1. A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus gordonsp@farmingdale.edu fgordon@nyit.edu BauldryWC@appstate.edu

2. College Algebra and Precalculus Each year, more than 1,000,000 students take college algebra, precalculus, and related courses.

3. 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!

4. 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.

5. 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.

6. 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.

7. 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.)

8. 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.

9. 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.

10. 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.

11. 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

12. 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.

13. 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.

14. 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.

15. Why Students Take These Courses Required by other departments Satisfy general education requirements For a handful, to prepare for calculus

16. 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

17. 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. 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.

23. 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.

24. 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.

25. 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.

26. 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.

27. 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.

28. 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).

29. 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

30. 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

31. 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.

32. 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

33. 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

34. 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

35. 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?

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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.

45. 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

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

47. 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.

48. 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

49. CRAFTY’s College Algebra Guidelines 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

51. 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

52. 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

53. 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

54. 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

55. 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

56. 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?

57. 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.

58. 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?

59. 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 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.

60. Responses in Traditional Class 10. No response 11. No response 12. No response 13. No response 14. The change in the x-coordinates over the change in the y- coordinates. 15. This is the rise in the number of students. 16. The slope is the average amount of years it takes to get 156 more students enrolled in the school. 17. Its how many times a year it increases. 18. The slope is the increase of students per year.

61. 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.

62. 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.

63. 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.

64. 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.

65. 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?

66. 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.

67. 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.

68. 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.

69. 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.

70. Some Illustrative Problems to Develop or Test for Conceptual Understanding And Mathematical Modeling

71. Identify each of the following functions (a) - (n) as linear, exponential, logarithmic, or power. In each case, explain your reasoning. (g) y = 1.05 x (h) y = x 1.05 (i) w = (0.7) t (j) q = v 0.7 (k) z = L (-½) (l) 3U – 5V = 14 (m) xy (n) xy 03 0 5 15.1 1 7 27.2 2 9.8 39.3 3 13.7

72. 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?

73. The following table shows world-wide wind power generating capacity, in megawatts, in various years. Year 19801985198819901992199519971999 Wind power 1010201580193025104820764013840

74. (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?

75. 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.

76. 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. IslandAreaN Redonda13 Saba45 Montserrat409 Puerto Rico345940 Jamaica441139 Hispaniola2941884 Cuba4421876

77. (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.

78. Write a possible formula for each of the following trigonometric functions:

79. 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 30 th day and the hottest day on the 214 th. (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  ?

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

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

82. 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

83. 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.

84. 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.

85. 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.

86. 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

87. 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.

88. The Research Component The following data is being collected: Grades; Retention information; Performance on common test items; Student retention and grades in subsequent courses

89. 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.

90. Still To Be Determined Performance on common exams; Performance in future courses.

91. The Challenges Ahead 2. Convincing college administrators to support (both academically and financially) efforts to refocus the courses below calculus.

92. 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.”

93. The Challenges Ahead 3. Convincing academic bodies outside of mathematics to allow alternatives to traditional college algebra courses to fulfill general education requirements.

94. 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.

95. 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.

96. 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.

97. 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.

98. 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.

99. 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.

100. 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.

101. 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.

102. 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?

103. 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.

104. 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.

105. 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

106. 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

107. 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

108. 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

109. 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

110. Concluding Thoughts For years, we have used the metaphor of the mathematics curriculum being a pipeline. But what is a pipeline?

111. 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.

112. 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!

113. 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.

114. 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.

115. 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.

116. 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.