Debunking misconceptions about mathematics in the

early years

Overview

1. Common misconceptions

2. Critical mathematical ideas underpinning number sense

3. The role of the early childhood educator

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.

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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

Five proficiencies

1. Conceptual understanding

2. Procedural fluency

3. Strategic competence- idea of choosing what is going to be an appropriate method to solve problems.

4. Adaptive reasoning - reason to other contexts

5. Productive disposition- most important

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)

Critical concepts underpinning number sense

More – lessCounting SubitisingPart part whole

More-less relationships

More-less relationships are not easy for young children.

Which group has more?

How many more?

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.

More – less relationships

• To help children with the concept of less, frequently pair it with the word more and make a conscious effort to ask “which is less?” questions

Make sets of more/less/same

For all three concepts, more, less and the same children should construct sets involving counters as well as make comparisons or choices between two sets.

Diffy Towers• Organise students into pairs and provide each

pair with a die and a supply of Unifix blocks. The first student rolls a die, takes a corresponding number of Unifix blocks from a central pile and builds a tower with them. The second student rolls the die and repeats the process. They then compare the two towers to see who has the most blocks and determine the difference between the two towers. The player with the larger number of blocks keeps the difference and all other blocks are returned to the central pile.

• The activity continues until one student accumulates a total of ten blocks

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

One and two more, one and two less

The two more and two less relationship involve more than just the ability to count on two or count back two. Children should know for example that 7 is 1 more than 6 and also 2 less than 9.

When Harry was at the circus, he saw 8 clowns come out in a little car. Then 2 more clowns came out on bicycles. How many clowns did Harry see altogether?

Ask different students to explain how they got their answer of ten. Some will count on from 8. Some may need to count 8 and 2 and then count all. Others will say they knew that 2 more than 8 is 10. The last response gives you an opportunity to talk about the 2 more than idea.

Early counting

The meaning attached to counting is the key conceptual idea on which all other number concepts are developed

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.

To develop their understanding of counting, engage children in

any game or activity that involves counts or comparisons.

Trusting Counting

There are many children who know the number string well enough to respond correctly to ‘how many’ questions without really understanding that this is telling them the quantity of the set.

1, 2, 3, 4, 5, 6, 7,…

2, 4, 6, 8, 10, …

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?”

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.

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.

Using number lines

“I was first to school today”

Ordinal number

Counting on and back

• The ability to count on is a “landmark” in the development of number sense.

Fosnot and Dolk (2001)

“Real Counting on”

First player turns over the top number card and places the indicated number of counters in the cup. The card is placed next to the cup as a reminder of how many are there. The second player rolls the die and places that many counters next to the cup. Together they decide how many counters in all.

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)

The calculator provides an excellent counting exercise for young children because they see the numerals as they count

Anchors or ‘benchmarks’ of 5

and 10Since 10 plays such a large role in our numeration system and because two fives make ten, it is very useful to develop the relationships for the numbers 1 to 10 to the important factors of 5 and 10.

Race to five/ten on a ten frame

 

• Roll the 3 sided or 6 sided die and count the dots.

• Collect the corresponding number of counters and place them on the five/ten frame.

• The exact number needed to complete the ten frame must be rolled to finish.

These early number ideas are basic aspects of number. Unfortunately, too may traditional programs move directly from these beginning ideas to addition and subtraction, leaving students with a very limited collection of ideas about number to bring to these topics. The result is often that children continue to count by ones to solve simple story problems and have difficulty mastering basic facts.

Subitising(suddenly

recognising)• Seeing how many at

a glance is called subitising.

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

• A fundamental skill in the development of students understanding of number.

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.

Conceptual subitiser to 5

• Verbally labels all arrangements to about 5 when only shown briefly

5easy

medium

Difficult

Conceptual Subitiser to 20(6 yrs)

• Verbally labels structured arrangements up to 20, shown only briefly, using groups.

“I saw three fives, so five, ten, fifteen”

Conceptual subitiser with place value and skip counting (7 yrs)

Verbally labels structured arrangements shown only briefly using groups, skip counting and place value. “I saw groups of tens and twos,

so 10, 20, 30, 40, 42, 44 …44!”

Conceptual subitiser with place value and multiplication (8 yrs)

Verbally labels structured arrangements shown only briefly using groups, multiplication and place value.

“I saw groups of tens and threes, so I thought 4 tens is 40 and 3 threes is 9, so 49 altogether”

Part whole relationships

To conceptualise a number as being made up of two or more parts is the most important relationship that can be developed about numbers.

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

Missing Cubes

Partitioning with bead strings

Move 8 beads to the end of the string. How many ways can you partition the beads in the next minute?

Record your findings so that you can describe them to others

How many different ways can you partition 8 dots in one minute on a ten frame?

Record your findings so that you can describe them to others

How many different ways are there for 5 frogs to be, in and out of the water?

What if there were 7 frogs? Can you find a pattern?

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

• Understanding part-part-whole relationships will enhance children’s flexibility, enabling them to represent problems in different ways, so they can choose the most helpful.

Relationships for numbers 10 to 20

A set of ten should play a major role in children’s initial understanding of number between 10 and 20. When children see a a set of six and a set of ten, they should know without counting that the total is 16

3. The role of the Early childhood educator

Role of the educator

Model mathematical language.Ask probing 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.

Probing Questions

A crucial part of a teacher’s role is to develop students’ ability to think about mathematics. To develop thinking processes teachers need to ask higher-order questions that require students to interpret, apply, analyse and evaluate information.

Encourage students to ask questions of each other so that they begin to develop maturity of thought.

The pedagogy• … less teacher talk, with the learning coming as

a result of the experience with the task and children sharing their insights.

• A culture of “not telling”· Listening to children· Encourage persistence· Probing questions• Learning involves struggle. They are not

learning if it’s not a struggle· Long ‘wait’ time· Time to reflect on their actions

“We use the word struggle to mean that students expend effort to make sense of mathematics, to figure something out that is not immediately apparent.  We do not use struggle to mean needless frustration...  The struggle we have in mind comes from solving problems that are within reach and grappling with key mathematical ideas that are comprehendible but not yet well formed”Hiebert and Grouws, 2007

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

References

AAMT & ECA. (2006). Position paper on Early Childhood Mathematics.www.aamt.edu.auwww.earlychildhoodaustralia.org.au

DEEWR. (2009). Belonging, Being & Becoming: The Early Years Learning Framework for Australia.http://www.deewr.gov.au/earlychildhood/policy_agenda/quality/pages/earlyyearslearningframework.aspx

Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. Second handbook of research on mathematics teaching and learning, 1, 371-404.

Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. Routledge.