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Part I: Recommendations for departments, programs and
all courses in the mathematical sciences
General Resources
for the CUPM Curriculum Guide
David Bressoud, chair of CUPM,
has written a series of “Launchings”
describing aspects of implementation of the CUPM Curriculum Guide. Topics
through August 2007 are Introduction, Who are we teaching?, Teaching Students
to Think, Only Connect!, Math & Bio 2010, Computational Science in the
Mathematics, On Sustaining Curricular
Innovation and Renewal, Targeting the math-averse, The Challenge of College
Algebra, Avoiding Dead-End Courses, How to find more majors, Teaching for
Transference, Keeping the Gates Open, Preparing K-8 teachers, Transition to
Proof, Statistics for the Math Major, Writing to Learn Mathematics, The Role
of Technology, Geometry in the Mathematics Major, Learning to Think as a
Mathematician, Expanding the Boundaries of the Mathematics Curriculum,
Attracting and Retaining Majors, Preparing Secondary Teachers, Preparing Our
Majors, What has happened to Modern Algebra and Real Analysis?, Return to
College Algebra, The Crisis of Calculus, Holding on to the Best and
Brightest, Reform Fatigue, The Dangers of Dual Enrollment, and What You Test
is What They Learn.
Many of the articles from PRIMUS: Problems,
Resources, and Issues in Mathematics Undergraduate Studies can be found
on the Find Articles
website.
1: Understand the student population and evaluate
courses and programs
Mathematical sciences departments should:
- Understand the
strengths, weaknesses, career plans, fields of study, and aspirations of
the students enrolled in their courses;
- Determine the
extent to which the goals of courses and programs offered are aligned
with the needs of students and the extent to which these goals are
achieved;
- Continually
strengthen courses and programs to better align with student needs, and
assess the effectiveness of such efforts.
Assessing
Programs, Courses, and Blocks of Courses
Supporting Assessment of
Undergraduate Mathematics (SAUM) is an MAA sponsored project to support
mathematics departments in strengthening their courses and programs based on
assessment information. The project supports faculty members and departments
across a variety of institutions in efforts to assess student learning in
individual courses, coherent blocks of courses, and
entire degree programs, using a range of assessment tools. (Madison,
2001) Blocks of courses targeted by the project include the major in
mathematics, courses for future teachers, college placement programs, and
general education courses, including those aimed at quantitative literacy.
Recognizing that much mathematics is learned outside mathematics courses,
this last block addresses the mathematical and quantitative literacy achieved
in entire degree programs. Institutions that use assessment for program
improvement, including research on learning, are of special interest to the
project.
A
great deal of information to support assessment efforts can be found in the project’s initial
volume, Assessment
Practices in Undergraduate Mathematics, edited by Bonnie Gold, Sandra
Keith and William Marion, which is available in its entirety on the website.
The volume includes a series of articles that address assessment and
evaluation from many perspectives, and it contains over seventy case studies
of assessment at institutions across the U.S.
Since publication of this
volume, the SAUM project has supported workshops to assist teams of faculty
in developing and implementing assessment plans. Reports on the projects are
posted on the SAUM website. One study, Undergraduate
Mathematics Program Assessment – A Case Study, from American University
examined the complete departmental mathematics program. Faculty found
that their learning goals were difficult to assess and were not leading to
informed program change. In response, they sent a team to the SAUM
PREP workshop in 2003. This led to productive conversations within the
department, revised learning goals that concentrated on results rather than
process, and multiple means of assessment.
Another
study, “Assessing
Allegheny College’s introductory calculus and precalculus
courses ” focused on a block of courses. This study assessed the
effectiveness of the introductory calculus and precalculus
courses using analyses of grade data, conversations with client departments,
and information about such courses at similar institutions. The initial assessment
led to substantial revisions in the department’s course offerings.
The study “Assessing
Written and Oral Communication of Senior Projects” from Saint Mary’s University
of Minnesota
evaluated the success of a single course. Through this assessment process,
the mathematics department learned that while its majors perform well in oral
presentations and write well on general topics, their communication about
technical aspects of mathematics could be improved. The report
describes the changes made to the Senior Seminar course in response to the
assessment findings, and changes that were incorporated into the mathematics
major program as a whole.
Faculty in the Oberlin
College
mathematics department developed a set of objectives for the mathematics
major that addresses both content and attitudes. To assess the success of
their majors in accomplishing these objectives, and to inform the effort to
improve their program, the department maintains a file of syllabi and major
assignments for each course, conducts an annual survey of post-graduate
Activity, collects a statistical profile of the mathematics majors in each
class, and administers sequenced surveys to mathematics majors when they
declare a major, when they graduate, and 5 years after graduation. For
further information, contact Michael
Henle.
St. Olaf's mathematics department conducted a faculty retreat as part of a
department self-study. To prepare for the retreat, faculty completed a
questionnaire covering the department's mission, its curriculum and programs,
staffing, students and assessment, faculty, communication, "fun
stuff," and technology, resources, and facilities. Questions included:
What should the goals of the department be in the next 5 years? How well does
the mathematics curriculum serve (a) math majors; (b)
students completing general education requirements; (c) students majoring in
other areas? How can we best gauge how well we serve our students?
What kind of support do you need to engage in your own professional
activities? Are you receiving it? Which department activities do
you value most? Results of the questionnaire were included with other data in
a summary document used by outside reviewers and by the department in
connection with its self-study. Outcomes of the self-study
continue to develop, but they include new emphasis on undergraduate research
and interdisciplinary collaboration. Significant structural changes to
the mathematics major are also under consideration. These changes would
encourage students to view mathematics, its applications, and its connections
with allied areas more broadly than before. For further information, contact Paul Zorn.
External
Support for Assessing Undergraduate Mathematics
The
National Science Foundation has supported a variety of professional
development opportunities that provide hands-on experience in assessment for
both faculty and graduate students, including the series of workshops
organized in connection with the SAUM project. Although these workshops
are no longer available to new faculty, there is an online guide
that can be freely used and adapted. Another NSF-funded project is
investigating the long-term impact that the use of technology in introductory
college mathematics courses has on students in STEM (Science, Technology, Engineering,
and Mathematics) disciplines. Individuals at six institutions collect data
locally while working as a team with experts who provide training and
support. For more information, contact project directors Susan Ganter
and Jack Bookman.
As
a result of a recent reorganization, the assessment of student achievement,
including research on assessment and the development of assessment tools and
practices, has been designated as one track of the National Science
Foundation’s Division of Undergraduate Education (DUE) Course,
Curriculum, and Laboratory Improvement (CCLI) program.
The
Colorado School of Mines has produced a website, Assessment Resource
Page, for departments developing their departmental assessment plans.
Links are provided to departmental assessment plans that are publicly
available on the worldwide web.
Assessment
Tools for the Classroom
Part II of SAUM’s publication, Assessment Practices in
Undergraduate Mathematics, entitled Assessment in the
Individual Classroom, provides examples of specific classroom
assessment practices that can be useful in attempting to understand
students’ thinking and determining what they understand. As David Bressoud writes in an article in this
section, “No matter how beautifully prepared our classroom presentation may
be, what the student hears is not always what we think we have
said.” A widely used technique based on Angelo and Cross’s Classroom
Assessment Techniques (Angelo & Cross, 1993), is the One-Minute
Paper, in which students are given the last few minutes of class to write the
answer(s) to one or two questions, such as “What
was the most important point in the lecture?” or “What is the slope of the
graph of a function at a point?” or “How comfortable do you feel asking
questions?” or “How clear was today's lecture for you?”
Additional classroom assessment techniques are organized into the categories
Testing and Grading, Classroom Assessment Techniques, Reviewing Before
Examinations, What Do Students Really Understand?, Projects and Writing to
Learn Mathematics, Cooperative Groups and Problem-Centered
Methods, Special-Needs Students, and Assessing the Course as a Whole.
The monograph Keeping Score by Ann
Shannon discusses a variety of issues involved in designing assessment tasks,
especially those that aim to evaluate a broad range of mathematical skills
and abilities. An executive
summary is available from the publisher, The National Academy Press. The
article “Mathematics performance assessment: A new game for students” by Ann
Shannon and Judith S. Zawojewski (Mathematics Teacher, 88(9), 752–757)
considers how to teach students to understand and benefit from new forms of
assessment that may initially seem strange to them.
Placement
Exams
Colleges and universities
frequently use placement exams to gather information about entering students’
mathematical abilities. Assessment Practices in
Undergraduate Mathematics edited by B. Gold et al. contains two
articles on placement exams. One describes the placement procedures
at St. Olaf College, and the other describes the methods at the University
of Arizona.
At St. Olaf College there are three levels of
placement exam (basic, regular and advanced) to respond to the varied
mathematical backgrounds of incoming students. Each exam includes subjective
questions about students’ mathematical motivation, background, calculator
experience, and plans for college mathematics study. At the University
of Arizona
there are two levels of placement exam (intermediate algebra and college
algebra/ trigonometry). Both schools report that a significant commitment is
needed from the department and its faculty to complete the placement testing
and assign students to appropriate courses. But they also report that these
efforts bear dividends, as students enrolled in appropriate courses tend to
be more successful. At the University
of Arizona
placement exam data have also been used to analyze the mathematics program,
inform future decisions on course offerings, and improve testing procedures.
Norma G. Rueda & Carole Sokolowski,
Merrimack College, wrote Mathematics
Placement Test: Helping Students Succeed, which describes their study
comparing the performance of students who took the course recommended by
their mathematics placement exam score and students who did not take the
recommended course. They found that students who followed the recommendation
did much better than those who took a higher-level course or did not take the
placement exam. Because of the careful statistical nature of the study, its
results have been useful in convincing students to follow placement exam
guidelines.
Assessing student background is especially important in “open-door”
institutions, where any high school graduate can be admitted. For
example, the CBMS2000 survey
(p. 141) found that in two-year colleges, “diagnostic or placement testing
[was] … almost universal in availability.”
The Transition Mathematics
Project is a collaborative project of K-12 schools, community and
technical colleges and baccalaureate institutions to assist with the
transition from high school to college/university mathematics in the State of
Washington.
Its Resource Center
contains information about placement tests and placement test issues, as well
as much other information.
Alvin Baranchik and Barry Cherkas
conducted a study of placement exams taken by more than 1000 students at an
urban four-year college and published the results in “Differential Patterns
of Guessing and Omitting in Mathematics Placement Testing” (in Dubinsky et
al. (Eds.), Research in Collegiate Mathematics Education, II,
1996, AMS, 177-193). They found that the tendency to guess, omit answers, or
not finish the exam varied by gender, ethnicity, first language, and
birthplace, in ways that could not be fully explained by prior mathematical
skill. The authors concluded that scoring by number of correct answers
reduces the representation in gateway courses of certain cultural and gender
groups for reasons unrelated to mathematical skill. They recommend that
colleges either employ “formula scoring,” which assigns a small penalty for
guessing or a small bonus for omitting questions, or that they provide
the following instructions at the beginning of the test: “Because of the way
this test is scored and interpreted, to be fair to yourself you must
answer all questions, even if you must guess. You will not be penalized
for guessing or incorrect answers.”
In January 2005 Derek Bruff, Harvard University,
posted a list
of resource articles about placement exams on the discussion list for the
Special Interest Group of
the MAA on Research in Undergraduate Mathematics Education (SIGMAA-RUME).
Mathematical
Autobiography
A mathematical
autobiography is a written assignment in which the writer relates and
reflects on memories of his or her experiences with mathematics. Reading
students’ mathematical autobiographies can help instructors understand that
their students’ mathematical experiences may be very different from their
own. An instructor’s awareness and acknowledgment of these differences may
facilitate classroom communication, especially in lower-division
courses. According to M. A. Conway, the assignment is most productive
when it prompts a purposeful, extended, memory-search and is rewarded (Conway,
1990). For example, it might receive a grade or extra credit or be used to
replace a test score. A sample of such an
assignment is from Shandy Hauk, University of Northern
Colorado
Advising
Interacting with
students through the advising process can help faculty understand students
and their goals and improve the effectiveness of a department’s program. The University
of Southern Mississippi
has developed a comprehensive
advising program from providing mentors for freshmen to conducting a
survey of graduating students.
Mount Holyoke College provides a webpage, Beginning the
Study of Mathematics and Statistics at Mount Holyoke: A "User's
Guide" to selecting a first course, to advise students who are
starting their collegiate mathematics study. The page explains that most
students begin with a calculus course, a quantitative reasoning or data
analysis course, a seminar course, or a computer science course. The webpage
then describes the options within each type of course and helps each student
choose the most appropriate one for his or her mathematical background and
interests.
The website for the mathematics department at
the University
of North Texas
includes links for academic advising and placement, course information,
student resources, and career information. The site provides
information for students on choosing their first math class, preparing for
the placement exam, and fulfilling the general education requirement. It also
includes course descriptions and sequencing information, tips for success
with mathematics courses, information about the mathematics laboratory and
tutoring help, description of math club activities, and extensive program and
career information for mathematics majors, including links to outside
resources.
Students at Kenyon
College
are offered “Advice to New
Students,” about how their choice of a first mathematics course could fit
their academic plans. For example, students who want only an
introduction to mathematics or a course to satisfy a distribution requirement
can select from Elements of Statistics, Surprises at Infinity, Quantitative
Reasoning, Pre-Calculus, Calculus A, and Introduction to Computer Science.
The University of Michigan at Dearborn
publishes an “Advising Newsletter” each semester. The newsletter is sent to
all declared mathematics majors and contains information on courses, careers,
competitions, scholarships, and on-campus jobs for math majors. It is
available through the mathematics department's link Information for
Students.
Redesigning
Courses and Programs in Response to Information about Students
High drop-out and failure rates for certain
groups of minority students in calculus at the University of California at Berkeley,
despite their relatively high mathematics
entrance scores, prompted Uri Treisman to
investigate the study habits of both successful and unsuccessful students.
Based on his findings, he introduced a mathematics workshop supplement for
the Berkeley
calculus course, recruiting African-American and Latino students who had
relatively high SAT Mathematics scores and/or intended to major in a
mathematics-based field. Key elements of the workshop involved:
* the commitment to help minority
students excel at the University
rather than merely avoid failure;
* collaborative learning and the use
of small-group teaching methods; and
* faculty sponsorship, which has
nourished the program and enabled it to survive. (Treisman,
1985, pp. 30–31)
The program at Berkeley
“replaces regular calculus discussion sections with workshop-style discussion
sections, in which the students collaborate on non-textbook, non-routine
problems.” During these work sessions (which meet for larger blocks of time
than traditional classes), “[students] begin working the problems
individually, then, when things get tough, in collaboration with one another.
These experiences lead to a strong sense of community and the forging of
lasting friendships” (Gillman, 1990, p. 8). The Berkeley
program has been so successful that it has been adapted by universities
and colleges throughout the country. Treisman has
stressed that the program is not remedial – and should not be – and
care is taken to prevent implementations from reverting to remedial
programs.
Eric Hsu’s website at San Francisco State University contains a number of
resources for Treisman-style workshops, including a
database of workshop problems and a handbook developed for instructors and
teaching assistants in the Berkeley
program. Additional information available on the Internet includes a history
of the programs, summaries of
evaluations (p.18), and details
of one evaluation. Another detailed evaluation is in An Efficacy Study of the Calculus Workshop
Model (Bonsangue, 1994). A few websites for existing Treisman-style programs are University of Wisconsin-Madison, Wayne State University, and University
of Texas at Austin.
At
Carnegie Mellon University
before 1995, between 7 and 10 percent of students enrolled in the computer
science program were female and their dropout rate was far higher than that
of men. Educational researcher Jane Margolis and computer scientist Allan
Fisher redesigned the computer science program, and the percentage of women
receiving degrees increased considerably. Margolis and Fisher’s study of the
Carnegie Mellon program and how the program was redesigned are described in Unlocking the Clubhouse:
Women and Computing. Further
changes are described in the webpage Women in
Computer Sciences: Closing the Gender Gap in Higher Education. Although the situation at Carnegie Mellon
concerned women and computer science, the book may be useful for those
thinking of studying and redesigning mathematics programs.
The
Mathematical
Thinking and Problem Solving course offered by Alan Schoenfeld, University of California Berkeley,
was designed in response to research findings about student beliefs,
attitudes, and problem solving abilities. More discussion of his course is in
Part 1, Section 2.
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