Response to Hill, Ball and Schilling

In this article, the authors attempt to develop measures of teachers’ combined knowledge of content and students by writing, piloting, and analyzing results from multiple choice items.

To be honest, this articles’ attempts to conceptualize and measure teacher knowledge did not resonate strongly with me. A 30 page paper and detailed questionnaires allowed the authors to come up with conclusions I consider trite such as “teachers have skills and insights and wisdom beyond that of other mathematically educated adults” (pg. 395) and “Teachers know that students often make certain areas in particular areas or that some topics are likely to be difficult… but teachers often reason about students’ mathematics:  They see student work, hear student statements, and see students solving problems. Teachers must puzzle about what students are doing or thinking, using their own knowledge of the topic and their insights about students.” (p. 396) While these conclusions are valid, they seem quite obvious, and the task of trying to measure and quantify something that so clearly would be extremely different for every teacher seems disingenuous. Certainly, I’m a better Math teacher now than I was in my first year – while my mathematical knowledge has not greatly increased, my knowledge of how students learn, understand, and make errors has increased, allowing me to teach more effectively.

The article also challenges the use of Multiple Choice testing. This has been a regular struggle for me – when teaching in Ontario, my colleagues and I almost never used multiple choice testing. When I wrote my first test for my first job teaching in BC, I showed it to the department head for input, and he said “it looks good, but where’s the multiple choice?”  Having now taught and subbed at multiple schools in Vancouver, I notice that multiple choice testing is the norm, and often can even make up students’ entire grades in some classes. As per the authors, I think it’s important that we “think carefully about the multiple-choice format” (p. 396).  This form of testing can lead or mislead students to correct or incorrect answers, and does not test skills in the same way that a full solution allows us to evaluate process and thinking. I believe the long tradition of provincial exams, along with bigger classes and less prep time encourage multiple choice testing in BC classrooms, but that the Ministry should be more involved with standardizing evaluation so that students are not being primarily evaluated in a multiple choice format.

Question: Do you think that Multiple Choice testing accurately evaluates students?  Do you think the Ministry of Education has a role in dictating the types of assessments in BC classrooms?

Response to Thurston

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? 

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