MAA/NCTM Joint Position on Calculus

I am very pleased to report that MAA and NCTM (National Council of Teachers of Mathematics) have just approved the joint position statement on calculus that comes at the end of this column. It was crafted in response to the recognition of three serious problems:

1. The first problem is the general perception among high school students and their parents that preparation for college should, if at all possible, include studying calculus in high school. The result is that too many students short-change their preparation in algebra, geometry, trigonometry, and other mathematical topics in order to stay on a fast track to calculus. Too many otherwise talented students arrive at university without the mathematical foundation that is needed to succeed in the college-level mathematics required for their intended major.

This problem is documented particularly clearly in two national longitudinal studies conducted by the US Department of Education. From the high school class of 1992, 31% of those who completed a course of calculus in 12th grade or earlier then enrolled in precalculus when they got to college.[1] From the high school class of 2004, 17% (one in six) of those who had completed a calculus course in secondary school reported taking remedial mathematics when they got to college.[2]

The solution is to insist that the first priority should be to establish a solid foundation in mathematics, rather than to get into calculus while still in high school.

2. The second problem is related to the first. It is that too often the calculus course that is taught in high school is a version that trains students in techniques of differentiation and integration and in procedures for solving certain standard problems without developing their understanding of calculus. While this builds familiarity with the language of calculus, it misrepresents the true nature of college-level mathematics and creates a false sense of confidence. One of the most dramatic findings of the MAA’s national study of Calculus I instruction in college and university is the high confidence level of entering students and how precipitously it drops as a result of encountering the reality of college-level expectations for calculus.[3]

The solution is to insist that when calculus is taught in high school, it is taught as a course whose content really is equivalent to a mainstream college course.

3. The third problem is an outgrowth of the first two. Today’s reality for most students headed into STEM careers is a double dose of introductory calculus, once on each side of the high school to college divide. At the very least, this wastes student time, especially when the college course is taught as if this material were being experienced for the first time. For those students who, for whatever reason, have had a bad or simply uninspiring experience of calculus in secondary school, the prospect of repeating their experience can dissuade them from continuing their study of mathematics. Even for those who enjoyed their secondary school calculus, repeating this course can lead to boredom and poor study habits, often resulting in poor performance and a move away from mathematics intensive disciplines.

The solution is to insist that college faculty recognize their audience and deal with it, modifying how calculus is taught and offering alternatives to calculus for entering students.

Recommendations that address these problems lie at the heart of this position statement. Most satisfying from my point of view is the recognition that the issue of when and how calculus is taught is important and requires the efforts of both MAA and NCTM.

I want to thank all those who have worked to make this joint position statement possible. First is Mike Shaughnessy, President of NCTM, who agreed to make the adoption of such a statement one of his personal priorities. Next is Gail Burrill, former NCTM President and current Chair of the MAA-NCTM Joint Committee on Mutual Concerns who created the framework in which it was possible to craft this statement and who provided much useful feedback during its creation. And finally I want to thank the three people who worked with me to actually write the statement: Mike Boardman at Pacific University, Tom Kilkelly at Wayzata High School in Minnesota, and Dane Camp at New Trier Township High School in Illinois.

Question: How should secondary schools and colleges envision calculus as the course that sits astride the transition from secondary to postsecondary mathematics for most students heading into mathematically intensive careers?

MAA/NCTM Position Although calculus can play an important role in secondary school, the ultimate goal of the K–12 mathematics curriculum should not be to get students into and through a course in calculus by twelfth grade but to have established the mathematical foundation that will enable students to pursue whatever course of study interests them when they get to college. The college curriculum should offer students an experience that is new and engaging, broadening their understanding of the world of mathematics while strengthening their mastery of tools that they will need if they choose to pursue a mathematically intensive discipline.

In particular, the fact that calculus is a college mathematics course that increasingly is taught in high school requires that—

1. Students who enroll in a calculus course in secondary school should have demonstrated mastery of algebra, geometry, trigonometry, and coordinate geometry;
2. The calculus course offered in secondary school should have the substance of a mainstream college-level course;
3. The college curriculum should acknowledge the ubiquity of calculus in secondary school, shape the college calculus curriculum so that it is appropriate for those who have experienced introductory calculus in high school, and offer alternatives to calculus.

Faculty in our colleges and secondary schools should work together to re-envision the role of calculus in secondary and postsecondary mathematics education.  Faculty on both sides of the transition from secondary to college mathematics should work together to strengthen the mathematics curriculum so that students who intend to pursue a mathematically intensive career can acquire the mathematical knowledge and capabilities needed for such a career. College faculty and secondary teachers should define what it means for a student to be ready for college-level mathematics. After a student has matriculated in college, they should assess the effect of college-level mathematics offered in secondary school.  They also should clarify and broaden what is meant by college-level mathematics for secondary school.  They should also work to achieve a better understanding of the mathematical strengths and weaknesses of matriculating students, assess the effectiveness of placement programs for collegiate mathematics, and clarify and broaden what the first year of college mathematics can and should entail.

MAA and NCTM are committed to taking appropriate action within the structure of their organizations to assist in guiding the implementation of these recommendations.

[1] US Department of Education. 2008. National Education Longitudinal Study of 1988 (NELS:88). nces.ed.gov/surveys/NEL


[2] National Science Board. Science and Engineering Indicators: 2010. National Science Foundation. Arlington, VA. Appendix Table 1-20. www.nsf.gov/statistics/seind10/appendix.htm


[3] The MAA study Characteristics of Successful Programs in College Calculus found that 61% of the students in college Calculus I had studied calculus in high school, and 58% of all the students starting Calculus I expected to earn an A in the course. Confidence was the greatest single casualty of the course, dropping by almost half a standard deviation from the start to the end of the course.

On Engaging to Excel

The President’s Council of Advisors on Science and Technology (PCAST) has just released its report to President Obama on undergraduate Science, Technology, Engineering, and Mathematics (STEM) education: Report to the President, Engage to Excel: producing one million additional college graduates with degrees in Science, Technology, Engineering, and Mathematics.

As the title suggests, the bottom line is to increase the number of undergraduate degrees (Bachelor’s or Associate’s) earned in STEM fields by one million. As with budget savings, the numbers are made more impressive by making this a ten-year goal. Even so, that amounts to an extra 100,000 STEM majors per year. From the most recent year for which we have data, 2009, there were 241,000 Bachelor’s degrees awarded in STEM fields and 70,000 Associate’s degrees. We are thus looking to increase the number of STEM degrees by about one-third.

This is a report that matters. Last fall PCAST did a thorough survey of all federal grant programs that support undergraduate work in STEM fields, focusing on the National Science Foundation (NSF) but including programs of the Departments of Education and Labor. This report carries specific recommendations on how this money should now be directed. The NSF programs Transforming Undergraduate Education in STEM (TUES), STEM Talent Expansion Program (STEP), and Widening Implementation and Demonstration of Evidence-Based Reforms (WIDER) have come in for particular attention.

There are four principal recommendations:

1. Catalyze widespread adoption of empirically validated teaching practices.
2. Advocate and provide support for replacing standard laboratory courses with discovery-based research courses.
3. Launch a national experiment in postsecondary mathematics education to address the mathematics-preparation gap.
4. Encourage partnerships about stakeholders to diversify pathways to STEM careers.

While there is much that I could say about each of these, I want to focus this column on Recommendation 3 because it directly impacts the mathematics community.

After acknowledging the central role of mathematics in preparing students for STEM careers, the report then asserts that “introductory mathematics courses often leave students with the impression that all STEM fields are dull and unimaginative” [p. vi] and later complains that “Discipline-based education on effective undergraduate mathematics teaching also appears less developed when compared with other STEM fields.” [p. 27] The report describes at some length the problem of mathematically under-prepared students and recommends a sustained effort to increase the production of high school mathematics teachers by 100,000 (presumably over ten years) and a joint effort by NSF and the Departments of Labor and Education to “support a national experiment in mathematics undergraduate education.”

This national experiment has four components:

1. Summer and other bridge programs for high school students entering college;
2. Remedial courses for students in college, including approaches that rely on computer technology;
3. College mathematics teaching and curricula developed and taught by faculty from mathematics-intensive disciplines other than mathematics, including physics, engineering, and computer science; and
4. A new pathway for producing K-12 mathematics teachers from undergraduate and graduate programs in mathematics-intensive fields other than mathematics.

I have bolded and italicized the last two.  I would like to be able to read them generously as suggesting that the mathematics community could benefit from working with and learning from the experience of other STEM fields. But the nature of these recommendations combined with the other previously mentioned statements from this report suggest that PCAST does not trust the mathematics community to get right undergraduate mathematics education either in support of other STEM fields or in the preparation of K-12 mathematics teachers. In this report, there is a clear sense of frustration that despite its central role in STEM education, the mathematics community appears to have been slow to rethink its undergraduate curricula or pedagogy on a truly national scale.

In fact, there have been and continue to be many promising, sustained, and national efforts to reform undergraduate mathematics education. The mathematical community is learning from the successes that have occurred in physics, chemistry, and biology, but the lessons learned are not always easy to apply. Because of its central role across the disciplines, the sheer number of students involved, and the complexity of the issues that reach right down into pre-kindergarten, accomplishing measurable improvement in undergraduate mathematics education on a national scale is extremely difficult. We welcome the help of the other STEM disciplines, but ultimately this is the responsibility of the mathematics community.

I will return to this report in future columns. For now, I’d like to close with one observation and a question. In the fall of 2007, 276,000 of the entering freshmen in full-time four-year undergraduate programs declared that they intended to major in a STEM field. This number had remained essentially unchanged since 2001. Thanks to the economic downturn, by this past fall the number of entering freshmen intending to major in a STEM discipline had grown to 424,000 [data from The American Freshman]. There are our extra 100,000 STEM majors and then some. They are out there if the incentives are right. Now, can we keep them?

Trends in Race/Ethnicity and Gender Representation in the Mathematical Sciences

Each year, the US Department of Education’s National Center for Education Statistics publishes its Digest of Education Statistics. Among its many annual tables, easily accessible back to 1990, is one that describes the number of Bachelor’s degrees by sex, race/ethnicity, and field of study. I’ve put together my own longitudinal table for the science, engineering, and mathematical majors. These have been the source of my tables on trends for women (see We Are Losing Women from Mathematics, September 2009) and for members of various racial or ethnic groups (see MAA Speaks Out on Capitol Hill, October 2009).

The disturbing trends that I identified then have continued. In particular, the number of women majoring in the mathematical sciences has continued to stagnant while the number of men has seen very healthy growth (see Graph 1). I will present the graphs of the number of Bachelor’s degrees in the mathematical sciences by race/ethnicity and gender and then conclude with some observations.

By racial or ethnic group, we see the same pattern reflected among Asian and Pacific Islander majors and, to a lesser extent, among Hispanic students (Graphs 2 and 3).

The situation is very discouraging for Black non-Hispanic students. The absolute decline over the past decade in the total number of Bachelor’s degrees in the mathematical sciences earned by Black non-Hispanic students is almost entirely the result of a decline in the number of Black women earning math degrees.

The following graph shows what has been happening to the number of non-resident aliens majoring in the mathematical sciences in the United States.

Observations: The most striking feature of Graph 1 is the sharp decline in the number of math majors during the 1990s, followed by an equally sharp rebound by men that is not reflected in nearly so steep a recovery for women. There are several factors that have been at play. One is that the college-age population decreased during the 1990s, bottoming out around 1997. Another is that the second half of the ‘90s was a time of economic prosperity. Unemployment rose in the early 2000s, peaking in 2003. By the fall of 2008, it was clear that we were heading into an economic disaster. As I showed in A Benefit of High Unemployment (October 2010), the number of students choosing to enter mathematically intensive majors is highly correlated with the economic situation: the harder students expect it to be to find a job when they graduate, the more likely it is that they will choose a mathematically intensive major. The year 2000 also saw the collapse of the dot.com bubble and with it a huge decline in the number of students heading into computer science. It is not unreasonable to assume that many of them switched to mathematics.

During the last decade, the growth in mathematics majors has occurred entirely in universities with graduate programs in mathematics and has decreased sharply at colleges only offering a bachelor’s degree in mathematics (see Good News from CBMS, November 2011). It is these undergraduate institutions that have traditionally had the largest representations of women and students from under-represented groups. I believe that this has contributed to both the increasing discrepancy between men and women and to the decline in the number of Black mathematics majors. Less clear is what has caused the decline in mathematics majors at undergraduate colleges.

It is noteworthy that the growth in the number of Hispanic mathematics majors has been very strong throughout the past two decades. This reflects the tremendous growth in the number of Bachelor’s degrees earned by Hispanic students: from 33,000 in 1990 to 130,000 in 2009, from 3% of all Bachelor’s degrees in 1990 up to 10% in 2009.

Graph 5 is interesting because it illustrates very clearly the post-911 effect on student enrollments from outside the United States. The first two years of sharply stricter student visa requirements, 2002 and 2003, were reflected in a sharp drop in the number of non-resident alien math majors 2007. The good news is that by 2009 we had almost fully recovered. But the most interesting feature of Graph 5 is that there has not been a widening gap between the number of non-resident men and the number of non-resident alien women choosing to go into mathematics. This may be entirely explainable by the fact that women are still catching up to men in this category (see Graph 6).

Note: I have not included data for Native American/Alaskan Native students because the numbers are so small, usually between 60 and 80 combined per year. As a result, there is very high year-to-year variation, obscuring clear trends. The NCES tables do not separate Mathematics from Statistics majors. Mathematics Education majors are not separated from other Education majors, so are not included.

First, Do No Harm

Does remediation work? Do entering college students with weak mathematical skills benefit from pre-college mathematics courses? Or are such courses merely a waste of their time and tuition dollars and the institution’s resources? Worse, might they actually impede these students’ chances of successfully completing a two-year degree? The answer, of course, depends on how such courses are taught, but the results of a 2008 study by Calcagno and Long [1] imply that—in at least one large state system—directing students to pre-college mathematics has decreased their likelihood of completing an Associate’s Degree.

One must be careful about extrapolating from a single study. The Calcagno and Long study tells us about students who entered public two-year colleges in the state of Florida during the years 1997–2000. Nevertheless, it forcefully makes the point that we cannot assume that merely offering pre-college mathematics is any guarantee that we are actually providing a service to students. In view of the fact that pre-college mathematics constitutes a substantial majority of the mathematics taught in two-year colleges, all colleges offering such courses have a responsibility to determine whether they are helping or hurting their students. The Calcagno and Long paper also offers a useful statistical approach to answering this question.

Assessing the effectiveness of remediation is difficult. The students in pre-college courses are, by definition, those with weaker mathematical backgrounds. One can expect that they are less likely to complete a two-year degree regardless of the quality of the intervention. The question is whether students who are assigned to pre-college mathematics are benefiting from the program. Would they have just as well, or perhaps even better, if they had simply gone straight into College Algebra? One approach to answering this question is to randomly assign students near the cut-off either to go straight into the college-level course or to first take the pre-college course. This has two difficulties. One is in getting a sufficient number of students so that the results are statistically significant. The other difficulty is ethical. Students who otherwise would have been allowed to proceed directly to a college-level course are held back and/or students who need help are denied it.

This is a statistical design problem that medical research encounters when studying the effectiveness of an existing treatment for which there is no alternative. Denying treatment to those who need it would be unethical, but testing the treatment on those whose need is marginal may not produce meaningful results. The statistical method that has been developed for such situations is called regression discontinuity design (RDD). It is particularly effective when there is a test to determine whether treatment is needed and when the cut-off is sharp: Those who score at or below a given level receive the treatment; those above do not.

Such is the case in the state of Florida where there is a common mathematical placement exam used by all public two-year colleges, the Florida College Entry Level Placement Test (CPT), part of the College Board’s ACCUPLACER system. There is a common cut-off score used by all colleges. On a scatterplot of test scores versus some desirable outcome, such as completion of a two-year degree, Calcagno and Long constructed a quadratic regression curve for those at or below the cut-off and a separate regression curve for those above. If the intervention is beneficial, there should be a discontinuity at the cut-off score, with the limit of the regression curve from the left higher than the limit of the curve from the right. This is illustrated in the following scatterplot that demonstrates the unsurprising insight that assigning students to remedial mathematics increases the total number of credits that they earn, including those credits that do not count toward the degree.

image 1

However, the discontinuity is reversed on the three outcomes that are meaningful: completion of a college credit bearing course in mathematics (specifically, College Algebra, MAC 1105, required for the Associate’s Degree), completion of the Associate’s Degree, and transfer to a four-year undergraduate program. The scale on the left represents the fraction of students who succeed. Each circle represents the mean value for all students with a given CPT score.

One of the problems with RDD is that the statistical power is very low. It requires a large sample size to find statistically significant results. Calcagno and Long used the records of 100,000 Florida students who enrolled in an Associate’s Degree program and took the CPT from fall 1997 through fall 2000. They tracked these students through spring 2006. They were able to adjust their analysis to account for non-compliance with the results of the placement test, which is relatively uncommon in this system, and to investigate the effect of students who took the CPT more than once.

It is worth noting that the negative effects of remediation are small: 1.4 percentage points on completion of College Algebra, 0.6 percentage points for completion of an Associate’s Degree, and 0.1 percentage points for transfer to a four-year program. In none of these cases is the difference from zero statistically significant. Nevertheless, the fact that the remediation is not demonstrably beneficial and likely is harmful should be deeply disturbing.

In an earlier column, The Problem of Persistence (January, 2010), I discussed the issue of students who arrive in college unprepared for college-level mathematics. I am encouraged by the work that is being done by organizations such as the American Mathematical Association of Two-Year College (AMATYC) and the Carnegie Foundation for the Advancement of Teaching to reconceive what mathematics these students need, to create new sequences for them such as Statway, and to assess the effectiveness of these new trajectories. The Calcagno and Long study is one more sign of the importance of this work and the need to assess the effectiveness of what we are doing.

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[1] Calcagno, J. C. and B. T. Long. 2008. The Impact of postsecondary remediation using a regression discontinuity approach: Addressing endogenous sorting and noncompliance. National Bureau of Economic Research working paper no. 14194. http://www.nber.org/papers/w14194

Faculty Trends from CBMS

Last month, in Good News from CBMS, I reported on the preliminary student enrollment numbers from the Conference Board of the Mathematical Sciences (CBMS) survey of Mathematics Departments in the United States. This month I will be looking at preliminary data on the distribution of faculty in mathematics departments by status: full-time or part-time, and whether tenured or tenure-eligible. I’ll also look at the distribution of women at the three types of institutions: research universities, comprehensive universities, and undergraduate colleges. As with last month’s numbers, these are preliminary results that are subject to adjustment, but I expect any changes to be minor and that the trends will hold up.

There are two competing pressures that affect the number of faculty positions in mathematics. As we have seen, enrollments at the level of calculus and above are up by over 30% over the past five years. On the other hand, university budgets are severely strained. The good news is that more faculty have been hired and they have been full-time faculty. In fact, the number of part-time faculty has decreased over the past five years. The bad news is that the additional faculty positions have been entirely non-tenure-eligible. The number of tenured and tenure-eligible faculty declined slightly, from 17,256 to 16,362, over the past five years.

CBMS data on full-time versus part-time employment of faculty in mathematics departments goes back to 1970. Counting the number of tenured faculty within the ranks of the full-time faculty began in 1975. In 1995, CBMS began to record the number of tenure-eligible faculty. Almost all of the drop in tenured and tenure-eligible faculty from 2005 to 2010 has been in the ranks of the tenure-eligible. The number of tenured positions dropped by only 127, but there was an 18% drop (from 4381 to 3614) in the number of tenure-eligible faculty. As the graph indicates, much of this drop can be attributed to the fact that 2005 was an anomalous year with a large number of tenure-eligible positions. The number of such positions in 1995 and 2000 was around 3300. The data for 2010 seems to reflect a correction back toward these levels.

Different types of institutions have seen very different patterns of faculty status since 1995, the first year that CBMS began tracking department responses according to the highest mathematics degree offered: PhD (which I refer to as research universities), Masters (which I categorize as comprehensive universities), or Bachelors (undergraduate colleges).

We see that it is especially at the research universities where the growth in non-tenure-eligible full-time faculty has occurred. Research universities are also losing their tenured positions. The comprehensive universities have taken the biggest hits across the board. Even though the number of part-time faculty declined here also, they constitute 30% of the faculty positions at the comprehensive universities. Perhaps the brightest picture is at the undergraduate colleges, though the increase in the number of tenured faculty was more than offset by the decline in the number of faculty in tenure-eligible positions.

Turning to the number of women, we see that their representation within mathematics faculty has continued to grow and that they are much better represented in undergraduate colleges and comprehensive universities than in research universities.

The next graph, women as a percentage of tenure-eligible faculty, shows a sharp decline in their representation at undergraduate colleges from 1995 to 2005. If we look at where women were teaching in 1995, it was predominantly at undergraduate colleges. In that year almost 2/3rds of the women in tenure-eligible positions were at undergraduate colleges, compared to 52% of all faculty in tenure-eligible positions. Just five years later, 55% of women in tenure-eligible positions were at undergraduate colleges, where it has stayed since then. The decrease in women as a percentage of the tenure-eligible faculty at undergraduate colleges appears to be a reflection of increased opportunities at research and comprehensive universities.

Good News from CBMS

Every five years, the Conference Board of the Mathematical Sciences (CBMS) conducts a survey of Mathematics Departments across the United States to determine Fall course enrollments and gather other data. The latest survey was conducted in 2010, and the data collected last year are just starting to trickle in. I have been able to get a peek at some of the preliminary data.[1] They show a very encouraging trend in course enrollments in Mathematics. From Fall 2005 until Fall 2010, the number of students enrolled in Mathematics at the level of calculus or above rose by over 30%, from 699,000 to 915,000.

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 has been a strong upward trend toward mathematics, the sciences, and engineering over the past decade. It has recently accelerated. In the past five years, the number of students intending to major in Mathematics has risen by 31%, in the Physical Sciences by 37%, in Engineering by 44%, and in the Biological Sciences by 67%. Most of this growth has occurred in just the past three years, since 2007. This is most dramatic within Engineering, which went from 102,000 freshmen intending to major in this discipline in the Fall of 2007 to 156,000 in Fall 2010. This has resulted in a huge increase in the number of students taking mathematics at the level of Calculus and above. The primary beneficiaries have been institutions with strong engineering and science programs, i.e. the large state universities.

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.

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[1] Precise numbers are subject to final revision, but the adjustments should be small and the trends are clear.

Quantitative Literacy versus 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.

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The Economist recently published an article that is directly relevant to August’s column, The Best Way to Learn. See “The Great Schools Revolution”, The Economist, Sept 14, 2011.