Transcript Some Data on Mathematics Education

A Fresh Start for
Collegiate Mathematics:
Rethinking the
Courses below Calculus
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College Algebra and Precalculus
Each year, more than 1,000,000 students
take college algebra, precalculus, and
related courses.
The Focus in these Courses
Most college algebra courses and certainly all
precalculus courses were originally intended
and designed to prepare students for calculus.
Most of them are still offered in that spirit.
But only a small percentage of the students have
any intention of going on 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, calculus enrollment in
college has been steady, at best.
Calculus and Related Enrollments
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, 240,000 took AP Calculus exams
Reportedly, about twice as many students take
calculus in high school, but do not take an AP
exam.
Some Implications
Today more students take calculus in high
school than in college.
We should expect:
• Fewer college students taking these courses
•The overall quality of the students taking
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.)
Enrollment Flows
Several studies of enrollment flows into calculus:
• Less than 5% of the students who start college
algebra courses ever start Calculus I
• About 10-12% of those who pass college algebra
ever start Calculus I
• 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
• Only about 10-15% of students in college algebra
are in majors that require calculus.
Some Interesting Studies
In a study at eight public and private universities
in Illinois, Herriott and Dunbar found that,
typically, only about 10-15% of the students
enrolled in college algebra courses had any
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.
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.
Why Students Take These Courses
• Required by other departments
• Satisfy general education requirements
• For a handful, to prepare for calculus
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.
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.
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
from the Partner Disciplines, edited by Susan
Ganter and Bill Barker.
What the Physicists Said
• Students need conceptual understanding first,
and some comfort in using basic skills; then a
deeper approach and more sophisticated skills
become meaningful. Computational skill
without theoretical understanding is shallow.
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
• 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 Business Faculty Said
• Courses should stress conceptual
understanding (motivating the math with the
“why’s” – not just the “how’s”).
• Students should be comfortable taking a
problem and casting it in mathematical terms.
• Courses should use industry standard
technology (spreadsheets).
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
CRAFTY & College Algebra
Confluence of events:
• Curriculum Foundations Report published
• Large scale NSF project - Wm Haver, VCU
• Release 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)
• Formally endorsed 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?
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 using 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’s College Algebra Guidelines
1. Course Goals
• Develop logical reasoning skills needed by informed
and productive citizens;
• Strengthen algebraic and quantitative abilities useful
in the study of other disciplines;
• Develop mastery of those algebraic techniques
necessary for problem-solving and math’l modeling;
• Improve ability to communicate mathematical ideas
clearly in oral and written form;
• Provide opportunities to analyze, synthesize, and work
collaboratively on explorations and reports;
•Develop students’ ability to use technology for
understanding and doing mathematics.
CRAFTY’s College Algebra Guidelines
2. Student Competencies
• Problem Solving in the context of solving real world
situations with emphasis on model creation, interpretation
and revision of the model, if necessary;
• Understanding the concepts of function and rate of
change by effectively using multiple perspectives
(symbolic, numeric, graphic, and verbal) to investigate and
apply elementary functions, particularly linear, exponential,
power, polynomial, logarithmic, and periodic functions
•Data Analysis including collecting, displaying,
summarizing, and interpreting data; transforming data to
linearize it; fitting an appropriate function to the data; using
the function for prediction and analysis
CRAFTY’s College Algebra Guidelines
3. Emphasis in Pedagogy
•
Developing students’ competence and confidence in their problemsolving abilities;
• Utilizing algebraic techniques needed in the context of problem
solving
• Emphasizing the development of conceptual understanding.
• Improving students’ written and oral communication skills in math;
• Providing a classroom atmosphere that is conducive to exploratory
learning, risk-taking, and perseverance;
• Providing student-centered, activity-based instruction, including
small group activities and projects;
• Using technology (computer, calculator, spreadsheet, computer
algebra system) appropriately as a tool in problem-solving and
exploration
A Fresh Start to Collegiate Math
• Background
• New Visions for Introductory Collegiate Mathematics
• The Transition from High School
• The Needs of Other Disciplines
• Student Learning and Research
• Implementation
• Influencing the Mathematics Community
• Ideas and Projects that Work
Distribution Plan
With support from the NSF, the MAA has
developed a distribution plan to provide
one free copy to any department that
requests one.
Announcements were sent to all
department chairs informing them of the
details.
Common Themes
• Conceptual Understanding is more important
than rote manipulation
• The Rule of Four: Graphical, Numerical,
Algebraic and Verbal Representations
• Realistic Applications via Math Modeling
• Non-routine problems and assignments
• Algebra in Context – Not Just Drill
Common Themes
• Families of Functions – Linear, Exponential,
Power, Logarithmic, Polynomial, and
Sinusoidal
• The significance of the parameters in the
different families of functions
• Limitations of the models developed –
the practical significance of the domain
and range
Common Themes
• Data Analysis
• Connections to Other Disciplines
• Writing and Communication
• More Active Classroom Environment –
Group Work, Collaborative Learning,
Exploratory Approach to Mathematics
• Use of Technology in Teaching and Learning
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?
Results of One Study
My department offered 2 modeling-based and 2
traditional, algebraic-oriented precalculus sections.
The study involved 10 common questions (mostly
computational in nature) on the final exams.
The students in the modeling-based sections
outscored those in the algebraic-oriented sections
on 7 of the 10 questions.
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 Modeling 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 Modeling 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 Modeling 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 slope, they won’t be able to create
meaningful interpretations on their own for more
sophisticated 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?
• What is the significance of a definite integral?
Follow-Up Results in Calculus
The students involved in the precalculus study
were then followed in Calculus I the next term.
The calculus course was a reform course with
emphasis also on conceptual understanding, not
just manipulation.
Follow-Up Results in Calculus
On every weekly quiz, on every class test, and on
the final exam, the students from the modeling
sections of precalculus consistently scored higher
than the students from the traditional sections.
On an attitudinal survey, the students from the
modeling section had significantly better attitudes
toward mathematics, its usefulness, and the
importance of technology for problem solving.
Follow-Up Results in Calculus
77% of the students who had been in a
modeling section of precalculus ended up
receiving a passing grade in Calculus I.
41% of those who had been in a traditional
section of precalculus received a passing
grade in Calculus I.
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 and
will not be able to transfer the mathematical ideas to
courses in other disciplines.
Some Illustrative Problems
to Develop or Test for
Conceptual Understanding
And
Mathematical Modeling
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) w = (0.7)t
(j) q = v 0.7
(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?
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 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) Predict the high temperature on March 15.
(f) What are all the dates on which the high temperature is
most likely 80?
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
The Challenges Ahead
1. Convincing the math community
The MAA’s Committee on Service Courses has
agreed to assist CRAFTY by conducting a
project to identify and highlight “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 (both 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.”
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. 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 schools.
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.