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

  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.

Mathematical Proficiency

 Strategic Competence  Conceptual Understanding  Procedural Fluency  Adaptive Reasoning  Productive Disposition

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

Conceptual Understanding “When knowledge is learned with understanding it provides a basis for generating 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?

1/3 + 2/5 = 3/8 16 - 8 12 9.83 x 7.65 = 7,519.95

Discussion Questions

 What is Conceptual Understanding?

 How do we teach for Conceptual 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, and less prone to forgetting Using procedures can help to strengthen and develop understanding

Does Practice Make Perfect?

 Understanding concepts helps recall procedures correctly  Mastering concepts fosters the ability to choose appropriate math tools and strategies

How Do You Know They Got It?

 What are some successful strategies you 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 Competence 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.

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 Strategic Competence

 Ability to solve mathematical problems  Comprehension of mathematical concepts

Procedural Fluency

 Knowledge of algorithms

Adaptive Reasoning

 Capacity for logical thought, reflection, explanation and justification

Productive Disposition

 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.”