This guide is for new K-12 teachers, the focus is on middle and high school. It is meant to either get you ready quickly, and up to a point, or fill in or enrich some gaps, but there are only 50 books I can add to this guide. These are sufficient for a fairly thorough understanding of K-12 math. Consider this guide a jumpstart, or simply look at the section that is of interest to you, as not all are equally useful to each aspiring math teacher.
I made this for anyone that is currently where I was, trying to find a way to learn more, gain a sensible perspective of the whole, without wasting time and money. There are seven sections:
Section 1: General Mathematics Section 2: Math Education Section 3: Fun and Enriching Math Section 4: Resources Section 5: Classroom Management Section 6: The California Subject Examinations for Teachers (CSET) Section 7: More comprehensive texts
Small (BIG) note, if you are looking into becoming a teacher: You will likely be looking at obtaining a credential. Credentials are not limited to university programs. There are alternatives, see http://en.wikipedia.org/wiki/Alternative_teaching_certification For example, in California, you can thru Project Pipeline (projectpipeline.org) enroll in classes and work as an intern at low performing schools, both as required by state law, but... you can also get PAID while working toward a credential. If you feel confident that you can work full time, you do not need to go thru a university program, working for free while paying for tuition!
Hierarchy key: Section > Roman Numeral > A > i
Section 1: General Mathematics These are written by mathematicians and are broad in their overview. A) What Is Mathematics? An Elementary Approach to Ideas and Methods by Richard Courant (see below for more by him) and Herbert Robbins' classic. There's good reason it's first on this list. Very much content at a low cost. (Not that order is important hereafter.) B) Mathematics: A Very Short Introduction by Timothy Gowers (Fields Medalist) Consider also reading his essays, or chapters unpublished in this book, at his website: http://www.dpmms.cam.ac.uk/~wtg10/vsipage.html Recommended to me by a mathematician. Cheap but small C) The Mathematical Experience This is more personal and philosophical. A good guide to the major perspectives in and philosophy of mathematics. D) The Princeton Companion to Mathematics Aims to familiarize some of modern mathematics, even for professional mathematicians in unfamiliar fields, by its well outlined structure. Also edited by Timothy Gowers.
A) The Myth of Ability: Nurturing Mathematical Talent in Every Child For grades 1-8, or remedial work for older students. Definitely see also the accompanying website with workbooks to be used and many pages of samples to download at jumpmath.org Unlike the following book, this one includes explicit examples of how to teach and interact with a student and the strategies to use.
and "Lockhart's Lament - The Sequel" http://www.maa.org/devlin/devlin_05_08.html Major flaw with this book and the essays is the lack of models. But it is inspiring and rings a true bell about genuine learning. Cheap, but not a substantial amount more than the online essays.
III) THE 'GELFAND CORRESPONDENCE PROGRAM IN MATHEMATICS' The first three were originally part of the 'Library School of Mathematics' series, though oddly enough, the third book wasn't included in the later Correspondence Program list:
I know altogether these are ridiculously expensive, and could probably have been put into one volume for less than $50, but it's a good series.
IV) MATHEMATICS STANDARDS
A) The Mathematics Framework for California Public Schools K–12 The Framework is structured around the Content Standards and includes, beyond clarification and elaboration on the standards, many ideas and useful discussion a new teacher will need to consider and find useful. Free to download.
H) Basic Calculus: From Archimedes to Newton to its Role in Science (Textbooks in Mathematical Sciences) This is a most unique textbook, as it combines Greek and modern astronomy, history, science, economics, and calculus, as well as many applications. By far the most context–rich, complete math textbook I have yet seen, at least at this level! Another feature I particularly value is the many instances in the narrative the author asks "why" or subtly nudges the reader to consider a result, find it, or find the reason for it, with just the right touch of tact, not overtly, nor too challenging to leave the reader at a loss, with many results very well explained, but sometimes with a few nudges to think into the problem, rather than being handed all of it on a silver platter. The exercise problems range from simple to very difficult, requiring innovative and multiple steps to arrive at solutions. Don't let Basic in the title suggest easy or shallow. A nice collaborative effort with the physics teacher might be to teach math using this textbook, and the physics course using "The Mechanical Universe" videos and textbooks mentioned below in Section 4 III A. The author has on his website solutions, corrections, and links for historical enrichment: http://www.nd.edu/~hahn
I) Visual Complex Analysis Wow! Simply wow! I don't think you really need this book on this guide, but it's so amazing, so well written, such good graphics, so much you can learn, and has been compared to Feynman's Lectures on Physics in style of surprising ability to convey ideas that I simply had to include it. There are a few other great resources for visual mathematics books, e.g. Proofs without Words, Math Made Visual, Conics by Kendig, or Basic Calculus by Hahn. For my most complete list, check out: Visual Mathematics @ http://www.amazon.com/Visual-Mathematics/lm/R1PYBF6BBHX71P/ref=cm_lmt_fvsy_f_1_rysdsd1 Visual Mathematics II @ http://www.amazon.com/gp/richpub/listmania/fullview/R19JR46D6GWP9D/ref=cm_pdp_lm_title_1
I) MEMBERSHIPS & MAGAZINES Both of the following two organizations offer discounts to students or teachers. All of these publications offer insightful articles on either math, teaching a particular concept, or teaching tips, besides interesting problems, calendars, puzzles, etc. Subscription includes online access to archived articles.
A) The Mathematical Association of America (MAA.org) offers two journals I recommend, Mathematics Magazine and The College Mathematics Journal, though if you want to only get one, go with the first. I also recommend reading the many articles online, whether or not you have a membership, such as Devlin's Angle (read Lockhart's Lament at http://www.maa.org/devlin ), Math DL, MAA Writing Awards, Loci: Resources, etc. NB: The MAA publishes many books. Some of their series are mentioned below in this section 4, VI) Publishers
B) The National Council of Teachers of Mathematics (NCTM.org) offers two magazines, for high school it is the Mathematics Teacher and for middle school, Mathematics Teaching in the Middle School. For their "Principles and Standards", go to http://standards.nctm.org
C) The Math Forum @ Drexel http://mathforum.org/teachers Excellent resources. Much is free, but some things require membership.
II) MAGAZINES (without memberships)
A) + Plus Magazine http://www.plus.maths.org.uk
B) Pi in the Sky http://www.pims.math.ca/resources/publications/pi-sky This is produced for high school students.
III) WEBSITES A) Annenberg Media at http://www.learner.org has some of the best quality videos I've yet seen online in a wide range. Registration is free, and you can watch their videos online for free (or buy hardcopies).
My favorite videos are the Caltech produced "The Mechanical Universe" http://www.learner.org/resources/series42.html The following textbook accompanies the series, and could be used to teach high school physics: The Mechanical Universe: Introduction to Mechanics and Heat An ideal collaborative course might be high school physics using this text, and the calculus class using Hahn's Basic Calculus text mentioned above in Section 3 H. (Volume 1 of 2. Note: there is an "Advanced Edition" that condenses both books into one; I prefer the two book version.)
Important, also look at this site's resources for teachers. I'd add a specific link, but there's so much just for teachers, that exploring their resources might be most benificial.
B) I downloaded the following two movies and burned these to a dvd that I showed in part to kids: Both are produced by mathematicians and can be watched online or downloaded, in low or high quality, for free: i) Möbius Transformations Revealed http://www.ima.umn.edu/~arnold/moebius This is less than 5 minutes, and what you see is what you get. In other words, kids will get the essential idea. ii) Dimensions http://www.dimensions-math.org/Dim_E.htm This movie is over 2 hours long, divided into nine chapters. Much of it can be viewed by young children, the complex numbers section onwards will have a few sections with nice animations, but the concepts are too deep, except to advanced high school students, maybe.
C) Wolfram Education Portal (K-12) http://www.wolfram.com/edu Free small demonstrations to download. Download the free Mathematica Player to use these (link is in the right column on this webpage).
or Wolfram Demonstrations Projects (beyond K-12) http://demonstrations.wolfram.com
D) Math Education http://www.reddit.com/r/matheducation A place to share internet resources for teachers (and students).
E) Better Explained http://betterexplained.com/articles/category/math Kalid has done wonderful work explaining common math topics ... better.
IV) SOFTWARE
A) FREE SOFTWARE
i) Geogebra http://www.geogebra.org Very much like Geometer's Sketchpad. For many files that others have already created go to http://www.geogebra.org/en/wiki For tutorial videos and some demonstrations, see http://math247.pbworks.com/Learn+and+Use+GeoGebra
ii) SAGE http://www.sagemath.org Open-Source alternative to Mathematica and Maple. Can download or use in browser at http://www.sagenb.org For quick video tutorials, see http://www.sagemath.org/help-video.html
iii) Grapher Preinstalled on Mac
iv) Winplot for PC
B) $$$. Discounts available for Mathematica and Maple for teachers and students.
i) Geometer's Sketchpad http://www.dynamicgeometry.com Not as powerful as the following, though definitely sufficient for most of your needs, and, I believe, much easier. Free alternative is Geogebra. Check out the links to see what some have done with it: http://www.dynamicgeometry.com/General_Resources/Links.html
ii) Mathematica http://www.wolfram.com Used by professionals and universities. The difference between this and the Mathematica Player, mentioned above, among other things, is this is a creator, the Player is just a reader, able to read what others have created (besides the fact that the Player is also free).
iii) Maple http://www.maplesoft.com Used by professionals and universities.
V) MANIPULATIVES, KITS, TOOLS
A) Zome tools http://zometool.com
To learn about it, follow the links Brain Food > Golden Geometry and The Zometool System > and then click on the first on this page's list, "Zometool System Overview" http://www.zometool.com/brainfood/Golden-Geometry-and-The-Zometool-System
Watch "2, 3, 5, Infinity!" to see a lecture about Zome. http://video.google.com/videoplay?docid=-9040167214635732060&hl=en
B) Nienhuis Montessori – Montessori philosophy, as I understand it, is essentially using all the modalities, sight, sound, touch. http://montessori.nienhuis.com/html/01_products_cat_index.php?pcid=34&fluxmenu=m4____m25
These are ridiculously expensive. I don't own any, but have seen and would like some of them. Anyway, a place to get ideas and get started. Besides, it appears other companies make very similar items for far less money. So, browse the catalog at the following:
C) eNasco.com – to buy those manipulatives and a host of other things. By the way, my new favorite compass/protractor/ruler (all-in-one and no needle or awkward pencils) is the GeoTool Compass http://www.enasco.com/product/TB18640T They also offer a subscription to their catalog.
VI) PUBLISHERS Only those less well known or focused on mathematics
A) MAA Their series include Dolciani Mathematical Expositions (e.g. Conics), MAA Textbooks (e.g. Geometry Revisited by Coxeter), Classroom Resource Material (e.g. Math Made Visual and Proofs Without Words I & II), and the Anneli Lax New Library (aka New Mathematical Library, e.g. Introduction to Inequalities, Geometric Transformations I, II, & III, and Numbers: Rational and Irrational)
B) Springer (the yellow books) Undergraduate Texts in Mathematics (UTM), Graduate Texts in Mathematics (GTM), Readings in Mathematics (RIM, a subset of UTM and GTM), and Classics in Mathematics. There are others, but I'm not familiar enough with those to know their categories and descriptions.
C) Birkhäuser e.g. the Gelfand books mentioned above
D) American Mathematical Society (AMS) e.g. Geometry and the Imagination by David Hilbert and S. Cohn-Vossen
E) AK Peters e.g. Yearning for the Impossible by John Stillwell, Excursions into Mathematics mentioned above
F) Dover I must mention Dover because they republish well-regarded books FAR CHEAPER than all other publishers. (Shame on all publishers that charge more for books that do not require color or glossy prints, and also on university professors for using those expensive books if they have the freedom to choose!)
D) Rookie Teaching For Dummies Gathers all the things you need to know beyond how to teach your subject, such as how to deal with parents, administration, extracurricular duties, finessing relations with all of these people, their expectations, the students, etc., etc.. Again, lots of information your teaching program probably didn't teach you.
E) Conscious Classroom Management by Rick Smith, unavailable new from amazon, go to consciousteaching.com
F) Beyond Discipline: From Compliance to Community I wish to add Alfie Kohn as a general author, not so much for classroom management, but as an ideal. The ideal is basically loving and inspiring, rather than positive and negative conditionals to promote motivation and obedience. However, I believe that the latter makes more sense to most, so it is necessary for many teachers to do what they can, namely, classroom management. But it is the future, inevitable and hopeful, to live, love, and learn inspiringly. Many of his articles argue against popular forms of education. Go to alfiekohn.org
Section 6: The California Subject Examinations for Teachers (CSET) A good online tutorial can be found thru UC Irvine: http://ocw.uci.edu/courses/?cat=5
This list of books is to help you prepare for the CSET quickly. Followed by a more comprehensive list, if you're not in a rush. The Schaum's Outline series are good because these are like textbooks with only what you need to know, that is, without bells and whistles.
Richard Courant's text, What is Mathematics?, the first in this guide, contains enough abstract algebra and number theory for the CSET, and if you need to supplement any material, you could pinpoint it on the internet somewhere for free.
III) SUBTEST III: Calculus (including Trigonometry) and History of Mathematics I wish I could recommend some particular book just for its fine exposition on trigonometry, but there are none that really stand out for me. Consider simply getting a textbook that includes trigonometry, such as an "Algebra and Trigonometry" or simply "Trigonometry" textbook or, of course, Schaum's Outline.
B) "Short Calculus" by Serge Lang. See Section 7, Calculus, below for Serge Lang's "A First Course in Calculus" that is the full version of this Short Calculus.
The history section is basic enough that you'll come across the most important turning points in history simply by reading widely. However, here's a good overview to start from: http://www-history.mcs.st-andrews.ac.uk/HistTopics/History_overview.html
C) A History of Mathematics, Second Edition Some major turning points include, Babylonian use of Pythagorean triples, e.g. 3, 4, 5, and higher integers, irrational number, Hindu-Arabic numerals aka decimal system (including 0), analytic geometry or the Cartesian coordinate system (Descartes), etc.
Section 7: More comprehensive texts, if you're not rushed to pass tests, but seek to understand each subject more thoroughly:
Basic Mathematics Basically a precalculus text. It's written honestly and with no fluff. If I had one book to recommend only, for anything below calculus or linear algebra, this would be it.
Abstract Algebra: A Survey of Modern Algebra (Akp Classics) Recommended to me by Hung-Hsi Wu. Yes there are more advanced and thorough texts, e.g. Dummit and Foote. Another abstract algebra book that may be of interest that has more context and bridges the gap between algebra as most people know it and abstract algebra, there is Roger Cooke's "Classical Algebra: Its Nature, Origin, and Uses".
Linear Algebra: Introduction to Linear Algebra, Fourth Edition The author, Gilbert Strang, has video lectures and more available online either thru MIT's Open Course Ware at http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/CourseHome or thru iTunes U. This new 4th edition has some sample sections at http://math.mit.edu/linearalgebra
Number Theory: The Higher Arithmetic: An Introduction to the Theory of Numbers If you want a different introductory book, try "A Friendly Introduction to Number Theory" by Silverman; recommended to me by Hung-Hsi Wu. If you want a more serious and thorough book, try either G.H. Hardy's or Ivan Niven's "An Introduction to the Theory of Numbers", the same title for either author's book.
Calculus: Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra (volume 1 of 2) The second edition has minor differences from the first (improvements, I had the first and sold it), except the second includes linear algebra. The same Linear Algebra chapters are in both volumes, for whatever reason.
Introduction to Calculus and Analysis, Vol. 1 (Classics in Mathematics) Volume I of II. Note Volume II is divided into two books, II/1 and II/2. This set is Courant's expanded version of his classic Differential and Integral Calculus (2 Volume Set) that, as you can see, is prohibitively expensive, but so is Apostol's 2 volume Calculus (if you really want it, get it used, even an old printing is the same material). I prefer the three book set that is cheaper and double the material! Good as a reference or for more thorough explanations. Solutions are in Volume II/2, but only to Volume II. There is an out of print book with solutions and extras to Volume I. If your library has access to university system libraries, e.g. Link+, then look for Albert A. Blank's Problems in Calculus and Analysis.
A very interesting book, though more basic, is Hahn's text mentioned above in Section 3 H (see above for link to solutions). Unless you plan on reading only one calculus book, then I cannot recommend it more, as it is ideal for high school students, even advanced, and many problems in the book are very difficult.
Calculus by Gilbert Strang is also a great book. Free to download from the author, including a Study Guide and Instructor's Manual at http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm Do not assume that a free book is a poor book!!!
Another great free Calculus book is H. Jerome Keisler's "Elementary Calculus: An Infinitesimal Approach" at http://www.math.wisc.edu/~keisler/calc.html It is one of the few complete calculus texts based not on analysis and limits, but on infinitesimals, as based on Abraham Robinson's "Nonstandard Analysis".
Calculus, 4th edition is a popular among mathematicians. A separate "Answer Book" is sold that is very worthwhile as it contains many worked out solutions and some of the problems in this book are very, very tough, as much that is often explained in books is instead left to be discovered in these problems.
History: Mathematical Thought from Ancient to Modern Times, Vol. 1 (volume 1 of 3) Extremely thorough. Extremely well written is John Stillwell's "Mathematics and Its History". Other options include "A History of Mathematics" by Boyer and Merzbach, and "An Introduction to the History of Mathematics" by Howard Eves.
Of course there are many, many other books that could be placed in these categories, especially in more advanced areas, but 50 is the limit in guides, and this guide has reached its limit.