This slideshow presentation was created by Sheldon P. Gordon Farmingdale State University of New York firstname.lastname@example.org with contributions from Nancy Baxter Hastings (Dickinson College) Florence S. Gordon (NYIT) Bernard Madison (University of Arkansas) Bill Haver (Virginia Commonwealth University) Bill Bauldry (Appalachian State University) Permission is hereby granted to anyone to use any or all of these slides in any related presentations. We gratefully acknowledge the support provided for the development of this presentation package by the National Science Foundation under grants DUE-0089400, DUE-0310123, and DUE-0442160. The views expressed are those of the author and do not necessarily reflect the views of the Foundation.
College Algebra and Precalculus Each year, more than 1,000,000 students take college algebra and precalculus courses. The focus in most of these courses is on preparing the students for calculus. We know that only a relatively small percentage of these students ever go on to start calculus.
Some Questions How many of these students actually ever do go on to start calculus? How well do the ones who do go on actually do in calculus?
Some Questions Why do the majority of these 1,000,000+ students a year take college algebra courses? Are these students well-served by the kind of courses typically given as “college algebra”? If not, what kind of mathematics do these students really need?
Enrollment Flows • Based on several studies of enrollment flows from college algebra to calculus: • Less than 5% of the students who start college algebra courses ever start Calculus I • The typical DFW rate in college algebra is typically well above 50% • Virtually none of the students who pass college algebra courses ever start Calculus III • Perhaps 30-40% of the students who pass precalculus courses ever start Calculus I
Some Interesting Studies In a study at eight public and private universities in Illinois, Herriott and Dunbar found that, typically, only about 10-15% of the students enrolled in college algebra courses had any intention of majoring in a mathematically intensive field. At a large two year college, Agras found that only 15% of the students taking college algebra planned to major in mathematically intensive fields.
Some Interesting Studies • Steve Dunbar has tracked over 150,000 students taking mathematics at the University of Nebraska – Lincoln for more than 15 years. He found that: • only about 10% of the students who pass college algebra ever go on to start Calculus I • virtually none of the students who pass college algebra ever go on to start Calculus III. • about 30% of the students who pass college algebra eventually start business calculus. • about 30-40% of the students who pass precalculus ever go on to start Calculus I.
Some Interesting Studies • William Waller at the University of Houston – Downtown tracked the students from college algebra in Fall 2000. Of the 1018 students who started college algebra: • only 39, or 3.8%, ever went on to start Calculus I at any time over the following three years. • 551, or 54.1%, passed college algebra with a C or better that semester • of the 551 students who passed college algebra, 153 had previously failed college algebra (D/F/W) and were taking it for the second, third, fourth or more time
Some Interesting Studies • The Fall, 2001 cohort in college algebra at the University of Houston – Downtown was slightly larger. Of the 1028 students who started college algebra: • only 2.8%, ever went on to start Calculus I at any time over the following three years.
The San Antonio Project The mayor’s Economic Development Council of San Antonio recently identified college algebra as one of the major impediments to the city developing the kind of technologically sophisticated workforce it needs. The mayor appointed a special task force with representatives from all 11 colleges in the city plus business, industry and government to change the focus of college algebra to make the courses more responsive to the needs of the city, the students, and local industry.
Why Students Take These Courses • Required by other departments • Satisfy general education requirements • To prepare for calculus • For the love of mathematics
What the Majority of Students Need • Conceptual understanding, not rote manipulation • Realistic applications and mathematical modeling that reflect the way mathematics is used in other disciplines and on the job in today’s technological society
Some Conclusions Few, if any, math departments can exist based solely on offerings for math and related majors. Whether we like it or not, mathematics is a service department at almost all institutions. And college algebra and related courses exist almost exclusively to serve the needs of other disciplines.
Some Conclusions If we fail to offer courses that meet the needs of the students in the other disciplines, those departments will increasingly drop the requirements for math courses. This is already starting to happen in engineering. Math departments may well end up offering little beyond developmental algebra courses that serve little purpose.
Four Special Invited Conferences • Rethinking the Preparation for Calculus, • October 2001. • Forum on Quantitative Literacy, • November 2001. • CRAFTY Curriculum Foundations Project, • December 2001. • Reforming College Algebra, • February 2002.
Common Recommendations • “College Algebra courses should stress conceptual understanding, not rote manipulation. • “College Algebra” courses should be real-world problem based: • Every topic should be introduced through a real-world problem and then the mathematics necessary to solve the problem is developed.
Common Recommendations • “College Algebra” courses should focus on mathematical modeling—that is, • – transforming a real-world problem into mathematics using linear, exponential and power functions, systems of equations, graphing, or difference equations. • – using the model to answer problems in context. • – interpreting the results and changing the model if needed.
Common Recommendations • “College Algebra” courses should emphasize communication skills: reading, writing, presenting, and listening. • These skills areneeded on the job and for effective citizenship as well as in academia. • “College Algebra” courses should make appropriate use of technology to enhance conceptual understanding, visualization, inquiry, as well as for computation.
Common Recommendations • “College Algebra” coursesshould be student-centered rather than instructor-centered pedagogy. • - They should include hands-on activities rather than be all lecture. • - They should emphasize small group projects involving inquiry and inference.
Important Volumes • CUPM Curriculum Guide:Undergraduate Programs and Courses in the Mathematical Sciences, MAA Reports. • AMATYC Crossroads Standards and the Beyond Crossroads report. • NCTM, Principles and Standards for School Mathematics. • Ganter, Susan and Bill Barker, Eds., • A Collective Vision: Voices of the Partner Disciplines, MAA Reports.
Important Volumes • Madison, Bernie and Lynn Steen, Eds., Quantitative Literacy: Why Numeracy Matters for Schools and Colleges, National Council on Education and the Disciplines, Princeton. • Baxter Hastings, Nancy, Flo Gordon, Shelly Gordon, and Jack Narayan, Eds., A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus, MAA Notes.
CUPM Curriculum Guide • All students, those for whom the (introductory mathematics) course is terminal and those for whom it serves as a springboard, need to learn to think effectively, quantitatively and logically. • Students must learn with understanding, focusing on relatively few concepts but treating them in depth. Treating ideas in depth includes presenting each concept from multiple points of view and in progressively more sophisticated contexts.
CUPM Curriculum Guide • A study of these (disciplinary) reports and the textbooks and curricula of courses in other disciplines shows that the algorithmic skills that are the focus of computational college algebra courses are much less important than understanding the underlying concepts. • Students who are preparing to study calculus need to develop conceptual understanding as well as computational skills.
AMATYC Crossroads Standards • In general, emphasis on the meaning and use of mathematical ideas must increase, and attention to rote manipulation must decrease. • Faculty should include fewer topics but cover them in greater depth, with greater understanding, and with more flexibility. Such an approach will enable students to adapt to new situations. • Areas that should receive increased attention include the conceptual understanding of mathematical ideas.
NCTM Standards • These recommendations are clearly very much in the same spirit as the recommendations in NCTM’s Principles and Standards for School Mathematics. • If implemented at the college level, they would establish a smooth transition between school and college mathematics.
CRAFTY College Algebra Guidelines These guidelines are the recommendations of the MAA/CUPM subcommittee, Curriculum Renewal Across the First Two Years, concerning the nature of the college algebra course that can serve as a terminal course as well as a pre-requisite to courses such as pre-calculus, statistics, business calculus, finite mathematics, and mathematics for elementary education majors.
Fundamental Experience College Algebra provides students with a college level academic experience that emphasizes the use of algebra and functions in problem solving and modeling, provides a foundation in quantitative literacy, supplies the algebra and other mathematics needed in partner disciplines, and helps meet quantitative needs in, and outside of, academia.
Fundamental Experience Students address problems presented as real world situations by creating and interpreting mathematical models. Solutions to the problems are formulated, validated, and analyzed using mental, paper and pencil, algebraic, and technology-based techniques as appropriate.
Course Goals • Involve students in a meaningful and positive, intellectually engaging, mathematical experience; • Provide students with opportunities to analyze, synthesize, and work collaboratively on explorations and reports; • Develop students’ logical reasoning skills needed by informed and productive citizens;
Course Goals • Strengthen students’ algebraic and quantitative abilities useful in the study of other disciplines; • Develop students’ mastery of those algebraic techniques necessary for problem-solving and mathematical modeling; • Improve students’ ability to communicate mathematical ideas clearly in oral and written form;
Course Goals • Develop students’ competence and confidence in their problem-solving ability; • Develop students’ ability to use technology for understanding and doing mathematics; • Enable and encourage students to take additional coursework in the mathematical sciences.
Problem Solving • Solving problems presented in the context of real world situations; • Developing a personal framework of problem solving techniques; • Creating, interpreting, and revising models and solutions of problems.
Functions & Equations • Understanding the concepts of function and rate of change; • Effectively using multiple perspectives (symbolic, numeric, graphic, and verbal) to explore elementary functions; • Investigating linear, exponential, power, polynomial, logarithmic, and periodic functions, as appropriate;
Recognizing and using standard transformations such as translations and dilations with graphs of elementary functions; • Using systems of equations to model real world situations; • Solving systems of equations using a variety of methods; • Mastering those algebraic techniques and manipulations necessary for problem-solving and modeling in this course.
Data Analysis • Collecting, displaying, summarizing, and interpreting data in various forms; • Applying algebraic transformations to linearize data for analysis; • Fitting an appropriate curve to a scatterplot and use the resulting function for prediction and analysis; • Determining the appropriateness of a model via scientific reasoning.
An Increased Emphasis on Pedagogy and A Broader Notion of Assessment Of Student Accomplishment
CRAFTY & College Algebra • Confluence of events: • Curriculum Foundations Report published • Large scale NSF project - Bill Haver, VCU • Availability of new modeling/application based texts • CRAFTY responded to a perceived need to address course and instructional models for College Algebra.
CRAFTY & College Algebra • Task Force charged with writing guidelines • - Initial discussions in CRAFTY meetings • - Presentations at AMATYC & Joint Math Meetings with public discussions • - Revisions incorporating public commentary • Guidelines adopted by CRAFTY (Fall, 2006) • Pending adoption by CUPM (Spring, 2007) • Copies (pdf) available at • http://www.mathsci.appstate.edu/~wmcb/ICTCM
CRAFTY & College Algebra • The Guidelines: • Course Objectives • College algebra through applications/modeling Meaningful & appropriate use of technology • Course Goals • Challenge, develop, and strengthen students’ understanding and skills mastery
CRAFTY & College Algebra • The Guidelines: • Student Competencies • - Problem solving • - Functions and Equations • - Data Analysis • Pedagogy • - Algebra in context • - Technology for exploration and analysis • Assessment • - Extended set of student assessment tools • - Continuous course assessment
CRAFTY & College Algebra • Challenges • Course development • - There are current models • Scale • - Huge numbers of students • - Extraordinary variation across institutions • Faculty development • - Who teaches College Algebra? • - How do we fund change?
Conceptual Understanding • What does conceptual understanding mean? • How do you recognize its presence or absence? • How do you encourage its development? • How do you assess whether students have developed conceptual understanding?
What Does the Slope Mean? Comparison of student response on the final exams in Traditional vs. ModelingCollege Algebra/Trig Brookville College enrolled 2546 students in 2000 and 2702 students in 2002. Assume that enrollment follows a linear growth pattern. a. Write a linear equation giving the enrollment in terms of the year t. b. If the trend continues, what will the enrollment be in the year 2016? c. What is the slope of the line you found in part (a)? d. Explain, using an English sentence, the meaning of the slope. e. If the trend continues, when will there be 3500 students?
Responses in Traditional Class • 1. The meaning of the slope is the amount that is gained in years and students in a given amount of time. • 2. The ratio of students to the number of years. • 3. Difference of the y’s over the x’s. • 4. Since it is positive it increases. • 5. On a graph, for every point you move to the right on the 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. • Every year the enrollment increases by 78 students. • The slope here is 78 which means for each unit of time, (1 year) there are 78 more students enrolled.
Responses in Traditional Class 10. No response 11. No response 12. No response 13. No response 14. The change in the x-coordinates over the change in the 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 Reform Class • 1. This means that for every year the number of students increases by 78. • 2. The slope means that for every additional year the number of students increase by 78. • 3. For every year that passes, the student number enrolled increases 78 on the previous year. • As each year goes by, the # of enrolled students goes up by 78. • This means that every year the number of enrolled students goes up by 78 students. • The slope means that the number of students enrolled in Brookville college increases by 78. • Every year after 2000, 78 more students will enroll at Brookville college. • Number of students enrolled increases by 78 each year.
Responses in Reform Class • 9. This means that for every year, the amount of enrolled students increase by 78. • 10. Student enrollment increases by an average of 78 per year. • 11. For every year that goes by, enrollment raises by 78 students. • 12. That means every year the # of students enrolled increases by 2,780 students. • 13. For every year that passes there will be 78 more students enrolled at Brookville college. • The slope means that every year, the enrollment of students increases by 78 people. • Brookville college enrolled students increasing by 0.06127. • Every two years that passes the number of students which is increasing the enrollment into Brookville College is 156.
Responses in Reform Class 17. This means that the college will enroll .0128 more students each year. 18. By every two year increase the amount of students goes up by 78 students. 19. The number of students enrolled increases by 78 every 2 years.
Understanding Slope Both groups had comparable ability to calculate the slope of a line. (In both groups, several students used x/y.) It is far more important that our students understand what the slope means in context, whether that context arises in a math course, or in courses in other disciplines, or eventually on the job. Unless explicit attention is devoted to emphasizing the conceptual understanding of what the slope means, the majority of students are not able to create viable interpretations on their own. And, without that understanding, they are likely not able to apply the mathematics to realistic situations.