. International Journal of Mathematical Education in Science

and Technology.

Abramovich, S. & Leonov, G.A. (2008, to appear). Fibonacci numbers revisited:

Technology-motivated 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.

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 remarksConcluding 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.

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.