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

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.

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

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

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

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.

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

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.

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."

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.

Text Structure Structure of word problems in mathematics handout

Vocabulary

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

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

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

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

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.

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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Who can relate to this?

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

Authentic Engagement Ritual Compliance Passive Retreatism Rebellion

Authentic Engagement Ritual Compliance Passive Retreatism Rebellion

Authentic Engagement Ritual Compliance Passive Retreatism Rebellion

Story Connection Attitudes

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

Use of low level procedural tasks (75%)

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

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

Calculate mentally 6 3 2 2 7 + =

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

Making Connections

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 Quantity Shape Dimension Change Uncertainty

Multiplicative thinking Quantity Multiplication facts

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

Adapting Reading Strategies 49

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?

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? 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?

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? 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?

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.

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?

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.

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

+ add addition sum total altogether plus

+ add addition sum total altogether plus

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!

Why might this problem be difficult for some children?

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

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.

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

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

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

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?

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.

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

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?

How many sheep? 18

How many sheep?

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

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. 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)

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?

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

Representations Move from realistic to gradually more symbolic representation

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

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

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

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

Visualising & Automaticity

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

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

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

Supporting Visualisation

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

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.

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

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

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.

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?

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 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?

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

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…”

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?

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

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

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.