Differentiating Math Instruction:

Project EQUAL

Small Group InstructionResponding to Learners’ Needs

Big Ideas of Mathematics~Number & Operations~Algebra~Geometry~Measurement~Data analysis & probability

Processes for Doing Mathematics~Problem Solving~Reasoning & Proof~Connections~Communications~Representation

Responsive Teaching Framework for Differentiating Mathematics Instruction

How programs are designed is critical!

-Spiral vs. Strand

-Traditional vs. Explicit

-Scaffolding to increase mastery & generalization of skills/strategies vs. demonstrate & replicate

-Prior knowledge: Instruction vs. Assumption

-Examples & non-examples

-Sequencing of skills

-Progress monitoring vs. “wait and see”

Adapted from: Allsopp, D., Teaching Mathematics Meaningfully, 2007

Making mathematics

accessible through responsive teaching

Understanding & teachingThe big ideas in math ANDThe big ideas for DOING math

Understanding learning characteristics/ barriers

for students with difficulties In mathematics

Continuously assessing learningTo make informed instructional

decisions

Model for Meaningful Mathematics Instruction

Pre-Assessme

nt

Formative

Assessment

Summative Assessment

Making mathematics

accessible through responsive teaching

Educators must create meaningful learning experiences for students with persistent math difficulties.

This is accomplished by considering learner profile/characteristics and creating a match for the student through careful selection/use of:

MethodsPracticesProcedures

Making Mathematics Accessible Through Responsive Teaching

Readiness Interest Learning preferences

Differentiation of Instruction

based on students’

teachers can differentiate

Tomlinson, The Common Sense of Differentiation, ASCD, 2005 OPTIONS, FDLRS Action Resource Center

Differentiated Instructionis

A teacher’s response to a learner’s needs

clearlearning goals

respectful tasks

flexible grouping

ongoing assessment and

adjustment

positive learning environment

Content Process Product

guided by general principles of differentiation, such as

Forming Flexible Small Groups

Teachers may group students in order to:

o Provide more explicit, intensive, well-scaffolded instruction

o Increase a students’ stage of learning from initial acquisition to advanced acquisition

o Provide independent learner experiences to build proficiency

o Provide learning extension experiences

Using data to flexibly group students

What math data do you currently gather on:

ALL students?Some students?Few students?

Small Group Responsive Teaching FrameworkStep 1: Revisit Summary of MDA Results

SAZDJDADRFFJ

RJ ?SKNMJMXMTRJT

TW

FFFMIMMFIIIFMM

IIIMMMMIMMMIMM

IIIIMIIIMMMIII

MMMMMMMMMMMIMM

IIIMMMMIMMMIMM

MMMMMMMMMMMMMM

NameAbstract

Expressive Receptive

Representational

ExpressiveExpressive Receptive Receptive

Concrete

Key: M=Mastery 95%, I=Instructional 70/75%-95% F=Frustrational below 70%

Step 2: Consider additional Flexible Student Interviews

Who might we want to have an additional conversation with?

What would we want to know?

SA, ZD, JD, TR

Ask them to look at & create concrete & representational examples of 2 fractions, and explain how they know which one is greater…

What evidence do they show re: fractions = area? Do they understand that fractions represent area between 0 and 1 on the number line?

Step 2: Consider additionalFlexible Student Interviews

Ideas for your “Conversation”:

o “Listen” for student’s mathematical thinking

o Ask them to describe to you how they solved the problem

o Ask them to “teach” you how to solve the problem

o Ask them to watch you solve the problem and generate question(s)

o Use concrete objects or drawingsto further explore student understandings

So…What if I have a small group problem or an individual student problem… What do I do to increase student achievement?

How do I meet the needs of studentswho are struggling with mathematics?

Make In

structi

on

Explic

it!

Step 3: Develop Hypothesis for Small Group(s)

A group of my students can…demonstrate receptive concrete and representational

understanding when comparing fractions with like denominators

Using… manipulatives and drawings

However, they do not demonstrate an understanding of this concept…

At the expressive concrete or representational levels

I think this is because…They do not have conceptual understanding what

fractions represent (area) or the part : whole relationship.

Step 4: Form Flexible Small Group(s) for Mathematics Instruction

Who would be in our small, skill-based group?

SA, ZD, JD, TR

A group of my students can…demonstrate receptive concrete and representational understanding when comparing fractions with like denominatorsUsing… manipulatives and drawings However, they do not demonstrate an understanding of this concept…At the expressive concrete or representational levelsI think this is because…They do not have conceptual understanding what fractions represent (area) or the part : whole relationship.

How can teachers reach & teach

these students?Differentiating Mathematics Instruction:

An Explicit, Intensive Small Group Lesson Sequence

How does this instructional

sequence differ from the whole group

lesson?

Variables that influence students’ acquisition of

mathematics:

Instructional Design

Instructional Delivery

Classroom Organization & Management

Make Instruction Explicit!

Instructional DesignDecisions affecting What to Teach

Sequence of skills and concepts

Explicit instructional strategies

Pre-skills

Example selection

Practice and Review

Sequence of Skills and Concepts:

The order of which skills and strategies are introduced affects the difficulty students have in learning them.

Pre-skills of a strategy are taught before the strategy.Easy skills are taught before more difficult ones.Strategies and information that are likely to be confused are not introduced consecutively.Stein, Kinder, Silbert & Carnine, Designing Effective Mathematics Instruction: A

Direct Instruction Approach, 2006

Instructional Strategies:

Clear, accurate & unambiguous (Gersten, 2002)

Strategies must be well-designed and generalizable

Should draw focused attention to the relationship between and among math concepts and skills

Pre-skills:

Component skills of a strategy are taught before the strategy itself is introduced

Ensure students have mastered the pre-skills before introducing a new

instructional strategy

Assess preskills! This helps in determining where instruction should begin!

Example Selection:

Include only problems that students can solve by using a strategy that has been previously taught

Include both examples of the currently introduced type of problem as well as previously introduced problem types that are similar (discrimination)

CAUTION! Many commercial programs do not contain sufficient numbers of examples in their initial teaching to reach mastery, and rarely employ and adequate number of discrimination problems.

Systematic Practice

is necessary for those with low performance rates; without it, students may confuse or forget earlier taught strategies.

Facilitates retention over time

As students reach their goal, gradually decrease the amount of practice of that skill

Practice should never entirely disappear!Kinder, Silbert, Carnine, 2006

Make Instruction Explicit!

Instructional Delivery:How to Teach

Initial assessment and progress monitoring

Presentation techniques

Error-correction procedures

Diagnosis & Remediation

Instructional Delivery:Initial Assessment & Progress Monitoring

Initial (Pre) Assessment:

Answers the Question:

Which skills must be taught?

Dynamic Mathematics AssessmentFocus: Accuracy

Progress Monitoring:Answers the Question:

How are my students

responding to instruction?

On-going / continuous Progress Monitoring

Focus: Rate of Learning + Accuracy

Instructional Delivery:Presentation Techniques

Adequate Time

Appropriate Pacing

Frequent Student Responding

Responsive Scaffolding

Precise Monitoring

Positive, Corrective, & Immediate Feedback

When an error occurs:

Model/Lead

Test

Retest

Instructional Delivery:Error Correction Procedures

Instructional Delivery:Diagnosis & Remediation

Diagnosis: Determining the cause of a pattern of errors

“Can’t do… or … won’t do?”

If you need deeper diagnostic informationregarding a student’s knowledge of fractions, consider administering a Fraction Inventory

Remediation: Process of re-teaching a skill

Provide Instruction! Address Motivation!

Make Instruction Explicit!

Classroom Organization & Management

Elements of Daily Math Lessons

Teacher-Directed Instruction-Introduction of new skills-Remediation of previously taught skills

Independent Work -Exercises students complete without assistance

-Never assign independent work involving problem types/skills that have NOT been

instructed

Workcheck-Designed to correct errors students make on independent work-Provides insights into nature of student errors -Enables teacher to make necessary instructional adjustments

What evidence did you

observe of explicit & intensive

instruction?

I. Teacher explains taskII. Teacher models task “I do…”III. Teacher & students practice task

together—guided practice “We do…”

IV. Students (independently) practice tasks “You do...”

Step 6: Plan & Deliver Instructional Routine

Guided Practice!

Math Lab:Explicit Instructional Routine

Step 7: Develop & implement

student monitoring plan

Determine:

…who will be monitored?

…on what skill?

…using which tool?

…frequency (how often)?

Precision Teaching or Progress Monitoring?

Indicator: Progress Monitoring (Uses Equal Interval Chart)

Precision Teaching (Uses semi-logarithmic chart)

Uses one-minute timings to analyze data to determine student progress toward specific skills

Yes Yes

Helpful in making instructional decisions based on the review and analysis of student data

Yes Yes

One-minute timings always on grade level skills

Yes Not necessarily. If a student is working below grade level, the timing would be on the student’s instructional level.

One minute timings conducted three-four times per year on grade level skills

Yes No. The precision teaching fluency-building timings are conducted daily or three to four times per week.

SA, ZD, JD, TR

Differentiating Math Instruction:

Project EQUAL

Ongoing Assessment

What teachers need to know…

How are my students responding to mathematics instruction?

Adding it Up, National Research Council, p. 117, 2007

In order for students tobe successful in Mathematics, each of these intertwined strands must work together, so…

What do we measure to determine if they are making progress?

Continuously Assessing LearningEducators should evaluate what students know

and can do before, during and after instruction

Before Instruction: evaluate student prerequisite knowledge, skills, experiences and interests that relate to the target concept. Know where to begin instruction.

During Instruction: Are students gaining understanding? Are they able to use knowledge and skills proficiently? Results allow for immediate adjustment, if necessary.

After Instruction: Where are students in terms of their conceptual learning and skill proficiency? Assessing for the purpose of answering this question will help in planning future instruction.

Types of Assessment:Mathematics

Before: Screening

During: Formative

Diagnostic

(if necessary)

After: Summative

Pre-Assessment

Formative

Assessment

Summative

Assessment

Adapted from: Allsopp, D., Teaching Mathematics Meaningfully, 2007

Making mathematics

accessible through responsive teaching

Understanding & teachingThe big ideas in math ANDThe big ideas for DOING math

Understanding learning characteristics/ barriers

for students with difficulties In mathematics

Continuously assessing learningTo make informed instructional

decisions

Model for Meaningful Mathematics Instruction

Pre-Assessme

nt

Formative

Assessment

Summative Assessment

Big Ideas of Mathematics~Number & Operations~Algebra~Geometry~Measurement~Data analysis & probability

Processes for Doing Mathematics~Problem Solving~Reasoning & Proof~Connections~Communications~Representation

Responsive Teaching Framework for Differentiating Mathematics Instruction

Most Intensive

Few

Some

ALL

• Daily or Weekly, precise

• Graphically represented

EX. : Precision Teaching Great Leaps CBM probes

Least Intensive

• Weekly or Bi-weekly, more precise• Graphically

representedEX: CBM Probes

The more intensive the instruction…

the more intensive the monitoringshould be!

• District math benchmark assessments (as indicated)• Curriculum embedded assessments• Flexible student

interviews• Error Analysis• Student Work Samples• Rubrics for Problem

Solving (Applied Math)

MonitoringProgressorProgress Monitoring?

ProgressMonitoring

Monitoring Progress for

How can I measure student knowledge levels?Thinking Differently…

CRA Level of Understandin

gMethod Criterion

Abstract 1-5 minute timings(depends on nature of

target concept)

Fluency (Rate & Accuracy)

Representational(Drawing) 8-10 tasks Accuracy

90-100% 3 times

Concrete 3 tasks Accuracy100% 3 times

Start

INVESTIGATE!Review the continuous, formative assessment tools you have at your table.

As a Table Group, prepare to explain what it is, & how it might be used.

Think in terms of the intensity equalizer. Is it more intensive, or less intensive?

Progress Monitoring Probe

Abstract Level

Procedural:

See/Write 2 digit addition without

regrouping (sums < 20)

Measure:# of digits correct

Examples of Concrete and Representational/Drawing Probe Tasks

Use circle pieces and string to solve the following equations.

1. 3 x 4 = 12

Concrete

Representational/ Drawing

Making Instructional Decisions

Create a visual display of student performance data• Chart• Graph• Think of this visual display as a

“picture” of your students’ learning

Evaluate what the learning picture reveals about student learning

“goal line”

Visual Display

“corrects”

“incorrects”

What does this learning picture show?

Allsopp, 2008

Problem Solving Learning Picture Example: Student Use of Strategies

Allsopp, 2008

Algebraic Thinking Standard: Represent, describe, and analyze patterns and relationships using tables, graphs, verbal rules, and standard

algebraic notation.

NCTM Process: Representation

NCTM Process: Communication

0 1 2 3

Algebraic Thinking Standard: Represent, describe, and analyze patterns and relationships using tables, graphs,

verbal rules, and standard algebraic notation. Allsopp, 2008

Some General Guidelines for PM

Incorporate at concrete, drawing & abstract levels

Use short, easy to evaluate “probes”

Pinpoint key concepts for monitoring

Teach students to chart their learning

Use as a way to engage students in setting learning goals

At least 2-3 times weekly, more often if needed

CELEBRATE SUCCESS!!

Learn More About Continuous Progress Monitoring

at the MathVIDS Website:

http://fcit.usf.edu/mathvids/

Other Ongoing Monitoring Resources include:www.interventioncentral.org www.aimsweb.com

A Word About Follow Up

Choose an instructional practice you learned about in Project EQUAL

Implement in your Classroom

Join our Project Equal Wiki

Post a reflection about your action research.

With Gratitude…

Dr. David Allsopp

University of South FloridaFor his

Insightfulness andCollaborative Spirit

Thank You!

Contact Information:Donna Crocker 407.217.3679 donna.crocker@ocps.netKaren Geisel 407.317.3681 karen.geisel@ocps.netMarcia Levy 407.317.3677 marcia.levy@ocps.net