Article: Shulman 1987

Lee S. Shulman: Knowledge and Teaching: Foundations of the New Reform, Harvard Educational Review Vol. 57, No. 1, February 1987

As in his 1985 article, Shulman here is on a crusade to change the discussion of teacher knowledge, which he thought was too little concerned with "comprehension and reasoning, transformation and reflection". Instead of only looking at "the management of students in classrooms" he is concerned with "the management of ideas within classroom discourse".

A powerful example of the importance of content knowledge is Colleen, a teacher student with a master's degree in English. The contrast between her confident and highly interactive teaching of literature and her lack of confidence and her "didactic" style when teaching grammar, is highly illustrative. (The example is actually from Grossman (1985).)

Shulman tries to show what are the sources of the knowledge base for teaching. One of the headings is "Wisdom of practice", and he mentions one of the main problems with teaching as a profession, in my view - I will quote the paragraph in full:

One of the frustrations of teaching as an occupation and profession is its extensive individual and collective amnesia, the consistency with which the best creations of its practitioners are lost to both contemporary and future peers. Unlike fields such as architecture (which preserves its creations in both plans and edifices), law (which builds a case literature of opinions and interpretations), medicine (with its records and case studies), and even unlike chess, bridge or ballet (with their traditions of preserving both memorable games and choreographed performances through inventive forms of notation and recording), teaching is conducted without an audience of peers. It is devoid of a history of practice.

Article: Ball, Thames, Phelps

Ball, Thames, Phelps: Content Knowledge for Teaching - What Makes It Special?

What I particularly like about this article, is how convincingly it shows that there exists some "special" mathematical knowledge that teachers have (and need) that other professions do not. In discussions about teacher education, people often say that mathematics teachers need to know "the mathematics" and they need to now pedagogy, and often "the mathematics" is thought of as the same mathematics as their pupils are to learn, only more.

The authors gives the example that many professions need to be able to subtract two numbers, and some professions need to be able to check a subtraction to see if the answer is correct. However, only teachers need to be able to see or investigate whether the method used is valid in general or to pinpoint the error in the algorithm.

This is *not* pedagogical content knowledge, because this is not, in itself, concerned with students or learning. It is pure mathematics, but mathematics of a different kind than what is normally taught in universities. In addition to this (which the authors call "specialized content knowledge") comes pedagogical content knowledge, which is concerned with students and learning - for instance knowledge of which typical misconceptions students may have, knowledge of different ways of representing a certain mathematical idea, knowledge of how to choose examples that provide exactly the right progression etc.

I think the concepts "common content knowledge", "specialized content knowledge" and "pedagogical content knowledge" will be useful in further discussions on what we should teach in our teacher education courses.

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.

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."