Corbin Platti
Calculus is one of the most highly regarded subjects within K–12 education, with those wanting to enter fields of STEM often expected to have a strong foundation in the subject. With this pressure in mind, it is important to ensure that these classes are taught effectively.
In 1986, mathematics professors gathered at Tulane University in response to the growing concern over a failure rate of nearly 50% in many college calculus courses. Professors eventually agreed that the courses were teaching a limited selection of techniques that did not foster deep mathematical reasoning. This was due to a recent shift toward finite mathematics, which disincentivized faculty to make improvements to their calculus courses.
During the five day conference, professors discussed new approaches to teaching calculus that would primarily benefit students. The educators ultimately produced a set of principles to help improve their calculus courses, which were recorded in a book, “Toward a Lean and Lively Calculus” by the Mathematical Association of America. The movement to integrate these new principles into calculus courses would come to be known as Calculus Reform, which aimed to center instruction around four main aspects: verbal, graphical, analytical, and numerical. Implementation of these fundamentals into calculus courses led to higher reliance on visual aids, technology, written explanations of answers, and projects.
TO SOME, INCREASING RELIANCE ON TECHNOLOGY AND VISUAL AIDS TO TEACH CALCULUS MAY SOUND LIKE AN ATTEMPT TO AVOID THE RIGOR THAT COMES WITH LEARNING DIFFICULT CONCEPTS.
Based on a survey sent to students taking Advanced Placement (AP) Calculus AB up to multivariable calculus, all respondents indicated seeing aspects of Calculus Reform in their classes and having to utilize verbal explanations. Nearly all of the students said that while teaching, their teacher utilized visual aids and intuition based explanations. Furthermore, 67% reported that their teacher presented varying approaches to solving problems and had their students talk in groups.
According to AP Calculus BC teacher Dave Deggeller, the focus of the curriculum has shifted away from being highly procedural when he took the exam in 1989, to being more visually and verbally based. For instance, he noted how slope fields—a graphical representation of solutions—are now used while teaching differential equations. He also mentioned how technologies like graphing calculators have become an essential part of the course. With the release of the TI-82 calculator in 1993, the 1994 AP exam was the first that allowed students to use this technology, leading toward a more conceptual curriculum.
To some, increasing reliance on technology and visual aids to teach calculus may sound like an attempt to avoid the rigor that comes with learning difficult concepts. In reality, however, this is rarely a problem. Think back to early elementary school when you were first learning how to add and subtract numbers. You may recall your teacher using groups of physical objects to demonstrate the mechanics of these operations, or maybe you used your fingers to do something similar. For a moment, imagine that these visual demonstrations were not available to you, and that you were forced to learn by only looking at the numbers and symbols in an equation—the task would become significantly harder.
While most people can now add and subtract easily without any supplementary tools, they were essential in getting to this point of mastery. The use of technology and visual aids to teach calculus is no different: While students may rely on them as a crutch in the beginning, concepts become clearer through continued practice.
STUDENTS ARE EXPECTED TO “CULTIVATE THEIR UNDERSTANDING OF DIFFERENTIAL AND INTEGRAL CALCULUS THROUGH ENGAGING WITH REAL-WORLD PROBLEMS.”
The calculus reform movement has been successful in implementing its fundamental ideas in the curricula, as seen through College Board’s course descriptions of AP Calculus AB and BC. They note how students are expected to “cultivate their understanding of differential and integral calculus through engaging with real-world problems represented graphically, numerically, analytically, and verbally.”
Descriptions of college calculus courses are similar. For instance, the Massachusetts Institute of Technology describes their Calculus 1 offering as teaching “the mathematical notation, physical meaning, and geometric interpretation of a variety of calculus concepts” and giving students “insight into real-world applications of these mathematical ideas.” Calculus 1 at Carnegie Mellon University also places emphasis on real-world applications, specifically in regards to finance and economics.
As more schools have implemented reforms, various studies have compared student retention between traditional and reformed courses. For instance, when Boise State University’s Calculus 1 course was redesigned to have students spend more time solving problems in the contexts of other disciplines, retention rates improved by 3.4%, which—while modest—is still a statistically significant result.
IT IS IMPORTANT TO CONTINUE SEEKING FURTHER DEVELOPMENT USING THE DECADES OF NEW RESEARCH AND TECHNOLOGY AVAILABLE.
Since the emergence of calculus reform in 1986, the world has dramatically changed. Technology is more advanced and widely available than ever before, making calculus education accessible to nearly everyone. As advancements continue, the necessity for a well-constructed calculus curriculum will become increasingly important. While Calculus Reform has made notable progress, it is important to continue seeking further development using the decades of new research and technology available.
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