The Mathematics Curriculum

1. Mathematics for the Laboratory Sciences:College Algebra, Precalculus,and UpSheldon P. Gordongordonsp@farmingdale.edu

2. The Mathematics Curriculum At most schools, the mathematics curriculum is focused on moving students up the mathematics pipeline: either to become math majors or to serve the traditional needs of engineering and physics curricula. But these students are only a small minority of the students we see and whose needs we should be serving.

3. Bachelor’s Degrees in Mathematics In 2005, P There were 1,439,264 bachelor’s degrees P Of these, 14,351 were in mathematics This is less than one percent! (There are 23,000 degrees in recreation and leisure!)

4. The Needs of Our Students The reality is that virtually none of the students we face are going to be math majors. They take our courses because of requirements from other disciplines. What do those other disciplines want their students to bring from math courses?

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

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. What the Biologists Said • New areas of biological investigation have resulted in an increase in quantification of biological theories and models. • The collection and analysis of data that is central to biology inevitably leads to the use of mathematics. • Mathematics provides a language for the development and expression of biological concepts and theories. It allows biologists to summarize data, to describe it in logical terms, to draw inferences, and to make predictions.

13. What the Biologists Said • Statistics, modeling and graphical representation should take priority over calculus. • The teaching of mathematics and statistics should use motivating examples that draw on problems or data taken from biology. • Creating and analyzing computer simulations of biological systems provides a link between biological understanding and mathematical theory.

14. What the Biologists Said The quantitative skills needed for biology: • The meaning and use of variables, parameters, functions, and relations. • To formulate linear, exponential, and logarithmic functions from data or from general principles. • To understand the periodic nature of the sine and cosine functions. • The graphical representation of data in a variety of formats – histograms, scatterplots, log-log graphs (for power functions), and semi-log graphs (for exponential and log functions).

15. What the Biologists Said Other quantitative skills: • Some calculus for calculating areas and average values, rates of change, optimization, and gradients for understanding contour maps. • Statistics – descriptive statistics, regression analysis, multivariate analysis, probability distributions, simulations, significance and error analysis. • Discrete Mathematics and Matrix Algebra – graphs (trees, networks, flowcharts, digraphs), matrices, and difference equations.

16. What the Biologists Said • The sciences are increasingly seeing students who are quantitatively ill-prepared. • The biological sciences represent the largest science client of mathematics education. • The current mathematics curriculum for biology majors does not provide biology students with appropriate quantitative skills. • The biologists suggested the creation of mathematics courses designed specifically for biology majors. • This would serve as a catalyst for needed changes in the undergraduate biology curriculum. • We also have to provide opportunities for the biology faculty to increase their own facility with mathematics.

17. What the Chemists Said • Six themes for the mathematical preparation of chemistry students emerged. Mathematicians are asked to keep these six themes in mind as courses in mathematics are redesigned.

18. What the Chemists Said • Multivariate Relationships: Almost all problems in chemistry from the lowly ideal gas law to the most sophisticated applications of quantum mechanics and statistical mechanics are multivariate. • Numerical Methods: Used in a host of practical calculations, most enabled by the use of computers

19. What the Chemists Said • Visualization: Chemistry is highly visual. Practitioners need to visualize structures and atomic and molecular orbitals in three dimensions. • Scale and Estimation: The stretch from the world of atoms and molecules to tangible materials is of the order of Avogadro’s number, about 1024. Distinctions of scale, along with an intuitive feeling for the different values along the scales of size, are of central importance in chemistry.

20. What the Chemists Said • Mathematical Reasoning: Facility at mathematical reasoning permeates most of chemistry. Students must be able to follow algebraic arguments if they are to understand the relationships between mathematical expressions, to adapt these expressions to applications, and to see that most specific mathematical expressions can be recovered from a few fundamental relationships in a few steps.

21. What the Chemists Said • Data Analysis: Data analysis is a widespread activity in chemistry that depends on the application of mathematical methods. These methods include statistics and curve fitting.

22. Health-Related Life Sciences • “Many participants put special emphasis on the use of models.” “Models are a way of organizing information for the purpose of gaining insight and providing intuition into systems that are too complex to understand any other way”. • “Students should master a higher level interface, e.g.: spreadsheet, symbolic/numerical computational packages( e.g. Mathematica, Maple, Matlab), statistical packages. • BE FLEXIBLE: package topics creatively thru long-term interaction between mathematics and the life sciences.

23. 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.

24. 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.

25. 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.

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

27. Conceptual Understanding Everybody talks about emphasizing Conceptual Understanding, but • 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?

28. 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?

29. 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.

30. 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.

31. 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.

32. 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.

33. 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.

34. 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.

35. 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?

36. Further Implications If we focus only on developing manipulative skills without developing conceptual understanding, we produce nothing more than students who are only Imperfect Organic Clones of a TI-89

37. 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.

38. 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.

39. Should x Mark the Spot? When students see 50 exercises where the first 40 involve solving for x, and a handful at the end 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.

40. Some Illustrative Examples and Problems for Conceptual Understanding and Mathematical Modeling

41. 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

42. 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?

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

44. (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?

45. A Temperature Experiment An experiment is conducted to study the rate at which temperature changes. A temperature probe is first heated in a cup of hot water and then pulled out and placed into a cup of cold water. The temperature of the probe, in ̊C, is measured every second for 36 seconds and recorded in the following table. Time 1 2 3 4 5 6 7 8 42.3 36.03 30.85 26.77 23.58 20.93 18.79 17.08 31 32 33 34 35 36 8.78 8.78 8.78 8.78 8.66 8.66 Find a function that fits this data.

46. A Temperature Experiment The data suggest an exponential decay function, but the data do not decay to 0. To find a function, one first has to shift the data values down to get a transformed set of data that decay to 0. y = T – 8.6 = 35.439(0.848)t Then one has to fit an exponential function to the transformed data. Finally, one has to undo the transformation by shifting the resulting exponential function. T = 8.6 + 35.439(0.848)t.

47. Modeling the Decay of a Drug Every drug is washed out of the bloodstream, usually by the kidneys, but at different rates. For example, in any 24-hour period, about 25% of any Prozac in the blood is washed out, leaving 75% of the amount. This suggests an exponential decay function. If the initial dosage is 80 mg. then the model is D(t) = 80 (0.75)t, t in days.

48. Predictive Questions What will the level of Prozac (or any drug) be after 7 days (or any given number of time periods)? How long will it take until the level of Prozac is down to 10 mg (or to any given level)? What is the half-life of Prozac in the blood?

49. Repeated Doses of a Drug 25% of the Prozac in the blood is washed out each day, leaving 75% Typical dose is 40 mg each day

50. Level of Prozac D0 = 40 D1 = .75(40) + 40 = 30 + 40 = 70 D2 = .75(70) + 40 = 92.5 D3 = .75(92.5) + 40 = 109.375 {40, 70, 92.5, 109.375, 122.031, 131.523, …}