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.

Welcome!

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• 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.

FALL 2011 MATHEMATICS SOL INSTITUTES

Kyle Schultz, Asst. Professor of Mathematics Education, James Madison UniversityJ. Patrick Lintner, Director of Instruction, Harrisonburg City Su Chuang, Mathematics Specialist K-12, Loudoun County Michael Traylor, Secondary Mathematics Consultant 6-12, Chesterfield County

GRADE BAND: 9-12

Our Goal

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Promoting Students’ Mathematical Understanding

through Problem Solving, Communication,and Reasoning

Mathematical Communication

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

Mathematical Reasoning

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• 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.

Communication and Reasoning

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Solve: • What strategies could you use to solve each

problem?• What strategies would you expect your

students to consider?• What mathematical connections and

representations did you consider?

2 25 2 3x xand x

Classroom Video of these tasksVideo Link

• This video shows a lesson where students explore the same exponential equations.

• While watching the video, look for examples of the five process standards in action.

• We will pause the video at different points during the video to allow you to record your observations.

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Examining Differences between Tasks

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Examine the three algebra 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?

Examining Differences between Tasks

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What is cognitive demand?

thinking required

Task Sort Activity

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• Sort the provided tasks as high or low cognitive demand.

• List characteristics you use to sort the tasks.

Discussing the Task Sort

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Task A B C D E F G H I J

Low

High

Criteria for a low level task

Criteria for a high level task

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

12Adapted 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

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• Require critical analysis of the task and solutions

13Adapted 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

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)

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Thinking About Implementation• A mathematical task can be described

according to the kinds of thinking it requires of students, it’s level of cognitive demand.

• 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.

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The Challenge of Implementation• 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.

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Factors Associated with Lowering High-level Demands• Shifting emphasis from meaning, concepts, or

understanding to the correctness or completeness of the answer

• Providing insufficient or too much time to wrestle with the mathematical task

• Letting classroom management problems interfere with engagement in mathematical tasks

• Providing inappropriate tasks to a given group of students

• Failing to hold students accountable for high-level products or processes

17Adapted 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

Factors Associated with Promoting Higher-level Demands• Scaffolding of student thinking and reasoning• Providing ways/means by which students can

monitor/guide their own progress• Modeling high-level performance• Requiring justification and explanation through

questioning and feedback• Selecting tasks that build on students’ prior knowledge

and provide multiple access points• Providing sufficient time to explore tasks

18Adapted 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

Lesson StructureTo foster reasoning and communication focused on a rich mathematical task, we recommend a 3-part lesson structure:

1. Individual thinking (preliminary brainstorming)

2. Small group discussion (idea development)

3. Whole class discussion (idea refinement)

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Organizing High-Level Discussions: 5 HabitsPrior to the lesson, 1. Anticipate student strategies and responses

to the task

More on 5 Habits can be found in: “Orchestrating Discussions” by Smith, Hughes, Engle, & Stein in Mathematics Teaching in the Middle School, May 2009

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Organizing High-Level Discussions: 5 HabitsWhile students are working, 2. Monitor their progress,3. Select students to present their work, and4. Sequence the presentations to maximize

discussion goals

More on 5 Habits can be found in: “Orchestrating Discussions” by Smith, Hughes, Engle, & Stein in Mathematics Teaching in the Middle School, May 2009

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Organizing High-Level Discussions: 5 HabitsDuring the discussion,5. Ask questions that help students connect the

presented ideas to one another and to key mathematical ideas

More on 5 Habits can be found in: “Orchestrating Discussions” by Smith, Hughes, Engle, & Stein in Mathematics Teaching in the Middle School, May 2009

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Practicing the 5 Habits: AnticipateWork on the following geometry task and think about possible student strategies/solutions:

Triangle ABC has interior angle C measuring 105°. The segment opposite angle C has a measure of 23 cm. Describe the range of values for the measures of the other sides and angles of triangle ABC. Explain your reasoning.

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Practicing the 5 Habits: Select and SequenceLet the provided samples work on the triangle task represent the work your students observed while monitoring their work.

1. Select 4 to 5 students who you would call on to present their work.

2. Sequence these students to optimize the class discussion of this task.

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Practicing the 5 Habits: ConnectingFor the student work you selected and sequenced,

3. Identify connections within that work that you would hope to highlight during the class discussion.

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Next StepsDistrict-Level Implementation• How could the ideas presented today be

structured for implementation with teachers in your district?

• What classroom artifacts could your teachers bring with them to be incorporated into sessions?

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Next StepsSupport Documents• Facilitator Guide for this sesssion• Process Standards “Look Fors” handout

• Questions?

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