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Integrating Statistics
into Modeling-Based
College Algebra
Sheldon P. [email protected]
Florence S. [email protected]
Accessing the Talk
This PowerPoint presentation and the
DIGMath Excel files that will be used can
all be downloaded from:all be downloaded from:
farmingdale.edu/~gordonsp
College Algebra and Precalculus
Each year, more than 1,000,000 students take
college algebra and precalculus courses.
The focus in most of these courses is on The focus in most of these courses is on
preparing the students for calculus.
We know that only a relatively small percentage
of these students ever go on to start calculus.
Some Interesting Studies
In a study at eight public and private universities
in Illinois, Herriott and Dunbar found that,
typically, only about 10-15% of the students
enrolled in college algebra courses had any enrolled in college algebra courses had any
intention of majoring in a mathematically
intensive field.
At a large two year college, Agras found that only
15% of the students in college algebra planned to
major in mathematically intensive fields.
Some Interesting Studies
Steve Dunbar has tracked over 150,000 students taking
mathematics at the University of Nebraska – Lincoln for
more than 15 years. He found that:
• only about 10% of the students who pass college
algebra ever go on to start Calculus I
• virtually none of the students who pass college algebra
ever go on to start Calculus III.
• about 30% of the students who pass college algebra
eventually start business calculus.
• about 30-40% of the students who pass precalculus
ever go on to start Calculus I.
Some Interesting Studies
William Waller at the University of Houston –
Downtown tracked the students from college algebra in
Fall 2000. Of the 1018 students who started college
algebra:
• only 39, or 3.8%, ever went on to start Calculus I at • only 39, or 3.8%, ever went on to start Calculus I at
any time over the following three years.
• 551, or 54.1%, passed college algebra with a C or
better that semester
• of the 551 students who passed college algebra, 153 had
previously failed college algebra (D/F/W) and were
taking it for the second, third, fourth or more time
Some Interesting Studies
The Fall, 2001 cohort in college algebra at the University
of Houston – Downtown was slightly larger. Of the 1028
students who started college algebra:
• only 2.8%, ever went on to start Calculus I at any time
over the following three years. over the following three years.
The San Antonio Project
The mayor’s Economic Development Council of
San Antonio recently identified college algebra as
one of the major impediments to the city
developing the kind of technologically
sophisticated workforce it needs.sophisticated workforce it needs.
The mayor appointed special task force including
representatives from all 11 colleges in the city plus
business, industry and government to change the
focus of college algebra to make the courses more
responsive to the needs of the city, the students,
and local industry.
Some Questions
Why do the majority of these 1,000,000+
students a year take college algebra courses?
Are these students well-served by the kind of Are these students well-served by the kind of
courses typically given as “college algebra”?
If not, what kind of mathematics do these
students really need?
Another Question
As calculus rapidly becomes (for better or
worse) a high school subject, what can we
expect of the students who take the courses
before calculus in college?before calculus in college?
Hard as it may be to believe, I expect that
they will be more poorly prepared for these
courses, which even more dramatically will
not serve them well.
Why Do Our Students Fail?
They have seen virtually all of a standard
skills-based algebra course in high school.
They do not see themselves ever using any of
the myriad of techniques and tricks in the the myriad of techniques and tricks in the
course (and they are right about that).
They equate familiarity with mastery, so they
don’t apply themselves until far too late and
they are well down the road to failure.
The Needs of Our Students
The reality is that virtually none of the students
we face in these courses today or in the future
will become math majors.
They take these courses to fulfill Gen Ed They take these courses to fulfill Gen Ed
requirements or requirements from other
disciplines.
What do those other disciplines want their
students to bring from math courses?
Mathematical Needs of Partners
• In discussions with faculty from the lab
sciences, it becomes clear that most
courses for non-majors (and even those
for majors in some areas) use almost no for majors in some areas) use almost no
mathematics in class.
• Mathematics arises almost exclusively in
the lab when students have to analyze
experimental data and then their weak
math skills become dramatically evident.
Curriculum Foundations Project
CRAFTY held a series of workshops
with leading educators from 17
quantitative disciplines to inform the
mathematics community of the current mathematics community of the current
mathematical needs of each discipline.
The results are summarized in the MAA
Reports volume: A Collective Vision:
Voices of the Partner Disciplines, edited
by Susan Ganter and Bill Barker.
What the Physicists Said
• Students need conceptual understanding
first, and some comfort in using basic
skills; then a deeper approach and more
sophisticated skills become meaningful. sophisticated skills become meaningful.
• Conceptual understanding is more
important than computational skill.
• Computational skill without theoretical
understanding is shallow.
What the Physicists Said
• The learning of physics depends less
directly than one might think on
previous learning in mathematics. We previous learning in mathematics. We
just want students who can think. The
ability to actively think is the most
important thing students need to get
from mathematics education.
What the Biologists Said
• New areas of biological investigation have resulted in an increase in quantification of biological theories and models.
• The collection and analysis of data that is central to biology inevitably leads to the use of to biology inevitably leads to the use of mathematics.
• Mathematics provides a language for the development and expression of biological concepts and theories. It allows biologists to summarize data, to describe it in logical terms, to draw inferences, and to make predictions.
What the Biologists Said
• Statistics, modeling and graphical representation should take priority over calculus.
• The teaching of mathematics and statistics should use motivating examples that draw on problems or data taken from biology.or data taken from biology.
• Creating and analyzing computer simulations of biological systems provides a link between biological understanding and mathematical theory.
What the Biologists Said
The quantitative skills needed for biology:
• The meaning and use of variables, parameters, functions, and relations.
• To formulate linear, exponential, and logarithmic functions from data or from general principles.functions from data or from general principles.
• To understand the periodic nature of the sine and cosine functions.
• The graphical representation of data in a variety of formats – histograms, scatterplots, log-log graphs (for power functions), and semi-log graphs (for exponential and log functions).
What the Biologists Said
Other quantitative skills:
• Some calculus for calculating areas and average values, rates of change, optimization, and gradients for understanding contour maps.
• Statistics – descriptive statistics, regression analysis, multivariate analysis, probability distributions, simulations, significance and error analysis.
• Discrete Mathematics and Matrix Algebra –graphs (trees, networks, flowcharts, digraphs), matrices, and difference equations.
What the Biologists Said
• The sciences are increasingly seeing students who are
quantitatively ill-prepared.
• The biological sciences represent the largest science
client of mathematics education.
• The current mathematics curriculum for biology
majors does not provide biology students with majors does not provide biology students with
appropriate quantitative skills.
• The biologists suggested the creation of mathematics
courses designed specifically for biology majors.
• This would serve as a catalyst for needed changes in
the undergraduate biology curriculum.
• We also have to provide opportunities for the biology
faculty to increase their own facility with mathematics.
What Business Faculty Said
• Courses should stress problem solving,
with the incumbent recognition of
ambiguities.
• Courses should stress conceptual • Courses should stress conceptual
understanding (motivating the math with
the “why’s” – not just the “how’s”).
• Courses should stress critical thinking.
• An important student outcome is their
ability to develop appropriate models to
solve defined problems.
What Business Faculty Said
Mathematics is an integral component of the business
school curriculum. Mathematics Departments can help
by stressing conceptual understanding of quantitative
reasoning and enhancing critical thinking skills.
Business students must be able not only to apply
appropriate abstract models to specific problems, but appropriate abstract models to specific problems, but
also to become familiar and comfortable with the
language of and the application of mathematical
reasoning. Business students need to understand that
many quantitative problems are more likely to deal
with ambiguities than with certainty. In the spirit that
less is more, coverage is less critical than
comprehension and application.
What Business Faculty Said
• Courses should use industry standard
technology (spreadsheets).
• An important student outcome is their • An important student outcome is their
ability to become conversant with
mathematics as a language. Business
faculty would like its students to be
comfortable taking a problem and casting
it in mathematical terms.
The Common Threads
• Conceptual Understanding, not rote manipulation
• Realistic applications via mathematical
modeling that reflect the way mathematics
is used in other disciplines and on the job
• Statistical reasoning is the primary mathematical
topic needed for all other disciplines.
• Fitting functions to data/ data analysis
• The use of technology (though typically Excel,
not graphing calculators).
Implications for College Algebra
Students don’t need a skills-oriented course.
They need a modeling-based course that:
• emphasizes realistic applications that mirror
what they will see and do in other courses;what they will see and do in other courses;
• emphasizes conceptual understanding;
• emphasizes data and its uses, including both
fitting functions to data and statistical methods
and reasoning;
• better motivates them to succeed.
Further Implications
If we focus only on developing
manipulative skills
without developing
conceptual understanding,conceptual understanding,
we produce nothing more than students
who are only
Imperfect Organic Clones
of a TI-89
Should x Mark the Spot?
All other disciplines focus globally on the entire universe of a
through z, with the occasional contribution of αααα through ωωωω.
Only mathematics focuses on a single spot, called x.
Newton’s Second Law of Motion: y = mx, Newton’s Second Law of Motion: y = mx,
Einstein’s formula relating energy and mass: y = c2x,
The Ideal Gas Law: yz = nRx.
Students who see only x’s and y’s do not make the connections
and cannot apply the techniques learned in math classes when
other letters arise in other disciplines.
Should x Mark the Spot?
Kepler’s third law expresses the relationship between the
average distance of a planet from the sun and the length
of its year.
If it is written as y2 = 0.1664x3, there is no suggestion of If it is written as y2 = 0.1664x3, there is no suggestion of
which variable represents which quantity.
If it is written as t2 = 0.1664D3 , a huge conceptual
hurdle for the students is eliminated.
A Modeling-Based Course
1. Introduction to data and statistical
measures.
2. Behavior of functions as data and as 2. Behavior of functions as data and as
graphs, including increasing/decreasing,
turning points, concave up/down,
inflection points (including normal
distribution function).
A Modeling-Based Course
3. Linear functions, with emphasis on the
meaning of the parameters and fitting
linear functions to data, including the
linear correlation coefficient to measure linear correlation coefficient to measure
how well the regression line fits the data.
A Modeling-Based Course
4. Nonlinear families of functions:
• exponential growth and decay, applications
such as population growth and decay of a
drug in the body; doubling time and half-life;
• power functions;• power functions;
• logarithmic functions;
• Fitting each family of functions to data
based on the behavioral characteristics of
the functions and deciding on how good the
fit is.
A Modeling-Based Course
5. Modeling with Polynomial Functions:
Emphasis on the behavior of polynomials
and modeling, primarily by fitting
polynomials to datapolynomials to data
A Modeling-Based Course
6. Extending the basic families of functions
using shifting, stretching, and shrinking,
including:
• applying ideas on shifting and stretching • applying ideas on shifting and stretching
to fitting extended families of functions
to sets of data
• statistical ideas such as the distribution
of sample means, the Central Limit
Theorem, and confidence intervals.
A Modeling-Based Course
6a. Functions of several variables using
tables, contour plots, and formulas with
multiple variables.
A Modeling-Based Course
7. Sinusoidal Functions and Periodic
Phenomena: using the sine and cosine as
models for periodic phenomena such as
the number of hours of daylight, heights the number of hours of daylight, heights
of tides, average temperatures over the
year, etc.
Some Illustrative
Examples and Problems
The following table shows world-wide average
temperatures in °°°°C in various years.
Year 1880 1900 1920 1940 1960 1980 1990 2000
Temp 13.80 13.95 13.90 14.15 14.00 14.20 14.40 14.50
(a) Decide which is the independent variable and which is the
dependent variable.dependent variable.
(b) Decide on appropriate scales for the two variables for a
scatterplot.
(c) State precisely which letters you will use for the two
variables and state what each variable you use stands for.
(d) Draw the associated scatterplot.
(e) Raise some predictive questions in this context that could
be answered when we have a formula relating the two
variables.
The following table shows world-wide wind power
generating capacity, in megawatts, in various
years.
Year 1980 1985 1990 1995 1997 2000 2002 2004
Wind
power 10 1020 1930 4820 7640 13840 32040 47910power 10 1020 1930 4820 7640 13840 32040 47910
0
10000
20000
30000
40000
50000
1980 1985 1990 1995 2000 2005
(a) Which variable is the independent variable and which
is the dependent variable?
(b) Explain why an exponential function is the best model
to use for this data.
(c) Find an exponential function that models the
relationship between power P generated by wind and the
year t.
(d) What are some reasonable values that you can use for (d) What are some reasonable values that you can use for
the domain and range of this function?
(e) What is the practical significance of the base (1.1373) in
the exponential function you created in part (c)?
(f) What is the doubling time for this function? Explain
what it means. Solve: 52.497(1.1373)t= 2× 52.497.
(g) According to your model, what do you predict for the
total wind power generating capacity in 2010?
A Temperature Experiment
An experiment is conducted to study the rate at which
temperature changes. A temperature probe is first
heated in a cup of hot water and then pulled out and
placed into a cup of cold water. The temperature of
the probe, in ̊̊ ̊̊C, is measured every second for 36 the probe, in ̊̊ ̊̊C, is measured every second for 36
seconds and recorded in the following table. Time 1 2 3 4 5 6 7 8
42.3 36.03 30.85 26.77 23.58 20.93 18.79 17.08
31 32 33 34 35 36
8.78 8.78 8.78 8.78 8.66 8.66
Find a function that fits this data.
A Temperature Experiment
5
10
15
20
25
30
35
40
45
Te
mp
era
ture
(d
eg
ree
s C
)
time (1 - 36 seconds)
The data suggest an exponential
decay function, but the points
don’t decay to 0.
To find a function, one first has
to shift the data values down to
get a transformed set of data get a transformed set of data
that decay to 0.
Then one has to fit an exponential function to the
transformed data. Finally, one has to undo the
transformation by shifting the resulting exponential
function. T = 8.6 + 35.439(0.848)t.
The Species-Area Model
Biologists have long observed that the larger the area of a
region, the more species live there. The relationship is
best modeled by a power function. Puerto Rico has 40
species of amphibians and reptiles on 3459 square miles
and Hispaniola (Haiti and the Dominican Republic) has and Hispaniola (Haiti and the Dominican Republic) has
84 species on 29,418 square miles.
(a) Determine a power function that relates the number
of species of reptiles and amphibians on a Caribbean
island to its area.
(b) Use the relationship to predict the number of species
of reptiles and amphibians on Cuba, which measures
44218 square miles.
The accompanying table and associated
scatterplot give some data on the area (in
square miles) of various Caribbean islands and
estimates on the number of species of
amphibians and reptiles living on each.
Island Area N80
100
Nu
mb
er
of
Sp
ec
ies
Redonda 1 3
Saba 4 5
Montserrat 40 9
Puerto Rico 3459 40
Jamaica 4411 39
Hispaniola 29418 84
Cuba 44218 76
0
20
40
60
80
0 15000 30000 45000
Area (square miles)
Nu
mb
er
of
Sp
ec
ies
A Tale of Two Students
The Next Challenge: Statistics
Based on the Curriculum Foundations
reports and from discussions with
faculty in the lab sciences (and most
other areas), the most critical other areas), the most critical
mathematical need of the partner
disciplines is for students to know
statistics. How can we integrate
statistical ideas and methods into math
courses at all levels?
The Curriculum Problems We Face
• Students don’t see traditional precalculus or
college algebra courses as providing any useful
skills for their other courses.
• Typically, college algebra is the prerequisite for • Typically, college algebra is the prerequisite for
introductory statistics, especially at universities.
• Introductory statistics is already overly
crammed with far too much information.
• Most students put off taking the math as long as
possible. So most don’t know any of the statistics
when they take the introductory courses in bio or
other fields.
Integrating Statistics into Mathematics
• Students see the equation of a line in pre-
algebra, in elementary algebra, in intermediate
algebra, in college algebra, and in precalculus.
Yet many still have trouble with it in calculus.
• They see statistics ONCE in an introductory • They see statistics ONCE in an introductory
statistics course. But statistics is far more
complex, far more varied, and often highly
counter-intuitive, yet they are then expected to
use a wide variety of the statistical ideas and
methods in their lab science and other courses.
Integrating Statistics in College Algebra
Data is Everywhere! We should capitalize on it.
1. A frequency distribution is a function – it can be
an effective way to introduce and develop the
concept of function.concept of function.
2. Data analysis – the idea of fitting linear,
exponential, power, polynomial, sinusoidal and
other functions to data – is already becoming a
major theme in some college algebra courses. It
can be the unifying theme that links functions,
the real world, and the other disciplines.
Integrating Statistics in College Algebra
But, there are some important statistical issues that need to be addressed. For instance:
1. Most sets of data, especially in the sciences, only represent a single sample. How does the regression line based on one sample compare to regression line based on one sample compare to the lines based on other possible samples?
2. The correlation coefficient only applies to a linear fit. What significance does it have when you are fitting a nonlinear function to data?
Integrating Statistics in College Algebra
2. The correlation coefficient only applies to a linear fit. What significance does it have when you are fitting a nonlinear function to data?
Integrating Statistics in College Algebra
3. The z-value associated with a measurement xis a nice application of a linear function of x:
xz
µ
σ
−=z
σ=
It can provide the source of many algebra problems that have a simple underlying context.
Integrating Statistics in College Algebra
4. The normal distribution function is
2 2( ) / 21
2( )
xN x e
µ σ
σ π
− −=2σ π
It makes for an excellent example involving
both stretching and shifting functions and
a function of a function.
Match each of the four normal distributions (a)-(d) with one of the corresponding sets of values for the parameters µ and σ. Explain your reasoning.
(i) µ = 85 , σ = 1
(ii) µ = 100, σ = 12
0
0.1
50 100 150
(a)
0
0.1
50 100 150
(b)
0.1(c)
0.1(d)
(ii) µ = 100, σ = 12
(iii)µ = 115 , σ = 12
(iv) µ = 115 , σ = 8
(v) µ = 100 , σ = 6
(vi) µ = 85 , σ = 70
50 100 150
0
50 100 150
Integrating Statistics in College Algebra
5. The Central Limit Theorem can be
interpreted in terms of stretching and shifting
functions -- the mean of the distribution of
sample means is a shift and the standard
deviation σdeviation
produces a stretch or a squeeze, depending on
the sample size n.
x
n
σσ =
Some Conclusions
Few, if any, math departments can exist
based solely on offerings for math and
related majors. Whether we like it or not,
mathematics is a service department at
almost all institutions.almost all institutions.
And college algebra and related courses
exist almost exclusively to serve the needs of
other disciplines.
Some Conclusions
If we fail to offer courses that meet the
needs of the students in the other
disciplines, those departments will
increasingly drop their requirements for
math courses. This is already starting to math courses. This is already starting to
happen in engineering.
Math departments may well end up offering
little beyond developmental algebra courses
that serve little purpose.
For More Information
This PowerPoint presentation and the DIGMath
Excel files demonstrated can all be downloaded:
farmingdale.edu/~gordonsp
Sheldon P. Gordon
Florence S. Gordon
