Assessing Understanding Through Reasoning Books

Sally K. Roberts and Carla Tayeh

March 2010, Volume 15, Issue 7, Page 406
http://www.nctm.org/eresources/view_media.asp?article_id=9178

In this article, teacher candidates are challenged to make a booklet of reasoning and proof which show that they know how to communicate mathematical concepts and ideas to a larger audience. Their instructors do not allow them to say, "You know what I mean" even if it is obvious that they do know what the student means. It is important for teacher candidates to be able to understand a problem at a deeper level than by simply giving the correct answer. Later in the article, middle school students were challenged to use reasoning and proof to solve a problem without a clear-cut answer. Some of the students found an incorrect answer using faulty reasoning. Some of the students came up with the correct answer using good reasoning, but without providing adequate defense or proof to show how they found the answer. Finally, some of the students used good reasoning and proof to find and convey their answers to a wide audience.

I think that reasoning and proof can be the most challenging, yet important, component in mathematics. Because communicating our reasoning takes time and practice, teachers should begin to help students communicate their thoughts at a young age. I hope that I can resist the temptation to nod at a student and say, "Yep. You have the right idea. I know what you are trying to say. Good job." Students need to be given time to sort through their reasoning - to justify their thinking. This is perhaps as important as coming up with the correct answer.