The trends are also very encouraging when we break enrollments out into four basic categories: Precollege (usually not for college credit), Introductory (including college algebra, precalculus, mathematics for liberal arts, and business mathematics), Calculus Level (including most sophomore courses including linear algebra, discrete mathematics, and differential equations), and Advanced (junior and senior courses). These numbers do not include enrollment in statistics courses.
Precollege courses rose only 4%, Introductory rose 22%, Calculus Level saw an increase of just over 30%, and Advanced course enrollments went up by almost 33%.
Most of the teaching of Precollege mathematics has shifted to two-year colleges, as the following graph illustrates. The category Other is dominated by elementary statistics, but also includes finite math, business math, math for liberal arts, and math for elementary teachers. Here, Calculus refers only to calculus courses, but includes Several Variable Calculus.
While Precollege mathematics continues to dominate 2-year college enrollments, accounting for over half of all the students in mathematics classes, its numbers rose by only 15%. Introductory enrollments rose by 14%, Calculus by 27%, and Other by 35%.
Parsing the data by type of institution, we see strong growth across undergraduate colleges (characterized as highest degree offered in mathematics is the Bachelor’s), comprehensive universities (highest mathematics degree is the Master’s), and research universities (which offer a doctorate in Mathematics).
The one disappointing bit of news is that the number of Bachelor’s degrees awarded by mathematics departments went up by only 6% over the past five years, from 14,611 to 15,499. These numbers include degrees in Operations Research, Actuarial Science, and joint degrees awarded by the mathematics department, but exclude degrees in Mathematics Education, Statistics, or Computer Science as well as degrees that might be considered mathematical science but were awarded by other departments. Here, there was a great deal of variation by type of institution.
At undergraduate colleges, the number of Bachelor’s degrees in Mathematics dropped by almost 9%, while it rose by 27% at comprehensive universities and by 14% at research universities.
This raises three obvious questions: Why have course enrollments risen so fast over the past five years? Why hasn’t this surge been reflected in increased numbers of majors in Mathematics? Why, despite the increase in mathematics enrollments, is the number of Bachelor’s degrees from undergraduate colleges moving in the opposite direction from the number at other types of institutions?
I believe that the answers to all three questions are related and are indicated by the pattern of intended majors of in-coming full-time students. These data are gathered by the Higher Education Research Institute at UCLA from most four-year undergraduate programs at the time of freshman orientation. They are reported annually in The American Freshman.
There is no mystery about what changed after 2007. As I reported in my Launchings column of a year ago, A Benefit of High Unemployment, there is a very high correlation between the economic situation as reflected in the unemployment rate and the attractiveness of scientific and technical majors.
The surge of students who have arrived in Mathematics departments because of the current economic downturn have not been with us long enough to significantly impact the number of majors. The fact that they are swelling enrollment not just in Calculus-level but also Advanced mathematics courses is an encouraging sign. It also presents a challenge for our departments to take advantage of this increased interest in Mathematics.
On August 25, Sol Garfunkel and David Mumford ignited a firestorm with their provocative piece in the New York Times on "How to Fix our Math Education.” I’d like to use this column to respond to one of their conclusions, the comment near the end of their article that, “In math, what we need is `quantitative literacy,’ …”
I’ve written many of my columns about Quantitative Literacy (QL), including as recently as this past January (Mathematics & Democracy + 10). I was one of the leaders in developing a QL program at Macalester, have served on the board of the National Numeracy Network, and continue to promote QL whenever and wherever I can. Yet I’m very bothered by the suggestion that QL is what we need in math.
This past winter, I visited Lehman College in New York to consult on the creation of a program in QL that is being developed by faculty from several disciplines. Their mathematics department was viewing this emerging course with some uneasiness. Would it replace their developmental courses in algebra? Were their students even ready for QL? In response to their concerns, I wrote:
A QL requirement should be independent of a mathematics requirement. If your students need algebra, QL should not replace that. In the other direction, algebra is not a substitute for QL. The mathematical and statistical skills needed for QL are basic. Algebra need not be a pre-requisite. What makes this college-level material is that these skills are applied and interpreted in messy, real-world situations, using quantitative approaches to aid analysis of complex social issues. In many respects, the natural home for QL is in the social sciences, but I believe that math and stat departments have an important role to play in keeping the mathematics of QL honest and encouraging quantitative thinking as one of the important tools for studying social issues.
This has been the guiding principle behind Macalester’s QL program. I am very proud of the strong inter-disciplinary nature of the QL program that we have created here, and I am suspicious of any program that claims to be QL but is taught exclusively by mathematicians. I also should add that Macalester has no mathematics requirement for graduation, but it does have a QL requirement. I heartily endorse this choice. I do not see a need for all students to study college-level mathematics, but I do see a need for improving their ability to apply quantitative reasoning.
This past summer, I had the chance to review a new textbook in QL that uses ratio and proportion as the unifying theme. The book presents a well-written course that can help students gain understanding of the power and uses of these basic mathematical tools. Many college students, some graduates of Macalester included, never achieve this level of understanding of ratio and proportion and would benefit from such a course. Yet I would hate to see this classified as a course in mathematics. It is not just that the mathematics is what should have been mastered in middle school. It is that the only reason this is a legitimate college-level course is that it transcends the concerns of the mathematics classroom.
While I feel strongly that we need to draw a clear distinction between mathematics and QL, I am not saying that mathematics should be taught without regard for the world beyond the classroom. All mathematics should be taught with the goal of promoting student ability to use these tools and ideas in ways that transcend the specific circumstances under which they have been learned. But we also need to recognize how very difficult it is to accomplish this. The research that I have seen suggests that the most effective means of reaching this goal is to lead students through an alternation of theory building and a variety of applications, combined with plentiful opportunities for personal experimentation and reflection. This goal is much more than quantitative literacy. It is the development of mathematical ability.
I want to take advantage of the new blog format of this column to generate a discussion that I hope will help us identify the greatest problems facing mathematics departments. The intent is to assist the MAA’s Committee on the Undergraduate Program in Mathematics (CUPM) in determining the kind of information and advice, especially with regard to the major in the mathematical sciences, that would be most helpful.
As I’ve travelled the country for the MAA over the past few years, I have seen that the greatest problem facing most departments is simply lack of resources. These include the inability to replace tenured faculty who have retired or left, the need to rely on adjunct and part-time faculty to meet expanding student demand for mathematics classes, and inadequate facilities. The MAA will not be able to help directly with any of these. It can help by providing access to what is known about the effects of large classes and part-time faculty on mathematics instruction and by illustrating best practices that have enabled departments to do more with less. It also can help indirectly by assisting departments in improving the major while attracting more students. I have observed in many settings that, especially in these times of tight budgets, what resources are available tend to flow toward those departments that the administration sees as having a healthy and successful majors’ program that is clearly tied to the core mission of the college or university.
Ten years ago, CUPM was engaged in preparing the CUPM Curriculum Guide 2004. That Guide looked broadly at the role of mathematics departments and made a number of recommendations. I began this column in February, 2005 for the purpose of drawing attention to and further explicating these recommendations. It is now time to think about the next iteration of the Guide. A writing group, chaired by Martha Siegel and including Carol Schumacher, James Sellers, Michael Starbird, Alan Tucker, Betsy Yanik, and myself, has begun the task of identifying the issues and problems that the new Guide should address. This time, the focus will return to the mathematics major.
We have collected suggestions from many people about the issues and questions for this iteration of the Guide. I am interested in your reactions: What have we missed? Where should our priorities lie as we prepare the next Guide? What is going to be most useful and how can it be made useful to more people? I do not promise that the next issue of the Guide will address all of the important issues. Some of them would take us too far afield. For some, the relevant information does not exist or is simply too difficult to obtain. Even so, knowing that something is a burning issue for many departments will help the MAA decide future areas of focus and may help in getting funding to seek out the needed information.
I encourage you to respond via a comment on this blog or by sending me an email, firstname.lastname@example.org. I am also interested in what effect, if any, the CUPM Curriculum Guide 2004 had for your department. Has your department used the Guide? If so, what in it or about it was helpful? What did you find inadequate or lacking?
As you think about the issues listed below, it would be helpful for us to know what you and your department would find most useful, such as
a. Examples of models that work (with evidence of why they are believed to work),
b. Examples of what different departments are attempting or experimenting with,
c. Descriptions of current common practice organized according to size and mission of the department,
d. Specific recommendations with evidence for the recommendation.
Potential Issues and Questions Relevant to the Major
1. Core courses. What are/should be the core courses that every math major takes? What is commonly required that is not essential for all majors? How many credit hours of mathematics should a math major be required to take?
2. Curricular goals. What are/should be the goals of the courses in the core curriculum?
3. Tracks and programs. What tracks or programs should or could be made available to students? How does the size and mission of the institution affect this choice? What courses belong in, say, a track in financial mathematics or in biological mathematics? How does a department go about building an interdisciplinary track or program?
4. Double majors, concentrators, and minors. What can/should departments do to attract students to a minor or concentration in the mathematical sciences, or to pick up a second major that is in the mathematical sciences?
5. Career opportunities. What are the options for students who graduate with a major in mathematics? How can these be communicated more effectively to our students? How can they be used to help shape a student’s course through the major?
6. Preparation for upper division courses. What are examples of the variety of ways in which departments prepare their majors for proof-based upper division courses? What strategies are successful, and for whom are they successful?
7. Preparation of future teachers. What mathematics should future teachers of mathematics know (elementary, middle-school, or secondary teachers)? For prospective secondary teachers, should their preparation differ from that of other mathematics majors? (Note: An update to the MAA/AMS publication The Mathematical Education of Teachers, with guidelines that address these questions, is currently underway.)
8. Preparation for work or graduate study. What do employers of mathematicians in business, industry, or government (BIG) need their employees to have learned as undergraduates? Which courses are going to be most useful for students heading into a career in BIG and why? What do graduate programs in the mathematical sciences need their students to have learned as undergraduates? Which courses are going to be most useful for students heading into graduate studies and why?
9. Statistics and computer science. What statistics and what computer science should be required of all mathematics majors? What about those majors headed toward specific careers? What software should all majors be able to use (Excel, LaTeX, MatLab, Mathematica/Maple, R)? What are the different ways in which familiarity with these packages can be incorporated into the mathematics major? Should operations research be required of all majors or of all majors in particular tracks or programs?
10. Reading and communication. What is being done to improve students’ ability to read, write, and orally communicate mathematics? What should be done? What expectations should we have of all mathematics majors and how can we know whether they have these skills?
11. Proof. What is being done across the curriculum to improve student ability to read, understand, and construct proofs? What expectations should we have of all mathematics majors regarding the ability to construct proofs and how can we determine whether they have these skills?
12. Internships. What are examples of successful internship programs? How were they developed and how are they run? What expectations for internships should there be for all departments?
13. Undergraduate research. What does undergraduate research mean and how is it interpreted at various institutions? Should there be guidelines for what it means and how it is implemented? What are examples of programs that successfully incorporate undergraduate research?
14. Capstones. What are the various capstone experiences that departments have in place for mathematics majors? Which are the most successful programs, and for whom are they successful?
While the next iteration of the Guide will be concerned with issues of the major, this seems an opportune time to identify other issues, some of which—such as attracting talented students to courses that could lead them toward a major in mathematics—are tangentially relevant. While we do not expect that the general issues listed below will be treated in the next Guide, CUPM still wants to know which of these are of concern so that it can help coordinate and publicize the work of MAA on these issues.
15. Placement programs. What’s available? What works? Who does it work for? How do departments identify and properly place students with uneven preparation?
16. Students requiring developmental mathematics. What are the successful programs in developmental mathematics? If developmental mathematics is currently outside the department’s purview, should the department become involved in developmental education?
17. Students with weak or uneven preparation. How do departments meet the needs of students who are almost ready for calculus or may even enter with credit for some calculus, but who still have significant gaps in their mathematical preparation for college?
18. Talented entering students. How do departments attract talented students to their courses and keep them engaged in mathematics?
19. Calculus instruction. What kinds of calculus courses are now being offered? What should be offered? What works for which groups of students?
20. Articulation with two-year colleges. How can/should this be made as seamless as possible? What are the issues?
21. Working within state requirements and articulation agreements. How can a department be creative within imposed strictures?
22. Online courses. Should they be encouraged or discouraged? Should there be guidelines? When are they appropriate?
23. Workforce. Who is teaching which courses (full-time vs. part-time, tenured vs. adjunct vs. graduate student)? Does it make a difference? What can be done to assist instructors to improve the learning that takes place in their classes?
24. Pedagogy. What pedagogical approaches are most effective under which circumstances? What professional development should be available for college faculty, and who is responsible for supplying it?
25. Technology. How and when is it used most effectively? Is it ever essential, and if so when?
26. Departmental self-assessment. What are the strategies being used by departments to assess their effectiveness? What do we know about which strategies are most effective, and for whom?
“The best way to learn is to do; the worst way to teach is to talk.” —Paul Halmos 
Last month, in The Worst Way to Teach , I wrote about some of the problems with instruction delivered by lecture. It stirred up a fair amount of discussion. Richard Hake started a thread on the MathForum . He added several references to my own list and sparked a discussion that produced some heat and a lot of light. I do want to clarify that I recognize how important what I say in the classroom can be, as I will expound a bit later in this column. Nevertheless, I stand by my statement that “sitting still, listening to someone talk, and attempting to transcribe what they have said into a notebook is a very poor substitute for actively engaging with the material at hand, for doing mathematics.”
I also want to respond to a number of people who stated that reliance solely on lecture is not the real problem with mathematics instruction today; the real problem is … It was not my claim that moving away from pure lecture would solve all of our problems or even our greatest problems in mathematics instruction, merely that there are better ways to teach.
One high school teacher asked me for practical suggestions of things he could do to more actively engage his students. Fortunately, there are a number of resources. The one I pointed him to and that I want to talk more about in this column is the Academy of Inquiry Based Learning, a clearinghouse of information about and resources for Inquiry Based Learning (IBL) .
What is IBL?
IBL is a descendant of the method of instruction made famous by R.L. Moore at the University of Texas at Austin. Moore would give his students basic definitions together with statements of theorems that used those terms. Students were forbidden to draw on any sources other than own intellect to prove the theorems. Class time was spent entirely in student presentation of proofs, which would be critiqued by the class. It was a demanding regimen that produced many research mathematicians and six presidents of the MAA: R.H. Bing, R.L. Wilder, E.E. Moise, G.S. Young, Jr., Richard Anderson, and Lida Barret were all students of R.L. Moore.
The pure Moore Method was taught in small graduate classes with hand picked students, but Moore also adapted his method for teaching calculus, and many others since have modified his approach to fit the needs of their own students. The core of what we have learned from Moore is that the teacher needs to talk less and the students need to do more. This is the essence of IBL. On the website of the Academy of Inquiry Based Learning, IBL is described as follows:
Boiled down to its essence IBL is a teaching method that engages students in sense-making activities. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate... all those wonderful skills and habits of mind that Mathematicians engage in regularly. Rather than showing facts or a clear, smooth path to a solution, the instructor guides students via well-crafted problems through an adventure in mathematical discovery. 
As this quote indicates, there is a very “big tent” approach to IBL today. Each year, the Educational Advancement Foundation sponsors a Legacy of R.L. Moore conference. In addition to the enthusiasm of the several hundred participants, what I find most impressive is the variety of ways in which people plug into the basic idea of IBL. At the latest conference, held in Washington, DC, June 2–4, Ted Mahavier talked about how to just get started by using one day per week for students to go to the board to present their homework solutions, Eric Hsu explained the connections to Triesman’s Emerging Scholars Program, Angie Hodge showed how this approach is expressed in the Math Teachers’ Circles, and Tom Banchoff explained his own take on IBL, which uses his course management software to enable his students to critique and learn from each others’ proofs.
My own IBL experience
Ever since I taught calculus at Penn State in 1993–94 with David Smith and Lang Moore’s Project CALC materials, I have recognized the importance of using at least some of my class time to engage students in the creative activity of doing mathematics: exploring, conjecturing, proving, and—most important—communicating. But such activities always were in conjunction with a fair dose of my own explanation of what is important and how to think about the mathematics. I am not prepared to give up that role. It is an important part of what it means for me to teach. But I am learning how to cede more of my control over what happens in the classroom.
This past year, I taught our junior/senior Number Theory course using Number Theory through Inquiry by Marshall, Odell and Starbird . The great advantage of this book is that it presents all of the theorems of elementary number theory but none of the proofs. As I used this book, I talked more than others might. Students were required to read and think about the theorems before we met in class. Class would start by answering questions they had about the reading, followed by student presentations of the results they were able to prove. Most of the class was spent discussing the more challenging proofs. Here, when the silence stretched too long, I would step in and explain how to think about this proof, perhaps even sketch a possible outline. For each class period, I identified several key proofs that each student would be required to write up and submit as homework. I did write out complete proofs in the first class or two, to explain the difference between the sequence of personal insights that convinces oneself that a complete proof has been found and the way one writes up a proof for public consumption. After that, I never again wrote out a complete proof. That was their responsibility, and that was the basis for their grade.
Unlike Moore, I encouraged students to work together and critique each other’s proofs outside of class. But each student had to produce his or her own complete written proof of each of the important theorems. At the end of the semester, I was pleasantly surprised at how much the students appreciated this experience. They greatly preferred creating their own proofs over trying to learn from someone else’s.
IBL: The study
Over the past several years, the Educational Advancement Foundation has sponsored a study of the effectiveness of IBL at four universities with established centers for the support of IBL courses: University of California, Santa Barbara; University of Texas, Austin; University of Michigan, Ann Arbor; and the University of Chicago. Sandra Laursen and her team of sociologists at the University of Colorado, Boulder conducted the study .
This was a large, complex undertaking that was complicated by the fact IBL was being implemented in a wide variety of types of courses, from mathematics for prospective elementary teachers through upper division mathematics, and in only a few cases were there comparable IBL and non-IBL sections. Nevertheless, there was a real difference in the way IBL and non-IBL courses were taught. On average, student-centered activities made up over 60% of class time in IBL courses. In non-IBL classes, the instructor talked an average of 87% of the time.
What Laursen found was that IBL made a difference across many areas. It produced higher cognitive gains, including understanding of mathematical concepts and improved thinking and problem-solving skills; higher affective gains, including increased confidence, improved attitude, and greater persistence; and higher social gains, including ability to collaborate and explain mathematical ideas to others. Laursen also found that the percentage of time that the instructor spent on student-centered activities was the single best predictor of student gains.
The strongest gains were observed among women and students with weak prior achievement. These gains appeared not just in the IBL class but also continued through subsequent required mathematics courses, whether or not they were taught using IBL. This happened without decreasing the achievement levels of men and students with strong prior achievement.
The fact that traditionally underrepresented groups of students benefit most from IBL should not be surprising. Those of us now teaching in our colleges and universities succeeded in the existing system because we knew or managed to learn how to convert the lectures into active engagement with the mathematics. Lecturing worked for us. But it does not work for the many students who have never learned how to study mathematics. Perhaps the best news from this study is that pulling time away from lecture does nothing to decrease the learning of those who best know how to benefit from that style of teaching. There is hope that by changing how we teach we can increase the population of students who can do mathematics.
 P. R. Halmos, E. E. Moise, and George Piranian. May, 1975. The Problem of Learning to Teach. The American Mathematical Monthly. Vol. 82, no. 5, 466–476.
Bressoud, D. July, 2011. The Worst Way to Teach, Launchings
Hake, R. Re: Lecture Isn't Effective: More Evidence. The Math Forum @ Drexel. July 15, 2011 12:56 PM
 What is IBL? Academy of Inquiry Based Learning.
 Marshall, D.C., E. Odell, M. Starbird. 2007. Number Theory through Inquiry. The Mathematical Association of America. Washington, D.C.
 Laursen, S., M.L. Hassi, M. Kogan, A.-B. Hunter, T. Weston. 2011. Evaluation of the IBL Mathematics Project: Student and Instructor Outcomes of Inquiry-Based Learning in College Mathematics. University of Colorado, Boulder.
Find more Launchings columns.
"The best way to learn is to do; the worst way to teach is to talk." —Paul Halmos 
"A good lecture is usually systematic, complete, precise—and dull; it is a bad teaching instrument." This common belief is also contradicted by the evidence that we have, the most recent and dramatic of which comes from the Carl Wieman Science Education Initiative (CWSEI) at the University of British Columbia (UBC). The CWSEI study compared lecture format with interactive, clicker-based peer instruction in two large (267 and 271 students, respectively) sections of introductory physics for engineering majors. The results were published in Science  and made enough of a splash that they were reported in The New York Times , The Economist , and ScienceNOW . What is most impressive is how well controlled the study was—ensuring that the two classes really were comparable—and how strong the outcome was: The clicker-based peer instruction class performed 2.5 standard deviations above the control group.
|Histogram of student scores for the two sections.|
“When given by such legendary outstanding speakers as Emil Artin and John von Neumann, even a lecture can be a useful tool—their charisma and enthusiasm come through enough to inspire the listener to go forth and do something—it looks like such fun.”But how many calculus classes resemble an inspiring lecture by Artin or von Neumann? Is that really what we are doing when we stand at the front of the room? In the CWSEI study, Instructor A was a much more animated and intense speaker than the regular lecturer for the experimental section. Yet by the end of week 11, his students performed no better than those in the other section.
Access pdf files of the CUPM Curriculum Guide 2004 and the Curriculum Foundations Project: Voices of the Partner Disciplines.
Find links to course-specific software resources in the CUPM Illustrative Resources.