Thinking and Working Mathematically
Loddon Mallee
1
Numeracy
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
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
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
5
The Mathematical Modes
[The Singapore Multi-modal Think Board]
6
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
e5
Story- apply
Numbercalculate
e5
Wordcommunicate
Thinking/Working
Mathematically
e5
e5
Real Thing- do
[eg: manipulative
materials]
Diagram- visualise
e5
Symbolmanipulate
[algebra]
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).
8
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 Mathematically
Using 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 Mathematically
Using 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 – apply
Write problems with an authentic context using the equations that
result
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 +
( increase, change unknown)
3.
= 56
+ 17 = 56
( increase, start unknown)
39
17
?
17
56
?
?
17
56
f
4.
56 - 17 = =
56
( decrease, result unknown)
17
5.
- 39 = 17
( decrease, start unknown)
?
?
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- Visualise
Key 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=
6½
=
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 skills
Processes
Algorithms
‘working out’
Strategies
1/3
1/3 students
Diagram - visualise
60
60
180
180 people
180 ÷ 3 = 60
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?
2
Differentiation- calculate in
different ways
0
0
6
del
45 x 25 =
1
7
x
3
10
8
8
9
1
3
Lattice
method
2 4
2
25
x
800
200
4
5
2
800
100
900
5
200
25
225
1 125
x
45
100
25
[40x20]+[40x5]+[5x5]+[5x20]=1 125
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?
Using the Multi-Modal Think Board
Topic – [might be ?]
del
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
Using a Multi- Modal Think Board
del
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 Board
Content specific mathematics through questions/tasks
Which fraction is smaller?
3
4
4
or
5
A corresponding open question/task is:
Level 5
What are some fractions smaller than
4
?
5
Discuss – in pairs, work through the 6 modes. How might this task
be addressed using each of the modes?
Discuss
Peter Sullivan 2003
Addition of Fractions
Closed question
1
3
7
+
4
?
=
12
[Level 5.6] Open task
Progressively remove numbers replacing them with blanks gets us to
a task like this
1
?
+
1
?
=
?
?
27
A Balanced Mathematical Program
Charles 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 Program
Charles 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 =
divided by two]
[zero point four
Task [a] and [c] – most students no
difficulty
Task [d] to [f] – more difficult
When the task is changed to
0.4 ÷ 0.2 =
even [a] and [b] become difficult
and most students would not be
able 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 they
demonstrate mastery, is a powerful and much more effective tool for
differentiating’
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 2003
Andrew Fuller
Instruction Model for Long Term Memory Input- Andrew Fuller
Approximate Times [arbitrary]
5 mins
Maximum10
minutes
10-15
mins
10 mins
10- 15 minutes
5 mins
Andrew Fuller Creating Resilient Learners- The Get It! Model of Learning 2003
Instruction Model- Andrew Fuller
‘Window of
Opportunity’- Long
Term Memory
Ritual
Trying out new behaviours – new
knowledge and understanding
2nd Memory
Peak
Tying it together.
Further exploration of new knowledge
and understanding
5
mins
Maximum
10
minutes
10-15
mins
10 mins
Suggested/arbitrary
10- 15 minutes
5 mins
Creating Resilient Learners- The Get It! Model of Learning 2003
Andrew Fuller
Instruction Model for Long Term Memory Input- Andrew Fuller
Closed Question [s]
Modelling/Explicit teaching
Open question [to differentiate a task]
Whole Group- reflection
Exploration of the task
Whole Group discussion
Target Group
Skills
practice
5 mins
Maximum10
minutes
10-15
mins
Demonstrate understanding/new knowledge
‘another way’ and or a new open question
around the key understanding for the session
10 mins
Approximate Times [arbitrary]
10- 15 minutes
5 mins
Andrew Fuller Creating Resilient Learners- The Get It! Model of Learning 2003
Modelled, Shared, Guided Mathematics
Whole Group/Small Group
Whole Group
Small groups
Whole group
[pairs/individual]
Small groups- pairs/individual
Whole group
Small group and Independent -Skills Practice
[target]
5 mins Maximum10
minutes
10-15
mins
10 mins.
Suggested/arbitrary times
10- 15 minutes
5 mins
Andrew Fuller Creating Resilient Learners- The Get It! Model of Learning 2003
and
John Hattie- Visible Learning
Direct Instruction Model and the Get it Model
Modelling
Checking for understanding
Success Criteria
Closure
Independent
Practice
Guided Practice
Intention of the lesson- focus
Checking for understanding
Independent Practice and or Guided Practice
Modelling
Learning Intentions
5
mins
Maximum10
minutes
10-15
mins
10 mins
Suggested/arbitrary times
10- 15 minutes
5 mins
Andrew Fuller Creating Resilient Learners- The Get It! Model of Learning 2003
E5 and Instruction- how it might look
Engage/Explain
Explore/Explain/Engage
Evaluate
Engage
Explain
Elaborate/Engage/Explore
Arbitrary times
5 min
Maximum
10
minutes
10-15 10 mins
mins
10- 15
minutes
5 mins
Questions
• ‘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
What are ways to create good questions?
• Peter Sullivan and Pat Lilburn
– Working backwards
– Adapt a standard question
How to Create Good Questions
Peter Sullivan/Pat Lilburn
Open-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 Lilburn
Open-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
What are ways to create good questions?
• The Question Creation Chart- Education Oasis
2006
Question Creation Chart (Q-Chart)
Is
Did
Can
Would
Will
Might
Who
What
Where
When
How
Why
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 Mathematically
Using Multi-modal Think Boards Khoon Yoong Wong 2004
Story- Apply
Story- Apply
Linking real world
mathematics to ‘text
book mathematics
reinforces concepts and
skills and enhances
motivation for learning
• 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 Mathematically
Using 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 Mathematically
Using Multi-Modal Think Boards
A Suggested Sequence
Real Thing
Virtual Manipulative
Word
Number
Diagram
Symbol
Academic Group , Khoon Yoong Wong 2004
National Institute of Education,
Nanyang Technological University, Singapore,
July 2004
Story
Working/Thinking Mathematically
Using 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 Education
Academic Group , National Institute of Education, Nanyang
Technological University, Singapore, July 2004 -paper
Peter Sullivan and Pat Lilburn, Open-ended Maths Activities Oxford
University Press 2000
Andrew Fuller Creating Resilient Learners- The Get It! Model of Learning 2003 –
Paper
John Hattie, Visisble Learning Routledge 2009
George Booker, Denise Bond, Len Sparrow and Paul Swan, Teaching
Primary Mathematics 3rd Edition Pearson Prentice Hall 2004
Doug Lemnov, Teach Like a Champion, Jossey – Bass 2010