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RtI + Math = Student Achievement for ALL
Aligning work Systemically and Systematically
Wendy Strickler, Ph.D and Holly Sampson , M.Ed
Hamilton County Educational Service Center
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Logistics
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Mathematics = ALL Students
Why do we care?
Do you know…
According to recent survey data,
What percent of the US population
are unable to calculate a 10% tip
on a lunch?
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What percent of eighth-graders
can not correctly shade 1/3 of a
rectangle?
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What percent can not solve a word
problem that required dividing
fractions?
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Here’s a little help…
27%
58%
45%
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Answers
*58%- Tip
*27%- Rectangle
*45%- Fractions in
Problem Solving
Student Exhibiting Mastery in
Algebra Readiness Components
Category
Entering
Math Tech 1
Entering
Algebra 1
Fractions and their Applications
3.6%
44.8%
Decimals their Operations and
Applications: Percent
13.1%
66.7%
Measurement of Geometrical Objects
23.8%
58.3%
Graphical Representation
15.5%
61.5%
Integers their Operations & Applications
32.1%
86.5%
Total number of students per course
84
96
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Overall Objectives



Learn and discuss components of curriculum,
instruction and assessments necessary for tiered
intervention support.
Review and further understand the systems within the
schools/district we work to further enhance the
support to our students.
Engage participants in an examination of effective
mathematic instructional strategies and begin sharing
possible resources for interventions and best practices
in mathematics instruction.
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Quick Math
Try this:
101 - 102 = 1
Make one move to create
a true statement.
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101 =102 - 1
101- 10²= 1
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There’s more than one
way to skin a cat…

Important to acknowledge how
individual students work and to
continue to build on what we know so
we can help students make gains.

Working systemically and
systematically can help us achieve
this.
Systemic / Systematic ???
Systemic / Systematic
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Turn and Talk
What systems are in place within
your district/school/team to
support mathematics?
An Integrated Systems Approach…
RtI- Response to Intervention
Academic Systems
1-5% Intensive Individualized
Interventions
5-10% Targeted Interventions
80-90% School-Wide
Interventions
Adapted from OSEP
Effective School-Wide
Interventions
Behavioral Systems
1-5% Intensive Individualized
Interventions
5-10% Targeted Interventions
80-90% School-Wide
Interventions
Decisions about tiers of
support are data-based
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Academic Systems
1-5% Intensive Individualized
Interventions
5-10% Targeted Interventions
80-90% School-Wide
Interventions
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3 Tiers of Support
 Tier 1: Core standards-based mathematics curriculum
for all students grounded in scientifically-based
research.
 Tier
2: Targeted or secondary resources and/or
processes to address the skills of students who need
further interventions supplemental to their core
mathematics instruction.
 Tier
3: Individualized support designed for students who
need intensive math skill support when provided with
primary and secondary intervention.
Adapted from: RTI- Response to Intervention pamphlet, HCESC
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Tier 1- The Core

Curriculum

Instruction

Assessment
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Tier 1-The Core

Key Characteristics




All students receive this instruction
Research-validated core curriculum
Instructional gaps filled with
supplemental research-based
materials
Differentiated support
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The Core: Math Curriculum

A Balanced Approach
Number sense
Computation fluency
Algebraic thinking
Math language
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Source: Ohio Department of Education Mathematics Content Standards, 2001
Ohio Standards
NCTM Standards
Number, Number Sense and
Operations Standard
Students demonstrate number
sense, including an understanding of
number
systems and operations and how
they relate to one another. Students
compute
fluently and make reasonable
estimates using paper and pencil,
technology supported
and mental methods.
Number and Operations Standard
• Understand numbers, ways of
representing numbers, relationships
among numbers, and number
systems;
• Understand meanings of
operations and how they relate to
one another; and
• compute fluently and make
reasonable estimates.
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Ohio Standards
NCTM Standards
Measurement Standard
Students estimate and measure to a
required degree of accuracy and
precision by selecting and using
appropriate units, tools and
technologies.
Measurement Standard
• Understand measurable attributes
of objects and the units, systems,
and processes of measurement; and
• Apply appropriate techniques,
tools, and formulas to determine
measurements.
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Ohio Standards
NCTM Standards
Geometry and Spatial Sense Standard
Students identify, classify, compare and
analyze characteristics, properties and
relationships of one-, two- and threedimensional geometric figures and
objects. Students use spatial reasoning,
properties of geometric objects, and
transformations to analyze mathematical
situations and solve problems.
Geometry Standard
• Analyze characteristics and properties
of two- and three-dimensional geometric
shapes and develop mathematical
arguments about geometric
relationships;
• Specify locations and describe spatial
relationships using coordinate
geometry and other representational
systems;
• Apply transformations and use
symmetry to analyze mathematical
situations; and
• Use visualization, spatial reasoning,
and geometric modeling to solve
problems.
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Ohio Standards
NCTM Standards
Patterns, Functions and Algebra
Standard
Students use patterns, relations and
functions to model, represent and
analyze problem situations that involve
variable quantities. Students analyze,
model and solve problems using various
representations such as tables, graphs
and equations.
Algebra Standard
• Understand patterns, relations, and
functions; represent and analyze
mathematical situations and structures
using algebraic symbols; and
• Use mathematical models to represent
and understand quantitative
relationships; analyze change in
various contexts.
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Ohio Standards
NCTM Standards
Data Analysis and Probability
Standard
Students pose questions and collect,
organize, represent, interpret and
analyze data to answer those
questions. Students develop and
evaluate inferences, predictions and
arguments that are based on data.
Data Analysis and Probability
Standard
• Formulate questions that can be
addressed with data and collect,
organize, and display relevant data to
answer them;
• Select and use appropriate statistical
methods to analyze data;
• Develop and evaluate inferences and
predictions that are based on data; and
• Understand and apply basic concepts
of probability.
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Ohio Standards
NCTM Standards
Mathematical Processes Standard
Students use mathematical processes
and knowledge to solve problems.
Students apply problem-solving and
decision-making techniques, and
communicate mathematical ideas.
Problem Solving Standard
Reasoning and Proof Standard
Communication Standard
Connections Standard
Representation Standard
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Tier 1: Math Core Curriculum
Checking the Research Base
Try:
 What Works Clearinghouse:
http://ies.ed.gov/ncee/WWC/
 Johns Hopkins Best Evidence
Encyclopedia:
www.bestevidence.org
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Quick Math
The answer is 42.
What is the question?
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Tier 1 – The Core:
Math Instruction

Explicit instruction

Pre-skills are taught to mastery

Strategies, including problem-solving
and graphic organizers
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Tier 1 – The Core:
Math Instruction

Concrete-representational-abstract
sequence

Authentic contexts

Multiple opportunities to practice with
guidance and feedback
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The Core: Instruction

Communication

Continuous progress monitoring

Cooperative learning such as peer
tutoring
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Recap of the Big “9”
Instruction at the “Core”









Explicit instruction
Pre-skills are taught to mastery
Strategies, including problem-solving and graphic
organizers
Concrete-representational-abstract sequence
Authentic contexts
Multiple opportunities to practice with guidance and
feedback
Communication
Continuous progress monitoring
Cooperative learning such as peer tutoring
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Video 1
Good Morning Miss Toliver
Table Talk
“Round and Round”
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Take a Break..
10 minutes

How do we acquire the information
necessary to make instructional
decisions for our students?
Assessment
What question are we trying to
answer?
Purposes of Assessment
SCREENING
DIAGNOSTIC
School-wide
Individual student
Broad index
Specific academic domains
Yearly / 3x / Monthly
As needed
Determine health of system, ID at-risk
ID specific student deficits
System focus
Student focus
Class / school instruction and curriculum
decisions
1st step for instructional planning
Selecting curriculum and instructional
methods
Planning or specifying interventions
Purpose of Assessment
PROGRESS MONITORING
EVALUATION
Class / small group / student
School wide
Specific academic skills or behavioral
targets
Broad index
<3 wks / weekly / daily
Yearly (or more frequently as needed)
Monitor, regroup students
Determine program needs
Student focus
System focus
Intervention effectiveness
Class / school instruction and curriculum
decisions
Continue or revise support
Core and intervention planning / revising
What Question are we trying to
answer within an RtI Framework?
Tier
Questions
Tier 1
How healthy is our overall system
(district/school/grade/class)?
Which students should be targeted
for additional
support/instruction/enrichment?
Are students progressing at
expected rates?
Are our students progressing at
expected rates? Closing
achievement/behavior gaps?
Meeting expected benchmarks?
Tier 2
Tier 3
Types of
assessment
Universal
screener
Universal
screener
Progress
Monitoring
More frequent
progress
monitoring;
Diagnostic
tests; others
Examples
Aimsweb, CBM,
OGT
Aimsweb, CBM,
YearlyProgressPro,
OGT
Aimsweb, CBM,
YearlyProgressPro,
Diagnostic
measures, OGT
*On-going assessments (e.g. formative, mastery, short-cycle, and informal
assessments) are part of all 3 tiers and used in meaningful ways
Tier 1 Assessment
Universal Screening





Connected to key academic content or
behavior
Conducted at least 3 times per year on a
regular basis, using comparable test forms
Administered school wide to all students
Used to determine if additional examination
is warranted
Features: short, few items, focus on critical
indicators
Tier 1 Assessment
2 Critical Pieces of Information

Is the Tier 1 core effective?
 For all students (aggregated)?
 For each subgroup of students
(disaggregated)?

Who are the ~20% of students needing
additional support?
Tier 1-The Core:
Math Assessment

Math Universal Screeners

4 considerations (Fuchs, 2006)
 Feasible to implement
 Strong predictor of high stakes tests
 Developmentally appropriate
 Accurate cut-score for determining
need for additional support
(note: because assessment impacts instruction, make sure
there is balance)
Tier 1-The Core:
Math Assessment
Universal Screening
Possible areas of focus:






Number Sense and Computation Fluency
Measurement
Geometry
Patterns, Functions, algebra
Data Analysis and Probability
Problem Solving
Tier 1: Math Assessment
Universal Screening

Possible resources:




Aimsweb
Yearly ProgressPro
Intervention Central (Numberfly, Math
Worksheet Generator)
Using Curriculum Based Measurement
for Progress Monitoring in Math, 2007
(Fuchs, Fuchs, and colleagues)
Tier 1: Math Assessment

Decision Rules

Deno & Mirkin Instructional Mastery criteria
 Grades 1-3: 20+ digits correct per minute
 Grades 4-6: 40+ digits correct per minute



AIMSweb norms
Support lowest 25% of class/grade
Ongoing Research In This Area

Important to collect data; if all students
struggling with skill should have mastered
based on ODE, need Tier 1 instruction
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Tier 1: Math Assessment

Curriculum-Based Assessment
(CBA)
 Creating your own tools
 Sample grade-level screening tools
CBA Example
Kindergarten Number and Operations
3 Questions For Each Indicator

=

Draw 5 dots
Which is bigger: 7 2
Fill in the blank: 2 3 __ 5
Count backward 10  1



CBA Example for High School
3 questions for each indicator

Number sense: Identify subsets of
the real number system
1) Which number is an irrational
number?
 A) -2
8
 B)
 C) 3 22/8
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Turn and Talk

Based on your grade level,
discuss a keystone skill that all
students need to master.

How might a CBM look?
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Video 2
Secondary Example
Connecting Knowledge
www.PD360.com
Support- Tier 1-The CORE
4 Key Characteristics
Curriculum/Resources
Instruction
What system is in place to support the work at Tier 1?
What initial or additional supports need to be in place to support the Core- Tier 1?
How does this look in other schools within your district (or other districts)?
Assessment
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Levels of Support Chart




Describe “The Core” within your
classroom/ school/district.
What system is in place to support the
work at this level?
What initial or additional supports need to
be in place to support the Core-Tier 1?
How does this look in other schools within
your district (or other districts)?
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Tier 1-The Core

Key Characteristics




All Students receive this instruction
Research-Validated Core Curriculum
Instructional Gaps filled with
Supplemental Research-based
Materials
Differentiated Support
Support - Tier 2- Targeted Instruction
3 Key Characteristics-
Resources
Instruction
What system is in place to support the work at Tier 2?
What initial or additional supports need to be in place to support tier 2?
How does this look in other schools within your district (or other districts)?
Assessment
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Tier 2
Targeted Supports

Key Characteristics:



Small group or individual instruction in
addition to the core
Research-based
strategies/interventions
Regular progress monitoring for
efficient changes as needed
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Tier 2 : Math Support
Focused Targets:

Number Sense

Fluency in Computation

Algebraic Skills
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Tier 2 : Math Support
Tiered Interventions in Development
 PALS for Math (Peer-Assisted Learning
Strategies) (Fuchs, Fuchs & Karns, 2001)
 3-Tier Math Intervention for Grades K-2
(Bryant & Bryant, 2007; focus on number
sense and basic skill knowledge and
application)
 Developing Algebraic Literacy (K-3) (D.
Allsopp and colleagues)
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Tier 2
Math Support
Problematic Skills

Language of Math

Word Problems

Multi-step Problems
Bryant
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Tier 2
Math Support:
Instruction




Increased support and explicitness
Increased modeling
Increased opportunities for guided
instruction
Corrective feedback (More
opportunities)
Explicit and Systematic
Instruction

Model: Provide explicit examples of new
material.

Practice: Provide ample opportunities for
students to practice new material. Ample is
defined by the individual needs of each student.

Assess (ongoing): Check students’
understanding of the new material throughout
the lesson.

Feedback: Immediately correct any incorrect
student responses by repeating the teacher
63
model.
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Explicit Instruction

Break down the skills into manageable and
deliberately sequenced steps.

Provide overt instruction in the skills and
opportunities to practice (Roshenshine & Stevens,
1986).



2008
Step by step manner
Clear and detailed explanations
Mastery of each step is assured before moving on to the
next
Futures Task Force on
Academic Outcomes
64
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Explicit Instruction

Modeled, Guided, and Independent Practice
“I do” (presentation of materials),
“we do” (guided practice),
“you do” (independent practice).

Feedback and Ongoing Assessment
Uses a high number of teacher questions and
student responses with frequent checks for
understanding.
Sample Intervention
Increased Explicit Instruction
Steps:
1: Teacher selects problems from student work
that warrant explicit instruction,
2: Teacher demonstrates how to perform the
algorithm “thinking aloud” the steps,
3: Students imitate the process on similar
problems,
4: A completed model remains as a referent on the
student’s paper as a permanent.
(Rivera & Smith, 1987; 1988; Bryant, Hartman, & Kim, 2003)
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3rd grade OAT Math Released
(Spring 2008)
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Quick Math

BUDDY UP!!!
4th grade OAT Math Released
(Spring 2008)
4th grade OAT Math Released
(Spring 2008)
OGT Example Spring 2006
Question 4
Standard: NNS Benchmark: G. Estimate, compute and solve problems involving real numbers,
including ratio, proportion and percent, and explain solutions.
Adam was going to buy a new lawn mower from Lawn Care Depot
for $169, less a 10% discount. He saw the same mower on sale at
Tractors-R-Us. Their mower originally cost $210 and was on sale
1
for 3 off.
Determine the sale price of the mower at each store. Show your work or
provide an explanation to support your answer.
Identify which store would be the most economical place to purchase the
mower. (2 points)
Points
Student Response
2
The focus of the item is to calculate discounts of
original prices. The response states that the sale
price of the lawnmower at Lawn Care Depot is
$152.10 and the sale price at Tractors-R-Us is $140.
The student identifies Tractors-R-Us as the most
economical place to purchase the mower.
1
The response provides evidence of a partially
correct answer and/or solution process. The
response shows understanding of some key
elements of the task but contains gaps or flaws.
0
The response indicates inadequate understanding
of the task.
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Lunch

List of suggested places on back
table…
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Tier 2
Math Support:
NCTM recommends 5 processes
for strugglers:





Communication
Problem solving
Reasoning/proof
Connections (graphic organizer)
Representations (draw, diagram)
Process Articles and
Jigsaw Discussion

Small group
•
Each person chooses one of the articles to
read and capture important points of interest.
Fill in “Process Standards in A NUTSHELL”
Sheet
Each person take turns sharing out
important aspects of their article. Fill in the
sheet provided.
•
•
Process Standards in a Nutshell
Communication
Problem solving
Reasoning/proof
RtI Assessments
Tier 1: Universal Screening
Tier 2:
Tier 3:
Progress Monitoring (Keystone
Skill) and additional information
-More frequent progress monitoring
-More assessments based upon
individualized needs (e.g. prerequisite skills, diagnostics)
*On-going assessments (e.g. formative, mastery, short-cycle, and informal
assessments) are part of all 3 tiers and used in meaningful ways
Tier 2:
Math Assessment

Progress Monitoring-current focus
mostly fluency




Aimsweb
Yearly ProgressPro
Intervention Central (Numberfly, Math
Worksheet Generator)
Using Curriculum Based Measurement
for Progress Monitoring in Math, 2007
(Fuchs, Fuchs, and colleagues)
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Tier 2:
Math Assessment

Decision Rules

Fuchs and Fuchs (2009) recommendations
 CBM end levels (after 10-20 weeks of
Tier 2 tutoring)
 Slope of student’s progress
 “typical” end-of-year benchmarks
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Tier 2:
Math Assessment
Curriculum-Based Assessment

If progress monitoring probes are
not available, can you create a
measure of progress monitoring?
CBA Example: 6th grade Algebra



3 questions for each indicator
Write math equation
 Six people are bringing 12 cupcakes each to a
party. How many cupcakes will be at the party?
Solve these equations
 150/3=
Write a word problem for these equations


Write an equivalent formula


12 / 4
2(L + W) = 2L + 2W
Write the formula


Input
Output
1
3
2
6
3
9
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Chart




Describe the Tier 2 interventions within
your classroom/ school.
What system is in place to support the
work at Tier 2?
What initial or additional supports need to
be in place to support Tier 2?
How does this look in other schools within
your district (or other districts)?
Support- Tier 2 – Targeted Instruction
3 Key Characteristics
Resources
Instruction
What system is in place to support the work at Tier 2?
What initial or additional supports need to be in place to support tier 2?
How does this look in other schools within your district (or other districts)?
Assessment
Tier 3: Individualized Instruction

Key Characteristics:




Small group or individualized
instruction in addition to the core
Research-based
strategies/interventions
Individualized to students needs
based on collaborative problem
solving
Frequent progress monitoring
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Tier 3: Individualized
Instruction

Curriculum


Content should align with core
curriculum to which student still
has access
Content should fill in gaps in
knowledge/skills as determined
through further diagnostics
Tier 3: Individualized
Instruction

Focused targets:




**Fractions and other concepts relating
to rational numbers
Fluency with standard algorithms
 Commutative, Associative, Distributive
laws
Translation of word problems into
symbols
Basic measurement concepts
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Tier 3: Individualized
Instruction

Instructional strategies:




Increased opportunities for direct
instruction, practice, and feedback
Re-teaching skills not yet mastered
Using visuals and verbalization
Teaching metacognitive strategies
Critical Components of Effective Instruction for Students Struggling in Math
Instructional Component
Explicit Instruction (n=11)
Multiple heuristics (n=4)
Verbalizations (n=8)
Visuals for teacher and student (n=7)
Visuals for teacher only (n=5)
Visuals combined (n=12)
Range and Sequence of examples (n=9)
Teacher feedback (n=7)
Teacher feedback with recommendations (n=3)
Teacher feedback combined (n=10)
Student feedback (n=7)
Student feedback with goals (n=5)
Student feedback combined (n=12)
Cross-age tutoring (n=2)
Within class peer-assisted learning (n=6)
Random Effects Mean
1.22
1.56
1.04
0.54
0.41
0.47
0.82
0.21
0.34
0.23
0.23
0.13
0.21
1.02
0.12
Gersten, R., Chard, D., Jayanthi, M., Baker, S., Morphy, P., & Flojo, J. (2008). Mathematics Instruction for Students with
Learning Disabilities or Difficulty Learning Mathematics: A synthesis of the intervention research. Portsmouth, NH: RMC
Research Corporation, Center on Instruction
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Tier 3: Individualized
Instruction

Assessment



Curriculum Based Assessment to
specify skills and needs
Additional diagnostics
Frequent monitoring of progress
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Tier 3
Each student participating in Tier 3
should have a collaborativelydeveloped written plan outlining:



Intervention plan and logistics
Assessment plan and logistics
Goals
Support- Tier 3 – Individualized Instruction
4 Key Characteristics
Resources
Instruction
What system is in place to support the work at Tier 2?
What initial or additional supports need to be in place to support tier 2?
How does this look in other schools within your district (or other districts)?
Assessment
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Ideas
for
Supporting
Learning
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I Forget…..
Forgetting happens very quickly…
47% of forgetting occurs in the first 20 min.
62% of forgetting occurs in the first day.
82% of forgetting occurs in the first 3 weeks. Forgetting
slows down after 2 weeks….. But then, there is not much
left to forget.
Therefore, the prime time to process, discuss and
reflect is immediately after new information is
presented ---- before the forgetting begins.
Source: Instruction for All by Paula Rutherford
What Math Students Need Across the Tiers
Focused Targets
1.
2.
3.
Langua Numb Computa
ge of
er
tion
Math Sense Fluency
Vocabul
ary and
Writing
4.
Word
Proble
ms
Strategy/ Activity/ Other
Building background /
providing relevance
Quick Math
Turn and Talk
Models
5.
Algebraic
skills
(Including
fractions and
all rational
numbers)
Process Standards
6.
7.
8.
9.
10.
11.
Multi- Geometry Reasoni Representa Communic Connectio
step
and
ng/
tions
ation
ns
proble Measurem Proof
(draw,
(graphic
ms
ent
diagram)
organizer)
Brief Description
Need(s) Addressed
(see list above)
12.
Problem
Solving
Application:
How has it been
used/could it be used
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Students With
Math Difficulties…

Struggle with Math Language

Problem Solving

Computational Fluency
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Students with math
difficulties…
Struggle with Math Language
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Strategies

Vocabulary
- Frayer Model
- Marzano’s 6 Steps

Journaling

Talk Moves
(Prime - Even - Percent )

Choose one of the mathematical
words above. (Don’t tell anyone your choice.
Don’t write it on your paper)


Fill in the outer boxes to help
explain your word.
Be ready to share your information
to your group.
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?
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Share
Why is teaching vocabulary
important?
Why We Should
Teach Vocabulary
•
•
•
•
•
Learning is fundamentally and profoundly
dependent on vocabulary knowledge.
Vocabulary knowledge is highly correlated with
overall reading/math achievement.
Vocabulary deficiencies are a primary cause of
academic failure in Grades 3–12.
Vocabulary knowledge affects a student’s ability
to participate fully in both social and academic
activities.
Significant disparities exist in word knowledge
among students.
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
Knowledge of important terms is
critical to understanding any
subject.

The more terms we know about a
subject the more skilled we are in
that subject.
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A Six-Step Process for
Teaching New Terms
Step 1: Provide a description, explanation, or example of
the new term
Step 2: Ask students to restate the description,
explanation, or example in their own words
Step 3: Ask students to construct a picture, symbol, or
graphic representing the term or phrase
Step 4: Engage students periodically in activities that will
help them add to their knowledge of the terms in their
notebooks
Step 5: Periodically ask students to discuss the terms
with one another
Step 6: Involve students periodically in games that allow
them to play with terms
Building Academic Vocabulary, Marzano and Pickering
A Six-Step Process for Teaching New Terms
Step 1: Provide a description,
explanation, or example of the new
term
A function is a relationship between two things like
Example: Function
height and weight. As one goes up, the other goes up.
Isn’t it generally true that as kids grow in height over
the years, their weight has also gone up? We could
describe this relationship by saying, “Your weight is a
function of your height?”
Term: ______________________
Describe: ___________________________
___________________________________
___________________________________
___________________________________
Draw:
Level of Understanding 1 2 3 4
A Six-Step Process for Teaching New Terms
Step 2: Ask students to restate the description,
explanation, or example in their own words
Term: Function
Describe: _
A Six-Step Process for Teaching New Terms
Step 2: Ask students to restate the description,
explanation, or example in their own words
Term: Function
Describe: It’s when one thing makes
another happen or one thing goes up the
way that another goes up.
A Six-Step Process for Teaching New Terms
Step 3: Ask students to construct a picture,
symbol, or graphic representing the term or
phrase.
Types of pictures:

Draw the actual thing 
diameter
=

Use a symbol

Draw an example
Represent the idea with graphics
Dramatize the drawing with cartoon bubbles


Picture drawing needs to be
explicitly taught.
Symbols
Peace
Space
Examples
Conservation
Algebra
Graphics
Centralization
De-Centralization
Building Academic Vocabulary
Marzano
Dramatize /Cartoon Bubbles
Dialogue
Self-Evaluate
After Step 3 is completed, ask students to selfevaluate by circling 1 2 3 4 at the bottom of
their vocabulary page for each word they’ve
learned.
4 - I understand even more about the term
than I was taught
3 - I understand the term and I’m not confused
about any part of what it means
2 - I’m a little uncertain about what the term
means, but I have a general idea.
1 - I’m very uncertain about the term. I really
don’t understand what it means.
Term: ______________________
Describe: ___________________________
___________________________________
___________________________________
___________________________________
Draw:
Level of Understanding 1 2 3 4
A Six-Step Process for Teaching New Terms
Steps 4-6 Reinforce Learning of the Terms
Step 4: Engage students periodically in activities
that will help them add to their knowledge of
the terms in their notebooks
Step 5: Periodically ask students to discuss the
terms with one another
Step 6: Involve students periodically in games
that allow them to play with terms
A Six-Step Process for Teaching New Terms
Step 4: Engage students periodically in
activities that will help them add to their
knowledge of the terms in their notebooks
Solving Analogy Problems
Free Association
Classifying Terms
Comparing Terms
Solving Analogy Problems
Inch
Foot
As
millimeter
centimeter
An inch is part of a foot and a
millimeter is part of a
centimeter.
Free Association
Oral:
 Call out a term and ask students (as a class, in small
groups, or in pairs) to say any word they think of that
is related to the term
 After a few seconds say “stop”. The last person to
say a word must explain how it is related to the
target
Written:
 Students write terms in notebook.
 When you say “stop” students exchange with
partner and explain how the words are related.
Classifying Terms
Description: Classifying is the process of
grouping items on the basis of similar attributes.
Two Types of Classification Tasks:
1.
2.
Structured – students are given the categories
and place the terms into the correct categories
Open-Ended – students are given terms and
they come up with categories OR they are
given categories and come up with terms
A Six-Step Process for Teaching New Terms
Step 5: Periodically ask students to
discuss the terms with one another
Think
Pair
Share
A Six-Step Process for Teaching New Terms
Step 6: Involve students periodically in games
that allow them to play with terms
What is the Question?
Vocabulary Charades
Name that Category
Draw Me
Talk a Mile a Minute
Journaling
Using Journals in Mathematics



Vocabulary Building
Allows students to show rationale
for work.
Captures thinking over time.
“Talk Move” Strategies
Talk Move: Revoice
-Teacher restates student’s idea then verifies its accuracy
Used to: Highlight particular idea for discussion
Talk Move: Repeat/Rephrase
-Student restates what another said
-Then move into having them rephrase and check for accuracy
Used to: Encourage listening and understanding what
others are saying
Talk Move: Adding On
-Teacher asks students to expand upon ideas already stated
Used to: Validate original ideas
Talk Move: Agree/Disagree and Why?
-Students analyze ideas and defend their position
Used to: further understand and address misconceptions
What Math Students Need Across the Tiers
Focused Targets
1.
2.
3.
Langua Numb Computa
ge of
er
tion
Math Sense Fluency
Vocabul
ary and
Writing
4.
Word
Proble
ms
Strategy/ Activity/ Other
Building background /
providing relevance
Quick Math
Turn and Talk
Models
5.
Algebraic
skills
(Including
fractions and
all rational
numbers)
Process Standards
6.
7.
8.
9.
10.
11.
Multi- Geometry Reasoni Representa Communic Connectio
step
and
ng/
tions
ation
ns
proble Measurem Proof
(draw,
(graphic
ms
ent
diagram)
organizer)
Brief Description
Need(s) Addressed
(see list above)
12.
Problem
Solving
Application:
How has it been
used/could it be used
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Students with math
difficulties…

Struggle with problem solving





Cannot remember procedural steps
Fail to verify answers and settle on first answer
Making “borrowing” errors; fail to carry
Reach “unreasonable” answers
Misplace digit in multi-digit numbers
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Strategies

Heuristics

Analyzing the Type of Problem
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Heuristics
“General in Nature”

Serves different purposes such as helping the
child to:






understand the problem;
simplify the task;
identify possible causes;
identify possible solutions;
think or reason.
They are often used in combinations to solve the
problem


Breaking apart problem into manageable
pieces
Work Backwards


Act it out





(Video)
Simplify it


(What’s the question?)
(Make harder numbers easier at first)
State it a different way
Make suppositions
Use guess and check
Use before after concept
sc-math.com/math/heuristics.php
What Can I Use?
Make a Table
Look for a Pattern
Draw a Diagram
Compare and Contrast Data
Simplify the Problem
Write a Mathematical Sentence
Make a Graph or Table
Work Backward
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Memory Cues
Five Steps for Solving Word Problems
1.
2.
3.
4.
Determine what I need to find
Decide what information I need in order
to find it.
Do the math.
Check my work to see if it agrees with
the first step.
Determining the
Type of Problem
2
Common Underlying
Structure of Word Problems
Specific Type of Problem


Change (over time) - increase/decrease
Quantity (compare)-
(Change Problem)
A
B
C
(Compare Problem )
There are 21 hamsters and 32 kittens at the pet store.
How many more kittens are at the pet store than
hamsters?
32
21
?
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Problem 1
Brad has a bottlecap collection.
After Madhavi gave Brad 28 more
bottlecaps, Brad had 111
bottlecaps. How many bottlecaps
did Brad have before Madhavi
gave him more?
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Problem 2
Brad has a bottlecap collection.
After Brad gave 28 of his
bottlecaps to Madhavi, he had 83
bottlecaps left. How many
bottlecaps did Brad have before
he gave Madhavi some?
2 Problem Solving
Structures cont…
Teach students to recognize the common
structures when there are:
1. Problems that have superficial changes but
are really the same… (½ , half, one-half)
2. Irrelevant story information or additional
information.
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Mike wants to buy 1 pencil for each of his
friends. Each packet of pencils contains 12
pencils. How many packets does Mike have
to buy to give 1 pencil to each of his 13
friends?
Mike wants to buy 1 pencil for each of his
friends. Sally wants to buy 10 pencils. Each
box of pencils contains 12 pencils. How
many boxes does Mike have to buy to give
1 pencil to each of his 13 friends?
What Math Students Need Across the Tiers
Focused Targets
1.
2.
3.
Langua Numb Computa
ge of
er
tion
Math Sense Fluency
Vocabul
ary and
Writing
4.
Word
Proble
ms
Strategy/ Activity/ Other
Building background /
providing relevance
Quick Math
Turn and Talk
Models
5.
Algebraic
skills
(Including
fractions and
all rational
numbers)
Process Standards
6.
7.
8.
9.
10.
11.
Multi- Geometry Reasoni Representa Communic Connectio
step
and
ng/
tions
ation
ns
proble Measurem Proof
(draw,
(graphic
ms
ent
diagram)
organizer)
Brief Description
Need(s) Addressed
(see list above)
12.
Problem
Solving
Application:
How has it been
used/could it be used
To be a leader in advancing education and learning
To be a leader in advancing education and learning
Students with math
difficulties…

Struggle With Memory Problems




Poor long-term memory retrieval
skills
Poor working memory
Cannot recall number facts
automatically
Cannot remember procedural steps
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To be a leader in advancing education and learning
Students with math
difficulties…
Struggle with
Computational Fluency
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Strategies

Quick Math Facts

Individualized Checklists
Students Tracking their Own Work
Computational Fluency
( Automaticity of facts)
Quick Math Facts

Set a goal

The student is given a kitchen timer and instructed to set the
timer for a predetermined span of time (e.g., 2 minutes) for each
practice set.

The student completes as many problems as possible before the
timer rings.

The student then graphs the number of problems correctly
computed each day on a time-series graph, attempting to better
his or her previous score
.
Adapted from Interventioncentral.com
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‘Individualized SelfInstruction Checklist’

Explicitly identify pattern errors within an
individual’s work.
(Teacher does this WITH the student.)

Develop checklists.

Students use their checklists to analyze own work.
Adapted from Interventioncentral.com
What Math Students Need Across the Tiers
Focused Targets
1.
2.
3.
Langua Numb Computa
ge of
er
tion
Math Sense Fluency
Vocabul
ary and
Writing
4.
Word
Proble
ms
Strategy/ Activity/ Other
Building background /
providing relevance
Quick Math
Turn and Talk
Models
5.
Algebraic
skills
(Including
fractions and
all rational
numbers)
Process Standards
6.
7.
8.
9.
10.
11.
Multi- Geometry Reasoni Representa Communic Connectio
step
and
ng/
tions
ation
ns
proble Measurem Proof
(draw,
(graphic
ms
ent
diagram)
organizer)
Brief Description
Need(s) Addressed
(see list above)
12.
Problem
Solving
Application:
How has it been
used/could it be used
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Time to Share Ideas


Choose a strategy you use in
your classroom.
Share it with your partner or
small group.
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Ways to achieve
success include:




Explicit and Systematic Instruction
Showing the work in different ways.
(Metacognitive)
Embedding process standards within
the instruction.
Progress monitoring- includes
pre/post assessment, student
tracking of own work.
Resources

Building Academic Vocabulary- Teacher’s Manual
Marzano and Pickering, 2005

Reading and Writing to Learn Mathematics- A Guide
and Resource Book
Martinez and Martinez , 2001

Strategies to Help ELL Students Talk and Write About
Math
Javits Grant,
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Resources






www.sst13.org
www.interventioncentral.org
www.bestevidence.org
http://ies.ed.gov/ncee/wwc/
www.centeroninstruction.org
sc-math.com/math/heuristics.php
Resources

National Mathematics Advisory Panel FINAL REPORT

Assisting Students Struggling with Mathematics:
Response to Intervention (RtI) for Elementary and
Middle Schools (article)
NCEE, What Works Clearinghouse,
US Dept of Education
These (and additional resources) are listed on a page at
the back of your resource packet.
Reflections: RtI + Math = Success for All Students
Name __________________ School_________________ (optional)
3 things I want to remember from today
.
.
.
2 things I’d like to try
.
.
1 thing I’d like further information about in a follow- up session
.
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Thank you!
Please make sure you share your
reflection sheets with us!



Questions/Comments:
Wendy.strickler@hcesc.org
Holly.sampson@hcesc.org