Many college math and science students have difficulty developing and evaluating mathematical arguments and proofs. As a result, a number of colleges and universities have instituted introduction to proof transition courses to help prepare these students for more theoretical upper level math courses, but with limited success. Building on prior work, this project continues to develop a library of high quality videos of students constructing proofs that are appropriate for such a transition course, and develops methods for using these videos to help students strengthen their abilities at solving problems and constructing proofs. These videos are used for both professional development and to support student learning.
Using guided discussions with faculty, the videos help mathematics teachers develop insights into some of the mysteries of students' mathematical thinking, enabling them to hone their pedagogical approaches. For example, the videos have helped some faculty develop insight into three critical moments that play an important role in students' proof production: (1) finding a key idea that gives insight into why the claim is true and that helps guide proof development, (2) finding a technical handle that allows the student to communicate the key idea in a way that is precise and that insures the proof is correct, and (3) integrating all the necessary ideas and steps of the argument and writing a coherent, formal proof.
For student learning, the videos help students develop their reasoning abilities. Particular ways the videos help are: 1) Students learn the value of taking the time to unpack the statement to be proved, of using definitions, of stepping back and considering alternative approaches, of considering why their approach was not working in a particular situation, and of letting that understanding guide the use of other approaches. 2) Students learn to not only begin with examples, but to use examples in ways that generate insight, a key idea, or a technical handle as they learn to ask whether their examples are helping them, and if not, why not. 3) Students practice becoming more reflective on their own thinking. The students in the videos are only slightly more advanced than the students watching, so their thinking is more visible than that of the professor to the viewing students. Viewers are able to compare their ideas with those of the students in the videos, see the thinking of the students in the video unfold, see them make errors, and learn how these errors can be resolved.
