College students struggle with the switch from thinking of
mathematics as a calculation based subject to a problem solving
based subject. This book describes how the introduction to proofs
course can be taught in a way that gently introduces students to
this new way of thinking. This introduction utilizes recent
research in neuroscience regarding how the brain learns best.
Rather than jumping right into proofs, students are first taught
how to change their mindset about learning, how to persevere
through difficult problems, how to work successfully in a group,
and how to reflect on their learning. With these tools in place,
students then learn logic and problem solving as a further
foundation.Next various proof techniques such as direct proofs,
proof by contraposition, proof by contradiction, and mathematical
induction are introduced. These proof techniques are introduced
using the context of number theory. The last chapter uses Calculus
as a way for students to apply the proof techniques they have
learned.
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