Pedagogy and the development of
abstract concepts: the case of school
mathematics
Anne Watson
Oxford 2013
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?
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
• 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).
Addition actions:
combining, counting,
aggregating
Growth of understanding
• Aligning my understanding to the authority of
the number system
• Getting right answers
Addition: relations
a+b=c
b+a=c
c–a=b
c–b=a
c=a+b
c=b+a
b=c- a
a=c- b
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
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
Growth of understanding
• Recognising structures/relations
• Using properties to reason
• Knowing what has to be true
Which comes first?
• The underlying relation?
• The specific examples?
• 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)
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
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)
Growth of
understanding
(Pirie-Kieran)
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)
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
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)
• 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)
Negative numbers – manifestation
problems
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
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
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.
Functions
• A relation between variables in which each
input variable for which the function is
defined is related to a predictable output
variable
h=ρt
a
x
+5
x
0
1
2
3
4
y
2
3
6
11
18
Function
representations
x7
f(x) = 0 when x is
rational, f(x) = x when
x is irrational
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
Designing pedagogy
Enactive – iconic – symbolic
(Bruner; action -> representation)
Manipulating – getting a sense – articulating
(Mason; action -> relation -> expression)
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.
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)
Growth of understanding
• Agency
• Alignment of concept image (my messy
understandings) with concept definition
(formal presentation)
• Action
• (pupil voice)