Article: Rowland, Huckstep and Thwaites

Tim Rowland, Peter Huckstep and Anne Thwaites: Elementary teachers' mathematics subject knowledge: the knowledge quartet and the case of Naomi

This article is a precursor to the article Turner/Rowland treated a previous post.

I've often discussed with Norwegian mathematics colleagues in teacher education the problem of how students are supervised in their school-based placements. Too often, the discussions with students concern only "administrative" and pedagogical issues, and too little attention is given to the mathematical parts of the lesson. This article is an attempt to help this situation.

The point of the research was to "develop an empirically based conceptual framework for the discussion of mathematics content knowledge, between teacher educators, trainees and teacher-mentors, in the context of school-based placements". The "Knowledge Quartet" is the result:
- Foundations (Propositional knowledge and beliefs)
- Transformations (How the content knowledge is transformed "into forms that are pedagogically powerful" (Shulman))
- Connection (between different parts of a lesson, between lessons or between different parts of the curriculum)
- Contingency (How "to respond to children's ideas")

The authors cite Ma citing Duckworth that intellectual "depth" and "breadth" "is a matter of making connections". This quite deep insight suddenly got very visual for me: isolated points of learning can hardly be have "depth" or "breadth". (My visual image/metaphor soon breaks down, however, so I think I'll stop thinking about it...)

The second part of the lesson, discussing Naomi's lesson, showed how her lesson could be analysed from the four "points of view" given above.

The article is highly recommended.

Just an old sweet song keeps Georgia on my mind

Did you know that the official song of the state of Georgia, US is "Georgia on my mind"?

Actually, that song is the only thing I associate with Georgia. (Except, when realizing that the state capital is Atlanta, I suddenly also associate it with Coke and CNN, as well as olympics and a bomb.)

Why do I suddenly care about Georgia? Because I'm going there in June - for the InSITE 2009 conference, presenting a paper with a colleague.

Good ideas for things to see and do in Georgia are welcome!

Article: Turner/Rowland

Fay Turner and Tim Rowland: The knowledge quartet: a means of developing and deepening mathematical knowledge in teaching?

This is a exemplary article. It gives an overview of a theoretical framework and at the same time shows how this framework can be used in practice to provide "discussion points" in teacher education. This is so refreshing, far too many writers leave it to the reader to find practical consequences of the theories.

The Knowledge Quartet is a way of looking at the knowledge that a teacher needs to teach mathematics:
- foundation (what the teacher learned "in the academy")
- transformation ("knowledge-in-action", planning the teaching etc)
- connection (for instance the sequencing of material)
- contingency (the ability to change plans on the go)

I'm about to read another article about The Knowledge Quartet now, which will probably give me more insight into its use.

Article: Lee S. Shulman

For some reason, I haven't read anything by Lee Shulman before reading his short 1985 article "Those Who Understand: Knowledge Growth in Teaching" today. This does not mean that his thoughts haven't influenced me, however - last year I even wrote a short paper (for a course I'm taking) partly based on some of the ideas.

The main point in this article is perhaps distinguishing "among three categories of content knowledge":
(a) subject matter content knowledge
(b) pedagogical content knowledge
(c) curricular knowledge

There is a tension even in the Norwegian mathematics education field between people focussing on the mathematics and other focussing on the pedagogy. Shulman's categories introduced more nuances into that discussion.

An example of "pedagogical content knowledge" that I teach in my courses in mathematics in teacher education is students' typical misconceptions and how to overcome them. Another example is which bits of history of mathematics are useful to motivate students in particular parts of the curriculum. Examples of particular methods of teaching should also be included here.

"Curricular knowledge" includes both knowledge of different materials available (including digitally) and when they can usefully be used. Knowledge of what goes on in other subjects in the same grade or in mathematics in other grades are also included.

These three categories will probably prove to be useful for forming the discussions we will have now that the curriculum (in teacher education) will be changed again.

Shulman's article is written as an answer to George Bernard Shaw's "He who can, does. He who cannot, teaches." Shulman's version is: "Those who can, do. Those who understand, teach."