Guest post by: Bryan Penfound

Bryan Penfound is an instructor at the University of Winnipeg whose focus is teaching pre-service teachers mathematics content knowledge. He works jointly for the Department of Mathematics and the Faculty of Education. This post is the third in a three-part series on teacher training. The first, found here, explored evidence-based practices in teacher training programs. The second, found here, offered suggestions on entrance requirements and streamlining certification for teacher training programs in mathematics education.  In this post, Bryan discusses how re-structuring mathematics content and pedagogical content courses to include deep content knowledge and evidence-based practices may be of benefit for our future teachers.

HOW MATHEMATICS DEPARTMENTS CAN HELP: MATHEMATICS CONTENT COURSES

What can Mathematics and Education Departments do to help teachers be prepared to teach well as soon as they start working in the classroom? In my previous post, I suggested that teachers should be certified into one of three streams (Primary, Junior and Intermediate) because each stream requires quite different content knowledge. Assuming this certification structure,  Mathematics Departments could determine what content knowledge was important to each stream, and how to ensure that future educators connect deeply with this content. I have suggested some topics on the diagram below (of course this is not meant to be an extensive list).

Yes, I am suggesting that future teachers re-connect with the mathematical content that they will be using in the future. I am not convinced that K-2 teachers need to take Calculus at the university/college level, for example (if they want to, we shouldn't dissuade them, though). These individuals may not even have the required prerequisite for many university/college level courses in mathematics! So care needs to be taken to develop a content course that is both relevant and challenging at their certification level. This course could potentially draw on ideas of Calculus to help explain concepts, but needs to be done in a way that is accessible to our pre-service teachers, and helps them understand the mathematical concepts at a higher level than would be expected from their future students (for example, see Appendix I in the NCTQ No Common Denominator Report).

The University of Winnipeg has developed two math content courses that are meant to serve as degree distribution requirements for Early/Middle Years teachers. These courses take into account that certified teachers in Manitoba, Canada may be teaching anywhere from K-8, so the topics are distributed across the full spectrum of curriculum content. Mathematical content courses such as these may be a good place to start in designing more narrow-focused content courses should certification become more streamlined.

HOW EDUCATION DEPARTMENTS CAN HELP: PEDAGOGICAL CONTENT COURSES

Our Education Departments need to pick up exactly where the mathematical content courses leave off. Ideally, now that pre-service teachers have deep content knowledge in mathematics, pedagogical content courses need to develop ideas that are pertinent to teaching mathematics. This course should not be a course in preaching about traditional versus constructivist methods; strong educators that understand each method has its time and place in the context of learning would be the ideal candidates as instructors of such a course. Ideally, a pedagogical content course would cover topics such as presenting mathematical ideas in a clear and concise way, selection of purposeful representations, understanding and directing common student misconceptions, sequencing of mathematical ideas, modifying tasks to make them easier or more challenging, and evaluating mathematical explanations and notation. This set of topics, taken from this article by Deborah Ball, is not intended to be an exhaustive list, but a place to start conversation. It is important to include in-service teachers in this conversation, as they often have specific ideas that would be beneficial to heed when it comes to training our future teachers.

In addition, a good pedagogical content course might cover the cognitive processes behind learning; or, at least, the direct applications of cognitive psychology to education. Teachers could make more informed choices about how to present information to students if they received instruction on the basics of learning, including but not limited to: the lack of evidence for learning styles; the benefits of spacing practice over time; different ways of implementing retrieval practice; and how to train students to self-explain while problem-solving. Perhaps this could even be its own stand-alone course!

As I did some searching through some highly respected education programs in the country, I noticed that one of them did not have a specific mathematics pedagogy requirement. Rather, what was offered was a 500-level course in general science that doesn't even mention mathematics at all and looks like a vested-interest course (mention of inquiry-based methodology only and 21st century society claims). Also, although there are pedagogy courses that mention “learning” in the title, a quick scan of the course descriptions reveals that none of them actually address the cognitive mechanisms according to which learning happens! Instead, “learning” is discussed in a much more abstract and philosophical manner – for example, referring somewhat cryptically to “the ways in which teachers work within the parameters of prescribed curricula and assessment requirements”, and “current realities in society”.

Will those taking these courses realistically be prepared to meet the needs of their future mathematics students? Or would a more targeted approach that combines solid mathematical content with practical pedagogical strategies be more helpful? I would love to hear your thoughts.