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Perspectives on Transition Courses

Perspectives on Transition Courses. Alex M. McAllister alex.mcallister@centre.edu Mathematics Department, Centre College. MAT 290: Foundations of Mathematics.

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Perspectives on Transition Courses

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  1. Perspectives on Transition Courses Alex M. McAllister alex.mcallister@centre.edu Mathematics Department, Centre College

  2. MAT 290: Foundations of Mathematics Foundations of Mathematics develops the abstract thinking and writing skills necessary for proof-oriented mathematics courses and surveys various areas of mathematics. Important mathematics concepts and questions are studied from several areas, including mathematical logic, abstract algebra, number theory, and real analysis. Further topics include complex analysis, statistics, graph theory, and/or other areas of mathematics according to the interests of instructor and students.

  3. Two Motivating Questions • How can we help our majors succeed? Particularly in their upper-level courses? • What should every math major “know” when they graduate?

  4. How can we help ??? Transition from computationally-oriented lower level courses to the more theoretical upper level courses. The hallmarks of a mathematician: • Sound reasoning • Communicating with precise language • Asking probing questions about mathematics

  5. Two Motivating Questions • How can we help our majors succeed? Particularly in their upper-level courses? A Transition-Bridge course… in what context??? • What should every math major “know” when they graduate?

  6. What are the fundamental ideas and questions of mathematics ???

  7. What are the fundamental ideas and questions of mathematics ??? • Pythagorean Theorem • Fermat’s Last Theorem • Division Algorithm • Fundamental Theorem of Arithmetic • Uniqueness of objects/factorization • There exist infinitely many primes • Goldbach’s Conjecture • Riemann Hypothesis • The square root of two is irrational • Fundamental Theorem of Algebra • Graphs of Complex Functions • Mean Value Theorem • The definition of the derivative • Basic properties of functions • Basic properties of derivatives • The definition of the integral • Fundamental Theorem of Calculus • Taylor’s Theorem • Fourier Series • Proofs by Induction

  8. What are the fundamental ideas and questions of mathematics ??? • The definition of number/cardinality • |Z| = |Q| • Cantor’s Theorem • Pascal’s Triangle • The Binomial Theorem • Basic Combinatorics • Basic Probability • Euclidean geometry • Non-Euclidean geometries • The Gödel Incompleteness Theorems • Basics of Set Theory • Russell’s Paradox • Graph Theory • Attributes/adjectives of functions: • one-to-one • onto • increasing/decreasing • continuous • differentiable • integrable • Equal • Basics of logic • connectives • truth tables • arguments • quantifiers

  9. Two Motivating Questions • How can we help our majors succeed? Particularly in their upper-level courses? A Transition-Bridge course… in what context??? • What should every math major “know” when the graduate from Centre College? A Survey course…

  10. The Book…A Survey of Advanced Mathematics • Chapter 1: Mathematical Logic • Chapter 2: Abstract Algebra • Chapter 3: Number Theory • Chapter 4: Real Analysis • Chapter 5: Probability and Statistics • Chapter 6: Graph Theory • Chapter 7: Complex Analysis

  11. Lead up to Infinitude of the PrimesEmbedded Question 9 on pp 145 (a) Determine if the following integers are prime; if not, give a nontrivial divisor. • 2 + 1 • 2 • 3 + 1 • 2 • 3 • 5 + 1 • 2 • 3 • 5 • 7 + 1 (b) Formulate a conjecture about the number p1 • p2 •…• pn + 1.

  12. Lead up to Infinitude of the PrimesEmbedded Question 9 on pp 145 (a) Determine if the following integers are prime; if not, give a nontrivial divisor. • 2 + 1 = 3 • 2 • 3 + 1 = 7 • 2 • 3 • 5 + 1 = 31 • 2 • 3 • 5 • 7 + 1 = 211 (b) Formulate a conjecture about the number p1 • p2 •…• pn + 1.

  13. Lead up to Infinitude of the PrimesEmbedded Question 9 on pp 145 (a) Determine if the following integers are prime; if not, give a nontrivial divisor. • 2 + 1 = 3 • 2 • 3 + 1 = 7 • 2 • 3 • 5 + 1 = 31 • 2 • 3 • 5 • 7 + 1 = 211 (b) Formulate a conjecture about the number p1 • p2 •…• pn + 1. (c) Find two primes greater than 50 that divide 2 • 3 • 5 • 7 • 11 • 13 + 1 ( = 30031 = 59 • 509). (d) Reformulate your conjecture. What can you say about the primes p1, …, pn (not) dividing p1 • p2 •…• pn + 1?

  14. Prime Number Theorem Exercise #62 on pp 150 Complete the following table. The following data suggests lim (n->∞) π(n) / [n/ln(n)] = ??

  15. Prime Number Theorem Exercise #62 on pp 150 Complete the following table. The following data suggests lim (n->∞) π(n) / [n/ln(n)] = ??

  16. Some Success

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