Math 287, Spring 2013

Note: this is an archived version of the course page. The formatting has been simplified and student comments have been erased.

Welcome

Like Math 187, this course is intended to bridge the gap between computational mathematics and proof-based mathematics. Computational mathematics means solving with numeric or formulaic answers using established techniques, such as in trigonometry or calculus. However in deeper mathematical study we ask much broader questions and demand that the answers be accompanied by a proof. In this course we will not only practice proving theorems, we will explore the whole mathematical process. This means we will pay particular attention to the discovery, conjecture, and exposition of theorems as well.

This class satisfies the CID requirement in mathematics.

Course information

Course title: Communication in Mathematics
Meeting times: T,Th 3–4:15pm
Meeting place: ILC-402
Instructor office: MG 237-A
Office hours: M 9am, and Th 4:30pm
Textbook: Laboratories in Mathematical Experimentation (Mount Holyoke)

Grading

Attendance: Since a majority of time in class will be spent working on projects with a group, attendance and participation are crucial to your success. Documented absences are always excused, and you may have a couple of free unexplained absences, but after that absences will lower your grade rapidly.

Written work: The majority of your grade will be based on five or six written “lab reports”, each of which receives a letter grade.

Presentations: Together with your group, you will be asked to give at least one presentation during the course. More details about this will appear later on.

Lab reports

Most of the actual investigation and experimentation will be done during class time. For homework, you will be asked to document the results of your experiments in the form of a lab report. Make sure that you keep a notebook for your classroom experiments so you can still reference them when you go home to write up your report!

Lab reports consist of at least three sections, outlined below. In the Introduction, the textbook describes one possible layout for your write-ups. You may follow their advice, or alternatively, mine here:

Section 1: Begin by writing an introduction explaining the topic of the module. Assume that your audience is somebody who is familiar with mathematical language, but not necessarily with this particular topic. Be sure to include motivation for studying this topic, either from the text or from your own thoughts. Write careful definitions for any of the fundamental concepts used in the module (again, remember your audience). Finally, preview some of the questions you will be addressing in the upcoming sections.

Section 2: This section is a development of your results. Describe any important examples and counterexamples you came up with, and say what questions they answer. Summarize any other key experimental results. Write any general facts or theorems that you can prove, and give the proof.

Section 3: Finally, write down any questions that you worked on but were not able to fully answer. Are there any patterns that appear to hold, but which you cannot fully explain? Describe any conjectures you have made based on your results and experimentation, and explain how the evidence supports your conjectures.

Lab report grading rubric

Your labs will be graded on at least four criteria:

Exposition: The quality and effectiveness of your writing, not including proofs. Includes introduction and motivation, presentation of results, style, structure, and clarity throughout.

Proofs: The quality and clarity of your technical mathematical writing, that is, the statements of results and definitions and the proofs of results.

Depth: The strength and completeness of your experimental and theoretical results, scientifically speaking.

Synthesis: How well you conveyed understanding of the results and goals of the lab as a whole. This can be done in many ways, including: summarizing results, motivating and asking questions, making conjectures, and drawing connections between the results, evidence, and conjectures.

Technologies

In addition to an array of mathematical topics, we will learn to use a few of the most important technologies used in mathematical research and practice.

Sage and python

The most important technology in this class is Sage, a computer algebra system and programming language. While you are welcome to use any language you prefer, I recommend Sage because it is free to use online. You can read about it at sagemath.org; check out their video introductions! You can start using Sage very quickly at sagenb.org.

Here is the introductory code from the first day of class, on how to write functions, plot graphs, and work with arrays.

LaTeX

LaTeX and its TeX ancestor has been the primary tool for the dissemination of mathematics for 35 years, even though it has changed very little in that time. This long reign will likely come to a slow but sure end, as LaTeX is finally being replaced by more portable and web-aware technologies. Still, it is important for us all to master the LaTeX system, since the language it provides for expressing mathematics will certainly be the standard for many years to come.

A couple of references that I find very useful: The (not so) short introduction to LaTeX, the NASA guide to LaTeX commands, and The comprehensive LaTeX symbol list.

See also the introductory document from class on February 5.

Beamer

Beamer is a LaTeX documentclass designed especially for creating fancy powerpoint-style presentations. The output is a pdf file that you can click through page-by-page while you speak. In the code, you simply enclose each slide within \begin{frame}{My Title} and \end{frame}. Inside, simply use the LaTeX commands that you are used to. The ultimate Beamer reference is the full user guide, but this may contain too much information… you could also have a look at this short intro (and its source). Finally, see the sample beamer template from class.

Further technologies

Labs and due dates

Lab 5 (presentations at end of semester)
This lab will culminate with a group presentation. You will turn in an outline for the presentation as well as your Beamer slides. The content of the presentation is just the same as before: introduction to the concepts, statement of results, a proof or two, summary and questions. The only difference is you will explain it verbally. Of course, a presentation leaves much less time to explain things than a written work, so you will have to be choosy about which details to present.

Lab 4 (completed by April 16)
This lab is to be completed using WordPress. We will have a discussion session on April 9, so be sure to have some posts with plenty of mathematical content by then.

Lab 3 (due Tuesday, March 19)
Chapter 5 of Laboratories. A rough draft is due March 12 (bring 2 copies, name optional, to exchange). Here is some code to get started: graph

Lab 2 (due Tuesday, February 26)
Chapter 2 of Laboratories. A rough draft including at least the basic definitions should be turned in Thursday, February 14. A rough draft including the first two sections (plus?) is due on Thursday, February 21 (bring 2 copies, name optional, for trading with classmates). Here is some code to get started: cyclic

Lab 1 (due Tuesday, February 5)
Chapter 1 of Laboratories. A rough draft of the first two sections of your report should be turned in Tuesday, January 29. Here is some code for lab 1: iterfunc.