Common Misunderstandings Adrian Berenger 24 February 2010 Teaching & Learning Coach Moreland 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 Count Developing 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?

Place-value Moving 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 Thinking Understanding 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?

Partitioning Building 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 Reasoning Extending 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

Generalising Skills and strategies to support equivalence, recognition of number properties, and use of algebraic text Understanding Equivalence Number Properties Pattern Recognition Understanding Algebraic Language

Pattern Recognition Number of Squares Number of Matches n 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

Q4. SPEEDY SNAIL a. 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?