1. Thoughts about variation and example spaces
Anne Watson & Cecilia Kilhamn
University of Gothenburg
April 2017

2. Example space
Population: refers to the scarcity or density of available examples
Generativity: refers to the possibility of generating new examples within the space using given examples and their associated
Connectedness: refers to whether examples are disconnected, loosely connected, or well-connected.
Generality: refers to the extent to which the given example is specific or whether it is representative of a class of related examples
(Sinclair, Watson, Zazkis & Mason, 2011)

3. Conventional and personal ES
conventional example spaces: examples used by teachers and displayed in textbooks (IOL/EOL)
personal example spaces: may contain central, obvious, isolated, examples that come to mind; idiosyncratic; experiential (LOL/PES)

4. Comparing PES arising from different RoC
Group A (26) 1. We are going to learn about ‘3q’ 2. If q = 3 then 3q = 9 If q = 6 then 3q = 18 If q = 4 then 3q = 12 If q = 9 then 3q = Make 3q equal to 15 Make 3q equal to 6 Make 3q = What is 3q when q = 1? What is 3q when q = 10? What is 3q when q = 0? 5. Can you make up some questions like these for 5m and 2x?
Group B (25) 1. We are going to learn about ‘2x’ 2. If q = 3 then 2q = 6 If q = 6 then 2q = 12 Etc. 5. Can you make up some questions like these for 3m and 5y?

5. Outcomes
more students in group A used correct notation to give examples
the highest mean number of responses was for examples of ‘5m’ for group A
seven group A students and two group B students used numbers outside the range of small positive integers
over half of group B wrote either nothing, or seemingly irrelevant statements; only 7 out of the 25 students in group A offered this kind of response
on average students in group A produced about 40% more examples than group B.

6. Conjecture
3q as a teacher-given example was more successful than using 2x because it presented less potential for misinterpretation
the qualities of examples matter; what can be perceived from RoC?

7. PES as evidence for LOL
Population: refers to the scarcity or density of available examples
Generativity: refers to the possibility of generating new examples within the space using given examples
Connectedness: refers to whether examples are disconnected, loosely connected, or well-connected.
Generality: refers to the extent to which the given example is specific or whether it is representative of a class of related examples

8. Primary Mathematics Variation Study (with Cecilia & others)
lessons on NCETM website
variation used by teachers in lesson design
variation used by us as an analytical tool

9. Analysis of primary teaching videos from England
influence from Shanghai
teachers have a detailed conceptual focus (small steps; key/difficult/critical points)
we question whether the pupils' lived object of learning (LOL) matches the intended object of learning (IOL) so we analyse EOL using DoV, RoC, and INV

10. 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)

11. Our method
the NCETM videos were watched by the researchers, together and separately
4 researchers familiar with VT 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

12. What is varied? interplay of variation and invariance:
vary the feature that matters against an invariant background
vary features that do not matter so that the important features (invariants) can be recognised in different representations, contexts and formats
underlying dependency relationships do not vary; these are more than fusions of DoV – they are mathematical objects in themselves

13. Dependency relationships in the video clips
1. A pair of lines being perpendicular to each other (defining) 2. The ‘difference’ between two numbers can be represented in the same way as a ‘take away’ and calculated by subtraction because these are both transformations of the additive relationship

14. Excerpt 1 Year 6 (10/11 year olds) Lesson about reading from graphs
New class consisting of pupils who previously had been in attainment sets, therefore mixed prior knowledge and experience.
Teacher has identified confidence with a pair of perpendicular lines as an important prerequisite for reading from graphs (a technical, not conceptual prerequisite)

15. Discussion Relevance for current task
Limited RoC for ‘pair of perpendicular lines’ restricts example space
Non-defining aspects: similar line segments, joined end to end
Seems to reduce the meanings that some pupils already had (static angle; size of rotation) to static angle in L-formation.

16. DoVs necessary to extend example space
Relative length of lines
Position of intercept
Neither of these are defining aspects of ‘perpendicular pair of lines’ (optional?)

17. Enriching the PES Population: generate more examples
Generativity: extend RoCs of current DoVs; also open new DoVs (up to boundary)
Connectedness: ‘fusion’ across DoVs; relations within each example
Generality: what common features and relations do all examples have? Inductive, defining

18. Excerpt 2
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”)

19. In this clip …..
They have worked on ‘difference’ as a whole class by comparing the numbers of cars in a car park that are blue and red.

20. Invariants
part-part-whole diagram represents the additive relationship
use of p-p-w to find the difference

21. IOL, DoV, INV, EOL
IOL: to focus on part-part-whole; dependency relation between three quantities
DoVs: new contexts; new numbers; new layouts
INV: p-p-w diagram and language
EOL : a 'compare' situation can be represented using part-part-whole diagram
EOL: name missing part in diagram as 'difference'
EOL: represent and work out the differences in ‘matching’, ‘comparing’ and ‘sorting’ situations

22. Discussion
it took us several cycles to recognise a variation in meanings of ‘compare’
the word 'whole' in the phrase 'part-part-whole' no longer has meaning when the objects are different
this arose from considering enactive imagination associated with examples
the qualities of examples matter

23. Anne’s current thinking about learning from examples
Perceptual (seeing)
Enactive (doing/imagining)
Transformation (seeing/formatting differently)
Representation (translation/illumination)
Procedure (order of actions)
Structure (properties within objects and relations between objects)
Conceptual (patterns of behaviour and meaning among named objects)
Population; generativity; connectedness; generalisability; in PES (Sinclair et al.)
Contrast (similarity); separation; generalisation; fusion (does order matter?) (Marton et al.)
Underlying invariant relationship as a mathematical object (Venkat, Askew, Watson, Mason forthcoming)
What is the same/ what is different? (delineating/defining) (with the grain – expanding the population of PES to the boundary)
What varies/what stays the same?(relational/functional) (across the grain – connectivity and generalisation in the PES)
Defining/optional

24.
PMTheta.com