Teaching for Understanding and Engagement
Debbie Draper & Ann MacMillan
Maths & Comprehension
Module 4
Strategies that Work
Agenda
9:00
10:00 - 11:00
11:00 – 11:20
11:20 – 12:30
12:30 – 1:30
1:30 – 4:00
Mathematics and Integral Learning
Properties of Number –
vocabulary, practical applications
and problem solving activities
Break
Properties of Number contd.
Lunch
Comprehension Strategies applied
to Mathematics
http://www.thenetwork.sa.edu.au/
Comprehending Math:
Adapting Reading Strategies for Teaching Mathematics K-6
Arthur Hyde
http://www.braidedmath.com/
• Deal out 5 cards to each participant
• Arrange your five cards in order,
starting with the card that best
describes you, and ending with the
one that is least like you
• Now, you get a chance to discuss your
immediate response to the cards you were
dealt. Were there any you wanted to get rid of
immediately? Any you weren't willing to part
with? Would you be comfortable living your
life out with the hand you were dealt?
Negotiate…
Theory
Conceptual
Connections
Importance
of
Visualisation
Research
Overview
Practical
Strategies
Practice
Attitudes
Making
Connections
Whole brain processing
LEFT BRAIN
• Logical
• Sequential
• Rational
• Analytical
• Objective
• Looks at parts
RIGHT BRAIN
• Random
• Intuitive
• Holistic
• Synthesizing
• Subjective
• Looks at wholes
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.
Your story
• Consider your educational
experiences in mathematics
• Share with people at your table
• Be ready to share with the whole
group
Story
Connection
Attitudes
Over to Ann
Comprehension Strategies &
Scaffolding
Number
Algebra
Measurement
Geometry
Statistics
Probability
Discussing
Using objects
Re-enacting
Drawing pictures
Making lists / tables
Connecting
Questioning
Inferring
Visualising
Determining Importance
Summarising
Synthesising
Monitoring Understanding
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..
• What do I do?
• Why?
• What other ways are there?
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.
Story Problems
• Just look for the key word (cue word)
that will tell you what operation to use
Fundamental Messages
•
•
•
•
•
•
•
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!
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?
What do I know?
What do I want to work out?
Are there any conditions or constraints?
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
A small plane carrying three people makes
a forced landing in the desert. The
people decide to split up and go in three
different directions in search of an oasis.
They agree to divide equally the food
and water they have which includes 15
identical canteens, 5 full of water, 5 half
full of water and 5 empty. They will want
to take the empty canteens in case they
find an oasis. How can they equally
divide the water and canteens among
themselves?
Another way of looking at questioning in
mathematics....
Yet another way of looking at
questioning in mathematics....
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.
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
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?
An article about using Newman’s Prompts
Conceptual
Understandings
Patterns
• All branches of mathematics have
characteristic patterns
Mathematics - the Science of Patterns
Trusting the Count
Countable Unit: Ones
Place Value
Countable Unit: Tens
• Subitising
• Principles of counting
• Part part whole relationships
•
•
•
•
New unit – 10 ones is 1 ten
Number names regular,
irregular
Counting with new unit
Second place value system
Additive to
Multiplicative thinking
•
Countable Unit: Whole numbers
•
Concepts and strategies for
addition/subtraction
Factors, arrays, area models, Cartesian
products, mental strategies
Partitioning
Proportional reasoning
•
Countable Unit:
Rational numbers
Countable Unit:
Rational numbers
•
•
Generalising
Countable Unit: An unknown/variable
•
•
•
Fractions – concepts, naming,
recording
Decimal fractions
Relative proportions
Recognising patterns
Modelling, predicting
Expressing general case in
words and symbols
“Seeing” the
value of the
number by
subitising
“Naming” the
value of the
number by
counting
7 or seven
• Use of low level procedural tasks (75%)
x
Here it is
3 cm
• Find x
4 cm
• Leads to lack of conceptual understanding
Ma and Pa Kettle Maths 2:14
http://www.youtube.com/watch?v=Bfq5kju627c
Abbott And Costello 13 X 7 is 28 2:56 http://www.youtube.com/watch?v=Lo4NCXOX0p8
Concepts
Patterning
Models
History of Mathematics 7:04 http://www.youtube.com/watch?v=wo-6xLUVLTQ
Making Connections
Concepts are abstract ideas that organise
information
Multiplicative
thinking
Multiplication
facts
Traditional Approach
•
•
•
•
Explanation or definition
Explain rules
Apply the rules to examples
Guided practice
D
E
D
U
C
T
I
V
E
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?
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?
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?
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?
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?
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?
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?
Braid Model of Problem Solving
Understand the problem
• KWC
• Making Connections
Planning to solve the problem
• What representations can I use?
Solving the problem
• Work on the problem using a strategy
• Do I see any patterns?
Checking for understanding
• Does my solution make sense?
• Is there a pattern that makes the answer
reasonable?
• What connections link this problem to the big
ideas of mathematics?
Visualisation in Mathematics
• What does it mean to you?
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
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.
Make
Materials
Real-world, stories
Perceptual Learning
five
Name
Language
read, say, write
Record
5
Symbols
recognise, read, write
Making the Links
Are we giving students the opportunity to make the
links between the materials, words and symbols?
Materials
Symbols
Think Board
Picture
Words
MAKE TO TEN
Being able to visualise ten and combinations
that make 10
DOUBLES & NEAR DOUBLES
Being able to double a quantity then add or
subtract from it.
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?
Representations
• Move from realistic to gradually more
symbolic representation
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
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
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
Visualise a point in space
Visualise a different point in space
Now imagine joining these points
with an imaginary ruler and pencil
Now imagine removing these points as they
are stopping your “line” extending itself.
Remove one point.
Now he can escape but can only go in the
same direction he is currently heading. Let’s
call him Ray! Unfortunately Ray is a unidirectional character. He is still not happy.
Ray is still not happy. He wants to be free to
extend in both directions
Remove the other point.
Now Ray is happy but he has to change his
name. He is now known simply as Line.
Your turn
• Make up a visualisation script
about intersecting,
perpendicular and parallel
lines
Mathematics reasoning
requiring visualisation
What connections are you making?
What are you visualising?
Visualising & Automaticity
• An equilateral triangle is one in which all
three sides and all three angles are equal.
• An isosceles triangle has two equal sides and
two equal angles
• A right angled triangle has one of its angles
equal to 90°
• An acute triangle has each of its angles less
than 90°
• A scalene triangle has each side of a
different length.
• A triangle is considered an obtuse triangle if
it has one angle greater than 90°.
What other ideas do you have
for developing automaticity in
maths using visual techniques?
What if the world only had 100 people? 1:31 http://www.youtube.com/watch?v=QCSLjGnIfUc
Inference
Sometimes all of the information you need
to solve the problem is not “right there”.
What You Know
+ What you Read
______________
Inference
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.
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. Ask students to paraphrase the question
4. 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?
5. What might the answer be or NOT be? Why?
6. Ask students to agree or disagree and explain why.
7. 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.
8. What else do we know and not know?
9. What can we infer?
If she gives each friend 4 plums, she will have 6 plums left over
What can you infer from this?
Determining Importance
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
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 some tissues
at the register. He put 40 bottles of
water in the beverage aisle. He noticed a
bottle must had spilled earlier so he
cleaned it up. How many items did he
restock?
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
Summarising & Synthesising
Journaling as a closure activity 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…”
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?
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?
What is
the rule?
Draw it
What
connections
do I know?
Journal
Show an
example
How does it
relate to
my life?
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
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.
Considerations
• What is the concept or big idea I want
students to understand?
• To what prior knowledge should we try to
connect?
• Are there different models of the concept?
• Is there a sequence of understanding that
the students need to have?
• What other mathematical concepts are
related?
Considerations
• What are the different real life situations or
contexts in which students would
encounter the concept?
• Will they see it in other curriculum areas?
• How can I vary the contexts to build up a
more generalised understanding?
• What version of the situation can I present
to start them thinking about the concept ?
• What questions can I ask to engage and
intrigue them?