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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.

GRADE BAND: 9-12

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

Promoting Students’

Mathematical Understanding

through

Problem Solving,

Communication,

and Reasoning

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

- 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.

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?

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.

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?

thinking

required

What is cognitive demand?

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

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

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

- 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

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

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)

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.

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.

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

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

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

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

- Individual thinking (preliminary brainstorming)
- Small group discussion (idea development)
- Whole class discussion (idea refinement)

Prior to the lesson,

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

While students are working,

Monitor their progress,

Select students to present their work, and

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

During the discussion,

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

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.

Let the provided samples work on the triangle task represent the work your students observed while monitoring their work.

- Select 4 to 5 students who you would call on to present their work.
- Sequence these students to optimize the class discussion of this task.

For the student work you selected and sequenced,

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

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?

Support Documents

- Facilitator Guide for this sesssion
- Process Standards “Look Fors” handout
- Questions?