The Development of Mathematical Proficiency

Presented by the Math Coaches of LAUSD, District KPresented by the Math Coaches of LAUSD, District K

Based on: Based on: Adding It Up: Helping Children Learn MathematicsAdding It Up: Helping Children Learn Mathematics, National , National Research Council, National Academy Press, Washington D.C., 2001Research Council, National Academy Press, Washington D.C., 2001

Adding It Up: Helping Children Learn Mathematics

The research evidence is consistent and compellingThe research evidence is consistent and compellingshowing the following weaknesses:showing the following weaknesses: US students have limited basic understanding US students have limited basic understanding

of mathematical conceptsof mathematical concepts They are notably deficient in their ability to They are notably deficient in their ability to

solve even simple problemssolve even simple problems And, overall, are not given educational opportunity And, overall, are not given educational opportunity

they need to achieve at high levelsthey need to achieve at high levels In short, the authors tell us that US teachers In short, the authors tell us that US teachers

focus primarily on one area, computation.focus primarily on one area, computation.

Mathematical Proficiency

Strategic CompetenceStrategic Competence

Conceptual UnderstandingConceptual Understanding

Adaptive ReasoningAdaptive Reasoning

Procedural FluencyProcedural Fluency

Productive DispositionProductive Disposition

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

Conceptual Understanding

It is comprehension of concepts, It is comprehension of concepts, operations and relationshipsoperations and relationships

It helps students avoid critical It helps students avoid critical errors in problem solvingerrors in problem solving

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

“When knowledge is learned withunderstanding it provides a basis

forgenerating new knowledge.”

What do these say about the student’s Conceptual

Understanding?

1/3 + 2/5 = 3/81/3 + 2/5 = 3/8

9.83 x 7.65 = 7,519.959.83 x 7.65 = 7,519.95

1616 - 8- 8 1212

Discussion Questions What is Conceptual Understanding?What is Conceptual Understanding?

How do we teach forHow do we teach forConceptual Understanding?Conceptual Understanding?

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

Procedural Fluency

Skill in carrying out mathematical steps and Skill in carrying out mathematical steps and computationscomputations

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

Using procedures can help to strengthen and Using procedures can help to strengthen and develop understandingdevelop understanding

Does Practice Make Perfect?

Understanding concepts helps recallUnderstanding concepts helps recallprocedures correctlyprocedures correctly

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

How Do You Know They Got It?

What are some successful strategiesWhat are some successful strategiesyou use to develop procedural fluency?you use to develop procedural fluency?

How are procedural fluency How are procedural fluency and conceptual and conceptual understanding related?understanding related?

How would you solve this problem?

A cycle shop has a A cycle shop has a total of 36 bicycles total of 36 bicycles and tricycles in and tricycles in stock. Collectively stock. Collectively there are 80 wheels. there are 80 wheels. How many bicycles How many bicycles and how many and how many tricycles are there?*tricycles are there?*

**Adding It Up, Adding It Up, National Research Council, 2001, p.126 National Research Council, 2001, p.126

What is the problem?What is the problem?

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

Describe more than one Describe more than one way to solve this problem?way to solve this problem?

Questions to Consider

Strategic CompetenceThe ability to formulate, represent and solve mathematical problems.

Formulate problemsFormulate problems

Multiple strategiesMultiple strategies

FlexibilityFlexibility

Nonroutine problems vs. routine problemsNonroutine problems vs. routine problems

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

Adaptive Reasoning “…the glue that holds everything

together.”

Adaptive Reasoning is the capacity for:Adaptive Reasoning is the capacity for:

Logical thoughtLogical thought

ReflectionReflection

ExplanationExplanation

JustificationJustification

Conditions Needed Real-world, motivating tasksReal-world, motivating tasks

Utilizes the knowledge-base and Utilizes the knowledge-base and

experience that children bring to school experience that children bring to school

Rigorous questioningRigorous questioning

Students justify their Students justify their

work on a regular basis work on a regular basis

Questions How do you promote adaptive reasoning How do you promote adaptive reasoning in your in your classroom?classroom?

What is the evidence that What is the evidence that your students are regularly your students are regularly using adaptive reasoning? using adaptive reasoning?

What are the long-term benefits of What are the long-term benefits of students utilizing adaptive reasoning?students utilizing adaptive reasoning?

Productive Disposition

Mathematics makes senseMathematics makes sense

Mathematics is useful and worthwhileMathematics is useful and worthwhile

Steady effortSteady effort

Effective learners and doersEffective learners and doers

Key Points

Emotional development Emotional development Self-efficacy and self-imageSelf-efficacy and self-image

Stereotype threatStereotype threat Peer pressure to under-achievePeer pressure to under-achieve

“ “Wise educational environments”Wise educational environments” Affective filter - math as a “second” languageAffective filter - math as a “second” language

Application

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

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

Mathematical Proficiency

Ability to solve Ability to solve mathematical problemsmathematical problems

Comprehension of mathematical conceptsComprehension of mathematical concepts

Capacity for Capacity for logicallogical thought, reflection, thought, reflection, explanation andexplanation and justificationjustification

Knowledge of algorithmsKnowledge of algorithms

Views mathematics as Views mathematics as sensible, useful, & sensible, useful, & worthwhile, coupled worthwhile, coupled with a belief of abilitywith a belief of ability

Conceptual UnderstandingConceptual Understanding

Procedural FluencyProcedural Fluency

Strategic CompetenceStrategic Competence

Productive DispositionProductive DispositionAdaptive ReasoningAdaptive Reasoning

Bringing It All Together

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

How will you incorporate mathematical How will you incorporate mathematical proficiency into daily proficiency into daily

teaching teaching practice?practice?

In Conclusion The goal of instruction should be The goal of instruction should be

mathematical proficiencymathematical proficiency It takes time for mathematical proficiency It takes time for mathematical proficiency

to be fully developedto be fully developed Mathematical proficiency spans number sense, Mathematical proficiency spans number sense,

algebra & functions, measurement & algebra & functions, measurement & geometry, SDAP, and geometry, SDAP, and

mathematical reasoningmathematical reasoning

““All young Americans must All young Americans must learn to think learn to think

mathematically and must think mathematically and must think mathematically to learn.”mathematically to learn.”