livingcalculus.com

Tag: Rigor

  • Introducing the Power Rule

    The power rule poses an exceptional opportunity for students to explore a topic beginning with inductive pattern finding rather than deductive proof. I often hear math teachers talking about how finding patterns is an essential part of math. Which it is! But how often do we model this for our students? All too often, teachers present a formula (like the power rule) before giving students a chance to think and spot patterns for themselves. If we want students to see that spotting patterns is essential to mathematics, then this needs to be a core part of our teaching practice.

    After doing some work with numerical derivatives, I introduce calculus with limits by showing my students this picture and asking them to approximate the slope of this graph at x=3.

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  • On Teaching the Mean Value Theorem (part 1)

    The Mean Value Theorem (MVT) is a central piece of a first class in calculus. However, the role it plays in the class is often not clearly defined. In many classes and books, students are told that the MVT is the “backbone of calculus,” or some such thing: it is the foundational result on which many important things are built. From the standpoint of analysis, this is a reasonable claim: the MVT and extensions of it are essential to proving everything from the Increasing Function Theorem (IFT) to some of the more general versions of L’hospital’s rule. But how does this come across to a typical calculus student? Such a student has not yet taken (and may never take) a course in analysis and lacks the context to see why the MVT is so manifestly important.

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