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Math Tools for Unpacking Addressing the West

Virginia

Next Generation Math Standards

Elementary School Version

West Virginia RESAs 3 and 7Charleston and

Morgantown, WV

April, 2013

1

Essential Workshop Questions

- What is the relationship between the Common Core

Standards an the Next Generation Math Standards,

and why were they developed? - How are the Next Generation Math Standards

organized? - What are the Six Instructional Shifts and the

Eight Mathematical Practices What are their role

in the Next Generation Standards? - What processes are useful for unpacking the

standards? - What are the implications of the Standards on the

way we approach the teaching and learning of

mathematics?

2

VKR Math Vocabulary Activity

- Assess your Vocabulary Knowledge Rating (VKR) of

personal knowledge of these important workshop

words. - Consider each word and check the appropriate

column. Check 4 column, if you could explain and

teach others. Check 3 column if you know the

term well, but would not want to teach others.

Check 2 column if you have heard of the term.

Check 1 column if the word is new to you.

CCSS WV Institute 4 3 2 1

VKR

Standards for Mathematical Practices

Content Standards

etc.

VKR Math Vocabulary Activity

1 2 3 4

Common Core StandardsNext Generation

StandardsStandardClusterObjectiveTeaching

StrategyStudent Engagement ActivityFive Stages

of TL MathSix Instructional Shifts Eight

Mathematical Practices

Whats the connection between the Common Core

Standards and the Next Generation Standards, and

why were these standards developed?

5

What are the Common Core Standards?

The Common Core Standards are a product of a U.S.

education initiative that seeks to bring diverse

state curricula into alignment with each other by

following the principles of standards-based

education reform. The initiative is sponsored by

the National Governors Association (NGA) and the

Council of Chief State School Officers (CCSSO).

At this time, 45 U.S. States are committed to

implementing the Common Core Standards.

What are the Next Generation Standards?

The Next Generation Standards are West Virginias

education standards. These standards parallel

the Common Core Standards, and contain

modifications that meet the specific needs of

West Virginia. The Next Generation Standards

represent the next logical step in the

progression of the statewide movement called

EducateWV Enhancing Learning. For Now. For the

Future.

Why were the new Standards developed?

- The Next Generation Standards were developed

to provide a consistent, clear understanding

of what students are expected to learn, so

teachers and parents know what they need to

do to help them. be robust and relevant to

the real world reflect the knowledge and

skills that our young people need for success

in college and careers

Why were the new Standards developed?

- The Next Generation Standards were developed

to be sure that American students are fully

prepared for success in the global economy - help teachers zero in on the most important

knowledge and skills - establish shared goals among students,

parents, and teachers

Why were the new Standards developed?

- The Next Generation Standards were developed

to help states and districts assess the

effectiveness of schools and classrooms and

give all students an equal opportunity for

high achievement - help solve the problem of discrepancies

between States test results and

International test results - replace the discrepant array of curriculums

that existed across the country

How are the Next Generation Standards organized?

11

Common Core and Next Generation Organization

Terminology

Common Core Standards (CCS)

Next Generation Standards (NGS)

Domain (CCS only)

Standards (CCS and NGS)

Cluster (CCS and NGS)

Objective (NGS only)

Common Core Organization/Terminology

In the Common Core Standards, the terms domain,

standard, and cluster have the following

meanings. domain used for the broad math

strand or category name standard

more specific math category name (next

level beyond domain) cluster group of

specific learning objectives that

connect with the standard

Common Core Standards for MathExample of how

they are organized

- Grade 5 Standard Operations and Algebraic

Thinking - Write and interpret numerical

expressionsM.5.OA.1 Use parentheses, brackets

or braces in numerical expressions, and evaluate

expressions with these symbols. - M.5.OA.2 Write simple expressions that record

calculations with numbers.

Domain

Standard

Cluster

Next Generation Organization/Terminology

In the Next Generation Standards, the terms

standard, cluster, and objective have the

following meanings. standard used for the

broad math strand or category name

(replaces the CC word domain) cluster more

specific math category name (next

level beyond standard, replaces the CC

word standard) objectives specific things

that students should learn and be able

to do (listed in each cluster)

How are the Next GenerationMath Standards

organized?

- Grade 5 Standard Operations and Algebraic

Thinking - Write and interpret numerical

expressionsM.5.OA.1 Use parentheses, brackets

or braces in numerical expressions, and evaluate

expressions with these symbols. - M.5.OA.2 Write simple expressions that record

calculations with numbers.

Standard

Cluster

Objectives

The Next Generation Math Standards for Grades K-5

- The next five slides show the standards (broad

math categories or strands) for grades K-5. Note

the similarities and differences among the grade

levels.

The Next Generation Math Standards for Grades K-5

- Kindergarten Standards
- Counting and CardinalityQuestions and Algebraic

ThinkingNumbers and Operations in Base

TenMeasurement and DataGeometry

The Next Generation Math Standards for Grades K-5

- First Grade Standards
- Operations and Algebraic ThinkingNumbers and

Operations in Base TenMeasurement and

DataGeometry

The Next Generation Math Standards for Grades K-5

- Second Grade Standards
- Operations and Algebraic ThinkingNumbers and

Operations in Base TenMeasurement and

DataGeometry

The Next Generation Math Standards for Grades K-5

- Third Grade Standards
- Operations and Algebraic ThinkingNumbers and

Operations in Base TenNumbers and Operations

with FractionsMeasurement and DataGeometry

The Next Generation Math Standards for Grades K-5

- Fourth Grade Standards
- Operations and Algebraic ThinkingNumbers and

Operations in Base TenNumbers and Operations

with FractionsMeasurement and DataGeometry

The Next Generation Math Standards for Grades K-5

- Fifth Grade Standards
- Operations and Algebraic ThinkingNumbers and

Operations in Base TenNumbers and Operations

with FractionsMeasurement and DataGeometry

The Next Generation Math Standards for Grades K-5

- The next five slides show the breakdown of the

common Operations and Algebraic Thinking

(Questions and Algebraic Thinking for

Kindergarten) standard for grades K-5. Each

slide shows the clusters for the standard, and

the number of objectives associated with each

cluster.

The Next Generation Math Standards for Grades K-5

- Take note of the standard, cluster, and number of

objectives for each cluster. Work with a partner

from your grade level, and see if you can guess

what the objectives are for your grade-level

clusters.

How are the Next Generation Math Standards

organized?

- Kindergarten Standard and Cluster
- Questions and Algebraic Thinking Understand

addition as putting together and adding to, and

understand subtraction as taking apart and taking

from (5 objectives)

How are the Next Generation Math Standards

organized?

- First Grade Standard and Cluster
- Operations and Algebraic ThinkingRepresent and

Solve Problems Involving Addition and

Subtraction- (2 objectives) Understand and

Apply Properties of Operations and the

Relationship between Addition and

Subtraction- (2 objectives) Add and Subtract

within 20- (2 objectives) Work with Addition

and Subtraction Equations- (2 objectives)

How are the Next Generation Math Standards

organized?

- Second Grade Standard and Cluster
- Operations and Algebraic Thinking Represent and

Solve Problems Involving Addition and

Subtraction- (1 objective) Add and Subtract

within 20- (1 objective) Work with Equal

Groups of Objects to Gain Foundations for

Multiplication- (2 objectives)

How are the Next Generation Math Standards

organized?

- Third Grade Standard and Cluster
- Operations and Algebraic Thinking Represent and

Solve Problems Involving Multiplication and

Division- (4 objectives) Understand

Properties of Multiplication and the Relationship

between Multiplication and Division- (2

objectives) Multiply and Divide within 100- (1

objective) Solve Problems Involving the Four

Operations and Identify and Explain Patterns

in Arithmetic- (2 objectives)

How are the Next Generation Math Standards

organized?

- Fourth Grade Standard and Cluster
- Operations and Algebraic Thinking Use the Four

Operations with Whole Numbers to Solve Problems-

(3 objectives) Gain Familiarity with Factors

and Multiples- (1 objective) Generate and

Analyze- (1 objective)

How are the Next Generation Math Standards

organized?

- Fifth Grade Standard and Cluster
- Operations and Algebraic Thinking Write and

Interpret Numerical Expressions- (2

objectives) Analyze Patterns and Relationships-

(1 objective)

The Next Generation Math Standards for Grades K-5

- After guessing what the objectives are for each

cluster, work in grade-level teams and read the

objectives for each cluster identified in this

activity. For each objective, work together to

create a math problem that captures the essence

of the objective. The standard, clusters,

objectives and sample problems will be share with

the entire group to provide a K-5 vertical view

of the teaching and learning progressions

associated with the K-5 math program.

Six Instructional Shifts Associated with West

Virginias Next Generation Math Standards

33

Six Instructional Shifts in Math

Focus

Coherence

Fluency

Understanding

Applications

Dual Intensity

New Points of Emphasis for Teaching the Next

Generation Standards

Instructional Shifts

- Instructional Shifts within the common core are

needed for students to attain the standards.

Kelly L. Watts, RESA 3

6 Shifts in Mathematics

- Focus
- Coherence
- Fluency
- Deep Understanding
- Applications
- Dual Intensity

Kelly L. Watts, RESA 3

Focus

- In reference to the TIMMS study, there is power

of the eraser and a gift of time. The Core is

asking us to prioritize student and teacher time,

to excise out much of what is currently being

taught so that we can put an end to the mile

wide, inch deep phenomenon that is American Math

education and create opportunities for students

to dive deeply into the central and critical math

concepts. We are asking teachers to focus their

time and energy so that the students are able to

do the same.

Kelly L. Watts, RESA 3

Focus

- Teachers
- Make conscious decisions about what to excise

from the curriculum and what to focus on - Pay more attention to high leverage content and

invest the appropriate time for all students to

learn before moving onto the next topic - Think about how the concepts connect to one

another - Build knowledge, fluency, and understanding of

why and how we do certain math concepts.

- Students
- Spend more time thinking and working on fewer

concepts - Being able to understand concepts as well as

processes. (algorithms)

Kelly L. Watts, RESA 3

Coherence

- We need to ask ourselves
- How does the work Im doing affect work at the

next grade level? - Coherence is about the scope and sequence of

those priority standards across grade bands. - How does multiplication get addressed across

grades 3-5? - How do linear equations get handled between 8 and

9? - What must students know when they arrive, what

will they know when they leave a certain grade

level?

Kelly L. Watts, RESA 3

Coherence

- Students
- Build on knowledge from year to year, in a

coherent learning progression

- Teachers
- Connect the threads of math focus areas across

grade levels - Think deeply about what youre focusing on and

the ways in which those focus areas connect to

the way it was taught the year before and the

years after

Kelly L. Watts, RESA 3

Fluency

- Fluency is the quick mathematical content what

you should quickly know. It should be recalled

very quickly. It allows students to get to

application much faster and get to deeper

understanding. We need to create contests in our

schools around these fluencies. This can be a

fun project. Deeper understanding is a result of

fluency. Students are able to articulate their

mathematical reasoning, they are able to access

their answers through a couple of different

vantage points its not just getting the answer

but knowing why. Students and teachers need to

have a very deep understanding of the priority

math concepts in order to manipulate them,

articulate them, and come at them from different

directions.

Kelly L. Watts, RESA 3

Fluency

- Students
- Spend time practicing, with intensity, skills (in

high volume)

- Teacher
- Push students to know basic skills at a greater

level of fluency - Focus on the listed fluencies by grade level
- Create high quality worksheets, problem sets, in

high volume

Kelly L. Watts, RESA 3

Deep Understanding

- The Common Core is built on the assumption that

only through deep conceptual understanding can

students build their math skills over time and

arrive at college and career readiness by the

time they leave high school. The assumption here

is that students who have deep conceptual

understanding can - Find answers through a number of different

routes - Articulate their mathematical reasoning
- Be fluent in the necessary baseline functions in

math, so that they are able to spend their

thinking and processing time unpacking

mathematical facts and make meaning out of them.

- Rely on their teachers deep conceptual

understanding and intimacy with the math concepts

Kelly L. Watts, RESA 3

Deep Understanding

- Students
- Show, through numerous ways, mastery of material

at a deep level - Use mathematical practices to demonstrate

understanding of different material and concepts

- Teacher
- Ask yourself what mastery/proficiency really

looks like and means - Plan for progressions of levels of understanding
- Spend the time to gain the depth of the

understanding - Become flexible and comfortable in own depth of

content knowledge

Kelly L. Watts, RESA 3

Applications

- The Common Core demands that all students engage

in real world application of math concepts.

Through applications, teachers teach and measure

students ability to determine which math is

appropriate and how their reasoning should be

used to solve complex problems. In college and

career, students will need to solve math problems

on a regular basis without being prompted to do

so.

Kelly L. Watts, RESA 3

Applications

- Students
- Apply math in other content areas and situations,

as relevant - Choose the right math concept to solve a problem

when not necessarily prompted to do so

- Teachers
- Apply math in areas where its not directly

required (i.e. science) - Provide students with real world experiences and

opportunities to apply what they have learned

Kelly L. Watts, RESA 3

Dual Intensity

- This is an end to the false dichotomy of the

math wars. It is really about dual intensity

the need to be able to practice and do the

application. Both things are critical.

Kelly L. Watts, RESA 3

Dual Intensity

- Students
- Practice math skills with a intensity that

results in fluency - Practice math concepts with an intensity that

forces application in novel situations

- Teacher
- Find the dual intensity between understanding and

practice within different periods or different

units - Be ambitious in demands for fluency and

practices, as well as the range of application

Kelly L. Watts, RESA 3

The Next Generation Math Standards for Grades K-5

- The next six slides show the six instructional

shifts and short instructional scenarios that

each connect with one of the shifts. Read each

scenario and determine the instructional shift

that it represents.

Six Instructional Shifts in Math

Fluency

Mrs. Johnson, a fifth-grade teacher, delivered

two informational lessons on the concept of

parentheses, brackets, braces, and numeric

expressions. After two days of paper/pencil

practice, she decided to teach her students the

550 Game (demonstrated in the Corwin/Silver

Strong workshop) and to let them compete in

pairs. Her goal was to help her 5th graders to

sharpen their proficiency with numeric

expressions and math symbols, and to mentally

process numbers faster.

Six Instructional Shifts in Math

Focus

In planning a unit on Place Value, Mrs. Smith

used the Five Stages planning tool (demonstrated

in the Corwin/Silver Strong workshop) to ensure

that she would design lessons and student

engagement activities that would help her

students to develop a strong knowledge base,

understanding of concepts, proficiency of skills,

and the ability to solve a variety of related

problems.

Six Instructional Shifts in Math

Dual Intensity

Principal Joe visited several math classes and

noticed that the lessons all emphasized

procedures, skills, and practice. Joe met with

the teachers and complimented them on their

thorough approach to skill development. Joe also

encouraged them to work together and to devise a

plan to show students how those math skills are

used in the real world. The goal would be to

continue to strengthen students skills, and to

teach students how to use those skills in problem

solving.

Six Instructional Shifts in Math

Coherence

Several math teachers and administrators from

participated in a joint exercise where they

investigated several Next Generation math

objectives from grades levels K-5. The

participants developed sample math problems that

aligned with the K-5 objectives and shared their

work with each other, so they could all

understand how the curriculum pieces fit

together.

Six Instructional Shifts in Math

Understanding

Prior to learning the rules associated with

operations on fractions and mixed numbers,

students participated in the Fraction Paper

Cutting Activity (demonstrated in the

Corwin/Silver Strong workshop). The

student-centered activity allowed students to cut

paper, form fraction pieces, and use their paper

pieces to model and investigate a variety of

fraction problems.

Six Instructional Shifts in Math

Applications

Mr. Williams noticed that his fourth-grade

science curriculum presented a number of

opportunities to integrate math into several

science lessons, and vice versa. Mr. Williams

decided to create a simple correlation of science

concepts with math concepts that featured common

math concepts and skills, so they can be taught

together.

The Six Instructional Shifts

- Can you remember the Six Instructional Shifts?

The Great Coverup Strategy, shown on the next

slide, will challenge you to see how many shifts

you can recall and recite.

Six Instructional Shifts

Focus

Coherence

Fluency

Understanding

Application

Dual Intensity

57

Standards for the Eight Mathematical Practices

58

Making a case . . .

Work individually and investigate the result of

adding two even whole numbers. Is the sum always,

sometimes, or never even? Create a sensible rule

for adding two even whole numbers and the

expected result. Explain why your rule

works. Continue to work individually and

investigate the result of adding two odd whole

numbers. Is the sum always, sometimes, or never

odd? Create a sensible rule for adding two odd

whole numbers and the expected result. Explain

why your rule works.

Share your findings, rules, and explanations with

a learning partner. Will your rules always work?

Be sure to critique your partners argument.

Making a case . . .

In the preceding activity, participants had

opportunities to think about math, investigate

math, draw conclusions, communicate their

findings to other participants, and critique each

others thinking. This kind of math engagement

satisfies one of the 8 Mathematical Practices

shown below.

Mathematical Practice 3 Construct viable

arguments and critique the reasoning of others

The Eight Mathematical Practice are shown on the

next slide.

The 8 Mathematical Practices

Building insights about meaning, and learning how

to communicate those insights

- 1. Make sense of problems and persevere in

solving them. - 2. Reason abstractly and quantitatively.
- 3. Construct viable arguments and critique the

reasoning of others - 4. Model with mathematics.
- 5. Use appropriate tool strategically.
- 6. Attend to precision.
- 7. Look for and make use of structure.
- 8. Look for and express regularity in repeated

reasoning.

Eight Mathematical Practices Applied to a Real

Standard

Review the list of Eight Mathematical Practices.

How can they be applied to the standard and

objectives below?

- Grade 5 Standard Operations and Algebraic

Thinking - Cluster Write and interpret numerical

expressionsM.5.OA.1 Use parentheses, brackets or

braces in numerical expressions, and evaluate

expressions with these symbols. - M.5.OA.2 Write simple expressions that record

calculations with numbers.

Unpacking the Standards

63

Unpacking the Standards

- Many organization templates and tools exist and

can be used to unpack math standards. One such

tool is the Five Stages Unpacking Tool for Math

Standards. This tool is aligned with the Five

Stages of Teaching and Learning Mathematics. The

next three slides provide an explanation of the

Five Stages of Teaching and Learning Math.

Try this . . .

1. Write the numerical expression for the sum of

the interior angles of a polygon with n sides.

(n 2)180

2. Explain why this formula works.

3. Use the formula to calculate the sum of the

interior angles of an octagon.

(8 2)180 6(180) 600 480 1080 degrees

4. Knowing that 3 interior angles of home plate

are right angles, find the measures of the other

two.

Try this . . .

4. Knowing that 3 interior angles of home plate

are right angles, find the measures of the other

two.

(n 2)180

(5 2)180

(3)180

540

540 270 270

270 2 135o

Try this . . .

1. Write the numerical expression for the sum of

the interior angles of a polygon with n sides.

(n 2)180

Knowledge

2. Explain why this formula works.

Understanding

3. Use the formula to calculate the sum of the

interior angles of an octagon.

Proficiency of Skills

(8 2)180 1080 degrees

4. Knowing that 3 interior angles of home plate

are right angles, find the measures of the other

two.

Applications

Each angle 135 degrees

Retention

5. Now that you know how to solve this kind of

problem, what will help you to remember how to

solve the

problem for future applications?

The Five Stages of Teaching and Learning

Mathematics

- Success or failure associated with solving an

arbitrary math problem comes down to five

questions. 1. Did the student know the math

vocabulary, terms, formulas, and number facts

associated with the problem?2. Did the student

understand the math concepts, hidden questions,

and math connections in the problem?3. Was the

student fluent with respect to the math

procedures and skills needed to solve the

problem?4. Was the student able to apply the

knowledge, understanding, and skills in relation

to the real-world context of the problem?5. Was

the student able to retain or remember important

math facts, skills, and concepts needed to solve

the problem?

The Five Stages of Teaching and Learning

Mathematics

- The Five Stages of Teaching and Learning

Mathematics is a helpful framework for planning,

teaching, and assessing a math lesson or unit. - The Five Stages of Teaching and Learning

Mathematics can also serve as a model for

unpacking a math standard.

The Five Stages of Teaching and Learning Math

Knowledge

Understanding

Proficiency of Skills

Applications

Retention

Great Considerations for Unpacking a Math Standard

The Five Stages of Teaching and Learning

Mathematics

- The next three slides provide an example of how

the Five Stages of Teaching and Learning Math can

be used to unpack a math objective. A sample

objective is shown below. - Grade 4 M.4.NF4 Apply and extend previous

understandings of multiplication to multiply a

fraction by a whole number.

Unpacking Grade4 M.4.NF4

- Grade 4 M.4.NF4 Apply and extend previous

understandings of multiplication to multiply a

fraction by a whole number.

Knowledge

Teaching Strategies

product- answer to multiplication

problem The fractional equivalent to a whole

number n is n/1. 1 times any number is the

number itself. 0 times any number is zero n x

a/b na/b how to simplify an improper fraction

Mental Math Strings that feature these

facts The Great Cover Up Convergence

Mastery Proceduralizing

Understanding

Teaching Strategies

For any fraction a/b, a is the number of

times that 1/b occurs If n gt1, then n x a/b is

greater than a/b. The concept of n x a/b

expresses the idea of bringing the amount a/b to

the table n times. improper fraction and proper

fraction equivalencies

The hands-on/multiplication component of the

Fraction Paper Cutting Activity

Unpacking Grade4 M.4.NF4

- Grade 4 M.4.NF4 Apply and extend previous

understandings of multiplication to multiply a

fraction by a whole number.

Proficiency of Skills

Teaching Strategies

Multiply any whole number n times any of

the common fractions a/b where b 1, 2, 3, 4, 5,

6, 8, 10, and 12. Simplify problems of the

type n x a/b n x a/b m and n x a/b

c/b

Mental Math Strings that feature these

facts The Great Cover Up Algebra War Games

(modified) Timed Challenges (for fractions)

Convergence Mastery

Applications

Teaching Strategies

Work with Tangram pieces Solve problems

involving fractional pieces of Hersheys

chocolate bars Solve two-step word problems

Solve problems involving fractional parts of time

and money

Task Rotation applied to problem solving

Graduated Difficulty Modeling and

Experimentation

Unpacking Grade4 M.4.NF4

- Grade 4 M.4.NF4 Apply and extend previous

understandings of multiplication to multiply a

fraction by a whole number.

Retention

Teaching Strategies

General Math Facts Measurement

Equivalencies Properties of Fractions

Patterns

Review math facts using Timed Challenges

Incorporate measurement equivalencies in fraction

problems Create patterns based on whole numbers

x fractions

8 Math Practices that apply

1. Make sense of problems and persevere in

solving them. (All problems and experiences)2.

Reason abstractly and quantitatively. (Fraction

Paper Cutting Activity)3. Construct viable

arguments and critique the reasoning of others

(Is nxa/b always gt a/b?)4. Model with

mathematics. (Fraction Paper Cutting Activity,

Tangrams, Candy bars)5. Use appropriate tool

strategically.6. Attend to precision. (Computing

exact answers, not estimates)7. Look for and make

use of structure.8. Look for and express

regularity in repeated reasoning. (n x a/b always

equals na/b.)

The Five Stages of Teaching and Learning

Mathematics

- The next two slides provide a sample objective

for grades K-5. Work with a grade level partner.

Unpack the objective using the Five Stages

Unpacking Tool. Make connections between the

Eight Mathematical Practices and the things that

students will learn and experience as they learn

the math associated with the objective.

Unpacking the Common Core Math Standards

The Five Stages of Teaching and Learning

Mathematics

Grade K Solve addition and subtraction word

problems, by adding and subtracting within 10, by

using objects or drawings Grade 1 Apply

properties of operations as strategies to add and

subtract within 20 Grade 2 Use addition and

subtraction within 100 to solve one and two-step

word problems

Knowledge

Understanding

Proficiency of Skills

Applications

Retention

Work with a partner, choose a standard, and

unpack the standard using the Five Stages tool.

Unpacking the Next Generation Math Standards

The Five Stages of Teaching and Learning

Mathematics

Grade 3 Fluently multiply and divide within 100,

using strategies such as the relationship between

multiplication and division Grade 4 Solve

multi-step word problems, posed with whole

numbers, using the four operations Grade 5 Use

parentheses, brackets, or braces in numerical

expressions, and evaluate expressions with these

symbols

Knowledge

Understanding

Proficiency of Skills

Applications

Retention

Work with a partner, choose a standard, and

unpack the standard using the Five Stages tool.

Instructional Considerations

The 3- 4- 5- Math Instructional Model

78

The 3- 4- 5- Math Instructional Model

RVD

3

Repetition, Variation of Context, Depth of Study

4

The Four Learning Styles and Task Rotation

5

The Five Stages of Teaching and Learning Math

Teaching math associated with the Next Generation

Standards Mathematics

- The next slides provide important information

about The RVD Instructional Model, The Four

Learning Styles of students, and The Five

Stages of Teaching and Learning Math - Each of these have important roles in the

teaching and learning of mathematics.

R - V - D

RVD provides teachers with three important ideas

that can be applied to the teaching and learning

process. Repetition reminds us that practice is

an essential tool for developing fluency and

proficiency with math skills and procedures.

Variation reminds us that students need to

experience math in more than one context.

Different instructional and application contexts

give students opportunities to make important

connections and deepen their understanding of

math. Depth reminds us that students need to

learn and experience all aspects of a math

concept and not superficially engage in exercises

that only scratch the surface.

Introduction to the Four Learning Styles

Interpersonal Learner

Mastery Learner

Understanding Learner

Self-Expressive Learner

Introduction to the Four Learning Styles

Mastery Learners

Want to learn practical information and

procedures

Like math problems that are algorithmic

Approach problem solving in a step by step

manner

Experience difficulty when math becomes abstract

Are not comfortable with non-routine problems

Want a math teacher who models new skills,

allows time for practice, and builds in feedback

and coaching sessions

Introduction to the Four Learning Styles

Interpersonal Learners

Want to learn math through dialogue and

collaboration

Like math problems that focus on real world

applications

Approach problem solving as an open discussion

among a community of problem solvers

Experience difficulty when instruction focuses

on independent seatwork

Want a math teacher who pays attention to their

successes and struggles in math

Want a math teacher who pays attention to their

successes and struggles in math

Want a math teacher who pays attention to their

successes and struggles in math

Introduction to the Four Learning Styles

Understanding Learners

Want to understand why the math they learn works

Like math problems that ask them to explain or

prove

Approach problem solving by looking for

patterns and identifying hidden questions

Experience difficulty when there is a focus on

the social environment of the classroom

Want a math teacher who challenges them to

think and who lets them explain their thinking

Want a math teacher who pays attention to their

successes and struggles in math

Want a math teacher who pays attention to their

successes and struggles in math

Introduction to the Four Learning Styles

Self-Expressive Learners

Want to use their imagination to explore math

Like math problems that are non-routine

Approach problem solving by visualizing the

problem, generating possible solutions and

explaining alternatives

Experience difficulty when instruction focuses

on drill and practice and rote problem solving

Want a math teacher who invites imagination and

creative problem solving into the math classroom

Want a math teacher who pays attention to their

successes and struggles in math

Want a math teacher who pays attention to their

successes and struggles in math

The Four Learning Styles

Research shows that student learn in different

ways. The Four Learning Styles provide the basis

for a teaching and learning framework that

addresses the different ways students learn. By

providing rich learning experiences that reflect

the different learning styles, teachers can lead

more students to success in math. The Task

Rotation Teaching Strategy provides four tasks,

one for each type of learner. Students who study

math through the contexts of different learning

styles will increase their levels of success in

math.

The Five Stages of Teaching and Learning Math

Knowledge

Understanding

Proficiency of Skills

Applications

Retention

Great Considerations for Planning, Teaching, and

Assessing a Math Lesson

The Five Stages of Teaching and Learning

Mathematics

- Success or failure associated with solving a math

problem comes down to five questions. 1. Did

the student know the math terms, formulas, and

number facts associated with the problem?2. Did

the student understand the math concepts, hidden

questions, and math connections in the

problem?3. Was the student fluent with respect

to the math procedures and skills needed to solve

the problem?4. Was the student able to apply

the knowledge, understanding, and skills in the

context of the problem?5. Was the student able

to retain or remember important math facts,

skills, and concepts needed to solve the problem.

Cooperative Planning Activity

- Work together and talk about how you will use the

information and strategies, featured in this

workshop, to improve math instruction and

achievement in your classroom(s).

Workshop Reflections

Specific facts and ideas that I learned today

Things I learned that will really help me in my

classroom

Why the things I learned will help my students to

learn math

Creative modifications and extentions to the

things I learned today