In addition to the skills developed in programs for K-8 teachers, departments should ensure that mathematical sciences majors preparing to teach secondary mathematics
The Mathematical Association of America (MAA) has a multi-dimensional
Mathematicians to Educate Teachers (PMET) to help meet the need for
better preparation of the nation's mathematics teachers by mathematics
faculty. The PMET program has four major components:
The text, The Mathematics Pre-Service Teachers Need to Know, by James Milgram, Stanford University, is primarily intended for mathematicians and mathematics educators to help develop courses for pre-service teachers. It is currently available on the Internet and includes chapters for both elementary and secondary pre-service teacher education. The final chapter contains reports of experiences teaching mathematics to pre-service and in-service teachers by Sol Friedberg, H. Wu, Kristin Umland, Kathi King, Sybilla Beckman, Paul Wenston, David Benson, and Kenneth Gross. Two of the appendices discuss the way algebra and geometry are taught to sixth graders in one of the Russian school mathematics programs.
In his article “On the education
of mathematics teachers,” Hung-Hsi Wu, University of California Berkeley,
discusses the importance of content knowledge and provides a reexamination of
upper-division mathematics courses. He proposes offering an alternative
curriculum, or track, for prospective secondary school teachers, as well as
for others not intending to continue to graduate study in mathematics, and he
argues that such a track would not represent a watering down of the major. Wu
Gregory Hill of Portland University has developed a course for pre-service secondary teachers called Teaching Mathematics with Technology. In the syllabus he states: “Technology is pervasive in our schools, but is it being used well? What fundamental educational goal is being advanced by the use of technology? How can we use technology to advance our goals in teaching mathematics? To answer these important questions, we need to have an idea of what our goals are as mathematics teachers. It is important for you as a future teacher to realize that mathematics is a way of thinking rather than a collection of rules. It is the conceptual grasp that will help both you and your students learn and apply mathematics.” Four weeks of the course are devoted to algebra and 8 weeks to geometry.
Learning to Teach
Secondary Mathematics (LTSM) is an NSF-funded project based at the School
of Education, University of Colorado, Boulder and the Metropolitan State
College of Denver. The project is designed to produce case studies of
pre-service teachers’ development over a 5-year period. The LTSM project is
addressing questions about:
Information about using technology for visualization and exploration is contained in Part 1, Section 5.
Resources for teachers of geometry, especially courses aimed at least in part at prospective teachers, may be found at the following websites: St. Olaf College, the teachers’ page of “geometry: the online learning center,” the Geometry in Utopia section of Joe Malkevitch’s webpage at York College of the City University of New York, the homepage for Carl Lee’s course Geometry for Prospective Middle School Teachers at the University of Kentucky, and a set of notes, “Visualizing Mathematics,” written by Lee.
Colm Mulcahy at Spelman College has taught and offered mini and short courses on a hands-on, discovery-based approach to geometry based on the text Experiencing Geometry in Euclidean, Spherical and Hyperbolic Space by David Henderson, Cornell University. Mulcahy also uses as resources two books by Paulus Gerdes, Geometry from Africa: Mathematical and Educational Explorations and Women, Art and Geometry in Southern Africa.
J. Chris Fisher, University of Regina, reported that in his Euclidean Geometry course, which is primarily taken by prospective high school teachers, he emphasizes helping students learn to formulate and critique arguments, while avoiding unsurprising theorems, formalism and axiomatics. He supplements the text with additional exercises and projects using the software package Cinderella.
Dynamic geometry software packages, such as Geometer’s Sketchpad, Cabri Geometry, Cinderella, and GeoGebra (which is free), are now widely used in secondary level geometry courses. Some resources for using them effectively are available on the Math Forum Geometry Software website.
David Joyce, Clark University, developed a widely used website for Eulid’s Elements. It contains the complete text together with additional explanations, critiques, and accompanying Java applets. Jim Morey, University of British Columbia, wrote an applet that leads students through the logic of Euclid’s proof for the Pythagorean theorem. An annotated animation of the Chinese proof for the theorem, which is the oldest known, is from Davis Associates, Inc. A French website Le Kangourou des Mathématiques has animations illustrating the proof that ratios of corresponding sides of similar triangles are equal (which it refers to as Thales theorem), Euclid’s proof for the Pythagorean theorem, and Archimedes derivation of the formula for the area of a circle. The text and verbal explanation are in French, but all three visualizations can be viewed one after another by clicking on the website. The websites given in part I section 5 contain many other applets that can be used for teaching geometry, as well as other subjects, in courses for prospective teachers.
Jaime Carvalho e Silva, Arizona State University, maintains a reference site for information about courses on the history of mathematics, and Fredrick Rickey, United States Military Academy, maintains another such site. Some additional syllabi are from Fernando Q. Gouvêa, Colby College, Robert Rogers, State University of New York – Fredonia, and Sarah Greenwald and Gregory Rhodes, Appalachian State University.
Fred Rickey, U. S. Military Academy, has a History of Mathematics Page that includes information about his and other’s history of mathematics courses, articles about teaching the history of mathematics, and a page of additional Links to history of mathematics on the Web.
David Pengelley, New Mexico State University, wrote up advice for instructors preparing to teach a graduate course in the history of mathematics for teachers. Much of his advice could be used when preparing for an upper-level undergraduate course.
Two resources specifically developed for the use of history of mathematics in teacher education are Using History to Teach Mathematics: An International Perspective, edited by Victor Katz, University of the District of Columbia, and the CD modules Historical Modules for the Teaching and Learning of Mathematics, edited by Victor Katz and Karen Dee Michalowicz, the Langly School. (The modules are also discussed in Part 2, Section C.2.)
Mathematician Richard Hill and mathematics educator Sharon Senk, Michigan State University, wrote a article for the 2004 MER newsletter about their experience team-teaching a capstone course for prospective secondary teachers. In the report they discuss the content studied, how they shared instruction, the methods of assessment they used, and what they learned from their students and from each other.
A textbook that was developed specifically for a capstone course for pre-service secondary mathematics teachers is High School Mathematics: An Advanced Perspective by Z. Usiskin, et al. (2002). Following the guidelines of the MET document, the book is designed to help students understand the connections between the mathematics they will be teaching and the mathematics they have been learning in college. Topics include algebra, analysis with connections to geometry, geometry with connections to algebra and analysis, trigonometry, axiomatics and Euclidean geometry. Anthony Phillips, SUNY Stony Brook, has a syllabus for a course taught from the book, and the text is discussed in the article by Richard Hill and Sharon Senk mentioned in the previous listing.
Curtis Bennett, Loyola Marymount University in Los Angeles, developed a portfolio for the course Advanced Mathematics for Secondary Teachers, which he taught at Michigan State University in 2000. The portfolio includes the 114-page text he wrote for the course, a discussion of the desired course outcomes and assessment, and many other materials.
At The Preparation of Secondary Mathematics Teachers Conference held in 2002 at Columbus State University, Judy O’Neal, North Georgia College and State University, gave a presentation describing her experience teaching a capstone course for pre-service secondary mathematics teachers. Her course included 3 weeks on algebra and number theory, 4 weeks on geometry and trigonometry, 3 weeks on functions and analysis, 3 weeks on data analysis, statistics, and probability, and 2 weeks on discrete mathematics and computer science. The course was taught collaboratively by a mathematician and a mathematics educator.
The capstone course at Utah State University is intended to build on the competencies students have obtained in previous courses in the foundations of analysis, introduction to algebraic structures, and history of mathematics and introduction to number theory. The purpose is to convey a depth of understanding of the mathematical and statistical contents of middle and secondary school curricula so teachers will be able to interact with middle and secondary school students in response to students’ comments and questions, explain why various mathematical relationships exist or algorithms work, connect mathematics to real-world problem-solving, and lead their students to appreciate interconnections among mathematical topics.
The capstone course for prospective teachers at Montana State University Bozeman is Mathematical Modeling for Teachers, which uses pre-college mathematics to explore a variety of application areas. Topics include an overview of the modeling process, review of relevant technology, strategies to initiate modeling in the secondary classroom, modeling in the secondary curricula, and the classroom assessment of modeling activities.
The Mathematical Association of America (MAA) has a National Science
Foundation-funded, multi-dimensional program Preparing Mathematicians to Educate
Teachers (PMET) to help meet the need for better preparation of the
nation's mathematics teachers by mathematics faculty. The PMET program
has four major components:
As discussed under Recommendation B.4, the National Science Foundation supported several collaborative projects aimed at improving the preparation of future teachers.
The Mathematics with Emphasis in Secondary Teaching major at the University of Northern Colorado includes three courses that are based off campus in a Partnership High School. The program requires at least two hours per week working in the Partnership School during the three semesters students are enrolled in the courses.
In addition to the general recommendations for majors, programs for students preparing to enter the nonacademic workforce should include
See the entries under Reasoning with Data: Probability and Statistics in Part 2, Section C.1.
Leigh Lunsford, Athens State University, who has worked both in academia and industry, reflected on her experiences in “Back to Academia,” which appeared in Concerns of Young Mathematicians. She proposed the following list of skills/competencies needed by an undergraduate mathematics major planning to work in industry and elaborates on the list in the article. 1) Competency with computers: use of tools such as Maple, Excel and Power Point and two semesters of a programming language. 2) Basic competency with and the ability to use a broad range of mathematics, including exposure to both discrete and continuous mathematics, statistics, mathematical modeling, and simulation and the ability to understand mathematics well enough to apply it to solve problems. 3) The ability to learn new technical concepts and skills and especially to be able to figure out new things on one’s own. 4) The ability to work and communicate effectively with others, a skill many technical workers lack. Lunsford has found that since her return to academia she is “harder” on her students in the sense that she does not tell her students everything and expects them to discover and figure some things out for themselves. On the other hand, she believes in treating them with the respect owed to her own future coworkers. Ronald Guenther, Oregon State University, added “More Thoughts on Returning to Academia” about his own similar experiences in the next issue of the journal.
In “Should You Prepare Differently for a Non-academic Career?” Fan R.K. Chung from Bellcore suggested some basic guidelines to prepare for a mathematics career outside academia. Although his intended audience consisted of graduate students, the suggestions he gave in 1991 now apply to undergraduates as well: Development of good communication skills, maintaining balance between depth and breadth in one’s mathematical program, knowledge about the interconnections of mathematics and the use of technology. Chung also recommended acquaintance with technology-related areas of mathematics such as combinatorics, graph theory and number theory, and possibly geometry with a computational flavor (or with imagination), probability (or combinatorial probabilistic methods), numerical methods or even interdisciplinary courses in algorithms and data structures.
The mathematics department at Bryn Mawr College has gathered information from alumnae about their careers. The information appears on the department’s website and is also displayed on the walls in the department. Like many colleges and universities, Bryn Mawr informs students about the possible areas of work in industry through links to the career sites such as Mathematical Sciences Career Information, which is jointly sponsored by the AMS, the MAA, and SIAM. (Additional career information sites are given in Part 2, Section C.5.)
The American Mathematical Society career information website includes a webpage of “Early Career Profiles,” which gives information about job profiles of recent bachelors-level alumni from various colleges and universities.
Each fall a panel of Hiram College alumni discusses career options during the Hiram Mathematics Career Night and Ice Cream Social. All students at the college are invited to attend. The Mathematics Department also takes students on field trips to see mathematics in the workplace. The General Motors Technical Center in Warren, Michigan; the Biostatistics Department at the Cleveland Clinic Foundation, and Penske Logistics have been the sites of recent visits. A new program at Hiram is pairing current students with alumni who act as career mentors. (There are additional examples in Part 2, Section C.6 of inviting alumni back to speak about their career experiences.)
Stetson University has a Career Information website with listings of available jobs, places for students and alumni to post resumes, and information about job fairs etc. In addition the website has links to sites about careers in mathematics, opportunities in actuarial science, and Stetson University’s career services.
A few additional examples of the many schools that provide students with web-based information about entering the nonacademic workforce are University of Denver (click on “Prospective Students”), Middlebury College, and North Carolina State University.
Worcester Polytechnic Institute’s Center for Industrial Mathematics and Statistics (CIMS) has a summer research experience program designed to answer students’ questions about the role of a mathematician in business and industry, what it is like to work with technical experts on a problem that requires significant mathematics but also must satisfy real-world constraints, and what kind of mathematical and statistical tools are used to solve problems in business and industry.
The University of California, Los Angeles , hosts a program for exploration of applications of mathematics to industry through the Institute for Pure and Applied Mathematics (IPAM) (click on “Research in Industrial Projects for Students”).
The United States Department of Energy sponsors Science Undergraduate Laboratory Internships, which places students in paid internships in Science and Engineering at several Department of Energy facilities.
Additional programs are listed at the American Mathematical Society website Research Experience for Undergraduates.
At the University of Arizona, a Professional Master's Degree in Mathematical Sciences (PMMS) was initiated to prepare students for an occupational environment that requires interdisciplinary expertise, flexibility, and the ability to work collaboratively. The core mathematics requirement includes graduate courses in algebra, analysis, numerical analysis, methods of applied mathematics, probability theory, stochastic processes, and theoretical statistics. A breadth requirement adds three or four more thematically related advanced mathematics courses; a cross-disciplinary requirement typically consists of two courses from a scientific discipline outside mathematics; and the business requirement consists of one or two courses that were designed for the program. Students in the program take a research tutorial group, participate in a weekly industrial colloquium series, and are expected to have a high level of computational proficiency, taking courses to achieve this if necessary. Students without prior experience in industry take a summer internship in industry, a national laboratory, or some other setting outside the university during the first or second summer.
The Mathematics Association of America has established a Special Interest Group for Business, Industry and Government. Its purposes are to facilitate discussion among mathematicians at all levels - from students to established professionals - regarding issues of concern to mathematicians working in business, industry and government, encourage a greater role within the MAA for mathematicians working in business, industry, and government, enhance the resources and services available to business, industry, and government mathematicians, and provide a forum for input into curriculum development and reform from the perspective of mathematicians in business, industry, and government..
The Society for Industrial and Applied Mathematics was formed to advance the application of mathematics and computational science to engineering, industry, science, and society; promote research that will lead to effective new mathematical and computational methods and techniques for science, engineering, industry, and society; and provide media for the exchange of information and ideas among mathematicians, engineers, and scientists. In addition to online services offered at the site above, SIAM publishes eleven peer-reviewed research journals, including the all-electronic multimedia SIAM Journal on Applied Dynamical Systems; SIAM Review, a quarterly journal of peer-reviewed expository, survey, and education-oriented papers; Theory of Probability and Its Applications, a translation of a Russian journal; SIAM News, a news journal reporting on issues and developments affecting the applied and computational mathematics community; and approximately 25 books per year.
The American Mathematical Society (AMS), Mathematical Association of America (MAA) and the Society for Industrial and Applied Mathematics have joined in the Project for Nonacademic Employment to offer information on careers in the nonacademic workforce.
Mathematical sciences departments should ensure that
Students intending to pursue doctoral work in the mathematical sciences can supplement their course work through summer internships and research programs. The American Mathematical Society posts an extensive list of Research Experience for Undergraduates (REU) program information, with links to each program’s information web site. The following is a sampling of locations and topics of programs for 2007: California Polytechnic State University (Applied Mathematics), Carleton College and St. Olaf College (Summer Mathematics Program for Women Undergraduates), College of William & Mary (Matrix Analysis and its Applications), Iowa State University (Mathematical Biology, Discrete Mathematics, and Dynamical Systems), National Security Agency (The Director’s Summer Program), Oregon State University (Analysis of Algorithms, Geometry, Population Dynamics, and Topology ), Rose-Hulman Institute of Technology (Topics in Inverse Problems and Cwatsets), Texas A & M University (Matrix Analysis, Wavelets, Modeling Complex Ecocsystems, and Computational Algebraic Geometry), University of Illinois at Urbana-Champaign (Stochastic Modeling in Actuarial Science and Financial Mathematics; Evolutionary Games and Social Networks), University of Utah (Inverse Problems and Applications) and University of Wisconsin-Madison (Biostatistics, Number Theory, and Dynamical Systems).
University of Michigan offers its undergraduate students several opportunities for undergraduate research. Faculty engage with students in research either through the University of Michigan’s Undergraduate Research Opportunity Program, which creates research partnerships during the semesters for first or second year students, or through an REU fellowship in the summer.
The Mathematics Advanced Study Semesters (MASS) program at Pennsylvania State University, which began in 1996, is designed to provide a semester-long mathematical environment to prepare a group of talented undergraduate students recruited from throughout the United States for the research requirements and rigors of graduate school. Students take a weekly interdisciplinary seminar, attend weekly colloquium-type lectures by visiting and resident mathematicians, and take three core courses consisting of topics in algebra/number theory, analysis, and geometry/topology respectively, which are offered exclusively to MASS participants. Each core course involves a research project in the corresponding area of mathematics, often involving creative use of computers.
Initiated by Paul Erdos, László Lovász, and Vera T. Sos, the program Budapest Semesters in Mathematics provides an opportunity for North American undergraduates to enrich their mathematical experience. Through this program, junior or senior mathematics and computer science majors may spend one or two semesters in Budapest and study under members of Eötvös University and the Mathematical Institute of the Hungarian Academy of Sciences. Most instructors have had teaching experience in North America and are familiar with the cultural differences. All courses are taught in English, classes are small, and credits are transferable to North American colleges and universities. The website has a long list of reports from alumni on their activities after finishing the program.
One of the main goals of the Summer Undergraduate Mathematical Science Research Institute (SUMSRI) at the Miami University of Ohio is to address the shortage of minority mathematical scientists by encouraging minority students and women to become mathematical research scientists. Because of the shortage of minorities and women mathematical scientists, the program is especially interested in, but not limited to, African Americans and other underrepresented minorities and women. The program is designed to provide students with a research environment and improve their research abilities, improve their ability to work in groups and give them a long-term support group, provide them with professional role models, improve their technical writing skills, give them an opportunity to write a technical research paper and present a talk at a mathematics conference, inform them of available financial aid and opportunities for graduate school, make them aware of career opportunities in the mathematical sciences, and prepare them for the GRE.
Each year since 1999 the University of Nebraska has held a Nebraska Conference for Undergraduate Women in Mathematics. It is designed to give women mathematics majors considering graduate school a chance to meet other women who share their interest in the mathematical sciences, to learn about graduate school life from the perspective of current women graduate students representing math departments from across the country, and to hear talks by outstanding undergraduate women on their own research. Substantial financial support is provided for participants.
The University of California Leadership Excellence through Advanced Degrees (UC LEADS) program is designed to identify upper-division undergraduate students with the potential to succeed in science, mathematics and engineering, but who have experienced situations or conditions that have adversely impacted their advancement in their field of study. Once chosen as UC LEADS Scholars, the students embark upon a two-year program of scientific research and graduate school preparation guided by individual Faculty Mentors, who assist the students in designing a plan of research and enrichment activities fitted to their individual interests and academic goals. The "Action Plan" includes: academic year research; paid summer research experience; participation in the University-wide UC LEADS Symposium; attendance at professional or scientific society meetings; travel to another UC campus; and academic enrichment activities, including preparation for the Graduate Record Examination (GRE).
Two programs that support mentoring of underrepresented groups in mathematics are the Society for the Advancement of Chicanos and Native Americans in Science (SACNAS) and the Strengthening Underrepresented Minority Mathematics Achievement (SUMMA) Program of the Mathematical Association of America. The mission of SACNAS is to encourage Chicano/Latino and Native American students to pursue graduate education and obtain the advanced degrees necessary for research careers and science teaching professions at all levels. SUMMA was founded in 1990 to increase the representation of minorities in the fields of mathematics, science and engineering and improve the mathematics education of minorities.
The webpage Graduate School Information from Gustavus Adolphus College recommends (1) that students start thinking about graduate school during their junior year, and (2) that they use their departmental advisor is their best source of information and advice about graduate study. Questions addressed on the webpage include, “Should I go to graduate school?”; “How do I decide where to go?”, and “What do I need to do to apply?” The page also has information about alumni who went to graduate school and additional resources on the Internet such as the extensive collection of information in Graduate Student Resources on the Web! by Dan Horn, formerly at the University of Michigan.
The University of Washington mathematics department has a webpage with frequently asked questions about admission to its graduate program. The first question is “What math courses should I take to prepare for admission to the UW Math Graduate Program?” A webpage, Introduction to Advanced Degree Programs in Mathematics, at Virginia Tech University’s mathematics department contains advice for undergraduates about preparing for the graduate program at that institution. Colby College’s Fernando Gouvêa wrote “I want my P H D!” to help students at his college make plans for graduate study.
Oberlin College publishes a Handbook for Mathematics Majors to assist students in learning about the various areas of study and options, as well as to explore programs of study. The handbook includes the options and requirements for graduate study in mathematics.
Purdue University maintains a webpage Research and Prepare for Graduate Study with links to Beginning Your Research, Researching Your Options, Preparing for Graduate School, Testing Info, and Funding Your Graduate Study. A similar page, Preparing for Graduate School, is on the California Polytechnic State University career services website.
Students planning for graduate study are assisted by clear and complete information about the programs they are considering, such as may be contained in a graduate handbook prepared by a department. Making these handbooks available on the Internet is helpful to students considering a variety of different programs. Examples of graduate handbooks are those from the mathematics departments at the University of Kentucky, the University of Illinois at Urbana-Champaign, Kent State University, and Oklahoma State University. The University of Kentucky has an additional website for prospective graduate students.
The American Mathematical Society website Undergraduate Mathematics Majors contains information about Graduate School, Summer Programs, REUs, and Special Semesters, among other items.
Additional information about advising is contained in Part 2, Section C.6.
See Part 2, Section C for links and information on exemplary programs in mathematical sciences and for examples of programs at schools with a large number of majors in mathematics.