Mathematics 30 Introduction to Problem Solving

Optional Text: How to Solve It by Polya

This course will be unlike any mathematics class you have yet taken and probably any that you will subsequently take at UCSC. In this course you will learn mathematics by doing rather than listening. By learn to do mathematics I mean much more than become acquainted with some concepts and objects, though there will loads of those, drawn from elementary logic, set theory, discrete mathematics, number theory and abstract algebra. And it will be more than familiarity with theorems about these new concepts, though there will loads of these, too. In addition to these traditional aspects of learning mathematics you will also learn techniques which include strategies and tactics, including psychological strategies. We will learn how to get started and how to persist. We will learn how to monitor our thinking in order to gauge whether we are making progress and should continue along with the plan already formulated or whether what initially appeared fruitful is really barren and requires backtracking and the choice of a new approach. We will learn methods of argument such as direct proof, proof by contradiction and mathematical induction. We will learn how to investigate and formulate conjectures. We will learn fundamental tactics such as the search for symmetry, the extreme principle, the pigeonhole principle and the identification of invariants. By problems I mean novel problems, problems that you have not previously encountered and therefore you will be unable to fall back on some memorized template which requires little more than plugging in numbers or following a recipe. You will have to learn to orient yourself and figure out what is given and what is being asked. You will have to formulate a plan and implement it. It is one of the objectives of this course that your attitudes and beliefs about mathematics will change, among them: what it means to do mathematics, the role of effort and persistence, what constitutes a proof, who can do difficult mathematics.

The course will be different in other ways. Not only will I not lecture but I will not be the final authority in class. Rather, when a problem is being "solved" by a student in class I will lead a discussion on its merits. In particular, we will together decide such questions as: "Is the solution persuasive?" "Has the solution been communicated effectively?" "Has it been argued rigorously enough?" "Have all elements been taken into account?" "Can a more general problem be solved?" "Are there alternative approaches?" "Can the proof be simplified?" At other times I will lead the class in discussing the features of a problem that are general - the application of a new tactic or strategy that might be usefully employed in different contexts. Also, when new concepts or definitions arise or may be required I will introduce these, provide some historical context and indicate which areas of mathematics it comes from, e.g. congruence modulo an integer N is a concept in number theory, generating functions arise in enumerative combinatorics, groups in abstract algebra, etc.

During class we will always work in groups of three or four. These groups will change over the first two weeks but at the beginning of week three they will become permanent. In addition to class there will also be an opportunity to discuss the mathematics and anything else of relevance through the course website. There you can make postings about specific problems, problem techniques, psychological issues, the conduct of the class, etc. At any time any class member will be able to join the conversation.

In addition to the regularly scheduled classes we will also have a discussion section which will be devoted to putting solutions into TeX, a mathematical typesetting program. Of course, in that context we will also be working on problems.

Requirements: The evaluation for this course will be based on classroom and website contributions to our "problem solving conversation" problems solved, special annotated problems, a project, and a final.

The problems will be at three levels: free throws (one point), field goals (two points), and behind-the-arc problems (three points). The same problem may be submitted multiple times as long as the solutions are significantly different. The subsequent submissions of a problem will always be a valued as a free throw regardless of the original value since the most difficult barrier - the psychological - has already been hurdled with the knowledge that it can be solved. This is why priority is such an important thing in mathematics and other sciences.

Each week one problem of value at least two must be annotated, that is, the solution to the problem and your thinking about this problem (how you got oriented, how you decided what the problem is about, how you formulated a solution, why you chose one method over another, what investigations you undertook, how you monitored your thinking) is discussed through comments interspersed throughout the text. This need not be a problem you solved but can be a problem you worked on and made only partial (or even little) progress. You can annotate as many as two problem per week. The first annotation is worth two points, the second is worth one point.

The final will also consist of lots of one, two and three point problems and will be worth up to 30 points. You will also required to do a project and make presentations during our section time. These will be a longer investigation or study of a famous problem. Project are worth 10 points.

Satisfactory or passing work will require 70 points; good work will require 80 points; very good work requires 90 points; and excellent work requires 100 points.