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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. Adding It Up: Helping Children Learn Mathematics.

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The development of mathematical proficiency l.jpg

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 compellingshowing 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.

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Mathematical Proficiency

  • Conceptual Understanding

  • Strategic Competence

  • Procedural Fluency

  • Adaptive Reasoning

  • Productive Disposition

Let’s give kids something they can hold on to!

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Conceptual Understanding

“When knowledge is learned withunderstanding it provides a basis forgenerating new knowledge.”

  • It is comprehension of concepts, operations and relationships

  • It helps students avoid critical errors in problem solving

  • It is being able to represent mathematical situations in different ways

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What do these say about the student’s Conceptual Understanding?


- 8


1/3 + 2/5 = 3/8

9.83 x 7.65 = 7,519.95

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Discussion Questions Understanding?

  • What is Conceptual Understanding?

  • How do we teach forConceptual Understanding?

  • What does it look like when students have Conceptual Understanding?

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Procedural Fluency Understanding?

  • Skill in carrying out mathematical steps and computations

  • Understanding concepts makes learning skills easier, less susceptible to common errors, andless prone to forgetting

  • Using procedures can help to strengthen and develop understanding

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Does Practice Make Perfect? Understanding?

  • Understanding concepts helps recallprocedures correctly

  • Mastering concepts fosters the ability to choose appropriate math tools and strategies

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How Do You Know Understanding?They Got It?

  • What are some successful strategiesyou use to develop procedural fluency?

  • How are procedural fluency and conceptual understanding related?

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How would you Understanding?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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Questions to Consider Understanding?

  • What is the problem?

  • What do you need to know to solve this problem?

  • Describe more than one way to solve this problem?

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Strategic Competence Understanding?The ability to formulate, represent and solve mathematical problems.

  • Formulate problems

  • Multiple strategies

  • Flexibility

  • Nonroutine problems vs. routine problems

Allow nonroutine problems to be the vehicle to build Strategic Competence.

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Adaptive Reasoning Understanding?“…the glue that holds everything together.”

  • Adaptive Reasoning is the capacity for:

  • Logical thought

  • Reflection

  • Explanation

  • Justification

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Conditions Needed Understanding?

  • 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 Understanding?

  • 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 Understanding?

  • Mathematics makes sense

  • Mathematics is useful and worthwhile

  • Steady effort

  • Effective learners and doers

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Key Points Understanding?

  • Emotional development

    • Self-efficacy and self-image

  • Stereotype threat

    • Peer pressure to under-achieve

  • “Wise educational environments”

    • Affective filter - math as a “second” language

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Application Understanding?

  • How do teachers’ feelings/perceptions toward math affect productive disposition?

  • How can SDAIE teaching strategies increase productive disposition in math?

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Mathematical Proficiency Understanding?

Conceptual Understanding

  • Comprehension of mathematical concepts

Strategic Competence

  • Ability to solve mathematical problems

Procedural Fluency

  • Knowledge of algorithms

Adaptive Reasoning

Productive Disposition

  • Capacity for logical

    thought, reflection,

    explanation and


  • Views mathematics as

    sensible, useful, &

    worthwhile, coupled

    with a belief of ability

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Bringing It All Together Understanding?

  • How do the five strands of mathematical proficiency relate to standards-based instruction?

  • How will you incorporate mathematical proficiency into daily teaching practice?

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In Conclusion Understanding?

  • The goal of instruction should be mathematical proficiency

  • It takes time for mathematical proficiency to be fully developed

  • 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.”