Engaging All Students in Productive Mathematics Discussion and Problem Solving

FAME Follow-Up September 15, 2011

Melissa ChristieMathematics Coordinator

Santa Clara County Office of Education

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Outcome

Explore strategies and resources that can be used in the mathematics classroom to deepen conceptual understanding and promote problem solving for all students.

AgendaWelcome/Outcomes/AgendaWarm-UpInterpreting Algebraic Expressions Lesson

ExplorationCognitively demanding mathematical tasks

(CDMT) and the Cognitive Demand SpectrumInterpreting Algebraic Expressions Lesson

DebriefReflectionClosure

Warm-Up

Underlying Frameworks

Strands of Mathematical Proficiency

Strategic Competence

Adaptive Reasoning

Conceptual Understanding Productive

Disposition

NRC (2001). Adding It Up. Washington, D.C.: National Academies Press.

Procedural Fluency

1. Make sense of problems and persevere in solving them…start by explaining to themselves the meaning of a problem and looking for entry points to its solution

2. Reason abstractly and quantitatively…make sense of quantities and their relationships to problem situations

3. Construct viable arguments and critique the reasoning of others…understand and use stated assumptions, definitions, and previously established results in constructing arguments

4. Model with mathematics…can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace

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Standards for Mathematical PracticeStandards for Mathematical Practice

MatMathematically proficient students: hematically proficient students:

5. Use appropriate tools strategically…consider the available tools when solving a mathematical problem

6. Attend to precision…communicate with clear definitions in discussions with others

7. Look for and make use of structure…look closely to discern a pattern or structure

8. Look for and express regularity in repeated reasoning…notice if calculations are repeated, and look for both general methods and for shortcuts

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Lesson Exploration

Work in teams of three on the collaborative discussion task Interpreting Algebraic Expressions.

As you work through the lesson, be mindful of any mathematical misconceptions your students may have.

Capture all of your team’s work on poster paper.

A “Gallery Walk” of all posters will culminate the activity.

Mathematical “Big Ideas” of LessonUnderstand that different forms of an

expression may reveal different properties of the quantity in question.

See expressions in different ways that suggest ways of transforming them.

Understand that polynomial identities become true statements no matter which real numbers are substituted.

Transform simple rational expressions using the commutative, associative, and distributive laws.

MisconceptionsThis lesson will help to identify students who

have difficulty with the following math concepts:

recognizing the order of algebraic expressions

recognizing equivalent expressions understanding the distributive laws of

multiplication and division over addition

Mathematical Tasks:A Critical Starting Point for Instruction

Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.

Stein, Smith, Henningsen, & Silver, 2000

The level and kind of thinking in which students engage determines what they will learn.

Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997

Mathematical Tasks:A Critical Starting Point for Instruction

There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics.

Lappan & Briars, 1995

Mathematical Tasks:A Critical Starting Point for Instruction

If we want students to develop the capacity to think, reason, and problem

solve then we need to start with high-level, cognitively complex tasks.

Stein & Lane, 1996

Mathematical Tasks:A Critical Starting Point for Instruction

Outcome

Explore strategies and resources that can be used in the mathematics classroom to deepen conceptual understanding and promote problem solving for all students.