The document discusses observing systems and how we perceive things. It notes that unconscious thought shapes most of our conscious thought and that categorization is important for survival. The research project aimed to create a school culture where students see a need for algebra to express their ideas through collaboration between teachers and researchers. Over a year, lessons were observed and teachers and students were interviewed to understand how an "algebraic activity community of inquirers" could develop. The findings looked at patterns over time and contingencies between teacher and student behaviors as creativity and complex structures were supported.

1. Observing systems – how do we see what we see? Laurinda Brown University of Bristol Graduate School of Education

2. I KNOW YOU BELIEVE YOU UNDERSTAND WHAT YOU THINK I SAID, BUT I AM NOT SURE, YOU REALISE, THAT WHAT YOU HEARD , IS NOT WHAT I MEANT

3. What takes your attention? - There were some counterexamples to that. Remind me what that is. ~ One that does not fit the conjecture. - OK, Ben has done something very mathematical. He’s gone back and looked again and changed it [the conjecture]. ~ [Later in the same lesson.] All two digit numbers will add up to 99. [David’s conjecture is written on the board.] ~ I’ve got another counterexample to Ben’s. - This is how mathematicians work; are there counterexamples? Are two conjectures actually linked and so on.

4. Bigger questions, questions with more than one answer, questions without an answer are harder to cope with in silence. Once asked they do not evaporate and leave the mind to its serener musings. Once asked they gain dimension and texture, trip you on the stairs, wake you at night-time. (Jeanette Winterson, Written on the body , Vintage, 1993)

5. What is learning? How do we do it? So, what should I do as a teacher? (particularly teaching mathematics – another thing that keeps me awake nights  and as a mathematics teacher educator)

6. … conscious thought is the tip of an enormous iceberg. It is the rule of thumb among cognitive scientists that unconscious thought is 95 per cent of all thought and that may be a serious underestimate. Moreover, the 95 per cent below the surface of conscious awareness shapes and structures all conscious thought (Lakoff and Johnson, 1999:13)

7. Every living being categorizes. Even the amoeba categorizes the things it encounters into food or non-food, what it moves toward or moves away from ... We have evolved to categorize, if we hadn’t, we would not have survived. Categorization is, for the most part, not a product of conscious reasoning. We categorize as we do because we have the brains and bodies we have and because we interact with the world in the way we do (Lakoff and Johnson, 1999:18).

8. Minds make motions, and they must make them fast - before the predator catches you, before your prey gets away from you. Minds are not disembodied logical reasoning devices (Clark, 1997, p. 1).

9. One can only see what one observes, and one observes only things which are already in the mind. (Alphonse Bertillon, quoted in ‘A Hard Core of Wind’, Stuart Braham)

10. So, how do you see something new?

11. For me – that’s what the research process is for... seeing more and seeing differently

12. If possible, write down the title of your thesis

13. Title Developing algebraic activity in a ‘community of inquirers’

14. Brief Description Within this project researchers will work in collaboration with teachers to create a school mathematics culture in which pupils find a need for algebra to express their mathematical ideas. The results of the study will contribute to our understanding of the teaching and learning of algebra and will provide examples of teacher transformation and related teacher education.

15. Culture as distinct from nature 2) Culture as knowledge 3) Culture as communication 4) Culture as a system of mediation 5) Culture as a system of practices 6) Culture as a system of participation Duranti, Linguistic Anthropology, 1997, CUP (Ch 2)

16. Creating a learning culture As a faculty we try to have a bank of lots of different ideas for different activities and sometimes I try to write some of them down for other people to understand but it doesn’t work so it’s always when you hear someone talking about a topic saying ‘Oh you can do this little game or if you use a pyramid like this – it’s keeping those conversations going and at faculty meetings spending time doing that rather than there’s such and such a task coming up which you can put on a piece of paper and people do understand it. But doing practical activities is difficult to put on a piece of paper or a game or whatever you’re doing. (Interview, Teacher H, 1991/2)

17. Enactivist Methodology Two key features: 1) The importance of working with and from multiple perspectives 2) The creation of models and theories which are good-enough for, not definitively of as a continuing process throughout the research transforming the views of those taking part

18. Four themes: algebraic activity community of inquirers - ‘ becoming a mathematician ’ metacomments and purposes - understanding through writing

19. ‘ ... I have found that what I cannot say quite simply and without recourse to mystic jargon has not become sufficiently clear and concrete even to myself.’ ( Zen in the Art of Archery , Eugen Herrigel)

20. The Middle Way Abstract superordinate teachers’ images of mathematics and teaching Most useful ‘basic-level’ ‘purposes’ distinctions in the world Detail layer interactional behaviours and actions properties

21. What did we do? Research design Practicalities: 4 teachers and 4 researchers Twice a term for a year video-taped lesson Lesson observations at most once a fortnight Teachers interviewed 6 times in the year For each teacher 6 students interviewed 6 times in the year and their written work collected Children wrote about ‘what have I learnt’ at the end of an activity and encouraged to write whilst doing their mathematics.

22. Multiple views One researcher looked at each of algebraic activity; meta-commenting, pupil perspectives; similarities and differences across classroom cultures; similarities and differences in classroom cultures

23. This experiment confirmed our belief that different people reflect upon identical situations diversely. In these joint reflections, the diversity of individual reflections is of much benefit as it shows the participants other perspectives than just their own (Hošpesová et al., 2007). This resulted in the decision to compare our reflections with reflections of other people (teachers, teacher educators, researchers). Jana Macháčková, Group B

24. Good-enough theories - Meetings 6 whole day meetings of teachers and researchers in one year for joint planning, doing mathematics together, joint video reflection Pre- and post- meetings to the day meetings for the ‘researchers’ to share patterns emerging from our individual strands and allow strands to interact

25. And this was all thought through before putting in the bid!

26. What were findings? Patterns over time – being able to go back and look at data from the data set What visitors to the classrooms said Reduced to a ‘story’ – another brief summary of the outcomes...

27. The (A) story! The teachers’ behaviours are contingent on their pupils’ actions and responses ... ... as they work on mathematical activities that allow the pupils’ to explore similarity and difference often through classification. The pupils’ creativity is supported as they are encouraged in the asking of their own questions ... ...the process of working on which leads to the articulation of complex structures and patterns supporting algebraic descriptions.

28. - There were some counterexamples to that. Remind me what that is. ~ One that does not fit the conjecture. - OK, Ben has done something very mathematical. He ’ s gone back and looked again and changed it [the conjecture]. ~ [Later in the same lesson.] All two digit numbers will add up to 99. [David ’ s conjecture is written on the board.] ~ I ’ ve got another counterexample to Ben ’ s. - This is how mathematicians work; are there counterexamples? Are two conjectures actually linked and so on.

29. The power of the generic After so many years you’ve got this bank of things that you know work really well and you realise certain methods you can apply to so many different problems. So, simple things like drawing a grid on the board – yes you can do a multiplication square, but you can do hundreds of other things so you can do percentages of amounts and percentages of fractions or you can do something with algebra where you’re multiplying the terms together or you can have pyramids where you’re adding algebraic terms or you’re adding number terms and sometimes I know in my head banks of things that I know are going to work well and so when I plan them all I have to do is I write down one word. (Interview, Teacher H, 1991/2)

30. I feel that the children in this class might be challenged more, they seem to be able to do this work on symmetries quite easily. One idea I have is to ask questions ‘the other way round’ - for example, rather than ‘Draw the lines of symmetry on this shape’ … I might ask: ‘Draw a shape which has two lines of symmetry and has rotational symmetry of order four.’ In fact, ‘Which combinations of number of lines of symmetry and order of rotational symmetry are possible?’