 MAANotes
#49: Assessment Practices in Undergraduate Mathematics
Introduction
- Book
Dedication, Preface, and Acknowledgements
- Introduction:
Assessing Assessment, Lynn Arthur Steen
- Assessment
and the MAA, Bernard L. Madison
- How
to Use the Book
- Teaching
Goals Inventory, Self Scorable Version
The Teaching Goals Inventory is
reprinted from Classroom Assessment Techniques,
by Angelo and Cross. We suggest that you, the
reader, take this inventory (with a specific
course in mind, as the authors suggest) in part
for your own information, and also because it
can help you locate your own teaching styles
and priorities in the context of this book.
Part I: Assessing the Major
- Introduction,
William A. Marion
Portfolios, Capstone Courses, and Focus
Groups
- Assessing
a Major in Mathematics, Laurie Hopkins
At a small, private women's
liberal arts college in the South student
portfolios have become the principal means
for assessing the major. Unique to this
program, certain courses are designated
as portfolio development courses and in
these courses the student is asked to reflect
in writing on the connection between the
material included in the portfolio and the
department goals.
- Portfolio
Assessment of the Major, Linda R. Sons
A Midwestern, comprehensive
university which has five different programs
in the mathematical sciences - General,
Applied Mathematics, Computational Mathematics,
Probability and Statistics, and Mathematics
Education - requires students to maintain
assessment portfolios in courses which are
common to all five of the emphases. The
portfolios of those who have graduated during
the year are examined by a department assessment
committee shortly after the close of the
spring semester.
- An
Assessment Program Built around a Capstone
Course, Charles Peltier
A full-year senior capstone
course has evolved at a small, private women's
liberal arts college in the Midwest to become
the principal tool for assessing the major.
Within this two-semester seminar each student
has to develop an independent study project,
known as the comprehensive project. Preliminary
work on the project begins in the first
semester and oral and written presentations
of the completed project are given in the
second semester.
- Using
a Capstone Course to Assess a Variety of Skills,
Deborah A. Frantz
At a mid-sized, regional university
in the East each student must complete a
one-semester senior seminar in which a variety
of assessment methods are used to assess
student learning in the major. These methods
include: a traditional final exam, a course
project, an expository paper, reading and
writing assignments, a journal and a portfolio
- all of which are designed to assess a
variety of skills acquired throughout a
student's four years.
- The
Use of Focus Groups within a Cyclic Assessment
Program, Marie P. Sheckels
An entirely different approach
to assessing the mathematics major has been
developed at a state-supported, coeducational,
liberal arts college in the Midsouth. Graduating
seniors participate in focus group sessions
which are held two days prior to graduation.
These are informal sessions with a serious
intent: to assess student learning in the
major.
Comprehensive Examinations
- Assessing
the Major Via a Comprehensive Senior Exam,
Bonnie Gold
At a small, private men's
liberal arts college in the Midwest, a comprehensive
examination, known as comps, has been a
tradition for seventy years. It has evolved
into the assessment technique which the
department uses to assess student learning.
Comps are taken by seniors over a two-day
period just prior to the start of the spring
semester. The exam consists of two parts:
a written component in the mathematics major
and an oral component over the liberal arts.
- A
Joint Written Comprehensive Examination to
Assess Mathematics Processes and Lifetime
Metaskills, G. Daniel Callon
A rather unique approach to
giving a comprehensive exam to seniors is
described in this article by a faculty member
at a small, private co-ed college in the
Midwest. The exam is taken by seniors in
their fall semester and lasts one week.
It is a written group exam which is taken
by teams of three to five students. Currently,
the exam is written and graded by a faculty
colleague from outside the college. As part
of a college-wide assessment program, the
Department of Mathematical Sciences developed
a departmental student learning plan, detailing
the goals and objectives which students
majoring in mathematics or computing should
achieve by the time of graduation. For mathematics,
there were three major goals. The first
goal related to an understanding of fundamental
concepts and algorithms and their relationships,
applications, and historical development.
The second centered on the process of development
of new mathematical knowledge through experimentation,
conjecture, and proof. The third focused
on those skills which are necessary to adapt
to new challenges and situations and to
continue to learn throughout a lifetime.
These skills, vital to mathematicians and
non-mathematicians alike, include oral and
written communication skills, the ability
to work collaboratively, and facility with
the use of technology and information resources.
- Undergraduate
Core Assessment in the Mathematical Sciences,
John W. Emert
In this article a "less" comprehensive
exam to assess student learning in the core
courses taken by all under-graduate mathematics
majors at a regional, comprehensive university
in the Midwest is discussed. We are guided
through the process involved in developing
the assessment instrument which is used
in all four tracks of the mathematical sciences
program: Actuarial Science, Mathematics,
Mathematics Education and Statistics.
Overall Programs
- Outcomes
Assessment in the B.S. Statistics Program
at Iowa State University, Richard A. Groeneveld
and W. Robert Stephenson
Although the statistics program
at this large Ph.D.-granting university
in the Midwest is not housed within a mathematical
sciences department, the wide variety of
measures used to assess the statistics program
can serve as one model for assessing the
mathematics major. All of the assessment
measures used are described with particular
emphasis on surveys of graduates and surveys
of employers of graduates.
- Assessment
of the Mathematics Major at Xavier: First
Steps, Janice B. Walker
In an evolving assessment program
at a private, medium-sized, comprehensive
university in the Midwest a variety of assessment
techniques are being developed to assess
student learning. How two of them - exit
interviews and the Educational Testing Service's
Major Field Test in Mathematics - are part
of the fabric of the mathematics program's
assessment cycle is described in this article.
- Assessing
Essential Academic Skills from the Perspective
of the Mathematics Major Mark Michael
At a private, church-related
liberal arts college in the East a crucial
point for assessing student learning occurs
midway through a student's four year
program. A sophomore-junior diagnostic project
which is part of the Discrete Mathematics
course, taken by all majors, is the vehicle
for the assessment. Each student in the
course must complete a substantial expository
paper which spans the entire semester on
a subject related to the course.
- Department
Goals and Assessment, Elias Toubassi
At a large, research university
in the West, a major effort over the past
10 to 15 years has been underway to reform
the entry level mathematics courses which
the department offers. Assessment has been
at the heart of this process. Focusing on
all first year courses through assessment
has had a positive effect on the undergraduate
mathematics major and the courses in that
major. This article describes the process
and the ongoing assessment of student learning.
Special Programs Within the Major
- Analyzing
the Value of a Transitional Mathematics Course,
Judith A. Palagallo and William
A. Blue
At a large, regional university
in the Midwest a specific course (Fundamentals
of Advanced Mathematics) at the sophomore
level provides a transition for student
majors from the more computationally-based
aspects of the first year courses to the
more abstract upper-division courses. Surveys
have been developed to measure the effects
of this course on upper level courses in
abstract algebra and advanced calculus.
In addition, these surveys provide information
about student learning in the major.
- Assessing
the Major: Serving the Needs of Students Considering
Graduate School, Deborah Bergstrand
The focus of this article is
on assessing student learning for a segment
of the undergraduate mathematics majors:
those talented students who are prospective
mathematics graduate students. At this private,
liberal arts college in the Northeast there
are three aspects of the undergraduate mathematics
experience outside the standard curriculum
which are described in this paper: a senior
seminar (required of all majors), an Honors
thesis and a summer undergraduate research
experience. All three combined are used
to assess how well the department is preparing
students for graduate school.
Part II: Assessment
in the Individual Classroom
- Introduction,
Bonnie Gold
Testing and Grading
- What
Happened to Tests?, Sharon Cutler Ross
For students to believe that
we expect them to understand the ideas,
not just be able to do computations, our
tests must reflect this expectation. This
article discusses how this can be done.
- Interactive
E-Mail Assessment, Michael D. Fried
This article discusses how internet
technology can be harnessed to give students
semi-automated individualized help. The
intended reader appreciates how little problem
solving guidance students get in class,
and how they are on their own to handle
giving meaning to learning an overstuffed
curriculum.
- Flexible
Grade Weightings, William E. Bonnice
Using a computer spreadsheet
to compute grades, it's easy to let
students (even in a large class) focus on
their strengths by choosing what percent
(within a range selected by the instructor)
of their grade comes from each required
activity.
Classroom Assessment Techniques
(A La Angelo & Cross)
- The
One-Minute Paper, David M. Bressoud
This quick technique helps the
instructor find out what students have gotten
out of a given day's class, and works
well with both large and small classes.
- Concept
Maps, Dwight Atkins
By drawing concept maps, students
strengthen their understanding of how a
new concept is related to others they already
know.
-
If You Want to Know What Students Understand,
Ask Them John Koker
To gain insight into how much
students actually understand, and what they
have learned by working the problems, have
them discuss the evolution of their ideas
as they work on homework sets.
- Improving
Classes with Quick Assessment Techniques,
John W. Emert
This article discusses three
quick techniques which can alert the instructor
to potential problems. The first helps students
understand course goals, the second evaluates
the effectiveness of group work, and the
third is a general way of finding out how
things are going in time to make changes.
As I continue to experiment with various
informal classroom assessment techniques,
I have come to favor assessment tools which
use no more than ten minutes of class time
and require no more than an hour after class
to tabulate and form a response. The three
techniques included here provide quick ways
to find out what is going well and what
is not, and allow me to address any problems
in a timely manner.
- Creating
a Professional Environment in the Classroom,
Sandra Z. Keith
This article describes four
activities which can help students develop
a more professional attitude toward their
coursework: looking directly at their expectations,
revising goals after a test, finding applications
of the course in their chosen field, and
preparing a résumé.
Reviewing Before Examinations
- True
or False? Explain! Janet Heine Barnett
One way to find out what students
understand is to ask them true/false questions,
but have them justify their answers. These
justifications bring out confusions about
the concepts, but are also the beginning,
for calculus students, of writing mathematical
proofs.
- Define,
Compare, Contrast, Explain ..., Joann Bossenbroek
These short writing assignments
help students clarify concepts, and show
the instructor where more work is needed.
- Student-Created
Problems Demonstrate Knowledge and Understanding,
Agnes M. Rash
Having students write problems
shows the instructor where the gaps in their
understanding are at the same time that
it has students review for an upcoming test.
Further, it can help students find the relevance
of the course to their own interests.
- In-Depth
Interviews to Understand Student Understanding,
M. Kathleen Heid
Students' answers on tests
don't always show their true level of
understanding. Sometimes they understand
more than their answers indicate, and sometimes,
despite their regurgitating the correct
words, they don't understand what they
write. This article discusses a method to
probe what they actually understand.
Projects and Writing to Learn Mathematics
- Assessing
Expository Mathematics, Annalisa Crannell
This article gives many helpful
hints to the instructor who wants to assign
writing projects, both on what to think
about when making the assignment, and what
to do with the projects once they're
turned in.
- Assessing
Modeling Projects in Calculus and Precalculus:
Two Approaches, Charles E. Emenaker
Once you've assigned a writing
project and collected the papers, how are
you going to grade it without spending the
rest of the semester on that one set? This
article outlines two grading scales which
can make this more efficient.
- Assessment
During Complex Problem-Solving Group Work
in Class,Brian J. Winkel
During student efforts to attack
and solve complex, technology-based problems
there is rich opportunity for assessment.
The teacher can assess student initiative,
creativity, and discovery; flexibility and
tolerance; communication, team, and group
self-assessment skills; mathematical knowledge;
implementation of established and newly
discovered mathematical concepts; and translation
from physical descriptions to mathematical
models.
- Student
Assessment through Portfolios, Alan P.
Knoerr and Michael A. McDonald
Reflective portfolios help students
assess their own growth. Project portfolios
identify their interests and tackle more
ambitious assignments.
- Using
Writing to Assess Understanding of Calculus
Concepts, Dorothee Jane Blum
Student write expository papers
in an honors, non-science majors' calculus
course to integrate the major ideas they're
studying.
- Journals:
Assessment without Anxiety, Alvin White
Using journals for formative
assessment encourages students to explore
topics they might have been intimidated
by if they were being graded.
- Assessing
General Education Mathematics Through Writing
and Questions, Patricia Clark Kenschaft
General education students can
learn to read mathematics more thoughtfully
and critically by writing questions over
the reading, and short response papers-without
enormous investment of faculty time grading.
Cooperative Groups and Problem-Centered
Methods
- Combining
Individual and Group Evaluations, Nancy
L. Hagelgans
One frequent concern of faculty
members who have not yet tried cooperative
learning is that giving the same grade to
a whole group will be unfair both to the
hard workers and the laggards. This article
addresses this issue.
- Group
Testing, Catherine A. Roberts
From reviewing before a test
to various ways for students to take tests
collaboratively, this article looks at ways
that groups can be used to evaluate student
learning while increasing that learning.
- Continuous
Evaluation Using Cooperative Learning, Carolyn
W. Rouviere
This article discusses several
cooperative learning techniques, principally
for formative assessment. These include
ways to help students learn the importance
of clear definitions and several review
techniques (comment-along, teams-games-tournaments,
and jigsaw).
- Collaborative
Oral Take-Home Exams, Annalisa Crannell
Giving Collaborative oral take-home
examinations allows the instructor to assess
how well students handle the kind of non-routine
problems we would all like our students
to be able to solve.
- Assessment
in a Problem-Centered College Mathematics
Course, Sandra Davis Trowell and Grayson
H. Wheatley
When using a problem-centered
teaching approach, the instructor needs
new methods of assessment. This article
explores one such approach.
Special Need Students
- Assessing
Learning of Female Students, Regina Brunner
The author discusses assessment
techniques which she has found to be particularly
effective with female students.
- Strategies
to Assess the Adult Learner, Jacqueline
Brannon Giles
Adult students are often better
motivated than traditional-age students,
but many have not taken an examination for
many years. Thus, finding appropriate methods
of assessment poses a challenge.
- The
Class Mission Statement, Ellen Clay
To set the tone of a course
at the beginning, develop with the class
a "class mission statement," which can be
revisited as the course progresses to assess
progress toward meeting course goals.
- Early
In-Course Assessment of Faculty by Students,
William E. Bonnice
Students learn to take responsibility
for their learning by giving input on how
the class is going and what needs to be
changed. This works in classes of all sizes.
- Student
Feedback Teams in a Mathematics Classroom,
David Lomen
A student feedback team is a
subset of the class which gives the instructor
feedback on how the class is doing with
new material.
- Early
Student Feedback, Patricia Shure
In introductory courses in mathematics
at the University of Michigan, an instructional
consultant visits the class one-third of
the way into the semester. This observer
holds a discussion with the class, in the
absence of the instructor, about how the
course is going, and provides feedback to
the instructor.
- Friendly
Course Evaluations, Janet Heine Barnett
By having students write a letter
to a friend about your course, you can get
useful information both on what the students
have learned and on what they thought of
your course.
- The
Course Portfolio in Mathematics, Steven
R. Dunbar
The teaching portfolio is an
alternative to the standard course questionnaire
for summing up a course. The instructor
collects data throughout the semester into
a course portfolio, which can then be used
by that instructor or passed on to others
teaching the course.
Part III: Departmental
Assessment Initiatives
- Introduction,
Sandra Z. Keith
Placement and Advising
- Administering
a Placement Test: St. Olaf College, Judith
N. Cederberg
A small college nurturing a
large calculus clientele in a flexible calculus
program recognizes the need for careful
placement, and studies the effectiveness
of its efforts withstatistics to derive
a formula for placement.
- A
Mathematics Placement and Advising Program,
Donna Krawczyk and Elias Toubassi
Placement was the issue at this
large, comprehensive school. This article
explains, among other initiatives the department
took to improve its accessibility to students,
a Mathematics Readiness Testing Program.
Statistics measuring the reliability of
the testing are included.
- A
Comprehensive Advising Program, Stephen
A. Doblin and Wallace C. Pye
A large, budding university
concentrates on the importance of advising
students at all levels. From providing mentors
for freshmen to surveying graduating students
and alumni, this department operates with
continuing feedback.
- A
Quantitative Literacy Program, Linda R.
Sons
The author helped create MAA
Guidelines for Quantitative Literacy and
here spells out how this document was used
at her large university. This school tested
students and graded results in a variety
of courses. Results led to curricular changes.
- Creating
a General Education Course: A Cognitive Approach,
Albert D. Otto, Cheryl A. Lubinski,
and Carol T. Benson
What is the point of teaching
students if they're not learning? Here
a general education course operates from
a learning-theoretic mold; mathematics education
instructors become involved to help students
construct their own learning.
- Using
Pre- and Post-Testing in a Liberal Arts Mathematics
Course to Improve Teaching and Learning, Mark
Michael
This pre- and post-testing system
in a liberal arts mathematics course raises
interesting questions about testing in general,
and asks why students may sometimes appear
to go backwards in their learning.
- Coming
to Terms with Quantitative Literacy in General
Education: or, the Uses of Fuzzy Assessment,
Philip Keith
This article presents an administrative
look at the ramifications of accommodating
various departments' views of how quantitative
literacy is to be defined. The issue is:
what are the students telling us-how do
we interpret the answers they provide to
the questions we've asked? The value
of "fuzzy" assessment is discussed in the
interpretation of a simple survey which
helps move a collective bargaining institution
on track.
- Does
Developmental Mathematics Work? Eileen
L. Poiani
An inner city school with a
diverse, multicultural clientele is deeply
committed to raising students' mathematical
abilities. The school has operated with
grants that are now drying up, but that
help authored some assessment studies over
a ten-year period. They ask: does developmental
mathematics help, or is it a hopeless cause?
Mathematics in a Service Role
- Let
Them Know What You're Up to, Listen to What
They Say, J. Curtis Chipman
Effective Relations Between
a Department of Mathematical Sciences and
the Rest of the Institution How can a mathematics
department please its client disciplines?
This department finds a solution in establishing
a web of responsible persons to establish
goals for students and instructors, course
leaders, committees, and faculty liaisons,
for placement of students into courses and
for the content of those courses.
- Have
Our Students with Other Majors Learned the
Skills They Need? William O. Martin
and Steven F. Bauman
A large university begins by
asking teachers in other disciplines not
for a "wish list" but for a practical analysis
of the mathematical knowledge required in
their courses. Pretests for students reflect
these expectations, and discussion of results
encourages networking.
- A
TEAM Teaching Experience in Mathematics/Economics
Marilyn L. Repsher and J. Rody Borg
Opening a course to both mathematics
and business faculty teaching as a team
creates public dialogue about problems that
straddle two departments.
- Factors
Affecting the Completion of Undergraduate
Degrees in Science, Engineering, and Mathematics
for Underrepresented Minority Students: The
Senior Bulge Study, Martin Vern Bonsangue
Commissioned by the California
State University's Chancellor's
Office, this study looks at transfer students
and suggests key areas for reform.
- Evaluating
the Effects of Reform, Richard West
West Point turned an entire
department around. Using an in-depth assessment
study with careful attention to the needs
of client disciplines, the department created
a brand new curriculum, and continues to
study it with the "Fullan model" which the
author investigated in his dissertation.
- A
Comprehensive, Pro-Active Assessment Program,
Robert Olin and Lin Scruggs
A large technical institute
using the author as assessment coordinator,
creates a broad new assessment program,
looking at all aspects of the department's
role. Statistical studies guide improvements
in curriculum, teaching and relations with
the rest of the university.
Assessing Reform Courses
- Assessment
in One Learning Theory Based Approach to Teaching,
Ed Dubinsky
In this discussion piece, the
author explains an approach to teaching
based on Learning Theory, particularly examining
a Calculus course to ask how assessment
can best feed back into the learning environment.
- An
Evaluation of Calculus Reform: A Preliminary
Report of a National Study, Susan Ganter
This is a preliminary report
of a study at NSF dealing with what NSF
has looked for, what it has found, and directions
for future study. One such direction will
be to shift from "teaching" to "student
learning" and the learning environment.
- Increasing
the Dialogue About Calculus with a Questionnaire,
A. Darien Lauten, Karen Graham,
and Joan Ferrini-Mundy
A practical survey is provided
here which might be found useful for departments
looking to initiate discussion about goals
and expectations in a Calculus course.
- Assessing
Student Attitudes and Learning Gains, Nancy
Baxter Hastings
This assessment program stresses
breadth - pre- and post-testing, journals,
comparative test questions, student interviews
and questionnaires, and more.
- Does
Calculus Reform Work? Joel Silverberg
This large study examines data
over a long period of time regarding a calculus
"reform" course. Grades and attitude are
studied, with advice for novices at assessment.
- Assessing
Student Attitudes and Learning Gains
This assessment program stresses
breadth - pre- and post-testing, journals,
comparative test questions, student interviews
and questionnaires, and more.
- Assessing
the Effectiveness of Innovative Educational
Reform Efforts, Keith E. Schwingendorf
This study explains the creation
of a calculus reform program, its objectives,
and philosophy and provides an in-depth
comparison of reform-trained students with
traditional students.
- The
Evaluation of Project CALC at Duke University,
1989-1994, Jack Bookman and Charles
P. Friedman
This in-depth study analyzes
a Calculus reform program. It looks not
only at analytical gains in student understanding
but affective gains as well.
Part IV: Assessing Teaching
- Introduction,
William A. Marion
Assessing Teaching
- Departmental
Assistance in Formative Assessment of Teaching,
Alan P. Knoerr, Michael A. McDonald,
and Rae McCormick
At a small liberal arts college
in the West a department-wide program has
been developed to help faculty assess and
improve their teaching while courses are
in progress. Descriptions of what led to
this effort, the steps already taken, the
resources involved and plans for the future
are presented.
- Assessing
the Teaching of College Mathematics Faculty,
C. Patrick Collier
At a regional, comprehensive
university in the Midwest the mathematics
faculty have been grappling with what it
means to be an effective teacher and how
to evaluate such effectiveness. Their conclusion
is that student evaluations should not be
the primary means of evaluating teaching.
Hence, the department is in the process
of articulating a statement of expectations
for teaching from which appropriate assessment
instruments will be developed.
- Using
Video and Peer Feedback to Improve Teaching,
Joel David Hamkins
This article discusses a program
at Berkeley of using videotaping of actual
classes, and peer feedback, to improve teaching.
While the program was aimed at graduate
students, it can be adapted to use with
faculty members.
- Exchanging
Class Visits: Improving Teaching for Both
Junior and Senior Faculty, Deborah Bergstrand
A peer visitation program for
both junior and senior faculty at a small,
liberal arts college in the East has been
put into place to help improve the quality
of teaching. Every junior faculty member
is paired with a senior colleague to exchange
class visits. This program is designed to
foster discussion of teaching, and the sharing
of ideas and to provide constructive criticism
about the teaching effectiveness of each
member of the pair.
- Peer
Review of Teaching, Pao-sheng Hsu
At a land-grant university in
the Northeast one mathematics faculty member
has begun experimenting with a peer visitation
program in which a team of faculty visits
her class at least twice a semester. What's
unique about this process is that the team,
usually three in number, consists of faculty
both within and without the department and
all visit the same class at the same time.
This technique provides the instructor with
a diversity of views on her teaching effectiveness.
evaluations, since these are often mandated
by university administrations. In an effort
to generate discussion and broaden the perspective
on evaluation of teaching, I initiated the
experiment of peer review of my classes.
I have experimented with inviting faculty
members who have experience in ethnographic
research1 and are from outside of my own
discipline, as well as colleagues from my
own department.
Appendix, Further Reading, and Bibliography
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The
Mathematical Association of America