Voices of the Partner Disciplines: CRAFTY’s Curriculum Foundations Project

1. Voices of the Partner Disciplines:CRAFTY’s Curriculum Foundations Project

2. College Algebra and Precalculus In 2000, between 1,000,000 and 2,000,000 students took college algebra and precalculus courses The focus in most of these courses is on preparing the students for calculus. But only a small percentage ever go on to start calculus.

3. Enrollment Flows • Based on several studies of enrollment flows into calculus: • Only about 10% of the students who pass college algebra courses ever start Calculus I • Virtually none of the students who pass college algebra courses ever start Calculus III • Perhaps 30-40% of the students who pass precalculus courses ever start Calculus I

4. Why Students Take These Courses • Required by other departments • Satisfy general education requirements • To prepare for calculus • For the love of mathematics

5. 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 • Fitting functions to data • Recursion and difference equations – the mathematical language of spreadsheets

6. 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.

7. Common Recommendations • “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.

8. Common Recommendations A primary emphasis in “College Algebra” should be Mathematical Modeling: – 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.

9. 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 emphasize small group projects involving inquiry and inference.

10. Common Recommendations “College Algebra” courses should make appropriate use of technology to enhance conceptual understanding, visualization, and inquiry, as well as for computation “College Algebra” coursesshould be student centered rather than instructor centered pedagogy: they should include hands-on activities rather than all lecture

11. Important Volumes • CUPM Curriculum Guide: Undergraduate Programs and Courses in the Mathematical Sciences, MAA Reports. • Ganter, Susan and Bill Barker, Eds., A Collective Vision: Voices of the Partner Disciplines, MAA Reports. • 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, Shelly Gordon, Flo Gordon and Jack Narayan, Eds., A Fresh Start for Collegiate Mathematics: Rethinking the Courses Below Calculus, MAA Notes.

12. 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.

13. 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.

14. 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.

15. 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?

16. 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?

17. What Does the Slope Mean? Comparison of student response to a problem on the final exams in Traditional vs. ReformCollege Algebra/Trig Brookville College enrolled 2546 students in 1996 and 2702 students in 1998. 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?

18. 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 1996. 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.

19. 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.

20. 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 1996, 78 more students will enroll at Brookville college. • Number of students enrolled increases by 78 each year.

21. 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.

22. 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.

23. 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.

24. 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?

25. 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

26. 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.

27. 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 = c2x, 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.

28. 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 y2 = 0.1664x3, there is no suggestion of which variable represents which quantity. If it is written as t2 = 0.1664D3 , a huge conceptual hurdle for the students is eliminated.

29. 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.

30. Some Illustrative Examples of Problems to Develop or Test for Conceptual Understanding

31. Identify each of the following functions (a) - (n) as linear, exponential, logarithmic, or power. In each case, explain your reasoning.(g) y = 1.05x (h) y = x1.05 (i) y = (0.7)x (j) y = x0.7(k) y = x(-½) (l) 3x - 5y = 14

32. 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?

33. 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”.

34. 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.

35. The following table shows world-wide wind power generating capacity, in megawatts, in various years.

36. (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 totalwind power generating capacity in 2010?

37. Biologists have long observed that the larger the area of a region, the more species live there. The relationship is best modeled by a power function. Puerto Rico has 40 species of amphibians and reptiles on 3459 square miles and Hispaniola (Haiti and the Dominican Republic) has 84 species on 29,418 square miles. (a) Determine a power function that relates the number of species of reptiles and amphibians on a Caribbean island to its area. (b) Use the relationship to predict the number of species of reptiles and amphibians on Cuba, which measures 44218 square miles.

38. 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.

39. (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.

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

41. 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 30th day and the hottest day on the 214th. (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) What is the most likely high temperature on March 15? (f) What are all the dates on which the high temperature is most likely 80?

42. 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.

43. The Need for Real-World Problems and Examples

44. Realistic Applications and Mathematical Modeling • Genuine data enables the development of data analysis concepts to be integrated with the development of mathematical concepts • Realistic applications illustrates 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.

45. The Role ofTechnology

46. 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.

47. 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

48. 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.

49. Changing the Learning and Teaching Environment

50. 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