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Welcome to Math 215!
Welcome to Math 215!
This course explores one of the most fundamental questions humans have pondered:
How is the future determined by the present and the past?
The course investigates this question mathematically by studying various types of equations involving an unknown function and its derivatives. These differential equations help us understand how quantities change over time, and how that change depends on current and prior phenomena. The course's 19 lessons discuss many, many applications of differential equations to real-world phenomena, including in the sciences and social sciences. The prerequisite for the course is Calculus 2 (Math 116 at Wellesley).
Course Content
Course Content
Broadly speaking, we will be studying ordinary differential equations, applied linear algebra, and the various applications of those two topics to understanding real-world phenomena in the sciences and social sciences. In slightly more detail, the course is divide into the following four units.
Unit 1: First-Order Differential Equations
Unit 1: First-Order Differential Equations
This unit covers the theory, techniques, and applications of first-order differential equations (hereafter, FODEs). These are the simplest differential equations to study and solve. They also happen to show up in many places, both in real-world applications associated with various disciplines (e.g., chemistry, physics) and in purely mathematical settings (e.g., a proof). After completing this unit we will have learned how to solve the most common types of FODEs, and FOLDEs in particular. We will draw on this knowledge time and again in the subsequent units, returning to the solution techniques covered in this unit to help us develop solution techniques for higher-order ODEs and systems of ODEs. For this reason, it is fair to say that this unit provides the foundational knowledge we will need to build up the rest of the mathematical edifice of this course.
This unit will also introduce, reinforce, and emphasize mathematical modeling. This is the process of translating a real-world problem into mathematics, solving it, and interpreting your solution in the original real-world context that initiated the process. You no doubt have experience doing this in calculus courses (e.g., in optimization problems). This unit will hone that skill and help you start to think more like an applied mathematician. This will, in turn, be useful for tackling your science courses and social science courses, many of which involve mathematical analyses and/or modeling.
Unit 2: Second- and Higher-Order Differential Equations
Unit 2: Second- and Higher-Order Differential Equations
This unit covers the theory, techniques, and applications of ordinary differential equations of order larger than one. All but the last lesson in this unit focus on second-order ODEs (hereafter SODEs). As in Unit 1, we will see that SODEs -- and linear SODEs, in particular (which we’ll define in Lesson 3), which I'll refer to as SOLDEs -- show up in many places, both in real-world applications associated with various disciplines (e.g., chemistry, physics) and in purely mathematical settings (e.g., a proof).
After completing this unit we will have learned the theory behind SOLDEs, how to solve the most commonly encountered SOLDEs, and have picked up some content on complex variables (in Lessons 5 and 7). In addition, Lesson 7 sets us up for the next unit in the course -- linear algebra -- by showing us that in order to find the general solution to higher-order ODEs we need to calculate determinants of matrices, a central topic in linear algebra.
Unit 3: Applied Linear Algebra
Unit 3: Applied Linear Algebra
After completing this unit we will have learned the theory behind SOLDEs, how to solve the most commonly encountered SOLDEs, and have picked up some content on complex variables (in Lessons 5 and 7). In addition, Lesson 7 sets us up for the next unit in the course -- linear algebra -- by showing us that in order to find the general solution to higher-order ODEs we need to calculate determinants of matrices, a central topic in linear algebra.
After completing this unit we will have learned the theory behind linear algebra as well as many computational techniques for solving linear systems of equations, finding bases of vector spaces, and calculating a matrix's eigenvalues and eigenvectors. We will make heavy use of these concepts in Unit 4 to help use study systems of first-order ODEs both analytically and qualitatively.
In this Unit we will also connect linear algebra to the ODE theory and content we've learned in the first two Units. In particular, in Lesson 9 we'll return to the notion of the Wronskian of n solutions to a NOLDE as a determinant, a concept I briefly teased in Lesson 7. We'll also recast much of our work in SOLDE theory in linear algebra terms and realize that the new linear algebra content we will learn is much more powerful than it seems at first.
Unit 4: Systems of Differential Equations
Unit 4: Systems of Differential Equations
This unit covers the theory, techniques, and applications of systems of differential equations. Our focus, for the most part, will be on systems of FODEs that are linear and have constant coefficients. As in the previous Units, we will see that such systems show up in many places, both in real-world applications associated with various disciplines (e.g., chemistry, physics) and in purely mathematical settings (e.g., a proof).
After completing this Unit we will have developed techniques for solving NODEs, which is where we left of in the previous Unit. We will also broaden our study of ODEs beyond studying just one ODE to studying a system of ODEs. Along the way we will leverage the linear algebra concepts we've learned about -- like vector spaces, their bases, and the eigenvalues and eigenvectors of a matrix -- to understand the dynamics that result from systems of ODEs. Finally, the new techniques we'll discuss to visualize the solutions to systems of ODEs will give us a way to understand those solutions without having to actually solve the ODEs.
Learning Goals
Learning Goals
This course has been designed to achieve the following learning outcomes by the time you complete the course.
Foundational Knowledge: You will recognize, understand, and develop intuition for new mathematical concepts rooted in differential equations and linear algebra.
Connecting Content to Real-World Situations: You will recognize how differential equations and linear algebra arise from real-world problems and contexts, and be able to interpret the real-world implications of their solution.
Application Skills: You will learn how to create mathematical models involving differential equations and linear algebra that describe a variety of real-world phenomena.
Teamwork: You will learn how to engage in and facilitate open dialogue with classmates and others about mathematics, in ways that are respectful of differences and establish equitable learning environments.
Learning How to Learn: You will learn about the latest research on cognitive science and how it can help you become a better student. You will also learn how to pinpoint your areas of academic struggles and develop a plan for resolving them. Finally, you will learn how to become a more independent learner.
Textbook
Textbook
Though the vast majority of the course's content will come from the lesson notes and videos I've prepared (these are accessible via the Unit links above), some of the practice problems and supplemental material comes from the excellent book Differential Equations and Their Applications, 4th edition, by Martin Braun. The ISBN-13 for this text is 9780387978949. The first chapter will be made available on the course's website, but I recommend getting a copy of the book as soon as you can if you would like to work through all of the practice problems in the course.
Syllabus
Syllabus
If you are currently enrolled in this course with me then you've received a copy of the course's syllabus. It details the additional course policies and the course structure. For everyone else, the short story is that this course is structured in a flipped classroom format with a mastery grading scheme for assessments. I wrote about this duo in detail in an article I published in 2020 in a mathematics education journal, but here are the takeaways:
Students read through the lesson notes and the accompanying videos before class sessions start (you can find these inside the Unit links above).
They then submit reflections on what they learned, which include questions about summarizing what they learned and about what things, if any, they are still confused on.
During class sessions I clarify the points of confusion submitted via the reflections and also add additional context and supplementary content, as needed. We then work together on the practice problems in small groups.
Students are assessed using a mastery grading scheme I call Second Chance Grading. In short, weekly quizzes test for understanding and rather than have midterms, we have Second Chance Assessments. On these assessments, students can re-attempt previous quiz exercises and if they score higher the second time around they receive that new (higher score).
This structure is backed by the latest research on growth mindsets, mastery learning, and the "testing effect." (I discuss all that research in my article on Second Chance Grading.)
Getting Started
Getting Started
If you're ready to get started with the course, click on the Unit 1 link above. That will take you to a preview of the first unit in the course, its goals, and how it connects to the rest of the course. That page will also contain links to the individual lessons comprising the unit. At the bottom of those lesson pages you'll find navigation buttons that will help you advance and go backward between lessons.
I hope you enjoy the course. If you happen to catch any errors or have other feedback, please feel free to email me: ofernand@wellesley.edu.
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