1. Analysis of some primary lesson segments using variation
Anne Watson
University of Witswatersrand
March 2017

2. Thanks to:
Thabit Al-Murani, Cecilia Kilhamn, Debbie Morgan

3. Exchange visits between English and Shanghai teachers of mathematics
Political intention: change methods to 'mastery' to improve learning for all
Background: major differences in training, workload, culture and funding
Major difference observed by teachers is the grain size of the focus of lessons on critical aspects of a mathematical idea
Input about variation to structure what is available to be learnt

4. Role of variation Design tool for teachers
Analytical tool for researchers

5. Mastery
the idea that all pupils should learn core mathematical concepts together, with appropriate extra support where necessary
no implications for particular methods

6. Variation theory or lens or principle?
learning by discerning variation against a background of invariance
teachers have a detailed conceptual focus
does pupils' lived object of learning (LOL) match the intended object of learning (IOL)?
enacted object of learning (EOL) – how variation is used to connect LOL to IOL

7. What is varied? dimensions of possible variation (DoV)
range of permissible change (RoC)
DoV and RoC provide the EOL (?)
interplay of variation and invariance:
Vary the feature that matters against an invariant background
Vary features that do not matter so that the important feature can be recognised in different contexts
Underlying dependency relationship does not vary

8. Enacted object of learning (EOL)
dimensions of possible variation
ranges of permissible change
what is compared to what and how that comes about
population and structure of the example space
density of population
connectivity
generality
generativity

9. The study lessons on NCETM website
variation theory as an analytical tool

10. Background: Chinese meaning of variation (Sun 2011)
One problem, multiple methods of solution: representations, actions, diagrams, notations Transformations/variations of the problem: same structure, different formats One solution method, multiple problems: the class of problems that can be solved this way Examples and non-examples (contrast) Procedural and conceptual (!) (Gu, Huang Marton, 2004)

11. Contested terms Procedural and conceptual What has to vary
Critical aspects (Marton and Tsui)
Points:
key
difficult (misconception?)
critical

12. Our research focus
to identify the enacted objects of learning when primary teachers in England make use of variation to achieve mastery
we look for what is varied and how is it varied, and evidence of LOL (very incomplete)

13. Our method
the NCETM videos were watched by the researchers, together and separately
researchers produced chronological reports on the lesson
reports and commentaries were compared and collated to generate a shared view of the EOL in segments of the lesson, and how this was enacted through variation

14. Our perspectives
Al-Murani: how, when and who introduces which DoV and RoC and exchange between teacher and students (Gothenburg and Oxford influence)
Kilhamn: identified the EOL from analysis of the DoV and RoC, separation, contrast, fusion etc. (Gothenburg influence)
Morgan: knowledge of teachers, their intentions and how VT had been introduced through NCETM and Shanghai exchange, lesson observation of the actual lessons and prepared the videos for website use (Shanghai and policy influence)
Watson: how conceptual content was managed through use of variation; what dependency relations are expected to be, or available to be, learnt

15. Excerpt 1
Year 1
IOL: to move pupils from understanding subtraction as 'take away' to understanding it as 'difference‘
children mainly know additive facts about numbers to 10 and also know that these can be expressed in a range of ways, including part-part- whole diagrams
2nd lesson on ‘difference’ (lesson 1 was about “more than, less than”)

16. Before this clip ….. ‘Take away’ Invariants so far (INV):
cars in a car park as context
use of counters to represent cars
only numbers 5,3,2 have been used
part-part whole diagram use for additive relationship
DoV/RoC from pupils:
transformations of the part-part-whole relationship
5 - 3 = 2; = 3; = 5; = 5; 5 = 2 + 3; 5 = 3 + 2

17. What teacher presents Variation analysis
"Now represent: there are five red cars and there are three blue cars.”
From teacher:
DoV: image; RoC: 'remove' or 'compare' numbers
Pupils told to use a row of 5 and a row of 3:
●●●●●
●●●
From pupils:
DoV: layout
Colours of the counters seen as irrelevant
DoV: different coloured counters were used and the irrelevance of this was discussed

18. What teacher presents Variation analysis
"Looking at this picture, what is the difference between the number of red cars and the number of blue cars. Tell the person next to you."
From the pupils: DoV: the ways the difference is worked out and expressed; RoC: counting on, using number facts, comparing numbers, one to one matching
T draws a part-part-whole diagram to represent the situation with the cars, and overlays that onto a picture of counters:
All chant "we can use part-part-whole to tell us about difference"
From the teacher: DoV: the ways in which the part-part-whole diagram can be drawn; RoC: either schematically or drawing around, and labelling, the material representation.

19. What teacher presents Variation analysis
T gives a new task: “There are 7 children and there are 4 dinner tokens. Represent that.”
T says: “What [one pupil] said is that 3 and 4 equals 7. And we could use that addition fact to help us find the missing part. To help us find the difference”
DoV: what a part-part-whole diagram can be used for; RoC: 'take away‘ and ‘difference’
DoV: numbers change; RoC small integers
DoV: actions associated with ‘compare’
DoV: meaning of 'difference'; RoC: compare quantities of two similar objects; do a one-to-one matching of different objects

20. Invariants counters to represent cars and numbers
linear layout of counters
part-part-whole diagram represents the additive relationship
use of p-p-w to find the difference

21. IOL, EOL, DoV, LOL
IOL: to focus on part-part-whole so they 'understand it in a way that they can apply it to a different structure’; dependency relation between three quantities through addition and subtraction
EOL: same numbers, same diagram, same representations, similar context
EOL : a 'compare' situation can be represented using two lines of counters, in a linear layout.
DoV: ways they find numerical difference
EOL: name missing part in diagram as 'difference'
EOL: represent and work out the difference in a new situation (7,4,3 and matching): LOL is not the same for all students as some of them do not set up the diagram for this new situation

22. Reflections on our observations
coherence of IOL: a whole lesson is devoted to developing a new meaning for subtraction
because they know number facts, the lesson is not experienced as being about counting but about additive structure
various DoV are opened up and then pinned down to become invariant so that eventually the relationship between 'difference' and p-p-w becomes the only change around, and is then emphasised through a chanted phrase
EOL can be discerned from looking at differences within invariant features; LOL is how pupils discern difference - this is not possible to know for sure but can be deduced from what they say and do, and how this relates to the EOL

23. Current thinking
Perceptual (seeing); enactive (doing); transformational (seeing differently); representational (translation); procedural (order of actions); structural (relations between objects); conceptual (patterns among objects)
Contrast (similarity); separation; generalisation; fusion (does order matter?) (Marton et al.)
Population; connectivity; generalisability; generativity of PES (Sinclair et al.)
What is the same/ what is different? (delineating/defining) (with the grain – expanding the population of PES and non-examples)
What varies/what stays the same?(relational/functional) (across the grain – connectivity in the PES)

24. Value of method Tool to make conjectures about difficulties:
it took us several cycles to recognise a variation in meaning of ‘compare’
the word 'whole' in the phrase 'part-part-whole' no longer has meaning when the objects are different
representing the missing number using p-p-w instead of imagination (?)

25. Excerpt 2
Year 6
Lesson about reading from graphs

26.
PMTheta.com