The case of Talia and the moment of a force

Talia is also an experienced mathematics teacher and had been working with a class of 16- and 17-year-olds on the moment of a force about a point. Talia was seeking to recap a general method for calculating a moment and Figure 7.5 is the publicly shared figure that was drawn during turn 9 of Extract 3, although the points А, В, C and D have only been introduced here in order to bring greater clarity to the analysis and were not labelled during the lesson. Line segments AD and CD have also been added, as they were during the course of the episode.

The length of line segment BC represents the magnitude of the force in this diagram. AD is perpendicular to BD.There are two ways to calculate the moment of the force acting along BC about the point A.The first is to multiply the magnitude of the force by the length AD and this is the approach privileged by Talia in this episode.The second is to calculate the component of the force acting perpendicular to AB and multiply this by the length AB. This is an approach suggested explicitly by one of the students.

The episode has been parsed into three extracts. Extract 3 starts with Talia’s first initiation turn of the episode. Largely the sequence of turns that follow alternate between Sacha and Talia, but Sarah also contributes two turns. Talia does not comment directly after Sarah’s turns so for the purposes of this analysis instances of Sacha-Sarah-Sacha-Talia turns are treated as a single couplet.

Extract 3. Sacha and Talia co-modelling the action of a wrench

1 TALIA: So if we actually go back to what you were talking about when you

calculate the moment and the turning effect. Can you say again what you said then?

2 SACHA: It’s the distance ... the perpendicular distance to the pivot or the

point. So, um ... for example if you have like a wrench you would measure from the end point to the middle and then depending on angle you would take the force out as well - that would change the moment. So the force is being applied perpendicularly as well, at 90 degrees, and it’s the furthest away from the ... so...

3 TALIA: So here’s my little nut and I’ve put my wrench around it ... (drawing on board) and I’m putting my force ... and you’re saying I could apply my force like this or like this or like this.

4 SACHA: Yeah. And then you would measure ... like rather than measuring the

length of the thing, you would measure the perpendicular distance thing like horizontally...

• 5 SARAH: Between the pivot and the point.
• 6 SACHA: ... and the point at which the force is being applied.
• 7TALIA: So here ... first of all, do we care about the width of the actual nut?
• 8 SACHA: No.
• 9TALIA: So we’re going to reduce that just down to a particle. And do I care

about kind of the thickness of the actual wrench? So let’s just turn this into a line. And then you’re saying at this point here I want to measure the distance from there to there.

• 10 SACHA: But you want to measure it like...
• 11 SARAH: Like a triangle at the bottom.
• 12 SACHA: ... it’s like the bottom triangle from the point. So you wouldn’t measure the length of the actual wrench, you’d measure like the distance from here to under the point, if you know what I mean.
• 13 TALI A: So I’m going to draw a vertical line down here and measure that distance there.
• 14 SACHA: Believe so, yeah.
• 15 TALI A: Even if the force is acting in this direction.
• 16 SACHA: No. If it’s not...
• 17 TALIA: Hang on ... Simeon?

Sacha’s opening statement (turn 2) includes a mix of technical language such as ‘perpendicular distance to the pivot’ and phrases such as ‘take the force out’ and ‘being applied perpendicularly’. But it is the material world exemplification of the wrench that is problematised by Talia in turn 3 in that a diagram is drawn in order to make sense of what Sacha has said.The diagram drawn during turn 3 evolves into Figure 7.1 in turn 9 but the ‘nut’ is at A and the ‘wrench’ is along AB but extends beyond B. In turn 2, Sacha explains that the force is not applied at the far end of the wrench (beyond B) in this context but in ‘the middle’. Whether Sacha meant strictly in the middle of the wrench or any point along its length other than one of the ends is unclear, but the point where the force is applied corresponds to point В in Figure 7.5.

Flaving constructed a publicly shared figure, Talia problematises again in turn

9 Sacha’s statement from turn 2 by reinterpreting it with reference to Figure 7.5. Talia is referencing AB when saying ‘the distance from there to there.’ This is neither the required length using the method Talia privileges in this episode, nor is it the length Sacha and Sarah have in mind as they try to explain further in turns 10 to 13 using informal language. Again Talia does not problematise their use of vocabulary but interprets what has been said referencing Figure 7.5. She constructs a line through C perpendicular to AB, which she describes as vertical, although this is a property of the particular figure rather than a label that also applies in the general case. Sacha ‘believes’ (turn 14) that the length of the line segment between A and the intersection of the line Talia drew with AB is the required length. Talia problematises by suggesting a further example where the force is orientated differently (turn 15) and Sacha acknowledges in turn 16 that his method does not apply to this further case. At this point Talia invites a different student, Simeon, to speak (turn 17). Extract 4 is the sequence of turns between Simeon and Talia that follow.