Virginia Tech

The Traditional Course

Linear Algebra is a two-credit, introductory course taken by first-year students in engineering, physical sciences, mathematics, and other majors at Virginia Tech. The course serves about 2,000 students per year. Approximately 1,500 students enroll in the fall and 500 in the spring semester. A mix of tenure-track faculty (10), instructors (13), and graduate teaching assistants (15) teach 38 sections of the course, with about 40 students in each section. Each section meets twice a week for 50-minute lectures. Students can receive individual assistance during one-to-one office hours or during review sessions for the tests (if the teacher chooses to offer review sessions). The section teacher writes two in-class tests during the semester. The final exam is common to all sections; it consists of multiple-choice questions and counts for 10% of the course grade. An individual teacher can supplement this exam with additional questions of any kind. As it is originally configured, the course requires 105 hours of faculty, instructor, and/or graduate teaching assistant (GTA) time per section.

The traditional course faces four specific academic problems:

• The course does not address the range of various students’ preparation and learning styles. Many students learn the material easily, while others have difficulty due to weak math backgrounds or problems with the lecture format. On the other hand, students who come to the course with good preparation—students who could accelerate their work—are locked into completing within a standard time frame.
• Some students drop out of the course, and others stay registered but essentially give up and stop working along the way, after the deadline to drop the course.
• Course grades across sections often bear surprisingly little statistical relation either to "input" (e.g., SAT profiles) or to scores on the common final exam. A specific goal of the redesign is to make a better and more uniform connection between what is learned (topical completeness and depth, as measured in the common final exam, for example) and the course grade.
• Students in more advanced math, engineering, and mechanics courses do not retain skills from this introductory course.

The redesign of Linear Algebra is an early step in a larger transformation project for all large-enrollment mathematics courses at Virginia Tech, a project made possible by the opening of the Math Emporium. The Emporium is a 500-workstation computer lab/learning center staffed by faculty, teaching assistants, and undergraduate peer tutors who provide one-to-one help. An overarching goal of this project is to help students, beginning with their very first college courses, to be active agents in choosing and using learning methods and resources, something that is not easily done in the way courses are usually configured.

The Redesigned Course

Linear Algebra will be redesigned to include more options for self-directed study in an effort to improve students' content learning, success rates, active learning skills, and retention of material for later use. The learning goals for the redesigned course will require students to

• understand and be able to use the core concepts of basic linear algebra;
• understand how to use appropriate software to complete calculations;
• choose and use different learning methods and resources appropriate to their individual needs;
• retain and use core concepts of linear algebra in subsequent courses, even years later; and
• rebuild their understanding of linear algebra topics without further use of faculty resources.

To achieve these learning goals, the redesigned course structure will completely eliminate the 40-student sections; all coursework will be conducted in the Math Emporium. Course content will be organized into units that a student, progressing normally through the course, will cover at the rate of one or two per week. Each unit will end with a short, electronically graded quiz. Students can work on a completely flexible time schedule because the Math Emporium is open 24 hours per day, 7 days per week, with staffing by the Math Department for 75 to 80 hours per week. Except for the quizzes, the electronic material will be available to campus and local users on the Web. To supplement the new learning structure, experienced course faculty will develop a program of optional lectures, Web-based and e-mail communication with enrolled students, and methods and procedures for on-site help by faculty, GTAs, and peer tutors.

The system is designed for active, student-directed learning, since the student chooses how to learn the material. The absence of a routine classroom procedure encourages students to take more initiative and responsibility. Interactive tutorials provide individualized feedback for students to direct their learning. The Math Emporium's peer tutors, who guide rather than instruct, will point students toward appropriate resources and strategies. At the same time, comprehensive data collection allows teachers to adjust instruction as the course proceeds, on the basis of continuously monitored performance, and to deliver individualized suggestions and active help where appropriate. In this way, the system offers a personalized dimension that cannot be maintained in the old format.

A variety of learning materials, accessible at all hours, together with the in-person help available at the Math Emporium, will allow student and teacher to match learning and teaching styles. The computer-based practice, quiz, and testing systems provide instant feedback (including customized suggestions) and the opportunity for multiple attempts at tests. Computer-based exercises and quizzes provide a natural, interactive introduction to current computational methods.

Active, student-directed learning, with prompt and ample feedback, will promote the specific goals of increasing understanding and developing core skills. In addition, content and skills will be retained and used more successfully as the student moves on to other courses in mathematics and related fields. The electronic system also allows students to rebuild and extend their understanding of linear algebra later on, when it is encountered in another course.

Finally, the new system lends itself to grade outcomes that maintain standards and reflect uniformly what the student has learned. Presentations and tests are the same for everyone and are open to supervision and review over time by the department's curriculum committee. As data accumulate and are evaluated, Virginia Tech will develop assessment and testing instruments more appropriate to the redesigned context.

Traditional Course Structure

• 15-week term
• 38 sections of 40 students each year
• 2 (50-minute) lectures per week
• Common final exam
  
• 10 tenure-track faculty, 13 instructors, and 15 GTAs each teach one section of the course. They prepare and present lectures; design, administer, and grade two written tests; and hold one-to-one office hours and review sessions at their discretion.

Redesigned Course Structure

• 15-week term
• 1 section each term (1,500 students in fall, 500 students in spring)
• Optional weekly lectures
• Time spent in the Math Emporium depending on the student’s need
  
• One faculty Course Coordinator gives weekly lectures, holds office hours, designs online tests from an existing question database, and provides overall coordination of the course.
• Instructors and TAs work in the Emporium for varying amounts of time, helping students as needed in all math courses. They also proctor online tests and supervise Emporium help staff.
• Peer tutors work in the Emporium 8–10 hours per week, helping students as needed in all math courses

Summary

In summary, the redesigned course will implement the following changes:

• Combine multiple sections and treat them as one course
• Provide students with 24/7 access to course materials and resources online and in the Math Emporium
• Replace lectures with Web-based resources (interactive tutorials, computational exercises, an electronic hypertextbook, practice exercises with video solutions to frequently asked questions, applications, online quizzes)
• Increase quizzes for student feedback at the end of each unit
• Provide one-to-one help from a mix of faculty, instructors, GTAs, and peer tutors
• Offer students the possibility of accelerating through the course and of earning course credit through self-study and exam

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