Pedagogy and the development of abstract concepts: the case - PowerPoint Presentation

Slide1

Pedagogy and the development of abstract concepts: the case of school mathematics

Anne WatsonOxford 2013Slide2

Plan

Examples of school mathematical concepts to illustrate the endeavourExamples of how processes of mathematical conceptualisation are described

Consideration of whether these are pedagogically useful

How is a focus on activity useful?Slide3

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; functionsSlide4

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

).Slide5

Addition actions: combining, counting, aggregating Slide6
Slide7

Growth of understanding

Aligning my understanding to the authority of the number systemGetting right answersSlide8

Addition: relations

a + b = c

c

= a + b

b + a = c

c

= b + a

c – a = b

b

= c - a

c – b = a

a

= c - bSlide9

Growth of understanding

Being able to adapt relations to create appropriate toolsBeing able to decide how to solve problemsGetting right answers to problemsSlide10

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 objectsSlide11

Growth of understanding

Recognising structures/relationsUsing properties to reasonKnowing what has to be trueSlide12

Which comes first?

The underlying relation?The specific examples?Slide13

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)Slide14

Theories

Piaget: action, invariant relations and representationVergnaud: 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 talkSlide15

Ways to describe mathematical concepts

Piaget: action and representation - > schemaConcepts 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)Slide16

Growth of understanding (Pirie-Kieran)Slide17

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)Slide18

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 learnerSlide19

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)Slide20

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)Slide21

Negative numbers – manifestation problemsSlide22

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 objectSlide23

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 invarianceSlide24

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.Slide25
Slide26

Functions

A relation between variables in which each input variable for which the function is defined is related to a predictable output variableSlide27

x

+ 5

x 7

x

y

0

2

1

3

2

6

3

11

4

18

h =

ρ

t

a

f(x) = 0 when x is rational, f(x) = x when x is irrational

Function representationsSlide28

Designing pedagogy

Enactive – iconic – symbolic (Bruner; action -> representation)

Manipulating – getting a sense – articulating (Mason; action -> relation -> expression)Slide29

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.Slide30

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)Slide31

Growth of understanding

AgencyAlignment of concept image (my messy understandings) with concept definition (formal presentation)Action

(

pupil voice)