Connecting Young Children’s Mathematics Learning:

Early Years (P-3)Mathematical Association of Victoria 53rd Annual

ConferenceJoanne Mulligan

Joanne.mulligan@mq.edu.auMacquarie University Sydney

How do we connect mathematical ideas?

• What’s common about mathematical concepts?• Awareness of Mathematical Pattern and Structure (AMPS)• Pattern and Structure Assessment (PASA)• Common processes: structural groupings• Pattern and Structure Mathematics Awareness Program

(PASMAP)

What’s the underlying connection?• Finding similarities and differences• Discovering, continuing, creating patterns• Looking for generalised thinking• Finding relationships• Noticing structural features• Early symbolisation of relationships

Developing ‘pattern and structure’• How do children develop

the process of noticing similarity and difference?

• What is the role of spatial reasoning in developing mathematical knowledge?

• How do children develop spatial structure?

• What is the role of colinearity (horizontal and vertical coordination)?

“I see 11 all over as I count.” “I see 11 as a square of 3s and 2 more.”

Early underlying awareness of pattern and structure

Imagine 11 dots as a pattern.Draw them and explain what you see.

Kindergarten (Foundation) children

Awareness of Mathematical Pattern and Structure (AMPS)

“An Awareness of Mathematical Pattern and Structure (AMPS) generalises across early mathematical concepts, can be reliably measured, and is correlated with mathematical understanding”

(Mulligan & Mitchelmore, 2009)

Background ResearchSuite of studies 2002-2014 with 4 to 8 year olds

•Development and longitudinal evaluation of PASA (n=112)•Whole school assessment and intervention (n=726)•Case studies/ intervention ‘special needs’ children (n=12; n=18)•Large longitudinal empirical evaluation NSW and Queensland: combined Rasch scale with ‘I Can Do Maths’ (ICDM) (n=319)•3- year longitudinal study of 24 gifted 5-7 year olds: statistical reasoning and pattern and structure (n=21)•Validation study of PASA and PAT Maths ACER 2011-2105 (n=818)•PASA and Pattern and Structure Mathematics Awareness Program (PASMAP): pilot studies in Europe, UK, New Zealand, US and Asia

Awareness of Mathematical Pattern and Structure (AMPS): research findings

• Children capable of ‘emergent’ mathematical generalisation• Pattern and structure underpins conceptual development and ability to

abstract and generalise• Spatial abilities linked with development of number• Awareness of Mathematical Pattern and Structure (AMPS): recognition

of common structures and a tendency to look for patterns• Early school achievement is highly correlated with level of AMPS• Diverse range of AMPS at any one age level• Children with weak/low AMPS do not progress on mathematical

‘continuum’—mathematics becomes increasingly ‘confused’ and ‘crowded’

• ‘Intervention’ can improve AMPS

What is a mathematical pattern?

• Simple Repetition: ‘unit of repeat’ ABC ABC ABC• Spatial Patterns: 2D and 3D designs, tessellations,

transformations• Growing Patterns: increase or decrease

systematically e.g. triangular number pattern 1, 3, 6, 10, 15

• Functions: relationship between variables e.g. table of values 1 dog - 4 legs; 2 dogs - 8 legs

“My pattern is about numbers going up and down. I chose this pattern because I like numbers and maths” (aged 6 yr 3 months)

Simple repetitions: towers

Copy ABABAB tower Continue ABABAB patternIdentify screened parts Copy and draw from memory

Unit of repeat: “chunking” and ordering

Emergent Generalisation: Same Structure/Different Objects

Repetition as emergent generalisation: Same structure/different symbols

Organising data: emergent functional thinking

“1 dog – 4 legs2 dogs – 8 legs3 dogs – 12 legs

… You put 4 with each dog … it’s four for every dog you have no matter how many”

What are mathematical structures?• Numerical structure: e.g. base ten system

(multiplicative), exponents, combinatorials• Spatial structure: e.g. row and column array;

similarity ‘same shape different size’ and congruence’ same shape same size’

• Transformations• Units of measure• Structural features that lead to abstraction and

generalisation e.g. a+b = b+a; equivalence• Structural features of mathematical representations

e.g. graphs

Structure of multiplication: Using four squares as a unit

“I made a pattern so 1 big square is 4 little squares so its 4 for each square and 6 or 8 for rectangles. Every time you use the square its a four.”

(Five year old student)

How many little squares fit in a big square?

Using patterns and relationships to visualise number facts

Structuring relationships: Equivalence1 + 2 = 2 + 1

1 + 3 = 3 + 1

1 + 4 = 4 + 1

1 + 5 = 5 + 1

1 + 6 = _ + 1

What is the same/different? How many different patterns can you find?

Make another pattern with the same structure.

Stages of structural development• Prestructural: representations lack evidence of relevant

numerical or spatial structure• Emergent: representations show some relevant elements,

but their numerical or spatial structure is not represented• Partial structural: representations show most relevant

aspects but representation is incomplete• Structural: representations correctly integrate numerical and

spatial structural features• Advanced: children show they recognise the generality of the

underlying structure

Stages of structural developmentSomeone has started to draw some small squares to cover this shape.

Can you finish drawing the squares?

Prestructural Emergent

StructuralPartial

“We need 100 numbers, so 10 rows and 10 in each row. All numbers on the end in the column has to end with the same number….it doesn’t matter how far I go with the numbers… cause it’s the tens column we are adding…one more ten each time” (Jon, 5 years 8 months)

Constructing hundreds chart from memory (structural response - emergent generalisation)

Pattern and Structure Assessment: Early Mathematics (PASA F; 1; 2) (ACER)

• PASA assesses key concepts/mathematical structures AMPS score

• Assesses visual memory and spatial structuring, representational processes

• One-on-one interviews: F, 1, 2 (responses in recording booklet).

• 15-16 tasks; verbal, modelled and drawn responses

• Assesses for stage of structural development.

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Pattern and Structure (PASA) items as structural groupings

• 1. Sequences: repeating and border patterns; spatial pattern continuation, visual memory triangular array; growing patterns

• 2. Structured counting: visual memory rectangular array; multiple count; grouping;ten frames; hundred chart

• 3. Shape and Alignment: grid completion ( other items on transformations, congruence, 2D-3D)

• 4. Equal Spacing: distance/number line; strucutre of ruler, clockface; barchart

• 5. Partitioning: length (thirds); comparing triangles ( embedding); money ( base ten); capacity

PASA 2 Assessment Items1. Partitioning length into thirds 2. Border Pattern 3. Triangular Array 4. Partitioning Money 5. Ten Frames 6. Counting by Threes: number track 7. Spatial Pattern Continuation 8. Square Array 9. Structuring/using Hundred Chart 10. The analogue Clock 11. Grid Completion 12. Comparing Triangles

13. Growing Pattern Continuation 14. Making a Ruler 15. Constructing/interpreting Bar Chart 16. Comparing Capacities

Shape and alignmentPASA-1 Q11: Visual memory (triangular array structure)

I’m going to show you a pattern of dots, but only for a short time.

Then I want you to draw it. Are you ready?[Uncover for 2 seconds only]Now draw exactly what you saw.

Student responses

Prestructural Emergent

Partial structural Structural Advanced structural

PartitioningPASA-2 Q1: Partitioning length (thirds)Look at this strip of paper. I want to cut it into three pieces exactly the same size.[Do NOT say “in thirds”.]Can you show me where I should cut it? [Indicate a scissors cut with two fingers.]

Student responses

Emergent

Structural

Prestructural Partial structural

Advanced structural

Conceptualising fractions through pattern and structure

• I want to divide the strip (length) into two parts, one piece each). How many pieces? How many cuts (breaks)?

• If we had three children (three parts). We need three pieces. How many pieces? How many cuts (breaks)?

• If we had four children…• What is the pattern? (the pieces are always one more

than the cut)• (the more parts the bigger the number in the fraction)• What is the same ?

PASA assesses key concepts (across strands) central to mathematics curricula (see connection to Australian Curriculum-Mathematics)

PASA also assesses spatial visualisation, visual memory and representation

Responses shown as continuum of development (descriptors on the scale)

PASA and curriculum outcomes

Pattern and Structure Mathematical Awareness Program (PASMAP) (5-8 yr +)• Research-based challenging

tasks connecting structural groupings: scaffolds that teachers can apply to problem-based contexts

• PASMAP model focused on structuring with the aim to encourage emergent generalisation

• Pathways with Learning experiences linked to central concepts

The Pattern and Structure Mathematics Awareness Program Approach

• Highlight and model pattern and structure• Draw attention to mathematical features – “sameness” and

“difference”• Explicit focus on one aspect of structure at a time• Make connections between components of pattern and structure• Visual memory activities• Measurement and spatial structuring as a basis for number

concepts• Explain and justify thinking• Translate and generalise pattern and structure• Tasks gradually become more complex and link to other concepts

PASMAP Pedagogical SequenceConnecting mathematics through• Modelling• Representing • Visualising

• Generalising

• Sustaining

Seeing connections through 3D structures• Does it matter if a 3D

cube is transparent or opaque?

• Does it have to be placed in horizontal alignment?

• What patterns does the child notice?

• What does she do to find the repetition?

• What are the structural features?

PASMAP pedagogical sequences (Book1)

Pathway 1: Repeating PatternsPathway 2: Structured CountingPathway 3: Grid StructurePathway 4: Structuring ShapesPathway 5: Partitioning and SharingPathway 6: Base Ten StructurePathway 7: Growing PatternsPathway 8: Structuring MeasurementPathway 9: Structuring DataPathway 10: Symmetry and Transformation

PASMAP pedagogical sequences (Book2)

Pathway 1: Multiplication Patterns (MP) Pathway 2: Fitting Shapes Together (FS) Pathway 3: Partitioning and Fractions (PF) Pathway 4: Place Value (PV)Pathway 5: Metric Measurement (MM) Pathway 6: Patterns in Data (PD) Pathway 7: Angles and Direction (AD)

Towers to borders task• Make a tower e.g. ABC pattern with coloured

interlocking cubes• Make your tower of four units of repeat (3 x 4)• How many chunks? How many altogether?• Visualise: is it possible to transform the tower into a

square of the same pattern?• Is it possible to transform the square into a rectangle

keeping the same pattern?

Border patterns• Use coloured tiles or cubes to build linear pattern borders in

square or rectangular frames and record pattern.• Complete partially completed pattern.• Describe pattern in various ways e.g., using ordinals:

“Every third block is blue so I have x blue blocks in my pattern”.

Encouraging generalisation: “ It doesn’t matter how far it goes, every third block is blue”

• Repeat process and record from memory.

BordersModel and represent AB

2D Border patterns: Tasks • Provide 6 x 6 template (or other sizes) and two/three different sets of

coloured multilink cubes.• Predict how many cubes you will need of each colour to complete a

pattern such as AB or ABB or ABC.• How many pattern “chunks” can you complete for different sized

templates?• If the borders have a different number of squares can you always

complete a pattern? Why? Why not?• Extend the border task to include other units of repeat and different

sizes and shapes. • What are the similarities and differences. • Record your generalisations.

Transforming 2D Towers to 3D Houses(McKnight, 2010)

Use several towers to make a house. It must show a pattern or several patterns. Imagine what your house looks like. Use the cubes to build it. Draw it. Write about your house. Explain how you made the pattern.

(Grade 1 mixed ability class)

3D-2D monochromatic border pattern“There are four sides, there are four in each row [i.e.,

column], you count by fours…it has 20 rows. ... it has 22 blues, 22 greens, 22 whites and 22 oranges”.

3D-2D Border pattern with towers

“It shows a pattern because red, green, green, red.”

3D-2D checkerboard pattern“The pattern is on the top as well.”

Growing patterns: Spatial and numerical connections

• Take one square (or cube) and visualise how to make a (growing) pattern

• Continue the pattern• Make, draw or symbolise your pattern• Describe your pattern• Students initially focused on one element rather than the

relationships e.g., adding a row or the shape.

Pattern of squaresWhat comes next in the pattern so the pattern gets bigger?

Drawing and explaining pattern of squares from memory

Pre-structural response

Emergent structure: pattern of squares using single units.

Partial structure: equal sized single units but lacks co-linearity and “growing square” properties.

Partial structure: pattern of squares limited to 5 x 5

Structural response showing pattern, square grid structure, pattern sequence

http://topdrawer.aamt.edu.au/Patterns/

‘Take home’ messages

• Assess children for AMPS • Look for pattern / structural features? • How are they connected?• What’s the connection between this concept and

others ? • Connect Space and Number• Promote representation, visual memory, explaining

and generalisation

Acknowledgements• A/Professor Mike Mitchelmore (Macquarie University) and

Professor Lyn English (QUT) and research team assistants• Dr Andrew Stephanou and Dr John Lindsey (ACER) for their

work on the development of the AMPS scale and equating with the PAT Maths scale.

• Macquarie University Research• Australian Research Council Discovery Projects• NSW Department of Education, Association for Independent

Schools NSW, and the Queensland Department of Education• Children, teachers, principals, education officers, and research

assistants who facilitated the research.

ReferencesEnglish, L. D. & Mulligan, J. T. (Eds.), (2013). Reconceptualising early mathematics learning. New York: Springer

Mulligan, J. T., & Mitchelmore, M. C., & Stephanou, A. (2015). Pattern and Structure Assessment (PASA): An assessment program for early mathematics (Years F-2) teacher guide. Australian Council for Educational Research. Melbourne: ACER Press.

Mulligan, J. T., & Mitchelmore, M. C., & Stephanou, A. (2015). Pattern and Structure Mathematics Awareness Program Books 1 and 2. Australian Council for Educational Research. Melbourne: ACER Press.Mitchelmore, M. (& MQ team) (2014) Top drawer teachers: Patterns. http://topdrawer.aamt.edu.au/Patterns

Mulligan, J. T., Prescott, A., Mitchelmore, M. C., & Outhred, L. (2005). Taking a closer look at young students' images of area measurement. Australian Primary Mathematics Classroom, 10(2), 4-8.

Mulligan J. T., Mitchelmore, M. C., Kemp, C., Marston, J., & Highfield, K., (2008). Encouraging mathematical thinking through pattern and structure: An intervention in the first year of schooling. Australian Primary Mathematics Classroom, 13(3), 10-15.