# Teaching and Learning Theorems

Theorems are carefully worded statements about mathematical facts that have been proved to be true. Important (and some not so important) ideas in calculus and all of mathematics are summarized as theorems. When you come across a theorem you need to understand it; the author of your textbook would not have included it and the…

Today we continue to look at some previous posts that I hope will help you and your students throughout the year. We begin with some posts on graphing calculator use and then a few general things in three posts on beginning the year, followed by some mathematics I hope students know before they start studying…

# Proof

When math books present a theorem they almost always immediately present its proof. I tend to skip the proofs. I assume they are correct. I want to get on with the ideas in the text. Later I may come back and read through them. Is this a good thing to advise students to do? I…

# Theorems and Axioms

Continuing with some thoughts on helping students read math books, we will now look at the main things we find in them in addition to definitions which we discussed previously: theorems and axioms. An implication is a sentence in the form IF (one or more things are true), THEN (something else is true). The IF part…

# Definitions

Definitions are similar to theorems, but are true in both directions; technically, this means that the statement and its converse are both true (). The double arrow is read “if, and only if.” Both parts are either true or both parts are false. Definitions usually name some thing or some property.  Definitions are not proved.…

# Theorems

Theorems are statements that summarize the results that are true in mathematics. Theorems are statements that have been proved true; but the emphasis in AP Calculus is not on proof. Rather, it is on what the theorems mean and how to use them. Theorems have two parts: the “if …” clause called the hypothesis and…