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The Development of Mathematical Proficiency Presented by the Math Coaches of LAUSD, District K Based on: Adding It Up: Helping Children Learn Mathematics, National Research Council, National Academy Press, Washington D.C., 2001

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Adding It Up: Helping Children Learn Mathematics The research evidence is consistent and compelling showing the following weaknesses: The research evidence is consistent and compelling showing the following weaknesses: US students have limited basic understanding of mathematical concepts They are notably deficient in their ability to solve even simple problems And, overall, are not given educational opportunity they need to achieve at high levels In short, the authors tell us that US teachers focus primarily on one area, computation. In short, the authors tell us that US teachers focus primarily on one area, computation.

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Mathematical Proficiency Strategic Competence Conceptual Understanding Adaptive Reasoning Procedural Fluency Productive Disposition Let’s give kids something they can hold on to!

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Conceptual Understanding It is comprehension of concepts, operations and relationships It is comprehension of concepts, operations and relationships It helps students avoid critical errors in problem solving It helps students avoid critical errors in problem solving It is being able to represent mathematical situations in different ways It is being able to represent mathematical situations in different ways “When knowledge is learned with understanding it provides a basis for generating new knowledge.”

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What do these say about the student’s Conceptual Understanding? 1/3 + 2/5 = 3/8 1/3 + 2/5 = 3/8 9.83 x 7.65 = 7,519.95 9.83 x 7.65 = 7,519.95 16 - 8 12

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Discussion Questions What is Conceptual Understanding? What is Conceptual Understanding? How do we teach for Conceptual Understanding? How do we teach for Conceptual Understanding? What does it look like when students have Conceptual Understanding? What does it look like when students have Conceptual Understanding?

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Procedural Fluency Skill in carrying out mathematical steps and computations Skill in carrying out mathematical steps and computations Understanding concepts makes learning skills easier, less susceptible to common errors, and less prone to forgetting Understanding concepts makes learning skills easier, less susceptible to common errors, and less prone to forgetting Using procedures can help to strengthen and develop understanding Using procedures can help to strengthen and develop understanding

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Does Practice Make Perfect? Understanding concepts helps recall procedures correctly Understanding concepts helps recall procedures correctly Mastering concepts fosters the ability to choose appropriate math tools and strategies Mastering concepts fosters the ability to choose appropriate math tools and strategies

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How Do You Know They Got It? What are some successful strategies you use to develop procedural fluency? What are some successful strategies you use to develop procedural fluency? How are procedural fluency and conceptual understanding related? How are procedural fluency and conceptual understanding related?

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How would you solve this problem? A cycle shop has a total of 36 bicycles and tricycles in stock. Collectively there are 80 wheels. How many bicycles and how many tricycles are there?* *Adding It Up, National Research Council, 2001, p.126

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What is the problem? What is the problem? What do you need to know to solve this problem? What do you need to know to solve this problem? Describe more than one way to solve this problem? Describe more than one way to solve this problem? Questions to Consider

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Strategic Competence The ability to formulate, represent and solve mathematical problems. Formulate problems Formulate problems Multiple strategies Multiple strategies Flexibility Flexibility Nonroutine problems vs. routine problems Nonroutine problems vs. routine problems Allow nonroutine problems to be the vehicle to build Strategic Competence.

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Adaptive Reasoning “…the glue that holds everything together.” Adaptive Reasoning is the capacity for: Logical thought Logical thought Reflection Explanation Justification

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Conditions Needed Real-world, motivating tasks Utilizes the knowledge-base and experience that children bring to school Rigorous questioning Students justify their work on a regular basis

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Questions How do you promote adaptive reasoning in your classroom? What is the evidence that your students are regularly using adaptive reasoning? What are the long-term benefits of students utilizing adaptive reasoning?

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Productive Disposition Mathematics makes sense Mathematics makes sense Mathematics is useful and worthwhile Mathematics is useful and worthwhile Steady effort Steady effort Effective learners and doers Effective learners and doers

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Key Points Emotional development Emotional development Self-efficacy and self-image Stereotype threat Stereotype threat Peer pressure to under-achieve “Wise educational environments” “Wise educational environments” Affective filter - math as a “second” language

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Application How do teachers’ feelings/perceptions toward math affect productive disposition? How do teachers’ feelings/perceptions toward math affect productive disposition? How can SDAIE teaching strategies increase productive disposition in math? How can SDAIE teaching strategies increase productive disposition in math?

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Mathematical Proficiency Ability to solve mathematical problems Comprehension of mathematical concepts Capacity for logical thought, reflection, thought, reflection, explanation and explanation and justification justification Knowledge of algorithms Views mathematics as sensible, useful, & sensible, useful, & worthwhile, coupled worthwhile, coupled with a belief of ability with a belief of ability Conceptual Understanding Procedural Fluency Strategic Competence Productive Disposition Adaptive Reasoning

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Bringing It All Together How do the five strands of mathematical proficiency relate to standards-based instruction? How do the five strands of mathematical proficiency relate to standards-based instruction? How will you incorporate mathematical proficiency into daily teaching practice? How will you incorporate mathematical proficiency into daily teaching practice?

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In Conclusion The goal of instruction should be mathematical proficiency The goal of instruction should be mathematical proficiency It takes time for mathematical proficiency to be fully developed It takes time for mathematical proficiency to be fully developed Mathematical proficiency spans number sense, algebra & functions, measurement & geometry, SDAP, and mathematical reasoning Mathematical proficiency spans number sense, algebra & functions, measurement & geometry, SDAP, and mathematical reasoning “All young Americans must learn to think mathematically and must think mathematically to learn.”

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