Below is a selection of books and journals I own for further reading, which have helped to shape my approach to sharing mathematics and creating this curriculum.
“Basic Complex Variables For Mathematics And Engineering”, John H. Mathews, Allyn and Bacon Inc. (1982) This is one of my undergraduate textbooks, which I use as a reference when working with complex numbers.
“Beauty of Fractals, The”, Peitgen and Richter, Springer-Verlag (1986) This book is for taking ideas about the Mandelbrot set and Julia sets deeper (for instance, this is the book I used to figure out the coordinates, where Mandelbrot bulbs are connected to the cardioid, and the centers of the bulbs).
“Book of Numbers, The”, Conway and Guy, Copernicus (1996) This book is filled with examples of numbers and patterns, from recreational mathematics to number theory. It was the first book in which I came across Newton’s formula.
“Calculus”, Howard Anton, John Wiley & Sons (1984) This is one of my undergraduate textbooks, which I use as a reference when working with ideas from calculus.
“Chaos and Fractals, New Frontiers of Science”, Peitgen, Jurgens and Saupe, Springer-Verlag (1992) This is a great book to learn more about fractal geometry and chaos. It is still full of notes and stickies left by me. The 953 pages covers a lot and ties it all together. Sometimes the math is a little heavy, but it’s well written and full of examples. The authors have a way of explaining things clearly without watering down the math. I took a course on fractals and chaos instructed by them at Fermilab (of particle accelerator fame), just outside of Batavia Illinois (about 40 miles west of Chicago).
“Chaos and Fractals, The Mathematics Behind the Computer Graphics”, American Mathematical Society (1989) This was a gift to me from the American Mathematical Society, for being a student presenter at one of their meetings in Rhode Island. The talk was on math modeling, where I designed a keyboard which would allow for faster typing than the qwerty and the Dvorak keyboards (at the time, the world record for speed typing was performed using a Dvorak keyboard).
“Chaos, Fractals, and Dynamics, Computer Experiments In Mathematics”, Robert L. Devaney, Addison-Wesley Publishing Company (1990) This is a great starter book for learning about iteration, orbits, Julia sets, and the Mandelbrot set. I used this book as a reference when I first began writing my own fractal geometry programs. I also used to take high school students to Boston University’s annual Mathematics Field Days, where Devaney hosted talks and activities related to fractals and chaos (I even got to have lunch with him and some of the other teachers who brought students).
“Chaos, Making A New Science, James Gleick, Penguin Books (1987) This book gives a great layman’s introduction to fractal geometry, chaos, and dynamical systems. It describes the big ideas with prose, examples, and pictures.
“Code Book, The Secret History of Codes & Code-breaking, The”, Simon Singh, Ted Smart (2000) This is a great book for learning about the history of codes and code-breaking. It includes a nice description of public-key cryptography, and how prime numbers and modular arithmetic are involved.
“Collins Web-linked Dictionary of Mathematics”, Borowski and Borwein, Collins (2002) I use this book as a reference for looking up definitions and formulas (mathematics is about understanding, not memorizing).
“Colors of Infinity, The Beauty and Power of Fractals, The”, Clear Books (2004) This is the book behind the documentary film. It’s a good layman’s introduction to fractals, and it’s full of pictures.
“Concise Oxford Dictionary Of Mathematics, A”, Christopher Clapham, Oxford University Press (1990) This is a good mathematical dictionary, with fewer entries, but with longer and more helpful definitions.
“Cryptological Mathematics”, Robert Edward Lewand, Mathematical Association Of America (2000) This is a good book for looking at the math behind codes and code-breaking. I used this book to understand the math behind the RSA public-key system, and to create my own RSA program for students to use.
“Curious and Interesting Geometry”, David Wells, Penguin Books (1991) This book is a nice collection of geometrical objects, including cycloids, fixed points, and spirolaterals.
“Curious and Interesting Numbers”, David Wells, Penguin Books (1997) This book gives the significance, beauty, patterns, and connections behind every day numbers and beyond.
“Elementary Number Theory”, Charles Vanden Eynden, Random House (1987) This book is one of my undergraduate textbooks, which I use as a reference when working with topics in number theory.
“Entertaining Science Experiments With Everyday Objects”, Martin Gardner, Dover Publications, Inc. (1981) It’s Martin Gardner, so it’s full of cute/interesting topics, some of which I have modified, and used in the classroom.
“e: The story Of A Number”, Eli Maor, Princeton University Press (1994) A nice book on the history of the number e = 2.7182818284..., and its beauty and power.
“Euler, The Master of Us All”, William Dunham, The Mathematical Association of America (1999) This book does a nice job of giving examples and explaining the math behind the methods used by Euler (pronounced “Oiler”).
“Fabulous FIBONACCI Numbers, The”, Posamentier and Lehmann, Prometheus Books (2007) This book includes a nice collection of ways to better mathematically understand the Fibonacci numbers (feel free to skip the couple of silly non-mathematical chapters).
“Fearless Symmetry, Exposing the Hidden Patterns of Numbers”, Avner Ash and Robert Gross, Princeton University Press (2006) This book is about patterns, symmetry, and the ideas which current mathematicians continue to research to this day. It’s a nice reminder that mathematics is alive and doing well. This book also has many nice problems to try (the pages and margins are full of my solutions).
“Finite Mathematics and Its Applications, Farlow and Haggard, Random House (1988) This is one of my old college textbooks on finite math topics, things like probability, linear programming, and graph theory, which might exist at the ends of textbook chapters, but usually never get to in school.
“First Course In Chaotic Dynamical Systems, Theory and Experiment, A”, Robert L. Devaney, Westview Press (1992) A good introductory look into dynamical systems, especially orbits of iterated functions.
“Fractals And Chaos, An Illustrated Course”, Paul S. Addison, Institute of Physics Publishing (1997) This book is full of diagrams, and the fractal dimension section is helpful.
“Fractals, A very Short Introduction”, Kenneth Falconer, Oxford University Press, (2013) This is as advertised, a nice introduction to fractal geometry. It has 132 pages, but is only 4.5 inches by 7 inches in size.
“Fractal Cosmos, The Art of Mathematical Design”, Lifesmith, Amber Lotus (1998) This book is just full of pictures of zooms of Julia sets and Mandelbrot sets. The nice thing is, the parameters used to create them are given in the back of the book (it’s up to you to create the coloring scheme for the escape sets however).
“Fractals Everywhere”, Michael Barnsley, Academic Press Inc. (1988) This is a good book to learn more about Iterated Function Systems (Barnsley is hard to read however, without a good mathematics background).
“Fractals, The Patterns Of Chaos, Discovering A New Aesthetic Of Art, Science, And Nature”, John Briggs, Touchstone (1992) This is a friendly introduction to fractals and chaos picture book.
“Gamma, Exploring Euler’s Constant”, Julian Havil, Princeton University Press (2003) This is a nice look at the number 0.5772156..., and its connections to the harmonic series, primes, and the Riemann conjecture.
“General Physics”, Douglas C. Giancoli, Prentice-Hall, Inc. (1984) This is one of my undergraduate textbooks, which I use as a reference when working with physics. It does a nice job of explaining problems, and is also calculus based for a deeper understanding of ideas.
“Geometry”, Harold R. Jacobs, W. H. Freeman And Company (1974) A well written humorous approach to planar geometry.
“Handbook of Mathematical Formulas And Integrals, 2nd Edition”, Alan Jeffrey, Academic Press (2000) As the title suggests, it’s full of mathematical formulas, definitions, and pictures, and has come in handy quite a few times.
“Handbook Of Model Rocketry”, Stine and Stine, John Wiley & Sons, Inc. (2004) Going beyond how to build model rockets, this book is full of the math behind it all.
“History Of Mathematics, A, 2nd Edition”, Carl B. Boyer, John Wiley & Sons, Inc. (1991) As the title suggests, a brief look into the history of mathematicians and their claims to fame, from the Egyptians to the 1900’s.
“History Of Pi, A”, Petr Beckmann, St. Martin’s Press (1971) A nice collection of examples and stories related to π, and some of the places you wouldn’t expect it to appear.
“Imaginary Tale, The Story of √-1, An”, Paul J. Nahin, Princeton University Press (1998) A great look at complex numbers and their connections. And in Hahin fashion, there are plenty of opportunities for the reader to dive in and fill in the details purposefully left out (the blank pages at the end of the book are filled with my derivations, it’s a fun book for anyone who has already had calculus).
“Introductory Statistics For The Behavioral Sciences”, Welkowitz-Ewen-Cohen, Academic Press (1982) This is one of my wife’s college textbooks, which does a nice job at explaining basic statistical ideas. I used it to help me derive an algebraic way of explaining linear regression in chapter 5.
“Journey Through Genius, The Great Theorems Of Mathematics”, William Dunham, Penguin Books (1990) A nice selection of well written biographies and explanations of mathematics.
“Keys To Infinity”, Clifford A. Pickover, John Wiley & Sons, Inc. (1995) A fun book for anyone interested in programming to create mathematical objects. This book was helpful in creating my line art and Ford sphere programs.
“Loom Of God, Mathematical Tapestries At The Edge Of Time, The”, Clifford A. Pickover, Plenum Trade (1997) This is a fun book for anyone interested in recreational mathematics (this book was the first book I came across to mention vampire numbers).
“MAA Focus”, Mathematical Association Of America, June/July (2011)
“Mathematical Bafflers”, Dover Publications, Inc. (1980) Some neat problems, my favorite being the four fours challenge where you are challenged to create the numbers 1 through 100, using exactly four fours each time (which I have included in “The Appendix”, under “order of operations”).
“Mathematical Connections, A Companion for Teachers and Others”, Al Cuoco, The Mathematical Association Of America (2005) This is a good book for delving deeper into a few connected ideas, polynomials, combinatorics, and complex numbers.
“Mathematical Handbook of Formulas and Tables, 3rd Edition”, Spiegel - Lipschutz - Liu, McGrawHill (2009) As the title suggests, it’s full of mathematical formulas, definitions, and pictures, and has come in handy quite a few times.
“Mathematical Statistics, Basic Ideas and Selected Topics”, Bickel and Doksum, Holden-Day, Inc. (1977) One of my graduate level textbooks, which I used to better understand the least squares method of finding a best fit line.
“Mathematical Tourist, New and Updated Snapshots of Modern Mathematics, The”, Ivars Peterson, Barnes & Noble(1998) A collection of some more recent mathematical ideas, which are hard, but easy and fun to introduce, such as knot theory. This is one of the first books I came across that mentioned John Conway’s game of Life.
“Mathematics Magazine”, Mathematical Association Of America, June (2009)
“Mathematics Teacher”, National Council Of Teachers Of Mathematics, December 2006/January (2007)
“Mathematics Teacher”, National Council Of Teachers Of Mathematics, March (2010)
“math Horizons”, Mathematical Association Of America, February (2007)
“math Horizons”, Mathematical Association Of America, April (2011)
“math Horizons”, Mathematical Association Of America, April (2012)
“Matrices And Transformations”, Anthony J. Pettofrezzo, Dover Publications (1966) A great book on linear transformations of the plane.
“Mechanics”, Smith and Smith, John Wiley & Sons (1990) This is a nice book for better understanding mechanical physics, such as statics and projectile motion.
“Millennium Problems, The”, Keith Devlin, Basic Books (2002) This book describes seven of the top unsolved conjectures which mathematicians are currently working on, and how the Clay Mathematics Institute will award one million dollars for their proofs. Since the book came out, the fifth one was proven by the Russian mathematician Grigory “Grisha” Perelman (he refused to accept the one million dollars). If you’re interested in knowing more about this story, the book, “Perfect Rigor”, by Masha Gessen is pretty good.
“Number Theory and Its History”, Oystein Ore, Dover Publications, Inc. (1988) This is another good book to use as a reference when working with topics in number theory. I actually have five books on number theory, and I’ve found, they’re all pretty good.
“Origami, Eleusis, and the Soma Cube”, Martin Gardner, Cambridge University Press (2008) Again, it’s hard to go wrong with Martin Gardner, the soma cube section was quite good.
“Our Mathematical Universe, My Quest for the Ultimate Nature of Reality”, Max Tegmark, Alfred A. Knopf New York (2014) An entertaining and thought provoking argument that our universe and everything in it (including us), are mathematical structures. His “bottom-up” approach to the meaning of life is a nice touch at the end as well.
“Penrose Tiles To Trapdoor Ciphers”, Martin Gardner, W. H. Freeman and Company (1989) Once again, Gardner is good, the unique distances section gave me the idea for creating the lattice challenge program in chapter 7.
“Physics, Principles and Problems”, Murphy and Smoot, Charles E. Merrill Publishing Co. (1982) An old but good, introductory, high school level, book on physics.
“π, A Biography of the World’s Most Mysterious Number”, Possamentier and Lehmann, Prometheus Books (2004) Another good book on the beauty and power of π, including the last 28 pages which shows the first one hundred thousand decimal places of π.
“Practical Conic Sections, The Geometric Properties Of Ellipses, Parabolas And Hyperbolas, J.W. Downs, Dover Publications, Inc. (1993) A good book full of diagrams to better understand the versatility of conic sections.
“Prime Obsession, Bernhard Riemann And The Greatest Unsolved Problem In Mathematics”, John Derbyshire, Joseph Henry Press (2003) A nice lead up to what the Riemann hypothesis is and means.
“Princeton Companion To Mathematics, The”, Princeton University Press (2008) This will be of interest to math majors, it gives brief but solid explanations of mathematical ideas from all branches of mathematics. One of the goals is to make it easier for students and mathematicians to know what’s out there, because breakthroughs in mathematics frequently come from connections to seemingly unrelated areas of mathematics. For students still in high school, the book should prove to be eye widening, due to the sheer number of different branches of mathematics, and the idea that it’s impossible for anyone in this day and age to understand everything mathematical.
“Recreations In The Theory Of Numbers, The Queen Of Mathematics Entertains”, Albert H. Beiler, Dover Publications, Inc. (1966) Another good book on number theory, it still has stickies marking many of its pages.
“Recursive Universe, The”, William Poundstone, Dover Publications (2013) A great book to learn more about John Conway’s game of Life (students should explore creating creatures and running the game of Life with the programs I have written for them in chapter 1 first however, before learning what others have discovered).
“Science of Fractal Images, The”, Barnsley, Devaney, Mandelbrot, Peitgen, Saupe, and Voss, Springer-Verlag (1988) This is a good book for learning more about the math and the programming behind creating fractal images.
“Simple Science of Flight, From Insects to Jumbo Jets, The”, Henk Tennekes, The MIT Press (1996) This is a great book on flight, and the book I used to help create index card glider and paper helicopter (auto-rotator) projects for my students.
“Simpsons And Their Mathematical Secrets, The”, Simon Singh, Bloomsbury (2013) This book mentions many of the above and beyond topics found in my Polynomials course material. It’s a fun read, and you don’t have to be a Simpsons watcher to enjoy it (I don’t watch the program myself).
“Symmetry, A very Short Introduction”, Ian Stewart, Oxford University Press, (2013) This is as advertised, a nice introduction to the idea of symmetry and group theory. It has 144 pages, but is only 4.5 inches by 7 inches in size.
“Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics”, Robert B. Banks, Princeton University Press (1998) This is a fun book for people who have already taken calculus, and would like to test their application skills. The margins and blank pages are filled with my solutions to many of the given challenges related to applied mathematics problems.
“Trigonometric Delights”, Eli Maor, Princeton University Press (1998) This is a good book for learning more about the history of trigonometry and better understanding the related formulas.
“Unknown Quantity, A Real And Imaginary History Of Algebra”, John Derbyshire, Joseph Henry Press (2006) This is a good book on the history of algebra (from the algebra we know from middle and high school, to the algebra seen in college, usually referred to as modern or abstract algebra, which is the real subject of the book). A good understanding of number and polynomials is needed to enjoy this book. The following is a quote from page 15. “Polynomial has a fair claim to being the single most important concept in algebra, both ancient and modern.”
“When Least Is Best, How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible”, Paul J. Nahin, Princeton University Press (2004) This is a good introduction to extrema or max/min problems, where there is a highest or lowest value solution possible.
“Words of Mathematics An Etymological Dictionary of Mathematical Terms Used in English, The”, Steven Schwartzman, The Mathematical Association of America (1994) This book is good for people who find it interesting to know where mathematical terms came from, which sometimes helps to better understand mathematical terms and the ideas they describe.
“Zen Of Magic Squares, Circles, And Stars, The”, Clifford A. Pickover, Princeton University Press (2002) This is a fun book for people who are interested in magic squares and related ideas.