1. Strategies that Work Teaching for Understanding and Engagement
Maths & Comprehension
Module 11
Debbie Draper

2. We recognise Kaurna people and their land
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.

3. 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

4. Overview Mathematics Teaching – research findings
Mathematics and Integral Learning
Comprehension Strategies applied to Mathematics

5. Adapting Reading Strategies for Teaching Mathematics K-6 Arthur Hyde
Comprehending Math:
Adapting Reading Strategies for
Teaching Mathematics K-6
Arthur Hyde
Building Mathematical Comprehension:
Using Literacy Strategies to Make Meaning
Laney Sammons

7. Readers draw upon Content knowledge Knowledge of text structures
Pragmatic knowledge
Contextual knowledge
Content knowledge – a foundation upon which to build
Pragmatic – how they have solved problems before
Contextual – culture of mathematics learning, teachers’ attitude etc.

8. 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

9. 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.

10. 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."

11. 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.

12. Text Structure
Structure of word problems in mathematics
handout

14. Vocabulary

15. Vocabulary
Words that mean the same in the mathematical context e.g. dollar, bicycle
Words that are unique to mathematics e.g. hypotenuse, cosine
Words that have different meanings in mathematics and everyday use e.g. average, difference, factor, table
handout

16. 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

20. Confusion... Move the decimal point Just add a zero Times tables
Our number system
Eleven (should be tenty one)
Twelve (

21. Overview Theory Conceptual Connections Importance of Research
Visualisation
Research
Overview
Practical
Strategies
Attitudes
Practice
Making
Connections

22. Understanding why is important to me. I need to visualise and connect.
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.

23. Your story Consider your educational experiences in mathematics
Share with people at your table
Be ready to share with the whole group

24. 10

26. Who can relate to this?

28. Math class is tough
Do you have a crush on anyone
I’ll always be here for you

30. Authentic
Engagement
Ritual
Compliance
Passive
Retreatism
Rebellion

31. Authentic
Engagement
Ritual
Compliance
Passive
Retreatism
Rebellion

32. Authentic
Engagement
Ritual
Compliance
Passive
Retreatism
Rebellion

33. Story
Connection
Attitudes

34. 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

35. Use of low level procedural tasks (75%)

36. Find x x
3 cm
4 cm
Here it is
Leads to lack of conceptual understanding

37. Ma and Pa Kettle Maths 2:14 http://www.youtube.com/watch?v=Bfq5kju627c

38. Calculate mentally 6 3 2 2 7 + =

39. 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.”

40. History of Mathematics 7:04 http://www.youtube.com/watch?v=wo-6xLUVLTQ

43. Making Connections

44. 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.

45. Shared:
Use a “schema roller” or brainstorm to elicit current understandings. Ask students to add their ideas.
Record on anchor chart.

46. Making Connections (Maths to Maths)
Concepts are abstract ideas that organise information
Quantity
Shape
Dimension
Change
Uncertainty

47. Multiplicative
thinking
Quantity
Multiplication
facts

48. Traditional Approach Explanation or definition Explain rules
Apply the rules to examples
Guided practice D E U C T I V
No connections made
No schema activated

49. Adapting Reading Strategies49

50. Imagine that you work on a farm
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?

51. 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?

52. What do I know for sure? K. W. C.
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?

53. 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?

54. Are there any special conditions, clues to watch out for? C.
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?

55. 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.

58. 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?

59. Death, Taxes and Mathematics
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.

60. Story Problems
Just look for the key word (cue word) that will tell you what operation to use

61. +
add
addition
sum
total
altogether
plus

62. +
add
addition
sum
total
altogether
plus

63. 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!

64. Why might this problem be difficult for some children?

65. 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

66. 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.

67. Reading the problem
Reading
Comprehending what is read
Comprehension
Carrying out a transformation from the words of the problem to the selection of an appropriate mathematical strategy
Transformation
Applying the process skills demanded by the selected strategy
Process skills
Encoding the answer in an acceptable written form
Encoding

68. 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

69. 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?

70. An article about using Newman’s Prompts

71. What do I want to work out, find out, do?C.
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?

72. 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 K. W. C.

73. 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

81. 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

82. 17

83. How many sheep?

85. How many sheep? 18

87. How many sheep?

88. 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.

89. Building Understanding
Make
Materials
Real-world, stories
Perceptual Learning 5
five
Record
Name
Language
Symbols
read, say, write
recognise, read, write

90. MAKE TO TEN
Being able to visualise ten and combinations that make 10

91. DOUBLES & NEAR DOUBLES
Being able to double a quantity then add or subtract from it.
Counting on from the largest is only useful for small collections
Doubling is one of the first strategies children develop
We can explicitly teach doubling as a strategy from early ages
Doubling can be used as a strategy to solve problems. (if I know five and five is ten how can I use this to solve seven and seven)

93. Imagine that you work on a farm
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?

94. Making the Links
Are we giving students the opportunity to make the links between the materials, words and symbols?
Materials
Symbols
Words
Think Board
Picture

96. Representations
Move from realistic to gradually more symbolic representation

98. Number of sheep in each pen
Number of pens
Number of sheep in each pen 1 24 2 12 3 8 4 6

99. Number of sheep in each pen
Number of pens
Number of sheep in each pen 1 24 2 12 3 8 4 6

100. 24 2 columns 12 rows equal factors row column arrays quantity total
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
One factor
The other factor 1 24 2 12 3 8 4 6

101. Julie bought a dress in an end-of-season sale for $49. 35
Julie bought a dress in an end-of-season sale for $ 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?

102. !

103. Just Fractions 03:04

108. Visualising & Automaticity

111. 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

114. Mathematics Scope and Sequence: Foundation to Year 6
Measurement & Geometry
Shape
Year 5
Connect three-dimensional objects with their nets and other two-dimensional representations

121. Refer to AC Achievement Standards
What ideas can you generate for developing automaticity and comprehension in maths using visual techniques?

122. Supporting Visualisation

126. Inference
Sometimes all of the information you need to solve the problem is not “right there”. What You Know + What you Read ______________ Inference

127. What I Read: There are 3 people. What I Know: Each person has 2 feet.
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.

128. 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

129. Read the question aloud
Ask students whether there are any words they are not sure of. Explicitly teach any words using examples, pictures etc.
Does Peta keep any plums for herself?
Ask students to paraphrase the question

130. 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?
What might the answer be or NOT be? Why?
Ask students to agree or disagree and explain why.

131. Re-read the information
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.
What else do we know and not know?
What can we infer?

132. If she gives each friend 4 plums, she will have 6 plums left over
What can you infer from this?

134. Determining Importance

136. 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

137. 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?

138. Strategy 42 cans of peas 52 cans of tomatoes tissues at the register
40 bottles of water
water that he cleaned up
important
not important

141. 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…”

142. K. W. C. 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?

143. 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?

144. Journal What is Draw it the rule? What connections do I know? Show an
example
How does it
relate to
my life?

145. Journal A = L X W 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
A = L X W
Area equals length
multiplied by width
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

147. 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.