STANDARDSBASED EDUCATION AND ITS IMPLICATIONS FOR
MATHEMATICS FACULTY
Robby Robson, Oregon State University & M. Paul
Latiolais, Portland State University
I. INTRODUCTION
Throughout this century, higher education has played an important role in
setting standards for high school graduation college admission. This is
manifested by the system of Carnegie units and standardized tests such as
the SAT
^{1}. Considerably less effort has been
expended on setting standards for higher education itself. This is
changing. Calls for accountability, globalization and online competition
in the higher education industry, and increases in the percentage of the
population seeking degrees all make it necessary to better define and
assess the outcomes of a college
education
^{2}.
What does this mean for mathematics departments and for those of us who
teach mathematics at the college or university level? First, we believe
that our profession will be called upon to define standards for degrees and
courses. We may well see a new structure for our degree programs. Second,
we feel we will see significant changes in classroom practices, especially
with regard to assessment. Third, we foresee structural changes that will
affect placement, advising, and the way students advance through a
major. Our purpose here is to discuss these changes. We will give a general
overview of standardsbased education and then detail some of its possible
implications in concrete terms.
II. WHAT IS STANDARDSBASED EDUCATION?
What are Standards? In education the word "standard"
is used in many different ways^{3}. Some of
the ones relevant to our discussion are:
Content Standards: Content standards describe the knowledge and
skills that students should attain in a class or course of
study. (Example:"understand the concept and be able to apply matrix
algebra to compute the affect of a linear transformation on a
finitedimensional real vector space.")
Curriculum Standards: Curriculum standards describe general goals
or ways in which classes or programs should be organized and taught.
(Example: "Recognize mathematical problems that can be linearized and
apply techniques of linear algebra to solve them.")
Competency Standards: Competency standards set required or
desirable levels of performance on specified activities or tests. These are
also called benchmarks, proficiencies, and performance standards. (Example:
"compute the matrix representation of an algebraically defined linear
transformation from R^{m} to R^{n} in a given
pair of ordered bases.")
Performance Standards: Performance standards include competency
standards but are more general. For example, measures of institutional
success are often called performance standards. (Example: "At least
80% of students should be able to demonstrate competency in computational
linear algebra by the end of their junior year.")

Voluntary national mathematics tests^{4}
and standardized tests such as the SAT's represent competency standards.
The NCTM standards provide examples of both content and curriculum
standards. Institutional and professional accreditation boards are
starting to request information on graduation rates and employer
satisfaction with graduates. These are performance standards in the more
general sense.
There is often confusion among different types of standards. This stems
in part from the way in which they interact. Tests like the SAT or GRE are
really competency standards, but they also define a body of knowledge that
must be mastered by a student wishing to score well. As such, they
implicitly define content standards. Similarly, standardized tests dictate
curriculum standards for teachers who wish to adequately prepare their
students. "Teaching to the test" may be viewed as the practice of
translating competency or performance standards into content and
curriculum, if not into content and curriculum standards.
Conversely, it makes little sense to set standards for what students
should learn in a class without having a reliable way to determine whether
students have learned it. The practice of writing exams is that of
translating content standards into assessments. But students appreciate
knowing more than just the time, place, and topics covered on an exam; they
also want to know what types and level of problems will be on the exam and
what constitutes satisfactory performance. Answering these questions
requires translating content and curriculum into competency and performance
standards.
Criteria and Norms. Standards are often linked to specific
assessment methods. When considering these we must consider how the output
of an assessment is interpreted. There is a significant difference between
comparing the results of an assignment or test to those of other students
and interpreting the results in terms of independent criteria. Standards
that reference population norms are called normreferenced while
those that define independent criteria for levels of performance are called
criterionreferenced.
Tests graded on "curves" are examples of normreferenced
assessments. Lower level undergraduate mathematics courses often appear to
have rules (written or unwritten) about the rough percentages of students
who may be awarded A?s and B?s. In this case, grades are
normreferenced standards. At the other end of the spectrum, a fundamental
standard for awarding a doctorate in mathematics is a demonstrated ability
to do independent and original research of publishable quality. This is a
criterionreferenced standard.
Standardsbased Education. Standardsbased education consists of
two interrelated practices. The first is that of explicitly defining
content and, if appropriate, curriculum standards. In more common language,
this means explicitly defining learning objectives. The second is that of
defining associated criterionreferenced performance standards and
assessments. In other words, it is necessary to explicitly say what it
means for students to have achieved the learning objectives and at least
suggest ways in which this can be measured.
A final important aspect of standardsbased education is that the
standards are public. They do not depend on individual teachers and
are supposed to be intelligible to all informed and relevant participants,
including the students. This means that every student should understand the
expectations for satisfactory performance. There is also a policy point to
be made. Educational policy makers often site wide disparities among the
curricula and content offered by public education to different
socioeconomic classes. An A from one high school might not be the same as
an A from another. Standardsbased education attacks this problem by
uniformizing and publicizing standards that apply to all students in the
same educational system^{5}.
III. STANDARDSBASED DEGREES.
Degree Programs. What does it mean to complete an undergraduate
major in mathematics? Most departments define their major in terms of
course work. It is common to have a series of different tracks or options,
a core curriculum, and/or a senior thesis, but the essential requirement is
the successful completion of a set of
courses^{6}.
The standardsbased approach demands that a
major be defined in terms of what students know and can do. It asks us to
define assessments and performance measures by means of which students will
demonstrate competence. And it requires that these standards and measures
be public. Colleagues from other institutions should be able to look at
our standards, examine the evidence collected for our students, and make
consistent determinations as to which students have fulfilled our
requirements.
What would it take to define such standards? Content and curriculum
standards usually start with objectives that are few in number and general
in nature. These are then refined in several stages until they reach the
level of performance standards and assessments. As an example, we might
decide that one of several requirements for obtaining a degree in
mathematics is that a student
demonstrate
the ability to communicate mathematics orally and in
writing. 
This sounds great, but what does it mean? One could argue about
practically every word: What does it mean to demonstrate, what counts as
mathematics, and why should we require both oral and written communication?
More fundamentally, what are we really trying to capture? Does this refer
to formal proofs? Is it enough to be able to describe mathematical ideas or
does this include correct use of notation? If so, at what level of
sophistication? Are we really trying to say that students should be able to
produce results and communicate them as part of a team?
One way to address these questions is to define what are called a set of
criteria for proficiency. Thus we might say that to demonstrate
the ability to communicate mathematics orally and in writing a student
should
 Correctly use mathematical notation to describe mathematical models
and state mathematical results;
 Convey the essentials of a mathematical theory, argument, or model
to peers;
 Write short proofs and/or calculations in an acceptable style.

This is just an example, and is not hard to imagine debating at length
whether these are appropriate criteria, to what extent they must all be
met, whether other criteria should be added, and so on. But even were we to
accept these as criteria, we would still be a long way from defining both
how students can demonstrate they have met these criteria and setting an
appropriate level of performance. The next step is to define what are often
called indicators that move towards performance criteria. For
example, we might say that a student who can state the basic theorems of
undergraduate real analysis on demand has given evidence that she has met
the first criterion, and we might then go on to give examples of acceptable
and unacceptable statements of some theorems. But this would undoubtedly
be only one of a long list of indicators intended to frame the meaning of
1, 2, and 3 in terms that can be consistently used to make judgments by our
colleagues in mathematics and consistently understood by our colleagues in
other departments and by the students themselves.
Benefits and Motivation. The process of defining standards for a
degree program is hardly simple but it has some tangible benefits. Assuming
that the process took place at the department level in a single
institution, it would necessarily engage a department and its constituents
in the important discussion of what we really want our students to take
away from a mathematics degree. We suggest that this discussion will
provide a starting point for rethinking what we need to do to get our
students to where we want them to go. That is not to say that what we do
now is terrible, but every program should undergo periodic review and we
feel that a mathematics major designed by starting with a clear set of
overall objectives and performance standards has the potential to be more
efficient and effective than one put together from a collection of courses.
But the real motivation for thinking about standardsbased degrees is
that the world is changing.
 Many states are implementing standardsbased secondary education
systems^{7}. Our future students will
expect to
measure their progress by means external to individual classes. If
we want to make our major attractive, we should take these
expectations seriously.
 Higher education is attempting to grant access to more people and to
fulfill a role as the educators of "knowledge
workers"^{8}. This demands more
clarity in
communicating with the private sector. A list of courses does not
mean all that much to a prospective employer. A statement that a
student can apply differential equations and computer modeling
techniques to solve problems from other disciplines means more. The
same statement backed by publicly available performance standards
means a lot more.
 Finally, we are embarking on an age in which classes will be
available from many institutions via the Internet. In an educational
economy where courses are available from many vendors, it makes more
sense to phrase requirements in terms of overall degree
outcomes. This helps us maintain the quality of our degree. If
our institution sees itself as competing with others, published
standards may be necessary to position ourselves in the market
place.
Accreditation. If we do see a move towards standardsbased
degrees, one possible mechanism is that of professional accreditation. The
idea of accrediting mathematics programs is one that surfaces
periodically. We do not wish to argue its merits or demerits point out that
the accreditation process itself can effect significant changes in
teaching and assessment practices. This is illustrated by ABET, the
accreditation board of engineering and technology, which switched from a
totally coursebased set of requirements to an outcomesbased set of
requirements for all engineering degrees (see references in note 10). This
is resulting in fundamental revisions of engineering programs
nationwide.
IV. STANDARDSBASED COURSES AND TEACHING
As far as we know, there are no large standardsbased degree programs
in mathematics. But there is experimentation going on with standardsbased
courses in mathematics^{9} and other
disciplines^{10}. What do we mean by a
standardsbased course and how is a standardsbased course designed?
Just as was described for a degree program, the design of a
standardsbased course starts with a definition of what students should
know and be able to do at its conclusion (content standards) and criteria
that define satisfactory performance (performance standards). In its
simplest form this is the practice of specifying a set of learning
objectives. These objectives, however, be "public" and must be measurable
on the basis of tests, projects, and other evidence produced by the
students.
To test if learning objectives are public, it is good to do two thought
experiments. The first is to imagine teaching your class, gathering
evidence of student performance, and having a colleague at another
university evaluate your students using only this evidence and your written
list of objectives and criteria. You should feel confident that your
colleague would make judgments consistent with your own. The second
experiment is to ask yourself to what extent students can reliably
evaluate their own work on the basis of your stated objectives and
criteria. Students should have a clear picture of where they are headed and
be able to recognize when they have
arrived^{11}.
The design of a standardsbased course continues by choosing topics,
activities, and a syllabus that target the objectives. It is important to
note that the topics covered in a course and the learning objectives of a
course are not necessarily the same thing. If the point of a course is
getting the right answer on problems of a specific form or type, then that
should be an explicit objective. Indeed, the students will appreciate
knowing what they need to do in order to succeed. But individual
techniques and derivations are often steps on the way to something more
synthetic and should be included in the course topics but not in the course
objectives.
Standardsbased Assessment. As we hope is by now clear, the
process of defining course objectives is one part of a feedback loop. The
other part is assessment. The requirement that objectives be assessed and
that students generate assessable evidence limits the number objectives we
can realistically expect to reach and demands rethinking what students do
in and outside of the classroom. This might mean more time spent on
projects, more writing assignments, a different style of lecture, using the
Web for drill type quizzing  we each must find what works best for us. It
also means taking a new look at some familiar things such as the way in
which we grade tests.
Standardsbased assessment focuses on the holistic determination of
whether or not a student has met the demands of a particular standard. The
practice of scoring individual test questions and totaling the scores does
not really address this; it does not separate the score into component
objectives and is largely justified by the need to create a class norm and
to compare students to each other. Partial credit is another example of a
practice that is frequently normreferenced and not
criterionreferenced. Students might appreciate knowing that they got a
problem partly right and seeing where they went wrong, but discussions of
partial credit tend to center around issues of fairness and accuracy for
the sake of comparative evaluation.
An alternative method of grading an exam is to review the entire exam
paper in light of each course objective. The idea is to assign a simple
score of "failed to meet", "met", or
"exceeded" for each objective and not grade individual problems.
This is called using a "scoring guide". One could argue for a finer scale,
but we find advantages in the discipline imposed by not being able to
equivocate by saying that a student has met some fractional part of an
objective . Does it really do much good for a student to be able to
almost get the right answers when he or she takes the next
course?
What about multiple choice tests? What about projectbased and other
means of assessment? Many alternative means of assessment, especially those
that involve student writing, seem naturally suited for a standardsbased
system. On the other hand, resources often limit or preclude their use. For
example, we looked at the college algebra course taught in a large
lecture/recitation format at one of our institutions. A quick computation
showed that there are less than 30 minutes of graduate assistant grading
time available for each student for an entire term. This appears to say
that computergraded multiple choice tests will remain a major part of the
grading scheme. This does not, however, mean that multiple choice tests
need to be graded by assigning points to each problem and computing the
total. It would make a lot more sense to identify clusters of problems that
address each course objective and report the scores on each objective. If
computers deliver the exams, it is possible to go beyond that to an
adaptive testing scheme. If a student misses a question on a certain
objective, another one is delivered, and if a student correctly answers
questions on a topic, the program either drops the topic or delivers more
challenging questions.
Example. As an example, learning objectives for a purely
computational matrix algebra course might be stated as
Learning Objectives
To successfully complete this course students must demonstrate the
ability to
 Use matrices to solve systems of linear equations: Translate
equations into matrix form and use Gaussian elimination to find the
complete set of real solutions.
 Compute the range and null space of a real matrix: Given
an m x n real matrix, parametrize its null space and
range, and decide if a given vector is in its null space or range.
 Compute and use properties of the determinant of a square real
matrix: Use expansion by cofactors to compute determinants, apply
the formula det(AB) = det(A)det(B), and determine
invertibility by computing the determinant.
 Find the eigenvalues and associated eigenspaces for a real
matrix: Given an n x n real matrix, compute its eigenvalues and
parametrize each associated eigenspace.
 Choose and apply the above techniques to (an appropriate class
of) problems. (A description of the types and level of problems
would go here.)
The student must demonstrate the ability to do these both by hand
and by using a computer when appropriate.

Note that other topics, such as matrix notation, vectors, row reduction,
the span of a set of vectors, and possibly complex numbers, will have to be
covered in class and might appear on the syllabus. The objectives only
include the overall skills that students are supposed to take away from the
course and that will be assessed. All of these could be tested in
fairly traditional ways or tested in the context of more involved
projects. In the language of standards, the content standards are in
boldfaced type and the statements in regular type are performance criteria
or indicators. The italicized statement at the bottom is a performance
criterion that applies to all of the objectives. To complete this
description, we should include examples of the types of problems we expect
students to accurately solve.
The course described above is computational. What if we also feel that
students should also understand the notions of a vector space, linear
independence, a vector space basis, and dimension, even just as applied to
subspaces of R^{n}? We would then need to add new learning
objectives, but it makes little sense to have "understanding" as
an objective without knowing how students will demonstrate their
understanding. Should students be able to define, compute, manipulate
definitions, derive easy consequences, give examples and counterexamples,
and prove theorems? Should they be able to do more or is this too much too
ask? We need to decide and make our decision explicit. A learning
objective that captures some ways in which a student can demonstrate an
understanding and says what a student knows upon successful completion of
the course might look like:
Define, compute, and derive first consequences from the definitions
of a vector space, basis, and the dimension of a vector space as applied
to subspaces of R^{n}: State the definitions of vector
space, basis, span, linear independence, and dimension; prove results
that follow directly from these definitions; determine if a set of vectors
is a basis for a given subspace of R^{n} ; compute bases
and dimensions for vector subspaces of R^{n} defined by
algebraic properties or as spaces associated with matrices; use dimension
to compare subspaces and draw conclusions about the linear dependence
or independence of a set of vectors. 
V. STANDARDSBASED ADMISSIONS, PLACEMENT, ADVISING, AND
ADVANCEMENT
The last set of implications we will discuss deals with the manner in
which students advance as they progress through the educational system.
This involves a use of standards more familiar from technology than from
education: the use of standards to define the interface and guarantee
interoperability between two parts of a system.
College Admissions. For us, the first point at which two parts
of the educational system meet is at the point of college admission.
Admission standards have been in place for a long time. The traditional
criteria used are high school grade point averages, subject requirements
(also called Carnegie units), and standardized test scores (such as the SAT
and ACT). The validity and biases of standardized tests is a more
controversial issue, but there can be little doubt that grades and subject
requirements do not mesh well with the standardsbased educational reform
occurring in our nation's school systems. Grades are inherently
normreferenced and subject requirements lose their relevance as progress
through school is measured by means external to classes.
Regardless of the reason, the mathematical community simply does not
believe that high school transcripts are adequate indicators of what
students know and can do. Our evidence for this statement is the ubiquitous
placement exam given to entering freshmen. It also seems clear that the
data attached to students entering college in the near future will be new
and different. For example, the new proficiencybased admissions
system in the state of Oregon promises to provide different information on
entering students. In that system students will need to demonstrate
proficiency in a variety of content areas using public criterionreferenced
standards.
Advising. Data coming out of standardsbased educational systems
is new and different and will need to be integrated into our placement and
advising systems. If it is sufficiently good, we may be able to eliminate
placement exams. In Oregon, mathematics comprises seven of thirtythree
individual proficiencies, so we might expect good data. By the same token,
if more emphasis is placed on integrative abilities such as critical
thinking and communication skills, we may have less data on specific
content knowledge and be faced with an increased need for placementtype
exams in mathematics or other areas.
Graduate Admissions. The second point at which two parts of the
educational system meet is at the point of graduation. We have already
discussed the notion of a standardsbased degree in mathematics, but we
have not talked about what this means for graduate admissions. In our
opinion, it means a lot of good things. If standards for mathematics
majors and especially for mathematics courses are public and professionally
viable, we should have a much easier time making admissions decisions and,
more importantly, advising new graduate students as to what classes they
need or do not need to take.
Prerequisites. The final point of interface the is internal
interface between courses and their prerequisites. It seems logical that
if we have standards for the completion of a course, we might also have the
same type of standards for the commencement of a course. If courses have
entrance standards then it should be possible to determine whether a
student is ready for a course by checking that the entrance standards for
the desired course are a subset of the objectives for the courses the
student has successfully completed.
Naturally, some flexibility would be needed, but taking this point of
view might accomplish some useful things. Currently, there is no way for a
transfer student to know if the courses she has taken at one institution
really satisfy the prerequisites for a course at a second institution. In
fact, a faculty member often has trouble making this
determination. Publicly defined standards for entering and completing
courses would alleviate this problem. They are also significant for a
future in which courses are available from multiple institutions via the
Internet and in which students follow nonlinear career paths that have them
stopping in and out of higher education.
VI. CONCLUSION
Might any of this happen or is this all a theoretical exercise? We
admit that we do not know. On the other hand, standardsbased reform is
happening now in secondary schools, virtual universities are touting
competencybased degrees, universities and professional programs are being
asked to define standards for learning outcomes, and other forces at work
in changing the nature of higher education point to standardsbased
education. Our educational system will most certainly need to transform as
we move from the industrial age into the information age and
standardsbased education seems a good candidate for one such
transformation. We therefore feel that the mathematical community should
seriously consider its and be prepared for it if and when it comes
along. We presume there will be no shortage of debate and differing points
of view. We hope that we have started that debate on a positive note by
emphasizing some of the more interesting and potentially helpful aspects of
standardsbased education.
NOTES
1 
In 1893 the Committee on Secondary Schools Studies (better known as
"the committee of ten") was chaired by Charles W. Eliot,
president of Harvard University and had five university professors among
its members. Its work defined what we would now recognize as a standard
academic high school curriculum. The Carnegie Foundation for the
Advancement of Teaching established the Carnegie unit in 1914 "when
colleges began to worry about how to assess high school transcripts"
and the Educational Testing Service was established in 1947 by the College
Board (established in 1900), the American Council of Teachers, and the
Carnegie Foundation with impetus coming from several Ivy League
colleges. See http://www.carnegiefoundation.org/history.htm.

2 
See Albjerg Graham, Richard W. Lyman, and Martin Trow , Accountability
of Colleges and Universities, Columbia University Office of the Provost,
Columbia University, New York, New York. Available online at
http://www.columbia.edu/cu/provost/acu/index.html
for a discussion of accountability and Strategic Planning for Tulane:
Executive Summary (August, 1988) by the Tulane University Strategic
Planning Framework Committee, available online at
http://www.tulane.edu/~strplan/scan.htm,
for a discussion of all of these factors. The many sides of globalization
and the Internet can be seen by looking at the online summary notes from
the American Association of College Registrars and Admissions Officers
1997 Virtual Learning Environments: World Conference and Summit
available online at
http://www.merit.edu/~lmp/aacraorep.html.

3 
See Standards: A Common Language at
http://www.mcrel.org/standardsbenchmarks/docs/chapter3.html.

4 
Information on proposed (but unfunded and unimplemented) voluntary
national mathematics tests may be found at
http://www.nagb.org/.

5 
See, for example: Lynn Olson, Achievement Gap Widening, Study
Reports, Education Week, Dec. 4, 1996, and Kathleen Kennedy Mannzo,
Weighted Grades Pose Dilemmas in Some Schools, Education Week, June
17, 1998. 
6 
Some online references that illustrate this are:
and many more.

7 
In 1997 the American Federation of Teachers reported that every state
with the exception of Iowa and the District of Columbia either had or were
planning to implement state standards. A more recent AFT report with
qualitative analyses of state standards is summarized in Making Standards
Matter 1998 available from the AFT Web site at
http://www.aft.org/edissues/standards98/index.htm. 
8 
Higher education is being called upon to serve an everincreasing portion
of the population. According to the Census Bureau, in 1940, 24.5 percent
of the population over the age of 25 had completed four years of high school
or more. By 1996 this had risen to 81.7 percent. In the same group, 4.6
percent had completed four years of college in 1940 and 23.6 percent had
completed four years of college in 1996. In his 1997 State of the Union
speech President Clinton listed as the one of his ten principles in his
"Call to Action for American Education" that "we must make the 13th and
14th years of education  at least two years of college  just as universal
in America by the 21st century as a high school education is today, and
we must open the doors of college to Americans." 
9 
There is at least one example that has been around for over 25 years,
Alverno College, a small urban catholic women's college in Milwaukee,
Wisconsin. Some writeups, together with some research about the
effectiveness of their "abilitybased program", can be found in the
American Association for Higher Education Bulletin, Volume 36, Number 6,
published February, 1984 by the AAHE, Washington, DC. Other information is
available directly from
http://www.alverno.edu/educators/educators.html.

10 
Such experimentation is occurring on the authors' campuses, and a few
hours of Web browsing turned up truly standardsbased courses in writing,
psychology, health sciences, and other disciplines at a number of
universities. The work that is probably most significant to mathematics is
the work being done in Engineering as a result of new accreditation
criteria being used by the Accreditation Board of Engineering and
Technology (ABET). ABET 2000 is outcomesbased and represents a radical
change from previous curriculum based criteria. See
http://www.abet.org/eac/eac.htm
and How Do You Measure Success? Designing Effective Processes for
Assessing Engineering Education. (1998). Professional Book, American
Society for Engineering Education, Washington, DC. 
11 
When asked to produce learning objectives, faculty typically phrase the
objectives in terms of what a student should "learn", "understand",
"appreciate", and so on. These are good descriptive terms and may even be
somewhat welldefined among professional colleagues, but how useful are
they without a clearly defined notion of how the student is supposed to
demonstrate understanding, learning, or appreciation? How many times have
we encountered a student who did poorly on the test but was nonetheless
confident that he or she had understood, learned or appreciated the
material? 
Robby Robson is an associate professor
of mathematics and the university education reform coordinator at Oregon
State University . He is working on a standardsbased university
admissions policy and supporting faculty efforts at standardsbased
education across the campus. His mathematics tends to be algebraic in
nature and his current other passion is Webbased pedagogy.
M. Paul Latiolais is a professor of
mathematics at Portland State University. His mathematical research focus
has been in Algebraic Topology/ Combinatorial Group Theory. More recently
he has been studying systemic change in Higher Education, as well as
working on an environmental Statistics book and trying to figure out how to
teach proficiencybased courses.
Both Paul and Robby have worked extensively with the Proficiencybased
Admissions Standards System
(http://passous.uoregon.edu)
in writing mathematics standards for college admissions and in
experimenting with standardsbased approaches in their own classes.
You can reach Paul and Robby by email at
paul@math.pdx.edu and
robby@orst.edu.
Copyright ©1999 The Mathematical Association of America
MAA Online is edited by Fernando Q. GouvÃªa
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