Early mathematical thinking

Louise Hodgson 2013

How many cups would you need to make a triangle building with four levels?

Allow free exploration with the cups for two weeks before posing the problem

Instead of limiting instruction to counting skills or writing numerals throughout the early years, allow youngsters multiple ways to represent quantity.

Overview

1. Sources and observations

2. Early mathematical ideas and processes

3. The role of the early childhood educator

4. Learning opportunities for numeracy - play spaces

2. Sources and observations

Belonging,Being &

BecomingEarly YearsLearning

Framework

Mathematics development

Recentresearchfindings

Perspectivesfrom earlychildhood

educators -beginning

andexperienced

Young children are not ready for mathematics education.

Mathematics is for some bright kids with mathematics genes.

Simple numbers and shapes are enough.

Language and literacy are more important than mathematics.

Teachers should provide an enriched physical environment, step back, and let the children play.

Common misconceptions

Mathematics should not be taught as stand-alone subject matter.

Assessment in mathematics is irrelevant when it comes to young children.

Children learn mathematics only by interacting with concrete objects.

Computers are inappropriate for the teaching and learning of mathematics.

http://www.earlychildhoodaustralia.org.au/australian_journal_of_early_childhood/ajec_index_abstracts/early_childhood_teachers_misconceptions_about_mathematics_education_for_young_children_in_the_united_states.html

Common misconceptions

About ‘intentional teaching’

•Intentional teaching is one of the 8 key pedagogical practices described in the Early Years Learning Framework (EYLF).

•The EYLF defines intentional teaching as ‘educators being deliberate purposeful and thoughtful in their decisions and actions’.

Intentional teaching is thoughtful, informed and deliberate.

Intentional teaching and the Early Years Learning Framework

Intentional educators:

•create a learning environment that is rich in materials and interactions

•create opportunities for inquiry

•model thinking and problem solving, and challenge children's existing ideas about how things work.

Intentional teaching and the Early Years Learning Framework

Intentional teaching and the Early Years Learning Framework

Intentional educators:

• know the content—concepts, vocabulary, skills and processes—and the teaching strategies that support important early learning in mathematics

• carefully observe children so that they can thoughtfully plan for children’s next-stage learning and emerging abilities

• take advantage of spontaneous, unexpected teaching and learning opportunities.

Numeracy or Mathematics?

“Numeracy is the capacity, confidence and disposition to use mathematics I daily life”

EYLF, 2009 p.38

“Outcome 4: Children are confident and involved learners. Children develop dispositions for learning such as curiosity, cooperation, confidence, creativity, commitment, enthusiasm, persistence, imagination and reflexivity. Children develop a range of skills and processes such as problem solving, inquiry, experimentation, hypothesising, researching and investigating”. (EYLF, 2009)

Disposition of children

Encourage young children to see themselves as mathematicians by stimulating their interest and ability in problem solving and investigation through relevant, challenging, sustained and supported activities (AAMT and ECA 2006)

Low mathematical skills in the earliest years are associated with a slower growth rate – children without adequate experiences in mathematics start behind and lose ground every year thereafter. (Clements and Sarama, 2009, p. 263)

Interventions must start in pre K and Kindergarten (Gersten et al 2005). Without such interventions, children in special need are often relegated to a path of failure (Baroody, 1999)

2. Early mathematical ideas

Outcome 5: Children areeffective communicators.

Spatial sense, structure and pattern, number, measurement, data, argumentation, connectionsand exploring the world mathematically are thepowerful mathematical ideas children need to become numerate. (EYLF, 2009 p38)Research….Perry, Dockett & Harley (2007) - powerful ideas and professional development

Critical concepts underpinning number understanding.Counting SubitisingMore – lessPart part whole

Children who understand number relationships develop multiple ways to represent them.

Interpreting quantity

Symbol

threeNumber word

3

Principles of Counting

• Each object to be counted must be touched or ‘included’ exactly once as the numbers are said.

• The numbers must be said once and always in the conventional order.

• The objects can be touched in any order and the starting point and order in which the objects are counted doesn’t affect how many there are.

• The arrangement of the objects doesn’t affect how many there are.

• The last number said tells ‘how many’ in the whole collection, it does not describe the last object touched.

Principles of Counting

• Which of the principles of counting does Charlotte understand?

Principles of Counting

• Each object to be counted must be touched or ‘included’ exactly once as the numbers are said.

• The numbers must be said once and always in the conventional order.

• The objects can be touched in any order and the starting point and order in which the objects are counted doesn’t affect how many there are.

• The arrangement of the objects doesn’t affect how many there are.

• The last number said tells ‘how many’ in the whole collection, it does not describe the last object touched.

Intentional opportunities for counting

• Model counting experiences in meaningful contexts, for example, counting girls, boys as they arrive at school, counting out pencils at the art table.

• Involving all children in acting out finger plays and rhymes and reading literature, which models the conventional counting order.

• Seize upon teachable moments as they arise incidentally. “Do we have enough pairs of scissors for everyone at this table?”

Seize teachable moments as they occur

Ten frames

Pick up chips :

• Take a card from the pile and pick up a corresponding number of counters.

• Play until all the cards have been taken.

• The winner is the person with the most chips at the end of the game.

Estimating

1 10

Round about what would this number be ?

Guess my number :• The leader thinks of a secret

number. The children may assist the teacher in drawing a line on the white board to indicate the range in which the secret number lies. The leader asks the group to try and guess the secret number. The group asks questions of the leader to try and ascertain the number. The leader may only answer yes or no to the questions. (A process of elimination)

Sandwich boards

• Add string to numeral cards so they can be hung around the students necks. Provide each student with a numeral card. Students move around the room to music. Once the music stops, the children arrange themselves into a line in a correct forward or backward number sequence.

Ask students why they lined up the

way they did.

Understanding is encouraged through

sharing ourthinking.

Listening to each otherfacilitates learning.It makes us think.Talk is vital in

building understanding.

More-less relationships

More-less relationships are not easy for young children.

Which group has more?

How many more?

More – less relationships

• How many?• What is two

more?• One less?

More-less relationships

Four-year-olds may be able to judge which of two collections has more, but determining how many more (or less) is challenging, even when they count.

More-less relationships

• Young children must arrive at the important insight that a quantity (the less) must be contained inside the other (the more) instead of viewing both quantities as mutually exclusive. The concept requires them to think of the difference between the two quantities as a third quantity, which is the notion of parts-whole.

Stages in comparison

1. There are more blue than red and there are less red than blue

2. There are seven more blue than red and seven less red

3. Ten is seven more than three and three is seven less than ten

© Catholic Education Office Tasmania 2012

Whilst students need many counting experiences, teaching should also emphasise equally decomposing or partitioning collections into parts.

Subitising(suddenly

recognising)• Seeing how many at

a glance is called subitising.

• Attaching the number names to amounts that can be seen.

Subitising(suddenly

recognising)• Promotes the part part

whole relationship.• Plays a critical role in

the acquisition of the concept of cardinality.

• Children need both subitising and counting to see that both methods give the same result.

10 bead string

• They enable children to subitise up to five and learn the number combinations which make ten.

Peek and say :

• Have a different number of containers with different numbers of objects under each. Ask the children to find the container with 2,5,3… objects.

• Take a number ticket and try to find the container hiding the matching number of objects.

Speedy dominoes:• Share the domino

pieces. Play the game in the same way as regular dominoes, except in this game there is no turn taking.

• As soon as players see the opportunity to place a domino in the game, they may do so. The winner is the first player to correctly place all the dominoes.

Part whole relationships

Partitioning numbers into part-part-whole forms the basis for children coming to understand the meaning of addition and subtraction.

Parts – whole relationships

• The parts – whole relationship refers to the notion that you can break up (partition) a quantity and move bits from one group to another without changing the overall quantity. (e.g. 5 can be thought of as 3 and 2 or 1 and 4 etc)

A ten frame is effective in teaching parts /whole relationships, as in this example of combinations that total six.

For a true understanding of number relationships,Teachers must encourage young children to work with quantity in a variety of situations using different math manipulatives over an extended period of time.

Mathematical concepts do not inherently lie in manipulatives. Children must construct the understanding.

3. The role of the Early childhood educator

Role of the educatorPlanning and resourcing challenging learning environments.

Supporting children’s learning through planned play activity.

Extending and supporting children’s spontaneous play.

Extending and developing children’s language and communication through play.

Role of the educator

Model mathematical language.Ask challenging questions.Build on children’s interests and natural curiosity.Provide meaningful experiences.Scaffold opportunities for learning & model strategies.Monitor children’s progress and plan for learning.

Assessment methodsCollect data by observation and or/listening to children, taking notes as appropriate

Use a variety of assessment methods

Modify planning as a result of assessment

Effective teachers are inclusive of all learners

4. Learning opportunities for numeracy - play spaces

Play spaces

Outdoors (climbing, tunnels,tents, riding, construction, sand & water, gardening, dance and gymnastics)

Puzzles (spatial puzzles, number games, sorting)

ICT (computer games, creative graphics software, programmable toys, digitalcameras, calculators, interactive whiteboard

Bobby Bear NCTM Illuminations

Calculator counting

Calculator counting contributes to a better grasp of large numbers, thereby helping to develop students number sense.

“It is a machine to engage children in thinking about

mathematics” (Swan and Sparrow 2005)

Cultivate an interest in number

“Is googolplex a number?Can you make the calculator count until it gets to googolplex?What other big numbers are there?”Harry aged 5

Play spacesRole play (home, shop, dress-up, puppets)

Construction (blocks, tracks, linking materials)

Display area (peg line, pinboards, magnet board)

Play trays (sand, water, multiple objects e.g. buttons, pasta, shells, leaves)

Mini-worlds (story/drama,cloth or sand tray environments, small toy animals, people, vehicles

Play spacesModelling & painting

Graphics (drawing,writing, recording,shapes)

Reading and listening

areas (story-telling,picture books, rhymes,songs, CDs, music &percussion

Charlotte aged 3

• (2010)• SAGE Books UK• Distributed in

Australia by• Footprint Books

References

AAMT & ECA. (2006). Position paper on Early Childhood Mathematics.www.aamt.edu.auwww.earlychildhoodaustralia.org.auDEEWR. (2009). Belonging, Being & Becoming: The Early Years Learning Framework for Australia.http://www.deewr.gov.au/earlychildhood/policy_agenda/quality/pages/earlyyearslearningframework.aspxPapic, M. & Mulligan, J. (2007). The Growth of Early Mathematical Patterning: An Intervention Study. In J.Perry, B, Dockett, S, Harley, E. (2007) Preschool Educators’ Sustained ProfessionalDevelopment in Young Children’s Mathematics Learning. Mathematics Teacher Education and DevelopmentSpecial Issue 2007, Vol. 8, 117–134. Available at: http://www.merga.net.au/documents/MTED_8_Perry.pdfTucker, K. (2010) (2nd. Ed.). Mathematics through play in the early years. London: Sage.Hunting, R. et al. Mathematical Thinking of Preschool Children in Rural and Regional Australia: Research andPractice. Report & video clips at: http://www.latrobe.edu.au/earlymaths/resources.html