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Extending High Expectations to ALL Students Glenda Anthony Massey University

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New goals, new challenges Raising the floor by expanding achievement for all. Lifting the ceiling of achievement to better prepare future leaders in mathematics. Mathematics is a key resource for building a socially just and diverse democracy (Ball, 2005) Critical lever for social and educational progress (Moses & Cobb, 2001)

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Changing discourse Raising teacher expectations Closing the gap Shortening the tail Extending high expectations to all students Inclusive pedagogies, responsive pedagogies Integrating the multiples voices of the classroom to orchestrate/occasion learning spaces/opportunities Teacher agency

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Complexity of occasioning learning Providing Appropriate challenge Establishing participation rights and understandings Task cognitive demands

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Participation Attention to the rights and obligations of mathematical participation Inclusiveness, positioning (positive mathematical identity) Culture of respect and care Valuing of students’ contributions We all felt like a family in maths. Does that make sense? Even if we weren’t always sending out brotherly/sisterly vibes. Well we got used to each other… so we all worked…We all knew how to work with each other…it was a big group…more like a neighbourhood with loads of different houses. (BES p. 58)

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Who can participate? Top and bottom sets (BES p. 120) Immigrants and locals (BES p.65) Fast and slow kids (BES, p. 123) Busy versus challenging work (BES, p. 127) See Empsons fraction CASE 2

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Differential access to the curriculum Do students in low streamed classes/groups have poorer access to mathematics. They follow a protracted curriculum. Their reduced social obligations and lesser cognitive demands placed on low streamed students had the effect of excluding them from full engagement in mathematics.

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Culture of mathematical proficiency Okay to make mistakes More than a climate of politeness Caring about the development of mathematical proficiency

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Mathematical thinking Annie and Sam, in Year 1 both know that = 6 and = 6. Is that good?

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Michael: In a CGI class How is he positioned and included in the classroom community? What sorts of understandings is he forming about learning mathematics? How is his mathematical identity developing? What are the key pedagogical practices?

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Tasks need to be purposeful and provide an appropriate challenge It’s safer—children feel more comfortable if they’re not made to think. I realise this is cynical—but for many children with low IQs and poor/non existent English language skills, the concept of problem solving is alien. Also it takes up too much time and there is great pressure to “get through” the curricula. So whilst in theory I acknowledge the potential of problem solving, in reality with some clientele it’s too hard. (Anderson, 2003, p. 76)

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Focus on what is learnt rather than what is completed Ms Summers:You’ve finished! Doesn’t it feel good when you’ve done it? (Late in Y 3) Mrs Kyle:How many finished? (Looking around at the show of hands) Most of you didn’t finish. You must learn to put ‘DNF’—did not finish, at the bottom. (Early in Y 4) Ms Torrance:We have some amazing speedsters who have got on their rollerblades and got their two sheets done already. (p. 206)

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Classroom Tasks

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Analysing Mathematical Tasks

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Tasks lead to different opportunities Low levels of cognitive demands Memorisation tasks Procedures without connections High level of cognitive demand Procedures with connections Doing mathematics

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Low-level demand tasks Involve reproducing previously learned facts, rules, formulae, or definitions OR committing facts, rules, formulae, or definitions to memory. Cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use a procedure. Involve exact reproduction of previously seen material and what is to be reproduced is clearly and directly stated. Have no connection to the concepts that underlie the facts, rules, formulae, or definitions being learned and reproduced.

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Memorisation task What are the decimal and percent equivalents for the fractions ½ and ¼ ? Expected Student Response: ½=.5=50% ¼=.25=25%

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Procedures without connections Are algorithmic. Procedure is either specifically called for or its use is evident based on prior instruction, experience, or placement of the task. Little ambiguity about what needs to be done and how to do it. Have no connection to the concepts or meaning that underlie the procedure being used. Focus on producing correct answers. If required, explanations focus solely on describing the procedure that was used.

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Procedures without connections task Convert the fraction 3/8 to a decimal and a percent. Expected Student Response: FractionDecimalPercent = 37.5%

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Procedures with connections Focus attention on the use of procedures for the purpose of developing deeper levels of understanding Suggest pathways (explicitly or implicitly) that are broad general procedures that connect to underlying conceptual ideas as opposed to algorithms. Usually are represented in multiple ways (e.g., diagrams, manipulatives, symbols, problem situations). Require cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with the conceptual ideas that underlie the procedures in order to successfully complete the task and develop understanding.

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Procedures with connections task Using a 10 x 10 grid, identify the decimal and percent equivalents of 3/5. Expected Student Response:

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‘Doing’ mathematics tasks Require complex and nonalgorithmic thinking (i.e., there is not a predictable, well-rehearsed approach explicitly suggested by the task. Require exploration and understanding of the mathematical concepts, processes, or relationships. Demand self-monitoring of one’s own cognitive processes. Require students to access relevant knowledge and experiences and make appropriate use of them. Require students to analyse the task and actively examine task constraints that may limit possible solution strategies and solutions. Require cognitive effort and may involve some anxiety due to the unpredictable nature of the solution process.

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‘Doing mathematics’ task Shade 6 small squares in a 4 x 10 rectangle. Using the rectangle, explain how to determine each of the following: a) the percent of area that is shaded, b) the decimal part of area that is shaded, and c) the fractional part of area that is shaded.

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One possible student response (a) One column will be 10%, since there are 10 columns. So four squares is 10%. Then 2 squares is half a column and half of 10%, which is 5%. So the 6 shaded blocks equal 10% plus 5%, or 15%. (b) One column will be 0.10, since there are 10 columns. The second column has only 2 squares shaded, so that would be one-half of 0.10, which is 0.05, which equals (c) Six shaded squares out of 40 squares is 6/40, which reduces to 3/20.

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What is the Task Cognitive Level? The cost of a sweater at I.M Wolly’s was $45. At the Waitangi day sale it was marked 30% off the original price. What was the price of the sweater during the sale? Explain the process you used to find the sale price?

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Beware of superficial task features Requires the use of a calculator or diagram Involve multiple steps to complete Requires a written explanation Has a real-world context Is the task worthwhile just because students find it difficult?

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Decline of high-cognitive demands The higher the demands that a task placed on students at the task-set-up phase, the less likely it was for the task to have been carried out faithfully during the implementation phase. Stein, M., Grover, B., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: an analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2),

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Slip in demands Students not held accountable for high-level products or processes. Shift in emphasis from meaning to completion Students press teacher to reduce complexity Time: too little or too much.

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To maintain high-level cognitive demands Provide scaffolding of student thinking Provide means for students’ to monitor their progress Yourself or your students model high performance Require justification and explanation and meaning Build on students’ prior knowledge Provide conceptual connections Allow sufficient time: not too much or too little!

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