A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus...

A Fresh Start for Collegiate Mathematics:

Rethinking theCourses below Calculus

[email protected]@nyit.edu

[email protected]

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 FlowsSeveral 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.


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.


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


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 ProjectThe 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.


• 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.


• 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.


• 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 athttp://www.mathsci.appstate.edu/~wmcb/ICTCM


The Guidelines:• Course Objectives College algebra through applications/modeling

Meaningful & appropriate use of technology

• Course GoalsChallenge, develop, and strengthen students’understanding and skills mastery

CRAFTY & College AlgebraThe 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 AlgebraChallenges

• 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?

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

CRAFTY College Algebra Guidelines

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.

Fundamental Experience

• 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.

Course Goals

• 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.

Problem Solving

• 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;

Functions & Equations

• 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.

• 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.

Data Analysis

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.


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


3. Emphasis in Pedagogy• Developing students’ competence and confidence in their problem-solving 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

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.

Distribution Plan

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 Class1. 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 x-

axis. You move up 78 points on the y-axis.6. The slope in this equation means the students enrolled in

2000. Y = MX + B .7. The amount of students that enroll within a period of time.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 response11. No response12. No response 13. No response 14. The change in the x-coordinates over the change in the y-coordinates.15. This is the rise in the number of students.16. The slope is the average amount of years it takes to get 156

more students enrolled in the school.17. Its how many times a year it increases.18. The slope is the increase of students per year.

Responses in Modeling Class1. 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 SlopeBoth 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 ImplicationsIf 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.


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.


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 Problemsto Develop or Test for

Conceptual UnderstandingAnd

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 3 0

5

1 5.1 1

7

2 7.2 2

9.8

3 9.3 3

13.7

For the polynomial shown,(a) What is the minimum degree? Give two different reasons for your answer.(b) What is the sign of the leading term? Explain.(c) What are the real roots?(d) What are the linear factors? (e) How many complex roots does the polynomial have?

The following table shows world-wide wind power generating capacity, in megawatts, in various

years.

Year 1980 1985 1988 1990 1992 1995 1997 1999

Windpower 10 1020 1580 1930 2510 4820 7640 13840

0

5000

10000

15000

1980 1985 1990 1995 2000

(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 relation-ship 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.

The accompanying table and associated scatterplot give some data on the area (in square miles) of various Caribbean islands and estimates on the number species of amphibians and reptiles living on each.

Island Area N

Redonda 1 3

Saba 4 5

Montserrat 40 9

Puerto Rico 3459 40

Jamaica 4411 39

Hispaniola 29418 84

Cuba 44218 76

0

20

40

60

80

100

0 15000 30000 45000

Area (square miles)N

umbe

r of S

peci

es

(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 32F and 94F. 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.


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.


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


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.


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.”


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.


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.


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.


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.


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.


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.


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.

• 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?

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.

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.

Primary Recommendations to NSF

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.

Large Scale Program to Enable Institutions to Refocus College Algebra

It is recommended that NSF fund two or three in-depth, 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.

Research on Impact of Refocused College Algebra on Student Learning

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 re-focused college algebra.

Electronic Library of Exemplary College Algebra Resources

It is recommended that NSF fund a long-term 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.

National Resource Database on College Algebra

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 second-

class 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.

A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus...

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