Mathematics for the Biosciences at Farmingdale State Sheldon Gordon

Major Premise “Biology will do for mathematics in the twenty-first century what physics did for mathematics in the twentieth century”

Major Premise Almost all math and bio projects start at the calculus level or above. But the overwhelming majority of beginning biology students, both majors and especially non-majors, typically are at the college algebra or precalculus level. Most of these students have avoided math as much as possible.

Our Project Our original plan was to begin developing the first stages of a new mathematics curriculum to serve the needs of biology students, both the bioscience majors and the non-majors who take introductory biology courses. This would also impact the level of quantitative work in the biology courses.

Our Project The project is a collaborative effort between Farmingdale State College and Suffolk Community College. Farmingdale State brings an exceptional history of successful efforts in reforming the mathematics curriculum. Suffolk brings an outstanding record of utilizing technology and quantitative methods in its introduction biology courses and labs.

The Mathematical Needs of Biology In discussions with the biology faculty at both schools, it became clear that most courses for non-majors (and even those for majors in some areas) make almost no use of mathematics in class. Math arises almost exclusively in the lab when students have to analyze data. This is where their weak math skills are dramatically evident.

The Farmingdale State Project Our first step was to develop an alternative to our modeling-based precalculus course that would focus almost exclusively on biological applications. The course would feature a lab component taught by the biology faculty, so that each week’s primary math topic would be accompanied by an experiment requiring the use of that mathematical method.

Course Topics Week 1Behavior of FunctionsIntro to Measurements and Measuring Week 2Families of Functions, Linear Functions Linear Growth – part 1 Week 3Linear Functions and Linear Regression Linear Growth – part 2 Week 4Exponential Growth and Decay Functions Exponential Growth Week 5Exponential Regression and Power Functions Exponential Decay part 1 Week 6Power Functions and Polynomials Exponential Decay – part 2

Course Topics Week 7 New functions from OldPower Function growth Week 8 Logistic and Surge Functions Logarithmic Functions Week 9 Matrix Models and Linear Systems Logistic Growth Week 10 Sinusoidal Functions and Periodic Behavior Surge Functions Week 11 Periodic Functions – part 2 Polynomial growth Week 12 Probability ModelsPeriodic Behavior Week 13 Probability Models and Difference Equations Probability Model (genetics)

What Happened Next To accommodate the lab component, we had to change the precalculus course from four to five credits. Because of that and conflicts with other courses (intro chemistry), the biology students did not register for the course and it did not run.

What We’ve Done Instead All the labs in the introductory biology course are in the process of being changed to dramatically increase the level of quantitative experience – the new labs will incorporate most of the experiments that were to be part of the precalculus course.

What We’ve Done Instead The math department has created a new four credit precalculus course to serve the needs of the biology students – the same math course, but no lab. The focus is on conceptual understanding, data analysis and mathematical modeling, rather than on algebraic manipulation (other than in a few special cases where needed to solve problems that arise naturally in context).

What We’ve Done Instead The math department has also created a new two-semester calculus sequence for biology students – it will emphasize concepts over manipulation and will stress biological applications. The math department has created a one- semester post-precalculus mathematical modeling in the biological sciences course for bioscience majors and applied math majors.

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 Some Sample Problems

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 44,218 square miles. Some Sample Problems

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.

The ocean temperature near New York as a function of the day of the year varies between 36  F and 74  F. Assume the water is coldest on the 40 th day and warmest on the 224 th. (a) Sketch the graph of the water 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 water temperature on March 15. (f) What are all the dates on which the water temperature is most likely 60  ?

The Next Challenge Based on the Curriculum Foundations reports and from discussions with faculty in biology (and most other areas), the most critical mathematical need of the other disciplines is for students to know more about statistics. How do we integrate statistical ideas and methods into math courses at all levels?

What Students Really Need from Math The ability to make sense of data – to interpret graphs and tables Statistical measures of data Estimating the mean of a population from a sample

The Curriculum Problems We Face Students don’t see college algebra or precalculus as providing any useful skills for their other courses. Typically, college algebra is the prerequisite for introductory statistics. Introductory statistics is already overly crammed with too much information. Most students put off taking the math as long as possible. So most don’t know any of the statistics when they take the courses in bio or other fields.

Some Ideas for College Algebra Data is Everywhere! We should capitalize on it. 1.A frequency distribution is a function – it is an effective way even to introduce and develop the concept of function. 2. Data analysis – the idea of fitting linear, exponential, power, polynomial, sinusoidal and other functions to data – is already becoming a major theme in some college algebra courses. It can be the unifying theme to link functions, the real world, and the other disciplines.

Some Ideas for College Algebra 3.The normal distribution function is It makes for an excellent example involving both stretching and shifting functions and a function of a function.

Some Ideas for College Algebra 4. The z-value associated with a measurement x is a nice application of a linear function of x:

Some Ideas for College Algebra 5. The Central Limit Theorem is another example of stretching and shifting functions -- the mean of the distribution of sample means is a shift and its standard deviation, produces a stretch or a squeeze, depending on the sample size n.

Some Conclusions Mathematics is a service department at almost all institutions. Few, if any, math departments can exist based solely on offerings for math and related majors. 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. Most math departments may well end up offering little beyond developmental algebra courses that serve little purpose.

Contact Information Sheldon P. Gordon Farmingdale.edu / ~gordonsp