Disconnected Mathematics: The Whole is Not Greater than the Sum of its Parts

Reflections of a Second-career Math Teacher

One of my calculus periods continues to struggle mightily while learning new concepts, especially if they need to recall prior knowledge, or worse connect it to the new material.  As an example, when starting to explore integration concepts, students learn to use finite methods to estimate areas under functions, as early mathematicians discovered.

The basic technique uses multiple rectangles to approximate the area under the curve of a function over the interval x = a to x = b as the figure below shows.  Since the area of a rectangle is equal to its base times its height, we simply need to determine a specific value of f(x) for each rectangle used, where the number of rectangles is driven by n.  In the figure below, n varies from 2 to 4 to 8.  Note that Δx = (b-a) / n, which determines each specific x

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About Dave aka Mr. Math Teacher

Independent consultant and junior college adjunct instructor. Former secondary math teacher who taught math intervention, algebra 1, geometry, accelerated algebra 2, precalculus, honors precalculus, AP Calculus AB, and AP Statistics. Prior to teaching, I spent 25 years in high tech in engineering, marketing, sales and business development roles in the satellite communications, GPS, semiconductor, and wireless industries. I am awed by the potential in our nation's youth and I hope to instill in them the passion to improve our world at local, state, national, and global levels.
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