____ THE _____________
_____ MATHEMATICS ___
________ EDUCATOR _____
Volume 13 Number 1
Spring 2003
MATHEMATICS EDUCATION STUDENT ASSOCIATION
THE UNIVERSITY OF GEORGIA
Editorial Staff
A Note from the Editor
Editor
Brian R. Lawler
Dear TME readers,
An interesting collection of papers has come together for Volume 14 of The
Mathematics Educator. Each article tells the story of mathematics educators wrestling
with efforts to make meaning of their work. In his final piece of a three part series of
case studies begun in Volume 13, Andy Norton peers into a mathematician’s spiritual
beliefs. In contrast to earlier findings, this mathematician’s spiritual beliefs are in
harmony with his attitudes toward and ways of knowing mathematics.
Rasmus Nielsen presents his personal theoretical difficulties in wrestling with the
goals of conducting research in mathematics education. What is our commitment to
society – reproduction or renewal? At what grain-level is our commitment to those
who are our society?
Carmen Latterell; Norene Lowery; and Amy McDuffie, Valarie Akerson, & Judith
Morrison provide a synergistic triad of reports of working with teachers implementing
new classroom practices. As mathematics classrooms change and as what we value as
mathematical learning evolves, new dilemmas arise for both teachers and researchers.
Each of these authors works to investigate sensible and meaningful assessment
practices among the manifestation of these new dilemmas. Their findings begin to
elaborate a research agenda in the era of larger scale adoption of reform curricula.
And finally, a graduating senior from The University of Georgia kicks off the
issue. She reflects on the tensions of meeting the needs of her students in the face of an
“alphabet soup” of curriculum standards, standardized tests, and the other stressors of
student teaching.
I hope I have sufficiently sparked your interest to read our journal. Next, I hope to
tempt you to consider an academic contribution to our efforts. TME is entirely a
student run and student funded, peer-reviewed journal dedicated to sharing theory,
research, and practices within the mathematics education community. In its thirteenth
year of publication, the journal has proven to be a resource for serving, connecting,
and learning for the writers, editors, and readers. Please contribute to TME by sharing
articles with students and peers, linking to our web site, submitting manuscripts, or
helping to review. You will find information on pp. 47-48 of this issue on how you can
be involved.
Associate Editors
Holly Garrett Anthony
Dennis Hembree
Zelha Tunç-Pekkan
Publication
Laurel Bleich
Advisors
Denise S. Mewborn
Nicholas Oppong
James W. Wilson
MESA Officers
2003-2004
President
Dennis Hembree
Vice-President
Erik Tillema
Secretary
R. Judith Reed
Treasurer
Angel Abney
NCTM
Representative
Holly Garrett Anthony
Undergraduate
Representative
Tiffany Goodwin
Brian R. Lawler
105 Aderhold Hall
The University of Georgia
Athens, GA 30602-7124
tme@coe.uga.edu
www.ugamesa.org
About the cover
The cover art was inspired by a presentation given by Harold Asturias while he worked on the New Standards Project,
a collaboration of the University of Pittsburgh and the National Center on Education and the Economy. In the mid
1990’s, these partners built an assessment system to measure student progress toward meeting national standards. In
his presentation, Harold reminded the audience that any single tool that we peer through to attempt to measure student
understanding has important limitations. Some provide reasonable clarity but offer too narrow a view; others open the
lens wider but blur what we see. His conclusion was that we as educators are bound to consider multiple and varied
assessments when making significant decisions about children. But more importantly, he reminds us that learners are
more complex than any combination of measurement tools could ever report.
This publication is supported by the College of Education at The University of Georgia.
________ THE ________________
________ MATHEMATICS ______
__________ EDUCATOR _________
An Official Publication of
The Mathematics Education Student Association
The University of Georgia
Spring 2003
Volume 13 Number 1
Table of Contents
2
Guest Editorial… A Learning Environment Crippled by Testing: A Student
Teacher’s Perspective
AMANDA AVERY
5
Testing the Problem-Solving Skills of Students in an NCTM-Oriented
Curriculum
CARMEN M. LATTERELL
15 Assessment Insights from the Classroom
NORENE VAIL LOWERY
22 Designing and Implementing Meaningful Field-Based Experiences for
Mathematics Methods Courses: A Framework and Program Description
AMY ROTH MCDUFFIE, VALARIE L. AKERSON, & JUDITH A. MORRISON
33 How to do Educational Research in University Mathematics?
RASMUS HEDEGAARD NIELSEN
41 Mathematicians' Religious Affiliations and Professional Practices: The Case of Bo
ANDERSON NORTON III
46 Upcoming conferences
47 Subscription form
48 Submissions information
© 2003 Mathematics Education Student Association
All Rights Reserved
The Mathematics Educator
2003, Vol. 13, No. 1, 2–4
Guest Editorial…
A Learning Environment Crippled by Testing:
A Student Teacher’s Experience
Amanda Avery
My student teaching experience was to begin in
January 2003. In November prior to this, I was
assigned to a mentor teacher who taught Applied
Problem
Solving
(APS)
and
Advanced
Algebra/Trigonometry. Because the state of Georgia
has adopted (criterion referenced) End of Course Tests
(EOCT) for Algebra I and Geometry, I will focus a
majority of this commentary on my experiences with
my APS class. This course and its complement,
Applied Algebra, satisfy the requirement for Algebra I
for technical career students (as opposed to those that
are college prep students). The intention is for the
technical career students to get more of a “hands on”
approach to Algebra, with phrases like “contextual
teaching and learning” being strongly emphasized and
encouraged.1 My intention is to illustrate to you what I
was up against concerning standardized testing in my
APS classes, to express to you how I feel about these
pressures, and to demonstrate to you how my students
wanted me to assess them in other methods that aren’t
necessarily measurable.
Because I was unfamiliar with the curriculum for
the APS course, I carefully reviewed the state of
Georgia’s educational standards, called the Quality
Core Curriculum (QCC) and compared them to the
CORE-Bridges APS textbook used in my school. I was
relieved to discover that I didn’t need to teach the
breadth of the entire textbook, but only about five key
concepts in depth. I e-mailed my mentor teacher about
his experience the previous year with this textbook: “I
was looking at the QCCs for the APS class and
discovered that we can most likely skip chapters 4 and
8 (along with a few sections from chapter 2) from their
textbook. Let me know what you think.”
My mentor teacher’s response: “Due to the EOCT
that follows the Applied Algebra, we do not follow the
Amanda Avery is a mathematics education student at the University
of Georgia. Upon completing her student teaching in a rural public
high school, she was recognized as the outstanding graduating
senior with the department’s Hooten Award. She will begin her
teaching career this fall at John McEachern High School in
Powder Springs, Georgia. She enjoys sailing, dabbles in desktop
publishing, and collects black & white photographs of dogs. Her
email address is ImagineALA@hotmail.com.
2
QCCs in this course as closely as we do others. We
really don’t skip those sections at all, but modify and
enrich as allowed by the students of the class. … Our
two APS classes are just 23 and 17 students each. They
range from special ed students to seniors who have
algebra one, algebra two, and just need a third math to
graduate under the Technical Career diploma.
However, the course is based on the slower students
(bottom 25th percentile of the nation), and the others
are used as tutors and mentors. Most like being the
smartest in the class, as they are used to struggling in
CP [College Prep] and failing.”
Just so you have a little more taste of what my first
impressions were, I asked my mentor teacher for some
more feedback: “I was hoping to skip a majority of
chapter one. Of course, I really have no idea how
appropriate this is because I don’t know what the
students understand and can apply. I do feel, though,
that I should give the students a little more credit than
what their textbook implies.”
My mentor teacher’s feedback: “Most of the
students need the reinforcement of the first chapter, but
there are things we can do to not approach it the same
way as the book. I agree that it seems elementary, but
these students are not like the ones you observed in
block four classes in the fall [Honors Advanced
Algebra/Trigonometry]. They have struggled in math
their whole life, and most math is forgotten between
courses. Part of the hope is to let some find success if
they have the skills, but look at it in an applied sense.
We may get lucky, but students have failed the first
unit test (46% last spring).”
Now, for a challenge from the state legislature…
The state of Georgia, as part of its A+ Educational
Reform Act of 2000 passed a law, O.C.G.A. §20-2-281
that “mandates that the State Board of Education adopt
end-of-course-assessments in grades nine through
twelve.” Each EOCT is “directly aligned with the
standards in the QCC and will consist of multiple
choice questions.” According to its 2001 Information
Brochure, the purpose of EOCT is to “improve student
achievement through effective instruction and
assessment of the standards in the QCC and to ensure
A Crippled Learning Environment
that all Georgia students have access to a rigorous
curriculum that meets high-performance standards.” If
students take the EOCT ; what, then, will teachers,
schools, school boards, and the state board of
education do with the data that is collected? “The
results of the EOCT will be used for diagnostic,
remedial, and accountability purposes to gauge the
quality of education in the state.” The interesting part
of all this is that all Georgia high school students take
the Georgia High School Graduation Tests (GHSGT)
as exit tests for getting a high school diploma: IB,
college prep, tech prep, and general ed are all included
(special ed seals allow for modifications, as
appropriate). In addition to a comprehensive test that
covers “up through Algebra I,” as a requirement for
graduation, legislators also subject students to EOCTs.
Let’s not forget, also, that legislators are submitting
taxpayers to the costs of creating, administering, and
scoring these tests.
Now you have the general idea of what I was to
encounter, through my mentor teacher’s expectations
and the state’s mandates. I taught in block scheduling,
where students complete four entire courses in a single
semester, sitting in four 90-minute classes each day.
Despite my mentor teacher’s outlook on keeping an
iron fist on the APS textbook, I looked for ways to
satisfy the National Council of Teachers of
Mathematics (NCTM’s) vision from the Principles and
Standards, the Georgia QCCs, the EOCTs, the
GHSGTs, the PSAT, the SAT, the ACT, my APS
students and ME.2 I don’t want to completely give you
wrong impression—While I was very overwhelmed by
the pull of all these outside forces, I did enjoy my
student teaching. How? Because of my students—the
interactions I’ve had with them, and the few sparks that
I could witness igniting if they encountered something
new and challenging.
Early on I was confronted with wondering how I
can arrange a pacing guide that requires my students to
be tested comprehensively as juniors in March for the
GHSGT, tested cumulatively in April (or November)
for Algebra I for the EOCT, tested state-wide for the
PSAT as sophomores in October, make time for
standardized testing “prep sessions,” athletic pep
rallies, student organization meetings, special seminars
for students, fire & tornado drills, and for, oh yea,
teaching. I understand that assessment is a part of
teaching, but within my classroom I also must place
into the pacing guide room for formal assessments for
finals, midterms, projects, presentations, quizzes, and
reviewing homework. I hope you are beginning to
understand that I feel as though my student teaching
Amanda Avery
experience was one big assessment. Sometimes I felt as
though I never got to teach, and more importantly, my
students rarely had the opportunity to learn significant
mathematics, to struggle, to dabble—they were always
being evaluated, whether informally or formally.
It is with this point that I am most disturbed by
Georgia’s A+ Education Reform Act of 2000. I didn’t
understand how multiple choice standardized tests
could “improve student achievement through effective
instruction and assessment of the standards in the
QCC,” especially if the end results were to be used for
“diagnostic, remedial, and accountability purposes to
gauge the quality of education in the state.” It seems to
me that standardized multiple choice questions are
inadequate for assessing the “quality of education,”
especially if you consider quality mathematics
education as the vision put forth by NCTM’s Principles
and Standards. Even more importantly, I was also
vexed with what the results will do for my
students—they sort, rank, and stigmatize my students
against other students, other schools, other districts,
and other states who have widely varying curriculum
standards, resources, administrators, parents, and
students than mine, especially if “holding schools
accountable” means that my school district may not get
additional funding. Ethically, this seems as a step
backward if my community truly believes that ALL
students can be successful and where ALL students can
and must learn mathematics, where “no child is left
behind.” Even though the idea that students must be
able to illustrate proficiency in skills x, y, and z before
moving on to the next course is quite valid, I don’t
believe that standardized testing is helping my
students, their parents, me, my administrators, my
school board electorates, and my state board
legislatures to provide a complete picture, even if my
students excel at multiple choice standardized testing.
Lack of providing a complete picture is only the “tip of
the iceberg” towards the argument that standardized
testing is not the way to “improve student
achievement.” While I don’t believe that assessment
isn’t a part of the teaching process, I believe that
students’ attitudes, behaviors, and oral and written
communication skills must be also considered.
Yet, I am unconvinced that students should be
continually assessed, every moment in every class. It
was an obstacle for me to realize that my students were
afraid to participate in class because they have learned
the game—they know that the teacher is always
listening, the teacher is always judging, the teacher is
always thinking, “right or wrong.” No, actually, the
obstacle was in trying to change the game, to let my
3
students experience different rules, where they weren’t
judged on everything they thought, on everything they
tried to start. This experience is not centered on my
actions alone; in fact, the interactions that my students
had with each other was the most crucial part of
changing the rules. It took my students about six weeks
to realize that when I ask for ideas that I (or other
students) wouldn’t ridicule or thwart each other’s
efforts. It was almost over night that my students
started trusting me and trusting each other with their
ideas and suggestions.
What happened? I was able to teach; they were
allowed to struggle, they were given opportunities to
test ideas, and they weren’t being graded. Correction: it
wasn’t that I was suddenly able to teach; my students
now taught each other. Through a problem solving day,
my students demonstrated to me and to each other that
they could effectively provide valid inferences,
evaluate each other’s ideas constructively (not
critically), and more importantly, they started
questioning their own thought processes. The most
significant comment I made that day is that I “will not
grade what you do today. Rather, I want you to think
about how you can evaluate how you think you
understand what you are doing.”
My students informed me that I could understand
what they were thinking if I watched them do
problems, and if I listened to how they explained the
concepts to each other. Even though I did these things
informally, somehow this discourse allowed my
students to become more relaxed with each other; it’s
4
almost as if I needed their “okay” for them to finally
trust me and each other. The judging environment had
been eased; the iron fist relaxed its grip. They
understood that I didn’t like lecturing and being the
only one doing the mathematics while they passively
copied my notes, and I understood that they didn’t like
this either—they want the challenge, they want to
think. And standardized testing cannot ever measure
that.
REFERENCES
A+ Educational Reform Act of 2000 O.C.G.A:§20-2-281 (n.d.).
Retrieved May 16, 2003, from http://www.ganet.org/cgi-bin
/pub/ocode/ocgsearch?docname=OCode/G/20/2/281
Georgia Department of Education. (n.d.). End of Course Test
algebra one content description guide. Retrieved May 16,
2003, from http://www.doe.k12.ga.us/_documents/curriculum
/testing/al1.pdf
Georgia Department of Education. (n.d.). Information on End of
Course Tests. Retrieved May 16, 2003, from
http://www.doe.k12.ga.us/_documents/curriculum/testing/eoct
-broc.pdf
National Council of Teachers of Mathematics. (2000). Principles
and standards for school mathematics. Reston, VA: Author.
1
You might understand why I’ve added the emphasis, but
that is not the point of this editorial.
2
I wanted to emphasize that my students and I are also
important in this alphabet soup of curriculum standards and
standardized tests.
A Crippled Learning Environment
The Mathematics Educator
2003, Vol. 13, No. 1, 5–14
Testing the Problem-Solving Skills of Students
in an NCTM-oriented Curriculum
Carmen M. Latterell
An interesting question concerns how well NCTM-oriented students do on standardized mathematics tests.
Another important question that has received less attention is: Are standardized tests truly measuring the skills
that NCTM-oriented students have? Would other tests reveal skills that differentiate NCTM-oriented students
from traditional students? Moreover, what are these skills? This paper contributes to the answers to these
questions, and finds that students in one NCTM-oriented curriculum displayed such qualities as engagement,
eagerness, communication, flexibility, and curiosity to a much higher degree than traditional students did. The
implication given is that not only should we use standardized tests, but we should revise them and/or
supplement them to measure the qualities that are not currently being measured.
The National Council of Teachers of Mathematics
(NCTM) has been promoting standards for teaching
secondary mathematics (NCTM, 2000; NRC, 1989). At
the same time, many curriculum projects have been
developed for the purpose of providing NCTMoriented curricula for teachers (e.g., the NSF-funded
curricula). However, the implementation of these
NCTM-oriented curricula has not been without
problems (Curcio, 1999). In fact, the term “math wars”
has been used to describe the controversies (Senk &
Thompson, 2003b; Schoen, Fey, Hirsch, & Coxford,
1999). Some of these controversies surround the issue
of standardized testing (Hancock & Kilpatrick, 1993).
In fact, some parents have indicated concern that if
their children are in NCTM-orientated curricula, they
will be at a disadvantage on standardized tests (Senk &
Thompson, 2003a). Of course, standardized tests might
change in time, but it is an interesting question whether
NCTM-oriented students actually are at a disadvantage
on standardized tests. Perhaps it is an interesting
question because standardized testing should change in
time. In other words, the question of interest is not
simply “Are students in NCTM-oriented curricula at a
disadvantage in standardized testing?” but rather “Are
standardized tests the best measure of everything that
students in NCTM-oriented curricula can do?” And
then if the answer is that other tests can reveal abilities
that standardized tests do not, what are these other
abilities?
Carmen M. Latterell is an assistant professor of mathematics at the
University of Minnesota Duluth. Her research interests include the
testing of mathematics. This interest includes the types of questions
seen in this article as well as such issues as placement testing into
undergraduate mathematics courses.
Carmen M. Latterell
Standardized achievement tests are viewed as
"blunt" measuring instruments by some (Kilpatrick,
2003, p. 479). Certainly issues are more complex when
standardized tests attempt to measure students'
problem-solving ability. Since problem solving is
regarded as a process (NRC, 1989; NCTM 2000) and
not a product, standardized tests must be well
constructed to measure problem solving adequately.
While researchers say that standardized tests cannot
measure problem solving, it is important to remember
that testing is "value laden and socially constructed"
(Gipps, 1999, p. 370). Are standardized tests
measuring the abilities that we value for students in
NCTM-oriented curricula?
Now that NCTM-oriented curricula have been put
into action for several years, researchers are able to
examine these issues with research studies. This study
contributes to the literature by examining some of the
issues involved in the testing of students in a particular
NCTM-oriented
curriculum,
the
Core
Plus
Mathematics Project Curriculum (CPMP). The
overarching research question is how to better assess
students who are in a NCTM-oriented curriculum to
measure as fully as possible their problem-solving
abilities.
To answer this, some comparisons between
NCTM-oriented students and traditional students are
made, as well as some comparisons between NCTMoriented students' results on different types of tests.
The reader is cautioned that the intent of this study is
not to compare NCTM-oriented students to traditional
students, per se. Many studies (Senk & Thompson,
2003a) have already been done (this paper will review
some of them as applies to CPMP in a later section),
giving evidence that students in NCTM-oriented
5
curricula can perform as well as students in traditional
curricula on standardized tests.
The current study attempts to examine the
subtleties of testing students in a NCTM-oriented
curriculum. For example, although these students
might do as well as others on standardized tests, are
there other tests that are a better measure of these
students' problem solving abilities? Or even, are there
other tests that will measure abilities that standardized
tests do not and that traditional students may not have?
And, perhaps most important, what is the nature of
these abilities that are not (if in fact they are not) being
measured by standardized testing?
The research questions are:
1. How do the scores of students in CPMP compare
to students in a traditional curriculum on a
standardized problem-solving test?
2. How do the scores of students in CPMP compare
to students in a traditional curriculum on a parallel
constructed-response problem-solving test?
3. Are there differences that qualitative data can
illustrate between the manners in which pairs of
students in CPMP and pairs of students in a
traditional
curriculum
solve
non-routine
constructed-response problem-solving items?
Method
The basic method was to give a standardized
problem-solving test to students in CPMP and to
comparable students in a traditional curriculum. Effort
was made to have similar ability students in both sets
of curricula before entering the curriculum, using Iowa
Test of Educational Development (ITED; Feldt,
Forsyth, Ansley, & Alnot, 1993) data. In addition, a
test parallel to the standardized test but with
constructed-response items was also given to both sets
of students. Finally, both sets of students were given a
performance assessment. This section describes the
curricula, the students, a survey of the classroom
environments, and the tests.
The CPMP Curriculum
As stated earlier, the NCTM-oriented curriculum
used CPMP, the Core Plus Mathematics Curriculum.
The author had access to students in the state of Iowa,
and a large number of schools in Iowa had
implemented CPMP for several years. Therefore,
CPMP was chosen as the curriculum to represent
NCTM-oriented curricula. Yet, no claim is made that
CPMP is better or worse than other NCTM-oriented
curricula, or that the results, therefore, would be the
same under other curricula. Regardless, the researcher
6
judged it would be better not to have more than one
NCTM-oriented curriculum and thus possibly
confound the results.
CPMP has developed student and teacher materials
for a three-year high school mathematics curriculum
for all students and a fourth-year course for college
bound students. The main theme of CPMP is
mathematics as sense making. Students investigate
problems set in real-life contexts within an integrated
curriculum that includes algebra and functions,
geometry and trigonometry, statistics and probability,
and discrete mathematics. The curriculum for each
year is seven units with a capstone section which is “a
thematic two-week, project-oriented activity that
enables students to pull together and apply the
important mathematical concepts and methods
developed in the entire course” (Schoen & Ziebarth,
1998b, p. 153). Mathematical modeling is emphasized
throughout the curriculum. Graphing calculators are
used. Additional characteristics of CPMP are that the
curriculum is designed to be accessible to all students,
to engage the students in active learning, and to
provide multidimensional assessment (Hirsch &
Coxford, 1997). Assessment opportunities are
embedded within the curriculum and include students’
answers to questions in class, groupwork, student
journals, quizzes, in-class and take-home end-of-unit
assessments, cumulative written assessments, and
extended projects (Hirsch, Coxford, Fey, & Schoen,
1995).
Inclusion of topics in the CPMP curriculum is
based on the merits of the topics themselves; that is,
the topics must be important in their own right (Schoen
et al., 1999). The instructional sequence follows a fourstep process labeled as launch, explore, share and
summarize, and apply. The “launch” sets the context
for what is to follow and consists of a class discussion
of a problem. The “explore” is usually a group or pair
activity in which students investigate the problems and
questions. “Share and summarize” brings the class
back together to discuss key concepts and methods.
“Apply” is time in which individual students practice
what they learned (Hirsch et al., 1995).
CPMP authors have conducted numerous and
extensive research into the effectiveness of CPMP. At
the end of Course 1, CPMP students' average score on
the Ability to Do Quantitative Thinking (a subtest of
the nationally standardized Iowa Tests of Educational
Development) was significantly higher (p < .05) than
algebra students in traditional curricula (Schoen &
Hirsch, 2003). At the end of Course 3, CPMP students
performed significantly better (p < .05) on concept and
Testing Problem-Solving Skills
application tasks but significantly poorer on algebraic
manipulation tasks when compared with Algebra II
students in traditional curricula (Huntely, Rasmussen,
Villarubi, Sangtong, & Fey, 2001). Using SAT 1,
mathematics scores of CPMP III students versus
Algebra II students showed no significant differences
(Schoen, Cebulla, and Winsor, 2001). When American
College Testing (ACT) Assessment Mathematics Test
means were used, the Algebra II students had
significantly higher scores (Schoen, Cebulla, and
Winsor, 2001). Using placement tests constructed with
items from a test bank from the Mathematical
Association of America, CPMP Course 4 (N=164)
versus Precalculus (N=177) students showed no
significant differences on algebraic symbol
manipulation skill but a significant difference in favor
of CPMP on concepts and methods needed for the
study of calculus (Schoen & Hirsch, 2003). Another
study examined proof competence, as well as
perceived need for proof using CPMP III students
versus traditional Algebra II students; no significant
differences were found (Kahan, 1999).
This research suggests CPMP is a curriculum that
is at least as effective as a traditional curriculum except
in some by-hand manipulation skills when these skills
are needed outside of application context and graphing
calculators are not allowed. In many areas (such as
problem solving in context and with graphing
calculators), CPMP students outperform traditional
students. The design of the CPMP curriculum suggests
a curriculum that is indeed in alignment with NCTM
recommendations. The research indicates CPMP is
effective in promoting the achievement of student
reasoning, communication, problem solving, and
representation. In traditional skills (such as by-hand
manipulation skills), the CPMP curriculum might be
less effective than traditional curricula. In less
traditional skills (such as problem solving in multiple
representations), the CPMP curriculum might be more
effective than traditional curricula (Schoen et al., 1999;
Schoen & Ziebarth, 1998a).
The Traditional Curriculum
There is, of course, no such curriculum as
“traditional curriculum” and as NCTM’s standards
become increasingly respected, even the most
traditional of curriculum begins to look like NCTMoriented
curriculum.
The
researcher
fully
acknowledges that there is no true dichotomy between
NCTM-oriented curricula and traditional curricula any
longer, but rather a continuum.
Carmen M. Latterell
For the purposes of this study, I chose to define the
traditional curriculum to not be one of the NSF-funded,
nor identify itself in any manner as reform. The
traditional curriculum should lack an emphasis on
groupwork
and
graphing
calculators,
while
emphasizing hand symbolic manipulations as the only
meaning of algebra. This approach moved the
researcher to the traditional end of the continuum, but
once there, could traditional curriculum be described in
and of itself?
The traditional curricula in this study have an
emphasis on separate units of mathematical content (in
our case, algebra), which includes an emphasis on
procedures (although conceptual understanding is
present as well). The teacher most often serves as the
"teller" of information (i.e., the students were not
engaged in discovery work). Students most often work
individually. Testing is usually easily accomplished
through short-answer or even multiple-choice items.
Computation is important. In our case, since these were
traditional algebra classes, the curriculum included
solving (by hand) algebraic equations with solution
processes not dependent on technology.
Students
A sample of five Iowa schools using CPMP agreed
to participate in this study. As previously mentioned,
within the state of Iowa, there were many schools that
had implemented CPMP. The researcher had access to
a list of all Iowa schools that had CPMP in place for
several years. A subset of these schools was identified
and contacted, so that the subset were schools spread
throughout Iowa. Five of these schools agreed to
participate. This resulted in 230 students.
The researcher also had access to existing ITED
data. This data was gathered for the schools that had
already agreed to participate. Another sample of Iowa
schools was gathered according to two characteristics:
The size and location of the schools had to match one
of the five already in the sample, and the mathematics
curricula had to be of a traditional nature. The existing
ITED data was used to narrow this sample to schools
whose average school score was the same as those
already in the sample. This resulted in a much smaller
set of schools, but out of this set, five did agree to
participate. This resulted in 320 students.
The Classroom Environment Form
At this point, the teachers from each classroom
completed a Classroom Environment Form (written by
the researcher) to provide data on such interests as the
textbook, availability of graphing calculators,
7
groupwork practices, and assessment practices (see
Table 1). The purpose of gathering this data was twofold.
One purpose was to supply background and
context for the discussion of the results. Although the
purpose of this study was not to compare CPMP
students to traditional students, it is through that
comparison that the researcher is able to distinguish
differences in testing results that may be created as a
consequence of NCTM-oriented curricula. However, if
the intended curriculum (whether that is CPMP or
traditional) is not the enacted curriculum, then the
worth of this comparison is questionable. (The
researcher acknowledges that there is also the achieved
curriculum.)
The Classroom Environment Form was also
intended to supply possible explanations for results. An
example might illustrate this. CPMP students work in
groups as part of the curriculum. If it turns out that
CPMP students work better under testing practices that
include groups, perhaps that is reasonable. However,
just because CPMP authors call for students to learn in
groups, do the students actually spend time in groups?
Further, perhaps the traditional students also learn in
groups, and thus groupwork is not really a difference
between the two curricula. Clearly the Classroom
Environment Form will not necessarily give a full
disclosure of a classroom. It will only give what the
teacher chooses to say.
The Standardized Test
A standardized problem-solving test, the subtest
from the Iowa Test of Educational Development
(ITED) titled Test Q: Ability to Do Quantitative
Thinking, was used. The stated purpose of the ITED is
to “provide objective, norm-referenced information
about high school students’ development in the skills
that are the long-term goals of secondary
education—skills that constitute a major part of the
foundation for continued learning” (Feldt et al., 1993,
p. 4). Test Q at level 15 (grade nine) consists of 40
multiple-choice items with five response options.
Students are given 40 minutes to complete the test.
The questions, based on realistic situations, are
“practical problems that require basic arithmetic and
measurement, estimation, data interpretation, and
logical thinking” ( Feldt, et al., 1993, p. 13). In
addition, some of the questions test more abstract
concepts. “The primary objective of the test is to
Table 1
Comparison of the Two Types of Classrooms
NCTM-oriented
Textbook
CPMP
Traditional
Algebra I Explorations and Applications, McDougal Littell,
1997 (3 classrooms)
Algebra I Applications and Connections, Macmillan/McGraw
Hill, 1992 (2 classrooms)
Graphing calculators
constantly available
One teacher answered constantly available. The remaining
teachers stated that they were available sometimes during
classes and on tests, but less frequently during homework.
Mean percent estimate for the amount of
class time that students are in groups
86%
22%
Groupwork outside of class
Four teachers encouraged students
to work in groups outside of class.
Three teachers encouraged their students to work in groups
outside of class.
Multiple-choice tests used
Never used
2 teachers sometimes used them.
Students receive partial credit for work
and explanations.
Students receive credit for writing out
reasons on graded assessments.
Students are encouraged to give reasons
and not just an answer.
All teachers said “always”.
All teachers said “frequently”.
All teachers said “always”.
All teachers said “frequently”.
All teachers said “always”.
All teachers said “frequently”.
8
Testing Problem-Solving Skills
measure students’ ability to use appropriate
mathematical reasoning, not to test computational
facility under pressure” (Feldt et al., 1993, p. 13).
The Constructed-Response Test
ITED has two parallel forms. One of the forms was
given intact as the previous test. To create the second
test, the remaining form was modified in the following
manner: The researcher converted each item to a
constructed-response item. For most items, this simply
meant eliminating the choices and keeping the stem.
For items in which the choices completed a sentence,
the item was changed to form a question. For a small
number of items, the item was completely reworded,
but the purpose of the item and the context remained
the same. The researcher scored the constructedresponse tests using a scoring key in which each
problem is worked including numerical calculations
and a verbal explanation.
The Performance Assessment
Due to time constraints, a subset of the sample
consisting of two classrooms of CPMP students and
two classrooms of traditional students solved problems
in pairs using non-routine items while the researcher
observed. This resulted in approximately 100 students.
The following two problems, each written out on
separate pieces of paper, were given to pairs of
students.
1. How many keystrokes are needed to put page
numbers on a paper of length 124 pages?
2. Three friends, returning from the movie Friday the
13th Part 65, stopped to eat at a restaurant. After
dinner, they paid their bill and noticed a bowl of
mints at the front counter. Sean took 1/3 of the
mints, but returned four because he had a
momentary pang of guilt. Faizah then took 1/4 of
what was left but returned three for similar
reasons. Eugene then took half of the remainder
but threw two that looked like they had been
slobbered on back into the bowl. (He felt no pangs
of guilt—he just didn’t want slobbered-on mints.)
The bowl had only 17 mints left when the raid was
over. How many mints were originally in the
bowl? (Herr & Johnson, 1994, p. 303)
There are various solution processes for each of the
questions. For example, on the first problem, the
student could count. Another method would be to
reason as follows. The page numbers 1 through 9 are
one digit each. Thus, nine keystrokes are needed. The
page numbers 10 through 99 are two digits each; thus 2
x 90 =180 keystrokes are needed. The page numbers
Carmen M. Latterell
100 through 124 are three digits each; thus 3 x 25 =75
keystrokes are needed. The sum of 9, 180 and 75 is
264, which is the answer.
For the second problem, one could use algebra,
producing a rather complicated, linear equation to be
solved. An easier approach is to begin at the end. There
were 17 mints left in the bowl. Just before that, Eugene
threw two back in, so there were 15. Before that,
Eugene took half of the mints and left 15 in the bowl.
There were 30 mints in the bowl before Eugene began.
Before that, Faizah put three mints back, leaving 30, so
that was 27. Faizah took 1/4, leaving 27. So, there must
have been 36 mints in the bowl before Faizah. Right
before Faizah, Sean returned four to the bowl. So, there
must have been 32. Sean took 1/3 of the mints, leaving
32, so there must have been 48. This brings us to the
beginning of the problem, in which there were 48
mints.
The CPMP students had been working in pairs
throughout the school year (this assessment was
conducted in May). The same pairs were used for this
assessment. The classroom teachers paired the
traditional students. The students were asked to work
on these problems and think out loud with each other
on how to solve the problems. In addition, the students
were asked to write out their solution process after they
were happy with their solution. It was emphasized to
the students that the researcher was not interested in a
numeric answer, but interested in the process that they
used when solving the problem. The students were
observed with the researcher taking notes, but not
intervening, and videotaped while solving these
problems. The field notes, annotated transcripts, and
student work were then analyzed for the quality of the
problem-solving strategies and processes, the overall
success of the students in solving the problems, the
quality of the cooperative work, and other emergent
themes in the process of problem solving.
Results with Discussion
Given that the students were members of intact
classes, it was determined that the individual student
should not be the unit of analysis in analyzing the
results from the standardized test and the constructed
response test. So, schools were used as the unit of
analysis. In comparing CPMP to traditional schools on
the standardized test, the traditional schools had a
mean proportion correct score of .52 (.06 standard
deviation) and the CPMP schools had a mean
proportion correct score of .52 (.05 standard deviation).
Clearly there is no significant difference between
whether a school was CPMP or traditional.1
9
The standardized test in the format of constructed
responses faired about the same. The mean proportion
correct score for CPMP schools was .65 (.13 standard
deviation) and traditional schools had a mean
proportion correct score of .67 (.14 standard deviation).
Again, there is no significant difference between
whether a school was CPMP or traditional.
On the performance assessment, considering only
the correctness of a solution, there again was no
difference between the CPMP students and the
traditional students. Most of the students, whether from
CPMP or traditional, were able to solve the typewriter
problem. A slightly higher number of CPMP students
than traditional were able to solve the mints problem.
This may be due to the fact that although the problem
was much easier solved just working backwards,
traditional students attempted to use algebra to solve
the mints problem. However, if one counts only a
numeric answer, on both problems, there was close to
equivalence between the two groups of students.
A performance assessment allows one to look at
other issues than just correctness of an answer. Five
themes emerged from the analysis of the performance
data (see Table 2).
For ease of reporting, the discussion of the themes
will be given in Polya's framework for viewing the
problem-solving process (Polya, 1945/1973). Polya
outlined these stages as: (1) getting to know the
problem, (2) forming a solution plan, (3) carrying out
the solution plan, and (4) looking back.
The first two themes occurred in Polya’s getting to
know the problem stage. CPMP students were more
engaged in the problems than the traditional students
were and CPMP students were more eager to work in
pairs than the traditional students were.
CPMP students immediately became engaged in
the problems. This was evidence in the words that
CPMP students used. “Do we get to work on these?"
was actually a question a CPMP student shouted with
excitement toward the researcher as soon as the
researcher handed her the problem sheets. "I bet I can
solve this" and "we are good at problems" were two
other comments that the researcher interpreted as
positive.
In addition to expressing interests and enthusiasm
for solving problems, the CPMP students talked with
each other about what the problems meant. For
example, on the typewriter problem, the majority of
CPMP pairs immediately began to act out what it
might mean to type numbers on pages of a book. One
CPMP student said to her partner, "See, I know, it is
that… well, 35, that would be a 2. See?", meaning that
the number 35 has 2 digits in it.
In analyzing the annotated transcripts for students
in the traditional curriculum, the researcher could not
find a single positive comment toward problem
solving. In addition, one traditional student told the
researcher, "We are really bad at math" which
promoted another to say, "He [the teacher] hates us."
Several pairs of traditional students stated that they did
not want to work the typewriter problem or that they
could not work the typewriter problem because they
did not understand what the problem was asking.
Several traditional pairs read the problem and then
stated, "I don't get it." However, on the mints problem,
many of the traditional students stated that they
probably could solve it, as it required algebra and they
were in an algebra course.
As students were getting to know the problems, the
cooperation aspect was the second theme that emerged.
Table 2
Five Themes of Student Problem Solving Activity
Theme
Description
Students’ engagement in problems
CPMP students were more engaged in the problems than the traditional students.
CPMP students engaged in the problem-solving process beyond a numeric answer.
CPMP students asked to discuss the problems after they were done solving them.
Students’ enthusiasm for working in pairs
CPMP students were more eager and able to work in pairs than traditional students.
Students’ use of symbol manipulation for their solutions
The traditional students considered and actually used algebraic techniques.
Students’ ability to communicate mathematically
CPMP students were able to write about their mathematical processes.
Students’ flexibility in their solutions
CPMP students looked for more than one solution path.
10
Testing Problem-Solving Skills
In terms of the cooperation aspect, the CPMP students
began cooperating as they were getting to know the
problem. The traditional students didn't cooperate until
possibly the looking back stage when they compared
answers. Traditional students became familiar with the
problems individually, and mostly inside their heads
(not readily observable behavior). Since, CPMP
students cooperated while getting to know the problem,
the researcher could observe some of the ways that
they interacted with the problem. For example,
sometimes they tried to act out the situation. Other
times they described the situations to each other and
tried to clarify by talking to each other what the
situation really meant. So, “getting to know the
problem” is trying to understand the situation, and
CPMP students did this cooperatively, while traditional
students did not.
To summarize, while the CPMP students discussed
the problem, the traditional students worked separately.
The only time the traditional students engaged together
as a pair was when they compared answers with each
other. This point was made obvious with many of the
traditional pairs literally telling each other to work on
the problem and "we will check our answers at the
end." A minority of traditional students was unwilling
to compare answers, however, saying, "Solve it
yourself."
The first two themes (CPMP students were more
engaged in the problems than the traditional students
were and CPMP students were more enthusiastic to
work in pairs than the traditional students were.)
continued during Polya’s (3) carry out the plan stage.
The CPMP students engaged in conversation with each
other about the problems, and actually worked on the
problems together. An excerpt from the annotated
transcripts might illuminate this. For the sake of this
example, the students will be labeled simply Student A
and Student B. The students are working on the
typewriter problem.
Student A: How many two-digit numbers are there?
Student B: I don't know. 11, 12, 13, … [mumbles] …
let's …
Student A: It will take too long to count them. 10, too.
Student B: Oh, yes. There are 10 through 100. 99.
There are 10 through 99.
Student A: So there are 89.
Student B: Yes, 89.
At this point, the students continued in the problem
and started to examine the number of 3-digit numbers
that existed. The reader might be worried about the 89
(knowing that the correct number is 90), but the reader
Carmen M. Latterell
is asked to be patient, as we will return to this same
pair later in this section.
In addition to the first two themes continuing,
another theme (Students’ ability to communicate
mathematically) emerged. The CPMP students were
able to write about their mathematical processes,
demonstrating
an
ability
to
communicate
mathematically. Only one-fourth of the traditional
students showed anything other than a numeric answer.
All of the CPMP students showed something other
than their answer, with most showing the steps and
thought processes of their solution. For example, on
the typewriter problem, many of the CPMP students
wrote out the number of 1-digit numbers, 2-digit
numbers, and 3-digit numbers, labeling each
respectively. On the mints problem, several CPMP
students wrote the words "work backwards" or just
"backwards" on their sheet. Other CPMP students
showed the backward progression of numbers, but did
not use the word "backwards." On the mints problem,
the traditional students who did write something down
(again this was about one-fourth of them) wrote out the
algebraic equation that they had developed. Although
some traditional students did work backwards and
solve the problem, they simply showed the answer in a
box. When the researcher asked them to show their
solution process, they said that there was nothing to
show.
A theme (Students’ use of symbol manipulation for
their solutions) emerged during (2) Polya’s forming a
plan stage. The traditional students were much more
likely than the CPMP students to think of using symbol
manipulation. In fact, the traditional students at times
seemed to skip the (1) getting to know the problem
stage and enter right into (2) forming a plan, with that
plan being to use algebraic symbols. The traditional
students felt that they could work the mints problem,
because they thought there would be manipulations of
symbols for doing so. The traditional students, unlike
the CPMP students, were very quick on the mints
problem to use symbol manipulation.
Polya’s final stage, (4) looking back, revealed the
final theme that emerged from the analysis of students
problem solving activity: student’s flexibility to have
different solutions. The CPMP students looked for
more than one solution path and they stayed engaged in
the process beyond a numeric answer. The fact that a
numeric answer did not signify "done" to the CPMP
students seemed important to the researcher. The
CPMP students seemed to value finding a variety of
solution paths, whether they lead to a numeric answer
or not. Yet, the numeric answer was viewed to be
11
valuable as well. In particular, the numeric answer was
viewed as a method of checking the various solution
paths.
So, the researcher was actually amazed to find that
after the CPMP students had written down a solution
strategy, the vast majority of them asked each other if
there was another way to solve the problem, and then
continued to work on the problem. Although some
wrote down more than one solution process, most did
not. Although the alternative solution paths did not
show up in writing, most CPMP students talked about
alternative solution paths. There was no evidence any
of the traditional students searched for more than one
solution. This process of finding a second solution path
on the part of the CPMP students seemed to serve two
purposes. It appeared to be a check on the numeric
answer, but it also appeared to be an aid for helping
each other if one of the pair did not understand the
previous process. In this manner, the numeric answer
was important, but not the end goal. CPMP students
repeatedly asked each other if they were in agreement
on a process.
An example returns us to the students who thought
that there were 89 two-digit numbers. This pair
actually decided to count the digits by starting with 1
and ending with 124. (Many traditional students
counted the digits, as well, but this was their only
solution process.) These students then discovered that
their first answer did not match their second answer.
The students then tried to go back over their solution to
find out where there might be a mistake.
Finally, when the CPMP students stated that they
were done, every pair asked the researcher to discuss
the problems with them. None of the traditional
students wished to see the solution processes for either
question even when the researcher offered to work the
problems. The CPMP students seemed to view part of
the problem-solving process as explaining to each
other and the researcher what they were thinking.
Thus, the problem continued even when the CPMP
students had exhausted their solution paths. One pair of
CPMP students asked the researcher if she knew
whether class time would be spent on the problems on
the following day. The student stated that she wanted
to show the class her solution and ask class members
what their solutions were. When the researcher told her
that actually she really did not know if more class time
would be spent on these problems, the student ran to
the chalkboard and began to act out what she would
tell the class if she was given the time. Her partner
watched patiently throughout this process, and even
put in comments at appropriate places. Then the
12
partner turned to the researcher and asked, "Should I
go to the chalkboard, too?" This enthusiasm for the
problem-solving process was present in all the CPMP
students to a more or lesser degree, and appeared to be
virtually absent in the traditional students.
While the paper-and-pencil tests show CPMP
students and traditional students being equivalent in
their problem-solving abilities, the performance
assessment paints a picture of CPMP students excelling
at problem-solving characteristics and traditional
students lagging considerably behind. Obviously, the
researcher is not suggesting that the results (especially
the performance assessment results, which were with
100 students and 4 classrooms, while the paper-andpencil results had considerably more students and from
10 schools) generalize to all CPMP students and all
traditional students. However, the reader is reminded
that the goal of this study is really not to compare these
two types of students, but rather to comment on what
aspects of students’ “problem solving ability” may not
be seen in testing. Clearly, the performance test
revealed aspects of problem solving that were present
in at least this sample of CPMP students that the other
tests did not reveal with the CPMP students. None of
the tests revealed these aspects of problem solving in
the traditional students. It could be that the traditional
students lacked these aspects of problem solving. Or, it
is possible that the traditional students had these
aspects of problem solving, but were unable to
demonstrate them under any of these testing situations.
This last case, however, seems unlikely to the
researcher.
This study suggests that it is time to examine how
we are testing. If indeed we test what we value, do we
value the problem-solving skills that are tested by the
paper-and-pencil tests in this study, for example? Or do
we value the problem-solving skills that became
apparent in the performance test? Do we value
engaging in the problem-solving process beyond a
numeric solution? Do we value solving a problem
more than one way? These abilities may indeed be
present while taking paper-and-pencil tests, but they
are difficult to measure. Of course, it is possible to give
a paper-and-pencil test with constructed-response
items, and tell students to find more than one solution
process. It is, however, an interesting observation that
without being told to find more than one solution
process, CPMP students were inclined to do so. An
implication to this study, then, is to call for an
adjustment to how researchers and even classroom
teachers
evaluate
NCTM-oriented
programs.
Standardized testing is not the best method when
Testing Problem-Solving Skills
seeking to document types of students’ mathematical
features (e.g., engagement) seen in this study.
However, the implications can take researchers in
the other direction, as well. Rather than a call to stop
using standardized tests, perhaps this study calls for a
revision of standardized tests. As stated in an earlier
section, there have been numerous studies on the
CPMP curriculum. Many of these use standardized
testing materials, perhaps because this area concerns
the public. Recall the mention of parental concerns in
the introduction to this paper. However, it might be
that it is time for testing and measurement experts to
create alternatives to the standardized tests, so that
researchers could attempt to evaluate CPMP and other
NCTM-oriented curricula from a new perspective. Of
course, researchers can write their own testing
instruments, but there are advantages (including
validity, reliability, and the standardization of the
testing itself) to using a standardized test. An
implication to this study, then, is to call for a massive
revision of standardized tests.
Although it may be an obvious point, it probably
should not go without saying that there is no question
the CPMP students had more experience with aspects
of problem solving than the traditional students. The
CPMP students were more used to working together,
writing about mathematics, discussing mathematics,
not necessarily valuing a numeric answer, and looking
for more than one solution. The CPMP students had
had more opportunity to work with a variety of
problems. If the traditional students had these
experiences, would they too have these skills? This
may be true, although it might be argued that if
traditional students had these experiences they would
not be referred to as traditional students. The
traditional students were quick to think of algebraic
techniques, and indeed the traditional students had
more experience with algebra. Perhaps this returns us
to a previous point: what it is that we value in problem
solving? This is not to suggest that we have an either/or situation, as there is nothing to prevent us from
including more traditional in reform, or including more
reform processes in traditional.
Summary
The problem-solving ability measured on the
standardized tests did not show differences between
CPMP and traditional schools. In spite of the absence
of statistically significant differences, there were
qualitative differences between CPMP students and
traditional students observed in the performance
assessment. For example, CPMP students appeared to
Carmen M. Latterell
be more engaged in the problem-solving process. They
looked for alternative solutions processes; they worked
together as a group; and they showed more steps on the
written work.
With this rather large sample, the CPMP students
were not at a disadvantage on standardized problemsolving tests, in the sense that they scored as well as
the traditional students. However, the alternative
assessment did reveal some aspects of the CPMP
students that the standardized tests did not. Regardless
of one’s interest in the qualities (such as students’
engagement, flexibility etc. discussed in emergent
themes) that the standardized test did not measure, it
appears to be the case that one can succeed on a
standardized test while being in the CPMP curriculum.
And this study leaves us with the implication that
perhaps it is time to reconceptualize standardized
testing, and what, as researchers and classroom
teachers, we really want to test.
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1
The use of school as the unit of analysis resulted in an
enormous loss of power. The question may be asked as to whether
the lack of significance observed here is due to this lack of power.
Two factors suggest that this is probably not the case. First, the
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14
Testing Problem-Solving Skills
The Mathematics Educator
2003, Vol. 13, No. 1, 15–21
Assessment Insights from the Classroom
Norene Vail Lowery
Reform efforts in mathematics education challenge teachers to assess traditional forms of assessment and to
explore and implement alternative forms of assessment. Empowering all students with mathematical literacy
demands methods of assessment that reflect and enhance the present state of knowledge about learning, about
teaching, about mathematics, and about assessment. This discussion highlights insightful perspectives on
assessment strategies and techniques currently being addressed and implemented. A cohort of middle school
mathematics teachers reveal their experiences and reflections in addressing current assessment practices and
ventures in innovative and alternative approaches to assessment.
The activity which is the subject of this publication was produced under a grant from the Texas Higher Education Coordinating Board and
the U.S. Education Department under the auspices of the Eisenhower Grants Program (ID# 97053UH), titled the University of
Houston/Southeast District (HISD) Mathematics Project for Teachers (Grades 6-8).
Assessment is the central aspect of classroom
practice that links curriculum, teaching, and learning.
(NCTM, 1995). In the Principles and Standards for
School Mathematics (NCTM, 2000) assessment is
designated as one of the six underlying principles of
mathematics education. The Assessment Principle
states: "Assessment should support the learning of
important mathematics and furnish useful information
to both teachers and students" (NCTM, 2000, p. 22).
The emerging theme in assessment reform is to do
more assessment than evaluation; to become assessors
rather than evaluators.1 The aim is better assessment,
not more. Standards were created to provide guidelines
to improve mathematics education and to value the
importance of alternative 2, as well as authentic3
assessment procedures and protocols.
Traditional forms of assessment have been utilized
in mathematics classrooms for many years. However,
reform efforts in mathematics education challenge
teachers to reconsider traditional forms of assessment
and to explore and implement alternative approaches.
Assessment of school mathematics is addressed in
some manner in all of the NCTM documents (1989,
1991, 1995, 2000). It is essential that mathematics
teachers be informed and proactive in addressing issues
of assessment in mathematics classrooms. In response
to the call for changes, a cohort of middle school
Norene Vail Lowery, Ph.D., is an Assistant Professor of
Mathematics Education in the Curriculum & Instruction
Department of the College of Education at the University of
Houston, Houston, Texas. Research interests include elementary
and middle school mathematics education, preservice and inservice
teacher education, assessment, and the integration of literature and
mathematics. Her email address is nlowery@uh.edu.
Norene Vail Lowery
mathematics teachers in a large metropolitan area in
Texas reflected on their current assessment practices
and ventures in alternative forms of assessment in the
classroom. These teachers were participants in a grant
focusing on the strengthening of mathematical content
knowledge, the improvement of instructional
strategies, and the implementation of new curricula
fulfilling national standards and state-mandated
guidelines. In the light of education reform along with
the looming accountability of state-mandated
guidelines, these teachers began to realize the vision of
achieving mathematical power for all students. This
discussion highlights middle school mathematics
teacher's new perspectives as they implemented
alternative assessment strategies and techniques.
Research Methodology
I have chosen to present my experiences with these
teachers as a case study, as it is better suited for this
context-specific inquiry (Lincoln & Guba, 1985). Data
collection from the grant's workshop participants
spanned an academic year. These participants were
fifteen middle school mathematics teachers from five
different middle schools in the same urban school
district. Grant workshops were held on an urban
university campus with site visits and observations
conducted at individual teacher campuses throughout
the year. The workshops were conducted as a graduate
level course with each teacher receiving three semester
hours of credit upon completion.
A hands-on/minds-on, standards-based (NCTM,
2000) approach to learning and teaching (model,
observe, discuss) was implemented in the workshops.
The goals of the course were to explore mathematical
representations
through
content,
instructional
15
strategies, and authentic assessment. In order to initiate
change in assessment strategies, workshop instructors
focused the middle school teachers on three objectives
while examining the following questions:
below were validated through multiple sources through
the data collection and analysis that occurred
throughout the year-long workshops.
1. Set goals for student learning: What are the
important learner goals? What types of problems
are students able to solve? What concepts and
principles should students be able to apply?
2. Build an instructional program that reflects
NCTM’s Principles and Standards (2000) and
appropriate state-mandated objectives, include
alternative and multiple assessments, and include a
system for documentation and reporting: What
strategies best assess student understanding and
achievement?
3. Continually review and share classroom
assessment and its effects: What collaborative
supportive system in the mathematics learning
community and within their school administrative
personnel must be developed to change, alter, and
improve the assessment in the classroom?
Assessment is sometimes viewed as simply a
numerical value, a scale of student achievement.
Values of grades may vary from state to state, from
district to district, and, yes, even from classroom to
classroom. What measures of learning are really
represented in the assigned and recorded values?
Reform efforts such as the Principles and Standards
for School Mathematics (NCTM, 2000) provide needed
guidance and direction in changing the face of
classroom assessment.
Implementing new ways of doing assessment is not
an easy task, but no longer can mathematics teachers
afford to rely strictly on traditional formats. Alternative
forms of assessment offer more opportunities to reveal
a student's perceptions and conceptions of
mathematical knowledge. Most forms of alternative
assessment ask students to perform, create, produce, or
do something; tap high-level thinking; and involve
problem-solving skills (see Table 1). These forms use
tasks that represent meaningful instructional activities,
involve real-world application, are scored qualitatively,
and require new instructional assessment roles of
teachers (Herman, Aschbacher, & Winters, 1992).
As the workshop instructor modeled new
assessment strategies during monthly workshops,
teachers had opportunities to explore the assessments
as learner and as teacher. Group activities involved
planning new classroom assessment and practicing
with peers before implementation in their own
classrooms. In these sessions, teachers reported that
change could be successfully implemented as they
appreciated the collaborative efforts involved in their
experiences.
The use of models and manipulatives in teaching,
learning, and assessment encourages alternative forms
of assessment. Teachers in this program learned that
alternative assessment encourages the use of active
hands-on learning. Learning experiences such as these
create classroom activities and learning environments
that are accessible to all students. The teachers in this
project investigated and implemented innovative
approaches to alternative assessment in their own
classrooms.
Participants engaged in tasks such as developing
mathematics, teaching mathematics, and designing and
implementing instructional materials as well as
alternative forms of assessment. As the workshop
instructor and researcher, the author developed
multiple perspectives through participant observations,
random interviews, journal entries, reflections, and
course artifacts. Required products from the workshops
were used as data sources, including written
assignments,
anonymous
session
evaluations,
assessment projects and surveys, classroom student
examples, and a presentation project as a final
assessment. Various workshop tasks included:
evaluation, synthesis, and implementation of teaching
strategies, learning strategies, and national standards;
participation in inquiry and discovery activities (as
learner and as teacher); and implementation of
alternative forms of assessment in participants’
classrooms.
Here, as instructor and lead researcher of the grant
program, I report findings of the research agenda as a
narrative to weave together the responses from the
cohort of teachers in a collective manner. This
narrative draws most directly on data taken from the
teacher responses in individual and small group
discussions on assessment topics. The trends reported
16
The Workshop Design
Assessment Insights
Table 1
Workshop Participants' Identification Of Alternative Forms Of Assessment
Product / Process
Journal writing /
writing prompts
Projects
Performance
assessments / use of
manipulatives
Purpose
to assess development of mathematical
concepts; writing activities show more of how
the students are thinking
to develop and apply concept/scoring rubrics
to demonstrate concept attainment through the
use and mastery of manipulatives
to provide insight into students’ mathematical
thinking
Problem solving
to motivate interest; to promote critical
thinking
Diagnostic activities to determine student readiness for learning a
particular concept
Class discussion
to assess learning informally
Student conferences / to assess student’s ability to relate subject to
conversations
areas outside classroom
Classroom challenges to motivate interest and assess learning
Integration with other to use projects to integrate other subjects and
subjects
involve a variety of math concepts
Rubrics
to customize assessment to individual needs of
tasks
Questioning
to understand depth of students’ understanding
combined with
through questioning during activities
instruction
Cooperative learning
groups
Workshop Participant Insights
Journals are quite informative, but very time consuming. … [Yet] more
informative than just a number on a piece of paper. Another method,
which I have not used, is self-evaluation.
Turning in performance assessments to mathematics department (in our
school district, this is a regular procedure). [This is done] so they can
analyze concepts being taught and relate them to curriculum standards.
Manipulatives, exploration through questions, make them work on
becoming independent thinkers, bring outside experiences into the
classroom. Show the application or necessity. Try to assure students they
can do the work if they just try. Not looking for 100% accuracy, but risk
taking. Talking to and challenging students, competitions – mini games.
Concept mapping
Portfolios
Warm-ups
Homework
Tests:
Term – Cumulative
Short answer
Standardized &
state-mandated
Practice
Quizzes – pre & post
Notebooks
Norene Vail Lowery
to represent learning through samples of
student work
to assess problem solving strategies; to get
students into a mathematical framework
to assess transfer of learning; to demonstrate
application; to look for understanding and
comprehension of objectives
to compare for growth from previous tests
to develop higher order thinking skills
to assess mastery/transfer of concept and
skills; to track yearly growth
as a diagnostic
to diagnose areas that cause difficulty
… allows the student and teacher to see the objectives mastered and
objectives not mastered, particularly TAAS objectives.
I believe a rubric is most informative, but as a new teacher, I find it very
difficult to manage that in a classroom of students vying for attention.
Cooperative grouping/learning encourage students to help each other,
games for competition, discovery learning/exploration, problem solving,
and logic thinking activities.
Cooperative groups [are being used]. When student do projects, they
really enjoy working in groups and like to do their best.
Problems [that I have] encountered [include] students do not study, do not
do homework, do not ask questions. These are addressed in parent
conferences and student-teacher conferences. [However], some kids
realize objectives, concepts or skills they need to work on. For “lower”
level kids, it [TAAS] may serve as a motivator, but for especially bright
students, TAAS tends to limit what a teacher teaches. It tends to bring the
high lower, and may frustrate the low into apathy.
We displayed TAAS benchmark results visually [bulletin board displays]
so students could see where they stand related to other students.
[I] try to align TAAS objectives with the TEKS. Post results of TAAS
benchmarks…the students love to compete with each other. Since I work
in a low-performing school, TAAS gets emphasized over preparing the
students for algebra, something which is hard to resolve. Sometimes
students don’t see any correlation between what they are learning in class
and what is tested. I think in a way it has discouraged studying.
to assess student’s study habits, completion of
classwork, and concept understanding
17
The Results: Insights from Teachers
The series of workshops was conducted over the
academic year and provided teachers opportunities to
implement, observe, and revise many assessment
strategies. The workshop design created assessment
strategy experiences for the teachers in planning,
practicing with peers, implementing in their own
classrooms, and collaborating through thoughtful
mathematical discourse – both positive and negative
experiences were shared. As the instructor, I worked to
create a community of mathematics learners and
leaders by emphasizing these communication
opportunities. This interactive professional dialogue
was created and supported by access to peers and the
instructor during monthly grant sessions as well as via
a website between sessions. At the end of the grant
program, each teacher responded individually and in
small groups to a variety of assessment topic issues.
These teacher’s responses are presented in this section
in italics, enclosed and in boxes. These reflections-inthe-moment are direct citations from the teachers'
responses.
What Assessment Tasks were Explored
Implemented by the Middle School Teachers?
and
Teachers used a variety of traditional and nontraditional approaches to student assessment. The
teacher-developed chart (Table 1) represents the span
of assessment strategies explored and currently in
practice in the classrooms of these middle school
teachers. The third column indicates the teacher’s
reflections on the uses of these assessment strategies.
Teachers found that short answer tests, journal
writing, manipulatives, projects, concept mapping, and
performance assessments revealed a broad range of
capability, understanding, and communication of
mathematical concepts. Many different tasks were used
to create a complete picture of the students'
mathematical knowledge. Strategies for evaluating
performance on assessment activities also varied.
Teachers used rubrics quite extensively, as they
became comfortable with this system through the
workshops. In addition, concept maps, journal entries,
textbook assignments, and worksheets were very
informative. Teachers identified sources of feedback
such as group grades, participation grades, praise, peer
evaluation, and self-evaluation.
18
What Mathematics was Assessed? How Did State and
National Guidelines and Accountability Affect
Assessment Strategies?
The mathematical skills and concepts assessed by
the middle school teachers in the workshop were
typical for grades 6-8. As with many other states,
Texas has state-mandated curricula objectives, as do
many districts. Texas guidelines are called the Texas
Essential Knowledge and Skills (TEKS). For each
grade level and each subject area, there are specific
learning objectives and goals for Pre-K through 12th
grade. These curricula guidelines are correlated to the
statewide student test, the Texas Assessment of
Academic Skills (TAAS).4 The TAAS test is taken
based on grade level and subject matter. The final
TAAS is an exit test that must be passed as a
prerequisite for high school graduation. Within this
framework of curricula are thirteen TAAS objectives
that are assessed in mathematics. These have been
determined by the state, but are also related to the
national standards identified by the NCTM (cf. 2000).
These objectives were created to help ensure quality
and consistency. Learning accountability, in some
school districts, is even more defined by specific
objectives and goals for the grade levels.
[I use a] mastery tracking sheet, standardized tests,
TAAS, computer programs, independent practice
manipulatives, projects, and worksheets. We also review
and practice test-taking strategies. I feel this has helped
students become a little more confident because they at
least know what to expect.
Direct test preparation for the TAAS is
widespread. Many teachers used the item analysis from
the previous year's TAAS test to determine the areas of
strengths and weaknesses to improve on the objectives
that were deficient. Practice tests, six-week tests,
quizzes, and a section of the student's daily homework
are formatted so students can practice on how the
questions are structured as well as practicing and
applying the objectives. Teachers and students review
and practice test-taking strategies to develop more
confidence. Many of the workshop teachers felt that
too much focus was placed on the standardized test,
thus limiting the time available for alternative
assessments. Even so, teachers valued the need for
change and explored the potential of other forms of
assessment. Tutoring, motivation techniques, and
parental involvement were common efforts.
Assessment Insights
How Can Information from Alternative Assessment be
Integrated into Grading and Reporting Progress?
Weekly reports to parents; scheduled progress reports;
promoting ways parents can help students at home;
tutorials after school; TAAS data used to group students
by abilities work with parents on skills students need;
math make-and-take sessions [as a] parent workshop;
parent conferences; phone conferences/conversations;
parent involvement in schools; display example work for
the school; student/teacher conferences; Saturday school;
students are able to track themselves by objective using
TAAS data; award certificates; honor pictures taken and
put on the wall; and, TAAS classes.
… kids grade in groups. [I use] class participation grades.
The teachers shared strategies to integrate
information from alternative assessment into grading
policy. Alternative assessments were sometimes
counted as a test grade and sometimes as a daily grade,
depending on how much time was required. For
example, some teachers used notebooks as test grades.
It was common for teachers to offer extra credit
opportunities when implementing new forms of
assessment. Extra points were given for creativity and
originality, hoping to build student confidence. Most of
these teachers used homework to determine the depth
of student understanding and which concepts needed
re-teaching. Projects and journals offered students
opportunities to express their ideas, understanding, and
concerns. Some students worked better with
manipulatives; others with pen and paper.
The teachers reported a creative variety of
alternative forms of assessment implemented into
traditional protocol. Each type of assessment
determined a certain percentage of the grade.
Discussion of the variety of assessment practices and
grade recording encouraged all teachers to try more
alternative forms of assessment as well as developed
increase confidence in this endeavor.
The teachers communicated the types and
importance of assessment strategies and approaches to
students and parents through many venues. Some of
the ways used by the teachers include weekly reports to
parents, scheduled progress reports, promoting ways
parents can help students at home, tutorials after
school, and Saturday school. As a result, parents and
teachers participated in workshops, conferences, and
conversations to encourage and support student
learning. Positive reinforcements included special
privileges at school and at home, award certificates,
and other classroom and school acknowledgements.
Norene Vail Lowery
Teachers reported that sometimes students do not
see any correlation between what they are learning in
class and what is tested. Teachers tried to address these
issues by using real-world problems and scenarios.
Typical problems encountered involved students that
do not study or complete homework, or that do not ask
questions. These were addressed in parent conferences
and student-teacher conferences. Through these many
approaches, students were able to ask questions about
concepts they had not mastered.
What Results Did the Teachers See as they Used
Assessment to Improve Curriculum And Instructional
Practices?
I use concept maps, journal entries, textbook assignments,
and worksheets. These methods are very informative [One
method I use is to have the] whole class solve their
problems, [but] only take one solution from [the entire]
class on chalkboard. This encourages total class
collaboration, a step beyond small group work.
I try to celebrate different learning styles. Let students
explain to me what was just taught, if they are having
difficulties this means that I have to use another strategy.
Assessment does alter instruction.
The teachers studied and shared strategies to
improve mathematics curriculum and instructional
practices. They found that different assessment
instruments helped to take the focus off the
"computation and accuracy" aspect of mathematics,
and helped to encourage mathematical thinking. New
sorts of tasks in classrooms created a more complete
picture of the students' mathematical knowledge. The
workshop teachers reported that assessment informed
re-teaching, addressed students with math anxiety, and
identified students' need for more instruction and/or
reinforcement. Students were able to see the objectives
mastered and not mastered, as well as their own
strengths and weaknesses. Alternative assessment took
the emphasis away from right/wrong answers and
concentrated students and teachers on thought
processes.
What Assessment Encouraged Mathematics Learning?
While addressing curricula objectives, the teachers
made high priority of planning relevant activities that
connect mathematics with the real world and creating a
rich learning environment. The teachers tried
innovative approaches and teaching strategies to
address the mathematical content in a hands-on, mindson manner. Teachers used a variety of assessment
approaches in a traditional and non-traditional manner
19
for student assessment. Different learning styles were
more easily addressed by alternative assessment.
These teachers developed and implemented some
effective approaches to alternative assessment that
fostered student learning and helped to address
motivation concerns. Some strategies that encouraged
students to learn math were: doing extra credit
assignments, using peer tutoring, valuing classroom
discourse, and finding ways to justify their answer.
Teachers used manipulatives and exploration through
questioning to assist students in developing as
independent thinkers. Showing the application and
necessity of mathematics while bringing in real-world
scenarios was also an effective and valuable strategy.
Teachers reassured students that they can do the work
and encouraged risk taking. Students developed selfconfidence as they were asked to provide their opinion
on problems in classroom discourse and in writing.
This created a safe learning environment more
conducive to learning. Motivation appeared to be the
ultimate goal for ensuring student encouragement and
interests. Challenging students with competitions and
games was a good motivator for the middle school
student. Teachers also reported that cooperative
grouping encouraged students in problem solving and
logic while they learned to help each other. Teachers
encouraged students to justify why they did what they
did, focusing on the thinking processes rather than just
the answer.
Teachers confronted their own perspective of the
nature of mathematics by participating in learning
activities that encouraged deep reflection and
discourse. Davis, Maher, & Noddings (1990) believe
that this perspective has a direct bearing on the ways
reform can be approached. Unveiling or developing
one's own conception of the nature of mathematics was
an enlightening experience that promoted a deeper
understanding of reflective teaching and learning
mathematics. Teachers developed a better conceptual
understanding as they explored mathematics topics as
learners and teachers to better inform instruction and
assessment. Teachers examined and explored reasons
for evaluating and assessing student achievement.
Being aware that teachers evaluate and assess in order
to enable decision-making about mathematics
instruction and classroom climate was a critical aspect
of these teachers' learning. The protocols presented
above communicate important tensions for the middle
school mathematics teachers among testing
expectations, assessment of student understanding, and
the need to assign grades.
Appreciating the need for reform was another area
of study for the teachers. For the teachers, this meant
acknowledging that current testing procedures are
inadequate and realizing the need for further research.
Through the workshop experiences and the teachers'
own personal classroom action research, teachers
discovered why there is a need for reform in
assessment. It was apparent that using multiple
assessment strategies was a significant step toward
creating a more complete picture of the student's
mathematical understanding and achievement. New
evaluation models and technologies that utilize
assessment procedures that reflect the changes in
school mathematics are needed. Ultimately, the middle
school teachers demonstrated a belief that classrooms
should be active learning environments where
instruction is interactive and multiple forms of
assessment are interwoven with teaching.
Conclusion
Looking Forward
The teachers’ learning experiences focused on
developing and promoting better classroom
assessment. Initially, the teachers explored the recent
trends in changes from behavioral to cognitive views
of learning and assessment, as well as changes to
authentic, multi-dimensional, and collaborative
assessment. Teachers learned about the constructivist
perspective of teaching and learning school
mathematics that is predominant in the NCTM
Standards documents (NCTM, 1989, 1991, 1995,
2000).
Multiple forms of assessment are being advocated
as we come to understand that traditional means of
assessment have not addressed the needs of all
learners. Richard Stiggins estimates that educators
spend about a third of their time involved in
assessment-related
activities
that
guide
the
instructional and classroom decisions which directly
affect learning (1993). A time investment such as this
demands that teachers examine their current
assessment practices. Simply testing student
[Using] assessments that make them have a feeling of selffulfillment, to develop confidence... [have] students write
their opinion on a problem, ... [and] let students show
different ways of solving a given problem, let them justify
why they did it.
It is a slow process to get our students to do in-depth
work. I have not quite figured out the right formula to
motivate them. Motivation is a real challenge for me.
Students don't seem to have the confidence to try. I'm
working at it...
20
Assessment Insights
achievement with traditional instruments and protocols
is insufficient. Empowering all students with
mathematical literacy demands methods of assessment
that reflect and enhance the present state of knowledge
about learning, teaching, mathematics, and assessment.
Implementing improved assessment in the mathematics
classroom begins with combining instruction with
assessment to better meet the needs of the learner.
In order to plan and implement new strategies for
assessment, mathematics teachers should have
opportunities for professional development, as did
these middle school teachers. It is crucial that a support
system in the mathematics learning community be
developed along with any efforts to change, alter, and
improve assessment in the classroom. Mathematics
teachers must personally explore alternative
assessment strategies. They should be involved in
creating and implementing tasks that are exemplars of
mathematics instruction as envisioned by the NCTM.
As part of this effort to develop tasks, teachers should
have opportunities to observe students doing
mathematics and to examine the their products. A solid
basis for mathematics teaching, learning, and
assessment is created when teachers value and
comprehend recent trends, perspectives towards
mathematics teaching and learning, evaluation and
assessment, and the need for reform. The informed
mathematics teacher has the ability and the tools to
offer the best learning environment for improving
student achievement and understanding.
In this paper, I have attempted to present a multiperspective approach toward understanding and
implementing assessment reform. The middle school
mathematics teachers encountered many problems on
this journey from traditional classroom assessment to
implementing alternative assessment strategies. Some
problems were unique, but many were common among
all teachers. Some problems were collectively
resolved, while others, such as student motivation,
remain as ongoing obstacles to address. These teachers
learned about assessment and implementing innovative
strategies in a collaborative environment. As a result,
the need for a strong support system to implement
change was revealed and valued. The experiences and
insights of these teachers may promote and encourage
other middle school mathematics teachers to move
outside the comfort zone of traditional assessment
protocols and begin implementing innovative and
alternative approaches to assessment.
Norene Vail Lowery
REFERENCES
Davis, R., Maher, C., & Noddings, N. (1990). Introduction. In R.
Davis, C. Maher, & N. Noddings (Eds.), Constructivist views
of the teaching and learning of mathematics. Journal for
Research in Mathematics Education Monograph no. 4 (1-3).
Reston, VA: National Council of Teachers of Mathematics.
Herman, J., Aschbacher, P., & Winters, L. (1992). A practical
guide to alternative assessment. Alexandria, VA: Association
for Supervision and Curriculum Development.
Lincoln, Y., & Guba, E. (1985). Naturalistic inquiry. Newbury
Park, CA: Sage.
National Council of Teachers of Mathematics (1989). Curriculum
and evaluation standards for school mathematics. Reston,
VA: Author.
National Council of Teachers of Mathematics (1991). Professional
standards for teachers of mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (1995). Assessment
standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (2000). Principles
and standards for school mathematics. Reston, VA: Author.
Stiggins, R. (1993). Teacher training in assessment: Overcoming
the neglect. In S. Wise (Ed.), Teacher training in assessment
and measurement skills (27-40). [Buros-Nebraska Series on
Measurement and Testing]. Lincoln, NE: Buros Institute.
1
Drawing upon NCTM's Assessment Standards for School
Mathematics (1995), I make the following distinction between
assessment and evaluation. Evaluation is the process of determining
the worth of, or assigning a value to, something based on careful
examination and judgment. Assessment (as a noun) is used to
emphasize understanding and description of both qualitative and
quantitative evidence in making judgments and decisions.
Assessment (as a verb) is the process of gathering evidence about a
student’s knowledge of, ability to use, and dispassion toward
mathematics and making inferences from that evidence for a
variety of purposes. Evaluation is one use of assessment
information.
2
“Alternative”, as well as “traditional”, forms of assessment
may be less clearly demarked. There are common characteristics in
alternative assessment, most ask students to perform, create,
produce, or do something; tap higher-level thinking and problemsolving skills; use tasks that represent meaningful instructional
activities; involve real-world application; are scored qualitatively;
and, require new instructional and assessment roles of teachers.
Traditional assessment efforts seem skill or process-oriented, such
as common practices of end-of-the-unit testing. These efforts
present a clear line of distinction between instruction and
assessment.
3
Authentic assessment is a type of alternative assessment,
emphasizing practices that are relevant, real-world and focused on
meaningful learning.
4
Beginning in 2003, this test has been renamed the Texas
Assessment of Knowledge and Skills (TAKS).
21
The Mathematics Educator
2003, Vol. 13, No. 1, 22–32
Designing and Implementing Meaningful
Field-Based Experiences for Mathematics Methods Courses:
A Framework and Program Description
Amy Roth McDuffie, Valarie L. Akerson, & Judith A. Morrison
Performance-based approaches to learning and assessment are consistent with goals for standards-based
instruction and show promise as a vehicle for teacher change. Performance assessment involves students
participating in an extended, worthwhile mathematical task while teachers facilitate and assess their learning.
We designed and implemented a project in an elementary mathematics methods course in which preservice
teachers developed performance assessment tasks and then administered these tasks in K-8 classrooms. We
present our guiding framework for this project, the project design, and the teaching and learning experiences for
project leaders and preservice teachers. Recommendations and reflections are included for others intending to
implement similar projects.
This article is based on a paper presented at the Association of Mathematics Teacher Educators Annual Conference, Costa Mesa, CA,
January 18 – 20, 2001.
The current reform movement in mathematics
education is based on national and state standards for
students’ learning (e.g., National Council of Teachers
of Mathematics [NCTM], 1989, 2000) and on the
perspective that students learn by actively constructing
their own knowledge and understandings. Within this
context, educators and researchers have identified two
different but complementary needs for preservice
teacher education in mathematics. First, preservice
teachers need to learn to use performance assessment
strategies to effectively meet and assess standardsbased learning objectives. Second, preservice teacher
learning should be situated in classroom practice to
facilitate their pedagogical knowledge constructions
and their enculturation into a community of practice.
Below we discuss each of these needs for preservice
teacher education, and then we describe the program
we have developed to meet these needs.
Call for New Forms of Assessment
Following the release of NCTM’s Curriculum and
Evaluation Standards (1989), many states and local
Amy Roth McDuffie is an assistant professor of mathematics
education at Washington State University Tri-Cities. Her research
focuses on inservice and preservice professional development
toward Standards-based practices. Her email address is
mcduffie@tricity.wsu.edu.
Valarie Akerson is an assistant professor of science education at
Indiana University. Her research focuses on elementary teacher
and student conceptions of nature of science.
Judith Morrison is an assistant professor of science education at
Washington State University Tri-Cities. Her research interests
focus on teachers' diagnosis of students' science conceptions.
22
school districts have developed standards for students’
learning in mathematics. Included in these standards
and in NCTM’s updated standards, the Principles and
Standards for School Mathematics (PSSM) (NCTM,
2000), are greater emphases on the processes of doing
mathematics (e.g., problem solving and reasoning) and
on communicating thinking and solution strategies
(NCTM, 1989, 2000).
Also included in these standards is a call for new
forms of assessment. Traditional paper and pencil
classroom tests and standardized multiple-choice tests
focused on recall of facts and basic procedures do not
effectively measure what is valued for standards-based
learning (Darling-Hammond & Falk, 1997; Shepard,
2000). While traditional measurement approaches to
assessment were once aligned with the instructional
practices of a century past, these approaches are not
consistent with current teaching and learning goals
from a social constructivist perspective (Shepard,
2000). This incongruity has resulted in an emerging
paradigm for assessment that involves teachers’
assessment of students’ understandings and students’
self-assessments as part of the social process of
knowledge construction (Shepard, 2000). Educators
and researchers argue that to align assessment with
standards-based learning, the following changes are
needed: (a) the form and content of assessments must
represent
higher
order
thinking,
reasoning,
communication, problem solving skills, as well as a
conceptual understanding of subject matter; and (b) the
focus of assessment policy needs to shift to using
assessment for learning (Borko, Mayfield, Marion,
Meaningful Field-Based Experiences
Flexer, & Cumbro, 1997; Darling-Hammond & Falk,
1997; Shepard, 2000).
Consistent with these views, in mathematics
education the PSSM state that the primary purpose of
assessment should be to “support the learning of
important mathematics and furnish useful information
to both teachers and students…. Assessment should be
more than merely a test at the end of instruction to see
how students perform under special conditions”
(NCTM, 2000, p. 22). To achieve this goal, the
Standards call for embedding assessment in instruction,
rather than keeping assessment as separate from
learning (NCTM, 1995, 2000). Indeed, this call is
supported by research that indicates use of formative
assessments, the continual assessment of learning
throughout an instructional sequence, in instruction
enhances student learning (Black & Wiliam, 1998).
Performance Assessment to Improve
Teaching and Learning
As a result of this call, attention has been directed
to more authentic forms of assessment, including
performance assessment (PA). Indeed, well-designed
PA tasks can assess student understanding as well as
teach concepts as a formative assessment (DarlingHammond & Falk, 1997; Shepard, 2000). While a
single definition for PA does not exist, Stenmark’s
(1991) definition for PA in mathematics education
seems to capture the important aspects of this
approach. Stenmark states, “A performance assessment
in mathematics involves presenting students with a
mathematical task, project, or investigation, then
observing, interviewing, and looking at their products
to assess what they actually know and can do” (1991,
p. 13).
Educators and researchers argue that the
advantages of classroom based performance
assessment are that they provide the opportunity to:
1. Examine the process as well as the product and
represent a full range of learning outcomes by
assessing students’ writing, products, and behavior
(Danielson, 1997; Shepard, Flexer, Hiebert,
Marion, Mayfield, & Weston, 1996).
2. Situate tasks in authentic, worthwhile, and/or realworld contexts (Stenmark, 1991).
3. Preserve the complexity of content knowledge and
skills (Shepard et al., 1996).
4. Assess higher-order thinking skills and deeper
understandings (Firestone, Mayrowtz, & Fairman,
1998).
5. Embed assessment in instruction, rather than
separating it from learning (Stenmark, 1991).
Amy Roth McDuffie, Valarie L. Akerson, & Judith A. Morrison
6. Apply criterion referenced as sessment approaches
based on important learning outcomes, rather than
norm-referenced (Stenmark, 1991).
Early research indicated that using performance
assessment in instruction can improve student learning
and teaching. Fuchs, Fuchs, Karns, Hamlett, and
Katzaroff (1999) studied the effects of classroom based
performance-assessment-driven instruction. They
found that students in PA-driven instruction classes
demonstrated stronger problem solving skills than
comparison groups that were not PA-driven. Shepard
et al. (1996) found that the teachers involved in their
study were beginning to show substantial changes in
practice. The changes included: greater use of
manipulatives; increased emphasis on the teaching and
learning of problem solving strategies; and increased
class time and focus on written explanations in
mathematics. Similarly, in Borko et al.’s (1997) study
of a professional development program on using
performance assessment strategies in mathematics
instruction, they found that their teachers changed their
instructional practices to incorporate more problem
solving activities, student explanations of strategies as
a central component of their programs, and scoring
rubrics for assessing students’ solutions of open-ended
tasks. These changes all represent a shift towards
standards-based instruction. Given that these studies
indicated that work with PA served as a vehicle for
change for inservice teachers, we pursued a program
for preservice teachers that focused on PA as a means
of building their understanding of standards-based
practices.
While it is possible to derive many instructional
benefits from PA strategies, it is not clear that teachers
can easily or quickly learn to implement these
strategies in practice. Firestone, Mayrowetz, &
Fairman (1998) studied teachers where state testing
programs included PA tasks, and therefore teachers felt
compelled to use PA in instruction, however, little
change in instructional strategies resulted. Firestone,
Mayrowetz, & Fairman identified two major barriers to
change: a lack of the sophisticated content knowledge
required in implementing PA approaches, and a lack of
rich tasks and problems in curricular materials to
support this approach to instruction. Firestone,
Mayrowetz, & Fairman concluded that to effectively
implement performance assessment and thereby realize
the potential for improved student learning, teachers
needed substantive training opportunities (not just new
policies requiring new assessment approaches) and
new curricular materials that are aligned with
23
performance assessment strategies and a standardsbased vision for teaching and learning.
In accordance with Firestone, Mayrowetz, and
Fairman’s (1998) research, Borko et al. (1997) found
that
substantive
and
sustained
professional
development is needed for teachers to effectively use
and realize the benefits for PA approaches. Their
research also indicated that time was a major obstacle
to implementing PA approaches. In particular, time
served as a barrier in planning for the implementation
of new strategies; applying more complex scoring
rubrics in assessment; administering the assessment
tasks; recording observations of students’ working and
thinking as part of the assessment; and interviewing
students before, during, and after the assessment. For
successful change to occur, teachers need time to
implement new assessment approaches.
Recognizing the value of PA and the complexity of
using these strategies, we decided to make PA a focus
of our mathematics methods course. This decision was
part of our effort to prepare our preservice teachers
from the beginning of their careers to use these
approaches and to implement standards-based teaching
and learning in their own instructional practice. While
we view performance assessment as one form of
alternative assessments (cf., Stenmark, 1991), it allows
the opportunity for preservice teachers to implement
other forms of alternative assessment (e.g., brief
interviews with students and systematically observing
students) while students perform a task. Additionally,
the nature of performance assessment (focusing on the
process and product of doing mathematics), pushes
preservice teachers to think deeply about how students
think about and do mathematics. Performance
assessment also provides an approach for preservice
teachers to use in which assessment is part of
instruction, a primary focus of the PSSM. That is, tasks
facilitate students’ learning of content and processes
through meaningful problems while teachers assess
their work and products. Moreover, as is described
throughout this paper, the process of designing and
implementing a performance assessment task provided
us, as teacher educators, the opportunity to assess the
performance of the preservice teachers; consequently
the preservice teachers experienced performance
assessment as students while they designed and
conducted performance assessment with their students.
Situated and Constructivist Perspectives
on Teacher Learning
With the goal of developing preservice teachers’
abilities to implement PA in their classrooms, we
24
considered a second need identified in teacher
education literature: a need to situate preservice teacher
learning in classroom practice. Borko et al (1997)
emphasized the importance of this approach for
professional growth. They found that a key component
of their program was their teachers’ ability to
experiment with and implement the ideas of the
professional development workshops in their own
classroom practice and then to reflect on these efforts
in follow-up workshops.
This finding is consistent with the perspective of
teacher learning put forth by Putnam and Borko
(2000). They argue that for teachers to construct new
knowledge about their practice the learning needs to be
situated in authentic contexts. First, learning needs to
be situated in authentic activities in classrooms to
support transfer to practice. For preservice teachers, a
combination of university learning for theoretical
foundations and school-based learning for a situated
perspective is needed (Putnam & Borko, 2000).
Second, preservice and inservice teachers should
participate in discourse communities as part of learning
and enculturation in the profession. In particular,
preservice teachers need to learn about and contribute
to a community’s way of thinking (Putnam & Borko,
2000). This process of enculturation is especially
important to future teachers of mathematics because
many come to their education program with limited
views of teaching, learning, and doing mathematics
(Roth McDuffie, McGinnis, & Graeber, 2000).
Putnam and Borko (2000) recognize that
implementing this perspective in teacher preparation
programs can be problematic. While we want to place
preservice teachers in schools to experience the
activities of teaching as part of their learning, K-12
placement classrooms may not embody the kind of
teaching and learning advocated in university
classrooms and/or these kinds of classrooms may not
be available. Moreover, the pull of traditional school
culture is strong, and these traditions make it difficult
for student teachers to implement different approaches
and views (Putnam & Borko, 2000).
Smith (2001) discusses specific approaches for
situating teachers’ learning in practice based on a
synthesis of the literature. We incorporated two of the
approaches she recommends: using “samples of
authentic practice” (p. 9), and designing our project
around “the cycle of teachers work” (p. 10). The first
approach involved selecting an example of a
mathematical task with a set of carefully chosen
student responses. Teachers complete the task and
engage in doing mathematics as learners. Next,
Meaningful Field-Based Experiences
teachers analyze the task and a range of students’
responses, examining understandings, approaches, and
misconceptions in students’ thinking and work. The
second approach is intended to mirror the nature and
cycle of teachers’ work. This cycle begins with
planning for instruction by targeting learning goals,
considering students’ prior knowledge, and selecting
and/or designing experiences that will promote
students’
construction
of
knowledge
and
understandings. The cycle continues as teachers enact
the plan, making adjustments in the plan and
instructional decisions to meet students’ needs while
formally and informally assessing students’ learning.
Teachers complete the cycle as they reflect on the
teaching and learning experience, and use their
reflections to guide future instruction. In the next
section we describe how these ideas were incorporated
in our program.
Program Description
We first implemented our PA program in an
undergraduate mathematics methods course at
Washington State University Tri-Cities in Spring 2000
and have continued the program in 2001 and 2002.
This description focuses on the initial implementation.
While the program has changed slightly each year with
changes in university faculty, most of the core
elements have remained the same, and the revisions
and adjustments made over the two years will be
discussed at the end of the article. This methods course
focused on mathematics teaching and learning at the
K-8 level. The PA program was included as part of a
one-semester mathematics methods course that met for
three hours, once each week of the semester. Twenty-
two preservice teachers were enrolled in the methods
course, with 18 being between the ages of 20 and 24
and the remaining 4 being second-career students. The
PA program aimed to provide a learning experience
with both a university component to build theoretical
foundations and a field-based component to situate
learning in the authentic context of the school
classroom, as recommended by Putnam and Borko
(2000) and Smith (2001). The primary goals for
preservice teacher learning in this program were:
1. To develop skills and habits of mind for assessing
and diagnosing students’ mathematical thinking,
skills, understandings, and lack of understandings;
2. To understand issues of and strategies for
implementing
classroom-based
performance
assessment;
3. To have a meaningful field-based experience
including an opportunity to collaborate with expert
inservice teachers and work with students.
A brief timeline of the PA program is provided in
Table 1, and a description of these activities is
provided below.
Planning the Program
A collaborative team planned the performance
assessment program prior to the beginning of the
semester, and continued to meet and adjust the
program as needed during the semester. The planning
team was composed of a mathematics educator (first
author), a science educator (second author), four
middle school mathematics teachers, a middle school
social studies teacher (for inter-disciplinary
Table 1
Performance Assessment Program Timeline.
Week of Semester
3
3-5
5
6
7
8
9
9-12
13
14
Activity
Introductory PA workshop conducted during regular class meeting (3 hours).
Preservice teachers began to research PA task topics and plan task outside of class.
Preservice teachers submitted their PA task planning guides and their journal article reviews on their selected PA task
topics.
Collaborative team met to match mentors with preservice-teacher-groups.
Mentors met with their assigned preservice-teacher-groups during class to provide advice and feedback on preservice
teachers’ initial plans for their PA tasks (1 hour).
Preservice teachers submitted first draft of PA tasks to their mathematics methods professor (first author) and to their
mentor teachers.
Preservice teachers received written feedback from their mathematics methods professor (first author) and from their
mentor teachers.
Preservice teachers revised their tasks and field-test tasks in their mentor teachers’ class.
Preservice teachers submitted their report of their PA tasks
Preservice teachers submitted a follow-up lesson plan based on PA findings.
Amy Roth McDuffie, Valarie L. Akerson, & Judith A. Morrison
25
connections), and a secondary program administrator
from a Washington State Educational Service District.
The middle school teachers were recognized regionally
as teacher-leaders for their expertise in performance
assessment strategies, and more generally, for
implementing standards-based approaches to teaching
and learning. The team worked together to develop the
preservice teachers’ understanding of PA, match
preservice teachers to mentors, and to support the
preservice teachers in their PA task design and
implementation. These efforts were aimed at ensuring
that our program was providing for meaningful
interaction between preservice teachers and inservice
teachers, as called for by Putnam and Borko (2000)
(Goal 3), and thereby facilitating the preservice
teachers’ growth in understanding students’ thinking
and learning (Goal 1) and in implementing the PA
strategies (Goal 2). It should be noted that a practicum
field component was not built into the semester for the
preservice teachers, and thus this field experience was
initiated and arranged entirely by the planning team.
Introductory Assessment Workshop
This workshop was conducted during the regular
methods class meeting time for a three hour period.
The collaborative team planned and facilitated the
workshop with team members leading different parts of
the workshop. It was conducted to address our second
goal by briefly discussing general assessment issues,
providing an overview of the standards-based
assessment program in Washington State (e.g., see
Washington Commission on Student Learning, 1998),
and introducing the preservice teachers to performance
assessment issues and strategies.
To introduce the preservice teachers to
performance assessment, we asked them to work in
groups on a sample performance assessment task that
was written and field-tested as part of an assessment
program in Washington State. The task required the
preservice teachers to design a cereal box that would
reduce the amount of cardboard needed and still
maintain a specific volume, and then to write a letter to
the cereal company describing and defending their
design. While we only provided approximately twenty
minutes for the preservice teachers to work on the task,
they had enough time to identify key issues of the task
and key components of task-design. Next, we
discussed some of the features and purposes of the
task. Consistent with our framing of the features and
purposes of PA, we examined the authentic context of
the tasks, the open-ended questions involved, the
descriptive and persuasive writing components, the
multiple entry points and various solution methods
possible in performing the task, and opportunities for
assessing higher order thinking. After this discussion,
we gave the groups scoring rubrics and samples of
ninth grade students’ work on the task at various
performance levels. Using the scoring rubrics, the
groups assigned scores to their sample students’ work.
Following this group work, we discussed the scoring
process, the rubrics, and the task as a class. This
component was designed to attend further to our first
goal regarding students’ thinking and understandings
by exploring a sample of “authentic practice” (Smith,
2001, p. 9) in that the task selected was used in local
classrooms and students’ work (in their own hand) on
this task was analyzed for understandings and
approaches.
Next, we worked to formalize their knowledge of
performance assessment (Goal 2) by discussing
defining characteristics of performance assessment, as
well as advantages and limitations. Additionally, a
middle grades language arts teacher-leader facilitated a
brief discussion of types of writing used in
performance assessment (e.g., descriptive, expository,
and persuasive). We concluded the workshop with an
introduction of the planning guide (described in detail
below) and provided a few minutes for generating
ideas for the preservice teachers’ PA projects.
Researching Topics and Generating a Plan for the PA
Task
The preservice teachers formed groups of two to
three to collaborate on the PA task project. Each group
chose a mathematics topic for the focus of their task.
The groups were restricted to middle school
mathematics topics because all of the mentor teachers
selected were teaching at the middle school level. This
restriction was due to the planning team’s decision to
select mentor teachers with experience in PA, and we
had difficulties finding such teachers at the elementary
level. Once the topic was chosen, each group member
found a minimum of two journal articles discussing
teaching and/or learning issues for that topic. The
preservice teachers submitted a brief summary of each
of their articles and an explanation on how the
information in the article contributed to their thinking
and plans for their PA project. The purpose of this
component of the project was to lay a foundation for
understanding students’ thinking and learning (Goal 2)
by ensuring that the preservice teachers had some
awareness of the pedagogical issues surrounding their
topic as reported in mathematics education literature.
Additionally, each group used a planning guide to
outline major features of their task and keep them
focused on goals and purposes of performance
assessment (versus other types of projects or
assessments). To show a clear and mathematically
important purpose for the task, the preservice teachers
described the concepts and processes targeted for
assessment. To demonstrate how the task would
engage learners, the preservice teachers explained the
task’s authentic and/or worthwhile context, the role the
learner would play in performing the task (other than a
student doing math for a class), and the audience for
the product (other than a teacher grading a project). To
ensure alignment with selected goals and define criteria
for quality performances, the preservice teachers
created a table showing connections among learning
standards, task products and/or performances, and
criteria for measuring whether learning goals had been
met. Because the Washington State Essential
Academic
Learning
Requirements
(EALRs;
Washington Commission on Student Learning, 1998)
were emphasized in this course, our students identified
appropriate EALRs for their task. However, PSSM
could have been used in lieu of the EALRs. Regardless
of which standards were applied, this component
focused preservice teachers’ thinking on the notion that
assessments need to be aligned with important
instructional goals (part of Goal 2).
From this point, the groups continued developing
their tasks outside of class time. While many groups
created original tasks, the preservice teachers were
permitted to use outside resources (e.g., activity books,
journal articles, their Van de Walle (1998) textbook,
etc.) in developing their tasks. We did not require that
their work be entirely original because we wanted the
process to mirror that of teachers’ planning (cf.,
Smith’s (2001) recommendations), and teachers often
draw from existing resources, rather than write their
own tasks. Even in the cases where a problem, activity,
or task was used from an outside source, significant
work was required to develop the problem into a
performance assessment task and meet the assignment
requirements.
Collaboration
Teachers
Between
Mentors
and
Preservice
Using the information provided in the preservice
teachers’ planning guides (i.e., grade level and topic
targeted), we matched each preservice-teacher-group to
one of four mentor teachers. Each mentor teacher was
responsible for advising two groups of preservice
teachers.
Amy Roth McDuffie, Valarie L. Akerson, & Judith A. Morrison
After the mentor teachers had been assigned to
groups of preservice teachers, the mentors attended one
hour of a methods class. The preservice teachers
brought their planning guides and drafts of their PA
tasks to this meeting. During this hour, the mentor
teachers met with each of their groups to discuss their
ideas and plans for implementing the PA tasks. We
provided the mentors and preservice teachers with
specific discussion prompts including individual
students’ learning needs, mentor’s typical teaching
practices, and classroom norms. The members of the
planning team circulated to assist groups in designing
their tasks and keep groups focused on objectives.
These meetings were planned to address our third goal
of facilitating meaningful collaboration with teachers,
and consequently, more authentically engage in the
planning phase of teachers’ work as recommended by
Smith (2001).
Submitting the First Draft and Field-Testing the PA
Task
Continuing on the theme of experiencing the cycle
of teachers’ work (Smith, 2001), in the eighth week of
the semester, the groups submitted their first drafts of
their PA tasks to their mathematics methods professor
and to their mentor teacher. This draft included a brief
overview of the task, a table showing alignment
between task items and the EALRs (revised and
developed further from their initial plans), instructions
for administering the task and a list of materials
needed, the task as it would be administered to
students, and rubrics for scoring the task. Within a
week, both parties provided written feedback and
comments for the groups to consider before
administering their tasks to middle school students.
As part of our third goal of situating the project in
the schools, each group arranged a time to field-test
their PA tasks in their mentor’s class. The tasks were
designed to be completed in two to three 50-minute
class periods. Each mentor teacher decided with his or
her groups who would facilitate the task. In some cases
the mentor teacher was the primary facilitator and in
other cases the groups facilitated the task
administration. However, in all cases, the preservice
teachers observed throughout the task administration,
talked with students, and in some cases, interviewed
students about their thinking, and recorded notes on the
process.
Analyzing Results and Reporting on the PA Task
Following the field-test, the preservice-teachergroups scored the students’ work and analyzed selected
27
students’ work in greater depth for the purpose of
understanding students’ thinking and learning (Goal 1).
Finally, they prepared a written report of their analysis
of students’ work and their reflections on the
performance assessment process and project to
examine the strengths and limitations of PA, as part of
Goal 3.
Writing a Follow-up Lesson Plan
To help preservice teachers understand the
teaching and learning cycle of using assessment to
inform instruction (Goal 2), the preservice teachers
were required to write a lesson plan based on their
findings in the performance assessment task
administration. In some cases the lesson plans were on
a topic closely related to their PA task topic, and in
other cases the preservice teachers identified
weaknesses in underlying skills and thinking through
the PA, and correspondingly chose topics that were
less obviously related to their PA topic. As part of the
lesson plan, the preservice teachers explained how the
lesson was motivated by their findings in the PA task
administration.
Providing Release Time and Compensation for the
Mentor Teachers
Throughout the semester inservice teachers played
a key role in the project. They attended two class
meetings during the school day, an evening meeting,
and provided written comments on the first draft of
each of their two groups tasks. For this project, we
were able to provide substitute teachers to release the
mentor teachers from their teaching responsibilities on
the days they attended the methods class. Additionally,
the mentor teachers were compensated for their time
during the evening meeting and for their reading of the
projects. This funding was available through the
Washington State Educational Service District. We
believe that this support enhanced the extent to which
the mentor teachers were committed to the program,
and contributed to our efforts to meet Goal 3, creating
a meaningful collaboration with inservice teachers.
Reflections on the PA Program
We found that all of our goals were achieved in
that students began to develop understanding in our
areas of focus (Goals 1 and 2) through careful
facilitation of field-based experiences (Goal 3), and
indeed we experienced some unanticipated benefits.
However, these achievements were not gained without
some significant challenges. In the process of
implementing this project, we also recognized areas to
preserve and to change, and have made changes in our
28
program in semesters following the initial
implementation. These reflections and changes are
described below.
Benefits of the Program
Our first goal of developing skills in assessing and
diagnosing student thinking was met in that the
preservice teachers provided substantive analysis and
interpretations of students’ thinking, understanding,
and lack of understanding in their reports on their PA
tasks and follow-up lesson plans. For example, in
Karen’s (all names used are pseudonyms) final report,
she reflected on her students’ work and remarked,
Although [the group’s] worksheets were not…
complete, … [they] added new insights to the final
group discussion by introducing conjectures to the
problem…they exhibited a higher level of
reasoning. … They argued various points and
brought up ideas that even [we] had not considered.
Their inferences and thought processes led others
to question their own conclusions.
These comments demonstrate how the preservice
teachers were observing and analyzing their students’
work on a deeper level than simply looking for correct
answers.
In regard to our second and third goals, we believe
that our preservice teachers cannot truly come to
understand performance assessment, its complexities,
its benefits for understanding students’ thinking and
learning, and its benefits for informing teaching
without experiencing the process of designing and
implementing performance assessment tasks first hand.
At the end of the semester, the preservice teachers
demonstrated their understandings of PA in their
reports and comments. Sarah’s explanation of PA was
typical of preservice teachers’ understandings when
she described PA as:
A task which has a real world problem to assess
students’ understanding of a topic. …[It can be
used] to assess what someone already knows, like
at the beginning [of a unit], … or at the end to
evaluate what they have learned and how your
teaching has helped them to understand that
concept.
While we intended for the preservice teachers to
consider worthwhile or meaningful contexts, not just
“real world,” it was clear that Sarah understood the
primary purposes and approaches of PA.
Our experience in this project and their work in
designing and implementing PA tasks suggest that the
preservice teachers meaningfully constructed ideas as
to what constitutes performance assessment. For
Meaningful Field-Based Experiences
example, one student designed a PA task entitled,
“City Park” in which middle school students worked as
landscape architects (the role) to design a park with
playground equipment and a sprinkler system (the
context). In this task the students had to construct a
budget, calculate the area of their design, and satisfy
various design criteria established by the city council,
represent their design visually with a scale, and write a
letter persuading the city council (the audience) that
their design proposal should be accepted. This task
exemplifies how the preservice teachers were able to
incorporate key elements of performance assessment in
tasks that involve several important mathematical
concepts and processes.
Perhaps an even greater benefit was that the
preservice teachers began to understand assessment as
a formative process, rather than merely a grade in the
grade book. They began to generalize the ideas from
performance assessment to understand and be
interested in other forms of authentic and alternative
assessment such as interviewing and observational
record keeping. Dora exemplified these understandings
for assessing in multiple ways when she said,
[This type of assessment] engages the students in
real-world problems, capitalizes on their prior
knowledge, requires them to think critically, and
allows the teacher to assess by observation.… As I
circulated throughout the room listening to
students, making mental notes about what was
going well and what changes need to be made, it
was obvious that the students were using their prior
knowledge.
Moreover, as is evident in the earlier example of
the “City Park” task, designing and administering a
performance assessment task also seemed to help the
preservice teachers construct a more sophisticated
notion of problem solving in mathematics and more
fully understand what is meant by an open-ended task,
consistent with Shepard et al.’s (1996) findings for
inservice teachers. Focusing on our third goal
specifically, the situated nature of the project (i.e.,
designing an open-ended task for actual students,
working with an experienced teacher, and
administering the task in a school classroom) seemed
to be the most important factor in bringing about the
preservice teachers’ interest in the project and learning
from the project. Robert’s reflections represented what
we heard from virtually all of the preservice teachers in
their final reports and/or course evaluations. He stated,
“The project was an excellent opportunity to work with
an actual math class. It gave me a good picture of what
the students know and how they can learn.” Thus, we
Amy Roth McDuffie, Valarie L. Akerson, & Judith A. Morrison
found that following Borko et al.’s (1997), Putnam and
Borko’s (2000), and Smith’s (2001) recommendations
for situating tasks in actual classrooms were an
important part of our program.
Additionally, we observed professional growth
opportunities for the mentor teachers. All of the mentor
teachers commented that they learned more about
performance assessment strategies and gained ideas for
their own teaching through their involvement in the
project. For instance, one mentor stated, “[Working
with a preservice teacher in this program] affirmed my
strong belief in observable assessment for young
learners. It gave me a chance to teach someone else
techniques I have developed.”
Challenges of the Program
Similar to the Borko et al. (1997) finding, time
emerged as a primary challenge in implementing our
project. Time was a challenge in the form of demands
on the methods professor, mentor teachers, preservice
teachers, and the mentor teachers’ class time. For the
methods professor (first author), this project certainly
demanded more time in planning and assessing. As a
new endeavor, more time was required to plan the
project, especially in regard to the time required to
meet with the project team. While collaboration often
produces better results for learning, it seems to take
more time than working independently in teaching.
Additionally, assessing and providing feedback on the
preservice teachers’ work throughout the project
required more time than is typically spent assessing
written work in a methods class.
As described earlier, the mentor teachers in this
project were provided release time and compensation
for their significant time committed to the project.
Certainly, we preferred to offer support to inservice
teachers who took on this responsibility. However, this
funding was not available to us after the initial
implementation and we have found that the program is
manageable without funding.
The preservice teachers also experienced
significant time demands. While most preservice
teachers commented (either orally or in course
evaluations) that the project was worth the effort, they
all seemed to feel that the workload for the class was
heavier than other classes due to the PA project. This
challenge is consistent with Borko et al.’s (1997)
finding for the increased planning time required in
using PA. In addition to the PA project, the preservice
teachers had several additional course assignments and
requirements. In semesters following the initial
implementation, we reduced other assignments
29
recognizing the time this project requires and the
multiple purposes it serves (i.e., we found that writing
the PA assignment could serve in lieu of a lesson plan).
In addition to challenges with the magnitude of the
project, some of the preservice teachers had difficulty
arranging for administering their tasks in classrooms.
This mathematics methods course did not have a field
experience as part of the course. As such, time to be in
the schools was not allocated in their schedules.
Moreover, given that the timing had to meet the needs
of the mentor teacher, scheduling was not simply a
matter of finding time in the preservice teachers’
schedules. In some cases, not all group members were
able to be present for each day of the task; however,
everyone managed to be present for some part of it.
Our teacher education program soon will include a
practicum experience as part of a methods block
scheduling structure. As this practicum is instituted, we
are hopeful that some of the logistical issues,
particularly the scheduling problems associated with
the field component will be mitigated.
Most tasks required more time than anticipated by
preservice and mentor teachers, and correspondingly
either the task was modified or the mentor teacher
allowed the preservice teachers to use more than three
days of class time. Consistent with Borko et al.’s
(1997) findings, PA requires a substantial investment
of class time, and it is not easy to predict how long the
students will need to complete their work.
In addition to time demands, we faced a challenge
identified by other researchers (Putnam & Borko,
2000; Sykes & Byrd, 1992): finding appropriate
mentor teachers. We wanted the mentor teachers to
have expertise in PA and to be able to provide the
needed support to the preservice teachers. We had
limited success in finding these candidates. The
teachers involved with our planning team were well
qualified and successful mentors; however, the other
two teachers that were recruited were not as informed
about PA strategies and did not seem to be as
committed to supporting our preservice teachers. The
preservice teacher groups working with these teachers
commented that they provided limited support in
designing and implementing the task, and it seemed
that the mentors did not feel qualified to discuss PA
strategies. While we initially perceived that all of the
mentor teachers were interested in the project and had
the necessary expertise to provide support to the
preservice teachers, these teachers needed more
experience with these approaches before they could
adequately advice our preservice teachers.
30
Additionally, two groups of preservice teachers
mentioned that they had difficulty communicating with
their mentor teachers (e.g., emails and phone messages
were not returned, minimal written comments on their
PA task draft was provided, etc.), and these groups
perceived that they did not receive the same level of
support as their classmates. One mentor teacher had
some health concerns during the semester, and the
other teacher seemed to have pressing issues in her
teaching that resulted in less time being devoted to the
project. While these cases could be called exceptions,
we believe these situations are to be expected when
asking inservice teachers to take on another
responsibility. Thus, accommodations for unexpected
situations with mentor teachers should be expected and
planned for as much as possible.
We have recruited more mentor teachers through
referrals from participating teachers, and are adding
teachers who have been involved in summer
workshops and/or graduate courses focusing on
assessment offered at our university. Even as we have
expanded our pool of mentors, challenges remain. As
with any field-based work, we have found that we need
to be flexible with project due dates while still trying to
structure the program through the three-part
assignment (planning guide, task draft, final report) to
keep the preservice teachers progressing during the
semester.
Features of the PA Program to be Preserved
In attempting a program for the first time, we
found that we made several decisions along the way,
some that were well conceived and others that were
quick solutions. In this section we reflect on some of
the key decisions that worked well for us. First, we
were asked whether the mentor teacher or the
preservice-teacher-group should facilitate the task.
Given that the preservice teachers did not necessarily
have any experience in the mentor teachers’ classes
prior to administering the PA task, we allowed the
mentors and the preservice teachers to decide on the
preservice teachers’ level of involvement in facilitating
the task. The preservice teachers had various levels of
classroom experience, and leaving this decision to the
mentor-preservice teacher groups enabled everyone to
make decisions within individual contexts. The
primary purpose of the field-based component of this
project was not to provide a student teaching
experience as much as it was for preservice teachers to
learn about performance assessment in a situated
context of the middle school classroom. For preservice
teachers and mentors that were not comfortable with
Meaningful Field-Based Experiences
the preservice teachers facilitating the tasks, this
flexibility seemed to enhance the preservice teachers’
abilities to focus on performance assessment and
diagnose students’ thinking and learning more than it
might have if they had the added stress of teaching
during the task.
In regard to the assignments of the project, two
non-field-based components were important to
preservice teacher learning: the initial research of the
mathematics topic and the follow-up lesson plan. By
requiring that the preservice teachers find journal
articles examining the teaching and learning issues
surrounding their topic, the preservice teachers gained
in-depth expertise on the theoretical foundations of
their topic beyond what is normally discussed in class.
While this type of assignment has been a part of our
methods courses in the past, connecting it to the
situated context of the PA project gave it more
meaning for the preservice teachers. Additionally, the
project provided some assurance that they were better
informed about the pedagogical issues surrounding
their topic as they designed tasks, and many preservice
teachers commented that the project helped them
anticipate and/or avoid potential problems in the
classroom. Likewise, the follow-up lesson plan helped
the preservice teachers to see what a classroom teacher
would do with the information gained from the
assessment. We found the preservice teachers to be
more invested in these lesson plans than in others
required for the course because they had their
classroom experience and real students as their referent
when they designed them.
Features of the PA Program to be Changed or Added
First, in attempting to find qualified mentor
teachers, we were able only to find middle school
teachers who seemed to have appropriate experience.
Correspondingly, we limited our preservice teachers to
writing PA tasks for middle school mathematics. Some
of the preservice teachers were unhappy with this
limitation because they intended to teach at the
elementary level and were not interested in the middle
school level. For these preservice teachers, the PA
project seemed less authentic because it was not
situated in the grades in which they intended to teach.
Another problem was that all of the mentor teachers
were not selected prior to the start of the semester. It
may not be a coincidence that the two less committed
mentors were called upon part way into the semester
and therefore were not included in early planning
efforts. We believe that we would have been more
successful if we had involved all of the mentors in the
Amy Roth McDuffie, Valarie L. Akerson, & Judith A. Morrison
PA project throughout the entire semester. Since this
first implementation, we have been more successful in
assembling a cadre of mentor teachers at all grade
levels to draw from each semester and to better match
the PA project requirements with our preservice
teachers’ interests. However, we continue to struggle
with having all of the mentors selected prior to the start
of the semester. Some teachers and school districts are
reluctant to commit to the program in advance,
especially for the fall semester when schools are still
organizing their own teaching assignments.
In addition to more mentors, we realize that our
preservice teachers need more opportunities to interact
with their mentors. For example, a final meeting
between preservice teachers and mentor teachers
would provide an opportunity for preservice teachers to
review their analyses and report on the students’
performances. This meeting would provide the mentor
teachers with a new perspective on their students’
thinking, learning, abilities, and skills. It also might
serve to help the mentor teachers improve their
mentoring skills by more carefully examining the
products of the preservice teachers’ work. Moreover,
this meeting would provide preservice teachers with
feedback on their analyses based on the teachers’
knowledge of their students, and this feedback and
perspective is not possible from their methods
professor. However, logistics with scheduling and the
need for substitutes have impeded these plans.
Next Steps
We are continuing to implement our PA program
in mathematics methods courses. Our current efforts
include offering this PA program in both the
mathematics and the science methods course with the
students using PA to make connections between the
disciplines. The benefits we have experienced compel
us to continue to develop this program. A study is
underway to empirically investigate the effects of our
program on our preservice teachers’ learning of PA,
and more generally, the teaching and learning of
mathematics
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(1997). Teachers’ developing ideas and practices about
mathematics performance assessment: Successes, stumbling
blocks, and implications for professional development.
Teaching and Teacher Education, 13 (3), 259-278.
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Danielson, C. (1997). A collection of performance tasks and
rubrics: Upper elementary school mathematics. Larchmont,
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Weston, T. (1996). Effects of introducing classroom
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Sykes, G., & Bird, T. (1992). Teacher education and the case idea.
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Van de Walle, J. (2001). Elementary and middle school
mathematics: Teaching developmentally. New York: Addison
Wesley Longman.
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Educational Researcher, 29, 4-15.
32
Meaningful Field-Based Experiences
The Mathematics Educator
2003, Vol. 13, No. 1, 33–40
How to Do Educational Research in University Mathematics?
Rasmus Hedegaard Nielsen
Situated as a Ph.D. student in university mathematics education, I present some of my considerations about my
identity as a researcher in this field. I discuss the larger global and local societal issues and their connections to
educational research in university mathematics. My discussion goes beyond personal considerations and
touches upon the structures and ideas that are both internal and external to university mathematics education. I
discuss the different political projects that I can identify from my personal experiences across the fields of
educational research, practice, and policy in university mathematics. I place myself firmly within the tradition
of critical education, but also draw on postmodern theories. The results of the discussion are the identification
of challenges for a postmodern critical mathematics education, with a focus on university mathematics.
This paper is a modified version of a paper presented at the Third International Conference on Mathematics Education and Society held in
Helsingør, Denmark in April 2002.
I live in Denmark, one of the richest countries in
the world, a small democratic kingdom in the western
world. I am a white male living in a welfare society
with all the commodities of the western way of life. I
have free access to education—even to all the
universities. I have free access to medical care. I live in
a peaceful neighborhood. I can walk on the streets in
the middle of the night without any fear of being
attacked in any part of my city. I will not have to sell
my home or change my life drastically if I lose my job.
I have never felt starvation. I have democratic rights to
vote and to participate in political life. I do not risk
discrimination or being arrested at random. I live in a
country where everybody gets what he or she needs
and deserves. This is a brief glimpse of the Danish
society from my perspective.
Mathematics Education and the Larger Society
Is this short story from the Danish society in any
way relevant for the question of how to do educational
research? Does it make sense to ask what kind of
educational research should be conducted in the
context of this society? These questions I have asked
myself as a Ph.D. student in university mathematics
education. 1 I have done so because it is important for
me to consider the role that my work and I will take in
this society and it is important for me to contribute to
my society. And my society is the one you just
glimpsed. In the following analysis I try to sketch some
of the answers that I have found.
Rasmus Hedegaard Nielsen is a Ph.D. student in the Department of
Education at Aalborg University, Denmark. His thesis focuses on
the social roles of mathematics and university mathematics
education. His email address is riege@ofir.dk.
Rasmus Hedegaard Nielsen
The first answer I listened to when I began my
work focused on the internal problems of university
mathematics education, and thereby deemed my
glimpse of the Danish society irrelevant. It suggested,
for example, that I could look at how to get students to
learn the concept of function space better, or how to
understand how students actually learn concepts.2
Another group of issues I could look at was how to get
the students motivated and how to get more students to
pass the exams. This answer almost totally ignores the
relation between the larger society and university
mathematics education and thereby makes the context
of this education irrelevant, but it certainly offers a
paradigm of research in university mathematics
education. This I call the conservative answer.
Another answer suggests that the institutions of
university mathematics education are very important
for the welfare of the Danish society, understood
mainly in an economic sense. This makes my opening
glimpse of the Danish society relevant in the sense that
we need the institutions of university mathematics
education for maintaining the things that we like which
are represented by this glimpse. But this answer also
suggests that these institutions are in crisis in the sense
that the very important link between these institutions
and the larger society (understood as the economic
system) is not close enough. Furthermore, they often
claim that the institutions of university mathematics
education are also in crisis in the sense that they are
inefficient. ‘We’, the taxpayers, do not get enough for
the money ‘we’ pour into these educations. This means
that the focus of educational research should be on the
learning of mathematical concepts by the individual
student, but also on what and how the students should
learn. What and how they should learn should be
33
relevant for the economy, which in this context
translates mainly to the industry. It also suggests that I
could research into how to measure the efficiency of
the institutions, in order to increase the efficiency of
the institutions by benchmarking them. This I call the
neoliberal answer.
Many additional answers that reflect moments of
both the conservative and the neoliberal answers I’ve
already offered can be found. For example, one answer
takes the internal concerns from the conservative
answer seriously, but also finds mathematics education
to be a crucial institution in the Danish society, though
these institutions are not seen to be in crisis because
the link to the larger society is not tight enough. On the
contrary, it sees mathematics education in crisis
because it is being challenged by the neoliberal project.
Yet another answer tells me it is not certain that
university mathematics education is one of the
institutions that helps guarantee all the good things in
Danish society. Also it is not certain that university
mathematics education has no influence on the larger
Danish society. It might be that these institutions are
also the source of some of the bad things that I mention
next. This I call the critical group of answers.
Another way to experience my country—an
alternate glimpse—is to take a walk in the red-light
district near the railroad station, or simply walk down
the main shopping street. There you will notice the
junkies and homeless people that nobody wants. They
get pushed around between different places in the city.
Some of the homeless and alcoholics used to drink
their strong beers near a traffic junction in the middle
of Nørrebro where many people pass by. They used to
sit there peacefully, or at least as peacefully as they
could while being an alcoholic and having many other
problems, until the authorities took their bench. They
literally removed it. Now the homeless are at another
place where they do not ‘disturb’ the people that have a
job and ‘contribute’ to ‘our’ society. These people we
often do not see, are near the bottom of our small
peaceful society. They are not alone down there, others
are just harder to notice. They are being rejected by the
peaceful, democratic, and just Danish welfare society;
they are not considered good for anything (other than
trouble and someone to spend ‘our’ money on).
As with the first gaze into the Danish society, there
are different opinions about the relevance of this
glimpse for research in mathematics education. The
conservatives would tell me that this picture has
nothing to do with mathematics education and
especially university mathematics education. The
neoliberals would tell me that what ‘we’ need is more
34
and better mathematics education in order to address
these problems (if they are seen as problems of society
and not only individual problems). The critical
educators try to tell something akin to what is
mentioned in the Aims of the Third Mathematics
Education and Society (MES) Conference:
Mathematics qualifications remain an accepted
gatekeeper to employment. Thus, managing
success in mathematics becomes a way of
controlling the employment market. Mathematics
education also tends to contribute to the
regeneration of an inequitable society through
undemocratic and exclusive pedagogical practices
which portray mathematics and mathematics
education as absolute, authoritarian disciplines.
(Skovsmose & Valero, 2002, p. 3)
This means that this second glimpse becomes relevant.
But this is not all. Both the neoliberal and critical
group tell me to look further than the Danish society, to
the global world. They urge me to look at different
parts of the global world and in different ways. Let me
try to share one glimpse into the larger society.
Yet another way to experience the world (and not
only Denmark) is to take a plane from inside to outside
the Western World (or more correctly from inside to
outside the industrialised world and its holiday resorts
in the sunny or “exotic” parts of the not-soindustrialised world). Here you can meet hunger, war,
serious environmental problems, cultures being
destroyed, peoples and countries being plundered,
torture, and disasters and crimes of any kind.3 (Well,
not personally, it is likely you would have a nice room
in a hotel and a return ticket to your home country in
your pocket.4)
The original voices, conservative and neoliberal,
are again answering me, whispering different
suggestions in my ear: “Mathematics education is the
hero of civilization,” or “mathematics education is
innocent.” They continue whispering seductively,
“these problems that you see are only small errors in
the system and if only people would not resist the
system these errors would be easily fixed.” The critical
group of answers would suggest that mathematics
education might play a role in producing all the
nightmares that haunt the world globally, and that
mathematics education might play a role in the unequal
distribution of wealth globally.
Mathematics and the Larger Society
Not only can university mathematics education be
seen as playing different roles in our local and global
society, mathematics can also be seen as playing
How to do Educational Research?
different roles. And how we see and understand
university mathematics education might depend quite a
lot on the role played by mathematics in our societies,
both globally and locally. The conservatives claim that
mathematics is objective and neutral in itself, and it is
only the use of mathematics that can lead to good or
bad things. This means that university mathematics
education is protected from considerations about the
role of mathematics in society, and the only problem
with mathematics education is that students have
problems learning it. Mathematics is of major
importance because it is an important part of our
culture. The neoliberals are saying that mathematics is
of major importance in the pursuit of economic growth
and thereby the success of our societies, and in this
way mathematics education is made important as well,
namely as a producer of competencies in mathematics
in the shape of a mathematically skilled work force.
But I can also hear other people, for example in the
field of ethnomathematics5, that try to tell me a more
critical story:
The critical strand [of ethnomathematics] is not just
interested in the mathematics of Angolan sand
drawings and their use in story telling, but also in
the politics of imperialism that arrested the
development of this cultural tradition and in the
politics of cultural imperialism that discounts the
mathematical activity involved in creating Angolan
sand drawings. (Powell, Knijnik, Gilmer, &
Frankenstein, 1998, p. 45)
These voices say that mathematics and mathematics
education might not be innocent and might not be our
hero and problems might not just be errors, but
mathematics and mathematics education might have
something more substantial to do with all these
problems.
Who is right? Which story should be believed and
on what grounds should the different kinds of answers
be judged?
The Larger Society and Hegemonic Projects
I think it should be clear by now that what I have
called the larger society or ‘our’ society are terms that
are highly disputed and that these terms play a crucial
role in the stories that I am offered when I ask about
how to do educational research in university
mathematics. Personally, I felt that I had no firm
foundation that I could stand on when I was to judge
the different ideas about ‘our’ society. I found that the
discourse theory of Laclau and Mouffe (1987; see also
Torfing, 1999) expressed theoretically just that feeling.
In discourse theory, society is not seen as something
Rasmus Hedegaard Nielsen
that is without conflicts or something that can be
described from a neutral and objective standpoint.
Theoretically put, what I have sketched above is my
experience of the struggle of different political projects
that all try to dominate society and to that end give
different interpretations of what is important in society.
They all try to make their descriptions look neutral and
objective—to look like the truth about our society. In
this sense, such theories are in the same vein as the
theory of Foucault. In the words of discourse theory,
these efforts are called hegemonic projects and they are
said to try to gain hegemony. Hegemony translates
roughly into leadership, including cultural and political
dimensions. My point is that these struggles also
extend to the arena of university mathematics
education, and that this arena is both used as a resource
and as a stake in the struggles. It is not the case that a
hegemonic project is always struggling in all arenas
and it might be that the project takes different forms in
different arenas. The hegemonic projects are not some
overarching ideology that structures everything. The
answers I sketched in the previous section are very
different in scope, but they are all more or less entering
the arena of university mathematics education.
It should also be clear that I could have chosen to
give you, the reader, quite different glimpses of
society. But as I am also situated in these struggles (on
the side of the critical group, which you might have
guessed) I want to obtain something, and to this end I
have chosen these particular glimpses. I do not have
hopes that I can show you that things necessarily must
be like I see them; I only hope to show you something
you might not have seen before. This might be seen as
an answer of how to judge the different answers; yet
there is no way to stand on a firm ground and be able
to judge. You are always a part of these hegemonic
projects; you always see the world from somewhere.
In my work I wanted to get a little closer to the
hegemonic projects in order to know the terrain that I
was entering. This implied that I took a closer look at
how the different hegemonic projects are connected to
university mathematics education. In the following, I
concentrate on the Danish context, but I am sure that
the discussion also extends to most Western countries,
though maybe with different emphasis caused by the
different contexts. The works of Michael Apple have
inspired the following discussion of hegemonic
projects.
The Field of Educational Research
Different projects are present in the field of
educational research, such as the conservative and
35
neoliberal. When it comes to general education, critical
education can be said to constitute a project, but as far
as I can see, critical education has mainly been
interested in the primary and secondary educational
system. I want to mention critical education anyway,
because it is a project that I have sympathy for and
because I think it is possible to extend it to university
mathematics. In the field of educational research the
conservative project is dominant when it comes to
university mathematics. However, the field itself is
quite young.
Critical Mathematics Education
In the Aims of the MES Conference, I see concern
for identifying structural problems that affect the
people who are learning mathematics: “Mathematics
qualifications remain an accepted gatekeeper to
employment”, and “mathematics education also tends
to contribute to the regeneration of an inequitable
society through undemocratic and exclusive
pedagogical practices” (Skovsmose & Valero, 2002, p.
3). Here we see a focus on democracy, and elsewhere
on the idea of citizenship. Mathematics is not seen as
unproblematic, but seen as a potential social actor that
supports the production of risks in society.
Ethnomathematics is also interesting, since it for me
has a completely different focus on mathematics than
what is usual in the field of research in mathematics
education.
Only in the very broadest sense can these concerns
and focuses be said to be a part of a hegemonic project.
Critical mathematics education is a movement that is
connected to practitioners of the teaching of
mathematics but might have weak links to fields
outside mathematics education. This should be seen as
a challenge to critical mathematics education. It is a
movement that has not entered the field of university
mathematics education in a substantial way. This is
something that I would like to change.
University Mathematics Education
The conservative project dominates the field of
educational research in university mathematics (Hart,
1999). Hart identified the dominating research
tendencies in post-secondary mathematics education,
“Except for a handful of studies, most research at this
level has focused on the student or on various
pedagogical methods…” (p. 3). She proposed a
research agenda that can be characterized as a postconservative agenda. It retains conservative
characteristics since it still sees mathematics and
mathematics education as disconnected to the larger
36
society. It is post-conservative since it clearly goes
beyond the conservative agenda by proposing
constructionism (Hart refers to Gergen, 1992; and
Phillips, 1995) as the epistemological foundation of
research within what she calls post-secondary
mathematics education.
The Field of Educational Practice
At many departments of mathematics in the ‘old’
universities in Denmark, the teaching is centred on
courses based on lectures and classes (where the
student are supposed to solve problems) with typically
large numbers of students attending. The pedagogy is
often authoritative, picturing mathematics as an
absolute discipline and teachers as holding the absolute
truth about mathematics. Mathematics is seen as
packages of knowledge that should be put into the
heads of the students. The students are seen as
individuals and their context is unimportant (unless to
the degree that their motivation is of interest). The
teaching of mathematics and mathematics itself are
seen as unproblematic.
At some departments of mathematics, the teaching
is centred on group-based project work, but also
lectures. Some of the project work focuses on links to
the larger society and the role played by mathematics
in society. These universities have become the best
suppliers of workers because they focus on project
work in groups while the old universities more or less
try to copy their ways of organising the educations.6
My impression of groups that are in these
environments is that they both more or less represent
different degrees of what I call the conservative
project.
In the context of one of the old universities, some
of the problems that I have heard talked about are
economic problems and pedagogical problems. The
first kind is caused by the decrease in the number of
students studying mathematics 7 and the second by the
fact that the student population is becoming more
heterogeneous 8 and that the students lack motivation.
The first problem is understandable so, since economic
problems will mean less funding for the researchers
and teachers in the department. The second problem
consists partly of increasing difficulties teaching at a
level where as many as possible benefit, and partly of
increasing exam failure.
What I think is characteristic of these problems is
that they focus on economics and on the individual
students—they are both more or less external problems
being imposed on the departments. This naturally puts
other problems on the sideline. For example, there is
How to do Educational Research?
not much attention on mathematics itself: No one asks
why there is such a thing as university mathematics
education, what kind of mathematics should be taught,
or the relation between mathematics and the larger
society. Or this view perceives such problems and
issues in a particular way. The focus is on how
mathematics should be taught so that more students
want to study mathematics, so that they complete their
study faster, and so that they become better
mathematicians. It seems like the conservative ideas of
mathematics education that prevail in the departments
of mathematics are under attack from the neoliberal
ideas, and that this attack comes mainly from the area
of national university policy.
This identification of certain problems is not
innocent. It has caused different actions to be taken.
That tells me a certain story of what research in
university mathematics education should be and what I
ought to do to be a ‘normal’ researcher. For example,
there is a suggestion of making elite courses alongside
a normal course to accommodate the problem of a
heterogeneous student population. A Centre for
Science Education 9 has been built to undertake research
and development of mathematics education to make it
more ‘sexy’ 10 and thereby attract and motivate more
students.
The Field of University Education Policy
National
I have tried to understand the kinds of arguments
and understandings of the universities and society. One
thing that is striking in these debates on university and
society is the use of a particular idea that is always
connected to the role of the universities—the idea of a
knowledge society. This idea is used to refer to the kind
of society that we live in (at least in the Western
World) and by connecting to the universities via the
idea of knowledge, a certain perspective on universities
is constructed that dominates the debates. The idea
comes in different versions—for example, the concept
of a “learning society” in Michael Young’s The
Curriculum of the Future (1998, p. 137-155). I think of
the knowledge society as a contested concept.11 This
means that different groups in society (not necessarily
political parties) with different interests try to gain the
power to define the idea of a knowledge society and to
connect it with other different ideas. This would help
the groups gain the power to define facts, problems,
and solutions concerning, among other things, the
university and the role of the university in the
knowledge society.12 In other words, they try to make
their ideology hegemonic. Some groups try to connect
Rasmus Hedegaard Nielsen
the idea of a knowledge society with the ideologies of
business and management using ideas as production,
competition, management, and markets. Other groups
try to connect the knowledge society with the idea of
democracy with emancipation, the risk society, and
ethics.13
The typical dominant argument goes like this: We
are in, or partly in, a knowledge society, therefore the
role of the universities have changed in a certain way
and we, as a society, have to react responsibly to these
new conditions. This is the general form of the
argument, and when it is presented like this it is
obvious that defining the knowledge society to some
extent determines the new conditions of the
universities and thereby the kinds of reactions there
are. What is also obvious is that in this form of
argument there are reactions—not actions. This
supposes that the universities have the role of reacting
to the conditions in the society, and not the other way
around. This makes the university a ‘service’
institution of society, making sure that the right amount
and kind of knowledge is produced, and not an
institution that can critically examine parts of the larger
society, including itself! This idea of a ‘service’
institution nicely fits with the idea that research in
university mathematics education is ‘efficiency’
research, that it never gets critical in any profound
sense, but only makes sure that the ‘service’ institution
is as efficient as possible.
As examples of different contested ideas of the
knowledge society, I will examine articles from
Universities for the Future14 (Maskell & Jensen, 2001)
and from Education15. Sometimes there is a small
description of what is meant by a knowledge society,
normally focusing on economic. There are no
discussions of the processes that have lead to this
development or the adequacy of the concept itself. It is
taken as fact that we live in a knowledge society and
that this is a fact that we have to adjust to and, in
particular, the universities have to adjust to. These
kinds of description and this kind of construction of
necessity are also found in the political policies on
education of most of the political parties in Denmark.
After establishing the fact that we have to react to
the emerging knowledge society, the writers draw
conclusions about the role of the university. These
writers agree that it is a very important institution and
much more important than it used to be. They see it is
an institution where knowledge is ‘produced’ mainly in
two forms: as research results and as academic
workers. Both are conceived as inputs to the private
corporations that are so important for our welfare
37
system. This means that universities, as a knowledge
society, are conceived only from an economic
perspective and not from a cultural or political
perspective.16 This means that changing the structure of
the universities will have effects on the economy, and
more importantly, it means that this is the only relation
that is conceived in the relation between university and
society.
International
In this context, economy connects to competition,
markets, and freedom. The economic description of the
knowledge society is typically followed by some kind
of description of a globalisation process, constructing a
link between the success of ‘our’ welfare system and
how competitive our country is. If knowledge is the
most important factor for competition between
countries, then the success of our welfare system is
dependant on the success of our society as a knowledge
society. The General Agreement on Trade and Services
(GATS) that is a part of the World Trade Organization
(WTO)17 contains a clear neoliberal approach to higher
education, including university mathematics education.
It sees mathematics and mathematics education as a
commodity that should be given the opportunity to be
traded on a free market. The Bologna Declaration can
be seen as an attempt to clear the way for a free market
in university education in Europe.18 This declaration
has also some “regionalistic” agendas, such as the
building of a common European culture. The
agreements and declarations are beginning to have
effects on both the thinking and the everyday life of
higher education, including university mathematics
education, in most European countries.
How to do Educational Research
in Mathematics Education?
I hope that I have sketched with some clarity the
different answers that have been given to me in my
search for an identity as a researcher in university
mathematics education. These answers make different
suggestions for a research paradigm. I have tried to
sketch how these possible research paradigms are
contained in different hegemonic projects with very
different scopes and identities. I leave it to the reader to
think about the names that I have given to the different
answers: conservatives, neoliberals and critical.
The conservative suggestions are mainly focused
on mathematics and try to ignore external relations,
though the conservatives have been under pressure
from the neoliberals. The neoliberals focus on the
economic link between university mathematics
38
education and the larger society, which they
understand in mainly economic terms.
One of my main points would be that whether you
like it or not, deciding how to do educational research
in university mathematics education makes you a part
of these struggles in one way or the other. It is not a
neutral realm that can refer to the pursuit of truth for
the legitimisation of work being done. In this way, the
three glimpses of society that I have given are relevant
to consider. I have found it is not an easy thing to
choose how to do research in this field (or any other
field for that sake); there is no firm ground to stand on
from where to make a neutral and necessary
judgement.
Challenges for Postmodern Critical Educational
Research in University Mathematics
As mentioned before, I can identify with the
concerns of critical education, though I also find some
of the ideas problematic. Therefore, I have chosen the
word “postmodern” from critical education; this
signals my flirtation with discourse theory. As
mentioned, both the ideas of discourse theory and those
of critical mathematics education have not been
especially interested in university mathematics,
therefore there are a manifold of challenges and
uncertainties for a research paradigm that is inspired by
these two approaches. The challenges I focus on here
are those that I find important. This does not mean that
I see them as the most important or the only ones, but it
means that they are those that I have found interesting
and within my reach as a researcher.
Theoretical Challenges
There is need of a theoretical framework that can
help:
1. Conceptualise the multiplicity of educational
research paradigms, practices, and policies; and the
way that they internally compete and struggle.
2. Conceptualise the relations between the different
fields. I have focused on educational practice,
educational research, and educational politics in
this paper, but many others exist.
3. Conceptualise key-concepts such as society,
politics, and mathematics.
4. Conceptualise political implications as to what
democracy, citizenship, and mathematics education
should be like.
These are the challenges that I feel to be urgent. I
have appropriated the theoretical framework of the
Laclau and Mouffe’s discourse theory (1987) as an
approach to the concepts of society and politics and as
How to do Educational Research?
an approach to understand the struggle for power
within the discipline of education research. I have also
drawn on the work of critical mathematics education,
especially the work of Ole Skovsmose (1994), to
conceptualise mathematics in society.
The theoretical framework I call for should not
only be descriptive; it should also provide directions
and strategies for research in university mathematics
education, but also in society and education at large.
Also, the framework should give us directions that we
can explore as to theorize how we would like the
educational institutions of university mathematics to
be. I have myself focused on concepts such as
democracy, citizenship, and the apparatus of reason.
Empirical Challenges
Parallel to such a theoretical framework, there are
also challenges that relate to the understanding of the
actual state of the fields and their connections. How is
educational research in university mathematics actually
being done? What are the conceptualisations of
students, of mathematics, of connections to the larger
society, and so forth, that are implicit or explicit in the
kinds of research that takes place? How are the
educational practices in the universities and what are
the connections to the larger society and other fields?
In other words, what kind of world is it that the critical
strand is a part of and in which it finds itself?
But this is not all. There are also empirical
challenges connected to the normative part of the
theoretical framework. We need to explore empirically
how the ideas such as democracy and citizenship can
be realized in a university mathematics education.
I have personally concentrated on the first part of
these empirical challenges, and I have done so by
focusing on the three fields that I also mention above. I
have looked at the educational practices at a certain
department of mathematics at a university. I have
looked at international policy on higher education. And
I have looked at educational research in university
mathematics.
Building a Hegemonic Project
One of the most important features that a
postmodern critical mathematics education should have
is that it should be able to form an alliance of different
groups in order to get enough momentum. It should be
able to connect to other fields as the neoliberal project
has done. In my opinion, some of the most important
groups to connect to are the teachers at all levels of the
educational system, the students, and others. It is not
clear to me how these connections could be made, but
Rasmus Hedegaard Nielsen
the idea of a radical plural democracy19 and a
democratic citizenship seem to be a concern that can be
traced throughout many fields.
REFERENCES
Apple, M.W. (2001). Educating the ‘right’ way. New York:
Routledge Falmer.
Danish Ministry of Education. (n.d.). Uddannelse. Retrieved May
16, 2003, from http://udd.uvm.dk/?menuid=4515
Gergen, K. (1992, February). From construction in contest to
reconstruction in education. Paper presented at the
Conference on Constructivism in Education. University of
Georgia, Athens, GA.
Hart, L. C. (1999). The status of research on postsecondary
mathematics education. Journal on Excellence in College
Teaching, 10 (2), 3-26.
Kjersdam, F., & Enemark, S. (1994). The Aalborg experience.
Aalborg: Aalborg University Press.
Laclau, E., & Mouffe, C. (1987). Hegemony & socialist strategy:
Towards a radical democratic politics. London: Verso.
Maskell, P., & Jensen, H. S. (Eds.) (2001). Universiteter for
fremtiden. Copenhagen: Rektorkollegiet.
Phillips, D.C. (1995). The good, the bad and the ugly: The many
faces of constructivism. Educational Researcher, 24 (7), 5-12.
Powell, A. B., Knijnik, G., Gilmer, G., & Frankenstein, M. (1998).
Critical mathematics. Proceedings of the First International
Mathematics Education and Society Conference, 45-48.
Powell, A.B., & Frankenstein, M. (Eds.) (1997).
Ethnomathematics. Albany: SUNY Press.
Skovsmose, O. (1994). Towards a philosophy of critical
mathematics education. Dordrecht: Kluwer.
Skovsmose, O., & Valero, P. (2002). Proceedings of the Third
International Mathematics Education and Society Conference.
Copenhagen: Centre for Research in Learning Mathematics.
Torfing, J. (1999). New theories of discourse. Oxford: Blackwell
Publishers Ltd.
Vithal, R., & Skovsmose, O. (1997). The end of innocence: A
critique of ethnomathematics. Educational Studies in
Mathematics, 34, 131-157.
Young, M. F. D. (1998). The curriculum of the future. London:
Falmer Press.
1
My focus is on educational research within the departments of
pure or applied mathematics at the universities, not on the teacher
training colleges. Although in Denmark, the departments of
mathematics at the universities are educating the upper secondary
school teachers (the high school/gymnasium level). These teachers
are educated in just the same way as the students that choose to be
a researcher in mathematics or choose to work in industry or
elsewhere. In fact, at the university of Copenhagen no course in
mathematics education is available for those who later want to be
teachers in high school.
2
I.e., a focus on learning theory, and a certain kind of learning
theory.
39
3
I would like to mention only two numbers, namely the number
of 1,200,000,000 and 7,000,000. The first in the number of persons
in the world today that daily have under 1 dollar to live on. The
second is the number of children that die every year of hunger.
Compare this number to the 5,000 people that died when the World
Trade Center was destroyed.
4
This comment could be applied to what I am doing here; in
some sense this paper is also exploiting the people in hopeless
situations.
5
See for example Powell & Frankenstein (1997), and for a
discussion of ethnomathematics see for example Vithal &
Skovsmose (1997).
6
In Kjersdam & Enemark (1994) there is a presentation of
Aalborg University as a success in the sense that they supply the
employment market with some of the best workers.
7
In Denmark some of the funding of the universities are partly
dependent on the number of students.
8
This means that there is a group of about 20% that finds
learning mathematics easy and a group of about 80% that has
difficulties, at least according to some of the lecturers in the
department.
9
Science includes mathematics in this context. The centre’s
homepage is http://www.naturdidak.ku.dk.
10
As I heard one of the speakers say at the opening of the centre
the 27th of March 2001.
40
11
More correctly I think the concept is partly contested because
the dominant part of the debates actually agrees to a large extent,
but there are also many disagreements.
12
I must admit I am a little uncertain about this formulation. I do
not want to think of the idea of a knowledge society as something
that one can apply or use like a tool to gain power. It is more like
something that is a part of the construction of the way one
perceives the society and one’s identity. I am not sure if I really end
up doing what I do not want in this paper.
13
These thoughts on how ideas, concepts and power interact are
to a large extend inspired by Michael Apple (e.g., 2001).
14
This is my translation of the title ‘Universiteter for fremtiden’.
This book consists of articles written by politicians and others.
15
This is my translation of the title of the Danish magazine
‘Uddannelse’ published by the Danish Ministry of Education (n.d.),
16
Young (1998, p. 156) has identified a similar trend in public
education.
17
This agreement is being negotiated continuously, and has been
the target of many protests especially for making education a
commodity that can be traded as bananas are traded.
18
See
http://www.unige.ch/cre/activities/Bologna%20Forum/
Bologne1999/bologna%20declaration.htm for the text of the
declaration. There is resistance to this declaration from for example
the Attac movement.
19
This is a hegemonic project proposed by Laclau and Mouffe.
See (Torfing, 1999, p. 247-261) for an introduction.
How to do Educational Research?
The Mathematics Educator
2003, Vol. 13, No. 1, 41–45
Mathematicians’ Religious Affiliations and Professional
Practices: The Case of Bo
Anderson Norton III
Bo’s case is the third of three case studies exploring relationships between the domains of religious belief and
mathematical practice among university research professors. As a Buddhist, Bo’s mathematics and religious
views are integrated in a surprising epistemology. His epistemology and other relationships are contrasted by
those presented in previous case studies of a Jewish professor and a Christian professor, at the same university.
While the previous cases highlighted the transfer of methods of practice across domains and the need to
reconcile potentially conflicting aspects of the two domains, Bo’s case reminds educators that each student
holds her own universe of thought and that mathematics plays a prominent role in developing that universe; or
is it “the way of knowing the universe?”
This paper reports on the third of three case
studies, all intended to investigate the implications of
religious affiliation in the professional lives of
mathematicians. These case studies offer contrasting
perspectives in answer to my research question: How
do strong religious convictions influence professional
mathematicians’ practices and their views of
mathematics? The previous cases revealed the need for
reconciliation of mathematical truth and professional
practice with religion in order to make mathematical
practice meaningful. Reconciliation can be difficult
because one realm may supercede the need for the
other (Norton, 2002b). However, in the case of Bo, the
two realms are fundamentally integrated so that,
together, they provide an epistemology.
I selected the three participants for my study
because they had reputations as devout representatives
of three distinct religious groups—Judaism,
Christianity, and Buddhism—among professors in the
mathematics department of a large southern university.
Before conducting one-hour interviews with each of
them I was not certain that I would be able to identify
more than a superficial influence. In fact, the
participants themselves were largely unaware of such a
relation, but as they recounted their personal histories,
evidence of significant connections emerged. For
Joseph, the Jewish participant, religion helped to define
and inform his professional practice of research and
teaching as “meritorious activity;” on the other hand,
Charles struggled through years of conflict before
Andy Norton is currently working on his doctoral dissertation in
mathematics education and master’s degree in mathematics at the
University of Georgia. His research interests include students’
mathematical conjectures and their role in learning. His email
address is anorton@coe.uga.edu.
Anderson Norton III
reconciling his early desire to do research mathematics
with his most fundamental Christian beliefs. Bo’s
situation was different in that he developed his
Buddhist beliefs and his mathematical career while
simultaneously exploring other possibilities in both
realms.
In my analysis of Bo’s interview I identified two
major themes: his belief in cause and consequence, and
his world of quantifiable objects with infinite
coordinates. In this paper I report on these themes
along with Bo’s background and relevant history,
which I use to contrast Bo’s unique perspective with
those of past mathematicians and with the other two
cases. I also include a poetic transcription in order to
give a flavor for Bo’s own language; though I
employed artistic license in the order of phrases, the
words are his (see Figure 1). A detailed account of my
methods for developing both the narratives and the
poetic transcription can be found in Norton (2002a).
Einstein and Bo
Because of their similarities in practice and belief,
I find it especially interesting to contrast Bo’s views
with those of Albert Einstein. I begin here with a brief
summary of Einstein’s philosophy on science and
religion, as reported in his bibliographies. I return to
these points in the discussion section following Bo’s
narrative.
Like Bo, Einstein was a mathematician with
Buddhist views. Though he was a Jew by heritage, he
did not believe in a personal god and instead referred
to a “cosmic religious feeling” (1990, I, p. 26). He
claimed that Buddhism had a strong element of this
feeling. Far from believing that science and religion
were at odds with one another, he claimed, “in this
materialistic age of ours the serious scientific workers
41
are the only profoundly religious people” (p. 28)
because they are able to think abstractly and
universally. In Out of My Later Years, Einstein noted
that “the realms of religion and science are clearly
marked off from each other” in that they answer
different questions (1990, II, p. 26). Still, he
proclaimed, “science without religion is lame; religion
without science is blind” (p. 26). Much of this thinking
is echoed in Bo’s story, though there are some notable
differences of viewpoint and profound differences in
background.
Bo’s Narrative
Bo is a 30-year old Chinese man who has been
living in the United States for about 10 years. He was
raised in a family without religious beliefs, but began
to explore his own beliefs as an undergraduate in
Shanghai. There he studied philosophy, the Bible,
Taoism, Buddhism, and other religions. He found that
Buddhism fit his nature: It offered him a “home for his
mind to rest.” His beliefs were strengthened when he
met a group of Buddhists in graduate school in the
United States.
When Bo was denied admittance for undergraduate
study in physics at the Shanghai University of Science
and Technology, he turned to his second choice:
mathematics. He found that he was better suited for
mathematical study because it offered him freedom
that physics did not—there were no experiments or
computer skills required in the study of pure
mathematics. He went on to receive a Ph.D. from the
State University of New York at Stony Brook.
His interests in mathematics were piqued even
before college, when he learned about infinity. The
infinite still plays a role in his post-doctoral research.
He studies operator theory, a branch of mathematics
that examines behaviors of objects in infinitedimensional space. He feels that this research should
occupy 80% of his time and energy, while the rest is
reserved for teaching.
Cause and Consequence
In Buddhism, there is no personal god controlling
things: “Everything is just cause and consequence.” In
fact, Bo believed in this universal phenomenon of
cause and consequence before learning of Buddhism.
His belief in the phenomenon contributed to his natural
inclination toward Buddhism. Since all is cause and
consequence, he cannot expect someone else to save
him, and this view countered a major tenet of many
western religions he had explored.
42
If you do bad things, you are going to be suffering
from that in the future. If you help other people,
you will be helped eventually. So, it’s a cause and
consequence kind of thing that I believe. And, I
also believe that by purifying one’s own soul…
you get rid of delusions to see your own nature.
You find a way to save yourself.
Bo refers to this purification as “a way to control your
own thoughts.” This is the central theme of his
religion, which provides him with a set of values.
Bo describes thoughts as clouds that come and go.
You use good thoughts to do good deeds and evil
thoughts to do evil deeds. If an evil or bad thought
enters your mind, you can just let it go. “Your mind is
like the sky. A cloud is like thought. They go and
pass.” This approach applies to mathematical study as
well. Any thought that distracts Bo from his research is
a bad thought. Letting go of distracting, bad thoughts
allows him to focus on his research.
Bo emphasizes the importance of being oneself.
This value is based on the nature of life. He believes
that he is defined as a mathematician because
mathematical thoughts are the most frequent thoughts
in his mind. In fact, on his failure to gain admission in
the physics program of his university he says that “life
made a correct decision for me.” Rather than ascribing
this decision to a mindful deity, he refers to the natural
consequence of his failure that suited his nature,
embodied by his decision to study mathematics.
“Being a researcher is a value of [one’s own]
spirit.” Bo finds freedom in mathematics, as he has in
Buddhism. This openness is common to Bo’s nature as
well. Perhaps his value of freedom offers further
explanation of his affinity for both mathematics and
Buddhism. While Buddhism offers him “a feeling of
[being] at home,” mathematics makes him happy. “If it
makes me happy, then I can make friends around me
happy.”
Making others happy is another important religious
value for Bo, and “teaching… is a happy thing to do.”
Bo describes teaching as “telling other people what
you understand” so that they can appreciate your ideas.
He likes teaching because it allows him to interact with
“vibrant students.” He calls teaching “a social value,”
and feels that it is important to practice patience in the
classroom. When students ask repetitive questions or
criticize him in his teaching, Bo keeps a peaceful mind.
Rather than letting negative remarks aggravate him, he
reminds himself “there is no target to be hit” by these
remarks and lets them pass by. This orientation, then,
is another influence of his religion upon his practice of
teaching.
Mathematician’s Religion: Bo
It seems that many aspects of Bo’s profession
make him happy. It is a part of his nature or Karma,
which he knows through a seventh sense—the sense of
being oneself (the sixth sense is consciousness). “Your
Karma was a seed. So, for example, a person may
develop talent for mathematics.” Karma is carried in
the eighth sense—an ever-present and immense store
of knowledge. Because the seed grows, the eighth
sense is the knowledge that “grows out of the seed”
and is like creation. Because the seed is eternal,
attaining knowledge is like discovery. “Many times,
we discover a thing that should be there…So I may
think it is a discovery or I may think it brings back
memory.”
In mathematics too, Bo found that it is difficult or
even impossible to distinguish between the discovery
and creation of knowledge. “Mathematics is like a tree.
It's already there, [but] grows different branches….
Only history can tell…. I don’t think it’s purely
creation or purely discovery. It is in between.”
Infinite Sequences of Coordinates
The strongest relation between Bo’s Buddhism and
mathematics exists in the intertwining of the two
realms resulting in an epistemology that stems from his
belief in cause and consequence. In Buddhism,
everything is an image in the mind that is given by
objects that we cannot otherwise know. “We cannot
say that the thing itself is ‘what what.’ We can only say
that the image it gives us is ‘what what.’” In other
words, we cannot know an object for what it is,
independent of our own unique perspective. Thus there
is already a strong epistemology embedded in
Buddhism. It includes the belief that our knowledge of
objects depends on the observer. “[Bo’s epistemology]
is an association of objects with numbers, because
ultimately we can process numbers in our minds – not
an object itself.” So, in Buddhism, “everything is
understood as a sequence of coordinates.” Bo explains
that because objects can be seen from infinitely many
perspectives, objects must be infinite sequences of
coordinates. These are quantitatives, and it is through
mathematics that people study quantitatives and their
relationships. Thus mathematics is “the way of
studying the universe.”
In mathematics objects are also viewed as
sequences of coordinates. In fact, this aspect of
mathematics is the central focus in Bo’s chosen branch
of study, operator theory. In operator theory,
mathematicians study objects and relations between
pairs of objects in infinite-dimensional space. Bo’s
decision to study operator theory may have risen from
Anderson Norton III
his initial high school interest in mathematics, infinity.
This initial interest in the infinite then may be a
common cause to both his religious and professional
pursuits.
Bo’s religion and mathematics seemed to grow
together in many ways. He was drawn to mathematics
and Buddhism for at least two common reasons: his
nature and his value of freedom. In fact, given Bo’s
theory of Karma, we can say that it was in his nature to
become a Buddhist mathematician. Certainly many
ideas and practices from one domain flow to the other.
In particular, Bo’s view of the world and his means of
understanding it are intertwined with his profession
and his religion.
Discussion
In discussing each of the three cases from my
study, I have used Charlotte Methuen’s four categories
of historical relationships between mathematics and
religion: conflict, independence, dialogue, and
integration (1998). These categories provide general
contexts from which to examine mathematicalreligious influence and to compare these influences
within and across cases (both historical cases and those
from my study.) Methuen recounted the life and
philosophy of the 16th century philosopher, Philip
Melanchthon. Melanchthon clearly fell into the last
category, claiming, “the study of mathematics offers a
vehicle by which the human mind may transcend its
restrictions and reach God” (p. 83). Bo is another
example of integration, where mathematics is the
vehicle to which we are restricted in reaching the
universe. Though some disciples of Buddha may have
been able to transcend this restriction and “know
without thought,” mathematics is his primary way of
knowing.
Bo seems to share the cosmic religious feeling of
which Einstein wrote. He might also agree with
Einstein that religion provides an avenue for abstract
thought that contributes to scientific study. However, it
is not clear for Bo that “the realms of religion and
science are clearly marked off from each other”. In fact
they seem to coalesce into a single realm of thought
that is uniquely mathematical and Buddhist, but can be
neither of these alone.
Mathematics educators can learn from Bo’s
example. Though they might stop short of promoting a
mathematical religion, there is an element of Bo’s view
that educators may want to instill in their students.
Mathematics may not be the way of knowing the
universe, but it certainly provides ways of
understanding it. Bo’s view is an admission that
43
A Feeling of Being at Home
There’s no ultimate consciousness that creates this world
And sets rules for other things to play.
Everything is just cause and consequence—the universal law.
Buddhism gives freedom and a home for my mind to rest.
I have a new feeling of being settled, and I am finding a way
To save myself by purifying my own soul.
I am responsible for my deeds and thoughts, but to be honest
Thoughts themselves are not distinguished by good and evil.
Fighting in battle, you use evil thoughts. Just be watchful of these thoughts.
My mind is like the sky. A cloud is like thought. They go and pass.
I am defined as a mathematician; I cling to mathematical thoughts.
Centering on mathematical problems, I am using a cloud in a drought.
To thoughts, there’s a deep part and a shallow part. The shallow part is
Given by the object that stands in front of you. What is the deep part?
I don’t know. Ability of dealing with image & thoughts is immense.
Limit, infinity, derivative - mathematics is a very freestyle subject.
Between creation and discovery, it’s like a tree branching out.
Only history can tell, but life made a correct decision for me.
Being a researcher is a value of the spirit, and it makes me
Happy. Teaching—interacting with energetic & vibrant
Students—is a social value. It’s a happy thing to do.
Buddhists believe everything is image in our mind.
We cannot really say the thing itself is what what.
See the cup? A fly may see this cup in a different way.
Buddhism gave me another way to look at mathematics.
Mutually, mathematics deepened my understanding of Buddhism.
Everything is understood as a sequence of coordinates.
Every element is described in infinite-dimensional space
Everything has the ability to be infinite: every particle,
Every human, every social event, and mathematics…
It turns out to be the way of studying the universe. Or is
Mathematics just one approach humans adopt to study this world?
Ultimately we can process numbers in our mind – not an object itself.
There was one disciple of Buddha who knew things without thought,
Like when I’m thirsty I know I’m thirsty without thought.
He just expanded this capability. It brings back memory and is there forever.
There are things that exist beyond human sensation and we will never know.
But we should have a peaceful mind and remember that, ultimately,
There is no self.
Figure 1: A poetic transcription of excerpts from Bo’s interviews.
44
Mathematician’s Religion: Bo
humans cannot know the universe for what it is (i.e.,
that an object is “what what”), but that mathematics
offers a myriad of lenses for viewing it—perhaps for
examining different subsets of the infinite coordinates
within it.
This characteristic of mathematics is recognizable
in its employment in the sciences. Chemistry, geology
and economics (to name just a few fields) all use
mathematics in order to explain the biological and
sociological environments of humans. By accepting
particular assumptions and adopting prescribed
methods associated with a field, in a sense one reduces
the study of the universe to a few measurable
coordinates. After all, these presumptions enable
ascription of a cause to a consequence and prediction
of phenomena, yet this pattern of assuming and
ascribing says nothing about truth except that humans
cannot directly perceive it.
There is at least one more aspect of Bo’s view
from which educators can learn: Each human being has
a different view of the universe. Since mathematics is
(at least in Bo’s view) the human way of understanding
the universe, each person might infer that she develops
her own mathematics. That is, people use mathematical
thoughts as they occur in them to satisfy their own
goals. The way people use those thoughts yields
consequences that determine their direction in future
development. In trying to foster development, teachers
must first recognize their students’ universes of
thoughts and then try to determine the causes and
consequences associated with the use of those
thoughts. Moreover, in teaching students, teachers
must understand what motivates student thinking, else
students may let pass the products of teachers’ best
intentions as clouds through the sky.
Because it admits observer-dependent truths (or at
least observer-dependent perceptions of Truth), Bo’s
religious philosophy for mathematics may be the most
desirable for establishing meaning for mathematical
activity without conflicting with religious views.
Clearly Bo’s and Einstein’s mathematical philosophies
were in harmony with their religions, but Einstein
could also have carried on the faith of his Jewish
Anderson Norton III
heritage without abandoning his philosophy. In fact, he
claimed that Judaism already had present in it an
element of this view (1990, I). Certainly in the case of
Joseph (the Jewish participant of my study), there was
a strong religious respect for science and its role in
humanity. In Charles’ Christianity, Charles made a
distinction between God’s knowledge and our own and
believed that man was capable only of “wavering
toward” divine knowledge through trial and error; thus
religious Truth and scientific thought need not conflict
and often compliment one another.
Whatever their religions, in all three cases the
mathematicians felt the necessity of making religious
meaning for their practice and defining the role of
mathematics in their spiritual lives. Charlotte
Methuen’s four categories provide contrasting
descriptors for the relationships between the two
realms in establishing this role. Though Joseph’s
Judaism stood independent of any mathematical truth,
his practices within the two realms overlapped, and he
held a religious value for mathematical study and
teaching as meritorious activity. In that sense, the
relationship between his two practices was also one of
dialogue. For Charles, there was no built-in religious
value for his mathematical practice, so he struggled to
integrate the conflicting domains and find some
religious value for his mathematical practice in serving
God. Bo’s case provides the strongest example of
integration – one that led to an essential meaning of
mathematical study in understanding the Universe.
REFERENCES
Einstein, A. (1990). The world as I see it; Out of my later years.
New York: Quality Paperback Books.
Methuen, C. (1998). Kepler's tübingen: Stimulus to a theological
mathematics. Sydney, Australia: Ashgate.
Norton, A. (2000a). Mathematicians’ religious affiliations and
professional practices: The case of Joseph. The Mathematics
Educator, 12(1), 17-23.
Norton, A. (2000b). Mathematicians’ religious affiliations and
professional practices: The case of Charles. The Mathematics
Educator, 12(2), 28-33.
45
Conferences 2003
CMESG/GCEDM
Canadian Mathematics Education Study Group
http://plato.acadiau.ca/courses/educ/reid/cmesg/cmesg.html
Acadia University
Nova Scotia, Canada
May 30–June 3
ICIAM
International Congress on Industrial and Applied Mathematics
http://www.iciam.org/iciamHome/iciamHome_tf.html
Sydney, Australia
July 7–11
CIAEM
Eleventh Interamerican Conference on Mathematical Education
http://www.furb.br/xi-ciaem
Blumenau-SC, Brazil
July 13–17
PME and PME-NA
Joint International and North American Conference on the Psychology of
Mathematics Education
http://www.hawaii.edu/pme27
Honolulu, HI
July 13–18
JSM of the ASA
Joint Statistical Meetings of the American Statistical Association
http://www.amstat.org/meetings
San Francisco, CA
Aug. 3–7
ISA RC04
International Sociological Association Research Committee on Sociology of
Education Midterm Conference Europe 2003
http://www.ucm.es/info/isa/rc04.htm
Lisbon, Portugal
Sep. 18-20
GCTM
Georgia Council of Teachers of Mathematics
http://www.gctm.org
Eatonton, GA
Oct. 16–18
SSMA
School Science and Mathematics Association
http://www.ssma.org
Columbus, OH
Oct. 23–25
RUME
Research in Undergraduate Mathematics Education
http://www.math.la.asu.edu/~hauk/arume
Phoenix, AZ
Oct. 23–26
AMTE
Association of Mathematics Teacher Educators
http://amte.net
San Diego, CA
Jan. 22-24
AERA
American Education Research Association
http://www.aera.net
San Diego, CA
Apr. 12-16
NCTM
National Council of Teachers of Mathematics
http://www.nctm.org
Philadelphia, PA
Apr. 22-24
PME-NA
North American Conference on the Psychology of Mathematics Education
http://www.pmena.org
Toronto
Ontario, Canada
Oct. 21–24
Conferences 2004
46
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In this Issue,
Guest Editorial… A Learning Environment Crippled by Testing: A Student Teacher’s
Perspective
AMANDA AVERY
Testing the Problem-Solving Skills of Students in an NCTM-Oriented Curriculum
CARMEN M. LATTERELL
Assessment Insights from the Classroom
NORENE VAIL LOWERY
Designing and Implementing Meaningful Field-Based Experiences for Mathematics
Methods Courses: A Framework and Program Description
AMY ROTH MCDUFFIE, VALARIE L. AKERSON, & JUDITH A. MORRISON
How to Do Educational Research in University Mathematics?
RASMUS HEDEGAARD NIELSEN
Mathematicians’ Religious Affiliations and Professional Practices: The Case of Bo
ANDERSON NORTON III