At WGU, the core mathematics content is lower division Analysis (Calculus I,II,III), Geometry, Statistics, and Algebra (Linear and Abstract Algebra). Generally, the statistics and geometry taught to preservice teachers is substantially different than what a math graduate would be expected to know, but the other topics differ primarily in depth and rigor. Research has repeatedly shown a correlation between a teacher’s course work in the discipline and student achievement gains specifically for secondary mathematics^{1}. That means going beyond the WGU courses of study is likely to have considerable value. Before embarking on anything here, please read the introduction to Milgram 2005^{2}. Milgram’s research focused on preparation for K-8 teachers, but I think the general ideas about mindset and preparation are likely to apply to secondary math teachers. Secondary teachers also have to work with students who have been given inadequate instruction before high school, students who have high levels of math anxiety, and students who display math aversion behaviors. The best thing about learning higher division math is that you should be able to express mathematical concepts in ways that avoid triggering anxiety around numbers or which appeal to individual interests.

The majority of what you will study at WGU and beyond will not be taught in high schools. Do not let that demotivate you from pursuing a deeper understanding of advanced math. The fact is, until you have some understanding of analysis, abstract algebra, and modern geometry, it is impossible to understand the structure of math as a discipline. And if you don’t understand the discipline, you will not be able to fully tailor your teaching to the aspirations, interests and abilities of your students.

I’ve gathered the resources that I think are the best for independent study, with a goal of closing the gap between courses available to WGU students and what would be expected of an undergraduate pursuing graduate study in mathematics.

By Topic:

- General/Preliminary
- Houston, How to Think Like a Mathematician. This book starts with set theory, then reading and writing math, and problem solving (using Polya’s structure below). After these preliminaries, Houston introduces logic and proof. An appendix that outlines general proof strategies is three pages of wonderful. The chapters are short, with relevant examples and good exercises. One of the unifying themes of the text is its emphasis on common misunderstandings acquired from life and high school, particularly from the procedural or ‘cookbook’ nature of high school math.
- Coursera: Introduction to Mathematical Thinking (Keith Devlin, Stanford). With the same aim as HTTLAM, it is in its second offering as I write this. The main advantage to this over a book is the forums.
- Velleman, How to Prove It. One of the more popular texts on proofs, and has many exercises. I recommend Houston first.
- Polya, How to Solve It. The classic treatise on systematic problem solving. Polya was an educator in the best possible sense. Math is very much about stating problems precisely and solving them, and this book provides a fantastic frame for that pursuit.

- General/Reference
- Gowers, Princeton Companion to Mathematics. Available on the WGU ebrary.
- Bronshtein, et al., Handbook of Mathematics. I have an older edition, and have never encountered a topic that is not thoroughly summarized here. I can consistently find a theorem or formula whenever I need it. Everything you need in a desktop reference as an undergrad.

- Algebra
- Burn, Groups: A Path to Geometry. Burn will be on this list three times, because his books are fantastic. The general strategy of these books is to use illustration or very short exposition to establish some definitions, and then teach concepts almost entirely through posing challenges to the reader, while periodically naming concepts after they are demonstrated or proved (by the reader!). I’m filling notebooks with drawings and proofs while working through this book. It is the second math textbook I have ever read that made me feel like I was “doing math” (Velleman, above, was the first). Each chapter has preliminary, supplemental, or concurrent readings listed at the beginning, and historical notes at the end.
- Harvard Extension: Abstract Algebra. Videos, lecture notes, and problem sets. Uses Algebra , which is the text written by Michael Artin for the Algebra courses at MIT, and which is also used for the first two algebra courses at Berkeley.

- Analysis
- Burn, Numbers and Functions: Steps to analysis. See the note above for Groups. This one has the advantage of being free for WGU students. ebrary
- Mattuck, Introduction to Analysis. Written in a conversational style, and intended for self-study, this text is also used in MIT’s introductory course in analysis.
- Rudin, Principles of Mathematical Analysis. Text used for Analysis I and II at both MIT and Berkeley. Read Burn and Mattuck first, especially since Burn is free for WGU students.
- Francis Su’s Real Analysis Course. The introductory Analysis course at Harvey Mudd, lectures recorded in 2010. Follows chapters 1-5 of Rudin (above).

- Geometry
- Hartshorne, Geometry: Euclid and Beyond. This book is one quarter history, one quarter theory, and half exercises. It requires a copy of Euclid (a copy of Hilbert wouldn’t hurt either). Chapters 1 and 2 set a solid foundation of axiomatic geometry, after which the rest of the chapters can be grazed out of order. Chapter 7 (Non-Euclidean Geometry) is highly recommended for WGU students. ebrary
*more coming soon*

- Statistics
- Coursera: Data Analysis From the syllabus:
*This course is an applied statistics course focusing on data analysis. The course will begin with an overview of how to organize, perform, and write-up data analyses. Then we will cover some of the most popular and widely used statistical methods like linear regression, principal components analysis, cross-validation, and p-values. Instead of focusing on mathematical details, the lectures will be designed to help you apply these techniques to real data using the R statistical programming language, interpret the results, and diagnose potential problems in your analysis. You will also have the opportunity to critique and assist your fellow classmates with their data analyses.***This course is over, and not yet scheduled for its next run, if you are interested, follow the instructor’s blog.** - 36-309: Experimental Design for Behavioral and Social Sciences This is the course that follows introductory data analysis at CMU. If you did well with the OLI, you’ll be able to handle this course with some effort. The textbook, homework, and labs used in the course are supplied from the public course homepage, along with lecture notes and previous exams. The course is designed to use SPSS, so if you use different software, it will require additional work. SPSS data can be imported to R using the
*foreign*package from CRAN. GNU PSPP exists as a free SPSS replacement, but I have not used it. - Software. IBM offers 12-month licenses for SPSS to students for about $100 through several retailers, including OnTheHub OnTheHub also offers a perpetual MiniTab license for about $100, and a presumably renewable 12-month license for Spotfire (S+) for free. R and Octave are free-as-in-FLOSS. WGU students can obtain student licenses of Matlab directly from Mathworks, and Mathematica directly from Wolfram.
*At the beginning of the 2013/14 academic year, there is still no way for WGU students to obtain student licenses of Stata.*

- Coursera: Data Analysis From the syllabus:
- Quantitative Methods/Educational Psychology
- Probability and Inference
- Regression and the General Linear Model
*EPsy 8262 is the second course of the two-semester Ph.D. level statistics sequence in Educational Psychology at the University of Minnesota. The course will cover a number of regression methods. Emphasis will be placed on viewing traditional statistical methods as special cases of multiple regression, which itself is a special case of the general linear model (GLM). Emphasis in the course is given to learning how to do data analysis. We will also devote considerable time to illustrating how to present regression results in prose, tables and figures. Instead of formal textbook readings, each unit is supported by lectures and course notes.*

- Resources to help with WGU course work:
- MIT’s OpenCourseWare Scholar courses: Full lecture videos, worked problems for each lecture, problem sets, and exams. Work through following sequence:

^{2} Milgram, R. J. (2005). The Mathematics Pre-Service Teachers Need to Know.