Case Studies for Quantitative Reasoning
A Casebook of Media Articles
by
Bernard L. Madison, Stuart Boersma, Caren Diefenderfer, and
Shannon Dingman
Fayetteville, Arkansas
Ellensburg, Washington
Roanoke, Virginia
Supported by the National Science Foundation
Grant # DUE-071503
ii
Contents
Table of Contents
page
Introduction
Section 1 Using Numbers and Quantities
1. Introduction to numbers and quantities
2. Case Study 1.1 Numbed by the Numbers & Three Bad Numbers
3. Case Study 1.2 Harps Supermarkets Advertising Poster
4. Case Study 1.3 What $1.2 Trillion Can Buy
Section 2 - Percent and Percent Change
1. Introduction to percent and percent change
2. Case Study 2.1 Tax Rates Case Study
3. Case Study 2.2 Other People’s Money
4. Case Study 2.3 Big Stink in Little Elkins
5. Case Study 2.4 Trainees fueling agency’s optimism
6. Case Study 2.5 More Mothers of Babies Under 1 Are Staying Home
Section 3 – Measurement and Indices
1. Case Study 3.1 Forbes hospital rates high in health care
2. Introduction to Indices
3. Case Study 3.2 Tell the Truth: Does this Index Make Me Look
4. Case Study 3.3 Market Gauges
5. Case Study 3.4 GOP Disputes and Is the glass half full …
6. Case Study 3.5 FOXTROT cartoon
7. Case Study 3.6 Real Estate Track and Median home prices fell
Section 4 – Linear and Exponential Growth
1. Introduction to interest on money
2. Introduction to weighted averages
3. Case Study 4.1 Credit Card Disclosure Statement
4. Case Study 4.2 Words of a Guru: Math, Plain and Simple
5. Case Study 4.3 Forcing fuel efficiency on consumers doesn’t work
Section 5 – Graphical Interpretation and Production
1. Case Study 5.1 Enrollment Rates Rise and UA Enrollment up
2. Case Study 5.2 Number of Students on Central Plaza
3. Case Study 5.3 Two Views of a Tax Cut graphic
4. Case Study 5.4 Sexual Betrayal Graph
5. Case Study 5.5 Decade After Health Care Crisis, Soaring Costs
Section 6 – Counting, Probability, Odds, and Risk
1. Introduction to counting
2. Introduction to probability, odds and risk
3. Case Study 6.1 Why Journalists Can’t Add
3
Contents iii
4. Case Study 6.2 The Odds of That
Additional Exercises
1. Section 1
2. Section 2
3. Section 3
4. Section 4
5. Section 5
6. Section 6
1
Case Studies in Quantitative Reasoning
Introduction – Case Studies for Quantitative Reasoning
Residents of the United States face critical quantitative reasoning (QR) challenges
more often than residents of any society in history. These challenges occur for two
major reasons:
 Personal prosperity in US society requires numerous QR-based decisions.
 Sustaining US democratic processes requires citizens who can reason
quantitatively.
These two demands for QR are required of all US citizens and are in addition to the
QR demands of the workplace, which vary depending on the area of work.
This casebook provides a tool for educational response to the enormous QR demands
that US residents face. It is the foundation for developing and delivering an everfresh, real-world-based course that starts or moves students down a path toward
quantitative literacy (QL). The terms QL and quantitative reasoning (QR) are used
interchangeably, and other terms such as numeracy essentially have the same
meaning.
The contents of this book are governed by two criteria:
 Every QR problem is a contextual problem, that is, the quantitative reasoning
is about a circumstance that is embedded in or grows out of a real-world
context.
 Every mathematical or statistical topic investigated is one that is contained in
or useful in critiquing a public media article.
This book contains twenty-four case studies of public media articles, mostly from
newspapers. Also included are introductory notes and exercises on the basic concepts
of understanding and comparing
Quantitative literacy (QL) is a habit of mind,
quantities; percent and percent
and, consequently, achieving QL requires both
change; indices; interest on money; extensive interaction between students and
weighted averages; counting; and
teachers and practice beyond school. At the
collegiate level, we are concerned with a high
probability, odds and risk. Each of
level of QL, befitting persons with baccalaureate
the articles contains quantitative
degrees, analogous to what Lawrence Cremin
information, analyses, or
(1988) termed liberating literacy, as opposed to
argument. These case studies are
inert literacy. Therefore, the QL we seek
meant to be both items of study
includes command of both the enabling skills
needed to search out quantitative information
and examples of case studies that
and power of mind necessary to critique it,
students and teachers can create
reflect upon it, and apply it in making decisions.
using public media articles from
the present day, keeping the
Cremin, L. A. (1988). American education: The
material fresh and more obviously
Metropolitan experience 1876-1980. New York.
NY: Harper & Row.
relevant. Students discovering and
presenting for class discussion
articles that illustrate QR increase essential student engagement in the course. By
creating case studies and discussing those in this book, students will develop
Introduction
2
reasoning skills and disposition toward continuing practicing those skills beyond a
course and beyond school. They will develop a habit of mind to reason quantitatively
in their everyday lives as citizens, consumers, and workers.
A course based on this book will differ in many ways from traditional courses in
mathematics or statistics. First, the book is not organized by mathematical topic,
rather by the reasoning domains required to understand and critique articles in public
media. Second, the exercises and study questions are not versions of template
problems organized by solution methods. Third, and most important, the exercises
and study questions are designed to address common, challenging quantitative
reasoning situations in daily life.
This approach to teaching toward quantitative literacy (QL) is based on the
assumption that there are canonical QL situations that students need to address and
resolve. These situations very often involve the following steps:





Encountering a challenging contextual circumstance, e.g. reading a newspaper
article that contains the use of quantitative information or arguments.
Interpreting the circumstance, making estimates as necessary to decide what
investigation or study is merited.
Gleaning out critical information and supplying reasonable data for data not
given.
Modeling the information in some way and performing mathematical or statistical
analyses and operations.
Reflecting the results back into the original circumstance.
These steps often require careful reading of both continuous prose and discontinuous
prose (such as graphical representations), using mathematics or statistics, and then
interpreting and critiquing the original prose in light of the mathematical results.
Critical reasoning is required throughout. In general, students are not expecting this
complicated process because their previous mathematics experiences have been
narrower and more precisely defined. Frequently, the fourth phase gets the most
attention because it is the process of traditional mathematics and statistics courses.
Many of the problems are ill posed and require reasonable assumptions to resolve,
and many of the problems have multiple reasonable responses. Consequently,
conclusions require explanations of reasoning that led to the conclusions.
Quantitative Reasoning Proficiency
The model of mathematical proficiency as described in the National Research
Council publication Adding It Up1 is helpful in understanding the challenges of QR.
In the model, mathematical proficiency has five intertwined strands (See the
1
Kilpatrick, J., Swafford, J., & Findell, B., Eds. (2001). Adding it up. Washington, DC: National
Academies Press.
3
Introduction
accompanying box). Whereas some traditional mathematics courses depended
strongly on the second strand, procedural fluency, all five of these strands are critical
for QR, and the last two listed, adaptive reasoning and productive disposition, take on
added importance for proficiency in QR. This stems from the fact that QR is a habit
of mind and requires adapting reasoning to numerous unpredictable contexts.
Productive disposition is often missing from students and non-students when it comes
to understanding and critiquing quantitative material. The plaints by students, “where
will I ever use this” and “I was never good at math,” run counter to productive
disposition because
Mathematical Proficiency from Adding It Up
they question the
 conceptual understanding – comprehension of
relevance of school
mathematical concepts, operations and relations
mathematics and
 procedural fluency – skill in carrying out procedures
one’s own efficacy
flexibly, accurately, efficiently, and appropriately
 strategic competence – ability to formulate, represent,
to understand and
and solve mathematical problems
use it.

adaptive reasoning – capacity for logical thought,
reflection, explanation, and justification
productive disposition – habitual inclination to see
mathematics as sensible, useful, and worthwhile, coupled
with a belief in diligence and one’s own efficacy.
Assessment

The nature of the
exercises and study
questions in this
casebook requires new thought about assessing student work. That which is valued in
a response to a QR problem situation differs from that which is valued in traditional
disciplinary-based courses, especially those in mathematics or statistics. Assessing
QR involves judging written analyses and reflections and the quality of evidence
given in support of arguments or conclusions. Because of these differences, frequent
reminders are necessary for students to supply the following:





Evidence supporting reasoning or assertions.
Calculations that produce numerical results.
Correct units on quantities.
Complete and correct sentences stating evidence and conclusions.
Precision of language in stating questions and results.
Content of Casebook
As mentioned above, there are three different types of materials in this book:
1)
2)
3)
4)
Introductory notes on basic concepts
Warm up exercises on the basic concepts involved in case studies
Articles that are the subject of case studies
Study questions on the articles
For students who need more practices with some of the types of problems
encountered in the six sections, there are additional exercises in the back of the book.
The questions and tasks take different forms, including the following:
Introduction
4
a) Identifying and reporting quantitative information and arguments from the
articles.
b) Developing additional quantitative information from information in the
articles.
c) Critiquing the arguments, analyses, and conclusions of the articles.
d) Extending the arguments beyond those in the articles.
e) Research and reporting on concepts related to the articles.
The questions and tasks in the case studies of the articles can be used in various ways:
discussed in class; some assigned for student responses; all assigned for student
responses; or discussed in groups of students in class to produce a group response.
The content is arranged in six sections, sorted by basic concepts that occur in the
articles, but various concepts recur throughout the case studies. Occasionally,
concepts that are unfamiliar to students are encountered without full explanation. For
example, it is assumed that students can produce graphs of linear and exponential
functions. Skipping sections will likely not mean that material that is needed later is
being omitted.
Prerequisites
Proportional, graphical, statistical, and algebraic reasoning are required to analyze the
cases in this book and numerous similar cases that can be developed by students and
teachers. Basic knowledge of algebra, descriptive statistics, and proportionality is
necessary, but there is little dependence on algorithms and complex mathematical
concepts. No knowledge of trigonometry, analytic geometry, or calculus is assumed,
but the ideas of all three (proportionality, geometric reasoning, rate of change,
approximation, etc.) are very helpful in fully developing the study of various cases.
In terms of course prerequisites, students need to have a working knowledge of
middle school mathematics and high school algebra (or college algebra). The
sophistication of the case studies derives mostly from the contexts that span
economics, sociology, politics, government policies, entertainment, health, and
measurement.
Acknowledgement
We gratefully acknowledge support of the National Science Foundation for the
further development and expansion of this course (DUE-0715039).
Bernard L. Madison
Shannon Dingman
Fayetteville, Arkansas
Stuart Boersma
Ellensburg, Washington
Caren Diefenderfer
Roanoke, Virginia
5
Introduction