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.