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Teaching Real Analysis an active approach

Real Analysis or Advanced Calculus?. Real Analysis is the branch of mathematics that allows us to describe limiting processes precisely.It is crucial that our students be able to make direct connections between the mathematical ideas they are studying in their real analysis course and the intuitio

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Teaching Real Analysis an active approach

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    1. Teaching Real Analysis—an active approach Session for 2011-2012 Project NExT Fellows Mathfest, 2011 Lexington, KY Presenter: Carol S. Schumacher Kenyon College And The Educational Advancement Foundation

    2. Real Analysis or Advanced Calculus? Real Analysis is the branch of mathematics that allows us to describe limiting processes precisely. It is crucial that our students be able to make direct connections between the mathematical ideas they are studying in their real analysis course and the intuition they developed about limiting processes in their calculus courses. At the same time, I believe it is incredibly important that the course not be a re-tread of calculus that does no more than “cross t’s and dot i’s.”

    3. First Definitions

    4. We start where the students are

    5. Intuition First

    6. Intuition First

    7. First Definitions

    8. Don’t just stand there! Do something.

    9. Don’t just stand there! Do something.

    10. Second “epsilonics” definition Your students have been thinking about sequence convergence for a while now and they are beginning to get the hang of this new way of thinking. You are ready to tackle continuity. How do you start with your students’ previous understanding of continuity (from calculus) and end up with the standard ?-? definition of continuity?

    11. “That’s obvious.”

    12. I Stipulate Two Things

    13. I Stipulate Two Things

    14. Negating Statements

    16. Thinking like an Analyst We have skills and practices that we use when we think like analysts. We hold presuppositions and assumptions that are unlikely to be shared by a student who is new to real analysis. We know where to focus of our attention and what can be safely ignored.

    17. Thinking like an Analyst We have skills and practices that we use when we think like analysts. We hold presuppositions and assumptions that are unlikely to be shared by a student who is new to real analysis. We know where to focus of our attention and what can be safely ignored.

    18. Great Versatility is Required

    19. Great Versatility is Required

    20. Great Versatility is Required

    21. “Epsilonics” “Organizing principle” for final proof

    22. “Epsilonics”---Some general principles “Organizing principle” for proof

    23. “Epsilonics” proofs General rules of thumb To get started, calculate the quantity that you want to make “small.” Must find a relationship between it and the quantity (or quantities) that you know to be “small.” “The sum of small things is small”---the triangle inequality. “The product of something small and something bounded is small” The “multi-task” delta.

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