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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

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Page 1: 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

Discovering New Knowledge in the Context of Education: Examples

from Mathematics.

Sergei Abramovich

SUNY Potsdam

Page 2: 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.

Page 3: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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).

Page 4: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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

Page 5: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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.

Page 6: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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

Page 7: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

Trapezoidal representations

for an=32n

Trapezoidal representations

for an=32n

Page 8: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

Trapezoidal representations for

an=52n

Trapezoidal representations for

an=52n

Page 9: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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

Page 10: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

Example 2.How to show one-fourth?

Example 2.How to show one-fourth?

Page 11: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

One student’s representationOne student’s representation

Page 12: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

Representation of 1/nRepresentation of 1/n

Page 13: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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).

Page 14: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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

Page 15: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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

Page 16: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

Equality as a turning pointEquality as a turning point

1

1+

4

2+

9

3+ ...+

n2

n= n2

Page 17: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

Surprise! From > through = to <

Surprise! From > through = to <

1

1+

5

2+

12

3+ ...+

n(3n −1) /2

n< n(3n −1) /2

Page 18: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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

Page 19: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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

Page 20: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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.

Page 21: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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, …

Page 22: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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

Page 23: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

ConvergenceConvergence

Page 24: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

CCCC

PROPOSITION 1.(the duality of computational experiment and

theory)

PROPOSITION 1.(the duality of computational experiment and

theory)

Page 25: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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

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

Page 26: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

Hitting upon a cycle of period three

Hitting upon a cycle of period three

Page 27: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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)

Page 28: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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?

Page 29: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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

Page 30: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

Factorable equations of loci

Factorable equations of loci

Page 31: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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

Page 32: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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.

Page 33: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

Spreadsheet modeling of d(k, i)Spreadsheet modeling of d(k, i)

Page 34: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

Spreadsheet graphing of Fibonacci Polynomials

Spreadsheet graphing of Fibonacci Polynomials

Page 35: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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

Page 36: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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.

Page 37: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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.

Page 38: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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.

Page 39: Discovering New Knowledge in the Context of Education: Examples from Mathematics. Sergei Abramovich SUNY Potsdam

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.