Quantitative ReasoningDr. Robert Mayes

Science & Math Teaching CenterUniversity of Wyomingrmayes2@uwyo.edu

Meta-Definition of Quantitative Reasoning

Hollins University Definition of QR (adjusted Mayes)

Quantitative reasoning is the application of mathematical and statistical concepts and skills to solve real-world problems. In order to perform effectively as professionals and citizens, students must become competent in reading and using quantitative data, in understanding quantitative evidence and in applying basic quantitative skills to the solution of real-life problems.

What is Quantitative Reasoning in STEM?

Think-pair-share: pair up and spend 5 minutes discussing characteristics of QR in your discipline. Speaker – 2 minutes on their view of QR with other

person listening Listener – can only ask clarifying questions, cannot

challenge the speaker Switch Speaker- Listener for 2 minutes, same rules One minute share out to reach consensus Share with whole group

What is Quantitative Literacy in STEM?

Three components of QR (MSP LTER, Mayes 2009): Quantitative Literacy, Quantitative Interpretation, Quantitative Modeling

Quantitative Literacy (QL): ability to manipulate and calculate, apply algorithms, possess number sense and estimation skills including understanding small and large numbers, employ proportional reasoning including working with ratios and percentages, and calculate basic descriptive statistics. These are primarily the arithmetic skills required to be a literate citizen.

QL Example

The U.S. consumed approximately British thermal units of energy in 2001. The U.S. population at that time was 285,000,000. What was the per capita energy consumption? In pairs solve the problem. Solution: What quantitative literacy skills does a

student have to apply to solve this problem?

161097 •

81040.3 •

Required QL skills

Numeracy/Number Sense Large numbers Orders of Magnitude (Powers of 10) Scaling – if we ask to put the answer in a

scale that gives better perspective Ratio

What is Quantitative Interpretation in STEM?

Quantitative Interpretation (QI): ability to interpret scientific and mathematical models, ability to interpret multiple representations of real world situations including tabular, graphic, analytic, and verbal representations, and determine correlation and causality. These are primarily the algebraic processes required to be a literate citizen, though it includes geometric, statistical, and discrete mathematical processes as well.

QI Example

Given the model below, what can you determine about carbon reservoirs?

Models in Science can take on a variety of forms. QI is about interpreting a given model, using it to

support an argument or make predictions

What is Quantitative Modeling in STEM?

Quantitative Modeling (QM): ability to logically reason with quantitative information, capacity to create a model representing a real world situation including geometric and analytic models, problem solving, and conducting statistical inference with hypothesis testing. These are primarily the pre-calculus, calculus, and statistics processes required to be a citizen scientist.

QM Example Heavy metals enter a watershed and are taken up by

fish. The table below provides data on lead and zinc concentrations in 10 fish from the Spokane River. Can you predict zinc levels based on lead levels? Create a model to answer this question. Fish Lead ppm Zinc ppm

Rainbow 0.73 45.3

Rainbow 1.14 50.8

Rainbow 0.60 40.2

Rainbow 1.59 64.0

Sucker 4.34 150.0

Sucker 1.98 106.0

Sucker 3.12 90.8

Sucker 1.80 58.8

Whitefish 0.65 35.4

Whitefish 0.56 28.4

QM Example

0

20

40

60

80

100

120

140

160

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Zin

c pp

m

Lead ppm

Spokane River

Series1

How can you use the graph below to estimate a model for the data? Does a linear model look feasible?Do the two variables appear to be related?

Required QM skills

QL – measure, small and large (ppm) numbers

QI – interpreting table and graph, correlation, causality

QM Eyeballing line of fit Determining relationship is linear Least squares model: z = 28.72 l +19.54

Components of QRCategories Quantitative

LiteracyQuantitative Interpretation

Quantitative Modeling

Component Numeracy• Number Sense• Small/large

Numbers• Scientific

NotationMeasurement • Accuracy-

precision• Estimation• Dimensional

Analysis• UnitsProportional Reasoning • Fraction• Ratio• Percents• Rates/Change• ProportionsBasic Prob/Stats• Empirical Prob.• Counting• Central Tendency• Variation

Interpreting• tables• graphs• equations• science models• statistical plotsLogarithmic ScalesStatistics • Normal

Distribution• Correlation• Causality

LogicProblem SolvingModeling• linear• polynomial• power• exponential• conceptual

models• table or graphStatistics• Least Squares Fit• Inference• Hypothesis

testing

Why Quantitative Reasoning?

STEM (Science, Technology, Engineering, Math) Remediation Crisis Significant number of high school graduates

need remediation at collegiate level STEM Pipeline

STEM majors not meeting national needs STEM course dropout rates excessive

Avoidance of quantitative disciplines due to lack of QR competence

Citizenship skills

Achieving QR

Cognitive development in QR appears to be an intractable problem for education Transition from High School to College:

courses are currency in which articulation is measured

QR is rarely explicit in courses across STEM and is often avoided to reduce student pain

Course by course articulation works against QR since interdisciplinary topic (Steen, 2004)

QR as Literacy QR is an interdisciplinary concern, so

Departments across schools and university need to share the burden Qualitative Literacy: writing across the

curriculum Quantitative Literacy/Reasoning

Collegiate leadership void impacts K-12, not articulated as entry requirement

Quantitative Reasoning

Defining QR Why Numbers Count: Quantitative Literacy

for Tomorrow’s America (College Board, 1997)

Mathematics and Democracy: The Case for Quantitative Literacy (National Council on Education and the Disciplines, 2001)

Quantitative Reasoning Components of QR

Algebra for All: modest ability in reading and interpreting formulas, understanding graphs, and solving simple equations

Civic Literacy: understanding need for data, ability to sort through conflicting claims, skepticism about the reliability or significance of data, recognizing the limits of computer models

Computer Mathematics: solving quantitative problems using standard computer packages

Cultural Literacy: recognize the contributions of mathematicians to society

Quantitative Reasoning Components of QR

Functional Mathematics: skills needed by ordinary people in life and work

Instrumental Mathematics: ability to interpret and apply mathematics and to understand, predict, and control relevant factors in a variety of contexts

Language of Science: support prospective scientists, engineers, life sciences, statistics

Mathematical Modeling: process of hypothesis-building and testing as in science, mathematicizing the problem, analyzing the mathematics, collecting data to verify a prediction of the model

Quantitative Reasoning Components of QR

Problem Solving: the problem and possible solutions are paramount, skills are secondary

Quantitative Practice: apprenticeship environments in which mathematics is used and learned by use but perhaps never explicitly exhibited in words and symbols

Quantitative Reasoning: emphasizes broad synthesis of logical, visual, verbal, and computational thinking; manipulative algebra is incidental to this goal

SCANS Skills: acquiring information, allocating resources, working with others, improving systems, and working with technology

National Numeracy Network

Extensive work in qualitative literacy done by Dartmouth College Babson College DePaul University Hollins University Macalester College Trinity College University of Neveda at Reno Washington Center at Evergreen State College

National Numeracy Network Babson College: eight competencies in

numeracy: Active listeners and readers, able to acquire,

organize, synthesize, evaluate, interrogate, and interpret information of all kinds including information from verbal, numerical, and visual sources

Able to formulate problems, identify opportunities, construct and test hypotheses, and apply to extend theory

Adept at establishing criteria, discovering and weighing alternatives, and using appropriate data to arrive at rational decisions

Comfortable with the creative process, can tolerate ambiguity, and are conscious of the limits as well as the value of rational and logical thought

National Numeracy Network Babson College: eight competencies in

numeracy: Able to communicate logically and persuasively in

spoken, written, and visual form, including graphs and tables

Intellectually confident and independent and are able to make responsible and thoughtful ethical decisions

Adept at doing company, industry, and competitive analyses; they are able to collect, interpret, and communicate information with insight and imagination in an increasingly global and fast-changing environment

Familiar with new technology and its impact on business environment and social change

Interdisciplinary Call

A Collective Vision: Voice of Partner Disciplines (Ganter & Barker, 2004) provides insight from other disciplines on appropriate outcomes for mathematics courses. stress mathematical modeling, conceptual

understanding, and critical thinking strategies increased emphasis on problem solving,

communication, and real world applications

QR for All MAA Committee on the Undergraduate Program

(CUPM) posed the question: What quantitative literacy requirements should be

established for all students who receive a bachelor’s degree?

Subcommittee on Quantitative Literacy Requirements (SQLR) was formed to study the question Quantitative Reasoning for College Graduates: A

Complement to the Standards (MAA, 1998) number of MAA publications concerning QR in higher

education (Steen, 2004; Madison & Steen, 2003).

QR Examples

Critical Thinking Is overpopulation a real problem? Modern

technology, especially in bioengineering, will enable scientists to develop far more efficient agriculture. In addition advances in irrigation technology, along with the development of crops that can grow in salt water, will enable the conversion of much of the world’s desert wastelands into productive farms. As a result agribusiness will be able to produce enough food for at least 50 billion people, about 8 times the current world population. Provide a logical argument related to balancing agribusiness and environmental issues.

QR Examples

Critical Thinking – visualization Analyze the Venn Diagram to determine what

percentage of the population has AB+ blood

A- 8%

A+ 34%

B+ 8%

A B

OO+ 35%

B- 2%AB-1%

O- 9%

QR Examples

Problem Solving – Unit Analysis Measurements of polar ice show that if all the

ice melts, about 25 million cubic kilometers of water will be added to the oceans, most of it coming from Antarctica. How much will sea level rise as a result, given that the total surface area of the Earth’s oceans is about 340 million square kilometers?

QR Examples

Problem Solving – Strategies & Heuristics In an effort to reduce population growth, in 1978

China instituted a policy that allows only one child per family. One unintended consequence has been that, because of a cultural bias toward sons, China now has many more boys than girls. To solve this problem some people have suggested replacing the one-child policy with a one-son policy – if a family’s first child is a boy, the family has reached its limit, but if it is a girl the family can have additional children until one is a boy. How would this affect the overall birth rate and the number of boys versus girls?

QR Examples

Number Sense – exponential growth According to the 2000 census, the population

of Albany county is approximately 50,000. A power company predicts the county’s population will increase 7% per year while the county supervisors predict that the population will increase by 7,500 each year. Which group predicts the largest population in 10 years?

QR Examples

Number Sense – percentages The percentage of students in a school

passing the PAWS test decreased by 15% from 2005 to 2006. After worried teachers redoubled their efforts, the percentage who passed increased by 15% from 2006 to 2007. In which of the years was the percentage of students who passed PAWS the highest, 2005, 2006 or 2007?

QR Example

Interpreting Visual Data The graph below shows the percentage

change in the value of a company’s stock. In 2000 did the stock reach its highest value or was the stock still increasing in value but declining after 2000?

Dr. Robert MayesUniversity of Wyoming

Science and Mathematics

Teaching Centerrmayes2@uwyo.edu