The 2014 Calculus Revision

Beginning in Spring 2011 The Mathematics Department undertook a comprehensive review and redesign of the sequence of calculus courses taken by students of mathematics, the physical sciences and engineering. This web page summarizes the history and outcome of that revision. (This page is adapted from documents prepared in the course of the deliberations which led to the new courses. Some of the links may no longer be active.)

History

Prior to 1988 the university used the quarter system. The calculus courses consisted of 1211, 1212, 1213 (5 hr.s each) for the freshman year and 2211, 2212, 2213 (3 hr.s each) for the sophomore year. With the change to semesters these became 1215, 1216 (5 hr.s each) and 2215, 2216 (3 hr.s each). In particular 2215 & 2216 comprised a year-long treatment of multivariable topics, linear algebra, and differential equations. That tripartite division of topics was natural under quarters but more awkward after the change to semesters. Many faculty were finding it hard to cover all the material on the syllabus. In 1989 the UPC was asked to consider the courses and make recommendations. The resulting report recommended a number of changes to the syllabi designed to improve efficiency. In 1991, on the recommendation of a committee chaired by Max Gunzburger, the department changed to the suite of courses used from 1991 until 2014. The courses were 1205-1206 (single variable calculus), 1114 (elementary linear algebra), 1224 (vector geometry), 2224 (multivariable calculus), and 2214 (differential equations). A separate web page provides more details on the content of those courses, their prerequisite structure. Concerns regarding the effectivenes of the courses and the sometimes unusual organization of topics began to emerge. In particular the organization of topics had made the assignment of credit for high school AP courses and courses transfered from other instutions akward. (More of the specific concerns are identified in the preceding link.)

In 2000, faculty concerns led to an effort to review the effectiveness of the calculus courses. An Ad Hoc Committee on Standards in the Freshmen/Sophomore Calculus was formed to review the concerns and issue a report. The task of completing the report was passed to the Calculus Committee. The resulting Report on Standards in Freshman/Sophomore Calculus (Feb. 2001) made a number of recommendations, including a call for a comprehensive review of the calculus courses. The UPC began the task of carrying out that review the same year. Excepting 2214, reports were prepared for each course based on CTE results, and a collection of proposals was presented to the faculty in April, 2002. Several faculty responded to the proposals, but no particular changes were instituted on the basis of those discussions.

In Spring 2011 the Mathematics Department began internal discussions leading toward a redesign. Input from other departments was solicited starting in the summer of 2011, culminating in an open meeting on Nov. 15, 2012 for representatives of all interested departments. In light of all the expressed concerns and suggestions the Mathematics Department decided to change to a new suite of courses described below.

The New Courses

The new courses as adopted in 2014 are described below. (Any changes made since then are not reflected here. See the current course catalogue for current versions.) In the descriptions below, chapter references are based on the initially-adopted textbook: Hass-Weir-Thomas, University Calculus: Early Transcententals (second edition), Addison-Wesley. (The table of contents is available from the publisher's web site.) Significant topical changes from the previous courses are highlighted in red.

Calculus for the Sciences: Math 1225-1226. These are 4 credit hours each, covering the standard material on calculus for functions of one variable.

Math 1225 covers Chapters 2-5. This includes limits, continuity, differentiation with applications, but also L'Hopital's rule, the definition of integral, and application to calculation of areas (both formerly in Math 1206).
Math 1226 covers Chapters 6-10 (with some omissions). This includes applications of integration to volumes and arc length, techniques of integration for trigonometric and rational functions, trapezoidal and Simpson's rules, but also infinite series, parametric curves, and polar coordinates (formerly in 1224 and 2224). There will also be lab work involving implementation with contemparary software (which would be Matlab at present). 1225 will be prerequisite.

Multivariable Calculus for the Sciences: Math 2204. A 3-credit hour course covering calculus of several variables covering Chapters 11-14. This includes basic material on vectors, planes and quadratic surfaces, analysis of space curves, functions of several variables, optimization including Lagrange multipliers, multiple integrals and applications, with software implementation. (Vectors and space curves were formerly in 1224.) Topics are sequenced to accomodate students taking ESM 2104 Statics concurrently. 1226 is prerequisite, but not 2114.

Introduction to Linear Algebra: Math 2114. A 3 credit hour classroom-based course including basic Gaussian elimination and introduction to matrices as in the current 1114, and additional linear algebra concepts including linear idependence, bases, rank of matrices, eigenvalues, diagonalization, symmetric and orthogonal matrices, and software implementation. 1225 will be prerequisite, but not 1226.

Introduction to Differential Equations, Math 2214, will remain unchanged. 1226 together with either 1114 or 2114 are prerequisite.

The new courses together with 2214 constitute 17 credit hours, one more than currently. Math 1224 was discontinued, its content having been moved into 2204. Math 1114 continued to be offered for those majors whose students cannot afford the extra credit hour of 2114 over 1114, allowing them to keep their total credit hours at 16.

The normal sequence in which students should take the new courses is

1225
1226
2204 and 2114
2214

However strong students may take 1226 & 2114 simultaneously, and 2204 & 2214 simultaneously, thereby completing the full sequence in 3 terms.

AP and Tansfer Credit

Under the old courses a student with a calculus AB score of 4 or 5, or a BC score of 3 will recieve 3 hours of credit for 1205. When taking 1206 here they found themselves repeating the introduction to integration that is part of the standard AB curriculum. Students with a BC score of 4 or 5 currently recieved 6 hours of credit for 1205 and 1206, but in 2224 found themselves repeating the infinite series material covered in their BC course. Under the new courses these two groups of students respectively will recieve either 4 hours credit for 1225 or 8 hours of credit for 1225 and 1226 and will spend less time repeating topics they studied in high school. (Complete information about credit currently awarded on the basis of AP, IB, or CLEP exam scores is provided by the Registrar's office.)

Many students transfer to Virginia Tech from Virginia Community College System. Descriptions of their mathematics courses are available on the VCCS web site, and the schedule of VT credit granted to transfer students who have taken these courses is provided by the Regisrar's office. We propose to award transfer credit for the new courses as follows.

VCCS Course(s)	VCCS Credits	Transfer to New VT Course(s)	VT Credits	Comments
173	5	1225	4
174	5	1226	4
174 & 275	9	1226 & 2204 & 1114	9
175	3	1016	3
176	3	2015	3
175 & 176	6	1225	4	alternative to 1016 & 2015
177	2	1114	2	this is unchanged
178	2	none		178 matched 1224, which is to be retired
273	4	1225	4
274	4	1226	4	presumes the inclusion of infinite series
275	4	1114	2 or 3
277	4	2204	3
285	3	2114	3

Transition Plan

The old courses were phased out and the new phased in over the 2014-15 academic year, according to the following proposed schedule.

	Summer 2014	Fall 2014	Spring 2015	Summer 2015	Fall 2015 and thereafter
1205	offered
1206	offered	offered
1224	offered	offered
2224	offered	offered	offered	offered

1225		offered	offered	offered	offered
1226		offered	offered	offered	offered
2114		offered	offered	offered	offered
2204		offered	offered	offered	offered

1114 and 2214 will continue to be offered every term.

Substitutions

We expect the following combinations of old courses will be accepted as substitutes for the new.

1205 and 1206 will substitute for 1225.
1206 and 2224 will substitute for 1226.
1224 and 2224 will substitute for 2204.

Upper Divison Mathematics Prerequisites and the Minor in Mathematics

We anticipate the following prerequisite changes for upper division mathematics courses. (For courses not mentioned the prerequisites would stay unchanged.)

A C or better in 2114 will be (the only) prerequisite for 3034, 3144
2204 and a C or better in 2114 would be the prerequisites for 3214.
A passing grade in 2114 will be the prerequisite for 4445; 2214 would be the prerequisite for 4446.

The minor checksheet will be revised to require the following.

1225, 1226, 2204, 2214;
Either 1114 or 2114;
Any 9 hours of math courses numbered 3000 or higher.

Note that this leaves at least one route to the minor for those students taking 1114 instead of 2114: 3414, 4564, 4574 (unless CS decides to make 2114 prerequisite for 3414). There will probably be other possibilities.