# Math Across the Curriculum 45

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Math Across
the Curriculum
Rob Kimball
Or
If we are not going
to use it outside of
math class, why do
we have to learn it?
Or
to a spreadsheet, calculated
some statistics, and used
those statistics to support
oral arguments….
In my English class!
That math stuff must really
be important.
Or
Innumeracy
It is everyone’s problem.
The simple fact is that many
students who enter college are
innumerate.
The simple fact is that many
students who enter college are
innumerate.
Judging from what is going on in our
society, you have to wonder how
many college graduates are as well.
Innumeracy
Innumeracy can make you poorer.
Couple’s numeracy skills linked to greater
family wealth, study finds
Innumeracy
Innumeracy can make you unhealthier.
Understanding Food Nutrition Labels
Challenging For Many People
Innumeracy
Innumeracy can make you
misunderstand important
information.
Numbers Are Just Numbers, But How You Grasp Them
Fills In Details
Innumeracy
Innumeracy can be passed down
through the generations
Parents should talk about math early and often
with their children — even before preschool,
report finds
National Center for
Educational Statistics
• 78% of adults cannot explain how to
compute the interest paid on a loan
• 71% cannot calculate miles per
gallon on a trip
• 58% cannot calculate a 10% tip for a
lunch bill
National Center for
Educational Statistics
• 78% of adults cannot explain how to
compute the interest paid on a loan
• 71% cannot calculate miles per
gallon on a trip
• 58% cannot calculate a 10% tip for a
lunch bill
National Center for
Educational Statistics
• 78% of adults cannot explain how to
compute the interest paid on a loan
• 71% cannot calculate miles per
gallon on a trip
• 58% cannot calculate a 10% tip for a
lunch bill
National Center for
Educational Statistics
• 78% of adults cannot explain how to
compute the interest paid on a loan
• 71% cannot calculate miles per
gallon on a trip
• 58% cannot calculate a 10% tip for a
If you order the “Onion Soup” and “The Lancaster
lunch
bill
Special” compute the 10% tip you should leave.
Mathematical Literacy
requires
Conceptual Understanding
Explaining MPG
a recent YouTube video that went viral
Adding It Up: Helping Children Learn
Mathematics, 2001
Mathematics is also an intellectual achievement
of great sophistication and beauty that
epitomizes the power of deductive reasoning.
For people to participate fully in society, they
must know basic mathematics. Citizens who
cannot reason mathematically are cut off from
whole realms of human endeavor. Innumeracy
deprives them not only of opportunity but also
of competence in everyday tasks.
The ability to perform some of the
basic operations of mathematics is a
necessary but not sufficient condition
of quantitative literacy.
Solve
{
x - y = -17
25x + 8y = 532
…tests showing that the demand for
the product is expected to decrease
3,125 units for every 1.00 increase in
price from the current level of 31.5
thousand units. The supply is
expected to increase 1,000 units for
each 1.00 increase in price from the
current 17,000 units.
Find the break-even point.
Richardson and McCallum
The Third R in Literacy
A goal for mathematics departments:
A goal for mathematics departments:
“Create a
mathematically
literate student.”
A goal for colleges:
A goal for colleges:
“Create a
quantitatively
literate student.”
Promote Math Across the Curriculum
Non SMET students
SMET students
Quantitatively
Literate
Teachers
Politicians
Voters
Consumers
Attorneys
QL: A habit of mind
Quantitative literacy describes a habit of mind
rather than a set of topics or a list of skills. It
depends on the capacity to identify
mathematical structure in context; it requires a
mind searching for patterns rather than
following instructions. A quantitatively literate
person needs to know some mathematics, but
literacy is not deﬁned by the mathematics
known.
Where is the Math?
Measurement
Statistics
Geometry
sampling
representations of
Estimation
data (numeric/graphic)
Reasonableness
Rates and Proportions variability
Models
Density
Optimization
Data Analysis
Predicting
numeric, graphic
Rate of Change
Where is the Literacy?
Dialogue
Debate
Problem Stating
Problem Solving
Investigating
Reasoning
Sense Making
Connecting
Bridging
Inferring
Curriculum Foundations Report (CRAFTY)
Life Sciences
…the definition of mastery of a mathematical concept
recognizes the importance of both conceptual
understanding at the level of definition and
computation.
…help prepare business students by stressing conceptual
understanding of quantitative reasoning and enhancing
critical thinking skills.
Where is the math?
The math found in other disciplines
is often transparent –
to students (and instructors?)
Persons Per Sq Mile
250-66,995
100-249
50-99
25-49
10-24
5-9
1-4
Population Densities (by county)
Diagnostic Tool
Math in context
means
math is meaningful
Teaching Math In Context
It is difficult to teach students to identify
mathematics in context – and many
teachers have no experience doing this.
It is much easier to teach an algorithm
than the insight needed to identify
quantitative structure.
Teaching in context poses a tremendous
challenge.
Teaching Math In Context
Problems in mathematics courses can be
contextualized – StatWay and QuantWay
http://www.carnegiefoundation.org/developmental-math
Problems in other disciplines are already
contextualized – we must help students
identify the math.
Programme for International Student
Assessment (PISA, 2000)
Mathematical Literacy: “An individual’s
capacity to identify and understand the
role that mathematics plays in the world,
to make well-founded judgments and to
use and engage with mathematics in ways
that meet the needs of that individual’s life
as a constructive, concerned, and reflective
citizen.”
A
P
P
L
Y
t
h
e
Other disciplines
M
A
T
H
National Center On Education
and the Economy
Math Panel – A review of first year
courses; “What are the quantitative
skills used in the course?”
Missing Math
The first-year courses students often take, in
addition to a math class, require little in the way
of mathematical thinking – numeracy –
quantitative reasoning.
Texts often focus on facts and procedures.
Tests are often even worse. They are often
computer - generated multiple choice questions
that don’t require reasoning or sense making.
(Rob’s review – not necessarily that of NCEE)
Math Across the Curriculum
Or
Interdisciplinary Studies
Models
Learning Communities
American
History
College
Algebra
English
Learning Communities
English
Arithmetic
Study
Skills
Impact Studies at Queensborough and Houston Community Colleges Feb, 2011
Aligned Interdisciplinary Studies
Communications
Psychology
Statistics
”…individuals must be broadly trained so that they can understand and
contribute to research that overlaps different fields .”
(National Research Council)
“Interdisciplinary learning is a 21st Century
imperative. We are continually faced with
societal and global challenges that require
interdisciplinary thinking to identify suitable
solutions, such as finding new energy sources,
dealing with the effects of our changing climate,
and ensuring populations across the globe have
adequate food and healthy living environments.”
Summary report from Project Kaleidoscope –
“What Works in Facilitating Interdisciplinary
Learning in Science and Mathematics”
Themed Studies
American History
Calculus
Economics
Energy Conservation, The Influence of Television, Exploration of Space,
Obesity Epidemic, The Changing Demographics of the U S, US Debt…
Even in English Class
Math across the
curriculum can occur in
many directions and
success can look very
different at different
institutions.
It takes a champion, or
two, to make it happen.
Quantitative Writing
In the increasingly complex, data-rich global
environments of the 21st Century, successful
students need to be equipped with flexible,
adaptive analytical higher-order strategies.
Quantitative writing addresses the need for these
higher-order thinking skills.
http://serc.carleton.edu/sp/carl_ltc/quantitative_writing/why.html
QW
Most issues of public
policy have a significant
quantitative dimension.
health care, energy usage, or immigration policy,
effective citizens must be able to interpret and
analyze numbers, read graphs, understand simple
statistics, and recognize the ways that numerical
data can be manipulated for rhetorical effect.
QW assignments help develop students for
responsible citizenship.
THE CURRICULUM
Spreadsheets are used throughout industry.
Educators, and especially mathematics
educators, seem reluctant to utilize this
ubiquitous tool.
http://www.serc.carleton.edu/sp/ssac_home
THE CURRICULUM
Criminal Justice – How long has the potato
Economics – Cost / Benefit Analysis of
driving across town for cheaper gas
Medicine – Examining the effect of dose,
time interval, and elimination rate on
attaining a therapeutic drug level
Math Across the Curriculum
HOW?
Friendly Conspiracies – collaboration between mathematics
faculty and faculty from other departments
(Hughes Hallett, 2001)
Gateway Testing – mathematics competency tests in
courses across the university (Bauman and Martin, 1995;
University of Nevada, Reno)
Instructional Support – provide support (equipment,
lessons, collaboration) for teachers
(Dartmouth College, 2001)
Workshops – face-to-face discussions that help faculty
outside of mathematics understand QL (Project
Kaleidoscope, PKAL 2002; CRAFTY (MAA) Haver & Ganter)
Quantitative Reasoning Requirements – a set of courses
designated to meet QR requirements for undergraduates
WHAT?
Lynn Steen
Mathematics is far more than just a tool for
research. In fact, its most common uses – and the
reason for its prominent place in school curricula –
are routine applications that are now part of all
kinds of jobs.
…If we look at these common uses of mathematics
from the perspective of the school curriculum, we
see that mathematics at work is very different
from mathematics at school.
Operations Research Laboratory
for Education (OREd)
Predicting
membership by
historical model
and the cohort
survival ratio.
Challenges to MAC
• Time – the density of the syllabus
• Ability – using mathematics may be
outside the comfort zone of some
• Effort – it is extra work to find/write
projects that use and promote
quantitative literacy
• What is Mathematics – too many people
characterize mathematics as computation
History
A Rock Used As A Doorstop
Changed North Carolina
Numeracy Through North
Carolina History
Kimball, 2011
History
San Marcos
Numeracy Through
Texas History
Kimball 2011
Mathematical proficiency, as we see it, has five strands:
• conceptual understanding—comprehension of
mathematical concepts, operations, and relations
• procedural fluency—skill in carrying out procedures
flexibly, accurately, efficiently, and appropriately
• strategic competence—ability to formulate, represent,
and solve mathematical problems
• 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.
Standards For School
Mathematics (NCTM)
Number and Operations
Algebra
Geometry
Measurement
Data Analysis and Probability
Change and Relationships
Some animals that dwell on grassy plains are safeguarded
against attacks by their large size; others are so small that
they can protect themselves by burrowing into the ground.
Still others must count on speed to escape their enemies.
An animal’s speed depends on its size and the frequency of
its strides. The tarsal (foot) bone of a horse is lengthened,
with each foot having been reduced to only one toe. One
thick bone is stronger than a number of thin ones. This
single toe is surrounded by a solid hoof, which protects the
bone against jolts when the animal is galloping over hard
ground. The powerful leg muscles are joined together at the
top of the leg so that just a slight muscle movement at that
point can freely move the slip lower leg.
The fastest sprinter in the world is the cheetah. Its legs are
shorter than those of a horse, but it can reach a speed of
more than 110 km/hr in 17 seconds and maintain that speed
for more than 450 meters. The cheetah tires easily,
however, whereas the horse, whose top speed is 70 km/hr,
can maintain a speed of 50 km/hr for more than 6 km.
A cheetah is awakened by a horse’s hooves. At the moment
the cheetah decides to give chase, the horse has a lead of
200 meters. The horse, traveling at top speed, still has
plenty of energy. Taking into consideration the data that has
been provided, can the cheetah catch the horse? Assume
the cheetah will need around 300 meters to reach its top
speed. Provide graphs to support your conclusions letting
the vertical axis represent distance and the horizontal axis
time. (Kindt, 1979)
Supporting Mathematics
Across the Curriculum
What should it mean
for the math curriculum?
1. The mathematics that is taught should be
embedded in the real world of the students.
2. Mathematical literacy will lead to different
curricula in different cultures.
3. The content of the mathematics curricula will
have to be modernized at least every ten years.
Reforming a Failing Curriculum
Student at the end of the semester: “Thank God
I’ll never have to take another math course in
my life!”
Arnold Packer: Quantitative literacy, in my
judgment, can save the day, not by being
added to the curriculum but by altering
required mathematics.
Specific Recommendations
1. QL has a strong partner and advocate in the science
community. (Science for All Americans (Project 2061
1989)
2. Consider engaging the social sciences in the quest for
QL.
3. Adopt detailed and specific goals with benchmarks for
progress.
4. Coordinate QL across disciplines by making QL part of
faculty development.
5. Promote changes (improvements) in pedagogy
advanced by national organizations.
6. Develop reliable and valid assessments of
experiments in curriculum and instruction that target
QL and publish the results.
Resources
• http://www.math.dartmouth.edu/~mqed/eBo
okshelf/
• http://www.mac3.amatyc.org
• http://www.therightstuff.amatyc.org
• http://www.maa.org/cupm/crafty
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