Tiered Math Instruction

OrRTI Project

November 20, 2009

Do not worry about your problems with mathematics,

I assure you mine are far greater.

-Albert Einstein

Objectives• 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 curriculum

• 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 Scientific Evidence

RTI Component

1. Universal screening (Tier I) Moderate Assessment: Screening

2. Focus instruction on whole number for grades k-5 and rational number for grades 6-8

Low Core/Tier 2/Tier 3

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 arithmetic facts

Moderate Core/Tier 2/Tier 3

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

Features• Short duration measures (1 to 5 minute(s) fluency

measures)– 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 sense– 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

administered

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 correct

• 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 answers)• 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 measuresDesigned by Foegen and colleagues assess pre-

algebra and basic algebra skills. Administered and scored similar to Math-CBM

Math CBM Computation and Concepts and ApplicationsConcepts and Applications showed greater

valdity in 6th, 7th, and 8th gradeBen 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

Algebra Probe A-31 Page 1

Solve: 9 + a = 15 a =

Solve: 10 – 6 = g g =

Evaluate: 12 + (– 8) + 3

Simplify: 9 – 4d + 2 + 7d

Simplify: 2x + 4 + 3x + 5

Simplify: 5(b – 3) – b

Solve: 12 – e = 4 e =

Solve: q • 5 = 30 q =

Simplify: 4(3 + s) – 7

Evaluate: 8 – (– 6) – 4

Simplify: b + b + 2b

Simplify: 2 + w(w – 5)

Solve:

18

12

6

b

b =

Solve: 1 foot =12 inches 5 feet = ____ inches

Simplify: 7 – 3(f – 2)

Simplify: 4 – 7b + 5(b – 1)

Evaluate: – 5 + (– 4) – 1

Simplify: s + 2s – 4s

Solve: 63 c = 9 c =

Solve: x + 4 = 7 x =

Simplify: 2(s – 1) + 4 + 5s

Simplify: – 5(q + 3) + 9

Simplify: 8m – 9(m + 2)

Evaluate: 9 + (– 3) – 8

Basic Pre-algebra skills

Ben Clarke, 2009

Math Screening & Monitoring • National Center on Student Progress Monitoring

(www.studentprogress.org)• Intervention Central’s Math Worksheet Generator

(www.interventioncentral.com)• AIMSweb

(www.aimsweb.com) • 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 ScreeningTTSD 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.

Suggestions• 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 Winter)

– Be aware of students who fall near the cut scores

Ben Clarke, 2009

Suggestions• 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

Suggestions• 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

Suggestions• 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

Roadblocks

• 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

Roadblocks

• 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

Roadblocks

• 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

Roadblocks

• 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

and– 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.

Suggestions

• 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 increments

– Look at rates of growth published by AIMSWeb

• Growth trajectories for responders/non responders can be based on local and class or grade performance

• 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

Suggestions

• 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 objectives

– Combined with progress monitoring provides a proximal and distal measuue of performance

Ben Clarke, 2009

Roadblocks

• Students within classes are at very different levels.

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

Ben Clarke, 2009

Roadblocks

• Insufficient time for teachers to implement progress monitoring.

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

Ben Clarke, 2009

Math

Instructional/Curricular Recommendations

• Recommendation 2: whole numbers/rational numbers

• 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.

Suggestions

• 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 properties,

• 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.

Suggestions

• 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 strategyinstruction (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

Suggestions

• Ensure that intervention materials are systematic and explicit and include numerous models of easy and difficult problems with accompanying teacher think-alouds.

• 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 class.”

National Mathematics Advisory Panel Final Report p. xxiii

Roadblocks

• 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

Roadblocks

• Those teaching in the intervention might not be experts or feel comfortable with the math content.

• Suggested Approach: Train interventionists to explain math content (including math concepts, vocabulary, procedures, reasoning and methods) using clear, student-friendly language.

Roadblocks

• 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 effectively.

Recommendation 4

•

Interventions should include instruction on solving word problems that is based on common underlying structures.

Suggestions

• 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 problems.

Roadblocks

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.

Roadblocks

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 another.

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.

Suggestions

• 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.

Roadblocks

• 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 students

Recommendation 6

Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.

Suggestions

• 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, 2004)

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 Disabilities, “Early Identification of Students with Math Disabilities,” July/August 2005 p 294

Recommendation 8

Include motivational strategies in Tier 2 and Tier 3 interventions.

Suggestions

• Reinforce or praise students for their effort and for attending to and being engaged in the lesson.

• Consider rewarding student accomplishment.

• Allow students to chart their progress and to set goals for improvement.

Mindset

• 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• Fluency and Automaticity• Problem Solving

Interdependent and

mutually reinforcing

Core curriculum and instruction

Depth BreadthFocus + Coherence =

Curricular Content

Linear proficiencyvs.

Spiraling

(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 learn.

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

automaticity• Gifted students should accelerate and receive

enrichment

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 correctly– 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

Interventions

• 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

• IES – http://ies.ed.gov/ncee/wwc/reports/Topic.aspx?tid=04#s=13

• Best Evidence– http://www.bestevidence.org/math/elem/elem_math.htm

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

approach

– Screening before progress monitoring

– Strategies for collecting data

Ben Clarke, 2009

Resources NMAP

http://www.ed.gov/about/bdscomm/list/mathpanel/index.html

Center On Instruction - Mathematics http://www.centeroninstruction.org/resources.cfm?

category=math

NCTM focal points http://

www.nctm.orfocalpoints.aspxlinkidentifier=id&itemid=270

PIR website (Best Practices/Articles) http://pacificir2.uoregon.edu:8100/

National Center Progress Monitoring http://www.studentprogress.org/

CA Intervention Standards http://www.cde.ca.gov/ci/ma/im/mathprogramnov2007.asp

Ben Clarke, 2009

Discussion

From where you sit in your current job, was the presentation consistent with how you

think about RtI in Math?

Why? Why not?

Contacts

• Dean Richards– drichards@ttsd.k12.or.us– 503-431-4135

• Jon Potter– jpotter@ttsd.k12.or.us– 503-431-4149

• Lisa Bates– lbates@ttsd.k12.or.us– 503-431-4079

Break Time