There’s More to Calculus Than Equations

We usually think of Newton and Leibniz as the inventors of calculus, but neither of them—for all their great brilliance—could create that extraordinary mathematical description of the world around us from nothing. In fact, the branch of mathematics we call calculus developed over many centuries in many different parts of the world, from ancient Greece to the Middle East, India, China, Japan, and Western Europe.

Newton and Leibniz pieced together a vast body of prior knowledge of topics in both differential and integral calculus and summed them up in the Fundamental Theorem of Calculus, thus shaping, connecting, and sharpening our understanding of calculus. But many of the problems that we study in calculus—areas and volumes, related rates, position/velocity/acceleration, infinite series, differential equations—had been studied well before their times (and often in a different chronological order from that in which they are presented in most of our calculus textbooks).

Incorporating the History of Calculus Into Your Lessons

This journey took 2,000 years and generations of students of mathematics from across the globe—although our own math students rarely know of this long process and often think less of themselves when they have difficulty mastering the abstract concepts of calculus.

Instead, the history of calculus can provide students with a richer and deeper understanding of those concepts. It shows them how and why such concepts were developed through centuries of hard work, partial success, sacrifice, trials, excitement, adversities, and delight.

Exploring calculus’s origins offers students the opportunity to trace the intellectual development of one of the most creative human endeavors. It places calculus in a clear and practical human context that demonstrates its astonishing usefulness. And, with its wealth of elegant concepts and proofs, colorful characters, and interesting stories, it may increase students’ interest and encourage a positive attitude toward mathematics.

The National Council of Teachers of Mathematics (NCTM) stated, “Mathematics is one of the greatest cultural and intellectual achievements of humankind, and citizens should develop an appreciation and understanding of that achievement, including its aesthetic and even recreational aspects.” The importance of including the history of mathematics in the school curriculum has been emphasized by professional councils such as the NCTM, the National Research Council, and the National Council for Accreditation of Teacher Education, and has been supported by research studies.

Furthermore, the teaching of the historical development of basic concepts has become central to the teaching of all the basic sciences and virtually all of the social sciences—so why not mathematics? In the words of James Whitbread Lee Glaisher, “I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history.”

Projects Can Encourage Deep Learning and Change Attitudes 

With this in mind, every year I invite my calculus students to read the book Infinite Powers: How Calculus Reveals the Secrets of the Universe, by mathematician Steven Strogatz. They divide themselves into groups of two or three. While all students have to browse through the whole book, each group selects three chapters on which to focus, research, take notes, understand, and summarize in a slide presentation.

Together, the students then present their findings from the book to a school audience of juniors, seniors, and faculty. The audience is usually fascinated by this presentation—unique in its kind for our school—because the subject matter is unknown to many, elicits preconceived notions from a few, and is not always fondly remembered by others. However, the student speakers exude such enthusiasm and confidence and have so much fun that their excitement is contagious, and the final applause is often thunderous.

Let Student Interest Lead the Way

This reading project is one of those low-floor, high-ceiling tasks where everyone in the group can begin and then work on at their level of interest and capacity. These tasks allow participants to do much more challenging mathematics, and in this particular project, there are many possible extensions that the students find out on their own. For example, some of my students studied one or more mathematicians and their work in depth; others dwelled on the mathematical details of a proof that interested them. Still others discovered and explored schools of mathematics they had never heard of before.

In my experience, students are thrilled to read a math book that is not a math textbook; they are excited to learn about this long history and finally understand what an incredible achievement calculus is for all of us; and they love learning about the many stories behind those mathematical processes and formulas.

I fondly remember some gossip exchanged about Kepler and Brahe’s intertwined lives and astronomical data, as well as taking sides between Team Leibniz and Team Newton—and so do the students, who reminisce about this project the following year or look forward to taking their turn at presenting their reading this time around.

This type of experience doesn’t have to wait until calculus class. There are many books that can be used at many different grade levels to inspire joy, awe, and love for mathematics. Learning comes from curiosity and interest in a subject; teachers providing an intellectual hook for that interest is key.