1. Reading Data The role of theory Stephen Lerman

3. Looking at ‘reading’ in relation to research What do data ‘mean’? Which tools are to be used to interpret data? Need to be –Systematic –Transparent –Replicable (recognition and realisation)

4. We will look first at a transcript of two boys working together on a problem of simplifying ratios. The teacher told us that M is more able than D, and that she likes more able students to help the less able. Previously the students had been shown cancelling and also substitution of numbers for letters

5. What would you say is happening here, what story can you tell of this problem- solving incident? Consider what grounds (justification) you are drawing upon to tell your story.

6. Simplify: (f)ab : ab =......... 1.M: What? Equals ab? [pause, D looks on M's page] Equals ab? 2.D: Yeah. 3.M: No, it equals one. 4.D: Wait a second... 5.M: 'Cause one, [punching calculator buttons] twelve times tw... no. One, look, look, look. One times two, divide one times two...it shouldn't equal four. [M appears to be substituting the values one and two for a and b] 6.D: [laughs]

7. 7.M: Um, yeah, it's, 'cause I'm doing [punching buttons] one times two, divide one times two, equals one. 8.D: So that's cancelled. The two b's are cancelled out. 9.M: Equals one. 10.D: Right? The two b's are cancelled out. 11.M: Hey, where'd my pen go? No come on, look, look, look, look. You've got to do BODMAS. Watch, watch, watch, watch. [punching buttons] One times two, divide one...come on, one times two. That's stuffed up. [with emphasis] One. 12.D:... I'm going to... this is... better.... 13.M: Look, look, look, look at this one, look at this one.

8. 14.D:... Hang on... 15.M: Divide. 16.D:... I'm going to do these, this one first. 17.M: Equals 1, it does equal 1. I've got to do this first.

9. Another example (Morgan, 1998; Morgan, Tsatsaroni & Lerman, 2002): Teachers evaluating students’ writing of mathematical investigations. The data: interviews with teachers

10. Investigate the relationship between the dimensions of a trapezium and the number of unit triangles it contains.

11. .. but again even that’s not, I mean he’s given.. one thing that I think they have to do is when they give a formula they should explain it using quite a few examples and show how it works. The thing that I always look for and I say to the kids is: you write it up as if you’re writing it for somebody who’s never seen this problem, […] I don’t think it’s clear enough for somebody to use it and then work out, I mean he hasn’t done even one example of how it works.

12. This is a major problem because he’s got these results but unless one is there in the class and you’re a teacher you don’t know whether this is his results or somebody else’s. He hasn’t shown any diagrams or where these results have come from.

13.  he’s also looked at the difference between each of the different piles. Now that straightaway will show him that there’s a pattern there as well as the initial pattern. So that’s something that would may come to when he’s investigating later. Okay, so he’s recognised that there is a pattern […] there is a limitation because he’s only gone up to ten units as the base; so that is something that he’s considered that it may be just for this number of units. So making predictions over a hundred or so base units may be something he could mention as well.

14.  it does say explain your working and it’s true that the candidate has got the answer and hasn’t just written down the answer and the explanation I find acceptable here. It’s done just as a mathematical explanation. I think sometimes that word ‘explain’ causes problems. [  ] So the fact that this student has used a simple calculation and left it at that actually at this stage makes me into an even more positive frame towards them because they see that a mathematical calculation can be sufficient explanation.

15. The formula is accurate, needs a bracket in it, but it’s quite clear that his intention and he’s given a nice example which clarifies his thinking, so although algebraically it’s not that strictly correct, it’s quite clear he knows what he’s doing.

16. I wonder, the fact that he’s drawn that dotted line across the middle makes me think he was looking at it in terms of two trapeziums but he hasn’t said that here [  ] so that seems like a very sensible idea.

17. Morgan (1998) Using critical discourse analysis, Morgan focused on linguistic features of teachers’ interviews to identify their positioning in discourses of assessment.

18. examiner, using externally determined criteria examiner, setting and using her own criteria teacher/advocate, looking for opportunities to give credit to students teacher/adviser, suggesting ways of meeting the criteria

19. teacher/pedagogue, suggesting ways students might improve their perceived levels of mathematical competence imaginary naïve reader interested mathematician interviewee

20. Morgan, Tsatsaroni & Lerman, (2002) Taking another look: developing a model based on two dimensions of voice and forms of practice, elaborated by two other dimensions, specialised/localised and focus on absence/presence

21. Four positions emerge (in place of 8): Examiner: using externally determined criteria Examiner: setting his/her own (professional) criteria Teacher-adviser Teacher-advocate

22. Different theoretical lenses, different sets of research tools, provide different ways of reading and writing research.