In the first part, we investigate Birman, Ko, and Lee's left canonical form of a braid and give a new diagrammatic approach. We use the left canonical form to characterize almost strongly quasipositive braids. In the second part, we investigate students' confidence in mathematics and problem solving skills. Every math instructor has heard students utter the phrase, "I'm bad at math." This attitude toward mathematics causes students to give up easily when working through a tough problem. With hard work, a positive attitude, and the right supports in place, every student can be successful at mathematics. This project implements new teaching methods to improve students' confidence, attitude, and motivation toward mathematics. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]