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MTH 210 Communicating in Mathematics
In the late 1970s, Grand Valley State Colleges decided to switch from the quarter system to the semester system. This took effect for the 1980-81 academic year. For that academic year, the College of Arts and Sciences added two new graduation requirements, which were the Basic Skills Requirements and the Supplemental Writing Skills Requirements. At that time, the Supplemental Writing Skills (SWS) Requirement was that “each degree candidate was required to complete two additional courses in which writing is emphasized.” These two courses could be courses that counted in the student’s distribution, major, or minor program.
The SWS Requirement in Mathematics
Because of the change to semesters, departments and programs were encouraged to review the requirements for their majors. Due to the inspiration and leadership of Dr. Virginia Muraski, the Department of Mathematics and Computer Science decided to institute a supplemental writing skills course that would be required of all mathematics majors and all mathematics minors seeking secondary certification. This course was MTH 235 Communicating in Mathematics. (Note: The course was renumbered as MTH 210 for the 1995-96 catalog.) Following is the description of this course from the 1980-81 catalog.
235 Communicating in Mathematics. A study of the logical and rhetorical techniques of exposition in the language of mathematics. The reading and discussion of selected mathematical writings. Intensive practice in communicating in the language of mathematics through analyzing and critiquing compositions based on the selected readings. Prerequisites: Mathematics 120 and a basic writing skills course. Offered fall semester of even-numbered years.
It is interesting to note that even though it was required of all majors, it was only offered in the fall semester of even-numbered years. This is undoubtedly why the prerequisite for the course was only a precalculus course. The course was changed to being offered every fall semester in the 1986-87 catalog and to being offered fall and winter semesters in 1991-92.
After being taught several times, the department began a discussion about the role of MTH 235 . Most faculty members liked the idea of having a course in mathematics that emphasized writing but some also felt it a “transition course” from the problem solving orientation of lower-level courses to the more theoretical and abstract upper-level courses was needed. In particular, many faculty members felt that before they took upper-level courses, students needed a course in which they would learn how to construct and write proofs. In the fall of 1987, the department had several discussions during department meetings about changing MTH 235 to a 300-level course, drop the SWS designation, and have the course focus more on mathematical proofs. (The course was to be called MTH 305 – Fundamental Concepts of Mathematics.) However, this proposal did not seem to gain a great deal of enthusiasm since many faculty members thought the course should remain an SWS course.
Because of these discussions, however, it seemed clear that many faculty members wanted MTH 235 to focus more on mathematical proofs. As a result, the department to begin a conversation about what the prerequisite for MTH 235 should be. For the 1989-90 catalog, the prerequisite was changed to MTH 201. The rationale for this change was that from the experience of the instructors, it was determined that students needed the mathematical maturity obtained by completing a formal course in the calculus in order to obtain a greater understanding and appreciation of the objectives of this course. In addition, most faculty members felt that it was not appropriate for students to take both MTH 235 and the first semester of calculus at the same time.
Until the Winter 1990 semester, Virginia Muraski was the only faculty member who taught MTH 235 when Karen Novotny taught the course. Fall 1990 was the first semester that two sections of MTH 235 were taught. (Taught by Drs. Muraski and Novotny). Eventually, Dr. Muraski convinced a few others (including Tom Gruszka, Ted Sundstrom, and Steve Schlicker) to teach the course, and more faculty who were involved with upper-level mathematics courses began teaching MTH 235.
Many of these faculty members began working with students to help them construct and write proofs since they felt that this was one of the most important aspects of communication in mathematics. As a result of this, those teaching the course began a discussion about whether certain content should be specified for MTH 235. Consequently, the department decided to look at the content for this course in light of the courses for which MTH 235 was a prerequisite. After many discussions, the department decided to include certain material for the course that was not adequately covered in other parts of the lower-level curriculum and would be of benefit in upper-level courses. Because of this, a new description for MTH 235 was approved by the department for the 1994-95 catalog. Following is this description.
MTH 235 Communicating in Mathematics. A study of proof techniques used in mathematics. Intensive practice in reading mathematics, expository writing in mathematics, and constructing and writing mathematical proofs. Mathematical content will be selected from the areas of logic, set theory, number theory, relations, and functions. Prerequisite: 201. Three credits. Offered fall and winter semesters.
The following two paragraphs are from the Course Change Proposal to change the description for MTH 235 that was approved by the department and university for the 1994-95 catalog.
This course has always been offered as a Supplemental Writing Skills course since one of the major objectives for this course is to improve the students' abilities to write in the language of mathematics. However, the course was also intended to serve as a transition from the problem solving approach of the lower level courses in vie major (such as calculus) to the more abstract, theoretical approach of the upper level courses in major (such as geometry and abstract algebra). One of the difficulties for students in mis transition has been in properly writing in the language of mathematics, and in particular, in the constructing and writing of mathematical proofs. Consequently, as the course has developed over the years, the emphasis in writing in the course has become constructing and writing mathematical proofs. This indeed is one of the most important aspects of communication in mathematics. We now want the course description for MTH 235 to reflect this emphasis on constructing and writing proof}.
As we started emphasizing proofs in the course, we realized mat we needed to specify some mathematical content for the course since the students had to write proofs in some areas of mathematics. Consequently, the department decided to look at the content for this course in the light of the courses for which MTH 23 5 is a prerequisite. After many discussions, the department decided on certain material for this course that was not adequately studied in other parts of the lower level curriculum and would be of benefit in the upper division courses. The new course description reflects our choice of content for this course.
The last change for this course was approved for the 1995-96 catalog. The course was renumbered from MTH 235 to MTH 210. The main rationale for this change was to reinforce the message that mathematics majors should take this course as soon as possible after MTH 201. In doing this, the faculty was attempting to produce more uniform mathematical backgrounds of the students enrolled in MTH 210. At this time, the department approved the following objectives for MTH 210.
1. To enhance the ability to construct and write mathematical proofs using standard methods of mathematical proof including direct proofs, mathematical induction, case analysis, and counterexamples.
2. To develop logical thinking skills and to develop the ability to think more abstractly in a proof oriented setting.
3. To improve the quality of communication in mathematics. This includes improving writing techniques, reading comprehension, and oral communication in mathematics.
4. To explore and understand the concepts described in the following course content: Elementary logic, sets, axiomatic systems, elementary number theory, relations, functions, and methods of mathematical proof including direct proofs, indirect proofs, mathematical induction, case analysis, and counterexamples.
For the 2000 – 2001 academic year, the course objectives were revised. The new course objectives approved then continue to be the course objectives, which are the following:
- To develop the ability to read and understand written mathematical proofs.
- To develop the ability to construct and write mathematical proofs using standard methods of mathematical proof including direct proofs, mathematical induction, case analysis, and counterexamples.
- To develop logical thinking skills and to develop the ability to think more abstractly in a proof oriented setting.
- To develop talents for creative thinking and problem solving.
- To improve the quality of communication in mathematics. This includes improving writing techniques, reading comprehension, and oral communication in mathematics.
- To explore and understand the concepts described in the following course content: Elementary logic, sets, axiomatic systems, elementary number theory, relations, functions, and methods of mathematical proof including direct proofs, indirect proofs, mathematical induction, case analysis, and counterexamples.
The Proofs Portfolio for MTH 210
Although there have been no other changes in MTH 210 that were sent to the curriculum committee for approval, there have been significant changes in the way that MTH 210 has been taught. One of the things that was difficult to deal with was the Supplemental Skills Writing requirement that the instructor must work with the students on revising drafts of their papers, rather than simply grading the finished pieces of writing. Perhaps the most common method of meeting this requirement was that some of the assignments required handing in draft versions for the instructor to make comments and suggestions to the student and then have the student hand in the final version of the assignment to be graded.
In his second year at Grand Valley (Winter 1997), Dr. Ed Aboufadel experimented with a new type of assignment in MTH 210 that is still being used in all sections of MTH 210. Inspired by an article in a mathematics journal by Stephen Post, “Teaching the Art of Problem Solving,” Dr. Aboufadel developed a so-called “Proof Portfolio” requirement MTH 210 as a way for students to learn how to do proofs, and to satisfy the “writing with revision” requirement for SWS courses. Although there have been many variations of this assignment developed by various instructors over the years, the basic idea, as introduced by Dr. Aboufadel, is as follows:
The portfolio consists of ten proofs or propositions to be proven or disproven. Students may hand in each proof to the professor two times to be critiqued. (Some instructors allow more than two submissions.) Most of the critique is directed toward the student's writing, but quite often, give some “mathematical direction” must be given to the student. However, most of the time this is quite general, such as, ``You have an algebra mistake here.'' The goal is that each student will have a completed a ``Portfolio of Proofs'' at the end of the semester. The proofs in the portfolio are chosen to illustrate the various proof techniques discussed in the course. Since students have the opportunity all semester to submit proofs for comments, in order to receive full credit, a proof or solution must be correct, complete, and well written with no spelling or grammatical errors. Hopefully, the students will be able to use their portfolios to provide examples of various proof techniques if they are required in later courses. The techniques are the usual ones discussed in this type of course, direct proof, proof of the contrapositive, proof by contradiction, proof using cases, and mathematical induction. For an example of a portfolio assignment used by Dr. Ted Sundstrom, please click here.
Textbooks and the Teaching of MTH 210
In the late 1990’s many faculty members were becoming more and more dissatisfied with the textbooks that were available for MTH 210. It was felt that most textbooks did not really address the issue of writing mathematical exposition. Although the instructors were trying to handle this during the course, it was felt that it would be nice if the textbook addressed this specifically and then used practices of good writing throughout the textbook. In addition, it was felt that most textbooks did not address the actual process of writing a proof. There were textbooks that did discuss how to construct a proof but for the most part, these textbooks did not provide any guidelines for writing a proof and did not cover material that the department wanted to be included in this course.
One other problem with textbooks was that they were not up-to-date with the way that the teaching of mathematics courses had changed during the 1990’s. Faculty at Grand Valley were requiring students to explore mathematical concepts rather than having the text or the instructor simply tell them about these concepts. That is, the faculty wanted students actively involved in the learning process. This often involved collaborative learning where students work in groups to brainstorm, make conjectures, test each others’ ideas, reach consensus, and hopefully, develop sound mathematical arguments to support their work. Most textbooks for this “transition” course were quite traditional and simply presented the material and then had the students attempt to work exercises or write proofs.
Because of these concerns with the available textbooks for MTH 210, the department approved a sabbatical leave proposal for the winter 2000 semester for Ted Sundstrom to develop text materials for MTH 210. This project was done with the assistance of Dr. Karen Novotny and Dr. Matt Boelkins and resulted in a textbook, Mathematical Reasoning: Writing and Proof, that was eventually published by Prentice-Hall. Some edition of this textbook (including preliminary editions) has been used as the textbook for MTH 210 since the 2000 – 2001 academic year. (Click on this link to see a list of some of the textbooks that have been used for MTH 210.)
MTH 210 and the Mathematics Minor
One interesting aspect of MTH 210 (MTH 235) has been its use within the mathematics minor. When it was first introduced, MTH 235 was required for the mathematics minor for secondary certification but not for the mathematics minor for those not seeking certification. The primary reason for this was that the department wanted to make it relatively easy for engineering and physics majors to obtain a mathematics minor. In 1995-96, a new minor was introduced. This was a mathematics minor for elementary certification and MTH 210 was required for this minor. For the 1996-97 catalog, the requirements for mathematics minor for those not seeking certification was changed. Prior to this, the requirements were MTH 201, MTH 202, MTH 227, and at least three mathematics or statistics courses at the 300 – 400 level. The new requirement added a requirement of MTH 203 Calculus III or MTH 210 and changed the number of 300 – 400 mathematics or statistics courses to from three to two. This still made it reasonable for engineering and physics majors to obtain a mathematics minor.