Developing a New Freshman Mathematics Course for Biology and Environmental Sciences Majors. Department of Mathematics December 2015

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1 Developing a New Freshman Mathematics Course for Biology and Environmental Sciences Majors. Department of Mathematics December 2015 Programs changing to allow Math 1044 as an alternate to Math 1042: GSTC General Science with Teaching BS GEOL Geology BA* NATS Natural Sciences BA** BIOL Biology BA,BS BITC Biology with Teaching BS ENVS Environmental Science BS*** GEOL Geology BS NSCM Neuroscience:Cell & Molecular BS ESTC Earth & Space Sci with Teachin BS *Note that the Geology BA used to have an option for Math 1042 among other math and statistics related courses within its Math Foundation Elective. Now more explicit that Math 1042 or Math 1044 is required. **Note that a second semester of Mathematics is not required, but optional for the Natural Science BA. ***Note that the Environmental Science BS used to have two required calculus courses and one statistics related course. Given the new course, students will be able to choose Math 1041 and Math 1042 and a third course in statistics at an introductory level or Math 1041 and Math 1044 and a third course in statistics at a more advanced level. Details: Course Concept: A one-semester course at the freshman level for majors in Biology and Earth and Environmental Sciences (EES), covering mathematical methods needed for empirical modeling and associated data analysis. Primarily for majors in Biology and EES. Title: Introduction to Probability and Statistics for the Life Sciences (In this document: Math 1042X.) Context and Rationale: The historically standard first-year calculus program concentrates on mathematics primarily used for applications in physics and engineering. In recent years, curricula have emerged that are better suited for

2 typical students in life and environmental sciences. These curricula generally shift the focus more to the modeling of biological systems. In Spring 2015 the Department of Biology here at Temple approached us about the possibility of devising a course sequence more suitable for their students. Over the summer, a committee comprised of members of the math, biology, physics, and EES departments met to discuss this subject. Biology faculty were particularly interested in having their students better prepared for biostatistics courses taken in sophomore or junior years. Several universities now offer a complete one-year calculus alternative for bio (and related) majors. There are also a number of textbooks available following this model. These courses are quite different than the standard first year calculus. In our summer discussions, we initially considered developing our own oneyear sequence, but after some time convinced ourselves that it would be too restrictive e.g., students who begin in the bio sequence would be essentially locked out if their interests shifted to say biophysics. So we settled on the idea that a standard Math 1041 Calc I would remain for all students, but that a second-semester course for biology and EES majors would be developed, focusing on better preparing these students for biostatistics and related courses. Brief course description: Probabilistic and statistical methods needed for empirical modeling and associated data analysis, with examples primarily taken from the life sciences. (A more precise description is given below.) Credits: 4 Prerequisites: Math 1041 Calculus I or Math 1941 Honors Calculus I with C or higher. Notes: 1. Students who take 1042X would not be able to go directly into Calculus III Math However, students could take both 1042X and 1042 for credit and then take Biology would allow students (in some of their programs) to substitute Math 1042X for Calc II. 3. We would likely have to exchange some topics between Math 1041 and Math 1042, so as to cover material on integrals necessary for Math 1042X. In

3 particular, some care will be needed to ensure that students completing Math 1041 and 1042X can still take Physics 2021 and Physics 2022, the one year physics sequence with applications in biology. (This task seems reasonable, in view of a list of calculus topics needed for Physics provided by Prof. Dziembowski in Physics.) 4. The proposed course sequence appears to be in line with recent changes in course requirements for admission to medical school. Implementation: Since Math 1042X will depend on a revised Math 1041, the earliest date for a large scale implementation would be Spring (With the revised Math beginning in Fall 2016.) However, a soft launch of the course could be begun in Fall 2016, for students who have completed both Math 1041 and Math 1042 in their current forms. A possible challenge to implementing this course is the preparation of our mathematics department instructors to teach it. In particular, finding funds (if necessary) for release time for mathematics faculty to prepare. Sample Detailed Course Syllabus Sample Textbook: Mathematics for the Life Sciences, Glenn Ledder, Springer. Sample course topics, as time permits. (Not necessarily in the order listed, and again as time permits.) I. Calculus and Modeling. 1.1 Working with Parameters Scaling Parameters Nonlinear Parameters Bifurcations * Properties of the Definite Integral * The Fundamental Theorem of Calculus * Computing Definite Integrals with the Fundamental Theorem * Computing Antiderivatives and Definite Integrals * Substitution * Constructing Antiderivatives with the Fundamental Theorem * Obtaining a Graph of f from a Graph of f! Mathematics in Biology... 85

4 2.1.1 Biological Data Overall Patterns in a Random World Determining Relationships Basic Concepts of Modeling Mechanistic and Empirical Modeling Aims of Mathematical Modeling The Narrow and Broad Views of Mathematical Models Accuracy, Precision, and Interpretation of Results Empirical Modeling I: Fitting Linear Models to Data The Basic Linear Least Squares Method(y=mx) Adapting the Method to the General Linear Model Implied Assumptions of Least Squares Empirical Modeling II: Fitting Semilinear Models to Data Fitting the Exponential Model by Linear Least Squares Linear Least Squares Fit for the Power Function Model y = Axp Semilinear Least Squares Empirical Modeling III: Choosing Among Models Quantitative Accuracy Complexity The Akaike Information Criterion Choosing Among Models Some Recommendations II. Probability 3.1 Characterizing Data Types of Data Displaying Data Measures of Central Tendency Measuring Variability Concepts of Probability Experiments, Outcomes, and Random Variables Probability Distributions Sequences and Complements Discrete Probability Distributions Distribution Functions Expected Value, Mean, and Standard Deviation

5 3.4 The Binomial Distribution Bernoulli Trials and Binomial Distributions Continuous Probability Distributions Cumulative Distribution Functions Probability Density Functions Expected Value, Mean,and Variance The Normal Distribution The Standard Normal Distribution Standard Probability Intervals The Poisson and Exponential Distributions The Poisson Distribution The Exponential Distribution Memory in Probability Distributions An Introduction to Statistical Inference Tests on Probability Distributions A Graphical Test for Distribution Type The Cramer von Mises Test Outliers Probability Distributions of Samples Sums and Means of Two Random Variables General Characteristics of Samples Means and Standard Deviations of Sample Sums and Means Distributions of Sample Means Inferences About Populations Inferences About Sample Means Inferences About Sample Proportions Estimating Parameters for Distributions Confidence Intervals Estimating Success Probability with a Likelihood Function Estimating Population Size by Mark and Recapture A Final Observation Conditional Probability The Concept of Conditional Probability Formulas for Conditional Probability The Partition Rule

6 4.7 Conditional Probability Applied to Diagnostic Tests The Standard Test Interpretation Problem * Part or whole may be contained in Calc I. Note: The above is adapted from a one-semester course developed by G. Ledder and used at University of Nebraska-Lincoln.