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Using key numeracy teaching principles as the basis of leading teaching improvement Peter Sullivan Numeracy keynote SA

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Abstract Supporting improvement in numeracy teaching is both demanding and complex. One way to manage the complexity is to have explicit goals for each step in the improvement process. After reviewing other similar lists, I identified six principles that can form the basis, individually and together, of improvement initiatives. Using the theme of the teaching of fractions, this session will elaborate each of the principles. Numeracy keynote SA

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What is being recommended about mathematics teaching? Numeracy keynote SA

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How does this connect to the AC? Numeracy keynote SA

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Describing the proficiencies Understanding – (connecting, representing, identifying, describing, interpreting, sorting, …) Fluency – (calculating, recognising, choosing, recalling, manipulating, …) Problem solving – (applying, designing, planning, checking, imagining, …) Reasoning – (explaining, justifying, comparing and contrasting, inferring, deducing, proving, …) Numeracy keynote SA

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The (brand) new UK National Curriculum …all pupils: become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils have conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language can solve problems by applying their mathematics to a variety of routine and nonroutine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions. Numeracy keynote SA

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https://www.education.gov.uk/schools/teachi ngandlearning/curriculum/nationalcurriculum 2014/a00220610/draft-pos-ks4-english- maths-science https://www.education.gov.uk/schools/teachi ngandlearning/curriculum/nationalcurriculum 2014/a00220610/draft-pos-ks4-english- maths-science Numeracy keynote SA

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From Impactful practices Imagine a world where students, in every mathematics classroom, are actively engaged with worthwhile tasks that promote mathematical understanding, problem solving and reasoning. Numeracy keynote SA

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Imagine classrooms where the interactions among students, and with their teacher, are focused on making sense of mathematics, alternative approaches to solving problems, and defending, confirming and verifying possible solutions. These are thinking and reasoning classrooms. Numeracy keynote SA

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An aside It is not a problem if we have told students what to do It is not reasoning if students are reproducing what we have told them It is not understanding unless students can explain in their own words with their own ideas Numeracy keynote SA

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What do you see as the most critical aspect of being a powerful learner of numeracy and literacy? Powerful learners connect ideas together, they can compare and contrast concepts, and they can transfer learning from one context to another. They can devise their own solutions to problems, and they can explain their thinking to others. Numeracy keynote SA

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Two task examples that we will use as the basis of the later discussion Numeracy keynote SA

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STRIPED RECTANGLE If the dotted (blue) rectangle represents what fraction is represented by the striped (red) rectangle? Work out the answer in two different ways. Numeracy keynote SA

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REPRESENTING A FRACTION If this represents 3, draw what represents 1 (work this out two different ways) Numeracy keynote SA

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What is the point of the six key principles ? We can all do these things better (although you will find many of them affirming of your current practice) Much advice is complex and hard to prioritise The principles can provide a focus to collaborative discussions on improving teaching The principles can be the focus of observations if you have the opportunity to be observed teaching Numeracy keynote SA

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AVAILABLE TO DOWNLOAD FREE FROM http://research.acer.edu.au/aer/13/ aer Numeracy keynote SA

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Key principle 1: Identify important ideas that underpin the concepts you are seeking to teach, and communicate to students that these are the goals of your teaching, including explaining how you hope they will learn Numeracy keynote SA

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Feedback - better when they know … Where am I going? – “Your task is to …, in this way” How am I going? – “the first part is what I was hoping to see, but the second is not” Where to next? – “knowing this will help you with …” Numeracy keynote SA

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In terms of learning intentions, we know It is difficult to describe the purpose of lessons and teachers benefit from discussions about intentions The learning intention should – not restrict – nor tell – nor lower the ceiling – but provide focus for the students – and the teacher Numeracy keynote SA

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What would you say is the learning intention for striped rectangle? Numeracy keynote SA

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This is taken from the lesson plan A fraction is a number. We can compare, add and multiply fractions just like we do for numbers, even if the calculation process is a little different. You will solve the problem for yourself and explain your thinking Numeracy keynote SA

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goals

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Key principle 2: Build on what the students know, both mathematically and experientially, including creating and connecting students with stories that both contextualise and establish a rationale for the learning Numeracy keynote SA

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Part 1: Using data Numeracy keynote SA

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It is as important to know what the students know as it is to know what they do not Learning mathematics is not a hierarchy of sequential steps on a ladder, but a network of interconnected ideas Students benefit from work on tasks that are beyond what they know – Students at GP 2 can work on GP 4 tasks Numeracy keynote SA

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Helen has 24 red apples and 12 green apples. What fraction of the apples are green? Numeracy keynote SA

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Year 5 93% Numeracy keynote SA

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Year 5 24%

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Helen has 24 red apples and 12 green apples. What fraction of the apples are green? 55% of year 7 students 67% of year 9 students Numeracy keynote SA

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What does that tell you? Numeracy keynote SA

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Part 2: Connecting with “story” Numeracy keynote SA

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A chameleon has a tongue that is half as long as its body... … how long would your tongue be if you were a chameleon? Numeracy keynote SA

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Part 3: Creating experience Numeracy keynote SA

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goals readiness

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Key Principle 3 Engage students by utilising a variety of rich and challenging tasks, that allow students opportunities to make decisions, and which use a variety of forms of representation Numeracy keynote SA

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Why challenge? Learning will be more robust if students connect ideas together for themselves, and determine their own strategies for solving problems, rather than following instructions they have been given. Both connecting ideas together and formulating their own strategies is more complex than other approaches and is therefore more challenging. We need strategies to encourage students to persist Numeracy keynote SA

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Related to those tasks above.. To what extent – Are they challenging? – Are they engaging? – Do they allow student decision making – Do they encourage different representations? Numeracy keynote SA

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What is 5 + 5 + 5 + 295 + 295 + 295 ? Number of students that completed this question 1343 Correct answers413 (30.8%) Numeracy keynote SA

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I prefer questions we work on in class to be I prefer learning how to do questions like this Much harderAbout the same Much easier TOTAL By myself 23929069598 Working with other students 78227162467 By listening to the teacher's explanations first 56140100296 TOTAL 3736573311361 Think about the question What is 5 + 5 + 5 + 295 + 295 + 295 ? Numeracy keynote SA

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I prefer questions we work on in class to be I prefer learning how to do questions like this Much harderAbout the same Much easier TOTAL By myself 239 118 290 103 69 21 598 242 Working with other students 78 15 227 64 162 13 467 92 By listening to the teacher's explanations first 56 17 140 36 100 18 296 71 TOTAL 373 150 657 203 331 52 1361 405 Think about the question What is 5 + 5 + 5 + 295 + 295 + 295 ? Numeracy keynote SA

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Quotes from PISA in Focus 37 When students believe that investing effort in learning will make a difference, they score significantly higher in mathematics. Teachers’ use of cognitive-activation strategies, such as giving students problems that require them to think for an extended time, presenting problems for which there is no immediately obvious way of arriving at a solution, and helping students to learn from their mistakes, is associated with students’ drive. Numeracy keynote SA

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goals readiness engage

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Key Principle 4: Interact with students while they engage in the experiences, and specifically planning to support students who need it, and challenge those who are ready Numeracy keynote SA

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Enabling prompt Numeracy keynote SA

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IF YOU ARE STUCK If this represents 7, draw what represents 2 (work this out two different ways) Numeracy keynote SA

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IF YOU ARE STUCK If this represents 11, draw what represents 5 (work this out two different ways) Numeracy keynote SA

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Extending prompt Numeracy keynote SA

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IF YOU HAVE FINISHED If this represents 8, draw what represents 2.4 (work this out two different ways) Numeracy keynote SA

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goals readiness engage difference

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Key Principle 5: Adopt pedagogies that foster communication, mutual responsibilities, and encourage students to work in small groups, and using reporting to the class by students as a learning opportunity Numeracy keynote SA

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A revised lesson structure In this view, the sequence – Launch (without telling) – Explore (for themselves) – Summarise (drawing on the learning of the students) … is cyclical and might happen more than once in a lesson (or learning sequence) Numeracy keynote SA

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CONSOLIDATING THE LEARNING If this represents 5, draw what represents 2 (work this out two different ways) Numeracy keynote SA

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goals lesson structure readiness engage difference

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Key teaching idea 6 Fluency is important, and it can be developed in two ways – by short everyday practice of mental calculation or number manipulation – by practice, reinforcement and prompting transfer of learnt skills Numeracy keynote SA

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One aspect is transfer This connects to the consolidating task Numeracy keynote SA

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Another aspect is fluency Numeracy keynote SA

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7 43 +

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? 3423 + Numeracy keynote SA

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35 ?13 + Numeracy keynote SA

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The unknowns are different ½ ?? + Numeracy keynote SA

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? ?1.7 + Numeracy keynote SA ??? ??? 0.2 + + + + +

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I prefer questions we work on in class to be I prefer learning how to do questions like this Much harderAbout the same Much easierTOTAL By myself 239 39.97% 290 48.49% 69 11.54% 598 100% Working with other students 78 16.7% 227 48.61% 162 34.69% 467 100% By listening to the teacher's explanations first 56 18.92% 140 47.3% 100 33.78% 296 100% TOTAL 373 27.41% 657 48.27% 331 24.32% 1361 100% Think about the question What is 5 + 5 + 5 + 295 + 295 + 295 ? (row) Numeracy keynote SA

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I prefer questions we work on in class to be I prefer learning how to do questions like this Much harderAbout the same Much easierTOTAL By myself 239 64.08% 290 44.14% 69 20.85% 598 43.94% Working with other students 78 20.91% 227 35.55% 162 48.94% 467 34.31% By listening to the teacher's explanations first 56 15.01% 140 21.31% 100 30.21% 296 21.75% TOTAL 373 100% 657 100% 331 100% 1361 100% Think about the question What is 5 + 5 + 5 + 295 + 295 + 295 ? (column) Numeracy keynote SA

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I prefer questions we work on in class to be I prefer learning how to do questions like this Much harderAbout the same Much easierTOTAL By myself 239 17.56% 290 21.31% 69 5.07% 598 43.94% Working with other students 78 5.73% 227 16.68% 162 11.90% 467 34.31% By listening to the teacher's explanations first 56 4.11% 140 10.29% 100 7.35% 296 21.75% TOTAL 373 27.41% 657 14.27% 331 24.32% Think about the question What is 5 + 5 + 5 + 295 + 295 + 295 ? ( % of the overall 1361 responses) Numeracy keynote SA

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I prefer questions we work on in class to be I prefer learning how to do questions like this Much harderAbout the same Much easierTOTAL By myself 239 (100%) 118 (49.37%) 290 (100%) 103 (35.52%) 69 (100%) 21 (30.43%) 598 (100%) 242 (40.47%) Working with other students 78 (100%) 15 (19.23%) 227 (100%) 64 (28.19%) 162 (100%) 13 (8.02%) 467 (100%) 92 (19.70%) By listening to the teacher's explanations first 56 (100%) 17 (30.36) 140 (100%) 36 (25.71%) 100 (100%) 18 (18%) 296 (100%) 71 (24%) TOTAL 373 (100%) 150 (40.21%) 657 (100%) 203 (30.90%) 331(100%) 52 (15.71%) 1361 (100%) 405 (29.76%) Think about the question What is 5 + 5 + 5 + 295 + 295 + 295 ? Numeracy keynote SA

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