Transcript STRATEGY MEETING

Math Tools for Unpacking
& Addressing the West Virginia
Next Generation Math Standards
Elementary School Version
West Virginia RESAs 3 and 7
Charleston and Morgantown, WV
April, 2013
1
Essential Workshop Questions
1. What is the relationship between the Common Core Standards
an the Next Generation Math Standards, and why were they
developed?
2. How are the Next Generation Math Standards organized?
3. What are the Six Instructional Shifts and the Eight
Mathematical Practices; What are their role in the Next
Generation Standards?
4. What processes are useful for unpacking the standards?
5. 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
VKR
Standards for Mathematical Practices
Content Standards
etc.
4
3
2
1
VKR Math Vocabulary Activity
1
Common Core Standards
Next Generation Standards
Standard
Cluster
Objective
Teaching Strategy
Student Engagement Activity
Five Stages of T&L Math
Six Instructional Shifts
Eight Mathematical Practices
2
3
4
What’s 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 Virginia’s
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
State’s 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 Math;
Example of how they are organized
Grade 5 Standard: Operations and Algebraic Thinking
Domain
Cluster
Write and interpret numerical expressions Standard
M.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.
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 Generation
Math Standards organized?
Grade 5 Standard: Operations and Algebraic Thinking
Standard
Objectives
Write and interpret numerical expressions Cluster
M.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.
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 Cardinality
Questions and Algebraic Thinking
Numbers and Operations in Base Ten
Measurement and Data
Geometry
The Next Generation
Math Standards for Grades K-5
First Grade Standards
Operations and Algebraic Thinking
Numbers and Operations in Base Ten
Measurement and Data
Geometry
The Next Generation
Math Standards for Grades K-5
Second Grade Standards
Operations and Algebraic Thinking
Numbers and Operations in Base Ten
Measurement and Data
Geometry
The Next Generation
Math Standards for Grades K-5
Third Grade Standards
Operations and Algebraic Thinking
Numbers and Operations in Base Ten
Numbers and Operations with Fractions
Measurement and Data
Geometry
The Next Generation
Math Standards for Grades K-5
Fourth Grade Standards
Operations and Algebraic Thinking
Numbers and Operations in Base Ten
Numbers and Operations with Fractions
Measurement and Data
Geometry
The Next Generation
Math Standards for Grades K-5
Fifth Grade Standards
Operations and Algebraic Thinking
Numbers and Operations in Base Ten
Numbers and Operations with Fractions
Measurement and Data
Geometry
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 Thinking
•Represent 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 Virginia’s Next Generation
Math Standards
33
Six Instructional Shifts in Math
Focus
Understanding
Coherence
Applications
Fluency
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
 Students
 Teachers
 Spend more time thinking and Make conscious decisions about
working on fewer concepts
 Being able to understand
concepts as well as
processes. (algorithms)
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.
Kelly L. Watts, RESA 3
Coherence
 We need to ask ourselves –
• How does the work I’m 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
 Teachers
 Build on knowledge from year
 Connect the threads of math
to year, in a coherent learning
progression
focus areas across grade
levels
 Think deeply about what
you’re 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; it’s 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
 Teacher
 Spend time practicing, with
 Push students to know basic
intensity, skills (in high
volume)
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
 Teacher
 Show, through numerous
 Ask yourself what
ways, mastery of material at a
deep level
 Use mathematical practices to
demonstrate understanding of
different material and
concepts
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
 Teachers
 Apply math in other content
 Apply math in areas where its
areas and situations, as
relevant
 Choose the right math
concept to solve a problem
when not necessarily
prompted to do so
Kelly L. Watts, RESA 3
not directly required (i.e.
science)
 Provide students with real
world experiences and
opportunities to apply what
they have learned
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
 Teacher
 Practice math skills with a
 Find the dual intensity
intensity that results in fluency
 Practice math concepts with
an intensity that forces
application in novel situations
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
Focus
Understanding
Coherence
Applications
Fluency
Dual Intensity
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
Understanding
Coherence
Applications
Fluency
Dual Intensity
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
Focus
Understanding
Coherence
Applications
Fluency
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
Focus
Understanding
Coherence
Applications
Fluency
Dual Intensity
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
Focus
Understanding
Coherence
Applications
Fluency
Dual Intensity
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
Focus
Understanding
Coherence
Applications
Fluency
Dual Intensity
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 partner’s
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 expressions
M.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.
Knowledge
2. Explain why this formula works.
(n – 2)180
Understanding
3. Use the formula to calculate the sum ofProficiency of
the interior angles of an octagon.
Skills
(8 – 2)180 = 1080 degrees
4. Knowing that 3 interior angles of home plate
are right angles, find the measures of the other
Applications
Each angle = 135 degrees
two.
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
Applications
Proficiency of Skills
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
• 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
Understanding
• For any fraction a/b, ‘a’ is the number of times
that 1/b occurs
• If n >1, 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
Teaching Strategies
• Mental Math Strings that feature these
facts
• The Great Cover Up
• Convergence Mastery
• Proceduralizing
Teaching Strategies
• 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
• 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
Applications
• Work with Tangram pieces
• Solve problems involving fractional pieces of
Hershey’s chocolate bars
• Solve two-step word problems
• Solve problems involving fractional parts of
time and money
Teaching Strategies
• Mental Math Strings that feature these
facts
• The Great Cover Up
• Algebra War Games (modified)
• Timed Challenges (for fractions)
• Convergence Mastery
Teaching Strategies
• 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
• General Math Facts
• Measurement Equivalencies
• Properties of Fractions
• Patterns
Teaching Strategies
• 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 > 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
Knowledge
Understanding
Grade 1: Apply properties of operations as
strategies to add and subtract within 20
Proficiency of Skills
Grade 2: Use addition and subtraction
within 100 to solve one and two-step
word problems
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
Knowledge
Grade 4: Solve multi-step word problems,
posed with whole numbers, using the four
operations
Proficiency of Skills
Grade 5: Use parentheses, brackets, or
braces in numerical expressions, and
evaluate expressions with these symbols
Understanding
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
3
4
5
RVD
Repetition, Variation of Context, Depth of Study
The Four Learning Styles
and Task Rotation
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
Mastery Learner
Understanding Learner
Interpersonal 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
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
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
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
Applications
Proficiency of Skills
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
Why the things I learned will help
my students to learn math
Things I learned that will really help
me in my classroom
Creative modifications and extentions
to the things I learned today