**Introduction to mathematical thinking**

Learn how to think the way mathematicians do-a powerful cognitive process developed over thousands of years.

The goal of the course is to help you develop a valuable mental ability-a powerful way of thinking that our ancestors have developed over three thousand years.

Mathematical thinking is not the same as doing mathematics-at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box-a valuable ability in today's world. This course helps to develop that crucial way of thinking.

The course is offered in two versions. The eight-week-long Basic Course is designed for people who want to develop or improve mathematics-based, analytic thinking for professional or general life purposes. The ten-week-long Extended Course is aimed primarily at first-year students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind. The final two weeks are more intensive and require more mathematical background than the Basic Course. There is no need to make a formal election between the two. Simply skip or drop out of the final two weeks if you decide you want to complete only the Basic Course.

Course Syllabus

1. Introductory material

2. Analysis of language-the logical combinators

3. Analysis of language-implication

4. Analysis of language-equivalence

5. Analysis of language-quantifiers

6. Working with quantifiers

7. Proofs

8. Proofs involving quantifiers

9. Elements of number theory

10. Beginning real analysis

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