Response to On Proof
and Progress by William P Thurston
Thurston wrote this article as a response to another (by
Jaffe and Quinn). In it, he discusses some of the positive aspects of the
article and criticizes some of the assumptions the authors of the article made.
In particular, he focuses on the questions of: 1) What is it that
mathematicians accomplish? 2) How do people understand mathematics? 3) How is
mathematical understanding communicated? 4) What is a proof? And 5) What
motivates people to do mathematics. After he considers different aspects of how
mathematicians understand these important questions, he discusses some personal
experiences that highlight how he gained his perspectives.
In particular, I was interested in Thurston’s discussions
about how the social interactions mathematicians partake in through their
research motivate them to continue researching particular aspects of the discipline.
He states: “most mathematicians don’t like to be lonely, and they have trouble
staying excited about a subject, even if they are personally making progress,
unless they have colleagues who share their excitement.” (Thurston, 48)
This made me consider how we teach math in schools,
particularly secondary schools. As featured in the video we watched in class
last week, math classes are often one of the last vestiges of the classroom
where students sit in rows in individual desks, and spend a great deal of time expected
to be silent either listening to the teacher explain concepts or trying concepts.
Group work and discussion regarding approaches to solving problems are often
discouraged in math classes, and are rarely evaluated, and students often feel
that if they share answers or approaches, they are cheating. Most, if not all
of students evaluations come from work which is done individually, and often
under pressure. This is in contrast to how mathematicians work in collaborative
environments, recognizing that colleagues can play a key role in seeing
problems in different ways and helping to maintain motivation and gain
understanding.
I try to address this in my class by encouraging group and
partner work while working on concepts, and providing assessments in class
where students are encouraged to work together and yet still receive marks for
their work. Yet, I have been criticized by colleagues for this approach, as they
feel it does not accurately reflect student abilities, and some students will
simply copy the answers of their classmates.
Question: Do you think it’s appropriate for student
evaluation in secondary school to be partly made up of work that is done in
collaboration?
Thanks for sharing your thoughts! I agree that it is interesting to learn that the social interactions mathematicians partake in through their research motivate them to continue researching - too often, the image we have in mind of mathematicians is that they are stand-alone advocates and experts of their fields, possibly further accentuated because as the article suggests, the mode of communication or language that many mathematicians tend to use is one that may be difficult to understand or relate to, if one is not familiar with the field concerned.
ReplyDeleteWhether a mathematician, a mathematics educator, or any educator for that matter, communication is an integral point of concern. I agree with the article on the need to pay more attention to communicating not only definitions, theorems and proofs, but also our ways of thinking. This is especially pertinent for conveying ideas to people who do not already know them, applicable both to the layman and also to our students. However, I recognise the difficulty involved in translating the encoding in one's own thinking, to something that can be conveyed to someone else - this will undoubtedly involve much time and effort dedicated to this purpose and process of translating thinking to communication to understanding. Nonetheless, it is heartening to know that, as the article suggests, it begins with the intent that more than the knowledge, people want personal understanding. It remains for us as mathematics educators to explore how we can apply this in our respective roles and contexts.
As Thurston challenges at the end of the article, have our actions done well in stimulating mathematics? This may be one of the essential questions to ask ourselves. Although a broad or indirect way of responding to the question you raised, I essentially believe that as educators, if we are clear about our intent, and are able to objectively justify how our actions serve to support our intent, that may be the most we can do, despite the views of others. So yes, I believe that there is room for student evaluation to be partly made up of work that is done in collaboration - after all, this means is crafted with the intent of encouraging deeper student understanding and thinking, and hopefully stimulating greater interest in the subject when challenges are collectively overcome in groups. Will there be loopholes where such a system may be exploited? As with all other means and systems, there is no full-proof solution. As a professional educator, we must be able to weigh the choice of this means over other means, and see how it may complement the rest of the package, so as to achieve the overall intent that we have set out to achieve for our students in our particular context.