1. Pedagogy and the development of abstract concepts: the case of school mathematics Anne Watson Oxford 2013

2. Plan Examples of school mathematical concepts to illustrate the endeavour Examples of how processes of mathematical conceptualisation are described Consideration of whether these are pedagogically useful How is a focus on activity useful?

3. What is a mathematical concept? A mathematician would say that mathematical concepts are abstract structures that encapsulate relations, properties and behaviours of quantitative, spatial and axiomatically-generated objects – e.g. addition; functions

4. At school level some concepts can be understood through their real-world manifestations, so conceptualisation can involve inductive generalisation from examples (Marton), language (Sfard), mental images (Greeno), sense-making by reflection on action (Piaget), in several situations (Vergnaud).

5. Addition actions: combining, counting, aggregating

7. Growth of understanding Aligning my understanding to the authority of the number system Getting right answers

8. Addition: relations a + b = cc = a + b b + a = cc = b + a c – a = bb = c - a c – b = aa = c - b

9. Growth of understanding Being able to adapt relations to create appropriate tools Being able to decide how to solve problems Getting right answers to problems

10. Relations within isosceles triangles Characteristics of defined class – equal edges (definition), learnt inductively through examples and definitions Properties – equal angles (theorems) observed inductively through dynamic diagrams/example -> conjecture; deductively reasoned from theorems, proofs -> facts about new objects, or new characteristics about old objects

11. Growth of understanding Recognising structures/relations Using properties to reason Knowing what has to be true

12. Which comes first? The underlying relation? The specific examples?

13. Generalisation is expressing an observed relation (ascent into talk from abstract) Generalisation as imposing a pattern on a set of examples or actions or experiences (ascent from experience to abstraction)

14. Theories Piaget: action, invariant relations and representation Vergnaud: fields/structures of Greeno: structured spaces Vygotsky: patterns of activity and talk in the figured world of mathematics Sfard: syntax of communication: talk, diagram, inner talk

15. Ways to describe mathematical concepts Piaget: action and representation - > schema Concepts in action and theorems in action (Vergnaud) Concept is altogether a field of: a set of situations, a set of operational invariants (contained in schemes), and a set of linguistic and symbolic representations (Vergnaud) Shift to: concepts ARE language and representations (Sfard, Janvier, Dorfman) Schemes as grounding metaphors of action (Lakoff, Nunez)

16. Growth of understanding (Pirie-Kieran)

17. APOS theory/description (Dubinsky) Piagetian ‘ascent’ model cf. historical development Action (some kind of transformation – reflection on input-output relation of action) Process (internal reconstruction that does not have to be performed – awareness of properties) Object (the action itself becomes a mathematical object that can be acted upon) Schema (principled linkages with other APO)

18. Concept image – concept definition (procept) (Tall) Examples, methods, words, symbols, situations, theorems.... etc. (experiences) Related to Vergnaud’s idea of conceptual field but more concerned with experience of learners in educational situations Formalisation: concept definition (convention) is alongside concept image – not above or below Spontaneous and scientific concepts (internal connections of ideas) Recognising that ‘tools’ remain to some extent perceived as tools, i.e. the contextual baggage that embodies the concept for a learner

19. AiC – RBC (Dreyfus et al.) Abstraction in context Davydovian; analysis -> synthesis; “vertical mathematisation”/progressive refining of abstract ideas Vertical reorganisation of students’ ideas through construction Outcomes of activity become tools for later activity; development of an abstraction can be tracked through successive activity. Relevant actions are epistemic actions – pertain to knowing – need to be observed to be pedagogically useful recognizing (relevant to this situation) (R) building-with (combine constructs to achieve goal) (B) constructing (new constructs) (C)

20. Kidron and Monaghan (2009) :... with Davydov’s dialectic analysis the abstraction proceeds from an initial unrefined first form to a final coherent construct in a two-way relationship between the concrete and the abstract – the learner needs the knowledge to make sense of a situation. At the moment when a learner realizes the need for a new construct, the learner already has an initial vague form of the future construct as a result of prior knowledge. Realizing the need for the new construct, the learner enters a second stage in which s/he is ready to build with her/his prior knowledge in order to develop the initial form to a consistent and elaborate higher form, the new construct, which provides a scientific explanation of the reality. (p. 86-87)

21. Negative numbers – manifestation problems

22. The ‘ascent to abstraction’ model Classification (what characteristics? what relations within objects?) (depends on variation offered) Generalisation (what is typical and essential?) (depends on variation offered) Definition and naming (necessary, sufficient and distinctive) Abstraction, treating as a new object

23. What is wrong with the inductive model? Many concepts are not obvious; inductive reasoning from experience often leads to error or limiting assumptions. Who decides what is worth generalising? Who decides what characteristics are important? Variation theory (Marton): the concepts we learn about are those that are presented through their variations, with learning involving discernment of variation and invariance

24. Language (Sfard) All we have is language and therefore conceptualisation is the use of language structures and syntax. The meaning of ‘multiply’ is how we focus on the similarities in a certain set of actions and phenomena. Cognition is communication.

26. Functions A relation between variables in which each input variable for which the function is defined is related to a predictable output variable

27. x + 5 x 7 xy 02 13 26 311 418 h = ρ t a f(x) = 0 when x is rational, f(x) = x when x is irrational Function representations

28. Designing pedagogy Enactive – iconic – symbolic (Bruner; action -> representation) Manipulating – getting a sense – articulating (Mason; action -> relation -> expression)

29. Constructionism All we can do is provide constructive tasks which have purpose and utility. The utility is the conceptualisation – hammering is what we do with hammers and what we use them for; graphing is what we use graphs for; functions are what we use them for.

30. Horizontal and vertical mathematisation (Freudenthal) Focus on actions and solution methods in context Provision of tasks that have similarity in underlying mathematical structure Recognition of structural similarity in solutions/situations Abstraction/ reorganisation/ “vertical mathematisation”/progressive refining of abstract ideas, then applying (ascent to concretisation)

31. Growth of understanding Agency Alignment of concept image (my messy understandings) with concept definition (formal presentation) Action (pupil voice)