The present expanded set of notes initially grew out of an attempt to flesh out the International Baccalaureate (IB) mathematics “Further Mathematics” curriculum, all in preparation for my teaching this during during the AY 2007–2008 school year. Such a course is offered only under special circumstances and is typically reserved for those rare students who have finished their second year of IB mathematics HL in their junior year and need a “capstone” mathematics course in their senior year. During the above school year I had two such IB math- ematics students. However, feeling that a few more students would make for a more robust learning environment, I recruited several of my 2006–2007 AP Calculus (BC) students to partake of this rare offering resulting. The result was one of the most singular experiences I’ve had in my nearly 40-year teaching career: the brain power represented in this class of 11 blue-chip students surely rivaled that of any assemblage of high-school students anywhere and at any time! After having already finished the first draft of these notes I became aware that there was already a book in print which gave adequate coverage of the IB syllabus, namely the Haese and Harris text1 which covered the four IB Mathematics HL “option topics,” together with a chapter on the retired option topic on Euclidean geometry. This is a very worthy text and had I initially known of its existence, I probably wouldn’t have undertaken the writing of the present notes. However, as time passed, and I became more aware of the many differences between mine and the HH text’s views on high-school mathematics, I decided
that there might be some value in trying to codify my own personal experiences into an advanced mathematics textbook accessible by and interesting to a relatively advanced high-school student, without being constrained by the idiosyncracies of the formal IB Further Mathematics curriculum. This allowed me to freely draw from my experiences first as a research mathematician and then as an AP/IB teacher to weave some of my all-time favorite mathematical threads into the general narrative, thereby giving me (and, I hope, the students) better emotional and intellectual rapport with the contents. I can only hope that the readers (if any) can find some something of value by the reading of my stream of-consciousness narrative. The basic layout of my notes originally was constrained to the five option themes of IB: geometry, discrete mathematics, abstract algebra, series and ordinary differential equations, and inferential statistics. However, I have since added a short chapter on inequalities and con- strained extrema as they amplify and extend themes typically visited in a standard course in Algebra II. As for the IB option themes, my organization differs substantially from that of the HH text. Theirs is one in which the chapters are independent of each other, having very little articulation among the chapters. This makes their text especially suitable for the teaching of any given option topic within the contextof IB mathematics HL. Mine, on the other hand, tries to bring out the strong interdependencies among the chapters. For example, the
HH text places the chapter on abstract algebra (Sets, Relations, and Groups) before discrete mathematics (Number Theory and Graph Theory), whereas I feel that the correct sequence is the other way around. Much of the motivation for abstract algebra can be found in a variety of topics from both number theory and graph theory. As a result, the reader will find that my Abstract Algebra chapter draws heavily from both of these topics for important examples and motivation. As another important example, HH places Statistics well before Series and Differential Equations. This can be done, of course (they did it!), but there’s something missing in inferential statistics (even at the elementary level) if there isn’t a healthy reliance on analysis. In my organization, this chapter (the longest one!) is the very last chapter and immediately follows the chapter on Series and Differential Equations.
This made more natural, for example, an insertion of a theoretical subsection wherein the density of two independent continuous random variables is derived as the convolution of the individual densities. A second, and perhaps more relevant example involves a short treatment on the “random harmonic series,” which dovetails very well with the already-understood discussions on convergence of infinite series. The cute fact, of course, is that the random harmonic series converges with probability 1.
I would like to acknowledge the software used in the preparation of these notes. First of all, the typesetting itself made use of the indus- try standard, LATEX, written by Donald Knuth. Next, I made use of three different graphics resources: Geometer’s Sketchpad, Autograph, and the statistical workhorse Minitab. Not surprisingly, in the chapter on Advanced Euclidean Geometry, the vast majority of the graphicsm was generated through Geometer’s Sketchpad. I like Autograph as a general-purpose graphics software and have made rather liberal use of this throughout these notes, especially in the chapters on series and differential equations and inferential statistics. Minitab was used primarily in the chapter on Inferential Statistics, and the graphical outputs greatly enhanced the exposition. Finally, all of the graphics were con- verted to PDF format via ADOBER ACROBAT R 8 PROFESSIONAL (version 8.0.0). I owe a great debt to those involved in the production of the above-mentioned products. Assuming that I have already posted these notes to the internet, I would appreciate comments, corrections, and suggestions for improve- ments from interested colleagues and students alike. The present version still contains many rough edges, and I’m soliciting help from the wider community to help identify improvements.
