Common Misunderstandings

Adrian Berenger24 February 2010

Teaching & Learning CoachMoreland Network

Scaffolding Numeracy

• Teachers need to know what each student knows and the next step that they may be able to achieve with support.

• Teachers need to know what individual responses might mean.

• This is particularly important in the area of Number without which students’ progress is seriously impacted.

COMMON MISUNDERSTANDINGS

Trusting the Count: Developing flexible mental objects for the numbers 0 to 10

Place Value: The importance of moving beyond counting ones, the structure of the base ten numeration system

Multiplicative Thinking: The key to understanding rational number and developing efficient mental and written computation strategies in later years

Partitioning: The missing link in building common fraction and decimal knowledge and confidence

Proportional Reasoning: Extending what is know about multiplication and division beyond rule based procedures to solve problems involving fractions, decimals, percent, ratio, rate and proportion

Generalising: Skills and knowledge to support equivalence of number properties and patterns, and the use of algebraic test without which it is impossible to engage with broader curricula expectations at this level

WARM UP ACTIVITY

Who done it!• Bugsy is taller than Spike.• Spike is taller than Bonnie.• Mugsy is shorter than Bonnie.• Lefty is taller than Bonnie, but shorter than Spike.• Fingers is standing somewhere between Shifty and

Happy.• Happy is shorter than Bugsy, but taller than Fingers.• Curly is standing exactly between Bugsy and Clyde.• Clyde is next to Happy.• There are seven suspects standing between Curly and

Mugsy.

MEET THE MUGS!

Responses?

Trusting The CountDeveloping flexible mental models for numbers 0 to 10

• Subitising– Being able to recognise objects grouped in ones, twos,

…fives.– To make, count, name and record small collections

• Ten frames• Open number lines

• Mental Models– Sophistication of counting strategies and being able to

deal with unseen number.• Mental object cards

How many?

How many?

How many?

How many?

How many?

How many?

How many?

Place-valueMoving beyond counting by ones, the structure of the base 10

numeration system

• Number Naming • Efficiency Counting • Sequencing • Renaming and Counting

Renaming & Counting

• Using concrete materials (MAB) to represent and rename numbers.– These activities specifically target ‘place-value’.

Multiplicative ThinkingUnderstanding rational number and developing efficient mental and written computation strategies in later years

• Countable Units• Additive Strategies • Sharing • Array and Region • Cartesian Product • Proportional Reasoning

Cartesian Product

• How do you work this out efficiently?

PartitioningBuilding common fraction and decimal knowledge and

confidence• Equal Parts • Fraction Naming • Fraction Making • Fraction Recording • Decimal Fraction Naming and Recording • Comparing and Ordering • Rational Number Sequencing

Comparing & Ordering

• Using Flexible Number Lines

Proportional ReasoningExtending what is known about multiplication and division

beyond rule-based procedures• Solving problems involving fractions, decimals,

percent, ratio, rate and proportion– Relational thinking– Sense of percent– Understanding scale factors– Relative proportion– Interpreting rational number– Understanding ratio– Working with rate– Using percent

Understanding Ratio

• CORDIAL RECIPE– How much cordial can John

make using 7 cups of concentrate?

• PARTY PUNCH RECIPE– Jean had 11 cups of

lemonade, how many cups of fruit juice does she need?

Mix 2 cups of Mango and Apple Juice with 3 cups of lemonade. Serve chilled.

Mix 1 cup of concentrate to 4 cups of water

GeneralisingSkills and strategies to support equivalence, recognition of

number properties, and use of algebraic text• Understanding Equivalence • Number Properties • Pattern Recognition • Understanding Algebraic Language

Pattern RecognitionNumber

of Squares

Number of

Matches1 42 73 104

31n

• What are the missing numbers?• Can you describe in your own words how to work out the number of matches if we know the number of squares?• Can you draw a model to describe what is happening?

GWPS Multiplicative Thinking Assessment FEB2010

Summary of Screening

1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

Multiplicative Thinking Feb2010 GWPS

ZONES

No

of S

tude

nts (

G5/6

)

Q4. SPEEDY SNAILa. Harry’s snail can travel at 15 centimetres per minute. • How far might Harry’s snail travel in 34 minutes? b. Samantha’s snail covered 1.59 metres in 6 minutes. • How far might Samantha’s snail travel in 17 minutes. Record your

answer in metres. c. Harry entered his snail in a race. Remember Harry’s snail can travel

at 15 centimetres per minute. Another snail entered in the same race, covered 3.71 metres in 24 minutes. Which is the faster snail?

• Show all your working so we can understand your thinking.

Conclusion• Responses to speedy snail.

• Questions?