FAME Follow-Up September 15, 2011

Melissa ChristieMathematics CoordinatorSanta Clara County Office of Education*

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 FrameworksStrands of Mathematical Proficiency

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

Procedural Fluency

1. Make sense of problems and persevere in solving themstart by explaining to themselves the meaning of a problem and looking for entry points to its solution2. Reason abstractly and quantitativelymake sense of quantities and their relationships to problem situations3. Construct viable arguments and critique the reasoning of othersunderstand and use stated assumptions, definitions, and previously established results in constructing arguments4. Model with mathematicscan apply the mathematics they know to solve problems arising in everyday life, society, and the workplace

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Standards for Mathematical PracticeMathematically proficient students: 5. Use appropriate tools strategicallyconsider the available tools when solving a mathematical problem6. Attend to precisioncommunicate with clear definitions in discussions with others7. Look for and make use of structurelook closely to discern a pattern or structure8. Look for and express regularity in repeated reasoningnotice if calculations are repeated, and look for both general methods and for shortcuts*

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 teams 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 expressionsrecognizing equivalent expressionsunderstanding the distributive laws of multiplication and division over addition

Mathematical Tasks:A Critical Starting Point for InstructionNot 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, 1995Mathematical 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.

*Turn and talk, what does the graphic communicate about the nature of the 5 strands**Finally, help students move from concrete manipulatives to representations including drawings or diagrams, and then to the abstract symbolic.

*****START is the operative word here. CDMT are a starting point but they provide no guarantee that students will actually engage in them as intended or that the desired mathematics will emerge from having done so.

It is the discussion around such tasks that is both critically important and incredibly difficult.