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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

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
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.
mathematical proficiency
Mathematical Proficiency
  • Conceptual Understanding
  • Strategic Competence
  • Procedural Fluency
  • Adaptive Reasoning
  • Productive Disposition

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

conceptual understanding
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
what do these say about the student s conceptual understanding

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

discussion questions
Discussion Questions
  • What is Conceptual Understanding?
  • How do we teach forConceptual Understanding?
  • What does it look like when students have Conceptual Understanding?
procedural fluency
Procedural Fluency
  • 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
does practice make perfect
Does Practice Make Perfect?
  • Understanding concepts helps recallprocedures correctly
  • Mastering concepts fosters the ability to choose appropriate math tools and strategies
how do you know they got it
How Do You Know They Got It?
  • What are some successful strategiesyou use to develop procedural fluency?
  • How are procedural fluency and conceptual understanding related?
how would you solve this problem
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

Questions to Consider
  • What is the problem?
  • What do you need to know to solve this problem?
  • Describe more than one way to solve this problem?
strategic competence the ability to formulate represent and solve mathematical problems
Strategic CompetenceThe 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.

adaptive reasoning the glue that holds everything together
Adaptive Reasoning“…the glue that holds everything together.”
  • Adaptive Reasoning is the capacity for:
  • Logical thought
  • Reflection
  • Explanation
  • Justification
conditions needed
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
  • 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?
productive disposition
Productive Disposition
  • Mathematics makes sense
  • Mathematics is useful and worthwhile
  • Steady effort
  • Effective learners and doers
key points
Key Points
  • Emotional development
    • Self-efficacy and self-image
  • Stereotype threat
    • Peer pressure to under-achieve
  • “Wise educational environments”
    • Affective filter - math as a “second” language
  • How do teachers’ feelings/perceptions toward math affect productive disposition?
  • How can SDAIE teaching strategies increase productive disposition in math?
mathematical proficiency19
Mathematical Proficiency

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

bringing it all together
Bringing It All Together
  • How do the five strands of mathematical proficiency relate to standards-based instruction?
  • How will you incorporate mathematical proficiency into daily teaching practice?
in conclusion
In Conclusion
  • 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.”