Understanding Young
Children's Difficulties in
Mathematics Learning
Joanne Mulligan
Australian Centre for Educational Studies
Centre for Research in Mathematics & Science
Education (CRiMSE)
Macquarie University, Sydney
Difficulties in Mathematics:
Key Aspects
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Research background
‘Pattern’ and ‘structure’
Common difficulties
Pattern & Structure Project
Count Me In Too (NSW DET)
Directions for teaching and learning
Research Agenda
• Identify early difficulties in mathematics
(abstraction): inability to see ‘pattern’ and
‘structure’
• Collect evidence from longitudinal, case
studies and classroom-based studies:
focus on ‘high’ and ‘low’ achievers
• Track children’s ‘structural’ development
rather than take ‘snapshots’ of specific
skills
• Design assessment and teaching
approaches to promote ‘structural’
development
Research Agenda
• Integrate mathematical concepts and
processes: what are the common
underlying features?
• What are the connections that we expect
children to make?
• Multi-focused approach:
attention
visual memory
information processing capacity
imagery and representation
language and semantic structure of mathematical
situations
! abstraction and symbolisation
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Related Research
• Neuroscientific advances
• Changes in brain activity from sensory at
back of the brain to the more generalised
frontal cortex when using logical thinking
• Human brain has visual areas that
perceive structure (colours, shades,
edges, outlines, quantitites)
• Clearer distinction between analytical
and visual processing; numerical
processing (Butterworth; Dehaene;
Seron)
Research Findings
Some children do not abstract ideas in a
way that promotes mathematical
development: pattern, structure and
relationships – that’s the essence of
mathematics.
Theoretical View
“The more a child’s internal images of
mathematical ideas develop structurally,
the more coherent and stable are their
external ideas (drawings, models,
diagrams, graphs, symbols, words,
explanations),
–and the more mathematically
‘competent’ the child will be”
(Thomas, Mulligan & Goldin, 2002)
Emergent Pattern & Structure
Grade 2
Partial Pattern & Structure
Grade 5
Pattern & Structure
Project
• Children’s perception and representation of
mathematical structure generalised across a
range of mathematical content domains and
contexts
• Early school mathematics achievement was
strongly linked with the child’s development
and perception of mathematical structure
(Mulligan, Prescott & Mitchelmore, 2004)
Triangular Pattern Task
(Flash card with pattern) Look carefully. Cover.
Draw exactly what you saw. Describe it.
o
o o
o o o
Pre-Structural
(Triangle of 6 dots)
Triangular Pattern:
Responses Over Time
Triangular Pattern:
Responses Over Time
Results:
‘Low ability’ children
• All low ability children made some progress
• No clear developmental pattern: Transition from
pre-structural to an emergent stage was
haphazard
• Some children reverted to earlier, more primitive
images after a year of schooling
• Some children did not progress because they
complicate or ‘crowd’ their images with
superficial idiosyncratic aspects
• More dissimilarity than similarity
responses, within and between cases
in
their
‘Pattern and Structure’
• A pattern may be defined as–
a numerical or spatial regularity, and the
relationship between the various components of
a pattern constitute its ‘structure’
Equal/Unequal Groups
Equal Groups Structure
• There are 2 tables with 3
children at each table.
How many children are
there altogether?
• Two groups of 3 children
but child partitions
groups using her image “4
girls and 2 boys”
• Shows equal groups but
child does not use the
multiplication structure
‘Structure’ of Concepts
• “The smaller the unit, the more you will
need to measure”
Jane’s scoop is twice as big as Peter’s scoop to
measure 1kg of rice
• The more (equal) parts, the smaller the
fraction( & larger denominator)
Which is smaller 1/5 or 1/6
Development of Units:
row and column structure
“Finish drawing the squares to fit into the rectangle”
Counting and ‘Base Ten’
• Counts and groups
in tens to form 100
• Visualises 100s
chart without base
ten, or row and
column structure
• Image imposed on
child
Structure of Clockface
Multiplication: Comparison
“factor” “times as many”
Sam has 3 pencils
and Rebecca has 9
times as many.
How many pencils
does Rebecca
have?
Multiplication:
Combinatorial
You can buy
icecreams in small
medium or large
cones in only four
different flavours.
How many
different choices
(types of
icecreams) can
you make?
Division: Quotition
There are 18
children and 3
children are
seated at each
table. How many
tables are there?
Common Difficulties
• Difficulties with number “discalculia”: failure to compress
counting procedures into flexible concepts can lead to
reliance on rote learning and meaningless concepts,
symbols and diagrams
• Semantic characteristics of mathematical word problems
• Lack of connectedness with prior concepts,within and
between mathematical ideas
• Dysfunction between short-term and working memory
• Perceptual and visualisation skills
• Uni-structural thinking: one aspect at a time
• Focus on idiosyncratic, superficial, ‘non-mathematical’
features
Common Difficulties:
Language
• Confusion with comparative terms: ‘more and
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less’ , greater than’, ‘less than’, ‘how many
more than’, ‘before’ and ‘after’,
Processes: ‘equal to’, share’,‘same as’,
‘difference’, ‘split and jump’, ‘regroup’
Distinguishing terms: ‘hundred’ & ‘hundredth’;
‘volume’( space) & ‘volume’ (sound)
Conceptual terms: ‘odd’, ‘even’, ‘prime’,
‘squared’, ‘cubed’, ‘mass/weight’; ‘ perimeter
and area’, ‘flip’, ‘turn’
Semantics: ‘divided by’ or into’, ‘times as many’
Using technology: time 1:40,13:40 or 20 to 2
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• Interpreting and constructing tables and graphs
Related Learning
Difficulties
• Behavioural disorders often associated with lack
of attention to mathematical structures and
‘rules’
• Oppositional children refusal to conform to
‘rigidity’ and accuracy of mathematical forms
• Attention Deficit with Hyperactivity: lack of focus
generally; intolerance of ‘abstract’ mathematics
• Perceptual difficulties linked to visualisation
• Capable intelligent children can be impulsive,
make errors, process images too quickly:
dynamic imagery can be detrimental
• Creative, natural development of mathematical
ideas often stifled by imposed ‘mathematics’
Common Difficulties
in Mathematics
• Reliance on counting by ones (unitary counting )
• Counting and additive methods persist through
to adulthood
• Structure of the base 10 number system:
multiplicative( x 10) not additive; decimal point
and number line confusion
• Multiplicative reasoning: equal groups, units,
ratio
• Conceptual confusion between measurement
and spatial concepts ie volume and mass
• Inability to apply facts to problems
• Fraction concepts: part-whole relations never
really developed; all numbers treated as wholes
Pattern and Structure
Project 2002-2005
• Pattern & Structure Research Program:
assessment, teaching & monitoring of
mathematical development Case studies of high
and low achievers 4-7 year olds
• Pattern & Structure Assessment (PASA):
Individual interviews: 30 tasks - Number Space
Measurement
• Pattern & Structure Program (PASP):
Framework of tasks linked to assessment and
key concepts
• PASP Professional Development: Teacher
mentoring; classroom-based development of
tasks and materials
Tasks: Number,
Space, Measurement
• Thirty tasks: common elements of mathematical
and spatial structure: number, measurement,
space and data
• Use or represent elements e.g. equal groups or
units; spatial structure e.g.rows or columns, or
numerical and geometrical patterns
• Number tasks: subitizing, counting in multiples,
fractions & partitioning, combinations & sharing
• Space and data tasks: triangular pattern,
visualising & filling a box; completing a picture
graph
• Measurement tasks: units of length, area,
volume, mass and time
Partitioning Task
Tell me how you would share the biscuits
fairly among 4 children.
Teaching/ Learning
PASP Approach
• Highlight pattern and structure
• Draw attention to mathematical features
“sameness’ and “difference”
• Explicit focus on one aspect of structure at a
time
• Visual memory activities: observe, screen,
record, revisit, apply to various situations
• Connections to other mathematical ideas
• Explain and justify
• Child must internalise mathematical features
Teaching Focus:
Triangular Pattern Task
• Child explains their initial inaccurate image (Intuitive
justification)
• Teacher shows pattern produced correctly by another
student with coloured discs (Modelling)
• How can we make your pattern the same as this one? Tell
me why we are making it the same? (Focus on ‘sameness’)
• Child’s attention is drawn to shape, size colour,equal sized
spaces. ( Focus on spatial or numerical structure)
• Screen each row or side of triangle successively then child
reproduces with counters. ( Successive screening)
• Child justifies that the pattern is the same(Justification)
• Child must reproduce from memory (Visual memory)
• Task repeated regularly ( Repetition)
Staircase of Twos
Image 1
Staircase of Twos
Image 2
Pre-Structural
Interview 1
Pre-Structural
Interview 2
Emergent Structure
Interview 3
Partial Structure
Interview 4
Structure:
Border Pattern
Benchmarking:
Empty Number Line
‘Count Me In Too’
• Research-based
Learning Framework
in Number (based on
work of Steffe, Cobb
& colleagues; Wright,
Mulligan &
colleagues;
Gravemeijer &
colleagues)
‘Count Me In Too’
Numeracy Framework
Structure of Ten Frames
Structure of Array
‘Screened’ Array
Using Patterns:
SENA tasks
Addition and Subtraction
Strategies
• Bridging to 10, 100
76 + 14 = 80 +10 = 90
• Building up and building down from 10s
100s
59 + 41 = 100
• Partitioning 28 + 37 = 20 +30 +8 +7
• Compensation 58 + 39 = 60 + 37
Semi-Formal Routines
• Partitioning (splitting) method
38 + 46 is calculated mentally as
30 and 40 = 70; 70 and 14 = 84
• Sequencing (jump) method
38 + 46 is calculated mentally as
38 + 40 = 78; 78 +6 =84
• Use of empty number line
Use of 100 Chart for
Addition and Subtraction
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Jump and split
Use of visualisation
Difficulties with
wrap around sequence
‘Jump and Split’
• Jump by tens
NSW Basic Skills Test
Bronte multiplied a
number by 3.
Her answer was 18.
She should have divided
the number by 3.
What should her answer
have been?
2
4
3
6
• What do you need to find
out in the first part of the
problem?
• What was the number that
Bronte multiplied by 3?
• Try to work out what 3
was multiplied by to get
18?
• Write a number sentence
to show where the missing
number is.
• Write a number sentence
that shows this same
number divided by 3.Why
will it be much smaller
than 18?
Teaching and Learning
• One ‘program’ does not fit all:
individualised and small group teaching
• Program of connectedness and revisiting
rather than lock step sequence
• Mix and match PASP concepts and tasks /
syllabus
• Child becomes familiar “comfortable”
through repetition and routine
• Focus on specific weakness identified by
assessment
Social and Emotional
Issues
• Develop positive attitudes towards
learning
• Focus on succeeding: avoid “maths is too
hard”, ‘math phobia’ and ‘math anxiety’
• Ease the fear of failure: Child becomes
familiar “comfortable”: understanding,
usefulness, repetition
Social and Emotional
Issues
• Child takes responsibility for their
mathematics learning: think
independently of the teacher (&
caregivers)
• Set specific limits and expectations:
• Scaffold and support child; healthy
positive learning relationships
Links with Home
• Caregivers participate in routine
activities at home eg ‘read a story
and counting patterns and puzzles
• Encourage small group games:
cards, board games, puzzles,
calculators
• Careful selection of technologybased games and programs
Resources
• NSW Department of Education & Training
Count Me In Too program and website
• Developing Efficient Numeracy Strategies
• Fractions Pikelets and Lamingtons
• Teaching Measurement Early Stage 1,
Stage 1; Stage 2/3
• Patterns and Algebra
• Basic Skills Testing: Enhanced Data on
Disk–Distractor Analysis; Teaching
Strategies