Mathematics has been described as an "invisible culture," shunned by many educated adults, yet exercising profound influence on all aspects of society--from engineering to economics, from strategic planning to political polls. The avoidance of mathematics extends also to campuses, where students, faculty, and administrators expend valuable educational energies arguing about multiculturalism and politically correct curricula, meanwhile ignoring less glamorous yet equally important educational policy concerning the nature of the university's mathematics curriculum.
Make no mistake about it: even as mathematics conveys the power of ideas, it also entails profound socio-political consequences. Success or failure in mathematics determines access to courses and curricula which lead to positions of influence in society. The increasing role of technology in the world of work amplifies the already strong signal sent by the scientific revolution that the language of mathematics is an essential component of literacy for our age.
At its best, education can be the equalizer of socio-economic differences. Yet more than any other subject, mathematics serves as a filter, enhancing or blocking access to professional careers in a manner that has disproportionately negative consequences for women and minorities. When students drop out of college for academic reasons, the culprit is often mathematics--not just because of one poor grade, but because failure in mathematics prevents further progress in so many other subjects.
For years, educators assumed that this was just in the nature of things--that mathematics learning was the result of genetic and cultural factors that predisposed certain people to success and others to failure. This belief endured--and still persists--despite significant evidence that it simply is not true. The records of many small colleges (especially women's colleges and historically black colleges), the success of special intervention programs at various universities, and evidence from educational research show that the traditional lecture style is effective only for students who arrive with uncommon levels of motivation and persistence; that most students learn better with more active, varied modes of instruction; and that virtually all students can succeed in mathematics provided they are placed in and supported by an appropriate community of learning.
There is no longer any excuse for excessive failure rates in college mathematics. Examples abound of ways to improve success rates for all students, even for those with poor mathematics preparation. Although effective programs may appear to cost more than ineffective ones, the benefits of success--in reduced repetition of courses, in improved retention and graduation rates, in increased opportunities for students--far outweigh the visible costs of these programs. In some cases, effective programs may actually be less expensive overall.
Deans and provosts who wish to improve mathematics education on their campuses need to recognize two realities:
About StudentsWho are your students? More fundamentally, do you know who your students are and why they are in your courses? Students who enroll in college mathematics courses arrive with amazing mixtures of aspirations and anxieties, often exaggerated, always intensely personal. Since student attitudes towards mathematics frequently have more influence on performance than do remembered skills or school-based learning, the first step to improved success is better understanding of the motivations of your students.
Do departmental priorities match the institutional mission? More pointedly, is your mathematics faculty committed to teaching the students you have? It is all too easy for faculty to covet students who fit an imagined mold of young scholars created in the faculty's image, or to view every first year student as a potential mathematics major. Instructional practice based on false assumptions yields disillusionment for both students and faculty. Effective instruction must harmonize the goals of the institution with the expectations of its students.
Do you believe that your department should educate all students? More concretely, do you believe that all students can learn mathematics? Do you offer appropriate and appealing courses that meet the needs of all students who enroll in your institution? Do you apply as much creative energy to improving the most elementary courses (those often termed "developmental" or, more derisively, "remedial") as to those that are more advanced? Most mathematics used in the world, after all, is just simple school mathematics applied in unusual contexts.
Do you have explicit goals for increasing the number of students from under-represented groups who succeed with mathematics courses? Vague intentions without explicit goals are too easily ignored. There are precious few departments of mathematics in the country whose records of success with Black, Hispanic, and other under-represented groups could not be significantly improved. Specific goals must relate to specific institutions, but surely one aspiration must be that mathematics becomes a pump rather than a filter for students who have been traditionally under-represented in the professional fields that build upon college-level mathematics.
What do your students achieve? More specifically, how do you know what your students have really accomplished? End-of-term exams generally reveal only short-lived mastery of procedural skills. Do you have any evidence concerning long-term goals or broad objectives? Do you ever ask students to solve authentic, open-ended problems, or to write, read, or speak about mathematics? Do courses provide opportunities for students to learn anything other than textbook-based template exercises? To what extent are course grades based on an examination of these broader goals of mathematics education?
Do you know what happens to students after they leave your courses? Do students who take mathematics courses go on to use their mathematics in subsequent courses? What about students who drop out or fail: do they give up on mathematics, or do they return and succeed in a subsequent course? Do those who receive good grades find that what they learned serves them well in subsequent courses? Do you have a mechanism for adjusting curricular emphases based on feedback from students who have taken your courses?
About CurriculumDo departmental objectives support institutional goals? Despite variations in rhetoric, widely shared goals for mathematics education are entirely consistent with broad goals of higher education: to develop students' capabilities for critical thinking, for creative problem solving, for analytic reasoning, and for communicating effectively about quantitative ideas. Yet the implicit objectives of many mathematics departments, as inferred from curricula, exams, and student performance, are often focussed on mastering relatively sophisticated yet intellectually limited procedural skills. Departments must express for themselves--and even more so, for their students--how their course objectives advance their institution's educational goals.
Do your courses reflect current mathematics? Since mathematics is such an old subject, it is all too easy for its curriculum to become ossified. Strong departments find that half of their courses are replaced or changed significantly approximately once a decade. As new mathematics is continually created, so mathematics courses must be continually renewed. Does the mathematics curriculum reveal to students a level of innovation and attractiveness that reflects the excitement of contemporary mathematical practice?
Are your faculty aware of the new NCTM Standards for school mathematics? More importantly, are you making plans to provide an appropriate curriculum that builds on the foundation of these Standards , following their spirit as well as their content? Colleges must be prepared for students arriving with increasingly disparate backgrounds--many from traditional authoritarian, exercise-based courses while an increasing number of others will enter college fresh from an active, project-centered approach that typifies the new school Standards. It would be ironic, indeed tragic, if intransigent college mathematics departments were to hold back reform of school mathematics by refusing to adapt to the new reality of a more diverse and powerful secondary school curriculum.
Are calculators and computers used extensively and effectively? Beginning with placement exams and continuing all the way through senior courses, calculators and computers should be used in every appropriate context. Since the mathematics used in the scientific and business world is a mathematics fully integrated with calculators and computers, the mathematics taught in college must reflect this reality. Anything less short-changes students, parents, and taxpayers.
Do you know what your majors do after graduation? How many take jobs in which they use their mathematics training? How many enter secondary or elementary teaching? What about graduate school--in mathematical sciences, in other sciences--or professional school? How well suited is your curriculum to the actual experience of your graduates?
Does your program help students see how mathematics connects to broad issues of human concern? Specifically, do your faculty and courses connect mathematics to student aspirations, to liberal education, to other disciplines? Do students who study mathematics emerge empowered to think and act mathematically in broad contexts beyond the classroom? Unless this happens, students feel cheated by lack of reward commensurate with effort required in typical mathematics courses.
About FacultyHow does the scholarship of your faculty relate to the teaching mission of your department? Does your faculty subscribe to a narrow view of research or to a broad perspective on scholarship? Does the department both expect and support professional development in its varied forms? Is there a departmental commitment to offer all majors suitable professional, scholarly, research, or internship opportunities? Traditional standards of mathematical research make direct connections to undergraduate teaching rather difficult, whereas a "reconsidered" view of scholarship eases constructive engagement in which faculty can thrive professionally and students can become junior colleagues.
What steps has your department taken to be sure that your faculty are well informed about curriculum studies and research on how students learn? Part of the professional responsibility of faculty is to know the scholarship that undergirds college teaching. Everyone has opinions about curriculum and pedagogy, but professionals need to support their opinions with evidence. Since graduate education in mathematics rarely provides any introduction to this arena of scholarship, departments must accept it as part of their responsibilities. Regular faculty seminars on issues of curriculum, teaching, and educational research help focus faculty attention on these important issues at the same time as they help create an environment for learning that is crucial to student success.
What are your priorities for teaching assignments? In particular, do you assign your best teachers to beginning courses? Are courses for non-majors given the same priority as those for majors? Do your faculty prefer students who learn without being taught or those who challenge teachers to teach effectively? How do faculty rewards reflect the teaching challenges they undertake? Is the quality of faculty teaching measured by the good students they attract to their courses or by the improvement of students in their courses?
Are your faculty fulfilling their responsibility for the preparation and continuing professional education of teachers? The new Standards for school mathematics include clear expectations for both content and pedagogical style in the mathematical preparation of school teachers at all grade levels. How many members of your mathematics department are familiar with these expectations? To what extent do your courses conform to these standards? What steps are you taking to ensure that all mathematics courses taken by prospective teachers meet appropriate professional expectations?
How are faculty resources allocated between courses that serve the major and those that serve general education? Typically 80% of the students in a mathematics department are enrolled either in service courses or in general education courses. Often these 80% of students command only 20% of faculty time and energy. Yet it is these students who will go on to be future policy leaders of society--members of boards of education and city councils, editors of local newspapers and leaders of Chambers of Commerce.
About CostsHave you calculated the true cost of the status quo? Courses staffed on the cheap (the "cash cow" approach to funding mathematics departments) result in students repeating courses, or failing related science courses, or dropping out of college altogether. Students who succeed in their first college mathematics course are far more likely to succeed in college than those who who do not. Cheap courses are not necessarily as cost-effective as they appear.
Are you aware that mathematics departments exercise disproportionate influence over an institution's graduation rate? A small increase in the percentage of students who complete mathematics courses with a well-earned sense of accomplishment can translate into increased graduation rates in many disciplines that depend on mathematics. Conversely, any decline in the success rate in elementary mathematics courses cascades into even larger drop-out rates by students who find themselves lacking prerequisites for key courses in their majors.
What resources are required to achieve your objectives? This is the most important question of all. Mathematics cannot be taught successfully without resources adequate to the task. Many mathematics departments suffer not only from insufficient resources, but also from inefficient distribution of existing resources. In return for a prudent self-study by a department of mathematics, the institution must be prepared to focus resources on promising new approaches in which the cost of success compares favorably with the cost of failure.
The spotlight of national attention that has been aimed at mathematics and science education has revealed not only weaknesses in the present system, but also outstanding examples of success. Mathematics need not remain a barrier to higher education. Investment in programs that make possible increased success in mathematics provides great leverage for any institution that wishes to improve the overall education of its students.
|Copyright © 1992.||Contact: Lynn A. Steen||URL: http://www.stolaf.edu/people/steen/Papers/aahe.html|