A Fresh Start for Collegiate Mathematics: Rethinking the Courses Below Calculus

Cover
Copyright page
Title page
Preface
Contents
Introduction
	1  The Conference: Rethinking the Preparation for Calculus  by Jack Narayan and Darren Narayan
		Rationale for the conference
		Overview of the conference
		Some conclusions
		References
	2  Twenty Questions about Precalculus  by Lynn Arthur Steen
		Introduction
		What?
		Who?
		Why?
		When?
		Where?
		How?
		Conclusion
		References
		Appendix A
Background
	3  Who are the Students Who Take Precalculus?  by Mercedes A. McGowen
		Introduction
		What is ``precalculus\"?
		What mathematics courses do undergraduate students enroll in?
		Mathematics course enrollment: 1980 {2000
		Precalculus enrollment: 1980 {2000
		Meeting the needs of our students: A two-year college profile
		Conclusions
		References
	4  Enrollment Flow to and from Courses Below Calculus by Stephen R. Dunbar
		The intent of the analysis
		The data source
		Description of the courses
		The data analysis
		Conclusions
		Appendix A  Description of the courses
		Appendix B  Diagram of course dependencies
		Appendix C  Tables
	5  What Have We Learned from Calculus Reform? The Road to Conceptual Understanding  by Deborah Hughes Hallett
		Goals
		Background to calculus reform: Rationale for change
		Changes in the teaching of calculus
		Cooperation with client disciplines
		Impact of calculus reform
		Summary
	6  Calculus and Introductory College Mathematics: Current Trends and Future Directions  by Susan L. Ganter
		Introduction
		What have we learned about the teaching and learning of introductory college mathematics?
		What questions still remain?
		What do we know about the interaction of introductory college mathematics with partner disciplines?
		How can current knowledge be used to improve introductory collegiate mathematics courses?
		Implications of the proposed changes
		Supporting QL in introductory college mathematics courses through networking
		Conclusions
		References
Theme 1. New Visions for Introductory Collegiate Mathematics
	7  Refocusing Precalculus: Challenges and Questions  by Nancy Baxter Hastings
		Introduction
		What is the problem?
		What is being done?
		Characteristics of alternative instructional materials
		Changes in pedagogy
		A dozen challenges, plus one, confronting refocusing precalculus
		References
	8  Preparing Students for Calculus in the Twenty-First Century  by Sheldon P. Gordon
		The student population
		Technology and its implications for mathematics education
		Changes in the mathematical needs of students
		The need for conceptual understanding
		The need for realistic problems
		Other topics that should be emphasized
		What can be removed
		Changes in pedagogy
		References
	9  Preparing for Calculus and Preparing for Life  by Bernard L. Madison
		Introduction
		Calculus: An important role
		Quantitative literacy: Need and responsibility
		Calculus: Domination and disappointment
		Calculus in high school
		References
	10  College Algebra: A Course in Crisis  by Don Small
		Introduction
		Traditional college algebra
		Vision for improved college algebra programs
		Improved college algebra: A base for quantitative literacy programs
		Summary
		References
	11  Changes in College Algebra  by Scott R. Herriott
		Introduction
		The DWF problem
		Who takes college algebra?
		Curricular models for college algebra
		elementary probability and statistics, and consumer mathematics.Local and national policy issues
		Conclusions
		References
	12  One Approach to Quantitative Literacy: Understanding our Quantitative World  by Janet Andersen
		Overview and philosophy of the course
		Sample section: Periodic functions
		Evaluation
		References
Theme 2.  The Transition from High School to College
	13  High School Overview and the Transition to College  by Zalman Usiskin
		Introduction
		Trend data on student performance
		Variability within the United States
		Curriculum
		Advanced placement calculus
		Calculus in high school?|Yes.
		Teachers and tests
		Calculus in college and the mathematics major
		Summary
		References
	14  Precalculus Reform: A High School Perspective  by Daniel J. Teague
		Introduction
		Precalculus reform at NCSSM
		Calculus reform at NCSSM
		Influencing the mathematics community
		My students are not me, your students are not you
		Proof and sense-making
		Conclusion
		References
		Appendix
	15  The Influence of Current Efforts to Improve School Mathematics on the Preparation for Calculus by Eric Robinson and John Maceli
		Introduction
		Should the primary driver of the K–12 curriculum, especially in the upper grades, be preparation for calculus?
		That\'s all well and good. But, don\'t we just need to examine the topic differences that there are in Standards-based school mathematics programs in order to understand the changes in students\'preparation for calculus?
		Content also is important and there is a sequential nature to the development of mathematicalcontent. How does the content in Standards-based school mathematics education lay a foundationfor calculus?
		What other things influence students\' preparation for calculus?
		References
		Appendix
Theme 3. The Needs of Other Disciplines
	16  Fundamental Mathematics: Voices of the Partner Disciplines  by William Barker and Susan L. Ganter
		The Curriculum Foundations Project
		The Curriculum Foundations Project and precalculus instruction
		Conceptual understanding and problem solving
		Modeling and Applications
		Communication
		Technology
		Instructional techniques
		Student assessment
		Whither college algebra?
		Conclusion
		References
	17  Skills versus Concepts at West Point  by Rich West
		Introduction
		Perspective
		Need
		Solution
		The gateway exam
		The standards
		Lessons learned
		Implications
		Conclusions
		References
		Appendix:  Required mathematical skills for entering cadets
	18  Integrating Data Analysis into Precalculus Courses  by Allan J. Rossman
		Introduction
		Using data to study functions
			Example 1: Scrabble Names
			Example 2: AIDS Epidemic
			Example 3: Margin-of-Error
			Example 4: Benford\'s Model for Leading Digits
		Data analysis principles for precalculus
			Example 5: Airfares
			Example 6: Televisions and Life Expectancy
		Data analysis in precalculus for general education
		Conclusion
		References
Theme 4. Student Learning and Research
	19  Assessing What Students Learn: Reform versus Traditional Precalculus and Follow-up Calculus  by Florence S. Gordon
		Introduction
		Background
		Student backgrounds prior to precalculus
		Assessing student performance in precalculus
		Assessing student attitudes in precalculus
		The follow-up study in calculus
		Assessing student performance and persistence on quizzes in calculus
		Assessing student performance on class tests in calculus
		Assessing student performance on the final exam in calculus
		Discussion of results
		References
	20  Student Voices and the Transition from Reform High School Mathematics to College Mathematics  by Rebecca Walker
		Introduction
		Design of the study
		Analysis of student beliefs and conceptions of mathematics
		Transition to college mathematics
		College mathematics placement
		Experiences in the classroom
		Experiences outside of the classroom
		College mathematics performance
		Students\' final reflections on their preparation
		Summary
		Limitations of the study and possibilities for further research
		References
Theme 5. Implementation
	21  Some Political and Practical Issues in Implementing Reform  by Robert E. Megginson
		Introduction
		Show respect for your colleagues\' teaching styles
		Keep your colleagues in the loop
		Get the backing of senior faculty
		Get the backing of client departments
		Get the backing of academic counselors
		Make sure skills are learned
		Know who takes the courses
		Prepare instructors for a changed classroom
		Finally, and perhaps most important . . .
		References
	22  Implementing Curricular Change in Precalculus: A Dean\'s Perspective  by Judy E. Ackerman
		Introduction
		Initiating curricular change
		Implementing curricular change
		Making a case to the dean
		Pilot first
		Support for faculty implementing change
		Summary
		References
	23  The Need to Rethink Placement in Mathematics  by Sheldon P. Gordon
	24  Changing Technology Implies Changing Pedagogy  by Lawrence C. Moore and David A. Smith
		Introduction
		Learning and working in an increasingly rich technological environment
		Making sense of mathematical information—using technology to check
		Student-to-student interactions
		Creation of interactive learning materials
		Intellectual demands of these new forms of learning
		References
	25  Preparing for Calculus and Beyond: Some Curriculum Design Issues  by Al Cuoco
		Introduction
		Goals for the program
		What habits? What topics? What students?
		The Evolution
		An example
		Lessons learned
		References
	26  Alternatives to the One-Size-Fits-All Precalculus/College Algebra Course  by Bonnie Gold
		Introduction
		Can one size fit all?
		An alternative: Splitting the course
		Advising issues
		Scheduling issues
		Articulation issues
		Another way to cut the pie
		Summary
		References
Theme 6. Influencing the Mathematics Community
	27  Launching a Precalculus Reform Movement: Influencing the Mathematics Community  by Bernard L. Madison
		Introduction
		The community
		Need for change
		Compatible with other priorities
		Empirical evidence
		Need a headline
		What are the issues?
		Stamp of approval
		Real support
		Top-down and bottom-up
		AP calculus and technology: the levers of calculus reform
		What about strategy?
		Conclusion
		References
	28  Mathematics Programs for the Rest-of-Us  by Naomi D. Fisher and Bonnie Saunders
		Introduction
		Note on ``rest-of-us\" terminology
		Profile of participating departments
		Workshop program overview
		Summing up
		Appendix: Participating departments
	29  Where Do We Go From Here? Creating a National Initiative to Refocus the Courses below Calculus  by Sheldon P. Gordon
		Background on subsequent activities
		The common elements
		Developing a national initiative
			A. Developing an organizational structure
			B. Collecting data
			C. Identifying and publicizing model projects and programs
			D. Influencing the mathematics community
			E. Forging connections
			F. Planned outcomes
		References
Ideas and Projects that Work: Part 1
	30  College Precalculus Can Be a Barrier to Calculus: Integration of Precalculus with Calculus Can Achieve Success  by Doris Schattschneider
		Introduction
		The problem and a response
		Some details on the course
		Student outcomes
		Teaching outcomes
		Challenges in establishing the course
		Other models for the integration of precalculus review in the calculus I course
		Some final comments
		References
		Appendix I  Comparative attrition and completion data from six institutions
		Appendix II  Topics outline: Calculus I with Review
	31  College Algebra Reform through Interdisciplinary Applications  by William P. Fox
		Introduction
		Our new courses
		Motivation through applications
		Student growth through projects
		Student responses
		Faculty growth and development
		Conclusions
		References
	32  Elementary Math Models: College Algebra Topics and a Liberal Arts Approach  by Dan Kalman
		Introduction
		Content and organization
		Students
		Classroom practice
		Student reactions and performance
		Conclusion
		References
	33  The Case for Labs in Precalculus  by Brigitte Lahme, Jerry Morris amd Elias Toubassi
		Background
		Why labs?
		Format of the labs
		Some advantages of the labs
		Some challenges associated with the labs
		Other resources for precalculus labs
		Other reform texts
		Conclusion
		served in a teaching postdoctoral position at Arizona from 1997 to 2000.References
	34  The Fifth Rule: Direct Experience of Mathematics  by Gary Simundza
		Introduction
		Experiencing mathematics
		A model for laboratory-based mathematics
		Conclusion
		References
Ideas and Projects that Work: Part 2
	35  Mathematics in Action: Empowering Students with Introductory and Intermediate College Mathematics  by Ernie Danforth, Brian Gray, Arlene Kleinstein, Rick Patrick, and Sylvia Svitak
		References
	36  Precalculus: Concepts in Context  by Marsha Davis
		Student background
		Project description/goals
		Modeling approach
		Lab activities
		Projects and explorations
		Classroom experiences
		References
	37  Rethinking College Algebra  by Benny Evans
		Course content
		Examples
			Running speed versus length
			Holling\'s functional response curve
		Conclusions
		References
	38  From The Bottom Up  by Sol Garfunkel
		Sample investigation
			Part 1: Exploration
			Part 2: Modeling
		Conclusion
		References
	39  The Functioning in the Real World Project by Florence S. Gordon and Sheldon P. Gordon
		Course contents
		Illustrative examples and problems
		Conclusion
		References
	40  The Importance of a Story Line: Functions as Models of Change  by Deborah Hughes Hallett
		References
	41  Using a Guided-Inquiry Approach to Enhance Student Learning in Precalculus  by Nancy Baxter Hastings
		The Workshop Mathematics Program
		The workshop approach
		Workshop Precalculus
		Sample activities
		Implementing the workshop approach
		References
	42  Maricopa Mathematics  by Alan Jacobs
		Features and content
		Results
		References
	43  College Algebra/Quantitative Reasoning at the University of Massachusetts, Boston  by Linda Almgren Kime
		Motivation
		Math/quantitative reasoning requirement
		QR course design
		Course topics
		Sample homework problems
		Conclusions
		References
	44  Developmental Algebra:The First Mathematics Course for Many College Students by Mercedes A. McGowen
		Examples
		Course structure
		Conclusions
		References
	45  Workshop Precalculus: Functions, Data, and Models  by Allan J. Rossman
		Course content
		Sample activity
		Sample materials and further information may be found at http://calc.dickinson.edu.References
	46  Contemporary College Algebra  by Don Small
		Features of the course
		Sample projects
		Summary
	47  Precalculus: A Study of Functions and Their Applications  by Todd Swanson
		Annotated table of contents
		Examples
		References
	48  Success and Failures of a Precalculus Reform Project  by David M. Wells and Lynn Tilson
		Where we started
		What we planned
		What we produced
		What we learned
		References
	49  The Earth Math Projects  by Nancy Zumoff and Christopher Schaufele
		References