Contents
Mission statement. I believe that one of the most important aspects of the ability to use one's mind is creativity. I also believe that the system of mass education does far worse in teaching creativity than it does most other skills. Indeed, students cannot be taught creativity as they are taught other skills; they must develop it largely on their own, in their own explorations of the world. I consider "research" to mean precisely "exploration of the world". In mathematics research, the world we are exploring is that of abstract structures which (usually) have counterparts in the real world but have been idealized for purposes of simplicity, elegance, and purity. The results of pure mathematics, therefore, are statements about the real world, but they are also statements about the idealized structures that mathematics considers.
The projects on this site were projects that arose out of my own curiosity. They have significantly deepened my appreciation of the world, and they have taught me how to accomplish research in mathematics. Thus I offer them to others as potential projects that will convey what it is like to do research in mathematics. Of course, none of the results that I found in these projects were new; many have been known for hundreds of years and can be found, for example, on numerous web pages. Yet they are nontrivial (and often challenging) problems that reveal many a structure.
Educational emphasis. The educational emphasis of higher mathematics courses is on the ability to write rigorous proofs. This is in a large sense backwards, since, in practice, new results usually are discovered experimentally or empirically before proofs are even attempted. While the approach to finding a result and the approach to proving it are similar in some ways, I believe that the former is a much more fundamental skill (and is often a lot more enjoyable) than the latter. Incidentally, it is also more difficult to teach. For these reasons, I have placed the emphasis of the projects not on proofs but on obtaining results. Often proofs are suggested (and in most cases they are given in the solutions), but students are not expected to be able to prove all of the results they obtain.
Arbitrariness of the projects. These projects were not assembled as a whole to provide a comprehensive survey of possible high-school accessible research avenues in mathematics; they simply arose out of my interest. Consequently, the variety of topics is heavily biased (as most topics fall within number theory or geometry). However, there are plenty of other fields in mathematics. If none of the projects here suit you, make one up yourself. One of the best ways to do this is to start with a mathematical result you are familiar with and ask about its generalizations. It has been my experience that the difficulty of such questions often becomes apparent rather quickly; if the question you have chosen is too difficult, try simplifying it or try another question altogether. The projects section lists the questions that motivated the projects on this site.
Time frame. I must state clearly that the projects are not mere worksheets that a student will be able to get through in a few hours. For students at the targeted experience levels, they may take several days, or in some cases (notably the Pythagorean Triples Project), several years before a student has come to understand all of the structure behind the topic. This is not to be daunting! Rather, it is suggestive of the depth to which some of these projects can proceed.
Levels of guidance. These projects are intended to be research projects — to convey what it is like to do research in mathematics. As such, they are constructed so as to impose as little direction as possible, and students are highly encouraged to ask their own questions and pursue their own lines of thought. If a question is vague, it is most likely to encourage independent thought; students should choose one interpretation and begin experimenting — not dwell on which direction is "right" (for often this only comes in hindsight). The questions in the projects are intended as guides only, and students should feel free to depart from the outlines at will.
I have set up the projects so that they are equally accessible and practical to students of various skills (and time allowances). There are effectively four levels on which students can work through the projects. The first is simply by taking the motivating questions as far as possible (and not bothering with the project page); students can formulate their own questions for investigation and seek the aspects that are particularly interesting to them. The second level is by following the project guides. Many of the questions in these guides are linked to hints, providing a third level of guidance. Finally, there are complete solutions to the projects. I would recommend that most students begin working at the second level and use the hints when they have difficulty. Highly motivated students will prefer to work on the first level and create the projects for themselves.
It is suggested that teachers shield their students (or that students shield themselves) from the complete solutions until they have worked as much as they want to on a particular problem, for it has been my experience that looking up the full solution to a recreational (or even research) question results in a sense of anticlimax and a significant decline in one's enthusiasm for the problem.
Solutions. The solutions to the projects are written in the formal style of mathematics. They are intended to be of use not only to students working through the corresponding projects but to the mathematical community at large. As a result, the expositions are not constrained to the outlines of their respective projects (which are themselves organized for didactic purposes), although most do follow their projects exactly. All notation is consistent. However, students may experience some frustration in "checking their answers," as they may not have used the same variables, initial values, representations, etc. that I use. This isn't entirely unfortunate, for it is a reality of any field of research that there will be differences in notation, and translating is a useful skill to learn.
Color schemes and hints. Project pages are coded green, exposition pages are coded orange, and other pages are coded blue. Links to hints are colored dark blue, and external links are colored purple.
Hints for selected questions can be found by clicking on the question. To avoid accidentally looking at the wrong hint, hints are greatly spaced out on the hint pages. Printer-friendly hint pages are available at the following links:
Accompanying programs. Many of the projects listed have accompanying Mathematica notebooks containing programs, tables, and graphics. If you do not have Mathematica, these notebooks can be viewed with the free Mathematica Player or a trial version of Mathematica. The notebooks are not integral to the projects, but they provide worthwhile supplements and may provide some further hints for students if they get stuck. If possible, students are highly encouraged to write their own progams (in any language — not necessarily in Mathematica) when they feel that a program will advance the project. (Mainly this applies to the Pythagorean Triples Project (for which programs and data are provided).) Since its invention, the mechanical computer has been an invaluable tool to mathematicians.
A note for teachers on grading. I strongly discourage grading students on these projects. The projects are likely to prove much more challenging than traditional coursework, and their open-endedness and demands on creativity really make it difficult, in my opinion, to assign a quantitative grade. Of course, these considerations hardly stop people from trying to impose grades in other subjects (toward which my views are similar). But I think that the level of success a student finds in these projects is somewhat dependent on their natural interest in mathematics and motivation based on that interest. Professional researchers put their hearts into research, and I doubt that much good research can be done without doing so. And, as ridiculous as it is to say, assigning math grades based on students' hearts is problematic at best.
Feedback. I would very much appreciate any and all feedback (especially corrections) you may have about these projects and their organization or presentation. I have already gone through these projects from the student's point of view, and I will not have an opportunity to teach them soon, so their subsequent improvement is dependent on feedback from others. What is successful about the projects? What is unsuccessful? How can they be improved?
Areas of mathematics. Areas of mathematics into which the projects fall (given in parentheses in the project listings) are briefly explained below.
Geometry is concerned with the arrangement of shapes in space and the measurements of these shapes.
Trigonometry refers to the study of the angles of a triangle. The sine and cosine functions are the core functions.
Algebra is the collection of techniques for solving algebraic equations. (In higher mathematics, the term "algebra" encompasses much more general structures, including groups and fields.)
Calculus includes the methods of differentiation and integration — the theories regarding the slopes of functions and the areas under functions.
Number Theory is the study of the integers and other structures like the integers. In particular, it largely concerns itself with prime numbers.
Graph Theory studies arrangements of connected points (known as graphs). Graphs provide an intuitive and visual language for discussing many problems that arise in mathematics and computer science.
Combinatorics, loosely speaking, is the branch of mathematics that is concerned with counting things. Finite, discrete probability falls into this category, for example.