Category Archives: Calculus

Reasoning abilities and conceptual knowledge needed for learning calculus

This post describes foundational reasoning abilities and mathematical knowledge students need to develop before beginning a course in calculus. Covariational reasoning – this involves recognition of two quantities that are changing together. A student who considers how two quantities in a dynamic situation change together is said to be engaging in covariational reasoning. This is… Read More »

Application of the Discriminant

The discriminant of a quadratic equation, ax2 + bx + c = 0 is D = b2 – 4ac. If D>0, the quadratic equation has two distinct roots; if D<0, then the equation has no real roots; and, if D=0, the we have two equal roots. Let’s apply it in the following problem. What is… Read More »

8 Different Ways to Think of the Derivative

In his paper The Transition to Formal Thinking in Mathematics, David Tall presents W.P. Thurston’s seven different ways to think of the derivative: Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function. Symbolic: the derivative of x^n is nx^n−1, the derivative of sin(x) is cos(x), the… Read More »

Teaching the derivative function without really trying

New mathematical ideas are usually built on another mathematical idea or ideas. Because of this, the teaching of mathematics if it is to make sense to students, should reflect this ‘building on’ process. Students should be able to see how the new idea is connected to what they already know. Good teaching of mathematics also demand… Read More »

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