Math Lectures: An Oxymoron?

by Carol A. Twigg

One of the most enjoyable parts of our work at the National Center for Academic Transformation is that we learn new things all the time. There’s nothing like spending most of your time engaging institutions of higher education in changing the way they think about teaching and learning to produce new ways of thinking.

Based on eight years of experience in working with a large number of colleges and universities as they seek to improve student learning while reducing instructional costs, we have identified a number of “models" and “principles” to guide the redesign of large-enrollment courses. We have learned that each of our Five Models for Course Redesign can produce improved student learning and reduced instructional costs if it embodies our Five Principles of Successful Course Redesign. Therefore, as part of the application process to both our national and state redesign programs, we have heretofore asked teams to select a redesign model and explain how they will embody the Five Principles within it as the first step in the planning process.

We haven’t wanted to prescribe a model for several reasons. First, we want each redesign team to “own” their redesign plan by making their own choices as they go through the planning process. Second, we are really interested in seeing variations on previous redesigns in different disciplines. We’d love to see the Emporium Model in fields other than math—a writing emporium, a chemistry emporium—or the Buffet Model in fields other than statistics. (We’ve seen the latter happen at Chattanooga State Technical and Community College where the psychology redesign team evolved their way from a Replacement Model into a Buffet Model, and we think this can happen in many other disciplines as well.) Third, we are also interested in seeing new models emerge as we work with greater numbers of institutions and greater numbers of disciplines.

But we have also learned that certain models seem to be appropriate to certain disciplines. For example, all of the foreign language projects that we have worked with have chosen the Replacement Model in which they move grammar instruction, practice exercises, testing, writing, and small-group activities to the online environment and use in-class time for developing and practicing oral communication skills. The nature of the discipline informs the choice of model. The Replacement Model has also been the model of choice in English composition for similar reasons.

In mathematics, the Emporium Model has consistently produced spectacular gains in student learning and impressive reductions in instructional costs. The Universities of Alabama and Idaho, LSU, Ole Miss, the University of Missouri-St. Louis, Virginia Tech and Wayne State have all replaced class meetings with a learning resource center that features student use of instructional software supported by on-demand personalized assistance. We more or less assumed that subsequent new projects in mathematics would want to emulate this success. Boy, were we wrong!

NCAT has now interacted with more than 50 institutions in their efforts to redesign introductory or developmental mathematics as part of formal course redesign programs. The majority of these institutions have initially chosen the Supplemental Model or the Replacement Model rather than the Emporium Model.

All of these institutions begin in more or less the same place. In the traditional format, the course meets three hours per week for 15 weeks and is taught in a didactic lecture format. Students often have access to a math help lab or tutoring center if they choose to take advantage of it. The course is taught by a combination of full-time faculty, temporary lecturers, adjunct faculty and graduate teaching assistants, depending on the institution. The department may choose textbooks or develop course outlines and content standards for the course, but typically there are no consistent efforts to ensure uniformity of content presentation or assessment across all sections. The result is unacceptably high failure rates and lack of student progress toward a degree.

Introductory math courses, including many that satisfy the general education requirements, are often problematic for several reasons. Many students are simply not interested in the subject. Others may lack adequate preparation. Still others may fear failure based on negative experiences in math classes in high school. The problem is magnified when course concepts, which students have already encountered in their high school coursework, are re-taught to them using the same instructional methods that did not work the first time.

Yet what do most institutions initially propose? Some propose to retain three hours of class per week supplemented by computer-based homework (the Supplemental Model). Others propose to replace one hour of class per week with one or two hours of lab work using instructional software while retaining two hours of class in the traditional format. All too often, this is what we hear: “Over the past year and a half, the math faculty have spent much time researching best practices in developmental math. They have examined such topics as assessment, reading in the math classroom, math tutoring labs, active learning strategies and technology. Results of this research were shared and, as a group, the faculty decided to maintain the current three hours of lecture and to supplement classroom instruction with lab time.” Didn’t Albert Einstein define insanity as doing the same thing over and over and expecting different results?

Does keeping lectures in a math course make sense? The opinion of all of the successful faculty project leaders who have redesigned math is unanimous: students do not learn math from going to lectures. The reason most success rates in math are so low, we believe, is because the three-lectures-per-week approach is simply not appropriate for introductory mathematics courses. Retaining two of the three lectures does not substantively change the model. Consequently, we do not believe that institutions taking this approach will be able to increase student learning.

Why Is the Emporium Model So Successful?

As the redesign team at Kingwood Community College in Texas has correctly observed, “The primary reason many students do not succeed in the [traditional math] course is that they do not actually do the problems. As a population, they generally do not spend enough time with the material, and this is why they fail at a very high rate.” This may seem obvious, but we can assure you that it is not.

The Emporium Model has been so successful for basically three reasons:

  • Students spend the bulk of their course time doing math problems.

By using an instructional software package such as MyMathLab, ALEKS or Hawkes Learning Systems, students are able to spend more time on task than when they simply watch or listen to a lecture given by someone else. Students find the software easy to use and achieve a comfort level with the technology in a short amount of time. Students especially like the instant feedback they receive when working problems and the guided solutions that are available when they do not get a correct answer. The software has been evolving and improving over the last five years, providing more reliable scoring and a better interface for students and instructors.

Attending lectures are a waste of student time and energy. The three hours spent listening to lectures are three hours that could be spent doing math. As one math professor has put it so well, "Students learn math by doing math, not by listening to someone talk about doing math."

  • Students spend more time on things they don’t understand and less time on things they have already mastered.

The traditional lecture format treats students as “one size fits all.” Some students are bored because other students’ questions result in repetition of conceptual material they have already mastered, while other students feel overwhelmed by the amount of material covered in one lecture session. In contrast, instructional software packages-- which include interactive tutorials, computational exercises, videos, practice exercises and online quizzes--can support auditory, visual and discovery-based learning styles.

Through diagnostic assessments for each student, areas of needed practice can be highlighted and individualized study plans developed. When a student understands the material, he or she can move quickly through it and demonstrate mastery. When a student gets stuck, he or she can ask for an example or a step-by-step explanation and take more time to practice.

  • Students get assistance when they encounter problems in doing math.

Traditional models increase the likelihood that students will get discouraged and stop doing the work because they must work without immediate support and admit before fellow students at the next class what they do not understand. Since most students would rather remain invisible than interact with the instructor in a public way, and will always protect themselves from embarrassment, they often do not resolve the questions they have. In addition, students typically turn in homework problems that are hand-graded and returned days after the students do the problems and make mistakes. By the time students see the graded homework, they are not sufficiently motivated to review their errors and correct the problem.

The Emporium Model provides help to students in a variety of ways. Students get help from the software’s online tutorials and guided solutions. Instant feedback lets students review their errors at the time they make them. Students also get help from fellow students. In several of the math emporia, computer stations are arranged in pods of four to six to encourage student collaboration. Students can also find help in the emporium where instructors, graduate teaching assistants and/or peer tutors are available to provide individual assistance when students encounter difficult concepts. Any problem areas that students encounter are addressed on an individual basis during lab time.

Overcoming the Most Common Objections to Using the Emporium Model

We find it hard to understand why faculty teams concerned about high failure rates in math would not embrace the Emporium Model. The logic of why it works seems impeccable.

Some of the objections are logistical:

  • We can’t afford to construct and equip an emporium. By redesigning the entire course using the Emporium Model, institutions have been able to produce substantial savings in annual operating costs, a portion of which can be re-directed to construct (which typically involves re-habbing existing space) and equip computer labs. Each of these universities produced the following annual savings by redesigning one course: LSU, $210,700; the University of Alabama, $60,000; Virginia Tech, $140,000; and Wayne State, $159,812. The University of Idaho produced an annual savings of $101,976 by redesigning three courses.
  • Our lab size limits our ability to use the Emporium Model. Some institutions say that their decision not to do an emporium stems from limits in the size of their lab. One applicant, for example, cites having only 85 workstations for 1500 students as an obstacle. The University of Idaho has been able to serve 2400+ students in a lab with 71 workstations in pods of four that are designed for as many as three students to work together at a single monitor. To accommodate this large number of students, the team has spread the load of student use more evenly by spreading assignment deadline dates across each day of the week. Thus 20% of students have deadline dates for assignments, tests and quizzes on Monday, 20% on Tuesday, and so on. The space is used more consistently, rather than just before a test or assignment is due, allowing more students to be accommodated in a smaller lab and reducing the lab downtime.

Other objections are a result of misunderstanding what the Emporium Model is. There is some tendency to confuse technology-enhanced learning such as online tutorials, homework, and quizzes with self-paced online courses in which students proceed at their own pace. We have all found that leaving students out on their own doing computer homework, without having some kind of face-to-face tutoring support available, is a recipe for disaster. Students need sufficient structure within a well-articulated set of specified requirements. A laissez faire, unstructured, open-entry/open exit model simply does not work.

For some faculty, getting rid of classroom meetings implies abandoning the human interaction side of a classroom and conjures up images of students working alone. Nothing could be further from what happens in a well-designed math emporium. As noted above, on-demand personalized assistance is a hallmark of the Emporium Model. At Virginia Tech, this assistance is available 24*7, and at most institutions using the Emporium Model, personal assistance is available far in excess of that offered by traditional formats.

In the overwhelming majority of cases, however, the decision to retain all or part of the lecture time stems from the reluctance of some or all of the faculty in the department to give up the traditional approach.

When this happens, we have several suggestions for those seeking to deal with faculty reluctance and move forward on a meaningful redesign.

  • Make lecture attendance optional for students. Several of our projects have done this as part of the transition from the traditional format to the fully redesigned format. The faculty will have the opportunity to see whether students want or need to go to lectures.
  • Offer different kinds of sections and compare the results. The idea here is to offer some sections with lectures and some sections without (or some with required lectures, some with optional lectures and some without) and then compare the learning results. In this way, disagreements among faculty can be settled based on empirical data.
  • Redesign what goes on in class. Keeping one class meeting per week and redesigning (rather than retaining) the activities that occur in that meeting is a successful strategy. Both the Universities of Alabama and Idaho have one weekly group meeting. Its purpose is to review past and future assignments, troubleshoot problems (technical or mathematical) that students are having with the course, and keep students on track to complete the course.

Of course, one of the best ways to overcome faculty resistance is to have a conversation with the faculty members who have led the effort to establish a math emporium on their campuses to learn more about what they did, how they did it and why they made particular choices. We especially recommend talking to or visiting the project leaders at LSU; the Universities of Alabama, Idaho and Missouri-St. Louis; Virginia Tech or Wayne State. Each has data to support the fact that students learn math by doing math, not by listening to someone talk about doing math. It’s really pretty simple, isn’t it?