Books for Foundations for Higher Mathematics


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Books for Foundations for Higher Mathematics

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Jul 10, 2009, 6:57 pm

We are culling some things off our bookshelves in preparation for moving and it got me to thinking about what books you all have used or recommend for making that transition from calculus and college algebra to upperclassman math courses such as modern algebra, topology, and analysis. One we had both used was D'Angelo and West's Mathematical Thinking. D also had Rotman's Journey into Mathematics and Schumacher's aptly named Chapter Zero.

What books have you run into for learning how to solve problems, write proofs, and the like, and what did you like or dislike about them?

Maybe this will also help the autodidacts in the group track down useful resources!

Jul 11, 2009, 6:59 pm

I'm too old to have taken a transitions course myself, but I've taught transitions courses 5 times. The first time I used D'Angelo and West. It had a lot of fun puzzles, but most of them were too hard for my students, and I didn't think content was presented systematically enough. The next 2 times I used Smith, Eggen, and St. Andre's A Transition to Advanced Mathematics. I loved that book, but my students didn't. The 4th time I ran the course as a Moore method course using my own set theory notes. That was a big miscalculation on my part, as the students really weren't mature enough for what I had prepared. Last time I taught out of Chartrand's Mathematical Proofs: A Transition to Advanced Mathematics. I find the book kind of boring, but the students like it, so I'm using it again this Fall. Topics that I consider standard but that aren't given sufficient attention in Chartrand are: total and partial orders, and images and preimages of sets.

Dez 18, 2009, 4:40 pm

I'm not sure exactly what you're after, but Polya's classic How to Solve It is an excellent source of tips and tricks of problem solving. It presents principles of reasoning that are essential for everyone -- from freshmen to researchers.

Editado: Jan 19, 2010, 1:59 pm

Proofs and Fundamentals by Ethan D. Bloch is recommended.

Fev 10, 2010, 1:01 pm

I really enjoy using Sundstrom's Mathematical Reasoning: Writing and Proof; it is written to be readable by students, there are "pre-lecture" exercises to prepare students, and there is a good amount on mathematical style, which is often overlooked in these types of books. The only thing it is missing, in my humble opinion, is a bit of introductory group theory and analysis (epsilon-delta proofs). I have written handouts for these topics, however, because I really like this text.