New blog

This blog has been used for mathematics-related postings. However, I have come to see the scope of this blog as too restrictive, as I also want to post on other subjects concerning teacher education. Instead of starting more blogs on each subject, I will instead start one blog covering all of my work-related activities. This will be called Teacher educator Bjørn. Please come over to my new blog!

Book: Legitimate peripheral participation

Jean Lave and Etienne Wenger: Situated learning - Legitimate peripheral participation. 1991.

In this interesting book, Lave and Wenger builds on five studies of apprenticeship situations (midwives, tailors, quartermasters, butchers and nondrinking alcoholics, to get to a theory of learning. The concept they create and see as key, is (as the title suggests) "legitimate peripheral participation". This has to do with learning by participating in a social practice. The "peripheral" part has to do with the way a person (for instance an apprentice) can have a small, but still significant, role.

There are at least two ways of reading this book. It could be seen as an attempt to discuss ALL learning. Some would, for instance, argue that what you can learn in a classroom is just how to be a student, and that it can not be transferred into the "real" world. By that kind of thinking, it would of course be hard to admit that I could possibly have learnt something from reading a book. More reasonably, I would like to see the book as an attempt to point out some features of ONE KIND OF learning.

Anyway, it is interesting to view the teacher education I am a part of from the point of view offered by this book. For instance, teacher students probably spend as much effort trying to learn the language neccessary to pass our exams (such as "zone of proximal development" or "cognitive conflict") as they do to learn the actual use of the concepts in everyday teaching.

A quick reading of this book of course did not make me fully grasp their ideas, but at least it is a point of departure for later readings.

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.

What happens when you add both the numerator and denominator

In an article (Rowland et al) is mentioned a kid who conjectures that the fraction you get by adding the numerators and denominators will be between the two original fractions. Example: you get a fraction between 1/2 and 3/4 by calculating (1+3)/(2+4) = 4/6 = 2/3.

This works, of course. One explanation I like is the following: Let's say that 2 persons together own 1 dollar and 4 other persons together own 3 dollars. If they decide to pool their resources, 6 persons will own 4 dollars. It's common sense that not all of them are now richer and not all of them are now poorer.

Another explanation concerns vectors. (Generally it's a terribly bad idea to illustrate fractions with vectors, but in this case it actually works.) If you add the vectors [2,1] and [4,3], you get the vector [2+4,1+3] = [6,4]. Obviously, the slope of the vector [6,4] (which is 2/3) is somewhere between the slopes of the two addend vectors (which are 1/2 and 3/4).

For some reason, some of my colleagues and I started to look at problems associated with this, such as: When is (a+c)/(b+d) exactly halfway between a/b and c/d? (Answer: only when b=d).

In a way, this whole thing illustrates why teachers need skills that other professions don't. While many professions need mathematics to get the right answer, mathematics teachers need to be able to see whether the solution a student proposes is actually correct. That may be with methods within the students' reach or with other methods.

In addition, the teacher of course needs to decide whether he/she should follow up on this - that is, whether the student will learn anything from going deeper into the problem. That is another (but related) kind of skill.

All of this relates to an article by Deborah Ball that I will blog about later...

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!