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Three starter tasks Dish washing If 8 students take 2 hours to wash dishes, how many hours would 12 students take to wash the same number of dishes? Multiples of 4 Is 36,924 a multiple of 4? Is there a shortcut for figuring this out? Always, sometimes or never true? The square of a number is greater than the number.

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CENTRE FOR EDUCATIONAL DEVELOPMENT Digging for Gold: Investigating rich tasks Anne Lawrence NCEA National Coordinator Mathematics & Statistics

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Setting the scene Lawrence was Otago's first gold-rush town, in the Tuapeka District, it was originally named The Junction, and then later renamed after the British war hero who defended Lucknow during the 1857 Indian Mutiny. At the height of the gold fever, it's population was 11,500; double that of Dunedin, making it one of the largest communities in the country. It is hard to believe that now with a current population of only

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Two Classrooms Explore tasks that provide the opportunity to develop reasoning and mathematical thinking. This is the gold we are digging for. What scaffolding is needed for students to access these tasks? How can we scaffold and still maintain levels of cognitive demand? Outline of this session

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Two Classrooms Learning environments in which teachers: encourage multiple strategies and ways of thinking, and support students to make conjectures and explain their reasoning are associated with higher student performance on measures of thinking, reasoning and problem-solving. Why?

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Levels of cognitive demand Sort into order of increasing levels of cognitive demand: Doing mathematics Memorisation Procedures with connections Procedures without connections

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Levels of demandCharacteristics of tasks Very Low Memorisation LowProcedures without connections HighProcedures with connections Very High Doing mathematics

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Levels of demandCharacteristics of tasks Very Low Memorisation LowProcedures without connections HighProcedures with connections Very High Doing mathematics Algorithmic Correct answer Explanation Understanding Algorithmic Correct answer Explanation Understanding

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Levels of demandCharacteristics of tasks Very Low MemorisationReproduction of facts, rules, formulae. No explanations LowProcedures without connections Algorithmic. Focused on producing correct answers. No explanations needed HighProcedures with connections Algorithmic. Meaningful /real-world context. Explanations required Very High Doing mathematics Non-algorithmic. Requires understanding and application of math concepts. Real-world context or mathematical structure. Explanations required Rich Tasks

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Levels of demandTasks Very Low Memorisation LowProcedures without connections HighProcedures with connections Very High Doing mathematics Dishwashing Multiples of 4 Always/sometimes/never Flying horseshoes Ostrich & seahorse Fuel for thought Disc-ness Dishwashing Multiples of 4 Always/sometimes/never Flying horseshoes Ostrich & seahorse Fuel for thought Disc-ness Rich Tasks

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Three starter tasks Dish washing If 8 students take 2 hours to was dishes, how many hours would 12 students take to wash the same number of dishes? Multiples of 4 Is 36,924 a multiple of 4? Is there a shortcut for figuring this out? Always, sometimes or never true? The square of a number is greater than the number.

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Flying horseshoes

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1. Which expression is the most useful for finding the maximum height of the horseshoe, and why is it the most useful expression? 2. What information can you conclude about the horseshoes flight from other equivalent expressions? Explain your answers.

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Ostrich and seahorse On the left is a picture of an adult ostrich. It is drawn to a scale of 1:48. On the right is a picture of a seahorse. It is drawn to a scale of 1:1.5. How many seahorses could you fit into the length of an ostrich neck? assessment.concord.org

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Which of the following would save more fuel? a)Replacing a compact car that gets 34 miles per gallon (MPG) with a hybrid that gets 54 MPG b)Replacing a sport utility vehicle (SUV) that gets 18 MPG with a sedan that gets 28 MPG c)Both changes save the same amount of fuel. Fuel for thought

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Discs-ness A gold coin is a disc, and an uncooked piece of spaghetti is a cylinder. If you think about it, a coin is also a cylinder and an uncooked piece of spaghetti is also a disc. Clearly the coin is more disc-like and the spaghetti more cylinder-like. Your task is to devise a measure of disc-ness. assessment.concord.org

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What support do students need to tackle these rich tasks? What scaffolding is appropriate? What prompts would you want to give? Supporting, Scaffolding, Prompting…

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Hold the demand Students of all abilities deserve tasks that demand higher level skills BUT teachers and students conspire to lower the cognitive demand of tasks! We want to prompt students to reason with mathematics and reason through problems rather than giving prompts that require little engagement and reasoning. How can we provide support without reducing the cognitive demand?

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Discs and cylinders A gold coin is a disc, and an uncooked piece of spaghetti is a cylinder. If you think about it, a coin is also a cylinder and an uncooked piece of spaghetti is also a disc. Clearly the coin is more disc-like and the spaghetti more cylinder-like. Your task is to devise a measure of disc-ness. How can we scaffold without lowering the demand?

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1.Given a coin, a tuna-fish can and a soup can: Devise a definition of disc-ness that allows you to say which object is the most disc-like and which is the least. 2.Given a mailing tube, a straw, and a piece of uncooked spaghetti: Use your definition of disc-ness to determine which object is the most disc-like and which is the least. 3.Write a formula (or algorithm or algebraic sentence) which expresses your measure of disc-ness. You may introduce any labels and definitions you like and use all the mathematical language you care to. 4.Make any measurements you need, and calculate a numerical value of disc-ness for each of the six items. 5.Discuss whether these numbers seem reasonable in light of your notion of disc-ness. 6.How would you change your answers to these questions if you were asked to write a formula for cylinder-ness rather than disc-ness?

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Scaffolding learning 1.Problem solving cycle 2.Warm up activity 3.Success criteria

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Two Classrooms Problem solving involves investigating a problem when the route to a solution is unclear. We need to use tasks which allow for student choice about the mathematics and the problem- solving strategies they use to model and investigate situations. Learning needs to be scaffolded so students are able to call on their mathematical knowledge and search for a pathway to the answer. Digging for gold

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Rich mathematical tasks are not enough. Students need rich mathematical experiences: The greatest learning occurs in classrooms in which mathematical tasks with high-level cognitive demand are used and the demand is consistently maintained throughout the teaching. Digging for gold

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Finding rich tasks

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May the force be with you as you dig for gold! Thank you

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