thinking and working mathematically

1

Thinking and Working Mathematically

Loddon Mallee

2010

Numeracy is the capacity, confidence and disposition

to understand and apply mathematical concepts,

problem solve, collect and analyse data and to make

connections within mathematics to meet the demands

of learning at school, work, home, community and

within civic life.

Literacy and Numeracy Statement

Blueprint Implementation Paper

DEECD 2009

Numeracy

Common Numeracy Situations in Everyday Life.

Activity

A look at some time in a normal day

eg: List the mathematical decisions you made from the time you woke up this morning until about 9.30?

eg: List the mathematical decisions you might be making while in a car on the way to work or school this morning

eg: List mathematical decisions you made in your lunch hour

An activity for maths coordinators for professional learning for teachers; and an activity to use with students

This module:Explicit Instruction:

– Thinking and Working Mathematically using the Multi- Modal Think Board

– Differentiation using the Multi- Modal Think Board– Using questions to differentiate tasks within mathematical

modes– An Instruction Model with differentiation in mind

5

Differentiating by:

• Asking frequent, targeted, rigorous questions of students as they demonstrate mastery

• Planning ,working, assessing and reflecting in different mathematical modes

• Working in different ways within a given mode• Using tasks that are open question based

6

The Mathematical Modes[The Singapore Multi-modal Think Board]

Thinking/Working Mathematically:A Think-Board [Multi-Model] to Teach Mathematics

Using Multi-Modal Think-Board to Teach Mathematics Khoon Yoong Wong,

Mathematics and Education Academic Group , National Institute of Education,

Nanyang Technological University, Singapore, July 2004

e5

Number-calculate e5

Word- communicate

e5

Diagram- visualisee5

Symbol- manipulate [algebra]

e5

Real Thing- do

[eg: manipulative materials]

e5

Story- apply

Thinking/Working Mathematically

8

Group Work in Mathematics:

Whole group- small group – whole group

Pairs is the most effective way to work in groups in

mathematical contexts. (Students also need to be given

the opportunity to work individually and practise being

able to work individually).

Using the Multi-Modal Think Board

Topic - Measurement and Subtraction

Real Thing – do [Level 1- 5]

What might students ‘do’ [action] to calculate the difference in height between 2 people?

Real Thing – do [Level 5-6]

For two ladders - ladder [a] 410cm in length and ladder [b] 420cm in length

Ladders [a] and [b] are leaning against a wall. They touch the wall 400cm above the ground. What is the difference in the distance between the foot of each of the ladders and the wall?

Working/Thinking MathematicallyUsing Multi-modal Think Boards Khoon Yoong Wong 2004

Real Thing- Do– the use of concrete manipulatives

– principle- learning by doing: I hear I forget; I see and I remember; I do and I understand [Piaget,Bruner]

– grounding mathematical ideas in concrete situations helps develop mental models that provide meaning to abstract symbols , hence reducing the chance of anxiety phobia towards mathematics

– without sufficient practical experience students have been found to lack numerical sense of measures about real objects and hence cannot determine whether their answers are reasonable or not in the real world

Working/Thinking MathematicallyUsing Multi-modal Think Boards Khoon Yoong Wong 2004

Real Thing- Do [continued]

– the transition from practical activities to formal abstraction, however, is not easy [Johnson 1989]

– poorly designed manipulatives or improper use can hinder rather than facilitate conceptual development

– ‘virtual manipulative’ – electronic technologies to support effective mathematics teaching

Using the Multi-Modal Think Board

* [Level 2- 4] Calculate the difference in number, between two groups of objects. One group of 39 and the other 17.

*[Level 3- 5] Calculate the difference between 2.48m and 11.48 m

Symbol – manipulate– whole - part concept comparison [ compare collections when one collection is

larger than another and with like and unlike objects]

– change concepts [ increase or increment problems and decrease or deficit problems]

– manipulate the equation [‘milk the equation for all it’s worth’]

Story – applyWrite problems with an authentic context using the equations thatresult

Using the Multi-Modal Think Board

Symbol Mode- manipulate

Real - apply

* 12 + 5 = 17

* 17 + 39 = 56

•7 x 6 = 42

*480 ÷  20 = 24

1.Rewrite the equations in as many ways as you can using only the numbers [values] provided. One of the numbers needs to be represented as an ‘unknown’ [variable] in each equation you write.

2. Write word problems (which have an authentic context) for some of the equations

The Change Concept

Try these for example – and there are more:

1. 39 + 17 = ( increase, result unknown)

2. 17 + = 56 ( increase, change unknown)

3. + 17 = 56 ( increase, start unknown)

4. 56 - 17 = = ( decrease, result unknown)

5. - 39 = 17 ( decrease, start unknown)

1739?

17 ?56

f5617 ?

56

17 ?

?

39 17

Using the Multi-Modal Think Board

Topic - Measurement and Division

86 220m of rope was divided into 6 equal lengths to be sold. How much rope was in each of the lengths? If 2/3 of the rope lengths were damaged in a fire how many metres of rope were not damaged?

Symbolic – manipulate

Syntactic- what do I need to know to work this out with a calculator?; division operation; fraction as an operator…

Diagram- visualise- How might we demonstrate this problem in a diagram?

Using the Multi-Modal Think Board Khoon Yoong Wong 2004

Diagram- VisualiseKey word: represent• pictures• diagrams• graphs • charts• figures• illustrations• come in varying degrees of abstraction eg: a picture of several

apples versus several dots

• can be a pictorial summary of work done in ‘real thing ‘ mode

• visual imagery ‘in the mind’s eye’

Using the Multi-Modal Think Board

Topic – Division of fractions

6 ÷  ½ =

6 ½

¼

Diagram – visualise

Story - application

=

Using the Multi-Modal Think Board

Topic – Fractions

One hundred and eighty people attended a school function. If 1/3 of them were students how many people were not students?

Number- Calculate Essential basic skillsProcesses Algorithms‘working out’Strategies

Diagram - visualise

180

60 60 60

1/3 180 people

1/3 students

180 ÷ 3 = 60

One third = 60

Using Multi-Modal Think Boards Khoon Yoong Wong 2004

Word – Communication– words are essential for communicating mathematical ideas and

thinking about them

– as a mode of representation, it also includes phrases and sentences

– as students often confuse the meaning of the same work when used in everyday situations and in mathematics

– more acute when students learn mathematics in a foreign language

– teachers should say mathematical terms precisely and consistently eg: x² is ‘x to the power of 2’ and not ‘x two’

Thinking and working mathematically

•6 modes for thinking and working mathematically

•Instruction in all modes regularly, consistently- ie: where appropriate

•Explicit instruction using closed questions

•Tasks design - using open questions

•Differentiation though using open questions

Using the Multi-Modal Think Board

Topic- multiplication [Level 4 - 5]

A closed question

Peter planted tomatoes seedlings in 35 rows with 20 in each row. If each plant produced [an average of] 43 tomatoes, what was the total crop?

Pairs/draw/discuss

In what ways might you represent this problem using a ‘diagram’ ?

Opening up the question/task

If Peter planted 375 tomatoes in rows and each plant produced 43 [on average] tomatoes, what might the planting in the rows look like? How many tomatoes did he have to sell?

If students were asked to represent this problem using manipulative materials/contexts what might that involve?

del45 x 25 =

45

25x800 200

25100

x 4 52

5

800 100

200 25

1 125[40x20]+[40x5]+[5x5]+[5x20]=1 125

900

225

2 3 x

3

4

0

91

60

281 0

28

1

7

Lattice method

Differentiation- calculate in different ways

Using the Multi-Modal Think Board

Topic- [might be?]

A closed task [Level 5]

Round off 1.29 to the nearest tenth

In what ways might you represent this problem using a ‘diagram’ ?

Pairs/draw/discuss

Opening up the question/task [Level 5]

What numbers when rounded off become 1.3?

What modes could you ask students to use to model / demonstrate understanding here?

del

Using the Multi-Modal Think Board

Topic – [might be ?]

Closed context/task [Level 5] 0.7 x 5 =

Open task could be: [Level 5]

The product of two numbers is 3.5. What might be the two

numbers be?

Pairs - What are activities you could ask students to do in each of the modes for this problem?

Diagram - visualise

Number - calculate

Story - apply

Real Thing - do

Symbol - manipulate

Word - communicate

del

Using a Multi- Modal Think Board

Topic – [might be ?]

Closed context/task [Level 6]

Circle the number which is closest to 5.4

5.3 5.364 5.46 5 5.6 5.453

Open task

Word – communicate

One of your friends ask you to explain the best way to decide which number is closest to 5.4. How would you explain how to work out which number is closest to 5.4?

Using the Multi – Modal Think BoardContent specific mathematics through questions/tasks

Which fraction is smaller?

A corresponding open question/task is:

Level 5

What are some fractions smaller than

Discuss – in pairs, work through the 6 modes. How might this task be addressed using each of the modes?

DiscussPeter Sullivan 2003

3 4

4 5or

4

5?

27

Addition of Fractions

Closed question

[Level 5.6] Open task Progressively remove numbers replacing them with blanks gets us to

a task like this

1 7

3 4 + =

1 1

? ?+ =

?

?

?

12

A Balanced Mathematical ProgramCharles Lovitt

A balanced mathematics program:

Will meet individual needs of students

AND

Ensure students are working mathematically

What do we mean by ‘working mathematically’?

[Turn and talk]

A Balanced Mathematics ProgramCharles Lovitt

Working mathematically simultaneously involves:

– essential skills practice

AND (of equal importance)

– thinking, reasoning and communication (Dimension of Structure)

AND

– meeting the demand of huge mixed ability in any given group- [potentially a 7 year spread in any class]

Using the Multi - Modal Think Board

Division of a decimal by an integer

0 . 4 ÷ 2 = [zero point four divided by two]

Task [a] and [c] – most students nodifficulty

Task [d] to [f] – more difficult

When the task is changed to

0.4 ÷ 0.2 = even [a] and [b] become difficultand most students would not beable to complete

Using Multi-Modal Think-Board to Teach Mathematics Khoon Yoong Wong

• [a] Read this aloud - word• [b] calculate its value [not with

a calculator] - number • [c] draw a diagram to illustrate

the operation - diagram• [d] demonstrate the operation

using real objects - real thing• [e] write a story or word

problem that can be solved using this operation – story

• [f] extend this operation to algebra – symbol - symbol

Think- Board

Use for:• planning• instruction• reflection• assessment

An Instruction Model [one of many…]

Andrew Fuller- The Get It! Model

http://www.lccs.org.sg/downloads/10Creating_Resilient.pdf

Link

‘We’re sometimes socialised to think we have to break students up

into different instructional groups to differentiate, giving them different

activities and simultaneously forcing ourselves to manage an

overwhelming amount of complexity.’

Doug Lemnov, Teach like a Champion 2010

‘Asking frequent, targeted, rigorous questions of students as theydemonstrate mastery, is a powerful and much more effective tool fordifferentiating’

Ask how or why.Ask for another answer.Ask for a better word.Ask for evidence.Ask students to integrate a related skill.Ask students to apply the same skill in a new setting.

Doug Lemnov, Teach like a Champion 2010

Creating Resilient Learners- The Get It! Model of Learning 2003Andrew Fuller

5 minsMaximum10 minutes

10-15 mins

10 mins 10- 15 minutes 5 mins

Approximate Times [arbitrary]

Instruction Model for Long Term Memory Input- Andrew Fuller

Andrew Fuller Creating Resilient Learners- The Get It! Model of Learning 2003

‘Window of Opportunity’- Long Term Memory

5

mins

Maximum10 minutes

10-15 mins

10 mins 10- 15 minutes 5 mins

Ritual

Tying it together. Trying out new behaviours – new knowledge and understanding

2nd Memory Peak

Instruction Model- Andrew Fuller

Suggested/arbitrary

Further exploration of new knowledge and understanding

Creating Resilient Learners- The Get It! Model of Learning 2003Andrew Fuller

5 minsMaximum10 minutes

10-15 mins

10 mins 10- 15 minutes 5 mins

Approximate Times [arbitrary]

Instruction Model for Long Term Memory Input- Andrew Fuller

Closed Question [s]

Modelling/Explicit teaching

Open question [to differentiate a task]

Exploration of the task

Whole Group discussion

Target Group

Skills practice

Demonstrate understanding/new knowledge ‘another way’ and or a new open question around the key understanding for the session

Whole Group- reflection

Andrew Fuller Creating Resilient Learners- The Get It! Model of Learning 2003

5 mins Maximum10 minutes

10-15 mins

10 mins. 10- 15 minutes 5 mins

Whole group

Small groups

[pairs/individual]

Whole group

Small group [target]

Small groups- pairs/individual

and Independent -Skills Practice

Whole Group

Whole Group/Small Group

Modelled, Shared, Guided Mathematics

Suggested/arbitrary times

Andrew Fuller Creating Resilient Learners- The Get It! Model of Learning 2003 and

John Hattie- Visible Learning

5 mins

Maximum10 minutes

10-15 mins

10 mins 10- 15 minutes 5 mins

Learning Intentions

Modelling

Intention of the lesson- focus

Success Criteria

Checking for understanding

Guided Practice

Modelling

Checking for understanding

Closure

Independent Practice and or Guided Practice

Independent Practice

Direct Instruction Model and the Get it Model

Suggested/arbitrary times

Andrew Fuller Creating Resilient Learners- The Get It! Model of Learning 2003

5 minMaximum10 minutes

10-15 mins

10 mins 10- 15 minutes

5 mins

Engage

Engage/Explain

Explore/Explain/Engage

ExplainElaborate/Engage/Explore

Evaluate

E5 and Instruction- how it might look

Arbitrary times

• ‘The sequence of learning does not end with a right answer; reward right answers with the follow-up questions that extend knowledge and test for reliability. This technique is particularly important for differentiating instruction’ [Doug Lemnov p41.]

• closed • open• ways to write good questions• using open questions to differentiate tasks

•

Questions

• Peter Sullivan and Pat Lilburn– Working backwards– Adapt a standard question

What are ways to create good questions?

How to Create Good Questions

Peter Sullivan/Pat LilburnOpen-ended Maths Activities, Peter Sullivan, Pat Lilburn, Oxford University Press 1997

Method 1: Working Backwards:

Step 1 Identify a topic

Step 2 Think of a closed question and write down the answer.

Step 3 Make up a question which includes [or addresses] the answer

eg:

Money

Total cost $23.50

I bought some items at the supermarket. What might I have bought

and what was the cost of each item?

How to Create Good Questions

Peter Sullivan/Pat LilburnOpen-ended Maths Activities, Peter Sullivan, Pat Lilburn, Oxford University Press 1997

Method 2: Adapting a standard question:

Step 1 Identify a topic

Step 2 Think of a standard question

Step 3 Adapt it to make a ‘good’ question

eg:

Subtraction

731-256=

Arrange the digits so that the difference is between 100 and 200

• The Question Creation Chart- Education Oasis 2006

What are ways to create good questions?

Is Did Can Would Will Might

Who

What

Where

When

How

Why

Question Creation Chart (Q-Chart)

Directions: Create questions by using one word from the left hand column and one word from the top row. The farther down and to the right you go, the more complex and high-level the questions.

Working/Thinking MathematicallyUsing Multi-modal Think Boards Khoon Yoong Wong 2004

Story- ApplyLinking real world

mathematics to ‘text

book mathematics

reinforces concepts and

skills and enhances

motivation for learning

Story- Apply• traditional word problems

related to everyday situations• reports in the mass media• historical accounts of

mathematical ideas • examples from other

disciplines

• students can and should generate their own

Working/Thinking MathematicallyUsing Multi-modal Think Boards Khoon Yoong Wong 2004

Using the multi–modal Think Board for Planning, Assessment

and Reflection– a series of lessons on a particular topic– a lesson

– consider carefully whether all or only some modes will be used in which sequence

– ie: determine the optimal combination

– perhaps begin with concrete manipulative materials and support/supplement with virtual [ICT]

– eg: students may be asked to explain why [a+b]² = a²+ b² using number, diagram and real thing

Working/Thinking MathematicallyUsing Multi-Modal Think Boards

A Suggested Sequence

Real Thing

Number Word

Diagram

Symbol

Story

Virtual Manipulative

Academic Group , Khoon Yoong Wong 2004National Institute of Education, Nanyang Technological University, Singapore, July 2004

Working/Thinking MathematicallyUsing Think Boards

Teachers:

For planning – day to day, weekly, units of work

For embedding the e5

For reflection

For assessment -encompassing a variety of approaches

For……

Students:

For reflection

For ways of demonstrating understanding/new understanding

[elaboration/explanation/reflection…]

For problem solving

For……..

Turn and talk.

Khoon Yoong Wong, Using Multi-Modal Think-Board to Teach Mathematics Khoon Yoong Wong,Mathematics and EducationAcademic Group , National Institute of Education, Nanyang Technological University, Singapore, July 2004 -paper

Peter Sullivan and Pat Lilburn, Open-ended Maths Activities OxfordUniversity Press 2000

Andrew Fuller Creating Resilient Learners- The Get It! Model of Learning 2003 –PaperJohn Hattie, Visisble Learning Routledge 2009

George Booker, Denise Bond, Len Sparrow and Paul Swan, TeachingPrimary Mathematics 3rd Edition Pearson Prentice Hall 2004

Doug Lemnov, Teach Like a Champion, Jossey – Bass 2010