Move…accelerate…reach the peak…bottom out…transform…
So many ways to look at change, to talk about change. Change is inescapable. And change can be scary, especially when you don’t understand what’s happening. Yet change can also be an opportunity for growth, for progress, for new insights.
The same could be said of the millions of students who take calculus classes every year. As teachers know, it’s a rich opportunity for student growth and insight, and also scary as they’re getting started. More than a math subject, calculus is fundamentally the language of change. It’s a beautiful and powerful expression of this universal phenomenon, and a great way to tackle a wide range of real-world problems.
MIT’s John Bush, professor of applied mathematics (and a fabulous photographer of fluid phenomena), offers some great calculus teaching tips in the above video. We hope you can motivate students by demonstrating how to revel in the beauty of the language, and all the things it can do, before diving into the nitty-gritty grammar of deltas and epsilons. Professor Bush suggests:
- Sharks hunt with calculus! They intuitively “follow the gradient” of scent, or the direction that gives the highest rate of increase, to take them toward their prey.
- Introductory physics students will have learned Snell’s law as an equation about the angles of light paths through different media. But calculus can show that it’s fundamentally an optimization problem of light following the fastest route; just like how a smartphone GPS map calculates your fastest route home through tangled streets and busy traffic. (See this explanation of route-finding by mathematician and writer Steven Strogatz.)
- Archimedes’ Principle is a great way to experience the concept and importance of volume, and also the way integrals work. (Again, see Steven Strogatz’s engaging explanation.)
As Professor Bush says, calculus “is a language that’s valuable in virtually all disciplines, from physics to biology, from finance to engineering.” Motivate deeper student engagement by demonstrating how beautiful it can be to describe their ever-changing world with the poetics of mathematics.
Calculus on OCW
OCW has a wealth of inspiring calculus teaching materials, all free and licensed for you to re-use and remix in your classroom. You’ll find complete courses, online textbooks, videos, many sample problems with solutions, and more. Here are some highlights.
These two courses, from the OCW Scholar series, provide the complete teaching materials comprising MIT’s undergraduate calculus requirement. They’re a great place to begin, and house some of OCW’s most popular calculus material.
These OCW sites include complete video lectures, selected problem solving videos by course teaching assistants, and problems and exams with solutions. You can start at the beginning and work through it all in sequence, or pick and choose your own topics of interest from the syllabus.
These open-licensed textbooks by respected authors are free to download and use in your classroom, and for your own learning and inspiration.
- Calculus for Beginners and Artists, by Daniel Kleitman
An overview of calculus in clear, easy to understand language designed for the non-mathematician.
- Calculus, by Gilbert Strang
In-depth treatment of single variable and multivariable calculus, with plenty of applications. Also has an online Instructor’s Manual and a student Study Guide.
- Calculus with Applications, by Daniel Kleitman
Detailed lecture notes (in the form of a textbook) that cover differential calculus in one and several dimensions.
- Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving, by Sanjoy Mahajan
From its MIT Press catalog description: “…an antidote to the rigor mortis brought on by too much mathematical rigor, teaching us how to guess answers without needing a proof or an exact calculation.”
Use these videos to inspire your own approach to teaching. Students can watch them for in-depth and engaging explanations of key concepts, and to supplement their classroom instruction time.
- Highlights of Calculus video series, by Gilbert Strang
Five videos provide an overview of the key topics and ideas of calculus and how they apply to real-life situations and problems. There are summary slides and practice problems complete with an audio narration by Professor Strang. The resource also includes a series of 12 videos, with slides and practice problems, that dig more deeply into derivatives.
- Single Variable Calculus lecture videos with PDF notes, by David Jerison
This sequence of videos and accompanying notes (beginning with the first class session) covers differentiation, applications of differentiation, the definite integral and its applications, techniques of integration, and exploring the infinite.
- Calculus Revisited: Single Variable Calculus, by Herb Gross
A revered series of videos and related resources covering the materials normally found in a freshman-level introductory calculus course. The series was first released in 1970, and has achieved something of a cult following in its second life on OCW and YouTube.
- Get problem solving tips in the recitation videos by course teaching assistants in 18.01SC Single Variable Calculus and 18.02SC Multivariable Calculus. Start with these videos at the end of the first 18.01SC session “Introduction to Derivatives“:
- Definition of Derivative, by Joel Lewis
- Graphing a Derivative Function, by Christine Breiner
Problem Solving and Assessment
Use questions and accompanying solutions from OCW’s worked examples, problem sets and exams directly with your students, or as a basis for your own instruction. Begin your exploration with these examples:
- A calculus view of checking account balances, problem (PDF) and solution (PDF), from the 18.01SC Single Variable Calculus session “Derivative as Rate of Change.”
- Differentiation problem set and differentiation exam, from 18.01SC Single Variable Calculus
- Comparing linear approximations and calculator computations, problem (PDF) and solution (PDF), from the 18.01SC Single Variable Calculus session “Linear Approximation.”
- Vectors problem set and vectors and matrices exam, from 18.02SC Multivariable Calculus
AP Calculus Exam Study
With OCW Highlights for High School, you can search for OCW calculus materials by topic and subtopic, to help students prepare for their AP Calculus exams.
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These material highlights are just a tiny fraction of all of the calculus content on OCW. In the words of Professor John Bush, with calculus “the possibilities are endless!”