Problem Solving Textbook Recomendations

[snoble]

I might be teaching a class next year simply called "problem solving." The idea of the class is to survey a several different general topics with a goal of teaching students how to mathematically and formally write up their ideas. Basically get students out of the Calc I habit of writing strings of equations and moving them towards a more prose discussion of their work.

Hopefully this will be the answer to students who complain "no one has ever told us what is a proof."

In the past this class has been taught without a textbook but I would be curious if anyone has any recommendations for a textbook on such a general topic.

Text book questions are not that important as I should have a wealth of questions to choose from. Does anybody remember reading a particular text when suddenly it clicked as to how this whole math language thing worked? I'm still waiting for it to click in my head how this whole english language thing works.

Steven

 

[whozum]

I wouldve never guessed english was your second language, but that epiphany happened to me outside of a textbook, but rather just at a general college experience.

However, I did use Stewart's 5th Edition MV calc with transcendentals for my calc 1-3 series.

 

[honestrosewater]

mathwonk just taught a similar class (here's the thread). You may want to PM him if he doesn't see this.
Everything just clicked for me recently when I understood what it meant for a propositional calculus to be sound and complete (in the sense that a formula F being a theorem implies F is a tautology and F being a tautology imples F is a theorem- apparently soundness and completeness have different meanings in different contexts). matt grime has said that a superficial skate through logic does more harm than good, but if you want to take the time, really understanding some basic logic and set theory would certainly help with problem solving and proof writing.
I've never taught a class before, but for what it's worth, I would want to have learned logic as follows. Read section 1.1 of "Mathematical Logic" by Joseph Shoenfield. Cover "Logic" by Wilfrid Hodges up to the introduction of syntactic sequents. Learn to use the natural deduction system presented here. Finish Hodges. Digest chapter 7 of "Set theory, logic and their limitations" by Moshé Machover. Add in some basic set theory, and I'd feel prepared for anything :)
You can read section 1.1 of Shoenfield online here. He sets the stage gently, and it really helps to be given those pieces straight away. "Logic" by Wilfrid Hodges is the best (and smallest) logic book I've ever read. He starts with arguments and validity, assuming no prior knowledge of logic, and covers all of the basics, with one exception: He doesn't introduce a proof procedure. But you can find several natural deduction systems online, like the one listed above, and they're easy to pick up. Because it's so well-written, the book is very tiny; I think I read it in about a week. A new edition is coming out in Decemeber and I can only find it used in the US- less than $20. The most recent edition is still available new in the UK - 7 pounds @Amazon- if you're not in the UK, it's a deal even with shipping.
Chapter 7 of "Set theory, logic and their limitations" by Moshé Machover formalizes Hodges. He constructs the language from scratch- and this is what I find makes such an enormous difference. The chapter concludes with a completeness proof. He uses the Definiton-Theorem-Remark format and doesn't waste a word, so it would be easy to write up your own notes (so your students wouldn't need the book). From Amazon- $33 new, $20 used. Worth its weight in gold. :)
You can incorporate problems from other branches of math along the way. And propositional logic is simple enough that you can start with almost nothing and build a whole system in a rather short time- and having a whole system in hand is when it clicked in a huge way for me. Plus, it's a great launchpad. Okay, I'm shutting up. Good luck with the class.