Developing a Hands-on Calculus Course

We hope that hands-on calculus classes will be developed around this book. We think it is particularly appropriate for students that might not have taken as much preparatory math as calculus students normally would have, like younger teens, or people in vocational or worker retraining programs. We emphasize active learning, first introducing a physical model and then deducing the math concept from that. This book develops intuition that enables starting on fundamentals of robotics, digital fabrication tools, electronics and the like.

Each chapter focuses on one or more central calculus topics, with associated 3D printable models or other activities. A fun K-12 summer or afterschool program might cover Chapters 1-4, Chapter 6, and some of the projects in Chapter 10. (Chapter 10 applies concepts from earlier chapters to creating electronic projects.) Chapters 5 and 7-9 cover somewhat more advanced concepts with 3D printable models. Chapters 11 through 13 are more algebra-heavy than the earlier chapters, and bridge between our intuitive approach and the tools to calculate values for practical purposes.

Advanced math students who need to learn differently, such as blind or visually-impaired students, are often not served well by a professor at a whiteboard. We hope this book and its associated projects embody enough universal design to act as a gateway to the elegant structure of calculus.

3D Printable Models

This book teaches calculus with materials like LEGO bricks, paper, and 3D prints. These last were developed with the free and open-source program OpenSCAD (openscad.org). Chapter 3 describes where and how to download OpenSCAD and the models, and has some suggestions on how to print them and (in most cases) modify them if you want to. The models are designed to print on a low-end consumer or hobbyist 3D printer with minimal fuss.

With that said, we have included photographs of the projects and described them in enough detail that you should be able to get by without actually creating them yourself. However, we find that it is a different experience to hold and move around a 3D model of a concept. We highly recommend trying out the LEGO brick exercises and creating the 3D prints, or at least manipulating them in 3D in OpenSCAD. The electronics demonstrations in Chapter 10 require a little more background, and are included for those who might be a bit more maker-inclined.

Chapter Layout

Each chapter begins with a calculus concept, and sometimes a little history about who came up with the idea in the first place. We then introduce a hands-on model and talk about how to use the model to think about the concept. Then we move into the relevant mathematical symbols and calculations. We end each chapter with a summary of the key points and a list of symbols and words you can look up, plus a few references.

In some cases, the calculations are beyond what we think is appropriate for this book, and we refer you to alternative resources you can use if you need to go that far. We have tried to keep algebra to a minimum, but we give you enough information and at least informal explanations of where things come from so that we do not have to resort to saying “just trust us.”

If you want to use a traditional text in tandem with this book, we highly recommend the work of the late George Thomas. Joan’s high school and undergraduate copy of the fourth edition of Thomas’ Calculus was her introduction to the field. The much-evolved 2018 fourteenth edition now written by Joel Hass, Christopher Heil, and Maurice Weir, and renamed Thomas’ Calculus, is an imposing but excellent work. We have a short list of websites, video courses, and books we found useful at the end of Chapter 13.

Each chapter ends with a list of words you can look up, as well as the names of symbols we use. We know that self-learners get stuck sometimes if they don’t know what a symbol is called and so can’t look things up. Where calculus symbols and terms are not used consistently by everyone, we try to capture the variations without getting so far into it that we cause more problems than we solve. We supply references that for the most part are free and easily accessible, such as Wikipedia. These sites are not peer-reviewed, but for someone who may have limited funds, time, and library access, they will be sufficient for getting started.

Many of the calculations covered in a basic calculus course now can be done by software programs like Mathematica and even some freely available utilities. The intent of this book is to give you enough background to be able to interpret the documentation for such tools and be able to walk your way into more sophisticated concepts there or in traditional textbooks. Computers are very good at algebra and calculations; people, not so much. Delegate to the machines where you can, but be skeptical of what they produce and read the documentation on limitations.

Acknowledgments

This book would not have been possible without the encouragement of many people and organizations. We first developed the concept as an entry in the 2017 Hackaday Prize and we are grateful to the many people who posted comments or made suggestions about our early efforts there.

We have appreciated reviewers and editors who made good suggestions along the way. We particularly want to thank our fearless technical reviewer, Dr. Niles Ritter, for catching us in many potential sins of omission and commission. Joan’s husband, astronomer Dr. Stephen Unwin, read and commented on early versions of key chapters, and provided encouragement and consolation when it seemed like this would never work.

We want to give a special shout-out to Michelle Lowman. In an amazing bit of symmetry, she was the editor for this book, but previously edited our first 3D printing book at another publisher. The production team at Make:, including Juliann Brown and Terisa Davis, saw to it that the book looks great. Finally, we appreciate Dale Dougherty and the Make: management team for taking a chance on this unconventional book. We also remember the late Patrick DiJusto, who edited our Make: Geometry book. Many of his critiques of our math-explaining style are reflected in this book as well.

Friends who teach in or are otherwise a part of the visually impaired community gave us insight into tactile-first learning. Notable among these were Dr. Yue-Ting Siu, Chancey Fleet, Lore Schindler, Mike Cheverie, Lindsay Yazzolino and researchers at the Smith-Kettlewell Eye Research Institute. Some of the material incorporated in this book was supported in part by grant number 90RE5024, from the U.S. Administration for Community Living, Department of Health and Human Services, Washington, D.C. 20201.

The Institute for Educational Advancement in Pasadena, particularly Alexis Hopper and Nicole Endacott, let us pilot this work with several cohorts of gifted pre-teens and young teens in their Academy program. This was an invaluable (and fun!) reality check that influenced many of the book’s projects.

Joan fondly remembers her own high school math teachers, particularly her calculus teacher, the late Sister Cecilia Madeline McCarthy, CSJ, at The Mary Louis Academy in Jamaica Estates, New York, and the other faculty there. The image of the school on the dedication page is courtesy of The Mary Louis Academy. She is also grateful to many members of the MIT faculty, including the late George Thomas, who taught her freshman calculus course just before he retired (hopefully coincidental timing).

And finally, we thank our families for putting up with 3D printed models all over the place, a 3D printer frequently whirring away all night, and endless pizza-fueled brainstorming sessions. We hope you will find your journey through calculus to be much easier because of all these contributions.