Teaching for Understanding and Engagement
Debbie Draper
Maths & Comprehension
Module 11
Strategies that Work
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Acknowledgement of Country
We recognise Kaurna people and their land
The Northern Adelaide Region acknowledges
that we are meeting on the traditional
country of the Kaurna people of the Adelaide
Plains. We recognise and respect their cultural
heritage, beliefs and relationship with the
land. We acknowledge that these beliefs are
of continuing importance to the Kaurna
people living today.
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NAR Facilitator Support Model – Team Norms
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Be prepared for meetings and respect punctuality
Be open to new learning
Respect others opinions, interact with integrity
Stay on topic, maintain professional conversation
Allow one person to speak at a time and listen actively
Enable everyone to have a voice
Discuss and respect diversity and differing views in a professional
manner and don’t take it personally
Accept that change, although sometimes difficult, is necessary for
improvement
Be considerate in your use of phones/technology
Be clear and clarify acronyms and unfamiliar terms. Ask if you
don’t understand.
Commit to follow through on agreed action
Respect the space and clean up your area before leaving
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Overview
• Mathematics Teaching – research findings
• Mathematics and Integral Learning
• Comprehension Strategies applied to
Mathematics
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Comprehending Math:
Adapting Reading Strategies for
Teaching Mathematics K-6
Arthur Hyde
Building Mathematical Comprehension:
Using Literacy Strategies to Make Meaning
Laney Sammons
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Readers draw upon
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Content knowledge
Knowledge of text structures
Pragmatic knowledge
Contextual knowledge
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Mathematics and Comprehension
In order to understand what the question is asking
students to do, reading and comprehension skills
need to be developed.
• Reading requires skills in code-breaking; i.e.,
knowing the words and how the words, symbols
and pictures are used in the test genre.
• Comprehension--i.e., making meaning of the
literal, visual and symbolic text forms presented-requires students to draw on their skills as a text
user, a text participant and a text analyst (Luke &
Freebody, 1997).
Cracking the NAPLaN Code, Thelma Perso
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Mathematics and Comprehension
There are many different codes in mathematics that children need to
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crack if they are to have success with the NAPLAN test genre.
These include:
English language words and phrases (e.g. wheels in the picture)
words and phrases particular to mathematics (e.g., number
sentence, total)
words and phrases from the English language that have a
particular meaning in the mathematics context but that may have
a different meaning in other learning areas (e.g., complete)
symbolic representations which for many learners are a language
other than English - these include "3" representing "three," "x"
representing "times," "multiply" and "lots of," "=" representing
"is equal to.“
images, including pictures/drawings, graphs, tables, diagrams,
maps and grids.
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Comprehension
There are many different codes in mathematics that children need to
crack if they are to have success with the NAPLAN test genre.
These include:
• drawings and images (such as the picture of the
tricycles) to help them visualise and infer what the
question might be asking
• words like "complete" and "total number of" to
infer what the question is asking; and they also
• translate from the drawing to the sentence and
then to the symbolic representation to understand
they need to "fill in the empty boxes."
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Comprehension
These comprehension strategies need to be explicitly
taught and deliberately practiced. Strategies
include:
• experiences with the test genre
• relating the text types (drawings, grids, word
sentences) to children's experiences
• asking children to retell the situation that is being
represented and describe or explain to others what
they are inferring and thinking about a situation.
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Text Structure
• Structure of word problems in mathematics
handout
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Vocabulary
Vocabulary
1. Words that mean the same in the
mathematical context e.g. dollar, bicycle
2. Words that are unique to mathematics
e.g. hypotenuse, cosine
3. Words that have different meanings in
mathematics and everyday use e.g.
average, difference, factor, table
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Effective Vocabulary Instruction
• does not rely on definitions
• relies on linguistic and non-linguistic
representations
• uses multiple exposures
• involves understanding word parts to
enhance meaning
• involves different types of instruction for
different words (process vs content)
• requires student talk and play with words
• involves teaching the relevant words
Marzano, 2004
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Confusion...
• Move the decimal point
• Just add a zero
• Times tables
• Our number system
Eleven (should be tenty one)
Twelve (.......................................
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Theory
Conceptual
Connections
Importance
of
Visualisation
Research
Overview
Practical
Strategies
Practice
Attitudes
Making
Connections
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The theory of
mathematics is
important to me. I
like to know what
experts know.
Understanding
why is important
to me. I need to
visualise and
connect.
I like knowing the
process and
practising
problems to get
better.
I need to know how
it is relevant to my
life. I like to be able
to discuss different
ways of solving the
problem.
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Your story
• Consider your educational
experiences in mathematics
• Share with people at your table
• Be ready to share with the whole
group
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Authentic
Engagement
Ritual
Compliance
Passive
Compliance
Retreatism
Rebellion
Authentic
Engagement
Ritual
Compliance
Passive
Compliance
Retreatism
Rebellion
Authentic
Engagement
Ritual
Compliance
Passive
Compliance
Retreatism
Rebellion
Story
Connection
Attitudes
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Making Connections
• Is affected by attitude
• Is unlikely to occur if maths is
taught as isolated strands
• Will be sketchy if maths is taught
using low level procedures
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• Use of low level procedural tasks (75%)
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x
Here it is
3 cm
• Find x
4 cm
• Leads to lack of conceptual understanding
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Ma and Pa Kettle Maths 2:14
http://www.youtube.com/watch?v=Bfq5kju627c
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Calculate mentally
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7+ 6
=
3 0 2
• Revoicing - “You used the 100s chart and
counted on?”
• Rephrasing - “Who can share what
________ just said, but using your own
words?”
• Reasoning - “Do you agree or disagree
with ________? Why?”
• Elaborating - “Can you give an example?”
• Waiting - “This question is important.
Let’s take some time to think about it.”
History of Mathematics 7:04 http://www.youtube.com/watch?v=wo-6xLUVLTQ
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Making Connections
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Modelled:
Using think alouds, talk to
students about the concept of
“schema”.
When I think about “million”
here are some connections I
have made:
• with my life
• with maths that I know about
• with something I saw on TV,
newspaper etc.
Shared:
Use a “schema roller” or
brainstorm to elicit
current understandings.
Ask students to add their
ideas.
Record on anchor chart.
Making Connections
(Maths to Maths)
Concepts are abstract ideas that organise
information
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Multiplicative
thinking
Multiplication
facts
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Traditional Approach
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Explanation or definition
Explain rules
Apply the rules to examples
Guided practice
D
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V
E
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Adapting Reading Strategies
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Imagine that you work on a farm. The owner
has 24 sheep tells you that you must put all
of the sheep in pens. You can fence the pens
in different ways but you must put the same
number of sheep in each pen. What is one
way you might do this? How many different
ways can you find?
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Making Connections
Maths to Self
• What does this situation remind me of?
• Have I ever been in a situation like this?
Maths to Maths
• What is the main idea from mathematics that is
happening here?
• Where have I seen this before?
Maths to World
• Is this related to anything I’ve seen in science,
arts….?
• Is this related to something in the wider world?
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What do I know for sure?
Imagine that you work on a farm.
The owner has 24 sheep tells you
that you must put all of the
sheep in pens. You can fence the
pens in different ways but you
must put the same number of
sheep in each pen. What is one
way you might do this? How
many different ways can you
find?
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What do I want to work
out, find out, do?
Imagine that you work on a farm.
The owner has 24 sheep tells you
that you must put all of the
sheep in pens. You can fence the
pens in different ways but you
must put the same number of
sheep in each pen. What is one
way you might do this? How
many different ways can you
find?
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Are there any special
conditions, clues to watch out
for?
Imagine that you work on a farm.
The owner has 24 sheep tells you
that you must put all of the
sheep in pens. You can fence the
pens in different ways but you
must put the same number of
sheep in each pen. What is one
way you might do this? How
many different ways can you
find?
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Students need to be able to
make connections between
mathematics and their own
lives.
Making connections across mathematical
topics is important for developing
conceptual understanding. For example,
the topics of fractions, decimals,
percentages, and proportions san usefully
be linked through exploration of differing
representations (e.g., ½ = 50%) or through
problems involving everyday contexts (e.g.,
determining fuel costs for a car trip).
Teachers can also help students to
make connections to real experiences.
When students find they can use
mathematics as a tool for solving
significant problems in their everyday
lives, they begin to view the subject as
relevant and interesting.
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Questioning
Common question in mathematics are...
• Why do I have to do this?
• What do I have to do?
• How many do I have to do?
• Did I get it right?
Common question in mathematics should be..
• How can I connect this?
• What is important here?
• How can I solve this?
• What other ways are there?
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There are two things in life we can be
certain of.....
Death, Taxes and Mathematics
At least 50% of year 5’s hate story problems. They
come to pre-school with some resourceful ways
of solving problems e.g. dividing things equally.
Early years of schooling – must do maths in a
particular way, there is one right answer, there is
one way of doing it. They are told what to
memorise, shown the proper way and given a
satchel full of gimmicks they don’t understand.
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Story Problems
• Just look for the key word (cue word)
that will tell you what operation to use
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Fundamental Messages
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Don’t read the problem
Don’t imagine the solution
Ignore the context
Abandon your prior knowledge
You don’t have to read
You don’t have to think
Just grab the numbers and compute!
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Why might this
problem be
difficult for
some children?
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Ben has 2 identical pizzas.
He cuts one pizza equally into 4 large
slices.
He then cuts the other pizza equally
into 8 small slices.
A large slice weighs 32 grams more
than a small slice.
What is the mass of one whole pizza?
grams
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Newman's prompts
• The Australian educator Anne Newman (1977)
suggested five significant prompts to help determine
where errors may occur in students attempts to solve
written problems. She asked students the following
questions as they attempted problems.
1. Please read the question to me. If you don't know a
word, leave it out.
2. Tell me what the question is asking you to do.
3. Tell me how you are going to find the answer.
4. Show me what to do to get the answer. "Talk aloud" as
you do it, so that I can understand how you are thinking.
5. Now, write down your answer to the question.
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1. Reading the problem
Reading
2. Comprehending what is read
3. Carrying out a transformation
from the words of the problem
to the selection of an
appropriate mathematical
strategy
4. Applying the process skills
demanded by the selected
strategy
5. Encoding the answer in an
acceptable written form
Comprehension
Transformation
Process skills
Encoding
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Ben has 2 identical pizzas.
He cuts one pizza equally into 4 large slices.
He then cuts the other pizza equally into 8
small slices.
A large slice weighs 32 grams more than a
small slice.
What is the mass of one whole pizza?
grams
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Read and understand the problem
(using Newman's prompts)
• Teacher reads the word problem to
students.
• Teachers uses questions to determine the
level of understanding of the problem e.g.
– How many pizzas are there?
– Are the pizzas the same size?
– Are both pizzas cut into the same number of
slices?
– Do we know yet how much the pizza weighs?
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An article about using Newman’s Prompts
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What do I know for sure?
What do I want to work
out, find out, do?
Are there any special
constraints, conditions,
clues to watch out for?
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Problem Solving Questions
• What is the problem?
• What are the possible
problem solving
strategies?
• What is my plan?
• Implement the plan
• Does my solution make
sense?
Up to 75 % of
time may need to
be spent on this
stage
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Ben has 2 identical pizzas.
He cuts one pizza equally into 4 large slices.
He then cuts the other pizza equally into 8
small slices.
A large slice weighs 32 grams more than a
small slice.
What is the mass of one whole pizza?
grams
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One of the main aims of
school mathematics is to
create in the mind’s eye of
children, mental objects
which can be manipulated
flexibly with understanding
and confidence.
Siemon, D., Professor of Mathematics Education,
RMIT 81
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Subitising
(suddenly recognising)
• Seeing how many at a glance is
called subitising.
• Attaching the number names
to amounts that can be seen.
• Learned through activities and
teaching.
• Some children can subitise,
without having the associated
number word.
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Make
Materials
Real-world, stories
Perceptual Learning
five
Name
Language
read, say, write
Record
5
Symbols
recognise, read, write
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MAKE TO TEN
Being able to visualise ten and combinations
that make 10
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DOUBLES & NEAR DOUBLES
Being able to double a quantity then add or
subtract from it.
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Imagine that you work on a farm. The owner
has 24 sheep tells you that you must put all
of the sheep in pens. You can fence the pens
in different ways but you must put the same
number of sheep in each pen. What is one
way you might do this? How many different
ways can you find?
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Making the Links
Are we giving students the opportunity to make the
links between the materials, words and symbols?
Materials
Symbols
Think Board
Words
Picture
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Representations
• Move from realistic to gradually more
symbolic representation
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Number of pens
1
2
3
4
6
8
12
24
Number of sheep
in each pen
24
12
8
6
4
3
2
1
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Number of pens
Number of sheep in
each pen
1
24
2
12
3
8
4
6
6
4
8
3
12
2
24
1
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equal
factors
row
column
arrays
quantity
total
24
2 columns
12 rows
One factor
The other factor
1
24
2
12
3
8
4
6
6
4
8
3
12
2
24
1
1 x 24 = 24
2 x 12 = 24
3 x 8 = 24
4 x 6 = 24
6 x 4 = 24
8 x 3 = 24
12 x 2 = 24
24 x 1 = 24
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• Julie bought a dress in an
end-of-season sale for
$49.35. The original price
was covered by a 30% off
sticker but the sign on the
rack said, “Now an
additional 15% off already
reduced prices”. How
could she work out how
much she had saved?
What percentage of the
original cost did she end
up paying?
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Just Fractions 03:04
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Visualising & Automaticity
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Mathematics Scope and Sequence:
Foundation to Year 6
Number & Algebra
Money and
financial
mathematics
Year 1
Recognise, describe and
order Australian coins
according to their value
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Mathematics Scope and Sequence:
Foundation to Year 6
Measurement &
Geometry
Shape
Year 5
Connect three-dimensional objects
with their nets and other twodimensional
representations
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Refer to AC Achievement Standards
What ideas can you generate for developing
automaticity and comprehension
in maths using visual techniques?
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Supporting Visualisation
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Inference
Sometimes all of the information you need
to solve the problem is not “right there”.
What You Know
+ What you Read
______________
Inference
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There are 3 people sitting at the lunch
table.
How many feet are under the table?
What I Read: There are 3 people.
What I Know: Each person has 2
feet.
What I Can Infer: There are 6 feet
under the table.
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Fact or Inference
• There are 0.3 g fat in 100 g of the soup
• The soup is 0.6 % protein
• One serve of the soup contains 450 kJ
• There is more fat than salt in the soup
• There are 3 fresh tomatoes in each can of soup
• In each serve of soup there is 20.7 g of carbohydrate
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1. Read the question aloud
2. Ask students whether there are any words they are
not sure of. Explicitly teach any words using examples,
pictures etc.
3. Does Peta keep any plums for herself?
4. Ask students to paraphrase the question
5. Ask students to make connections –have they shared
something out when they are not sure how it will
work out? Have you seen a problem like this before?
When might this happen in real life?
6. What might the answer be or NOT be? Why?
7. Ask students to agree or disagree and explain why.
8. Re-read the information. Peta has some plums – we
need to work out how many plums Peta has. Peta is
giving some plums to her friends . We don’t know
how many friends Peta has.
9. What else do we know and not know?
10. What can we infer?
If she gives each friend 4 plums, she will have 6 plums left over
What can you infer from this?
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Determining Importance
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Determining Importance
Some students cannot work out what
information is most important in the
problem. This must be scaffolded through
• explicit modelling
• guided practice
• independent work
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Solve this!
Nathan was restocking the shelves at the
supermarket. He put 42 cans of peas and
52 cans of tomatoes on the shelves on
the vegetable aisle. He saw 7 boxes of
tissues at the register. He put 40 bottles
of water in the drinks aisle. He noticed a
bottle must had spilled earlier so he
cleaned it up. How many items did he
restock?
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Strategy
42 cans of peas
52 cans of tomatoes
tissues at the register
40 bottles of water
water that he cleaned
up
important
important
not important
important
not important
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Summarising & Synthesising
Journaling gives students an opportunity
to summarise and synthesise their
learning of the lesson.
Use maths word wall words to scaffold
journaling. Include words like “as a
result”, “finally”, “therefore”, and “last”
that denote synthesising for students to
use in their writing. Or have them use
sentence starters like ”I have learned
that…”, “This gives me an idea that”, or
“Now I understand that…”
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What do I now know for sure?
How can I use this knowledge in
other situations?
What did I work out, find out, do?
How did I work it out?
Were there any special conditions?
What conclusions did I draw?
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What facts did I learn?
How did I feel?
What went well?
What problems did I have?
What creative ways did I solve the problems?
What connections did I make?
How can I use this in the future?
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What is
the rule?
Draw it
What
connections
do I know?
Journal
Show an
example
How does it
relate to
my life?
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A=LXW
Area equals length
multiplied by width
Multiplication facts
Arrays and grids
One surface of some solids e.g. cylinder
Same as 2 equal right angled triangles
Journal
A room has a
length of 4 metres
and width of
3 metres.
The area is
4m x 3m = 12 sq metres
Measuring material
for a tablecloth
Working out how many
plants for my vegetable
garden
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We now know a lot more about how children
learn mathematics.
Meaningless rote-learning, mindnumbing, text-based drill and
practice, and doing it one way, the
teacher’s way, does not work.
Concepts need to be experienced, strategies
need to be scaffolded and EVERYTHING needs
to be discussed.
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