Quantitative Reasoning Important to Assess?
Should a Basic Competency in QR Include?
About a Minimum QR Competence at College Entrance?
a QR Test on Current System
What Do We
Currently Know About the Quantitative Reasoning Skills of WWU
the Challenge: Some Models for Assessing Quantitative Reasoning
Marcia Davidson and Gary McKinney
WHAT IS QUANTITATIVE REASONING?
1997, the Washington State Legislature directed the Higher Education
Coordinating Board (HEC Board) to implement a budget-based
accountability system. From this directive, four assessment
initiatives were developed. Two of these - writing and critical
thinking - are familiar concepts to most educators. Two others -
information technology literacy and quantitative reasoning - may be
relatively new concepts, or at least relatively new terminologies.
The May, 2000, Dialogue (Issue No. 6) presented an overview of
information technology literacy. In this issue, an overview of
quantitative reasoning will be presented.
quantitative reasoning (QR) is assumed to be synonymous with
mathematics, and, indeed, the two are inextricably linked. Yet there
are differences, one of which is that while mathematics is primarily
a discipline, QR is a skill, one with practical applications. A
mathematician might take joy in abstraction, but the well-educated
citizen can apply QR skills to daily contexts: for instance,
understanding the power of compound interest or the uses and abuses
of percentages; using fundamental statistical analysis to gauge the
accuracy of a statistical study; or applying the principles of logic
and rhetoric to real world arguments.
Moreover, while mathematics is often exclusive, frequently
with a language of its own, QR is inclusive, its language plain and
everyday. In our information-rich - some might say
information-overloaded - society, QR skills are especially
important. We may no longer need to perform quantitative
calculations by hand, but we do need to interpret them and judge
their accuracy. Few people are trained to work with complex
mathematical concepts, but all educated citizens should be able to
understand mathematics well enough to develop informed opinions
about quantitative concepts.
illustrate the point, here are some test questions taken from a
freshman Quantitative Reasoning Study Packet at Wellesley College.
Answering them requires quantitative skills that most educators
would agree all educated citizens should
- Officials estimate that 320,000 Boston-area party-goers
attended the 1995 Independence Day celebration on the banks of the
Charles River. They also estimate that the party-goers left behind
40 tons of garbage. Given that a ton equals 2,000 pounds, how many
pounds of garbage did the average party-goer leave behind?
- One year ago, a person invested $6,000 in a certain stock.
Today, the value of the investment has risen to $7,200. If,
instead, the person had invested $15,000 one year ago instead of
$6,000, what would the investment's value be today? (Assume that
the investment would increase by the same proportion.)
- According to the Cable News Network (CNN), the number of
injured in-line skaters (or "roller-bladers") was 184% larger in
1994 than it was in 1993. Did the number of injured skaters almost
double, almost triple, or more than triple?
WHY IS QUANTITATIVE REASONING IMPORTANT TO
a state-mandated accountability measure
arguably not the most important reason to address QR as a component
of a complete education, it is one of four state-mandated student
learning outcomes. (As mentioned, writing, critical thinking, and
information technological literacy are the others.) Western
Washington University is leading the state effort in developing an
assessment of student learning in quantitative reasoning. We will be
developing a plan for assessing QR on our campus and will provide a
progress report to the Higher Education Coordinating Board later
a student learning outcome
For most educators the more important reason to assess QR is that
in order to become educated citizens students should graduate from
college with some level of competence in quantitative reasoning.
"For students in non-quantitative majors, the appropriate demand
is that QR instruction act as a basic element of the 'liberal arts'
curriculum; that it prepare graduates to function well as citizens
in modern society. Many students do not learn sophisticated math
skills, but all should be able to use simple math tools to reason -
to understand, interpret, critique, debunk, challenge, explicate,
and draw conclusions. In short, college graduates should be able to
evaluate the crush of quantitative data modern life throws at all
WHAT SHOULD A BASIC COMPETENCY IN QR
According to the Mathematical Association of America (MAA)3 ,
the following quantitative literacy (or QR) requirements should be
established for all students who receive a bachelor's
- Interpret mathematical models such as formulas, graphs,
table, and schematics, and draw inferences from them.
- Represent mathematical information symbolically, visually,
numerically, and verbally.
arithmetical, algebraic, geometric, and statistical methods to
- Estimate and check answers to mathematical problems in
order to determine reasonableness, identify alternatives, and
select optimal results.
- Recognize that mathematical and statistical methods have
example of what QR competency might look like is found in Wellesley
College's "Quantitative Reasoning Requirement". At Wellesley, all
freshmen are required to take a QR placement test. If they don't
meet minimum standards, they must enroll in QR 140, a course that
brings them up to competency. Once they have completed QR 140 (of if
they have already passed the QR placement test) students must, at
some point in their academic career, take a QR overlay course,
designed to "engage students in the analysis and interpretation of
data in a scientific or social context."4 The overlay course is
intended to provide students with a basic understanding of important
ways that numerical data are used in problem solving. Overlay
courses are offered in the humanities, sciences, and social
sciences. They have five basic goals (note that these goals echo the
Quantitative Literacy requirements established by the
- LITERACY: Topics and depth of coverage enough so
that students have the knowledge they need to function in
real-life situations involving quantitative data.
- AUTHENTICITY: Students use authentic numerical data
whenever possible. The experience should arise naturally from the
context of the course.
- APPLICABILITY: Examples should be adequate to
convince students that the methods of analysis they learn are of
general applicability and usefulness.
- UNDERSTANDING: It is important that student learning
go beyond rote application. They should be able to recognize when
they can apply what they have learned in the future.
- PRACTICALITY: Breadth and depth of topics should be
consistent with reasonable expectations of students when data
analysis is only part of a course requirement.
WHAT ABOUT A MINIMUM QR COMPETENCE AT COLLEGE
we already use test scores to establish the abilities of in-coming
students - for example, from the SAT and Math Placement Test - we
should be taking a measure of students' QR abilities. As mentioned
earlier, the connection between mathematics and QR is close, yet
also different - in all likelihood, different enough to warrant
taking stock. An additional test, however, does not necessarily mean
subjecting students to more tests; a slight remolding of the
existing testing framework would also work. Students anticipating
they will major in science or mathematics could continue to take the
Math Placement Test (MPT); students anticipating they will major in
non-science or mathematics could take a Quantitative Reasoning Test
Alternatively, all students, other than those excepted under
existing guidelines, could take a hybrid test that included elements
of the existing MPT, plus additional QR-related questions.
Regardless of the direction decided upon, new test questions - or
possibly an entirely new test - would need to be developed.5 This
logistical concern would need to be factored into any changes to the
current system. Most importantly, any revamped, or entirely
brand-new tests would need to reflect the anticipated curricular
IMPACT OF A QR TEST ON CURRENT SYSTEM
Logistically, the idea of two tests - MPT and QRT - would
create a fork: one leading to mathematics-intensive courses and/or
majors, the other leading to less mathematics-intensive courses and
majors. In circumstances where a student took the QRT, then later
decided to major in science or mathematics, the departments affected
could either accept the QRT results, or require the student to take
the MPT. Even if a second test were required, the amount of testing
for such students would have, practically speaking, a minimum impact
of current testing logistics.
idea of an integrated mathematics/QR test would not lead so
obviously to a fork and has the rhetorical advantage of creating the
sense of a more integrated curriculum, a sense that the University
values equally science and humanities, mathematics and quantitative
Whichever test were developed and implemented, the next
decision to make would be what to do with the scores - which would
depend entirely on the changes in curricular policy that are
adopted. If no curricular policy change is anticipated, then of
course the whole question of even having a test is moot.
curricular changes are anticipated, then the issue of testing
transfers arises, and has a very different impact. Currently,
transfers take the MPT only if they are going to enroll in a
mathematics course at Western. (Although transfers who have taken
calculus are not tested.) Depending on how QR is eventually woven
into Western's academic fabric, this practice might have to change.
If, for instance, courses with a strong QR component are developed
and/or recognized, or if certain courses get infused with a stronger
QR component, and these become required for graduation, transfers
not anticipating taking a mathematics course at Western might be
required to take the QRT for the same reason they take the MPT: to
find out if they are academically prepared for certain courses. If,
however, QR is concurrently woven into the curricular fabric in the
community colleges, then assessing a transfer's QR competence might
be addressed any number of ways, with the QRT being only the most
obvious. There could also be transcript reviews; articulation
agreements could be reviewed and revised, etc.
WHAT DO WE CURRENTLY KNOW ABOUT THE QUANTITATIVE
REASONING SKILLS OF WWU STUDENTS?
Currently, there is no quantitative reasoning general
education requirement (GER) at WWU. Students are offered a variety
of methods to satisfy their GER requirement in mathematics, and the
Math Placement Test (MPT) is administered to most students to assist
in placing them in appropriate courses, but there is no general
quantitative reasoning test or course available. Thus, we have no
data on the QR skills of our students.
MEETING THE CHALLENGE: SOME MODELS FOR ASSESSING
QUANTITATIVE REASONING SKILLS
been mentioned, some kind of QR education has been considered as an
alternative GUR to the current math requirement. In October, 2000, a
group of faculty from the six public baccalaureate institutions met
at a colloquy in Leavenworth, Washington, to discuss how to define
and assess quantitative reasoning in higher education. Based on
those discussions, but subjectively reinterpreted by the authors of
this paper, three models for the assessment of QR skills will be
presented below. In the first, there is a rubric, but no QR
pre-test; in the second, a QR pre-test is the model's centerpiece;
and in the third, aspects of the first two are combined into a
MODEL 1: This first model is based on a rubric that
examines the QR content that already exists in the curriculum of
major disciplines and/or departments. Each discipline and/or
department would need to be actively involved in developing rubrics
that worked within their area. As an example, faculty in the social
sciences who were attending the colloquy described seven
statistics-based research components that could be considered fairly
generic in their area, and that would need to be included in any QR
rubric applied to the social sciences. (It was also noted that not
all social science courses contain these research components;
nonetheless, the guiding principle of this model is that disciplines
and/or departments are the most appropriate base for QR assessment.)
This model does not use a pre-test of QR abilities or skills. It
rather takes an inventory of QR skills and/or abilities that already
exist within the curriculum, makes a vigorous public acknowledgement
of the existence and importance of QR, and encourages the emphasis
of QR in higher education. It may or may not stipulate that a
student must take one of the courses identified as having a strong
QR component in order to satisfy graduation requirements. The major
pitfall of this model is it may not meet the legislative mandate for
MODEL 2: This model borrows heavily from the Wellesley
model described earlier in this paper. The heart of this model is a
pre-test of QR skills, administered to all in-coming students,
either as an addition or an alternative to the MPT (math placement
test). Students unable to meet the minimum standards of QR
competency, as based on the pre-test, would enroll in a basic QR
course. Those meeting the standards would be required to enroll in
an "overlay" course in which QR concepts would be identified and
evaluated in courses in the major disciplines. Students required to
take the basic QR course would also be required to enroll in an
"overlay" course as a stipulation to graduation. As in MODEL 1, a
discipline and/or department QR rubric would be vital in identifying
appropriate "overlay" courses for students to take. When talking
about course logistics and the best use of resources, the idea of
using already existing courses as part of a QR requirement would be
far superior to creating a slew of new ones.
MODEL 3: This model would not require a pre-test in
QR, but would require all students to take a basic QR course in
addition, or as a supplement to the University's current mathematics
stipulations. This model may or may not include the additional
requirement of a QR "overlay" course, although a discipline and/or
department rubric would still be a sensible, probably necessary plan
of action, if for no other reason than to create a broader sense of
QR's current existence within the curriculum, as well as to respond
to legislative concerns.
of the above models, or parts of them, were to be adopted into
Western's higher education goals, how will we know if those goals
have been reached? Regardless of the domain - QR, writing, ITL,
critical thinking - this problem is central to all student learning
outcomes assessment. Are exit tests the solution? This strategy has
its appeals - straight-forwardness, the promise of quantifiable
results, a clean finality - but is also fraught with problems. For
example, who would pay for the costs? Do we pass them along to
students? What about the logistics of having 2500 people a year sit
down for such testing? Do we impact course/class work by setting
aside a week of testing days? What about the messy work of assessing
writing? No one yet has come up with a way of feeding a computer a
sample of student writing and have it spit out a meaningful rating.
And what, pray tell, happens if students don't meet the standards?
Do we force them to repeat courses? Do we hold them back? Yet of all
the legitimate concerns over exit testing, what educators fear most
is that exit tests lead inexorably to instructors teaching to the
test rather than their hearts, thereby stifling academic creativity,
flexibility, and innovation.
Certainly a more practical student learning outcomes
assessment tool are satisfaction surveys. Our current ones could be
modified to include questions about QR, Information Literacy, etc.
Yet such surveys never seem quite enough. They're a "soft" measure,
subjective and not as representative of our student population as
most researchers would like them to be. While the surveys themselves
could be more finely honed, and return rates could be increased,
satisfaction surveys would only ever be one part of a fuller student
learning outcomes assessment effort.
Probably QR assessment will need to tie in to the current
student learning outcomes assessment technique: allowing a
well-designed, thoughtfully considered curriculum to take care of
the end results by itself. Yet this approach, too - as steeped in
tradition and as well-intended as it may be - often feels "soft",
especially by today's higher academic standards, driven by the need
for objective quantifiable data.
Something relatively new that might help this student
learning outcomes conundrum are the performance standard rubrics
that have been developed, or that are being developed for the
domains established by the legislature. (To date, rubrics have been
developed for writing and critical thinking, while rubrics for
information technology literacy and quantitative reasoning are in
process.) What's useful about these rubrics is how they delineate
the expected performance abilities of students at various levels
along their academic careers. That these rubrics have been so
diligently thought out and produced is a very good thing - some
might even argue that it's a process long overdue. Content standards
have been debated and produced seemingly since the inception of
education; they are, at least currently, the bases for most
curricular decisions. On the other hand, clearly articulated
performance standards have been either sorely lacking or all but
invisible to those outside academia.
having the rubrics is good, but what is going to happen to them once
they are honed and ready for use? Will they be used as part of an
exit test? Oops. There's that again. But if not an exit test, what
about the idea of applying the standards of each domain's rubric to
the curriculum? What if from day-one to day-last of a student's
academic career each course contained some of, maybe even sometimes
all of the domain skills, with those skills then assessed according
to the standards proposed by the rubrics?
by way of promoting a point of discussion, we are going to suggest
in this Dialogue the idea of student learning outcomes assessment
serving as a matrix - and here we are defining matrix as that which
gives form to a thing. We are thinking specifically of calling it an
institutional matrix might work is this: a) here are some domains of
knowledge - writing, QR, ITL, critical thinking - that we want to
send our students into the world with as an educated citizenry; and
b) here, as laid out under each domain, are the rubrics establishing
the performance standards for those domains of knowledge. With the
domains and rubrics in place, c) each instructor incorporates into
his or her courses as many of these domains as is appropriate and
reasonable; and d) utilizes the rubrics (also as appropriate and
reasonable) to assess student learning outcomes.
might be helpful to think of this institutional matrix as an overlay
to most of the existing pedagogical techniques already practiced and
courses already taught. It's worth is as a curricular clarifier, an
intensifier, a focuser, the tool with which murky areas can be made
clear again, not just for those of us in academia, but for those
outside it - legislatures, taxpayers, the parents of our students.
For those folks especially clarity is of utmost importance, yet also
what many of them feel academia lacks.
current system of courses, grades, capstone experiences, et al., is
not necessarily bad or out-dated, but it is sorely lacking an
overlay of contemporary clarity. It may be that in the course of
applying such an institutional matrix there will be some fundamental
rethinking of our goals as higher educators; maybe there will need
to be some discussion about the balance between life-long learning
skills versus content knowledge in the various departments and
majors, but such discussions - and maybe even the reevaluations that
might come from them - will not hurt us, but rather might renew the
energies of all those involved in higher education.
math are inextricably linked, but while math is primarily a
discipline, QR is primarily a skill - one with practical
applications. Moreover, while mathematics is often exclusive,
frequently with a language of its own, QR is inclusive, its language
plain and everyday. Few people are trained to work with complex
mathematical concepts, but most can understand mathematics well
enough to develop informed opinions about quantitative concepts.
expectation that college graduates will demonstrate competence in
quantitative reasoning is broadly supported on both the state and
national levels; however, the assessment of QR skills in
undergraduates is complicated. For one thing, QR crosses domains.
For another, important assessment questions have yet to be
addressed: what QR skills should students enter with? what should
their QR skills look like when they graduate?
suggested three different assessment models that might achieve the
goal of determining whether our graduates meet acceptable standards
of QR competence. Moreover, we have suggested the idea of an
institutional matrix focused in the general education program that
would support, sustain, and encourage the development of student
learning outcomes from each of the four state initiatives -
quantitative reasoning, information technology literacy, writing,
and critical thinking - which form the core of a new and exciting
assessment arena for higher education in our state. Such a broad
scale approach to student learning outcomes has enormous potential
to impact how faculty think about student learning in all courses
Answers: 1. 1/4 lb/per party-goer; 2. Value now is $18,000; 3. More
than doubled, almost tripled.
2. Simpson, C. (November, 1999)
Quantitative Reasoning (QR) Progress Report (page 2). Office of
Institutional Research and Resource Planning. Western Washington
3. QR for College Graduates: A complement to the
Standards and the MAA's Subcommittee on Quantitative Literacy
Requirements (Committee on the Undergraduate Program in
4. Guidelines for the Quantitative Reasoning
Overlay Requirement, page 4.
5. To see more examples of QR
questions and/or tests, see contacts at the bottom of page 6 of this
Office of Institutional Assessment and
Dr. Joseph E. Trimble, Director; Gary R. McKinney,
technical assistance by Center for Instructional
Dr. Kris Bulcroft, Director; Web Design by Karen
For copies of Dialogue, OIAT technical reports, Focus
Research Summaries, or InfoFacts, please contact Gary McKinney,
Western Washington University, MS: 9010, Bellingham, WA 98225.
Telephone: (360) 650-3409. FAX: (360) 650-6893. E-mail: mailto://email@example.com. TTY:
(800) 833-6388. Join in discussions of Dialogue issues on the web
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