Brian J. Winkel

Parameter Estimates in Differential Equation Models for Population Growth

Abstract We estimate the parameters present in several differential equation models of population growth, specifically logistic growth models and two-species competition models. We discuss student-evolved strategies and offer Mathematica code for a gradient search approach. We use historical (1930s) data from microbial studies of the Russian biologist, G. F. Gause, and estimate growth rates, carrying capacities, and “coefficients for the struggle for existence.”

Parameter estimates in differential equation models for chemical kinetics

We discuss the need for devoting time in differential equations courses to modelling and the completion of the modelling process with efforts to estimate the parameters in the models using data. We estimate the parameters present in several differential equation models of chemical reactions of order n, where n = 0, 1, 2, and apply more general parameter estimation approaches to an optimization problem involving the production chemical reaction A → B → C.

In plane view: an exercise in visualization

We offer a problem in visualization which uses multivariable calculus concepts. The problem is essentially to describe (mathematically) what we can see on one mountain while sitting on an adjacent mountain. We present how our students, working in groups, attack the problem and the issues which surround the solution strategies produced. We have successfully used this problem for a number of years in our courses, devoting several class periods and about 2‐3 weeks outside of class to student development of a solution strategy. The problem serves to develop visualization skills, verbalization of mathematical concepts, and implementation of problem‐solving notions in mathematics including gradients, optimization, integration, surface area, and programming.

Lessons Learned from a Mathematical Cryptology Course

Abstract We present a description of a mathematical cryptology course taught to undergraduates in which cryptanalysis was a driving force. Historical discovery served to motivate student inquiry, reflection on personal analyses produced improved solutions, and projects permitted students to explore areas of personal interest.

Sourcing for parameter estimation and study of logistic differential equation

We present two simulation activities for students to generate real data and several data sources for the purpose of estimating parameters in the logistic differential equation model.

Computers Have Taken Us to the Brink in Mathematics ...and We Have Balked

Computers have been available to teach and do mathematics for decades. There have been spikes of interest, energy, and results in their use over the years. However, there is little sustained use of computers as they could be used to permit students to learn and do mathematics. To be effective computers need to be embedded in the learning process. They need to be ubiquitous and part of every students learning kit. Indeed, they should be a first tool of choice for most of what students do with mathematics; that is, plotting, algebraic manipulation, optimization, playing “what if” games, data analysis, etc. I outline a number of scenarios and experiences to show just how pervasive computer use can and should be. I put computer use in context with its most important applications for teaching mathematics—namely, modeling and inquiry or problem-based learning.

Birth and Death Process Modeling Leads to the Poisson Distribution: A Journey Worth Taking

Abstract This paper describes details of development of the general birth and death process from which we can extract the Poisson process as a special case. This general process is appropriate for a number of courses and units in courses and can enrich the study of mathematics for students as it touches and uses a diverse set of mathematical topics, e.g., probability, differential equations, difference equations, calculus, and infinite series. We guide the reader through the assumptions, derivation, and modeling applications which will permit the study of this useful subject in a number of settings. We offer illustrations of the Poisson process to demonstrate its applicability to interesting and real-life situations.

Population modelling with M&M's®

Several activities in which population dynamics can be modelled by tossing M&Ms® candy are presented. Physical activities involving M&Ms® can be modelled by difference equations and several population phenomena, including death and immigration, are studied.

Teaching Modeling with Partial Differential Equations: Several Successful Approaches

Abstract We discuss the introduction and teaching of partial differential equations (heat and wave equations) via modeling physical phenomena, using a new approach that encompasses constructing difference equations and implementing these in a spreadsheet, numerically solving the partial differential equations using the numerical differential equation solver in Mathematica, and analytically constructing solutions from reasoned building blocks. We obtain graphical feedback as soon as possible in each approach and permit “what if” modeling wherever possible. This approach is contrasted with the usual Fourier series development and series solution using boundary value solution strategies.

Fourier series optimization opportunity

This note discusses the introduction of Fourier series as an immediate application of optimization of a function of more than one variable. Specifically, it is shown how the study of Fourier series can be motivated to enrich a multivariable calculus class. This is done through discovery learning and use of technology wherein students build the sine Fourier series for the simple function f(x) = x and then generalize to the nth term sine Fourier series for a general function, f(x). It is shown how the students can then explore the power of the Fourier series to represent functions.