Refocusing the Courses Below Calculus A Joint Initiative of

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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
[email protected]
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 presentations.
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 Foundation.
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 need?
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 algebra:
• only 2.8%, ever went on to start Calculus I at any time
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.
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 today’s 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 modeling—that 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 studentcentered rather than instructor-centered
pedagogy.
- 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 Reports.
• AMATYC Crossroads Standards and the Beyond
Crossroads report.
• NCTM, Principles and Standards for School
Mathematics.
•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
contexts.
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 situations.
•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
NCTM’s 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 prerequisite 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, academia.
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 disciplines;
• 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
form;
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
situations;
• Solving systems of equations using a variety of
methods;
• 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 analysis;
• 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 texts
CRAFTY responded to a perceived need to address
course and instructional models for College Algebra.
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://www.mathsci.appstate.edu/~wmcb/ICTCM
CRAFTY & College Algebra
The Guidelines:
• Course Objectives
College algebra through applications/modeling
Meaningful & 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 students?
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 y’s over the x’s.
4. Since it is positive it increases.
5. On a graph, for every point you move to the right on the xaxis. 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.
8. Every year the enrollment increases by 78 students.
9. The slope here is 78 which means for each unit of time, (1
year) there are 78 more students enrolled.
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 ycoordinates.
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 year.
4. As each year goes by, the # of enrolled students goes up by 78.
5. This means that every year the number of enrolled students
goes up by 78 students.
6. The slope means that the number of students enrolled in
Brookville college increases by 78.
7. Every year after 2000, 78 more students will enroll at
Brookville college.
8. Number of students enrolled increases by 78 each year.
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.
14. The slope means that every year, the enrollment of students
increases by 78 people.
15. Brookville college enrolled students increasing by 0.06127.
16. 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 can’t make their own connections with a concept as
simple as the slope of a line, they won’t be able to create
meaningful interpretations and connections on their own for
more sophisticated mathematical concepts. For instance,
• 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 function?
• What do the parameters in a realistic sinusoidal model tell
about the phenomenon being modeled?
• What is the significance of the factors of a polynomial?
• What is the significance of the derivative of a function?
• 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.
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 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
which variable represents which quantity.
If it is written as t2 = 0.1664D3 , a huge conceptual
hurdle for the students is eliminated.
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
0
3
5
1
1
5.1
7
2
2
7.2
9.8
3
3
9.3
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)
0
1
3
1
0
1
2
3
0
3
2
2
f(x) - g(x)
f(x)/g(x)
f(g(x))
g(f(x))
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, f*g, f/g,
1.5
1
g(x)
0.5
0
0
-0.5
1
2
4
f(x)
-1
and g/f. Decide which is which.
3
-1.5
1
2
2
(b)
(a)
0
0
1
2
3
4
5
(c)
0
0
1
2
3
4
5
0
1
2
-2
-1
-2
10
10
5
5
0
0
0
-5
-10
(e)
(d)
1
2
3
4
0
5
-5
-10
1
2
3
4
5
3
4
5
5
The following table shows world-wide wind power
generating capacity, in megawatts, in various
years.
Year
1980 1985 1988 1990 1992 1995 1997
Wind
power
10 1020 1580 1930 2510 4820 7640
15000
10000
5000
0
1980
1985
1990
1995
2000
1999
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.
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
Number of Species
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.
100
80
60
40
20
0
0
15000
30000
Area (square miles)
45000
(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 < 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.
Functions
It is only in math classes that functions are given.
Everywhere else,
• The existence of functions is observed
• Formulas for functions are created
• Functions are used to answer questions about a
context
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 ideas
Develop mental images associated with abstract concepts
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 Approach
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 Approach
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
exams
• Take time to think
• Accept responsibility for learning
• First try to help each other
Student-Centered Learning:
The Role of the Instructor
• The instructor
–
–
–
–
–
Designs activities
Emphasizes learning
Interacts with students
Approaches ideas from the student’s 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 mathematics
But, if college algebra and related courses
change,
what happens to the next generation of math
and science majors?
Don’t they need all the traditional algebraic
skills?
But, if they don’t 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
Students Taking AP Calculus Exam
200000
150000
100000
50000
0
1991
1993
1995
1997
1999
2001
2003
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 expect:
• 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 cannonfodder, 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!
Bachelor’s Degrees in Mathematics
In 2002,
PThere were 1,292,000 bachelor’s degrees
POf these, 12,395 were in mathematics
This is under one percent!
Master’s Degrees in Mathematics
In 2002,
PThere were 482,000 master’s degrees
POf these, 3487 were in mathematics
This is 7 tenths of one percent!
PhD’s 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
Rethinking the Courses
Below Calculus
MAA Notes, 2005
Nancy Baxter Hastings, et al
(editors)
A Fresh Start to Collegiate Math
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
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
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
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
Implementation
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
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)
Doris
Schattschneider
An Alternate Approach: Integrating Precalculus into
Calculus
Bill Fox
College Algebra Reform through Interdisciplinary
Applications
Dan Kalman
Elementary Math Models: College Algebra Topics and a
Liberal Arts Approach
Brigette Lahme,
Jerry Morris and
Elias Toubassi
The Case for Labs in Precalculus
Ideas and Projects that Work (short papers)
Gary Simundza
The Fifth Rule: Experiential Mathematics
Darrell Abney and
James
Hougland
Reform Intermediate Algebra in Kentucky Community
Colleges
Marsha Davis
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:
1. National (standardized) tests
Not much we can do about them.
2. Home-grown tests
Rethinking Placement Tests
Four scenarios:
1. Students come from traditional curriculum
into traditional curriculum.
2. Students from Standards-based curriculum
into traditional curriculum.
3. Students from traditional curriculum into
reform curriculum.
4. Students from Standards-based curriculum
into reform curriculum.
One National Placement Test
1.
2.
3.
4.
5.
6.
Square a binomial.
Determine a quadratic function arising from a
verbal description (e.g., area of a rectangle
whose sides are both linear expressions in x).
Simplify a rational expression.
Confirm solutions to a quadratic function in
factored form.
Completely factor a polynomial.
Solve a literal equation for a given unknown.
A National Placement Test
7.
8.
9.
10.
Solve a verbal problem involving percent.
Simplify and combine like radicals.
Simplify a complex fraction.
Confirm the solution to two simultaneous linear
equations.
11. Traditional verbal problem (e.g., age problem).
12. Graphs of linear inequalities.
A Tale of Three Colleges in NYS
1. Totally traditional curriculum – developmental
through calculus.
2. Traditional courses – developmental through
college algebra, then reform in precalculus on up.
3. 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
didn’t learn.
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:
F (t )  1.656e0.03923t  299.5
2. They then create a sinusoidal model to fit the monthly
oscillatory behavior about the exponential curve
1 
 
S  t   3.5sin  2  t     0.5
  24  
3. They then combine the two components to get
C  t   F  t   S  t   1.656e
0.03923 t
1 
 
 3.5sin  2  t 
   299
  24  
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 tapped.”
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
requirements
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 today’s
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
CRAFTY’s 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 Barker.
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
reasonableness.
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 application.
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 “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
• Courses should use industry standard
technology (spreadsheets).
• 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 problems.
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
packages.
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 oral)
• 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:
1  tan x 
2
1
cos2 x
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 fractions–often all four
exhaustive (and exhausting) cases
• In Calculus I: Differentiating rational functions
• 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
Models
• Kime and Clark: Explorations in College Algebra
• 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 colleges.
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 rate.
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 M&D)
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 perform tasks where two or more numbers are needed to
solve the problem and they must be found in the material. The
operation(s) needed can be determined from the arithmetic
relation terms used in the question or directive.
NALS Examples
- Use a calculator to calculate the difference between the regular
and sale price
- Calculate miles per gallon from information on a mileage
record chart
- Use a calculator to determine the discount from an oil bill if
paid within 10 days
NALS Quantitative Paradigm
Skill Level 4 - Advanced
Approximate Educational Equivalence -Bachelor’s or
advanced degree
NALS Competencies Can perform two or more operations in sequence or a single
operation in which the quantities are found in different types
of displays, or where the operations must be inferred from the
information given or from prior knowledge.
NALS Examples
- Determine the correct change using information in a menu
-Calculate how much a couple would receive from Supplemental
Security Income, using an eligibility pamphlet
- Use information stated in a news article to calculate the
amount of money that should go to raising a child
NALS Quantitative Paradigm
Skill Level 5 - Superior
Approximate Educational Equivalence -High achieving collegeeducated populations
NALS Competencies Can perform multiple operations sequentially, and can also
find the features of problems embedded in text or rely on
background knowledge to determine the quantities or
operations needed.
NALS Examples
- Use a calculator to determine the total cost of carpet to cover a
room
- Use information in a news article to calculate the difference in
time for completing a race
- Determine shipping and total costs on an order form for items
in a catalog
Many College Graduates Demonstrate
Weak Quantitative Literacy Skills
Grads:
2 Yr. Colleges
Grads:
4 Yr. Colleges
Level 5: High
5
13
Level 4
30
40
Level 3
44
40
Level 2
17
10
Level 1: Low
4
3
Source: USDOE, NCES, National Adult Literacy Survey, 1992, in Literacy in the Labor Force: Results from the NALS,
September 1999, p. 61.
Quantitative Literacy and Job Opportunity, 1998-2008
Mathematical Literacy Level
Minimal
(Dropout)
Quantitative
Skill Level
15% of the Labor Force
12% of All Jobs in 2008
1998 Earnings: $20,300
9% of New Jobs, 1998-2008
24% of the Labor Force
Basic
(Below Average H. S. Graduate) 24% of All Jobs in 2008
1998 Earnings: $25,500
21% of New Jobs, 1998-2008
Competent
(Some Postsecondary)
Advanced/Superior
(Bachelor’s Degree)
35% of the Labor Force
37% of All Jobs in 2008
1998 Earnings: $31,600
36% of New Jobs, 1998-2008
26% of the Labor Force
27% of All Jobs in 2008
1998 Earnings: $45,400
33% of New Jobs, 1998-2008
Source: National Adult Literacy Survey; Current Population Survey; Bureau of Labor Statistics Employment Projections, 1998-2008.
Where Does College Algebra Fit In?
QL is something that should permeate the entire
mathematics curriculum, so that every student
develops these skills.
The one existing course that provides the best
opportunity to stress QL is college algebra:
• It has the largest enrollment
• It does not prepare or motivate large numbers
of students to go on to calculus
• It is taken to prepare students for courses in
other disciplines, and the themes of QL are the
mathematical topics needed in most other
disciplines today.
Some
Sample
Programs
A Sample Program
•
A Sample Program
•
A Sample Program
•
The Challenges
to Be Faced
The Challenges Ahead
• Convincing the math community
1. Conducting a series of extensive tracking
studies to determine how many (or how few)
students who take these courses actually go on
to calculus.
2. Identifying and highlighting “best practices”
in programs that reflect the goals of this
initiative.
CRAFTY’s Demonstration Project
All 1800 MAA Liaisons were asked if
their departments would be interested
in participating in a planned
pilot/research proposal.
Within 6 days, 211 departments
indicated that they were interested in
seriously considering this possibility.
CRAFTY’s Demonstration Project
Eleven colleges and universities
were selected to participate.
Each agreed to offer multiple pilot
sections of modeling based college
algebra courses as well as control
sections in order to determine the
effectiveness of these approaches.
CRAFTY’s Demonstration Project
University of Arizona
Essex Community College
Florida Southern University
Harrisburg Area Community College
Mesa State University
Missouri State University
North Carolina A&T
University of North Dakota
University of South Carolina
South Dakota State University
Southeastern Louisiana University
CRAFTY’s Demonstration Project
The 11 institutions agreed to pilot sections of
college algebra with the following features:
• Course organized around mathematical
modeling;
• Students assigned long-term project(s);
• Students assigned work to be completed in
collaboration with other students;
• Graphing calculators and/or computer utilities
utilized throughout;
• Algebraic skills deemed as critical will be
maintained, but deemphasized.
The Research Component
The following data is being collected:
• Grades;
• Retention information;
• Performance on common test items;
• Student retention and grades in subsequent
courses
Preliminary Findings
• 10 of 11 institutions offered sections as
planned;
• Great variation in extent to which planned
features were incorporated;
• Persistence in modeling sections was greater
overall;
• Institutions requested more professional
development.
Still To Be Determined
• Performance on common exams;
• Performance in future courses.
The Challenges Ahead
2. Convincing college administrators to
support (academically and financially)
efforts to refocus the courses below
calculus.
What Can Administrators Do?
When the University of Michigan wanted
to change to calculus reform, including
going from large lectures of 800 students
to small classes of 20 taught by full-time
faculty, the department argued to the dean
that by saving only 2% of the students who
fail out because of calculus, the savings to
the university would exceed the $1,000,000
annual additional instructional cost. The
dean immediately said “Go for it.”
What Can Administrators Do?
At Stony Brook University, all faculty applying
for promotion or tenure must supply evidence
documenting significant achievement in
instructional innovation in addition to research
and service achievements. The message is a
strong signal that a major research university
places a strong value on students’ education and
the undergraduate curriculum.
What Can Administrators Do?
At Farmingdale State University of New York,
all faculty who were involved in a major NSF
project to promote interaction between
mathematics and other quantitative fields were
rewarded with merit pay increases. These
activities were also counted highly in promotion
and tenure decisions.
The Challenges Ahead
3. Convincing academic bodies outside of
mathematics to allow alternatives to
traditional college algebra courses to fulfill
general education requirements.
An Example: Georgia
The state education department in Georgia
had a mandate for general education that
every student must take college algebra. A
group of faculty from various two and four
year colleges across the state lobbied for
years until they finally convinced the state
authorities to allow a course in
mathematical modeling at the college
algebra level to serve as an alternative for
satisfying the Gen Ed math requirement.
The Challenges Ahead
4. Convincing the testing industry to begin
development of a new generation of
placement and related tests that reflect the
NCTM Standards-based curricula in the
schools and the kinds of refocused courses
below calculus in the colleges that we hope
to being about.
The Challenges Ahead
5. Gaining the active support of a wide
variety of other disciplines that typically
require college algebra in the effort to
refocus the courses below calculus.
• CRAFTY and MAD (Math Across the
Disciplines) committee have launched a
second round of Curriculum Foundations
workshops to address this issue.
The Challenges Ahead
6. Gaining the active support of
representatives of business, industry, and
government in this initiative.
• Discussions are underway about
revisiting some of the participants in the
Forum on Quantitative Literacy.
The Challenges Ahead
7. Creating a faculty development
program to assist faculty, especially part
time faculty and graduate TA’s, to teach
the new versions of these courses.
This is a major focus of CRAFTY’s
demonstration project and AMATYC is
planning to extend its Traveling Workshop
program to encompass this.
The Challenges Ahead
8. Influencing teacher preparation
programs to rethink the courses they offer
to prepare the next generation of teachers
in the spirit of this initiative.
This would better prepare prospective
teachers to teach classes that are more
attuned to the spirit of the NCTM
Standards.
The Challenges Ahead
9. Developing a regional network through
the MAA sections and the AMATYC
affiliates to influence the local
mathematics communities and to provide
support at the local and regional levels to
faculty and departments who seek to
change these courses.
The Challenges Ahead
10. Influencing funding agencies such as
the NSF to develop new programs that are
specifically designed to promote both the
development of new approaches to the
courses below calculus and the widespread
implementation of existing “reform”
versions of these courses.
Influencing the Funding Agencies
The NSF recently requested the MAA and
the other national societies to provide
guidance about possible program efforts
that would promote both the development of
new approaches to algebra at all levels and
the widespread implementation of existing
“reform” versions of these algebra courses.
What is known about College Algebra?
• Annually 650,000 to 750,000 college students enroll
in College Algebra.
• Less than 10% of the students who enroll in College
Algebra intend to prepare for technical careers and a
much smaller percentage end up entering the
workforce in technical fields.
• Nationwide more than 45% of students enrolled in
College Algebra either withdraw or receive a grade
of D or F.
What is known about College Algebra?
• When given an opportunity, faculty from other
disciplines and representatives from business,
industry, and commerce have consistently called
for mathematics departments to make a major
change in the content of College Algebra.
• The curriculum committees of national
mathematics organizations have uniformly
called for replacing the current college algebra
course with one in which students address
problems presented as real world situations by
creating and interpreting mathematical models.
What is known about College Algebra?
• With support from NSF, a large
number of exemplary materials have
been developed and put in place,
although on a very small scale. The
materials address the areas stressed
by faculty from other disciplines and
representatives from industry and the
student success rate has increased.
Primary Recommendations to NSF
Based upon what is known concerning
college algebra, the working group
proposes an eight-year program of four
million dollars a year that would produce
a dramatic change in college algebra
nationwide.
Large Scale Program to Enable Institutions
to Refocus College Algebra
It is recommended that the NSF offer
extended change programs to large numbers
of institutions.
Each participating institution would engage in
a four year implementation project that would
include participation in an initial workshop
followed by on-going mentoring, site visits,
faculty development activities, material and
curriculum development, presentations,
publications and research.
Research on Impact of Refocused College
Algebra on Student Learning
It is recommended that NSF fund two or three indepth, multi-year, longitudinal research projects to
study all aspects of the development and
implementation of refocused college algebra with an
emphasis on determining the impact of well-designed
and well-supported refocused college algebra courses
on student achievement and understanding.
Electronic Library of Exemplary
College Algebra Resources
It is recommended that NSF provide
support to projects that would provide
departments and individual instructors
with resources (electronic and video) to
enable and equip them to teach refocused college algebra.
National Resource Database on College Algebra
It is recommended that NSF fund a longterm project to prepare and maintain a
national resource database that would
include (summary) information on funded
projects, textbooks, research articles, etc.
An evaluation component of the database
related to retention and other student
successes is recommended. This could be
based on a TIMSS-like model.
Concluding Thoughts
For years, we have used the
metaphor of the mathematics
curriculum being a pipeline.
But what is a pipeline?
Concluding Thoughts
Picture the Alaska pipeline that carries oil
from Prudhoe Bay to Valdez.
Every drop of oil lost en route is a valuable
commodity that is, at best, a complete loss,
and at worst, a potential threat to the
environment.
Concluding Thoughts
Do we really want to view the roughly 1,000,000
students who take college algebra and related
courses each year and do not end up majoring
in one of the SMET fields as a complete loss?
Maybe the pipeline metaphor has outlived its
usefulness!
Concluding Thoughts
The pipeline metaphor causes us to apply a
very negative psychological image to the
overwhelming majority of our students.
In turn, it leads many of us to think of the
courses we offer to these students as secondclass courses for students who are not
important to the mathematical enterprise.
Concluding Thoughts
The pipeline analogy is wrong!
The students who “leak out” are not losses. They
are simply going into other fields that require less
math or even different math.
That is the psychological image that this pipeline
metaphor causes us to apply to the overwhelming
majority of our students.
A Better Metaphor
Picture a river, particular one in the southwest.
Very little of the water from the headwaters ever
reach its end; many of these rivers eventually
peter out and all that remains are dry stream
beds.
But the water that doesn’t make it all the way
downstream is diverted to irrigate huge areas
and has been used to bring the desert to life.
A Better Metaphor
What a wonderful metaphor for how we should
view our students.
Those who only take college algebra or statistics
or finite mathematics should not be thought of
as losses; they should be thought of as valuable
commodities who, with the right emphases in
these courses, can irrigate all these other fields
and enrich them by bringing the value of
mathematics to bear.
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