Key understandings in
mathematics: synthesis of
research
Anne Watson
NAMA 2009
Research with Terezinha Nunes and Peter Bryant
for the Nuffield Foundation
Spatial reasoning
 Pre-school knowledge of space is relational, not
just descriptive
 Children learn the logic of measurement,
comparisons, transitive inferences, & 1-to-1
correspondence early
 Iteration of standard units has to be understood –
and is hard
 Relations between shapes should be understood
conceptually
Teaching spatial reasoning
 Build on spatial relational knowledge from
outside school
 The conceptual basis of measurement relates to
children’s knowledge of relations and
correspondence
 Children need lots of experience of iteration with
standard and non-standard units
Number
 Pre-school knowledge of quantities and
counting develop separately
 Additive understanding does not precede
multiplicative: very young children can reason
multiplicatively from everyday experiences
 Three principles relate to success in
mathematics: the inverse relation between
addition and subtraction; additive composition;
one-to-many correspondence
 It is difficult to think about relations rather than
quantities
Teaching number
 Children have to connect their knowledge of
counting with their knowledge of quantities
 Focus on inverse addition/subtraction,
additive composition and one-to-many
correspondence
 Multiplicative reasoning does not follow
additive in a linear fashion
Relations between quantities
 Numbers are used to represent quantities and
relations.
 Children tend to interpret statements about
relations as if they were about quantities
 Experts use diagrams, tables and graphs to
explore the relations before solving a problem
 Informal knowledge can obstruct formal learning
Teaching about relations
 Children need to learn to distinguish
between quantities and relations
 Include the study of relations explicitly
 Link mathematical representations in
teaching
 Focus on the relationships between informal
knowledge and formal learning
Fractions
 Knowledge of fractional quantities starts to develop
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before school
Two types of quantity are represented as fractions:
measurement and division
Transferring understanding between division to
measurement is really hard
Links between fraction concepts and procedures for
comparison and operation are not made
Fractions are usually taught as quantities, not as
relations
Teaching fractions
 Different types of reasoning in measurement
and division need to be linked
 Fractions need to be taught as relations
between quantities, as well as quantities
Algebra
 Success depends on (i) understanding underlying
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operations and relations and (ii) using symbolism correctly
It is important to learn how operations combine and relate
‘Letters stand for numbers’ in a variety of ways: constants,
unknowns, variables, parameters
Children interpret ‘=’ to mean ‘calculate’; but ‘=’ means
either ‘equal to’ or ‘equivalent to’
We all use ‘guess-&-check’ rather than algebra if it’s easy
‘Generalised arithmetic’, e.g. doing substitution, does not
give students reasons for learning to use algebra
Young students can use letters to express generalities
Teaching algebra
 Read numerical and algebraic expressions relationally
 Teaching should be based on describing generalisations
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based in properties (arithmetical rules, logical relations,
structures) as well as inductive reasoning from sequences
Hands-on ICT provides representations of relations
Support moves from expressing patterns to relating
properties
Understanding takes time, purpose & multiple experiences
Explore use of algebraic manipulation software
Learning higher mathematics
 Rules & exploration lead to different kinds of
knowledge; neither method necessarily enables
learning new concepts
 Students respond to familiarity, try to apply prior
knowledge, and generalise from their experience
 They need a repertoire of functions, operations,
representations, concepts and methods
 Choice of technology helps problem solving,
understanding of relations, identifying variables
and functions
Teaching higher maths
 Neither teaching to tests, nor problem-solving activities,
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lead to new abstract understandings on their own
Obvious perceptual reactions need to be recognised
Students need time for new meanings and to develop
from earlier methods
Students need to control switching between
representations and comparing to understand functions
Authentic, realistic, mathematical exploration uses
spreadsheets, graphing tools, etc. and deductive
reasoning
Developmental themes
throughout mathematics
 Number – not a unitary
idea
 Mathematical modes
of enquiry
 Logical reasoning
 Correspondence
 Inversion
 Inclusion
 Composition
 Transitivity
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 Tool use and implicit
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models
 Symbol systems
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Comparisons
Properties
Representations
Action and reflection
Direct and inverse
relations
Formal and informal
reasoning
Local and global
changes
Overarching messages for
primary and secondary
 Relating number, quantities, spatial properties,
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expressions …
Interplay of formal and informal understanding
Multi-strand approach to number and spatial reasoning
Multiple experiences over time, with several
representations, of new concepts
Rules & exploration do not necessarily enable new
concepts
Visual impact and recent similarity dominate over analysis
of meanings, variables and relations
Students rarely use mathematical ICT tools authentically