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### The Development of Mathematical Proficiency

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

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

- 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

- Conceptual Understanding

- Strategic Competence

- Procedural Fluency

- Adaptive Reasoning

- Productive Disposition

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

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

16

- 8

12

1/3 + 2/5 = 3/8

9.83 x 7.65 = 7,519.95

Discussion Questions

- What is Conceptual Understanding?
- How do we teach forConceptual Understanding?
- What does it look like when students have Conceptual Understanding?

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?

- Understanding concepts helps recallprocedures correctly
- Mastering concepts fosters the ability to choose appropriate math tools and strategies

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?

- 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 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 is the capacity for:
- Logical thought
- Reflection
- Explanation
- Justification

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

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?

Productive Disposition

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

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

Application

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

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

justification

- Views mathematics as

sensible, useful, &

worthwhile, coupled

with a belief of ability

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

- 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.”

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