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 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  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 Elementary and Middle School Mathematics (4 th edition), p =3; 1+2+3=6; =10; =15; 2+3=5; 2+3+4=9; =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 Elementary and Middle School Mathematics (4 th edition), p =3; 1+2+3=6; =10; =15; 2+3=5; 2+3+4=9; =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 integers  Polya, G 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= ; 15=4+5+6; 15=7+8; 15=15  Polya, G 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= ; 15=4+5+6; 15=7+8; 15=15

Trapezoidal representations for a n = 3  2 n

Trapezoidal representations for a n = 5  2 n

If N is an odd prime, then for all integers m≥ log 2 (N-1)-1 the number of rows in the trapezoidal representation of 2 m N 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 Spreadsheet modeling If N is an odd prime, then for all integers m≥ log 2 (N-1)-1 the number of rows in the trapezoidal representation of 2 m N 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 Spreadsheet modeling

Example 2. How to show one-fourth?

One student’s representation

Representation of 1/n

Possible learning environments (PLE)  Steffe, L.P The constructivist teaching experiment. In E. von Glasersfeld (ed.), Radical Constructivism in Mathematics Education.  Abramovich, S., Fujii, T. & Wilson, J Multiple-application medium for the study of polygonal numbers. Journal of Computers in Mathematics and Science Teaching, 14(4).  Steffe, L.P The constructivist teaching experiment. In E. von Glasersfeld (ed.), Radical Constructivism in Mathematics Education.  Abramovich, S., Fujii, T. & Wilson, J 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

From measurement to formal demonstration

Equality as a turning point

Surprise! From > through = to <

And the sign < remains forever: P(m,n) - polygonal number of side m and rank n

How good is the approximation?

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),

Example 3. FIBONACCI NUMBERS REVISITED

Spreadsheet explorations  How do the ratios f k+1 /f k 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 f k+1 /f k 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:

Convergence

CC PROPOSITION 1. (the duality of computational experiment and theory)

What is happening inside the parabola a 2 +4b=0?

Hitting upon a cycle of period three

Computational Experiment  a 2 +b=0 - cycles of period three formed by f k+1 /f k (e.g., a=2, b=-4)  a 2 +2b=0 - cycles of period four formed by f k+1 /f k (e.g., a=2, b=-2)  a 2 +3b=0 - cycles of period six formed by f k+1 /f k (e.g., a=3, b=-3)  a 2 +b=0 - cycles of period three formed by f k+1 /f k (e.g., a=2, b=-4)  a 2 +2b=0 - cycles of period four formed by f k+1 /f k (e.g., a=2, b=-2)  a 2 +3b=0 - cycles of period six formed by f k+1 /f k (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? Do there exist cycles with prime number periods? How could those cycles be computed?

Transition to a non-linear equation Continued fractions emerge

Factorable equations of loci

Loci of cycles of any period reside inside the parabola a 2 + 4b = 0

Fibonacci polynomials 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 a 2 =m s b 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 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: 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 ( ) 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.  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.