Thinking Mathematically

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1. nhatcherJan 2023
   Lot's of good books already here! In the same spirit as Polya's book "Thinking mathematically" by J. Mason, L. Burton and K. Stacey I learned a lot in the early days with Demidovich book on 5000 problems on mathematical analysis. Tom Apostol books on calculus, but for me his book on analytical number theory. Alongside alan baker's thin book on number theory. Gilbert Strang book(s) on linear algebra. Rudin book on functional analysis. Oh Hardy's book on divergent series! Ian Steward books on transcendental numbers and Galois theory. Elements of algebraic topology by Munkres is a fantastic book. So many books are invaluable to me in teaching not only math but mathematical thinking.

   I guess if you want to learn thinking but not necessarily math "thinking mathematically" above mentioned is your friend.

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2. matthewmorganJan 2023
   Came here to recommend Thinking Mathematically (text only) by J. Mason. If only every high school maths teacher had this book, the world would be a better place
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3. aethertapSep 2017
   I can second this, with a minor caveat. My daughter (third grade) loves these books, and we get a lot of good problem-solving into the curriculum as a result. The caveat, in my opinion, is that they don't provide enough cyclic review in their default configuration.

   Each chapter includes 80-100 problems, divided in to usually between 4-8 sections. The problems are great, but once a section is complete they're weak on later refreshes. I've been working around this by doing even-numbered problems the first time through a section, then half of the odds a few days later when we're a couple of sections downstream, then selecting randomly from all of the unfinished problems in the entire curriculum for just a couple of extra "old stuff" problems each day throughout the year. We also supplement with a number of other great resources, if you're looking to implement a more problem- and exploration-oriented math curriculum:

   1. Kitchen Table Math is great for selecting concepts to lead number talks with (for building number sense - this is the first part of our day)

   2. Saxon has excellent spaced-repetition exercises for shoring up the calculation side of things, and giving the student some easy wins for confidence building (we typically use Saxon's material as a warmup before Beast Academy)

   3. Thinking Mathematically (the one by J. Mason and L. Burton) has a unique and useful mental process for attacking hard problems when you're not handed a nice formula to plug things into. Once a week, we work through a hard problem using the method in this book.

   4. I haven't worked it in yet, but Arthur Benjamin's "Secrets of Mental Math" has a lot of stuff in it that will solidly connect arithmetic and algebraic thinking later.

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4. hdividerSep 2013
   I'd suggest that you also spend some time learning how to learn maths more effectively. That usually pays dividends over time and ought to give you a significant advantage.

   Two good books on this:

   How to Solve It

   http://www.amazon.com/How-Solve-It-Mathematical-Princeton/dp...

   Thinking Mathematically

   http://www.amazon.com/Thinking-Mathematically-J-Mason/dp/020...

   And if you want to get better at sitting maths-based exams, here's my own book, Exam Mastery: How to excel in maths-heavy exams :

   http://www.amazon.com/Exam-Mastery-excel-maths-heavy-ebook/d...

   (No need to buy though - if you want it, shoot me an email and I'll just send it over to you.)

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5. plinkplonkJul 2010
   Get the book "Thinking Mathematically" (http://www.amazon.com/Thinking-Mathematically-J-Mason/dp/020... non affiliate link) which exposes the process of how mathematicians think, using simple problems. You can then adapt the process to your preferred level of problems. Also learn proof technique (Velleman's How to Prove it" is a good book for this) and doesn't make any assumptions of mathematical knowledge.
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