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...