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.

FALL 2011 MATHEMATICS SOL INSTITUTES Kyle Schultz, Asst. Professor of Mathematics Education, James Madison University J. 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 3 Promoting Students’ Mathematical Understanding through Problem Solving, Communication, and Reasoning

Mathematical Communication 4 Students will use the language of mathematics, including specialized vocabulary and symbols, to express mathematical ideas precisely.

Mathematical Reasoning 5 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 6 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?

Classroom Video of these tasks Video 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. 7

Examining Differences between Tasks 8 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 9 What is cognitive demand? thinking required

Task Sort Activity 10 Sort the provided tasks as high or low cognitive demand. List characteristics you use to sort the tasks.

Discussing the Task Sort 11 TaskABCDEFGHIJ 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 12 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

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 13 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

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) 14

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. 15

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. 16

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 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

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 18 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

Lesson Structure To 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) 19

Organizing High-Level Discussions: 5 Habits Prior 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

Organizing High-Level Discussions: 5 Habits While students are working, 2.Monitor their progress, 3.Select students to present their work, and 4.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

Organizing High-Level Discussions: 5 Habits During 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

Practicing the 5 Habits: Anticipate Work 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. 23

Practicing the 5 Habits: Select and Sequence Let 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. 24

Practicing the 5 Habits: Connecting For the student work you selected and sequenced, 3.Identify connections within that work that you would hope to highlight during the class discussion. 25

Next Steps District-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? 26

Next Steps Support Documents Facilitator Guide for this sesssion Process Standards “Look Fors” handout Questions? 27