Effective Mathematics Programmes

8 July 2011

Overview of Today• The Number Framework • Key Mathematical Ideas for Number• An Effective Mathematics Class• Activities you can use • What does a lesson look like?• Planning

Target

Try to make today’s target in each of these ways --

1. Adding two numbers.2. Finding the difference of two numbers.3. Multiplying two numbers.4. Dividing one number by another.5. Adding three numbers.6. Multiplying three numbers.7. Multiplying and subtracting.8. Using a decimal.9. Using a fraction.10. Doing it an unusual way.

24

What’s been happening?Share your successes and challenges with others at your table.

•What has been going well in your numeracy programme?

•What questions do you still have?

•Is there anything in particular you want answered today?

Number Sense

• Having a good intuition about numbers and their relationships.

• Develops gradually as a result of exploring numbers, visualising numbers, forming relationships

• Grows more complex as children learn more.

Big ideas

• Numbers are related to each other through a variety of number relationships - more than, less than, composed of

• “Really big” numbers possess the same place-value structure as smaller numbers. Best understood in terms of real- world contexts

• Whole numbers can be described by different characteristics, even and odd, prime and composite, square. Understanding characteristics increases flexibility when working with numbers

Key Mathematical Ideas

Early number sense• Counting tells how many are in a set.

Ordinality leads to Cardinality• Numbers are related to each other through a

variety of number relationships more than, less than, connection to ten

• Number concepts are intimately tied to the world around us. Application to real settings marks the beginning of making mathematical sense of the world.

Van de Walle , Karp & WilliamsElementary & Middle School Mathematics: Teaching Developmentally

Allyn & Bacon 2010

Key Mathematical Ideas

Developing Meanings for the operations• Addition and subtraction are related. Addition names the

whole in terms of the parts, subtraction names a missing part• Multiplication is related to addition• Multiplication involves counting groups of like size and

determining how many there are in all. Multiplicative thinking• Multiplication and Division are related. Division names a

missing factor in terms of the known factor and the product. • Models can be used to solve contextual problems for all

operations, regardless of the size of the numbers. They can be used to give meaning to number sentences.

Van de Walle & LouvinTeaching Student Centred Mathematics,

NZ Curriculum Objectives and Number Framework- What to teach

Effective Pedagogy- How you teach it- How you respond to students

and their misconceptions

The Number Framework

• Embodies the achievement aims and objectives in Levels 1 - 4

• A distinction is made between strategy and knowledge

• Progress through the stages indicates an expansion in knowledge and the range of strategies that children have available to them.

Strategy

The mental processes children use to estimate answers and solve operational problems with

numbers.

Knowledge

The key items of knowledge that children need to learn.

Strategy Knowledge

creates new knowledge through use

provides the foundation for strategies

Three operational domains

Add/Sub

Mult/Div

Prop/Ratios

Four content domains

Number Identification

Number Sequence and Order

Grouping/Place Value

Basic Facts

Number Knowledge

• Book One - Page 14

• What are the key messages about knowledge from the framework.

• Work with a partner to create a thinking map

• Share back one key point.

Stages and Expectations

8 Advanced Proportional End of Year 9

7 Advanced Multiplicative

Early Proportional

End of Year 8

6 Advanced Additive

Early Multiplicative

End of Year 6

5 Early Additive End of Year 4

4 Advanced Counting End of Year 2

3 Counting from one by imaging

2 Counting from one on materials

1 1-1 Counting Emergent

The Frameworks/Thinking Stages

• Look at your number framework• What are the key progressions contained in

the strategy framework? Knowledge Framework?

• Thinking Stages (Peter Hughes) – observable behaviours, problem types.

Scenarios• Using your frameworks sort each of the

responses from the scenarios and put them at the appropriate stage in the number framework.

• There are scenarios for each domain.

How is this useful? What connections have you made?

Fifty and some more

• Say a number between 50 and 100. Children respond with “50 and ____.

• For 63, the response is “50 and 13”

• Use other numbers that end in fifty such as 350, 650 or 0.5

Write all the ways….

• How many ways can you make

36

• Show as many different ways as you can to make 36 – use materials, words, word stories, digits…

• After 1 minute you will pass your paper to the next person.

StrategiesMaraea has $37. She spends $9. How much money does she have left now?

Caleb has saved $165. He banks another $23 dollars. How much money does he have saved now?

Jody has scored 284 goals in netball this season. She gets another 67. How many goals has she scored altogether?

Anaru has 312 tropical goldfish in his aquarium. He sells 198 of them to the pet shop. How many tropical fish does he have now?

The electrician has 5.33 metres of cable. He uses 2.9 metres on a job. How much cable is left?

Solve these problems independently. When you have your answers talk with your table about how you solved them, are there some key strategies, can

you give them a name? Describe what you did, make a chart.

Key Mathematical Ideas

• Look at the key mathematical ideas for number knowledge and strategy

• What are the big ideas at each level?

• How does this impact on what you are helping the children in your class to learn?

Making connectionsBuilding conceptual understanding

Any lightbulb moments?

If I know then I know

• If I know 3 + 5 = 8 what else do I know?

• If I know 3 x 5 = 15 what else do I know?

• Work with a partner – after 2 minutes you will pass your chart to another group.

What is the purpose of this activity?How could you use it in your class?

Card Games

• Share your favourite mathematics card game

• What is it practising?

Add to tenTwo players

Deal all cards out between two players.

Take turns to turn over one card - state what else makes 10.

Also play by taking number off ten.

Working backwards - subtraction is harder. Children need lots of practise with subtraction

Make Ten, Two players

Deal out ten cards in a row.

First player looks across the row for combinations that make ten.

Aim is to collect as many cards as possible, so combinations that require more cards are best.

Continue playing until all the cards are used or until there are no more combinations that add to ten.

Winner has the most cards.

Make Ten again, Two players

Deal all cards out in 3x3 grid

Take turns to make 10 -

Continue playing until all the cards are used or until there are no more combinations that add to ten.

Winner has the most cards.

Salute• You need three players• A pack of playing cards (take out 10s and colour cards

• Two players collect one card each. Without looking at the card they put it on their forehead.

• The third player calls out the sum of the two cards• The two players then call out what card they hold on their forehead by

looking at the other player’s cards.• The player who calls out first wins those cards. • Continue playing until all the cards are used.

Variations• 10 more or ten less/ one more or one less• Multiply • Doubles

Speed (War)Two players

Deal all cards out between two players.

Place one card in middle. - e.g. 2 (add this number to card that is turned over)

Take turns to turn over one card - both players call out answer. First to call wins both cards.

If a tie, turn over another card. Highest card gets to keep all three cards.

Also for multiplication

Clear the Decks• You need a pack of cards, picture cards removed.• Decide on the target and discard any cards that are

equal to or greater than the target number.• Layout an array of cards face up so that the number of

cards on show is one less than the target number. • Play by clearing away any two cards that add to the

target number. • Replace the cards as soon as they are chosen. • Aim is to clear the whole pack of cards. • Read the number sentence as the cards are cleared

away.

An array if the target number is 6

Making Tens

• Deal out four cards

• Sort the cards from smallest to largest. Name the missing numbers

• Add the cards together to find the total– Use a range of strategies

– Make a ten 7+3, 5+5

– Use doubles and doubles + or – 1

Making Numbers

• Deal out the cards

• Turn over two cards - make the largest number you can.

• Make the smallest number.

• How many tens, how many ones?

• Repeat with three cards, put cards numbers in order – smallest to biggest.

Combinations

Aim: to record as many combinations using up to four cards

Deal out four cards.

Record as many combinations as you can - use all the operations.

E.g. 6, 4, 7, 9 is dealt

9+6+7- 4 =11 6 x 4 + 7 = 31 9 - 7 = 6 - 4

I Spy

Using Tens Frames/Dot cards

• Subitizing is the ability to instantly recognise a pattern or a number.

• Use the tens frames for instant recognition - what number is on the tens frame.

• How many more to make ten?• Tell the tens story - family of facts e.g.

– 6+4=1=, 4+6=10, 10-6=4, 10-4=6

• Make teen numbers - ten and four = fourteen

Triplets – Family of Facts

• Introduce triplets

• 10 , 6, 4

• Make chains of number triplets

10

6 4

1,2,3 Fists - Paper, Scissors, RockTwo players

Play as for Paper, Scissors, Rock

One or two hands

Count 1,2,3, put down some fingers - add/multiply together

True or False?Are the following statements true or false? Decide on

your answer – discuss with a partner and see if you agree.

93 - 38 = 91 – 40

Cain and Ryan went shopping at the $2 shop. Cain spent half of his pocket money, Ryan spent one

quarter of his pocket money. Cain spent the most.

True, False or Maybe

Some more true/false

Twenty + 30 + 3 tens = 8 tens

☐ = 3 + 4

2 + 3 = ☐ + 1

Challenge

• Think about some common misunderstandings you have noticed in your class.

• In your group write some true/false questions that you could pose to your class

• Use key mathematical ideas to help you if you are stuck.

Activities that reinforce place value knowledge• 100 or bust• Rocket - change numbers to suit the level.• Traffic Light - book 4 page 25

• Thousands book• Digit Cards - playing cards deal out three card , make the

largest, smallest number you can - order them in your group. Who has the highest? Lowest? How do you know?

• Staircase/Grid and a dice - strategy game. Make the highest lowest number

An effective mathematics class

• Look at the characteristics of an effective mathematics class. If a visitor walked into your classroom, what would they see, hear, feel.

• Highlight what you are doing on the chart.• Discuss with your table. • What are you not yet doing? • Identify one area that you want to work on?

Discuss with your table – ideas, places to start.

Using Thinking Groups

“If you teach without observing and reacting to the children,

you are teaching a programme, you are not teaching the children.”

Peter Hughes

The Lesson

• Warm up – knowledge• Group teaching – knowledge or strategy

– Pose a question, children talk to each other, listen and respond

– If they know it already, do something else!! If it’s too hard, fold back and repeat earlier learning

• Reflection, Plenary, Warm Down

Modeling Book to keep track of what has been learnt.

Lynne Petersen

• Teaching using the diagnostic question and using groups of four.

• Watch Lynne – what do you notice?

• What implications can you see for your own practise?

Planners

• Download from NZ Maths• Show you the problem progression, link

knowledge and strategy• Are long term, you will still need to have a

daily plan

Sharing Learning with children

• Student Profiles

• Strategy Progess

• Basic Facts

Key understandings

• Multiplication – 6 x 5 means 6 lots of five, there are 5 sweets in each bag and we have six bags altogether there are 30 sweets – important we do multiplication this way.

• If we know a fact we can turn it around to help us find out an answer we don’t know

Fraction Language

Use words before and use symbols with care.

e.g. ‘one fifth’ not 1/5

How do you explain the top and bottom numbers?

1

2

The number of parts chosen

The number of parts the whole has been divided into

Let the fraction do the talking!

What fraction of the shape is shaded?

3 sevenths 3 out of 7 7/3 7 thirds

5 views of fractions

€

3

73 over 73 : 7

3 out of 7 3 ÷ 7

3 sevenths

+ =

“I ate 1 out of the 2 sandwiches in my lunchbox, Kate ate 2 out of the 3 sandwiches in her lunchbox, so together we ate 3 out of the 5 sandwiches”

12

23

35

The problem with “out of”

23

x 24 = 2 out of 3 multiplied by 24 !!!!!

Resources

• Today’s presentation will be on the Lead teacher wikispace

• NZMaths is your “go to” place for planners, units, content support.

Two thoughts to leave you with

…listen to children’s mathematical explanations rather than listen for particular responses.

Fiona Walls, in Handling Number, p.27s, Teaching Primary School Mathematics and Statistics Evidence-based Practice

Averill & Harvey (Eds), NZCER 2010

If we were meant to talk more than listen we would have two mouths and one ear!

Reflection

• What will you go back to class and try next week?

• What has been a lightbulb moment for you?

• What connections have you made?

Picture title