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Welcome! 1 Conduct this experiment. –Place 10 red cards and 10 black cards face down in separate piles. –Choose some cards at random from the red pile and mix them into the black pile. –Shuffle the mixed pile. –Return the same number of random cards (face down) from the mixed pile to the red pile. Are there more red cards in the black pile or black cards in the red pile? Make a conjecture about the number of each type of card in each pile.

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MATH RIGOR FACILITATING STUDENT UNDERSTANDING THROUGH PROCESS GOALS Adapted from VDOE SOL Institutes GRADE BAND: 6-12 Summer 2012

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Introductions What is your name? Where do you teach? Which type of number (Natural, Whole, Integers, Rational, Irrational & Real) best matches your personality? 3

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Promoting Students’ Mathematical Understanding Five goals – for students to 1. become mathematical problem solvers that 2. communicate mathematically; 3. reason mathematically; 4. make mathematical connections; and 5. use mathematical representations to model and interpret practical situations 4

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Mathematical Problem Solving 5 Students will apply mathematical concepts and skills and the relationships among them to solve problem situations of varying complexities.

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Mathematical Communication 6 Students will use the language of mathematics, including specialized vocabulary and symbols, to express mathematical ideas precisely.

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Mathematical Reasoning 7 Students will learn and apply inductive and deductive reasoning skills to make, test, and evaluate mathematical statements and to justify steps in mathematical procedures. Students will use logical reasoning to analyze an argument and to determine whether conclusions are valid.

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Mathematical Connections 8 Students will relate concepts and procedures from different topics in mathematics to one another and see mathematics as an integrated field of study.

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Mathematical Representations 9 Students will represent and describe mathematical ideas, generalizations, and relationships with graphical, numerical, algebraic, verbal, and physical representations. Students will recognize that representation is both a process and a product.

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Mindstreaming 10 How do you choose the problems, tasks or projects that you plan for your students?

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Triplet Tasks 11 Complete the three tasks on your handout. In your group, discuss the following questions: What do students need to know to solve each task? How are the tasks similar? How are the tasks different? What process standards (problem solving, reasoning, representations, connections, communication) are used?

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Examining Differences between Tasks 12 What is cognitive demand? thinking required

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What is Rigor? Google it. Try –“instructional rigor” –“rigor and relevance” –“academic rigor” –“rigor mortis”

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Task Sort Activity 14 Directions: Individually, read each task. Sort and record the provided tasks as high or low cognitive demand. Record your answers in the google form. Discuss your results as a table group and come to a consensus. Describe the criteria you used to sort each task. How did you communicate your reasoning to your table group? List the criteria in your task analysis guide.

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Discussing the Task Sort 15 Task1/A2/B3/C4/D5/E6/F7/G8/H9/I10/J Low High Criteria for a low level task Computation, mechanical explanations One step, rote Skill oriented, Clear directions/task Basic concepts Criteria for a high level task Multi-task, reasoning, connections, steps Interpret/process info/analyze Justify Create ‘new’ item Extraneous info included

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Task Analysis Guide – Lower-level Demands Involve recall or memory of facts, rules, formulae, or definitions Involve exact reproduction of previously seen material No connection of facts, rules, formulae, or definitions to concepts or underlying understandings. Focused on producing correct answers rather than developing mathematical understandings Require no explanations or explanations that focus only on describing the procedure used to solve 16 Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

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Task Analysis Guide – Higher-level Demands Focus on developing deeper understanding of concepts Use multiple representations to develop understanding and connections Require complex, non-algorithmic thinking and considerable cognitive effort Require exploration of concepts, processes, or relationships Require accessing and applying prior knowledge and relevant experiences to facilitate connections Require critical analysis of the task and solutions 17 Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press DOING Mathematics

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What should be considered when selecting tasks? 18 Content Alignment Process Standards Cognitive Depth

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Content Alignment Alignment is based upon: SOL & Curriculum Framework Essential Understandings Essential Knowledge & Skills Essential Questions (middle schools) Vertical Articulation Pacing Guide - HCPS Purpose - student needs 19

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Vertical Articulation of Content Which related prerequisites did students have previously? How will they use this concept next year? Why is it important knowledge to have? Consistency Connections Relevance Vocabulary Building – Not Repeating! All these lead to deeper understanding and long-term retention of content 20

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Process Standards Communication: Talking and writing about math Problem-Solving Reasoning Multiple Representations Connections

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Cognitive Demand …students who performed best on a project assessment designed to measure thinking and reasoning processes were more often in classrooms in which tasks were enacted at high levels of cognitive demand (Stein and Lane 1996), that is, classrooms characterized by sustained engagement of students in active inquiry and sense making (Stein, Grover, and Henningsen 1996). For students in these classrooms, having the opportunity to work on challenging mathematical tasks in a supportive classroom environment translated into substantial learning gains. ---Stein & Smith, 2010 Understanding the “increased rigor” of the new SOL comes through analysis of the SOL and the Curriculum Framework

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Characteristics of Rich Mathematical Tasks High cognitive demand (Stein et. al, 1996; Boaler & Staples, 2008) Significant content (Heibert et. al, 1997) Require Justification or explanation (Boaler & Staples, in press) Make connections between two or more representations (Lesh, Post & Behr, 1988) Open-ended (Lotan, 2003; Borasi &Fonzi, 2002) Allow entry to students with a range of skills and abilities Multiple ways to show competence (Lotan, 2003) 23

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Thinking About Implementation In order for students to reason about and communicate mathematical ideas, they must be engaged with high cognitive demand tasks that enable practice of these skills. BUT! … simply selecting and using high-level tasks is not enough. Teachers need to be vigilant during the lesson to ensure that students’ engagement with the task continues to be at a high level. 24

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Let’s Recap! 25

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Process Standards 26 VIRGINIA (Process Goals) NCTM (Process Standards) CCSS (Mathematical Practices) Mathematical Problem Solving Problem Solving1) Make sense of problems and persevere in solving them. Mathematical Communication Communication3) Construct viable and critique the reasoning of others Mathematical ReasoningReasoning and Proof2) Reason abstractly and quantitatively Mathematical ConnectionsConnections7) Look for and make use of structure 8) Look for and express regularity in repeated reasoning Mathematical Representations Representations4) Model with mathematics 5) Use appropriate tools strategically 6) Attend to precision

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Key Messages We must not solely focus on multiple-choice assessments We must provide students with rich, relevant, and rigorous tasks that focus on more than one specific skill and require application and synthesis of mathematical knowledge We must connect mathematics content within and among grade levels and subject areas to facilitate long term retention and application We must reflect on our own teaching and resist the urge to blame students 27

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Juli K. Dixon, Ph.D. University of Central Florida

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