Presentation on theme: "Refocusing the Courses Below Calculus A Joint Initiative of MAA, AMATYC & NCTM."— Presentation transcript:
Refocusing the Courses Below Calculus A Joint Initiative of MAA, AMATYC & NCTM
This slideshow presentation was created by Sheldon P. Gordon Farmingdale State University of New York 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 , DUE , and DUE 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 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 Forum on Quantitative Literacy, November CRAFTY Curriculum Foundations Project, December 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 are needed 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” courses should 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.
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 use 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 & 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
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. Modeling College Algebra/Trig Brookville College enrolled 2546 students in 2000 and 2702 students in 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 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 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 The slope means that for every additional year the number of students increase by 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 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 Every year after 2000, 78 more students will enroll at Brookville college. 8. 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 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 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.
Further Implications If students can’t make their own connections with a concept as simple as the slope of a line, they won’t be able to create meaningful interpretations and connections on their own for more sophisticated mathematical 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 of a function? What is the significance of a definite integral?
Further Implications If we focus only on manipulative skills without developing conceptual understanding, we produce nothing more than students who are only Imperfect Organic Clones of a TI-89
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.
Should x Mark the Spot? All other disciplines focus globally on the entire universe of a through z, with the occasional contribution of through . Only mathematics focuses on a single spot, called x. Newton’s Second Law of Motion: y = mx, Einstein’s formula relating energy and mass: y = c 2 x, The ideal gas law: yz = nRx. Students who see only x’s and y’s do not make the connections and cannot apply the techniques when other letters arise in other disciplines.
Should x Mark the Spot? Kepler’s third law expresses the relationship between the average distance of a planet from the sun and the length of its year. If it is written as y 2 = x 3, there is no suggestion of which variable represents which quantity. If it is written as t 2 = D 3, a huge conceptual hurdle for the students is eliminated.
Should x Mark the Spot? When students see 50 exercises where the first 40 involve solving for x, and a handful at the end that involve other letters, the overriding impression they gain is that x is the only legitimate variable and the few remaining cases are just there to torment them.
Some Illustrative Examples of Problems to Develop or Test for Conceptual Understanding
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) y = (0.7) t (j) y = v 0.7 (k) z = L (-½) (l) 3U – 5V = 14 (m) xy (n) xy
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?
Two functions f and g are defined in the following table. Use the given values in the table to complete the table. If any entries are not defined, write “undefined”. xf(x)f(x)g(x)g(x)f(x) - g(x)f(x)/g(x)f(g(x))g(f(x))
Two functions f and g are given in the accompanying figure. The following five graphs (a)-(e) are the graphs of f + g, g - f, f*g, f/g, and g/f. Decide which is which.
The following table shows world-wide wind power generating capacity, in megawatts, in various years. Year Wind power
(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 relationship 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 does 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 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. IslandAreaN Redonda13 Saba45 Montserrat409 Puerto Rico Jamaica Hispaniola Cuba
(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 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) Estimate the high temperature on March 15. (f) What are all the dates on which the high temperature is most likely 80 ?
Some Conclusions We cannot simply concentrate on teaching the mathematical techniques that the students need. It is as least as important to stress conceptual understanding and the meaning of the mathematics. We can accomplish this by using a combination of realistic and conceptual examples, homework problems, and test problems that force students to think and explain, not just manipulate symbols. If we fail to do this, we are not adequately preparing our students for successive mathematics courses, for courses in other disciplines, and for using mathematics on the job and throughout their lives.
Functions It is only in math classes that functions are given. Everywhere else, The existence of functions is observed Formulas for functions are created Functions are used to answer questions about a context
The Need for Real-World Problems and Examples
Realistic Applications and Mathematical Modeling Real-world data enables the integration of data analysis concepts with the development of mathematical concepts and methods Realistic applications illustrate that data arise in a variety of contexts Realistic applications and genuine data can increase students’ interest in and motivation for studying mathematics Realistic applications link the mathematics to what students see in and need to know for other courses in other disciplines.
The Role of Technology
Technology allows us to do many standard topics differently and more easily. Technology allows us to introduce new topics and methods that we could not do previously. Technology allows us to de-emphasize or even remove some topics that are now less important.
Technology: How? Students can use technology as a problem-solving tool to –Model situations and analyze functions –Tackle complex problems Students can use technology as a learning tool to –Explore new concepts and discover new ideas –Make connections –Develop a firm understanding of mathematical ideas –Develop mental images associated with abstract concepts
Technology - Caution Students need to balance the use of technology and the use of pencil and paper. Students need to learn to use technology appropriately and wisely.
Changing the Learning and Teaching Environment
Traditional Approach vs. Student-Centered Approach With a traditional approach, students Listen to lectures Copy notes from the board Mimic examples Use technology to do calculations Do familiar problems in homework and on exams Fly through the material Hold instructor responsible for learning Go to instructor for help
Traditional Approach vs. Student-Centered Approach With a student-centered approach, students Participate in discussions Work collaboratively Find solutions and approaches Use technology to investigate ideas Write about and use new ideas in homework and on exams Take time to think Accept responsibility for learning First try to help each other
Student-Centered Learning: The Role of the Instructor The instructor –Designs activities –Emphasizes learning –Interacts with students –Approaches ideas from the student’s point of view –Controls the learning environment The instructor is a – Facilitator – Coach – Intellectual manager
Student-Centered Learning: Intended Outcomes Impel students to be active learners Make learning mathematics an enjoyable experience Help students develop confidence to read, write and do mathematics Enhance students’ understanding of fundamental mathematics concepts Increase students’ ability to use these concepts in other disciplines Inspire students to continue the study of mathematics
But, if college algebra and related courses change, what happens to the next generation of math and science majors? Don’t they need all the traditional algebraic skills? But, if they don’t develop conceptual understanding and the ability to apply the mathematics, what value are the skills?
The Link 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, however, calculus enrollment has been steady, at best.
Calculus and Related Enrollments In comparison, 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, about 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 And, as ever more students take more mathematics, especially calculus, in high school, we should expect: Fewer students taking these courses in college The overall quality of the students who take 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.)
Another Conclusion It is not conscionable for departments to treat students as mathematical cannon- fodder, by pushing them into courses they have little hope of surviving in order to increase the number of sections of calculus that are offered.
Associates Degrees in Mathematics In 2002, P There were 595,000 associate degrees P Of these, 685 were in mathematics This is one-tenth of one percent!
Bachelor’s Degrees in Mathematics In 2002, P There were 1,292,000 bachelor’s degrees P Of these, 12,395 were in mathematics This is under one percent!
Master’s Degrees in Mathematics In 2002, P There were 482,000 master’s degrees P Of these, 3487 were in mathematics This is 7 tenths of one percent!
PhD’s Degrees in Mathematics In 2002, There were 44,000 doctoral degrees Of these, 958 were in mathematics This is just over two percent! But less than half were U.S. citizens
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.
The Focus in these Courses But most college algebra courses and certainly all precalculus courses were designed to prepare students for calculus and most of them are still offered in that spirit. Even though only a small percentage of the students have any intention of going into calculus!
A Fresh Start for Collegiate Mathematics Rethinking the Courses Below Calculus MAA Notes, 2005 Nancy Baxter Hastings, et al (editors)
A Fresh Start to Collegiate Math Introduction Nancy Baxter HastingsOverview of the Volume Jack Narayan & Darren NarayanThe Conference: Rethinking the Preparation for Calculus Lynn SteenTwenty Questions Background Mercedes McGowenWho are the Students Who Take Precalculus? Steve DunbarEnrollment Flow to and from Courses below Calculus Deborah Hughes Hallett What Have We Learned from Calculus Reform? The Road to Conceptual Understanding Susan Ganter Calculus and Introductory College Mathematics: Current Trends and Future Directions
A Fresh Start to Collegiate Math Refocusing Precalculus, College Algebra, and Quantitative Literacy Shelly GordonPreparing Students for Calculus in the Twenty-First Century Bernie MadisonPreparing for Calculus and Preparing for Life Don SmallCollege Algebra: A Course in Crisis Scott HerriottChanges in College Algebra Janet Andersen One Approach to Quantitative Literacy: Mathematics in Public Discourse The Transition from High School to College Zal UsiskinHigh School Overview and the Transition to College Dan TeaguePrecalculus Reform: A High School Perspective Eric Robinson & John Maceli The Influence of Current Efforts to Improve School Mathematics on Preparation for Calculus
A Fresh Start to Collegiate Math The Needs of Other Disciplines Susan Ganter and Bill Barker Fundamental Mathematics: Voices of the Partner Disciplines Rich WestSkills versus Concepts Allan RossmanIntegrating Data Analysis into Precalculus Courses Student Learning and Research Florence Gordon Assessing What Students Learn: Reform versus Traditional Precalculus and Follow-up Calculus Rebecca Walker Student Voices and the Transition from Standards-Based Curriculum to College
A Fresh Start to Collegiate Math Implementation Robert MegginsonSome Political and Practical Issues in Implementing Reform Judy Ackerman Implementing Curricular Change in Precalculus: A Dean's Perspective Bonnie Gold Alternatives to the One-Size-Fits-All Precalculus/College Algebra Course Al Cuoco Preparing for Calculus and Beyond: Some Curriculum Design Issues Lang Moore and David SmithChanging Technology Implies Changing Pedagogy Shelly GordonThe Need to Rethink Placement in Mathematics Influencing the Mathematics Community Bernie Madison Launching a Precalculus Reform Movement: Influencing the Mathematics Community Naomi Fisher & Bonnie SaundersMathematics Programs for the "Rest of Us" Shelly GordonWhere Do We Go from Here: Forging a National Initiative
A Fresh Start to Collegiate Math Ideas and Projects that Work (long papers) Doris Schattschneider An Alternate Approach: Integrating Precalculus into Calculus Bill Fox College Algebra Reform through Interdisciplinary Applications Dan Kalman Elementary Math Models: College Algebra Topics and a Liberal Arts Approach Brigette Lahme, Jerry Morris and Elias ToubassiThe Case for Labs in Precalculus Ideas and Projects that Work (short papers) Gary SimundzaThe Fifth Rule: Experiential Mathematics Darrell Abney and James Hougland Reform Intermediate Algebra in Kentucky Community Colleges Marsha DavisPrecalculus: Concepts in Context
A Fresh Start to Collegiate Math Benny EvansRethinking College Algebra Sol GarfunkelFrom the Bottom Up Florence Gordon & Shelly GordonFunctioning in the Real World Deborah Hughes HallettImportance of a Story Line Functions as a Model Nancy Baxter Hastings Using a Guided-Inquiry Approach to Enhance Student Learning in Precalculus Allan JacobsMaricopa Mathematics Linda KimeQuantitative Reasoning Mercedes McGowan Developmental Algebra: The First Course for Many College Students Allan RossmanWorkshop Precalculus: Functions, Data and Models Chris Schaufele & Nancy ZumoffThe Earth Math Projects Don SmallContemporary College Algebra
A Fresh Start to Collegiate Math Ernie Danforth, Brian Gray, Arlene Kleinstein, Rick Patrick and Sylvia Svitak Mathematics in Action: Empowering Students with Introductory and Intermediate College Mathematics Todd SwansonPrecalculus: A Study of Functions and Their Applications David Wells Lynn TilsonSuccesses and Failures of a Precalculus Reform Project
The Need to Rethink Placement in Mathematics
Rethinking Placement Tests Two Types of Placement Tests: 1.National (standardized) tests Not much we can do about them. 2. Home-grown tests
Rethinking Placement Tests Four scenarios: 1.Students come from traditional curriculum into traditional curriculum. 2.Students from Standards-based curriculum into traditional curriculum. 3.Students from traditional curriculum into reform curriculum. 4. Students from Standards-based curriculum into reform curriculum.
One National Placement Test 1. Square a binomial. 2. Determine a quadratic function arising from a verbal description (e.g., area of a rectangle whose sides are both linear expressions in x). 3. Simplify a rational expression. 4. Confirm solutions to a quadratic function in factored form. 5. Completely factor a polynomial. 6. Solve a literal equation for a given unknown.
A National Placement Test 7. Solve a verbal problem involving percent. 8. Simplify and combine like radicals. 9. Simplify a complex fraction. 10. Confirm the solution to two simultaneous linear equations. 11. Traditional verbal problem (e.g., age problem). 12. Graphs of linear inequalities.
A Tale of Three Colleges in NYS 1.Totally traditional curriculum – developmental through calculus. 2.Traditional courses – developmental through college algebra, then reform in precalculus on up. 3.Totally reform – developmental through upper division offerings. All use the same national placement test.
A Tale of Three Colleges in NYS BUT New York State has not offered the traditional Algebra I – Geometry – Algebra II – Trigonometry curriculum in over 20 years! Instead, there is an integrated curriculum that emphasize topics such as statistics and data analysis, probability, logic, etc. in addition to algebra and trigonometry.
A Tale of Three Colleges in NYS So students are being placed one, two, and even three semesters below where they should be based on the amount of mathematics they have studied! And they are being punished: because of what is being assessed and what is not being assessed, because of what was stressed in high school and what was not stressed, because of what was taught, not what they learned or didn’t learn.
A Modern High School Problem Given the complete 32-year set of monthly CO 2 emission levels (a portion is shown below), create a mathematical model to fit the data. YearJanFebMarAprMayJunJulAugSepOctNovDecAvg
A Modern High School Problem 1. Students first do a vertical shift of about 300 ppm and then fit an exponential function to the transformed data to get: 2. They then create a sinusoidal model to fit the monthly oscillatory behavior about the exponential curve 3. They then combine the two components to get 4. They finally give interpretations of the various parameters and what each says about the increase in concentration and use the model to predict future or past concentration levels.
Placement, Revisited Picture an entering freshman who has taken high school courses with a focus on problems like the preceding one and who has developed an appreciation for the power of mathematics based on understanding the concepts and applying them to realistic situations. What happens when that student sits down to take a traditional placement test? Is it surprising that many such students end up being placed into developmental courses?
What a High School Teacher Said “If you try to teach my students with the mistaken belief that they know the mathematics I knew at their age, you will miss a great opportunity. My students know more mathematics than I did, but it is not the same mathematics; and I believe they know it differently. They have a different vision of mathematics that would be helpful in learning calculus if it were tapped.” Dan Teague
Rethinking Placement Tests What Can Be Done: 1. Home-grown tests: Develop alternate versions that reflect both your curriculum AND the different curricula that your students have come through. 2. National (standardized) tests Contact the test-makers (Accuplacer – ETS and Compass – ACT) and lobby them to develop alternative tests to reflect both your curriculum and the different curricula that your students have come through.
Why Students Take These Courses The vast majority of students take college algebra and related courses because: they are required by other departments or they are needed to satisfy general education requirements As a consequence, we have to pay attention to what the other disciplines want their students to gain from these courses.
Connecting with Other Disciplines All other disciplines are under pressure to teach more material to their students, and that material is much more than just the mathematical ideas and applications. If we do not provide courses that satisfy today’s needs of the other disciplines, they are likely going to drop the requirements for our courses and include the needed material in their own offerings.
Voices of the Partner Disciplines CRAFTY’s Curriculum Foundations Project
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 of the Partner Disciplines, edited by Susan Ganter and Bill Barker.
What the Physicists Said Conceptual understanding of basic mathematical principles is very important for success in introductory physics. It is more important than esoteric computational skill. However, basic computational skill is crucial. Development of problem solving skills is a critical aspect of a mathematics education.
What the Physicists Said Courses should cover fewer topics and place increased emphasis on increasing the confidence and competence that students have with the most fundamental topics.
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.
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.
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.
What Business Faculty Said Mathematics is an integral component of the business school curriculum. Mathematics Departments can help by stressing conceptual understanding of quantitative reasoning and enhancing critical thinking skills. Business students must be able not only to apply appropriate abstract models to specific problems but also to become familiar and comfortable with the language of and the application of mathematical reasoning. Business students need to understand that many quantitative problems are more likely to deal with ambiguities than with certainty. In the spirit that less is more, coverage is less critical than comprehension and application.
What Business Faculty Said Courses should stress problem solving, with the incumbent recognition of ambiguities. Courses should stress conceptual understanding (motivating the math with the “why’s” – not just the “how’s”). Courses should stress critical thinking. An important student outcome is their ability to develop appropriate models to solve defined problems.
What Business Faculty Said Courses should use industry standard technology (spreadsheets). An important student outcome is their ability to become conversant with mathematics as a language. Business faculty would like its students to be comfortable taking a problem and casting it in mathematical terms.
What the Engineers Said One basic function of undergraduate electrical engineering education is to provide students with the conceptual skills to formulate, develop, solve, evaluate and validate physical systems. Mathematics is indispensable in this regard.
What the Engineers Said The mathematics required to enable students to achieve these skills should emphasize concepts and problem solving skills more than emphasizing the repetitive mechanics of solving routine problems.
What the Engineers Said Students must learn the basic mechanics of mathematics, but care must be taken that these mechanics do not become the focus of any mathematics course.
What the Chemists Said Introduce multivariable, multidimensional problems from the outset Listen to the equations – most specific mathematical expressions can be recovered from a few fundamental relationships in a few steps. Of widespread use in chemistry teaching and research are spreadsheets to produce graphs and perform statistical calculations
Health-Related Life Sciences Put special emphasis on the use of models as a way to organize information for the purpose of gaining insight and to provide intuition into systems that are too complex to understand any other way. Students should master appropriate computer packages, such as a spreadsheet, symbolic/numerical computational packages (Mathematica, Maple, Matlab), statistical packages.
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
Some Implications Although the number of college students taking calculus is at best holding steady, the percentage of students taking college calculus is dropping, since overall college enrollment has been rising rapidly. But the number of students taking calculus in high school already exceeds the number taking it in college. It is growing at 8%.
Some Implications 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 Implications 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.
What Can Be Removed? How many of you remember that there used to be something called the Law of Tangents? What happened to this universal law? Did triangles stop obeying it? Does anyone miss it?
What Can Be Removed? Descartes’ rule of signs The rational root theorem Synthetic division The Cotangent, Secant, and Cosecant were needed for computational purposes; Just learn and teach a new identity:
How Important Are Rational Functions? In DE: To find closed-form solutions for several differential equations, (usually done with CAS today, if at all) In Calculus II: Integration using partial fractions–often all four exhaustive (and exhausting) cases In Calculus I: Differentiating rational functions In Precalculus: Emphasis on the behavior of all kinds of rational functions and even partial fraction decompositions In College Algebra: Addition, subtraction, multiplication, division and especially reduction of complex fractional expressions In each course, it is the topic that separates the men from the boys! But, can you name any realistic applications that involve rational functions? Why do we need them in excess?
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, 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
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
How Does the Quantitative Literacy Initiative Relate to College Algebra?
What is Quantitative Literacy? Quantitative literacy (QL), or numeracy, is the knowledge and habits of mind needed to understand and use quantitative measures and inferences necessary to function as a responsible citizen, productive worker, and discerning consumer. QL describes the quantitative reasoning capabilities required of citizens in today's information age -- from the QL Forum White Paper
QL and the Mathematics Curriculum In high school, the route to competitive colleges. The sequence is linear and hurried. No time to teach mathematics in contexts. Courses are routes to somewhere else. Other sequences are terminal and often second rate. The focus of the math curriculum is the geometry-algebra-trigonometry-calculus sequence.
Elements of QL Confidence with mathematics Cultural appreciation Interpreting data Logical thinking Making decisions Mathematics in context Number sense Practical skills Prerequisite knowledge Symbol sense
Two Kinds of Literacy Inert - Level of verbal and numerate skills required to comprehend instructions, perform routine procedures, and complete tasks in a routine manner. Liberating - Command of both the enabling skills needed to search out information and power of mind necessary to critique it, reflect upon it, and apply it in making decisions. Lawrence A. Cremins, American Education: The Metropolitan Experience New York: Harper & Row, (as quoted by R. Orrill in M&D)
How does the US compare to other countries?
NALS Quantitative Paradigm National Adult Literacy Survey Skill Level 1 - Minimal Approximate Educational Equivalence - Dropout NALS Competencies - Can perform a single, simple arithmetic operation such as addition. The numbers used are provided and the operation to be performed is specified. NALS Examples - Total a bank deposit entry
NALS Quantitative Paradigm Skill Level 2 - Basic Approximate Educational Equivalence - Average or below average HS graduate NALS Competencies - Can perform a single arithmetic operation using numbers that are given in the task or easily located in the material. The arithmetic operation is either described or easily determined from the format of the materials. NALS Examples - Calculate postage and fees for certified mail - Determine the difference in price between tickets for two shows - Calculate the total costs of purchase from an order form
NALS Quantitative Paradigm Skill Level 3 - Competent Approximate Educational Equivalence -Some postsecondary education NALS Competencies - Can perform tasks where two or more numbers are needed to solve the problem and they must be found in the material. The operation(s) needed can be determined from the arithmetic relation terms used in the question or directive. NALS Examples - Use a calculator to calculate the difference between the regular and sale price - Calculate miles per gallon from information on a mileage record chart - Use a calculator to determine the discount from an oil bill if paid within 10 days
NALS Quantitative Paradigm Skill Level 4 - Advanced Approximate Educational Equivalence -Bachelor’s or advanced degree NALS Competencies - Can perform two or more operations in sequence or a single operation in which the quantities are found in different types of displays, or where the operations must be inferred from the information given or from prior knowledge. NALS Examples - Determine the correct change using information in a menu -Calculate how much a couple would receive from Supplemental Security Income, using an eligibility pamphlet - Use information stated in a news article to calculate the amount of money that should go to raising a child
NALS Quantitative Paradigm Skill Level 5 - Superior Approximate Educational Equivalence -High achieving college- educated populations NALS Competencies - Can perform multiple operations sequentially, and can also find the features of problems embedded in text or rely on background knowledge to determine the quantities or operations needed. NALS Examples - Use a calculator to determine the total cost of carpet to cover a room - Use information in a news article to calculate the difference in time for completing a race - Determine shipping and total costs on an order form for items in a catalog
Source: USDOE, NCES, National Adult Literacy Survey, 1992, in Literacy in the Labor Force: Results from the NALS, September 1999, p. 61. Many College Graduates Demonstrate Weak Quantitative Literacy Skills
Quantitative Literacy and Job Opportunity, Quantitative Skill Level Mathematical Literacy Level 15% of the Labor Force 9% of New Jobs, Earnings: $20,300 12% of All Jobs in 2008 (Below Average H. S. Graduate) 24% of the Labor Force 21% of New Jobs, Earnings: $25,500 24% of All Jobs in % of the Labor Force 36% of New Jobs, Earnings: $31,600 37% of All Jobs in 2008 Minimal (Dropout) Basic Competent (Some Postsecondary) Advanced/Superior (Bachelor’s Degree) 26% of the Labor Force 33% of New Jobs, Earnings: $45,400 27% of All Jobs in 2008 Source: National Adult Literacy Survey; Current Population Survey; Bureau of Labor Statistics Employment Projections,
Where Does College Algebra Fit In? QL is something that should permeate the entire mathematics curriculum, so that every student develops these skills. The one existing course that provides the best opportunity to stress QL is college algebra: It has the largest enrollment It does not prepare or motivate large numbers of students to go on to calculus It is taken to prepare students for courses in other disciplines, and the themes of QL are the mathematical topics needed in most other disciplines today.
Some Sample Programs
A Sample Program
A Sample Program
A Sample Program
The Challenges to Be Faced
The Challenges Ahead Convincing the math community 1. Conducting a series of extensive tracking studies to determine how many (or how few) students who take these courses actually go on to calculus. 2. Identifying and highlighting “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.
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.
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
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.
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.
The Challenges Ahead 2. Convincing college administrators to support (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.”
What Can Administrators Do? At Stony Brook University, all faculty applying for promotion or tenure must supply evidence documenting significant achievement in instructional innovation in addition to research and service achievements. The message is a strong signal that a major research university places a strong value on students’ education and the undergraduate curriculum.
What Can Administrators Do? At Farmingdale State University of New York, all faculty who were involved in a major NSF project to promote interaction between mathematics and other quantitative fields were rewarded with merit pay increases. These activities were also counted highly in promotion and tenure decisions.
The Challenges Ahead 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.
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.
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.
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.
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.
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.
The Challenges Ahead 9. Developing a regional network through the MAA sections and the AMATYC affiliates to influence the local mathematics communities and to provide support at the local and regional levels to faculty and departments who seek to change these courses.
The Challenges Ahead 10. 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?
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 institutions. 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.