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Conference Board of the Mathematical Sciences (2001)

Integrating Historical Perspectives and Digital Technology in Mathematics Teacher Education Sergei Abramovich SUNY Potsdam, USA. Conference Board of the Mathematical Sciences. 2001. The Mathematical Education of Teachers . Washington, D. C.: MAA.

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Conference Board of the Mathematical Sciences (2001)

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  1. Integrating Historical Perspectives and Digital Technology in Mathematics Teacher Education Sergei AbramovichSUNY Potsdam, USA

  2. Conference Board of the Mathematical Sciences. 2001.The Mathematical Education of Teachers. Washington, D. C.: MAA. Appropriate historical contexts can provide “insight for teaching that [prospective teachers] are unlikely to acquire in courses for mathematics majors headed to graduate school or technical work” (p. 127).

  3. Conference Board of the Mathematical Sciences (2001) Capstone idea: helping prospective teachers “make insightful connections between the advanced mathematics they are learning and high school mathematics they will be teaching” (p. 39).

  4. 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

  5. Theodorus of Cyrene - a Pythagorean philosopher of the 5th century B.C. (a teacher of Plato) - studied line segments incommensurable in length using the spiral(drawn using a stick in the sand) Spreadsheet construction

  6. Spreadsheet embodies a mathematical definition (0,0) (x2,y2) 11 (x1,y1) (x0,y0)

  7. Square is the main geometric figure in the Sulvasutras - an ancient (6th - 3rd centuries B.C.) Indian geometry text known as the “rules of cord.”Example from the text: “The cord [sulva] stretched in the diagonal of an oblong produces both (areas) which the cords forming the longer and shorter sides produce.”

  8. “Atomistic” idea of numbers as geometric patterns (early Pythagoreans). Polygonal numbers can be found in the spiral as areas produced by the legs of the right triangles. Triangular numbers: 1, 3, 6, 10, 15, … Square numbers: 1, 4, 9, 16, 25, … Pentagonal numbers: 1, 5, 12, 22, 35, … …

  9. Developing polygonal numbers as a sieve process The Sieve of Eratosthenes (a 3rd century B.C. Greek mathematician, astronomer, geographer) Grassmann (a 19th century Germam mathematician, physicist, linguist, school teacher) – Axiomatization of Arithmetic; translation of the Rigveda – hymns dedicated to the gods in Hinduism

  10. PROBLEM POSING in the context of summation (the spiral of Theodorus) PP is “an activity that at the heart of doing mathematics” (NCTM,1989) Experience in problem posing for teachers: Why are some problems more difficult to solve than the others? Compare:to In general: to

  11. What is a hidden meaning of the equalities 1 + 2 + 3 + … + n = tn (43) 1 + 3 + 5 + … +2n-1 = n2? (64) These are special cases of PROPOSITION 1

  12. P(m, n) – polygonal number of side m and rank n; Tn - tetrahedral number of rank n; tn – triangular number of rank n.

  13. Pascal’ Triangle (17th century) China: Yanghui Triangle (13th century) Natural (RED); Triangular (BLUE); Tetrahedral (GREEN) numbers

  14. Fibonacci Polynomials x+1=0; x+2=0; x2+3x+1=0; x2+4x+3=0; x3+5x2+6x+1=0; x3+6x2+10x+4=0; x4+7x3+15x2+10x+1=0; …

  15. ALL FIBONACCI POLYNOMIALS HAVE ROOTS IN (-4, 0) ONLY! Modern application: Generalized Golden Ratios can oscillate with arbitrary large period (see Abramovich & Leonov, 2008, IJMEST).

  16. Summation vs. estimation In many cases summation formulas are difficult to find Transition from equalities to inequalities

  17. Numbers = twice the area of triangle E √4 D √3 O C √2 1 A B Measurement motivates the inequality

  18. Possible learning environment (PLE) – “a conceptual generalization a teacher can use in the creation of learning environments” • (Steffe, 1991). • Mathematical Induction Proof: • An agency for PLEs • Motivates problem posing

  19. Verifying the base clause PLE 1: Prove that a cord stretched in a leg of an isosceles right triangle is bigger that the one stretched in the altitude dropped on the hypotenuse.

  20. Verifying the base clause (continued) PLE 2: Interpret the base clause as the triangle inequality. 1

  21. Verifying the inductive transfer PLE 3: How are n and related? PLE 4: Prove the inequality

  22. PLE 4: Prove the divergence of the series Reduction to the harmonic series Convergence of the harmonic series: D’Oresme (French, 14th century); Mengoly (Italian, 16th century); Jacob Bernoulli (Swiss,17th century); Leibniz (German, 17th century); Euler (Swiss, 18th century).

  23. Concluding remarks Historical perspectives add context to mathematics teaching The geometrization of mathematical ideas has roots in history Technology use motivates mathematical learning Formal proof stimulates the development of new leaning environments The unity of history, mathematics and technology addresses the CBMS recommendations for the preparation of teachers

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