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Refocusing the Courses Below Calculus A Joint Initiative of MAA, AMATYC


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Title: Refocusing the Courses Below Calculus A Joint Initiative of MAA, AMATYC

Refocusing the Courses Below Calculus A Joint
Initiative of MAA, AMATYC NCTM
  • This slideshow presentation was created by
  • Sheldon P. Gordon
  • Farmingdale State University of New York
  • with contributions from
  • Nancy Baxter Hastings (Dickinson College)
  • Florence S. Gordon (NYIT)
  • Bernard Madison (University of Arkansas)
  • Bill Haver (Virginia Commonwealth University)
  • Bill Bauldry (Appalachian State University)
  • Permission is hereby granted to anyone to use any
    or all of these slides in any related
  • We gratefully acknowledge the support provided
    for the development of this presentation package
    by the National Science Foundation under grants
    DUE-0089400, DUE-0310123, and DUE-0442160.
  • The views expressed are those of the author and
    do not necessarily reflect the views of the

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 preparing
the students for calculus. We know that only a
relatively small percentage of these students
ever go on to start calculus.
Some Questions
How many of these students actually ever do go on
to start calculus? How well do the ones who do
go on actually do in calculus?
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 courses
typically given as college algebra? If not,
what kind of mathematics do these students really
Enrollment Flows
  • Based on several studies of enrollment flows from
    college algebra to calculus
  • Less than 5 of the students who start
    college algebra courses ever start Calculus I
  • The typical DFW rate in college algebra is
    typically well above 50
  • Virtually none of the students who pass college
    algebra courses ever start Calculus III
  • Perhaps 30-40 of the students who pass
    precalculus courses ever start Calculus I

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 intention of majoring in a mathematically
intensive field. At a large two year college,
Agras found that only 15 of the students taking
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 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
  • only 2.8, ever went on to start Calculus I at
    any time over the following three years.

The San Antonio Project
The mayors 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. The mayor
appointed a special task force with
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.
Why Students Take These Courses
  • Required by other departments
  • Satisfy general education requirements
  • To prepare for calculus
  • For the love of mathematics

What the Majority of Students Need
  • Conceptual understanding, not rote manipulation
  • Realistic applications and mathematical
    modeling that reflect the way mathematics is
    used in other disciplines and on the job in
    todays technological society

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. 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 the
requirements for 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.
Four Special Invited Conferences
  • Rethinking the Preparation for Calculus,
  • October 2001.
  • Forum on Quantitative Literacy,
  • November 2001.
  • CRAFTY Curriculum Foundations Project,
  • December 2001.
  • Reforming College Algebra,
  • February 2002.

Common Recommendations
  • College Algebra courses should stress
    conceptual understanding, not rote manipulation.
  • College Algebra courses should be real-world
    problem based
  • Every topic should be introduced through a
    real-world problem and then the mathematics
    necessary to solve the problem is developed.

Common Recommendations
  • College Algebra courses should focus on
    mathematical modelingthat is,
  • transforming a real-world problem into
    mathematics using linear, exponential and power
    functions, systems of equations, graphing, or
    difference equations.
  • using the model to answer problems in context.
  • interpreting the results and changing the model
    if needed.

Common Recommendations
  • College Algebra courses should emphasize
    communication skills reading, writing,
    presenting, and listening.
  • These skills are needed on the job and for
    effective citizenship as well as in academia.
  • College Algebra courses should make
    appropriate use of technology to enhance
    conceptual understanding, visualization, inquiry,
    as well as for computation.

Common Recommendations
  • College Algebra courses should be
    student-centered rather than instructor-centered
  • - They should include hands-on activities rather
    than be all lecture.
  • - They should emphasize small group projects
    involving inquiry and inference.

Important Volumes
  • CUPM Curriculum Guide Undergraduate Programs
    and Courses in the Mathematical Sciences, MAA
  • AMATYC Crossroads Standards and the Beyond
    Crossroads report.
  • NCTM, Principles and Standards for School
  • Ganter, Susan and Bill Barker, Eds.,
  • A Collective Vision Voices of the Partner
    Disciplines, MAA Reports.

Important Volumes
  • Madison, Bernie and Lynn Steen, Eds.,
    Quantitative Literacy Why Numeracy Matters for
    Schools and Colleges, National Council on
    Education and the Disciplines, Princeton.
  • Baxter Hastings, Nancy, Flo Gordon, Shelly
    Gordon, and Jack Narayan, Eds., A Fresh Start
    for Collegiate Mathematics Rethinking the
    Courses below Calculus, MAA Notes.

CUPM Curriculum Guide
  • All students, those for whom the (introductory
    mathematics) course is terminal and those for
    whom it serves as a springboard, need to learn to
    think effectively, quantitatively and logically.
  • Students must learn with understanding,
    focusing on relatively few concepts but treating
    them in depth. Treating ideas in depth includes
    presenting each concept from multiple points of
    view and in progressively more sophisticated

CUPM Curriculum Guide
  • A study of these (disciplinary) reports and the
    textbooks and curricula of courses in other
    disciplines shows that the algorithmic skills
    that are the focus of computational college
    algebra courses are much less important than
    understanding the underlying concepts.
  • Students who are preparing to study calculus
    need to develop conceptual understanding as well
    as computational skills.

AMATYC Crossroads Standards
  • In general, emphasis on the meaning and use of
    mathematical ideas must increase, and attention
    to rote manipulation must decrease.
  • Faculty should include fewer topics but cover
    them in greater depth, with greater
    understanding, and with more flexibility. Such
    an approach will enable students to adapt to new
  • Areas that should receive increased attention
    include the conceptual understanding of
    mathematical ideas.

NCTM Standards
  • These recommendations are clearly very much in
    the same spirit as the recommendations in NCTMs
    Principles and Standards for School Mathematics.
  • If implemented at the college level, they would
    establish a smooth transition between school and
    college mathematics.

CRAFTY College Algebra Guidelines
  • These guidelines are the recommendations of the
    MAA/CUPM subcommittee, Curriculum Renewal Across
    the First Two Years, concerning the nature of the
    college algebra course that can serve as a
    terminal course as well as a pre-requisite to
    courses such as pre-calculus, statistics,
    business calculus, finite mathematics, and
    mathematics for elementary education majors.

Fundamental Experience
  • College Algebra provides students with a college
    level academic experience that emphasizes the use
    of algebra and functions in problem solving and
    modeling, provides a foundation in quantitative
    literacy, supplies the algebra and other
    mathematics needed in partner disciplines, and
    helps meet quantitative needs in, and outside of,

Fundamental Experience
  • Students address problems presented as real
    world situations by creating and interpreting
    mathematical models. Solutions to the problems
    are formulated, validated, and analyzed using
    mental, paper and pencil, algebraic, and
    technology-based techniques as appropriate.

Course Goals
  • Involve students in a meaningful and positive,
    intellectually engaging, mathematical experience
  • Provide students with opportunities to analyze,
    synthesize, and work collaboratively on
    explorations and reports
  • Develop students logical reasoning skills needed
    by informed and productive citizens

Course Goals
  • Strengthen students algebraic and quantitative
    abilities useful in the study of other
  • Develop students mastery of those algebraic
    techniques necessary for problem-solving and
    mathematical modeling
  • Improve students ability to communicate
    mathematical ideas clearly in oral and written

Course Goals
  • Develop students competence and confidence in
    their problem-solving ability
  • Develop students ability to use technology for
    understanding and doing mathematics
  • Enable and encourage students to take additional
    coursework in the mathematical sciences.

Problem Solving
  • Solving problems presented in the context of real
    world situations
  • Developing a personal framework of problem
    solving techniques
  • Creating, interpreting, and revising models and
    solutions of problems.

Functions Equations
  • Understanding the concepts of function and rate
    of change
  • Effectively using multiple perspectives
    (symbolic, numeric, graphic, and verbal) to
    explore elementary functions
  • Investigating linear, exponential, power,
    polynomial, logarithmic, and periodic functions,
    as appropriate

  • Recognizing and using standard transformations
    such as translations and dilations with graphs of
    elementary functions
  • Using systems of equations to model real world
  • Solving systems of equations using a variety of
  • Mastering those algebraic techniques and
    manipulations necessary for problem-solving and
    modeling in this course.

Data Analysis
  • Collecting, displaying, summarizing, and
    interpreting data in various forms
  • Applying algebraic transformations to linearize
    data for analysis
  • Fitting an appropriate curve to a scatterplot and
    use the resulting function for prediction and
  • Determining the appropriateness of a model via
    scientific reasoning.

  • An Increased Emphasis on Pedagogy
  • and
  • A Broader Notion of Assessment
  • Of Student Accomplishment

CRAFTY College Algebra
  • Confluence of events
  • Curriculum Foundations Report published
  • Large scale NSF project - Bill Haver, VCU
  • Availability of new modeling/application based
  • CRAFTY responded to a perceived need to address
    course and instructional models for College

CRAFTY College Algebra
  • Task Force charged with writing guidelines
  • - Initial discussions in CRAFTY meetings
  • - Presentations at AMATYC Joint Math Meetings
    with public discussions
  • - Revisions incorporating public commentary
  • Guidelines adopted by CRAFTY (Fall, 2006)
  • Pending adoption by CUPM (Spring, 2007)
  • Copies (pdf) available at
  • http//

CRAFTY College Algebra
  • The Guidelines
  • Course Objectives
  • College algebra through applications/modeling M
    eaningful appropriate use of technology
  • Course Goals
  • Challenge, develop, and strengthen
    students understanding and skills mastery

CRAFTY College Algebra
  • The Guidelines
  • Student Competencies
  • - Problem solving
  • - Functions and Equations
  • - Data Analysis
  • Pedagogy
  • - Algebra in context
  • - Technology for exploration and analysis
  • Assessment
  • - Extended set of student assessment tools
  • - Continuous course assessment

CRAFTY College Algebra
  • Challenges
  • Course development
  • - There are current models
  • Scale
  • - Huge numbers of students
  • - Extraordinary variation across institutions
  • Faculty development
  • - Who teaches College Algebra?
  • - How do we fund change?

Conceptual Understanding
  • What does conceptual understanding mean?
  • How do you recognize its presence or absence?
  • How do you encourage its development?
  • How do you assess whether students have
    developed conceptual understanding?

What Does the Slope Mean?
Comparison of student response on the final exams
in Traditional vs. Modeling College
Algebra/Trig Brookville College enrolled 2546
students in 2000 and 2702 students in 2002.
Assume that enrollment follows a linear growth
pattern. a. Write a linear equation giving the
enrollment in terms of the year t. b. If the
trend continues, what will the enrollment be in
the year 2016? c. What is the slope of the line
you found in part (a)? d. Explain, using an
English sentence, the meaning of the slope. e.
If the trend continues, when will there be 3500
Responses in Traditional Class
  • 1. The meaning of the slope is the amount that
    is gained in years and students in a given
    amount of time.
  • 2. The ratio of students to the number of years.
  • 3. Difference of the ys over the xs.
  • 4. Since it is positive it increases.
  • 5. On a graph, for every point you move to the
    right on the x- axis. You move up 78 points on
    the y-axis.
  • 6. The slope in this equation means the students
    enrolled in 2000. Y MX B .
  • 7. The amount of students that enroll within a
    period of time.
  • Every year the enrollment increases by 78
  • The slope here is 78 which means for each unit of
    time, (1 year) there are 78 more students

Responses in Traditional Class
10. No response 11. No response 12. No
response 13. No response 14. The change in
the x-coordinates over the change in the
y- coordinates. 15. This is the rise in the
number of students. 16. The slope is the average
amount of years it takes to get 156 more
students enrolled in the school. 17. Its how
many times a year it increases. 18. The slope is
the increase of students per year.
Responses in Reform Class
  • 1. This means that for every year the number of
    students increases by 78.
  • 2. The slope means that for every additional
    year the number of students increase by 78.
  • 3. For every year that passes, the student
    number enrolled increases 78 on the previous
  • As each year goes by, the of enrolled students
    goes up by 78.
  • This means that every year the number of enrolled
    students goes up by 78 students.
  • The slope means that the number of students
    enrolled in Brookville college increases by 78.
  • Every year after 2000, 78 more students will
    enroll at Brookville college.
  • Number of students enrolled increases by 78 each

Responses in Reform Class
  • 9. This means that for every year, the amount of
    enrolled students increase by 78.
  • 10. Student enrollment increases by an average
    of 78 per year.
  • 11. For every year that goes by, enrollment
    raises by 78 students.
  • 12. That means every year the of students
    enrolled increases by 2,780 students.
  • 13. For every year that passes there will be 78
    more students enrolled at Brookville college.
  • The slope means that every year, the enrollment
    of students increases by 78 people.
  • Brookville college enrolled students increasing
    by 0.06127.
  • Every two years that passes the number of
    students which is increasing the enrollment into
    Brookville College is 156.

Responses in Reform Class
17. This means that the college will enroll
.0128 more students each year. 18. By every
two year increase the amount of students goes up
by 78 students. 19. The number of students
enrolled increases by 78 every 2 years.
Understanding Slope
Both groups had comparable ability to calculate
the slope of a line. (In both groups, several
students used ?x/?y.)
It is far more important that our students
understand what the slope means in context,
whether that context arises in a math course, or
in courses in other disciplines, or eventually on
the job.
Unless explicit attention is devoted to
emphasizing the conceptual understanding of what
the slope means, the majority of students are not
able to create viable interpretations on their
own. And, without that understanding, they are
likely not able to apply the mathematics to
realistic situations.
Further Implications
  • If students cant make their own connections with
    a concept as simple as the slope of a line, they
    wont be able to create meaningful
    interpretations and connections on their own for
    more sophisticated mathematical concepts. For
  • What is the significance of the base (growth or
    decay factor) in an exponential function?
  • What is the meaning of the power in a power
  • What do the parameters in a realistic sinusoidal
    model tell about the phenomenon being modeled?
  • What is the significance of the factors of a
  • What is the significance of the derivative of a
  • What is the significance of a definite integral?

Further Implications
If we focus only on manipulative skills without
developing conceptual understanding, we produce
nothing more than students who are only Imperfect
Organic Clones of a TI-89
Developing Conceptual Understanding
Conceptual understanding cannot be just an
add-on. It must permeate every course and be a
major focus of the course. Conceptual
understanding must be accompanied by realistic
problems in the sense of mathematical
modeling. Conceptual problems must appear in all
sets of examples, on all homework assignments, on
all project assignments, and most importantly, on
all tests. Otherwise, students will not see them
as important.
Should x Mark the Spot?
All other disciplines focus globally on the
entire universe of a through z, with the
occasional contribution of ? through ?. Only
mathematics focuses on a single spot, called
x. Newtons Second Law of Motion y mx,
Einsteins formula relating energy and mass y
c2x, The ideal gas law yz nRx.
Students who see only xs and ys do not make
the connections and cannot apply the techniques
when other letters arise in other disciplines.
Should x Mark the Spot?
Keplers 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 which variable represents which
quantity. If it is written as t2 0.1664D3 ,
a huge conceptual hurdle for the students is
Should x Mark the Spot?
When students see 50 exercises where the first
40 involve solving for x, and a handful at the
end that involve other letters, the overriding
impression they gain is that x is the only
legitimate variable and the few remaining cases
are just there to torment them.
  • Some Illustrative Examples
  • of Problems
  • to Develop or Test for
  • Conceptual Understanding

Identify each of the following functions (a) -
(n) as linear, exponential, logarithmic, or
power. In each case, explain your
reasoning. (g) y 1.05x (h) y x1.05
(i) y (0.7)t (j) y v0.7
(k) z L(-½) (l) 3U 5V 14
(m) x y (n) x y
0 3   0 5
1 5.1   1 7
2 7.2   2 9.8
3 9.3   3 13.7
For the polynomial shown, (a) What is the
minimum degree? Give two different reasons for
your answer. (b) What is the sign of the leading
term? Explain. (c) What are the real roots? (d)
What are the linear factors? (e) How many
complex roots does the polynomial have?
Two functions f and g are defined in the
following table. Use the given values in the
table to complete the table. If any entries are
not defined, write undefined.
x f(x) g(x) f(x) - g(x) f(x)/g(x) f(g(x)) g(f(x))
0 1 3        
1 0 1        
2 3 0        
3 2 2        
Two functions f and g are given in the
accompanying figure. The following five graphs
(a)-(e) are the graphs of f g, g - f, fg,
f/g, and g/f. Decide which is which.
The following table shows world-wide wind power
generating capacity, in megawatts, in various
Year 1980 1985 1988 1990 1992 1995 1997 1999
Wind power 10 1020 1580 1930 2510 4820 7640 13840

(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 the 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 the domain and range of this function? (e)
What is the practical significance of the base in
the exponential function you created in part
(c)? (f) What is the doubling time for this
exponential function? Explain what does it
means. (g) According to your model, what do you
predict for the total wind power generating
capacity in 2010?
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
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 species of amphibians and reptiles living
on each.
Island Area N
Redonda 1 3
Saba 4 5
Montserrat 40 9
Puerto Rico 3459 40
Jamaica 4411 39
Hispaniola 29418 84
Cuba 44218 76

(a) Which variable is the independent variable
and which is the dependent variable? (b) The
overall pattern in the data suggests either a
power function with a positive power p lt 1 or a
logarithmic function, both of which are
increasing and concave down. Explain why a power
function is the better model to use for this
data. (c) Find the power function that models
the relationship between the number of species,
N, living on one of these islands and the area,
A, of the island and find the correlation
coefficient. (d) What are some reasonable
values that you can use for the domain and range
of this function? (e) The area of Barbados is 166
square miles. Estimate the number of species of
amphibians and reptiles living there.

Write a possible formula for each of the
following trigonometric functions

The average daytime high temperature in New York
as a function of the day of the year varies
between 32?F and 94?F. Assume the coldest day
occurs on the 30th day and the hottest day on the
214th. (a) Sketch the graph of the temperature
as a function of time over a three year time
span. (b) Write a formula for a sinusoidal
function that models the temperature over the
course of a year. (c) What are the domain and
range for this function? (d) What are the
amplitude, vertical shift, period, frequency, and
phase shift of this function? (e) Estimate the
high temperature on March 15. (f) What are all
the dates on which the high temperature is most
likely 80??
Some Conclusions
We cannot simply concentrate on teaching the
mathematical techniques that the students need.
It is as least as important to stress conceptual
understanding and the meaning of the mathematics.
We can accomplish this by using a combination
of realistic and conceptual examples, homework
problems, and test problems that force students
to think and explain, not just manipulate
symbols. If we fail to do this, we are not
adequately preparing our students for successive
mathematics courses, for courses in other
disciplines, and for using mathematics on the job
and throughout their lives.
  • It is only in math classes that functions are
  • Everywhere else,
  • The existence of functions is observed
  • Formulas for functions are created
  • Functions are used to answer questions about a

The Need for Real-World Problems and Examples
Realistic Applications and Mathematical Modeling
  • Real-world data enables the integration of data
    analysis concepts with the development of
    mathematical concepts and methods
  • Realistic applications illustrate that data arise
    in a variety of contexts
  • Realistic applications and genuine data can
    increase students interest in and motivation for
    studying mathematics
  • Realistic applications link the mathematics to
    what students see in and need to know for other
    courses in other disciplines.

The Role of Technology
The Role of Technology
  • Technology allows us to do many standard topics
    differently and more easily.
  • Technology allows us to introduce new topics and
    methods that we could not do previously.
  • Technology allows us to de-emphasize or even
    remove some topics that are now less important.

Technology How?
  • Students can use technology as a problem-solving
    tool to
  • Model situations and analyze functions
  • Tackle complex problems
  • Students can use technology as a learning tool to
  • Explore new concepts and discover new ideas
  • Make connections
  • Develop a firm understanding of mathematical
  • Develop mental images associated with abstract

Technology - Caution
  • Students need to balance the use of technology
    and the use of pencil and paper.
  • Students need to learn to use technology
    appropriately and wisely.

Changing the Learning and Teaching Environment
Traditional Approach vs. Student-Centered
  • With a traditional approach, students
  • Listen to lectures
  • Copy notes from the board
  • Mimic examples
  • Use technology to do calculations
  • Do familiar problems in homework and on exams
  • Fly through the material
  • Hold instructor responsible for learning
  • Go to instructor for help

Traditional Approach vs. Student-Centered
  • With a student-centered approach, students
  • Participate in discussions
  • Work collaboratively
  • Find solutions and approaches
  • Use technology to investigate ideas
  • Write about and use new ideas in homework and on
  • Take time to think
  • Accept responsibility for learning
  • First try to help each other

Student-Centered Learning The Role of the
  • The instructor
  • Designs activities
  • Emphasizes learning
  • Interacts with students
  • Approaches ideas from the students point of view
  • Controls the learning environment
  • The instructor is a
  • Facilitator
  • Coach
  • Intellectual manager

Student-Centered Learning Intended Outcomes
  • Impel students to be active learners
  • Make learning mathematics an enjoyable experience
  • Help students develop confidence to read, write
    and do mathematics
  • Enhance students understanding of fundamental
    mathematics concepts
  • Increase students ability to use these concepts
    in other disciplines
  • Inspire students to continue the study of

  • But, if college algebra and related courses
  • what happens to the next generation of math and
    science majors?
  • Dont they need all the traditional algebraic
  • But, if they dont develop conceptual
    understanding and the ability to apply the
    mathematics, what value are the skills?

  • The Link to Calculus

Calculus and Related Enrollments
In 2000, about 676,000 students took Calculus,
Differential Equations, Linear Algebra, and
Discrete Mathematics (This is up 6 from
1995) Over the same time period, however,
calculus enrollment has been steady, at best.
Calculus and Related Enrollments
In comparison, in 2000, 171,400 students took one
of the two AP Calculus exams either AB or BC.
(This is up 40 from 1995) In 2004, 225,000
students took AP Calculus exams In 2005, about
240,000 took AP Calculus exams Reportedly,
about twice as many students take calculus in
high school, but do not take an AP exam.
AP Calculus

Some Implications
  • Today more students take calculus in high school
    than in college
  • And, as ever more students take more mathematics,
    especially calculus, in high school, we should
  • Fewer students taking these courses in college
  • The overall quality of the students who take
    these courses in college will decrease.

Another Conclusion
We should anticipate the day, in the not too
distant future, when college calculus, like
college algebra, becomes a semi-remedial
course. (Several elite colleges already have
stopped giving credit for Calculus I.)
Another Conclusion
It is not conscionable for departments to treat
students as mathematical cannon-fodder, by
pushing them into courses they have little hope
of surviving in order to increase the number of
sections of calculus that are offered.
Associates Degrees in Mathematics
In 2002, P There were 595,000 associate
degrees P Of these, 685 were in mathematics
This is one-tenth of one percent!
Bachelors Degrees in Mathematics
In 2002, PThere were 1,292,000 bachelors
degrees POf these, 12,395 were in mathematics
This is under one percent!
Masters Degrees in Mathematics
In 2002, PThere were 482,000 masters
degrees POf these, 3487 were in mathematics
This is 7 tenths of one percent!
PhDs Degrees in Mathematics
  • In 2002,
  • There were 44,000 doctoral degrees
  • Of these, 958 were in mathematics
  • This is just over two percent!
  • But less than half were U.S. citizens

Who Are the Students?
Based on the enrollment figures, the students who
take college algebra and related courses are not
going to become mathematics majors. They are
not going to be majors in any of the mathematics
intensive disciplines.
The Focus in these Courses
But most college algebra courses and certainly
all precalculus courses were designed to prepare
students for calculus and most of them are still
offered in that spirit. Even though only a small
percentage of the students have any intention of
going into calculus!
A Fresh Start for Collegiate Mathematics Rethinkin
g the Courses Below Calculus MAA Notes, 2005
Nancy Baxter Hastings, et al (editors)
A Fresh Start to Collegiate Math
Nancy Baxter Hastings Overview of the Volume
Jack Narayan Darren Narayan The Conference Rethinking the Preparation for Calculus
Lynn Steen Twenty Questions
Mercedes McGowen Who are the Students Who Take Precalculus?
Steve Dunbar Enrollment Flow to and from Courses below Calculus
Deborah Hughes Hallett What Have We Learned from Calculus Reform? The Road to Conceptual Understanding
Susan Ganter Calculus and Introductory College Mathematics Current Trends and Future Directions

A Fresh Start to Collegiate Math
Refocusing Precalculus, College Algebra, and Quantitative Literacy Refocusing Precalculus, College Algebra, and Quantitative Literacy
Shelly Gordon Preparing Students for Calculus in the Twenty-First Century
Bernie Madison Preparing for Calculus and Preparing for Life
Don Small College Algebra A Course in Crisis
Scott Herriott Changes in College Algebra
Janet Andersen One Approach to Quantitative Literacy Mathematics in Public Discourse
The Transition from High School to College The Transition from High School to College
Zal Usiskin High School Overview and the Transition to College
Dan Teague Precalculus Reform A High School Perspective
Eric Robinson John Maceli The Influence of Current Efforts to Improve School Mathematics on Preparation for Calculus

A Fresh Start to Collegiate Math
The Needs of Other Disciplines The Needs of Other Disciplines
Susan Ganter and Bill Barker Fundamental Mathematics Voices of the Partner Disciplines
Rich West Skills versus Concepts
Allan Rossman Integrating Data Analysis into Precalculus Courses
Student Learning and Research Student Learning and Research
Florence Gordon Assessing What Students Learn Reform versus Traditional Precalculus and Follow-up Calculus
Rebecca Walker Student Voices and the Transition from Standards-Based Curriculum to College

A Fresh Start to Collegiate Math
Robert Megginson Some Political and Practical Issues in Implementing Reform
Judy Ackerman Implementing Curricular Change in Precalculus A Dean's Perspective
Bonnie Gold Alternatives to the One-Size-Fits-All Precalculus/College Algebra Course
Al Cuoco Preparing for Calculus and Beyond Some Curriculum Design Issues
Lang Moore and David Smith Changing Technology Implies Changing Pedagogy
Shelly Gordon The Need to Rethink Placement in Mathematics
Influencing the Mathematics Community Influencing the Mathematics Community
Bernie Madison Launching a Precalculus Reform Movement Influencing the Mathematics Community
Naomi Fisher Bonnie Saunders Mathematics Programs for the "Rest of Us"
Shelly Gordon Where Do We Go from Here Forging a National Initiative

A Fresh Start to Collegiate Math
Ideas and Projects that Work (long papers) Ideas and Projects that Work (long papers) Ideas and Projects that Work (long papers)
Doris Schattschneider Doris Schattschneider An Alternate Approach Integrating Precalculus into Calculus
Bill Fox Bill Fox College Algebra Reform through Interdisciplinary Applications
Dan Kalman Dan Kalman Elementary Math Models College Algebra Topics and a Liberal Arts Approach
Brigette Lahme, Jerry Morris and Elias Toubassi Brigette Lahme, Jerry Morris and Elias Toubassi The Case for Labs in Precalculus
Ideas and Projects that Work (short papers) Ideas and Projects that Work (short papers) Ideas and Projects that Work (short papers)
Gary Simundza The Fifth Rule Experiential Mathematics The Fifth Rule Experiential Mathematics
Darrell Abney and James Hougland Reform Intermediate Algebra in Kentucky Community Colleges Reform Intermediate Algebra in Kentucky Community Colleges
Marsha Davis Precalculus Concepts in Context Precalculus Concepts in Context

A Fresh Start to Collegiate Math
Benny Evans Rethinking College Algebra
Sol Garfunkel From the Bottom Up
Florence Gordon Shelly Gordon Functioning in the Real World
Deborah Hughes Hallett Importance of a Story Line Functions as a Model
Nancy Baxter Hastings Using a Guided-Inquiry Approach to Enhance Student Learning in Precalculus
Allan Jacobs Maricopa Mathematics
Linda Kime Quantitative Reasoning
Mercedes McGowan Developmental Algebra The First Course for Many College Students
Allan Rossman Workshop Precalculus Functions, Data and Models
Chris Schaufele Nancy Zumoff The Earth Math Projects
Don Small Contemporary College Algebra

A Fresh Start to Collegiate Math
Ernie Danforth, Brian Gray, Arlene Kleinstein, Rick Patrick and Sylvia Svitak Mathematics in Action Empowering Students with Introductory and Intermediate College Mathematics
Todd Swanson Precalculus A Study of Functions and Their Applications
David Wells Lynn Tilson Successes and Failures of a Precalculus Reform Project

The Need to Rethink Placement in Mathematics
Rethinking Placement Tests
  • Two Types of Placement Tests
  • National (standardized) tests
  • Not much we can do about them.
  • 2. Home-grown tests

Rethinking Placement Tests
  • Four scenarios
  • Students come from traditional curriculum into
    traditional curriculum.
  • Students from Standards-based curriculum into
    traditional curriculum.
  • Students from traditional curriculum into reform
  • 4. Students from Standards-based curriculum into
    reform curriculum.

One National Placement Test
1. Square a binomial. 2. Determine a
quadratic function arising from a verbal
description (e.g., area of a rectangle whose
sides are both linear expressions in x). 3.
Simplify a rational expression. 4. Confirm
solutions to a quadratic function in factored
form. 5. Completely factor a polynomial. 6.
Solve a literal equation for a given unknown.

A National Placement Test
7. Solve a verbal problem involving
percent. 8. Simplify and combine like
radicals. 9. Simplify a complex fraction. 10.
Confirm the solution to two simultaneous linear
equations. 11. Traditional verbal problem
(e.g., age problem). 12. Graphs of linear

A Tale of Three Colleges in NYS
  • Totally traditional curriculum developmental
    through calculus.
  • Traditional courses developmental through
    college algebra, then reform in precalculus on
  • Totally reform developmental through upper
    division offerings.
  • All use the same national placement test.

A Tale of Three Colleges in NYS
BUT New York State has not offered the
traditional Algebra I Geometry Algebra II
Trigonometry curriculum in over 20
years! Instead, there is an integrated
curriculum that emphasize topics such as
statistics and data analysis, probability, logic,
etc. in addition to algebra and trigonometry.

A Tale of Three Colleges in NYS
So students are being placed one, two, and even
three semesters below where they should be based
on the amount of mathematics they have
studied! And they are being punished because
of what is being assessed and what is not being
assessed, because of what was stressed in high
school and what was not stressed, because of
what was taught, not what they learned or didnt

A Modern High School Problem
Given the complete 32-year set of monthly CO2
emission levels (a portion is shown below),
create a mathematical model to fit the data.

Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Avg
1968 322 323 324 325 325 325 324 322 320 320 320 322 323
1969 324 324 325 326 327 326 325 323 322 321 322 324 324
A Modern High School Problem
1. Students first do a vertical shift of about
300 ppm and then fit an exponential function to
the transformed data to get

2. They then create a sinusoidal model to fit
the monthly oscillatory behavior about the
exponential curve
3. They then combine the two components to get
4. They finally give interpretations of the
various parameters and what each says about the
increase in concentration and use the model to
predict future or past concentration levels.
Placement, Revisited
Picture an entering freshman who has taken high
school courses with a focus on problems like the
preceding one and who has developed an
appreciation for the power of mathematics based
on understanding the concepts and applying them
to realistic situations. What happens when that
student sits down to take a traditional placement
test? Is it surprising that many such students
end up being placed into developmental courses?

What a High School Teacher Said
  • If you try to teach my students with the
    mistaken belief that they know the mathematics I
    knew at their age, you will miss a great
    opportunity. My students know more mathematics
    than I did, but it is not the same mathematics
    and I believe they know it differently. They
    have a different vision of mathematics that would
    be helpful in learning calculus if it were
  • Dan Teague

Rethinking Placement Tests
What Can Be Done 1. Home-grown tests Develop
alternate versions that reflect both your
curriculum AND the different curricula that your
students have come through. 2. National
(standardized) tests Contact the test-makers
(Accuplacer ETS and Compass ACT) and lobby
them to develop alternative tests to reflect both
your curriculum and the different curricula that
your students have come through.

Why Students Take These Courses
  • The vast majority of students take college
    algebra and related courses because
  • they are required by other departments or
  • they are needed to satisfy general education
  • As a consequence, we have to pay attention to
    what the other disciplines want their students to
    gain from these courses.

Connecting with Other Disciplines
All other disciplines are under pressure to teach
more material to their students, and that
material is much more than just the mathematical
ideas and applications. If we do not provide
courses that satisfy todays needs of the other
disciplines, they are likely going to drop the
requirements for our courses and include the
needed material in their own offerings.
  • Voices of the Partner Disciplines
  • CRAFTYs Curriculum Foundations Project

Curriculum Foundations Project
A series of 11 workshops with leading educators
from 17 quantitative disciplines to inform the
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

What the Physicists Said
  • Conceptual understanding of basic mathematical
    principles is very important for success in
    introductory physics. It is more important than
    esoteric computational skill. However, basic
    computational skill is crucial.
  • Development of problem solving skills is a
    critical aspect of a mathematics education.

What the Physicists Said
  • Courses should cover fewer topics and place
    increased emphasis on increasing the confidence
    and competence that students have with the most
    fundamental topics.

What the Physicists Said
  • The learning of physics depends less directly
    than one might think on 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 Physicists Said
  • Students should be able to focus a situation
    into a problem, translate the problem into a
    mathematical representation, plan a solution, and
    then execute the plan. Finally, students should
    be trained to check a solution for

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. Computational skill without
    theoretical understanding is shallow.

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 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

What Business Faculty Said
  • Courses should stress problem solving, with the
    incumbent recognition of ambiguities.
  • Courses should stress conceptual understanding
    (motivating the math with the whys not just
    the hows).
  • Courses should stress critical thinking.
  • An important student outcome is their ability to
    develop appropriate models to solve defined

What Business Faculty Said
  • Courses should use industry standard technology
  • 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.

What the Engineers Said
  • One basic function of undergraduate electrical
    engineering education is to provide students with
    the conceptual skills to formulate, develop,
    solve, evaluate and validate physical systems.
    Mathematics is indispensable in this regard.

What the Engineers Said
  • The mathematics required to enable students to
    achieve these skills should emphasize concepts
    and problem solving skills more than emphasizing
    the repetitive mechanics of solving routine

What the Engineers Said
  • Students must learn the basic mechanics of
    mathematics, but care must be taken that these
    mechanics do not become the focus of any
    mathematics course.

What the Chemists Said
  • Introduce multivariable, multidimensional
    problems from the outset
  • Listen to the equations most specific
    mathematical expressions can be recovered from a
    few fundamental relationships in a few steps.
  • Of widespread use in chemistry teaching and
    research are spreadsheets to produce graphs and
    perform statistical calculations

Health-Related Life Sciences
  • Put special emphasis on the use of models as a
    way to organize information for the purpose of
    gaining insight and to provide intuition into
    systems that are too complex to understand any
    other way.
  • Students should master appropriate computer
    packages, such as a spreadsheet,
    symbolic/numerical computational packages
    (Mathematica, Maple, Matlab), statistical

Common Themes from All Disciplines
  • Strong emphasis on problem solving
  • Strong emphasis on mathematical modeling
  • Conceptual understanding is more important than
    skill development
  • Development of critical thinking and reasoning
    skills is essential

Common Themes from All Disciplines
  • Use of technology, especially spreadsheets
  • Development of communication skills (written and
  • Greater emphasis on probability and statistics
  • Greater cooperation between mathematics and the
    other disciplines

Some Implications
Although the number of college students taking
calculus is at best holding steady, the
percentage of students taking college calculus is
dropping, since overall college enrollment has
been rising rapidly. But the number of students
taking calculus in high school already exceeds
the number taking it in college. It is growing
at 8.
Some Implications
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. And college algebra and related
courses exist almost exclusively to serve the
needs of other disciplines.
Some Implications
If we fail to offer courses that meet the needs
of the students in the other disciplines, those
departments will increasingly drop the
requirements for 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.
What Can Be Removed?
How many of you remember that there used to be
something called the Law of Tangents? What
happened to this universal law? Did triangles
stop obeying it? Does anyone miss it?
What Can Be Removed?
  • Descartes rule of signs
  • The rational root theorem
  • Synthetic division
  • The Cotangent, Secant, and Cosecant
  • were needed for computational purposes
  • Just learn and teach a new identity

How Important Are Rational Functions?
  • In DE To find closed-form solutions for
    several differential equations, (usually done
    with CAS today, if at all)
  • In Calculus II Integration using partial
    fractionsoften all four exhaustive (and
    exhausting) cases
  • In Calculus I Differentiating rational
  • In Precalculus Emphasis on the behavior of
    all kinds of rational functions and even
    partial fraction decompositions
  • In College Algebra Addition, subtraction,
    multiplication, division and especially
    reduction of complex fractional expressions
  • In each course, it is the topic that separates
    the men from the boys! But, can you name any
    realistic applications that involve rational
    functions? Why do we need them in excess?

New Visions of College Algebra
  • Crauder, Evans and Noell A Modeling
    Alternative to College Algebra
  • Herriott College Algebra through Functions and
  • Kime and Clark Explorations in College
  • Small Contemporary College Algebra

New Visions for Precalculus
  • Gordon, Gordon, et al Functioning in the Real
    World A Precalculus Experience, 2nd Ed
  • Hastings Rossman Workshop Precalculus
  • Hughes-Hallett, Gleason, et al Functions
    Modeling Change Preparation for Calculus
  • Moran, Davis, and Murphy Precalculus
    Concepts in Context

New Visions for Alternative Courses
  • Bennett Quantitative Reasoning
  • Burger and Starbird The Heart of Mathematics
    An Invitation to Effective Thinking
  • COMAP For All Practical Purposes
  • Pierce Mathematics for Life
  • Sons Mathematical Thinking

How Does the Quantitative Literacy
Initiative Relate to College Algebra?
What is Quantitative Literacy?
Quantitative literacy (QL), or numeracy, is the
knowledge and habits of mind needed to understand
and use quantitative measures and inferences
necessary to function as a responsible citizen,
productive worker, and discerning consumer. QL
describes the quantitative reasoning capabilities
required of citizens in today's information age
-- from the QL Forum White Paper
QL and the Mathematics Curriculum
The focus of the math curriculum is the
geometry-algebra-trigonometry-calculus sequence.
  • In high school, the route to competitive
  • The sequence is linear and hurried.
  • No time to teach mathematics in contexts.
  • Courses are routes to somewhere else.
  • Other sequences are terminal and often second

Elements of QL
  • Confidence with mathematics
  • Cultural appreciation
  • Interpreting data
  • Logical thinking
  • Making decisions
  • Mathematics in context
  • Number sense
  • Practical skills
  • Prerequisite knowledge
  • Symbol sense

Two Kinds of Literacy
  • Inert - Level of verbal and numerate skills
    required to comprehend instructions, perform
    routine procedures, and complete tasks in a
    routine manner.
  • Liberating - Command of both the enabling skills
    needed to search out information and power of
    mind necessary to critique it, reflect upon it,
    and apply it in making decisions.

Lawrence A. Cremins, American Education The
Metropolitan Experience 1876-1980. New York
Harper Row, 1988. (as quoted by R. Orrill in
How does the US compare to other countries?
NALS Quantitative Paradigm National Adult
Literacy Survey
Skill Level 1 - Minimal Approximate
Educational Equivalence - Dropout NALS
Competencies - Can perform a single, simple
arithmetic operation such as addition. The
numbers used are provided and the operation to be
performed is specified. NALS Examples - Total
a bank deposit entry
NALS Quantitative Paradigm
Skill Level 2 - Basic Approximate Educational
Equivalence - Average or below average HS
graduate NALS Competencies - Can perform a
single arithmetic operation using numbers that
are given in the task or easily located in the
material. The arithmetic operation is either
described or easily determined from the format of
the materials. NALS Examples - Calculate
postage and fees for certified mail - Determine
the difference in price between tickets for two
shows - Calculate the total costs of purchase
from an order form
NALS Quantitative Paradigm
Skill Level 3 - Competent Approximate Educational
Equivalence -Some postsecondary education NALS
Competencies - Can pe