LEAN AND LIVELY CALCULUS REFORM

Lean and Lively

the history of calculus reform

Corbin Platti

CONTRIBUTOR

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 e ectively.

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.

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 rep-

TO SOME, INCREASING RELIANCE resentation of solutions—are now used while teaching di erential equations. He also mentioned how technolo-

ON TECHNOLOGY AND gies like graphing calculators have

VISUAL AIDS TO TEACH become an essential part of the CALCULUS MAY SOUND LIKE course. With the release of the TI-82 calculator in 1993, the 1994 AN ATTEMPT TO AVOID THE AP exam was the first that allowed

RIGOR THAT COMES WITH students to use this technology, LEARNING DIFFICULT CONCEPTS. 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 di cult 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 di erent: While students may rely on them as a crutch in the beginning, concepts become clearer through continued practice.

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 di erential 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 o ering 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, STUDENTS specifically in regards to finance and economics.

ARE EXPECTED TO As more schools have implemented reforms, various

“CULTIVATE THEIR studies have compared student retention between traUNDERSTANDING OF ditional and reformed courses. For instance, when Boise

DIFFERENTIAL AND State University’s Calculus 1 course was redesigned to INTEGRAL CALCULUS THROUGH ENGAGING 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 signif-

WITH REAL-WORLD icant result. PROBLEMS.” 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. IT IS IMPORTANT TO CONTINUE SEEKING FURTHER DEVELOPMENT USING THE DECADES OF NEW RESEARCH AND TECHNOLOGY AVAILABLE.

the basics of calculus reform explained

Instruction centered around verbal, graphical, analytical, and numerical learning Greater use of verbal explanations and visual representations Aims to develop a deeper conceptual understanding of calculus