Post on 16-Dec-2015
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Strategies that WorkTeaching for Understanding and Engagement
Mat
hs &
Com
preh
ensi
onM
odul
e 11
Debbie D
raper
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Acknowledgement of Country
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.
We recognise Kaurna people and their land
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NAR Facilitator Support Model – Team Norms• 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-6Arthur Hyde
Building Mathematical Comprehension:Using Literacy Strategies to Make Meaning
Laney Sammons
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Readers draw upon
• Content knowledge• Knowledge of text structures• Pragmatic knowledge• Contextual knowledge
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).
8Cracking the NAPLaN Code, Thelma Perso
Mathematics and 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:
• 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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ComprehensionThere 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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ComprehensionThese 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
Vocabulary
Vocabulary 1. Words that mean the same in the
mathematical context e.g. dollar, bicycle2. Words that are unique to mathematics e.g.
hypotenuse, cosine3. Words that have different meanings in
mathematics and everyday use e.g. average, difference, factor, table
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handout
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 16
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Confusion...• Move the decimal point• Just add a zero• Times tables• Our number systemEleven (should be tenty one)Twelve (.......................................
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Overview
Importanceof
Visualisation
Attitudes
ConceptualConnections
Making Connections
Theory
Research
PracticalStrategies
Practice
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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
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10 0
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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
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StoryConnectionAttitudes
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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• Find xx
4 cm
3
cm
Here it is
• Leads to lack of conceptual understanding
37Ma and Pa Kettle Maths 2:14 http://www.youtube.com/watch?v=Bfq5kju627c
2 7 6 3 0 2+ =
Calculate mentally
• 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.”
40History 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.
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Making Connections(Maths to Maths)
Concepts are abstract ideas that organise information
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Multiplicationfacts
Multiplicative thinking
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Traditional Approach
• Explanation or definition • Explain rules• Apply the rules to examples• Guided practice
DEDUCTIVE
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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 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?
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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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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?
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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?
Are there any special conditions, clues to watch out for?
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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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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..• How can I connect this?• What is important here?• How can I solve this?• What other ways are there?
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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.
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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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+add
addition
sumtotal
altogether
plus
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+add
addition sumtotalaltogetherplus
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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!
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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 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
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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
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How many sheep?
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How many sheep?
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How many sheep?
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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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Building Understanding
Materials
Real-world, stories
Language
read, say, write
Symbols
Make
recognise, read, write
NameRecord
Perceptual Learning
5five
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MAKE TO TENBeing able to visualise ten and combinations
that make 10
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DOUBLES & NEAR DOUBLESBeing 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 LinksAre we giving students the opportunity to make the links between the materials, words and symbols?
Think Board
Materials
Symbols
Picture
Words
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Representations
• Move from realistic to gradually more symbolic representation
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Number of pens Number of sheep in each pen
1 242 123 84 66 48 3
12 224 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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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
• 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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Recognise, describe andorder Australian coins
according to their value
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Mathematics Scope and Sequence: Foundation to Year 6
Number & Algebra Money andfinancialmathematics
Year 1
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Mathematics Scope and Sequence: Foundation to Year 6
Measurement & Geometry Shape
Year 5
Connect three-dimensional objectswith their nets and other two-
dimensionalrepresentations
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Refer to AC Achievement StandardsWhat ideas can you generate for developing
automaticity and comprehension in maths using visual techniques?
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
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 tomatoestissues at the register40 bottles of waterwater that he cleaned up
importantimportantnot importantimportantnot 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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Journal
Draw it
How does itrelate to my life?
What is the rule?
Show an example
What connections do I know?
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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
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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.