Recommendation for Math Teacher Job by lld99380


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									Tiered Math Instruction

       OrRTI Project
     November 20, 2009
Do not worry about your problems with
   I assure you mine are far greater.

           -Albert Einstein
• Look at IES
  recommendations for
  assessment and instruction
  in Mathematics
• Understand the major
  findings of the National
  Math Advisory Panel report
  and it’s implications to core
• Look at possible
  interventions to support
  struggling mathematicians
               The Math Caveat
• A lit search for studies on reading disabilities studies and
  math disability studies from 1996-2005 found over 600
  studies in the area of reading and less than 50 for
  mathematics (12:1)
• Specific RTI mathematics studies for a recent annotated
  bibliography totaled 9 studies
    IES Recommendation               Level of    RTI Component
1. Universal screening (Tier I)     Moderate        Assessment:
2. Focus instruction on whole
   number for grades k-5 and           Low        Core/Tier 2/Tier 3
   rational number for grades 6-8

3. Systematic instruction             Strong      Core/Tier 2/Tier 3

4. Solving word problems              Strong      Core/Tier 2/Tier 3

5. Visual representations           Moderate      Core/Tier 2/Tier 3

6. Building fluency with basic      Moderate      Core/Tier 2/Tier 3
   arithmetic facts
7. Progress monitoring                 Low          Assessment:
                                                 Progress Monitoring
8. Use of motivational strategies      Low        Core/Tier 2/Tier 3
 Assessment Recommendations
• Recommendation 1: Universal Screening
• Recommendation 7: Progress Monitoring
           Recommendation 1

    Screen all students to identify those at
    risk for potential mathematics difficulties
    and provide interventions to students
    identified as at risk.
       Coherent Assessment Systems

• Each type of assessment has a purpose

• The design of the tool should match the purpose
   – What are the implications for screening tools used with
     all students?

• Think purpose not tool

• How do each of these purposes fit together?
                                                 Ben Clarke, 2009
• Short duration measures (1 to 5 minute(s) fluency
   – Note many measures that are short duration also
     used in progress monitoring.

• Longer duration measures (untimed up to 20
  minutes) often examine multiple aspects of number
   – Issue of purpose is critical to examine

• Most research examines predictive validity from
  Fall to Spring.                        Ben Clarke, 2009
            Universal Screening
• The Math Measures:
  – K-1:
     •   Missing Number
     •   Quantity Discrimination
     •   Number Identification
     •   VanDerheyden: K-CBM
  – Grades 2-5:
     • Basic Facts
     • Concepts and Applications
     • Math Focal Points
  – -Secondary:
     • Prealgebra
Universal screener
           •   Missing Number
           •   K & 1 assessment
           •   One minute assessment
           •   Individually administered
Universal screener
           •   Quantity Discrimination
           •   K & 1 assessment
           •   One Minute assessment
           •   Individually
Universal screener
           •   Number Identification
           •    K & 1 assessment
           •   One Minute assessment
           •   Individually administered
VanDerheyden: K-CBM

               Ben Clarke, 2009
Universal screener
           • Computation
           • 5th grade example
           • 1-5 grade
           • Grows in complexity
             through the grades
           • Two to four Minute
             assessment (depending
             on grade)
           • Scored on digits
           • Group administered
Universal screener
         •   Monitoring Basic Skills
         •   4th grade example
         •   2-5 grade
         •   Grows in complexity
             through the grades
         •   Four to eight minutes
             (depending on grade)
         •   Scored on correct answers
             (some have multiple
         •   Group administered
         •   Fuchs, Fuchs and Hamlett
easy-CBM: Number and Operations

                          Ben Clarke, 2009
  Example: Reflecting critical math content

• easy-CBM

• Items created according to NCTM Focal
  Points for grade level

• 48 items for screening (16 per focal point)

• Ongoing research (not reviewed in practice
  guide)                            Ben Clarke, 2009
             Middle School
Algebra measures
  Designed by Foegen and colleagues assess pre-
   algebra and basic algebra skills. Administered
   and scored similar to Math-CBM

Math CBM Computation and Concepts and
  Concepts and Applications showed greater
   valdity in 6th, 7th, and 8th grade
                                        Ben Clarke, 2009
   Basic Skills (in Algebra)
• 60 items; 5 minutes
• Problems include:
   –   Solving basic fact equations;
   –   Applying the distributive property;
   –   Working with integers;
   –   Combining like terms;
   –   Simplifying expressions;
   –   Applying proportional reasoning
• Scoring: # of problems correct

                                             Ben Clarke, 2009
                     Basic Pre-algebra skills
Algebra Probe A-31                                     Page 1

Solve:                          Solve:
9 + a = 15                      10 – 6 = g
a=                              g=
Evaluate:                       Simplify:
12 + (– 8) + 3                  9 – 4d + 2 + 7d
Simplify:                       Simplify:
2x + 4 + 3x + 5                 5(b – 3) – b

Solve:                          Solve:
12 – e = 4                      q • 5 = 30
e=                              q=
Simplify:                       Evaluate:
4(3 + s) – 7                    8 – (– 6) – 4

Simplify:                       Simplify:
b + b + 2b                      2 + w(w – 5)
Solve:                          Solve:
    12                          1 foot =12 inches
6   18                          5 feet = ____ inches
Simplify:                       Simplify:
7 – 3(f – 2)                    4 – 7b + 5(b – 1)

Evaluate:                       Simplify:
– 5 + (– 4) – 1                 s + 2s – 4s

Solve:                          Solve:
63  c = 9                      x+ 4 =7
c=                              x=
Simplify:                       Simplify:
2(s – 1) + 4 + 5s               – 5(q + 3) + 9

Simplify:                       Evaluate:
8m – 9(m + 2)                   9 + (– 3) – 8

                                                           Ben Clarke, 2009
 Math Screening & Monitoring
• National Center on Student Progress Monitoring
• Intervention Central’s Math Worksheet Generator
• AIMSweb
• Monitoring Basic Skills Progress
  (Fuchs, Hamlet & Fuchs, 1998)
• The ABC’s of CBM (Hosp, Hosp,& Howell, 2007)
• DIBELS Math (2nd year Beta)
• Easy CBM
          Universal Screening
                     TTSD Decision Rules
–   K: Students receiving only ―o‖ and/or ―/‖ in the
    ―Progression of Mathematics Stages‖ on the Progress
    Report are screened using CBM.
–   1-2: Students receiving only ―1‖ and/or ―/‖ in ―math‖
    on the Progress Report are screened using CBM.
–   3-5: Students receiving only ―1,‖ ―2,‖ and/or ―/‖ in
    ―math‖ on the Progress Report AND scoring below the
    30th percentile on the OAKS, are screened using CBM.
–   Students who meet the above criteria are assessed using
    Curriculum Based Measurements (CBM: Missing
    Number for K/1 and Basic Facts for 2-5). Students
    scoring below the 25th percentile on CBMs are placed
    in Second Tier Interventions.
• Have a district level team select measures
  based on critical criteria such as reliability,
  validity and efficiency.

   – Team should have measurement expertise (e.g. school
     psychologist) and mathematics (e.g. math specialist)
   – Set up a screening to occur twice a year (Fall and
   – Be aware of students who fall near the cut scores

                                               Ben Clarke, 2009
• Use the same screening tool across a district
  to enable analyzing results across schools

  – Districts may use results to determine the
    effectiveness of district initiatives.
  – May also be used to determine systematic areas
    of weakness and provide support in that area
    (e.g. fractions)

                                        Ben Clarke, 2009
• Select screening measures based on the
  content they cover with a emphasis on
  critical instructional objectives for each
  grade level.

  – Lower elementary: Whole Number
  – Upper elementary: Rational Number
  – Across grades: Computational Fluency
    (hallmark of MLD)

                                        Ben Clarke, 2009
• In grades 4-8, use screening measures in
  combination with state testing data.

  – Use state testing data from the previous year as the first
    cut in a screening system.
  – Can then use a screening measure with a reduced pool
    of students or a more diagnostic measure linked to the
    intervention program for a second cut.

                                                  Ben Clarke, 2009
• Resistance may be encountered in
  allocating time and resources to the
  collection of screening data.

• Suggested Approach: Use data collection
  teams to streamline the data collection and
  analysis process.

                                         Ben Clarke, 2009
• Questions may arise about testing students
  who are ―doing fine‖.

• Suggested Approach: Screening all students
  allows the school or district to evaluate the
  impact of instructional approaches
  – Screening all students creates a distribution of
    performance allowing the identification of at-
    risk students
                                           Ben Clarke, 2009
• Screening may identify students as at-risk
  who do not need services and miss students
  who do.

• Suggested Approach: Schools should
  frequently examine the sensitivity and
  specificity of screening measures to ensure
  a proper balance and accurate decisions
  about student risk status.
                                     Ben Clarke, 2009
• Screening may identify large numbers of students
  who need support beyond the current resources of
  the school or district.

• Suggested Approach: Schools and districts should

   – Allocate resources to the students with the most risk
     and at critical grade levels
   – Implement school wide interventions to all students in
     areas of school wide low performance (e.g. Fractions)

                                                 Ben Clarke, 2009
       Recommendation 7

Monitor the progress of students
receiving supplemental instruction and
other students who are at risk.
• Monitor the progress of tier 2, tier 3 and
  borderline tier 1 students at least once a
  month using grade appropriate general
  outcome measures.

  – Same team that worked on screening can also
    work on progress monitoring
  – Need to carefully consider capacity to model
    growth in the context of instructional decision
    making                                 Ben Clarke, 2009
   TTSD Progress Monitoring
• CBMs are given every other week
  – Trained instructional assistants will complete
    progress monitoring
• Review trend lines every 12 weeks
  – We need a longer intervention period because
    growth on math CBMs happens in small
  – Look at rates of growth published by
• Growth trajectories for
  responders/non responders can be
  based on local and class or grade

• Or use projected rate of growth
  from national norms— e.g.
  AIMSweb 50th %tile
   – Grade 1, .30 digit per week growth
   – Grade 3, .40 digit per week growth
   – Grade 5, .70 digit per week growth
• Use curriculum-embedded assessments in
  intervention materials

  – Frequency of measures can vary - every day to
    once every week.
  – Will provide a more accurate index of whether
    or not the student is obtaining instructional
  – Combined with progress monitoring provides a
    proximal and distal measuue of performance

                                        Ben Clarke, 2009
• Students within classes are at very different

• Suggested Approach: Group students across
  classes to create groups with similar needs.

                                       Ben Clarke, 2009
• Insufficient time for teachers to implement
  progress monitoring.

• Suggested Approach: Train
  paraprofessionals or other school staff to
  administer progress monitoring measures.

                                      Ben Clarke, 2009
• Recommendation 2: whole numbers/rational
• Recommendation 3: systematic instruction
• Recommendation 4: solving word problems
• Recommendation 5: visual representation
• Recommendation 6: fluent retrieval of facts
• Recommendation 8: motivational strategies
           Recommendation 2

    Instructional materials for students
    receiving interventions should focus
    intensely on in-depth treatment of whole
    numbers in K-3 and on rational numbers
    in grades 4-8.
• For tier 2 and 3 students in grades K-3,
  interventions should focus on the properties of
  whole number and operations. Some older
  students would also benefit from this approach.

• For tier 2 and 3 students in grades 4-8,
  interventions should focus on in depth coverage of
  rational number and advanced topics in whole
  number (e.g. long division).
             Core curriculum content
• Whole number: understand place value, compose/decompose
   numbers, leaning of operations, algorithms and automaticity with facts, apply to
   problem solving, use/knowledge of commutative, associative, and distributive

• Rational number: locate +/- fractions on number line,
   represent/compare fractions, decimals percents, sums, differences products and
   quotients of fractions are fractions, understand relationship between fractions,
   decimals, and percents, understand fractions as rates, proportionality, and
   probability, computational facility

• Critical aspects of geometry and
  measurement: similar triangles, slope of straight line/linear functions,
   analyze properties of two and three dimensional shapes and determine perimeter,
   area, volume, and surface area
                                           Source: Ben Clarke & Scott Baker
                                           Pacific Institutes for Research
Difficulty with fractions is pervasive and
impedes further progress in mathematics
           Recommendation 3

    Instruction provided in math interventions
    should be explicit and systematic,
    incorporating modeling of proficient
    problem-solving, verbalization of thought
    processes, guided practice, corrective
    feedback and frequent cumulative review.
• Districts should appoint committees with experts
  in mathematics instruction and mathematicians to
  ensure specific criteria are covered in-depth in
  adopted curriculums.
   – Integrate computation with problem solving and
     pictorial representations
   – Stress reasoning underlying calculation methods
   – Build algorithmic proficiency
   – Contain frequent review of mathematical principles
   – Contain assessments to appropriately place students in
     the program
       Schema-based strategy
     instruction (Jitendra, 2004)
• Teach student to represent quantitative
relationships graphically to solve problems.
• Use Explicit Strategies:
   1. Problem Identification
   2. Problem Representation
   3. Problem Solution
• Be systematic: Teach one type of problem at
a time until students are proficient.
• Provide models of proficient problem solving.
                                               Kathy Jungjahann
• Ensure that intervention materials are systematic
  and explicit and include numerous models of easy
  and difficult problems with accompanying teacher

• Provide students with opportunities to solve
  problems in a group and communicate problem-
  solving strategies.

• Ensure that instructional materials include
  cumulative review in each session.
             Point of Discussion
―Explicit instruction with students who have
mathematical difficulties has shown
consistently positive effects on performance
with word problems and computations.
Results are consistent for students with
learning disabilities, as well as other student
who perform in the lowest third of a typical
National Mathematics Advisory Panel Final Report p. xxiii
• Interventionists might not be familiar with using
  explicit instruction and might not realize how
  much practice is needed for students in tier 2 and
  tier 3 to master the material being taught.

• Suggested Approach: Have interventionists
  observe lessons, practice with instructional
  materials, and provide them with corrective
  feedback on implementation
• Those teaching in the intervention might not
  be experts or feel comfortable with the math

• Suggested Approach: Train interventionists to
  explain math content (including math concepts,
  vocabulary, procedures, reasoning and methods)
  using clear, student-friendly language.
• The intervention materials might not incorporate enough
  modeling, think-alouds, practice or cumulative review to
  improve students’ math performance.

• Suggested Approach: Consider having a math specialist
  develop an instructional template which contains the
  elements of instruction identified above and which can be
  applied to various lessons.
   – If possible, have a math specialist coach new
     interventionists on how to use materials most
           Recommendation 4

    Interventions should include instruction on
    solving word problems that is based on
    common underlying structures.
• Teach students about the structure of various
  problem types, how to categorize problems, and
  how to determine appropriate solutions.

• Teach students to recognize the common
  underlying structure between familiar and
  unfamiliar problems and to transfer known
  solution methods from familiar to unfamiliar
 Math curriculum material might not classify the
  problems in the lessons into problem types

 Suggested Approach: Use a math specialist or a state or
  district curriculum guide to help identify the problem types
  covered in the curriculum at each level and the
  recommended strategies for solving them.
   • Students must be taught to understand a set of problem
      types and a reliable strategy for solving each type.
 As problems get more complex, so will the
  problem types and the task of
  discriminating among them.

 Suggested Approach: Explicitly and
  systematically teach teachers and interventionists
  to identify problem types and how to teach
  students to differentiate one problem type from
       Recommendation 5

Intervention materials should include
opportunities for students to work with
visual representations of mathematical
ideas, and interventionists should be
proficient in the use of visual
representations of mathematical ideas.
• Use visual representations such as number
  lines, arrays, and strip diagrams.

• If necessary consider expeditious use of
  concrete manipulatives before visual
  representations. The goal should be to
  move toward abstract understanding.
• Because many curricular materials do not include
  sufficient examples of visual representations, the
  interventionist may need the help of the
  mathematics coach or other teachers in developing
  the visuals.
• If interventionists do not fully understand the
  mathematical ideas behind the (representations),
  they are unlikely to be able to teach it to struggling
        Recommendation 6

Interventions at all grade levels should
devote about 10 minutes in each session to
building fluent retrieval of basic arithmetic
• Provide 10 minutes per session of instruction
  to build quick retrieval of basic facts.
  Consider the use of technology, flash cards,
  and other materials to support extensive
  practice to facilitate automatic retrieval.
• For student in K-2 grade explicitly teach
  strategies for efficient counting to improve the
  retrieval of math facts.
• Teach students in grades 2-8 how to use their
  knowledge of math properties to derive facts in
  their heads.
―Basic‖ math facts are important!
• Basic math facts knowledge
  – Difficulty in automatic retrieval of basic math
    facts impedes more advanced math operations
• Fluency in math operations
  – Distinguishes between students with poor math
    skills to those with good skills (Landerl, Bevan,
    & Butterworth, 2004; Passolunghi & Siegel,
             Point of Discussion
―the general concept of automaticity. . . is
that, with extended practice, specific skills
can read a level of proficiency where skill
execution is rapid and accurate with little or
no conscious monitoring … attentional
resources can be allocated to other tasks or
processes, including higher-level executive
or control function‖

(Goldman & Pellegrino, 1987, p. 145 as quoted in Journal of Learning
       Recommendation 8

Include motivational strategies in Tier 2
and Tier 3 interventions.
• Reinforce or praise students for their effort
  and for attending to and being engaged in
  the lesson.
• Consider rewarding student
• Allow students to chart their progress and to
  set goals for improvement.
• Incorporate social and intellectual support
  from peers and teachers
• Teach students that effort has a huge impact
  on math achievement
           Big Ideas from IES
• Provide explicit and systematic instruction in
problem solving.
• Teach common underlying structures of
word problems.
• Use visual representations.
• Verbalize your thought process.
• Model proficient problem solving, provide
guided practice, corrective feedback, and
frequent cumulative review.
Putting it all Together for Multi-
        tiered Instruction
• National Math Panel
• Process in TTSD
 Core curriculum and instruction
National Mathematics Advisory Panel Final
  Report, 2008
• Curricular Content moving toward algebra
• Teacher Proficiency
• Conceptual Understanding Interdependent
• Fluency and Automaticity          and
• Problem Solving               reinforcing
Core curriculum and instruction

        Curricular Content

 Focus + Coherence =   Breadth
  Linear proficiency

(Closure after Exposure)
           Learning Processes
• Conceptual understanding, computational fluency
  and problem-solving skills are each essential and
  mutually reinforcing.
• Effort-based learning has greater impact than the
  notion of inherent ability
• The notion of ―developmentally appropriate
  practices‖ based on age or grade level has
  consistently been proven to be wrong. Instead,
  learning is contingent on prior opportunities to
    Professional Development
• Teacher induction programs have positive
  effects on all teachers.
• Professional development is important-
  continue to build content knowledge as well
  as learning strategies.
• Teachers who know the math content they
  are teaching, including the content before
  and beyond, have the most impact on
  student achievement.
         Practices That Work
• Using formative assessments
• Low achievers need explicit instruction in addition
  to daily core instruction
• Technology supports drill practice and
• Gifted students should accelerate and receive
       So What? Now What?
• What information coincided with your
  understanding of effective math instruction,
  or practices in your district?
• What surprised you?
• What implications does the report have for
  this school year? Future years?
            Tier I in TTSD
•   45-90 minutes core instruction
•   K-12 curriculum alignment
•   Systematic instruction and feedback
•   Teach content to mastery
•   Focus on fractions!
     What about interventions?
• Emphasis on research-based instructional
  strategies (not ―programs‖)
• Increase opportunities to practice a skill
  – Guided practice (―I do, We do, You do‖)
  – Correction routine
  Tier II Interventions for Math in
      TTSD (Within the Core)
• Kindergarten
  – Increased teacher attention during math
• Grades 1-5
  – 10 minutes of additional guided practice per
    day OR
  – 10 minutes of Computer Assisted Instruction
    (CAI) per day
               Tier II & III:
         Research on Best Practices
               Baker, Gersten, and Lee, 2002

• Demonstrated, significant effects for:
  – Progress monitoring feedback, especially when
    accompanied by instructional recommendations
  – Peer Assisted Learning
  – Explicit teacher led and contextualized teacher
    facilitated approaches
  – Concrete feedback to Parents
• Emphasis on research-based
  instructional strategies (not programs)
• Increase opportunities to practice a
  skill correctly
  – Guided practice (―I do, We do, You do‖)
  – Correction routine
• There are few, but an increasing number
  of research based curricula available
                Intervention lists
• Best Evidence
     How to start and Next steps
• As you get started consider

   – Focus on one grade or grade bands
      • Long term trajectories suggest end of K critical benchmark
      • May have more expertise/comfort with whole number

   – Screening before progress monitoring

   – Strategies for collecting data

                                                        Ben Clarke, 2009

 Center On Instruction - Mathematics

 NCTM focal points
    http://www.nctm.orfocalpoints.aspxlinkidentifier=id&ite

 PIR website (Best Practices/Articles)

 National Center Progress Monitoring

 CA Intervention Standards
                                                     Ben Clarke, 2009

From where you sit in your current job, was
  the presentation consistent with how you
          think about RtI in Math?

             Why? Why not?
• Dean Richards
  – 503-431-4135
• Jon Potter
  – 503-431-4149
• Lisa Bates
  – 503-431-4079
Break Time

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