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Strategies that WorkTeaching for Understanding and Engagement
Mat
hs &
Com
preh
ensi
onM
odul
e 4 D
ebbie Draper &
Ann MacM
illan
Agenda
9:00 Mathematics and Integral Learning
10:00 - 11:00 Properties of Number – vocabulary, practical
applications and problem solving activities
11:00 – 11:20 Break11:20 – 12:30 Properties of Number contd.12:30 – 1:30 Lunch1:30 – 4:00 Comprehension Strategies
applied to Mathematics
http://www.braidedmath.com/
Comprehending Math: Adapting Reading Strategies for Teaching Mathematics K-6
Arthur Hyde
• 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…
Overview
Importanceof
Visualisation
Attitudes
ConceptualConnections
Making Connections
Theory
Research
PracticalStrategies
Practice
Whole brain processing
LEFT BRAIN• Logical• Sequential• Rational• Analytical• Objective• Looks at parts
RIGHT BRAIN
• Random• Intuitive• Holistic• Synthesizi
ng• Subjective• Looks at
wholes
Ana
lyse
Organi
se
Synthes
ise
Pers
onal
ise
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.
The theory of mathematics
is important to me. I like to know what
experts know.
Your story• Consider your educational
experiences in mathematics • Share with people at your table• Be ready to share with the whole
group
10 0
StoryConnectionAttitudes
Over to Ann
Comprehension Strategies & Scaffolding
NumberAlgebra
MeasurementGeometryStatistics
Probability
Content
DiscussingUsing objectsRe-enacting
Drawing picturesMaking lists / tables
Representing
ConnectingQuestioning
InferringVisualising
Determining ImportanceSummarisingSynthesising
Monitoring UnderstandingComprehension Strategies
QuestioningCommon 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?
Death, Taxes and Mathematics
There are two things in life we can be certain of.....
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
+add
addition
sumtotal
altogether
plus
+add
addition sumtotalaltogetherplus
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 Comprehension3. Carrying out a transformation
from the words of the problem to the selection of an appropriate mathematical strategy
Transformation
4. Applying the process skills demanded by the selected strategy
Process skills
5. Encoding the answer in an acceptable written form 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 Understanding
s
Patterns• All branches of mathematics have
characteristic patterns
Mathematics - the Science of Patterns
Trusting the Count
Countable Unit: Ones
• Subitising• Principles of counting• Part part whole relationships
Place Value
Countable Unit: Tens
• New unit – 10 ones is 1 ten• Number names regular,
irregular• Counting with new unit• Second place value system
Partitioning
Countable Unit: Rational numbers
• Fractions – concepts, naming, recording
• Decimal fractions• Relative proportions
Proportional reasoning
Countable Unit: Rational numbers
Additive to Multiplicative thinking
Countable Unit: Whole numbers
• Concepts and strategies for addition/subtraction
• Factors, arrays, area models, Cartesian products, mental strategies
Generalising
Countable Unit: An unknown/variable
• 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%)
• Find xx
4 cm
3
cm
Here it is
• 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
ConceptsPatterning
Models
History of Mathematics 7:04 http://www.youtube.com/watch?v=wo-6xLUVLTQ
Making Connections
Concepts are abstract ideas that organise information
Multiplicationfacts
Multiplicative thinking
Traditional Approach
• Explanation or definition • Explain rules• Apply the rules to examples• Guided practice
DEDUCTIVE
Making ConnectionsMaths 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 ConnectionsMaths 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?
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?
Braid Model of Problem SolvingUnderstand the problem• KWC• Making ConnectionsPlanning 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.
Building Understanding
Materials
Real-world, stories
Language
read, say, write
Symbols
Make
recognise, read, write
NameRecord
Perceptual Learning
5five
Making the LinksAre we giving students the opportunity to make the links between the materials, words and symbols?
Think Board
Materials
Symbols
Picture
Words
MAKE TO TENBeing able to visualise ten and combinations
that make 10
DOUBLES & NEAR DOUBLESBeing 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 Number of sheep in each pen
1 242 123 84 66 48 3
12 224 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
equalfactors
rowcolumnarrays
quantity total
242 columns
12 rows
1 x 24 = 242 x 12 = 243 x 8 = 244 x 6 = 246 x 4 = 248 x 3 = 24
12 x 2 = 2424 x 1 = 24
One factor The other factor
1 24
2 12
3 8
4 6
6 4
8 3
12 2
24 1
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 uni-directional 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 tomatoestissues at the register40 bottles of waterwater that he cleaned up
importantimportantnot importantimportantnot 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?
Journal
Draw it
How does itrelate to my life?
What is the rule?
Show an example
What connections do I know?
Journal
Measuring material for a tablecloth Working out how many plants for my vegetablegarden
A = L X WArea equals length multiplied by width
A room has a length of 4 metres
and width of 3 metres.
The area is 4m x 3m = 12 sq metres
Multiplication factsArrays and grids
One surface of some solids e.g. cylinder Same as 2 equal right angled triangles
We now know a lot more about how children learn mathematics.
Meaningless rote-learning, mind-numbing, 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?
