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Transform a traditional ODE course by incorporating a modeling-first method using SIMIODE projects. Learn what worked, student reactions, and proposed modifications for future courses.
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Incorporating a Modeling First Approach into a Traditional ODECourse Michael A. Karls
Abstract • In summer 2017, after teaching an introductory ODE course in a traditional format for many years, I decided to try incorporating a “modeling first” approach into my fall 2017 course, using projects from SIMIODE. • In this talk I will outline the projects chosen and look at how they fit into the overall course structure, how they were introduced to the students, student reactions and responses to the projects, and the impact of the changes to the course on the students. • I will also discuss what worked and what did not work and how the course could be modified for subsequent offerings.
What Has Been Done … • Since 1993, I have been teaching an introductory ODE course at Ball State. • The course, MATH 374 Differential Equations can be considered “classic” or “traditional” based on the material covered as well as the method of delivery.
Course Description (Catalog) • Introduction to nth-order ordinary differential equations, equations of order one, elementary applications, linear equations with constant coefficients, nonhomogeneous equations, undetermined coefficients, variation of parameters, linear systems of equations, and the Laplace transform. Use of standard computer software.
Course Objectives (Syllabus) • This course is an introduction to the study of differential equations. • After this course, a student will be able to solve standard ordinary differential equations using techniques learned in integral calculus. • Students will also learn how to interpret the behavior of solutions of differential equations. • In addition, the student will learn how to utilize a computer algebra package such as Mathematica in problem solving—a crucial skill for today’s applied mathematician.
Course Content (Syllabus) • After this course, you should be familiar with • definitions, • families of curves, • equations of order one, • elementary applications, • linear differential equations, • linear equations with constant coefficients, • nonhomogeneous equations, • undetermined coefficients, • variation of parameters, • linear systems of equations, • and the Laplace transform. • If time permits, we will also look at power series solutions and partial differential equations.
Course Grade (pre 2017) • Course grade will be based on the percentage of points earned out of a total of 600 possible points made up of the following: • Three in-class exams (100 points each): 300 points • Homework: 100 points • Final Project (includes a class presentation): 100 points • Comprehensive (Take-Home) Final Exam: 100 points • Actually, in-class portion of each exam is 80 points with a 20 point take-home portion. • This is due to the course switching from 75 min twice per week to 50 min three times per week. • This means that 40% of the course grade is based on in-class exams.
Textbook(s) Used • The first few times I taught the course, the textbook used was Elementary Differential Equations, by Rainville, Bedient, and Bedient. • I found that this book worked well, mainly because each section covers exactly one topic and the exposition is concise, easy to follow, and complete.
Textbook(s) Used • About 15 years ago, we switched to Boyce and DiPrima’sElementary Differential Equations and Boundary Value Problems text. • Textbooks for most lower-level courses are chosen by committee. • I’ve adapted textbook readings to my original notes. • Boyce and Diprima provides a nice amount of extra detail to supplement class lectures. • The only downside is that one has to “jump around” in the text to match class notes.
Ball State’s Mathematical Sciences Program (Undergraduate) • Five undergraduate majors (fall 2017): • Actuarial Science (100) • Mathematical Sciences (34) • Applied Mathematics • Mathematics • Mathematics Teaching (77) • Middle School • Secondary School • Minor in Mathematics (47)
MATH 374 • One ODE course offered each fall. • Usually about 20 – 30 students enroll. • Required for Mathematical Sciences majors. • Elective for Mathematics Teaching majors, Actuarial Science majors, and Mathematics minors. • For Fall 2017, • Half of the students were mathematical science or mathematics teaching majors. • The other half were majors in other programs, such as meteorology, physics, economics, and chemistry. • Some were also a Mathematics minor.
Typical Class Day (pre 2017) • The course format has been interactive lecture with questions posed to help with understanding of the concepts being discussed. • Probably some version of the “Socratic Method” (at least my attempt at this) …
Perhaps it is time for a change … • After recent experiences with this course, I felt that it would be a good idea to change how the course is taught. • One reason is that for any take-home exams based on textbook questions, all answers are easily available online (slader.com, chegg.com, etc.) • Another reason is based on student feedback …
Student Feedback (Fall 2016) • Not enough examples related to the real world. • Hard to see how differential equations will be used. • Instead of writing notes on board, use of PowerPoint or similar means could free up time for more example problems and applications.
My Plans for Revising the Course (Spring 2017) • Choose a new textbook – since the course is textbook independent, why not use one that is free? • Here’s one I found that is free and incorporates the CAS Sage into the web-based version of the textbook: • The Ordinary Differential Equations Project by Thomas W. Judson.
My Plans for Revising the Course (Spring 2017) • Spend less time in class lecturing. • Block out class time each week to work on examples or homework in class. • Incorporate real-world (SIMIODE) projects into the course. • Have a graduate assistant (GA) attend course and be available to assist students.
My Plans for Revising the Course (Summer/Fall 2017) • One concern about the choice of textbook was the fact that systems are introduced early and built upon for the rest of the text. • Most likely this would require a significant revision of how concepts are introduced. • Another issue is that we do not require Linear Algebra as a prerequisite for the course. • Just before the semester started, I discovered that the required textbook listed for the course had not been changed from Boyce and Diprima to Judson’s text.
My Plans for Revising the Course (Summer/Fall 2017) • Keep Boyce and DiPrimaas the textbook and focus on introducing SIMIODE projects into the course. • In order to tofree up more class time for projects, use PowerPoint slides for course notes. • Other instructors have used video lectures, but this was not an option for me (not enough time to create videos)!
SIMIODE Background Material • SIMIODE Starter Kit (https://www.simiode.org/starterkit) • Sample SIMIODE Course Syllabus • Testimonials and Reflections by Colleagues • DIFFERENTIAL EQUATIONS AT MANHATTAN COLLEGE PERSONAL ACCOUNT by Rosemary Farley • SIMIODE Workshop at MathFest 2017.
Choosing and Assigning Projects • My plan was to choose projects that would align with the material covered in class. • Assign one project each Friday, due the following week, with Friday’s class time used as a lab day (in lieu of working on homework examples). • Projects would be due the following Friday at midnight, submitted via Blackboard.
Choosing and Assigning Projects • No projects would be assigned in an Exam week or during the weeks with Final Presentations at the end of the semester. • This meant approximately eight project assignments! • Since I’d never used this approach, my plan was to see where we were each week and choose a project accordingly.
Course Grade (Fall 2017) • Course grade will be based on the percentage of points earned out of a total of 600 possible points made up of the following: • Three in-class exams (75 points each): 225 points • Homework: 75 points • Projects: 150 points • Final Project (includes a class presentation): 75 points • Comprehensive (Take-Home) Final Exam: 75 points • Note that this means that 37.5% of the course grade is based on in-class exams!
Project Assignments • After the students worked on the first SIMIODE project (Ant Tunneling) in class, I decided to write up an accompanying project assignment. • This was based on our discussions in class. • This assignment was designed to help guide students through the SIMIODE project.
MATH 374 Project 1, Fall 2017 • Read the Ant Tunnel Building Handout. • Answer parts (a) – (h) from the handout. For part (a), use T(x) = kx and T(x) = kx+, plus any other functions you created. • For the differential equation model obtained from the difference equation T(x+h) – T(x) = xh, does the solution to this this model seem reasonable? Why or why not? How could we verify this model is valid? • Show how one can arrive at the difference equation T(x+h) – T(x) = xh. Hint: Assume for fixed x, T(x+h) – T(x) is proportional to h. Also assume for fixed h, T(x+h) – T(x) is proportional to x.
MATH 374 Project 2, Fall 2017 • Read the M&M – Death and Immigration Handout. • Following the instructions in the handout on pages 1 and 3, perform the experiments for M&M Death and M&M Death and Immigration and record your results in Tables 1 and 2. Information from these tables will be put into your work turned in for this project! • Choose an integer from 6 – 16, record this number, and repeat the M&M Death and Immigration experiment with your chosen number instead of 10 for the number of M&M’s that immigrate into the population at each iteration. Record your results in a third table, Table 3, set up the same as Tables 1 and 2. Make a copy of this third table with the recorded data to share with a classmate – put your name on the table, but DO NOT let your classmate know your choice of immigration number. The information from Table 3 that you receive from a classmate will be put into your work turned in for this project!
MATH 374 Project 2, Fall 2017 • In class, I asked you to try to find functions a(n) and b(n) to model the data you collected for each of the first two experiments. If you were not able to do so, that’s ok – this is hard to do directly. As we discussed in class (and in the handout), another way to approach this is to find a relationship between the number of M&M’s at a given iteration in terms of the number of M&M’s at the previous iteration. We came up with the recurrence relationb(n+1) = 0.5 b(n) + 50 for n ≥ 1, as a suggested model for the second experiment. Note that this is equation (1) on page 5 of the handout. We also discussed the need for a starting point or initial condition for our model, which we agreed should be b(0) = 50.
MATH 374 Project 2, Fall 2017 • Find a recurrence relation of the form a(n+1) = _____________ for n ≥ 1, to model the first experiment, with appropriate initial condition for n = 0. • Create a table to display the information from Table 1 as well as a third column with the outputs for your model for each choice of integer n. Repeat for the information from Table 2. Include these tables you create in your answer for this question. Compare your models for Experiments 1 and 2 to the data you collected. How well do they agree? Justify your answer mathematically. • Solve the recurrence relation for the second experiment, using the suggested Mathematica command from the handout. This can also be done with Wolfram Alpha. Does your answer agree with that in the handout? Why or why not? • Solve the recurrence relation you found for the first experiment, using Mathematica or Wolfram Alpha.
MATH 374 Project 2, Fall 2017 • Repeat questions 3 and 4 by hand (i.e. without the use of the RSolve command). • On page 7 of the handout, there is a derivation of a differential equation associated with the second experiment, which leads to another model for the second experiment, namely the initial value problem (IVP), (9). Solve the IVP (9) using the suggested Mathematica command in the handout. This can also be done with Wolfram Alpha. Does your answer agree with the solution (10) in the handout? Why or why not? • Solve IVP (9) by hand (i.e. without the use of the DSolve command). • Using an argument similar to that used to obtain the differential equation (and IVP) for the second experiment, find an IVP to model the first experiment. Solve this second IVP.
MATH 374 Project 2, Fall 2017 • Set up a model for the data you get from a classmate for the third experiment, involving the function c(n). Note that there will be an unknown parameter in this model – use the symbol k for this parameter. Be sure to include the name of the classmate who gave you the data in the answer to this question! • Watch the following YouTube video on parameter estimation via Excel: https://www.youtube.com/watch?v=mEthE6Hia-k&feature=youtu.be. Use the Excel Solver to determine the parameter k for your model in question 9 to best fit the model for the data provided by your classmate. Include a copy of your work done in Excel – save this file as LastnameFirstname374Project2.xlsx. • Create a table to display the information from Table 3 as well as a third column with the outputs for your model found in question 10 for each choice of integer n. How well does your model fit the data? Justify your answer mathematically. Based on your model, guess the value of k chosen by your classmate!
My Observations … • One of the main obstacles for students (besides the mathematical concepts) was the software needed. • Students were more familiar with Excel (most had not encountered the Solver). • Mathematica was a challenge for many students. • This is typical for students in most courses.
My Observations … • Initially, students struggled with the projects. • As the semester progressed, they “got the hang” of the projects. • I had to provide liberal hints, especially with the projects involving Mathematica. • Overall, students seemed to do better in the course than in previous years.
Student Feedback (fall 2017) • While challenging, problems helped to engage in material and see real world uses of differential equations. • Some of the projects seemed to test students on Mathematica or Excel skills rather than differential equations. • Hard to find clear connections between projects and course material – especially when Mathematica was being used. • Projects were thrown at students in the beginning. Made more sense by the end of the course.
What to try next time … • Continue with the SIMIODE projects. • Spend more time on projects in class. • Include more projects. • Perhaps try video lectures posted ahead of time. • Ball State has Interactive Learning Spaces that may be more conducive to this setting than a traditional classroom. http://cms.bsu.edu/about/administrativeoffices/educationalexcellence/services/learningspacesinitiative
