The Mathematics and Computer Science Department offers two mathematics majors: the Bachelor of Arts in Liberal Arts Mathematics and the Bachelor of Arts in Secondary Education, Mathematics. The department has identified two required courses in each of the majors where writing is most emphasized. For the Liberal Arts Mathematics major the courses are Bridge to Advanced Mathematics (MATH 300) and Seminar in Mathematics (MATH 461), while in the Secondary Education Mathematics major the courses are Bridge to Advanced Mathematics and History of Mathematics (MATH 458).
Bridge to Advanced Mathematics is the course in the major which aims to help students in the transition from Calculus to upper-level proof-based mathematics courses. This course differs from other courses in the major in that the goal of this course is not to introduce students to new mathematics, but rather to teach students how to construct and write a sound mathematical argument. Students are introduced to various methods of proving statements and are taught to write logical, coherent proofs in paragraph form. In particular, students are taught that a good proof conveys to the reader what is being assumed, what is to be shown and what type of proof is going to be used.
Bridge to Advanced Mathematics serves as a gateway course in the major because once a student has successfully completed MATH 300, he or she can enroll in upper-level theoretical mathematics courses, such as Linear Algebra (MATH 315) or Introduction to Abstract Algebra (MATH 432), where students use proof techniques learned in Bridge to Advanced Mathematics.
The second course where writing is emphasized for our Liberal Arts Mathematics majors is our capstone course, Seminar in Mathematics, while for our Secondary Education Mathematics majors it is History of Mathematics. There are many places within the History of Mathematics and Seminar in Mathematics courses in which the skill of writing in the discipline comes to the fore. Naturally, these courses involve writing many mathematical proofs, the quintessential example of writing in the discipline of mathematics.
In the History of Mathematics course, writing comes up in other ways. For example a student might be asked to compare and/or contrast a modern proof or computation to one from an earlier era or to explain the significance or implications of some of the mathematics associated with a past culture.
Here are some recent examples of MATH 458 problems that involve writing in mathematics. It should be noted that there are a plethora of mathematical proofs assigned, none of which are noted here.
- In ancient Greece, there were three classical impossible geometric construction problems. Describe each.
- Explain carefully why the method of ancient Egypt known as “false position” must always give correct results for solving linear equations.
- In the famous Babylonian clay tablet known as Plimpton 322, the way that the rows have been ordered is a key reason why this tablet is of such historical importance. Describe how the rows are ordered and describe the implications about what mathematics was known to the Babylonians.
- Use Newton’s method of fluxions to find the slope of the tangent line to [EQUATION GIVEN HERE] and compare the work to a solution using modern calculus techniques. Explain the similarities and the technical differences.
In Seminar in Mathematics, students are asked to complete a project where they research and write about an appropriate historical and mathematical subject, along with at least one mathematician of significance in the area. Students write a paper on their topic and present their findings to the class. Examples of past projects are listed below:
- The Four Color Theorem: What does the Four Color Theorem say? What is unusual about the history of the proof of this theorem? Does the mathematical community accept the current “proof” of this result? Who are the key mathematicians involved in the history of this problem? Is there current work being done on this problem? Related problems?
- Penrose Tilings: What are the regular tilings of the plane, and how are Penrose tilings different? Who is Roger Penrose, and how is his work with these tilings significant? What are some of the interesting things that can be proved about Penrose tilings? What are some of their applications?
- π is irrational (indeed transcendental): What is the history of π? Who was the first to prove π is irrational? What is a transcendental number? Can we prove that π is transcendental? Are there other famous transcendental numbers?
- The Golden Ratio and its presence in history: Who first noticed the Golden Ratio? Where are some amazing places it arises in nature? How is the Golden Ratio linked to Fibonacci numbers? Who discovered this connection? Numbers in nature: How do the Fibonacci numbers and the Golden Ratio appear in flowers? Why is this so?
- The Brachistochrone Problem: Imagine two points A and B at different heights above the ground. There is an infinite number of smooth curves between A and B. Imagine a ball rolling from point A to point B. The time it takes the ball to roll from A to B depends on the curve. Find the curve that minimizes the time it takes the ball to roll from A and B. (Differential equations will be needed.) Who first introduced this problem? Who solved it? What sorts of mathematics has this led to?