Determinación del tamaño y capacidad del proyecto.

We identified 60 episodes in which the teacher produces a statement. These, however, are not all concerned with moving the lesson forward: some display expertise in other areas than the subject itself. Consistently with the focus of this chapter, we provide finer differentiation with the aims of identifying those episodes where: (a) the expertise displayed is mathematical or scientific, and (b) it is displayed by the teacher. This leads to the following four types of teacher statements, displayed in Table 6.1 below:

Subject expertise Different expertise

Teacher authority New elements

Non-mathematical (resp. non-scientific) statements

Other authority Reminders Summarising statements

Table 6.1 – Types of teacher statements

“New elements”, which feature the teacher displaying their own mathematical or scientific expertise, constitute our main interest, and will be

analysed in detail over the next sections. We now give a brief overview of the other types of statements.

Non-mathematical or non-scientific statements include descriptions of what a PhD is (to provide context for the research), statements about a road accident (given as an example in Excerpt 6.1 below) and general statements of fact (in particular, “we are not in America”). Organisational elements, for instance the title of a chapter or of a

competence are considered to belong to this category: through such statements, the teacher is displaying expertise as a teaching professional, but not as a subject specialist.

1 S1: “[S2] il vient pas aujourd'hui il s'est fait écraser”

“[S2] is not coming today, he got run over”

2 T: “Oui je [???] écraser. C'est pas grave. Il a eu un souci, oui j'étais la hier au collège quand ca s'est passé”

“Yes I [???] run over. It’s not serious. He’s had a worry, yes I was there yesterday at school when it happened”

Excerpt 6.1 (Varos Hill, second mathematics lesson)

Summarising statements start with a logical conjunctions (“So” or “Donc”) and summarise the work carried out directly before the episode in question23. In mathematics, they tend to imply a

23 _{It should be noted that the simple occurrence of the word “so” does not imply that what follows is}

a summary or a conclusion. Indeed, it is frequent for the teacher to use “so” as a synonym for “let’s move on”.

Page 165

generalisation of properties seen on examples. In such episodes, the teacher is displaying expertise in drawing conclusions or generalising such work, rather than content. The statement which is offered is itself the conclusion of a series of observations involving either the students (through IRE/IRF sequences) or, in physics at Skaro Motte, the physical environment. We give Excerpt 6.2 below as an example of such a statement, which takes place after an activity where students are asked to sort decimal numbers:

3 T: “So. All you have to do is go along, and you find the first column, where they're different. So you start by looking at this column, but they're all the same. So that's [no?] good. We need to look at the next column. Look at the next column. That's a zero, that's a one, so this one is bigger. Okay? And you don't care about the ones after the one. As soon as you get to that one, can you see this is zero, you say okay I know this is bigger. You look at this two. That one's even bigger still. The five. And the two five, and the and the four make no difference to that. Okay? Everyone happy with that? I bet you're all happy with that, cos you've used decimals since you were, very little, and you've seen pounds and pence. Yeah?”

Here, the teacher draws on common knowledge existing outside of school (“pounds and pence”), making it clear that he expects students to know the process for comparing decimal numbers. To that extent, it is also a reminder. However, this expectation is only made clear after the process is stated and after the teacher has checked student understanding, as though the link with everyday life were only an afterthought. It seems, rather, that the statement of the process for the comparison of decimal numbers is motivated by the students’ activities on specific examples directly before this excerpt. The link between these activities and the statement is made evident by the use of “So” at the start of the teacher’s turn. In particular, the teacher is using the students’ previously displayed knowledge as a source of authority. Therefore, the teacher is not displaying his own (factual) mathematical expertise; rather, he is an expert at description and generalisation.

Reminders are statements which, in their very presentation or context, carry the implication that the students already know what is being said, and which may later be used to support an argument. They are generally denoted by either the use of a tag question, or by a concise, official, rule-like formulation. It should be noted that the occurrence of an equivalent statement can confirm that students are expected to know this, it is not sufficient on its own to declare that a statement is a reminder. We give two examples (from mathematics) in Excerpts 6.3 and 6.4 below.

Page 167

4 T: “But. To keep fractions equivalent, whatever you multiply the bottom by you gotta multiply the top by the same thing. Yeah?”

Excerpt 6.3 (Gallifrey Vale, first mathematics lesson)

The tag question “Yeah?” at the end of this excerpt is seeking student approval, suggesting that they should agree or understand the preceding statement. In the case of this excerpt, the statement is not the logical conclusion of a worked argument: rather, it is a tool which the teacher wants to use for a worked example. This rupture with the flow of the argument is indicated by the “But”. Furthermore, “equivalent fractions” is used as a keyword in the rest of the lesson (including instructions mentioning “using equivalent fractions”), which confirms that this statement is, in fact, a reminder.

5 T: “La... Multiplication est prioritaire. Donc d'abord on effectue trois quarts fois un demi.”

“The… multiplication has priority. So first we do three quarters times one half”

Excerpt 6.4 (Varos Hill, fourth mathematics lesson)

Here, we can say that this is a reminder on two bases: firstly, notions of priority are not self-explanatory. In particular, there are no indications of what multiplication has priority over. This and the concise formulation heavily suggest that the teacher is quoting from an arsenal of rules the students should know. Other surrounding episodes (including student interrogation) confirm that the

students indeed know this rule. Secondly, it is introduced in order to be used on an exercise; rather than for its own sake.

Table 6.2 below gives the relative levels of occurrence of each type of statement:

Mathematics Science/Physics Total

GV VH GV SM Number of statements 11 20 4 25 60 Non-mathematical 3 5 1 6 15 Reminder 4 6 1 6 17 Summarising 3 6 0 5 14 New elements 3 4 2 14 23

Table 6.2 – Occurrence of each type of statement, by class

Thus, there are only 23 episodes (out of the total 1,291) where the teacher is the expert in charge of the introduction of new elements. It should be noted that the relative scarcity of summaries and reminders in Table 6.2 above is linked to the fact that some such utterances were not generally coded as statements (but, for instance, validating, references to the past, etc.), and therefore were not considered for this chapter.

Page 169