Rethinking College-Level Mathematics: A New Approach Below Calculus

1. A Fresh Start for Collegiate Mathematics:Rethinking theCourses below Calculusgordonsp@farmingdale.edufgordon@nyit.eduBauldryWC@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 special task force including 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. CRAFTY College Algebra Guidelines These guidelines are the recommendations of the MAA/CUPM subcommittee, Curriculum Renewal Across the First Two Years, concerning the nature of the college algebra course that can serve as a terminal course as well as a pre-requisite to courses such as pre-calculus, statistics, business calculus, finite mathematics, and mathematics for elementary education majors.

37. 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.

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.

39. 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;

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;

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.

42. 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.

43. 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;

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. 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.

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 Analysisincluding 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