responsiveness to instruction: mathematical understanding rowan-salisbury schools rti foundations...

Responsiveness To Instruction: Mathematical Understanding

Rowan-Salisbury Schools

RtI Foundations Training

August 2010

Erin Banks & Sarah Brown, School Psychologist and Program Specialist

When you need help with math…… Call Somebody!!! http://blutube.policeone.com/Clip.aspx?key=9A7EF76E53A5E67C

QuickTime™ and a decompressor

are needed to see this picture.

It’s not MY Fault!!!



How Our Students Feel About Numbers. . . . .



The Ultimate Test Anxiety. . .



And Why It’s All Worth It. . . . .

What’s Going on With Math?

NAEP 2009 results were released in March 2010

How do you think NC math results compare to: United States norms? Reading results?





NAEP (2009) Math Results:



In 2009, the average score in North Carolina was •lower than those in 4 states/jurisdictions•higher than those in 29 states/jurisdictions •not significantly different from those in 18 states/jurisdictions

(www.nces.ed.gov)

Test Yourself! Sample 4th grade NAEP Math test

question

It takes Mrs. Wylie 15 minutes to drive from her house to the store. What is the best estimate of the distance from her house to the store?

1. 5 feet2. 5 miles3. 20 feet4. 200 miles

Are You Smarter than an 8th Grader?



Which point is the solution to both equations in the graph?

1. (0,0)

2. (0,4)

3. (1,1)

4. (2,2)

5. (4,0)

National Mathematics Panel Report 2008: What do students need for success in Algebra?

Major Findings:•Proficiency with whole numbers, fractions and certain aspects of geometry and measurement are the critical foundations of algebra

•Explicit instruction for students with disabilities shows positive effects.

•Students need both explicit instruction and conceptual development to succeed in math.

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

Math Basics

International Research

TIMSS from Improving Mathematics Instruction (Ed Leadership 2/2004)

1995 Video Study• Largest and most carefully designed study of teacher instruction and mathematics

• More than 500,000 students (33,000 in US)• Grades 4, 8, 12• 41 nations participated

Results = 4th grade students scored ABOVE international average; BELOW average in measurement, estimation, and number sense!

8th grade students scored BELOW international average overall, including:• Measurement, geometry, and proportionality

Outperformed by 20 countries 12th grade students scored WELL BELOW international average (outperformed Cyprus and South Africa)

TIMSS from Improving Mathematics Instruction (Ed Leadership 2/2004)

1999 Follow-up Study Examined lessons in Grade 8 in United States, Germany, and Japan

US teachers teach student HOW to get answers while Japanese teachers teach for UNDERSTANDING

US teachers usually don’t develop math concepts and ideas

US teachers taught material with low-level math content vs. other teachers using high-level math content

What Do We Need to Do Differently? Help students develop understanding of math concepts

Teach lessons to challenge students (using high level content)

TIMSS (2007) Study U.S. students' average mathematics score was

529 for 4th-graders (500 is average) 508 for 8th-graders (500 is average)

Fourth-graders in 8 of the 35 other countries that participated in 2007 (Hong Kong, Singapore, Chinese Taipei, Japan, Kazakhstan, Russian Federation, England, and Latvia) scored ABOVE their U.S. peers

Among the 16 countries that participated in both the first TIMSS in 1995 and the most recent TIMSS in 2007, at grade 4, the average mathematics score increased in 8 countries, including in the United States, and decreased in 4 countries.

Higher than 1995!

Style vs.. Implementation

High Achieving countries use a variety of styles to teach (calculator vs.. no calculator, ‘real-life’ problems vs.. ‘naked’ problems)

High Achieving countries all implement connections problems as connections problems

U.S. implements connection problems as a set of procedures

NCDPI RtI Foundations Training

Exponents and GeometryWhat is 42 ?Why is it 4 x 4 when it looks like 4 x 2?

It means ‘make a square out of your 4 unit side’

Making Connections. . .


Exponents and GeometryWhat is 42 ?

--4 units--1111

You’d get how many little1 by 1 inch squares? 42 = 16


Some words about “Key Words”

They don’t work…


We tell them—more means add

Erin has 46 comic books. She has 18 more comic books than Jason has. How many comic books does Jason have.

But is our answer really 64 which is 46 + 18?


Sense-Making We need to notice if we are making sense of the math for our students, or if our discussion of the math contributes to ---

“The suspension of sense-making”Schoenfeld (1991)


Just Do It!


Content Standards Number & Operations Algebra Geometry Measurement Data Analysis & Probability

Process Standards Problem Solving Reasoning & Proof Communication Connections Representation



www.nctm.org

What does NC SCOS tell us? Number and Operations Measurement Data Analysis & Probability Algebra Problem-Solving

(uses the same 5 strands as NCTM Principles)



What do researchers tell us?Use Direct and Explicit Instruction to focus upon:

Number Sense Basic math operations Problem-solving skills(Kroesbergen & Van Luit, 2003)



Why Intervention in Math? Students have difficulty with:

Mastering math computation skills or Application of math

Speed and Accuracy are IMPORTANT! Students who cannot master basic computational skills are very unlikely to succeed at application of those skills.

(Shapiro, 2004)

Number Sense

Number Sense

ComputationPrerequisite

……in the same way as………

Phonemic Awareness Prerequisite

Reading Fluency

What is Number Sense? Manipulate Numbers

Adding on Counting up Skip counting

One-to-One correspondence Counting steps as they walk down/up

Understanding of how number systems work

NCTM Strands Without Number Sense

Number &

Operation

s

Measurement

Geomet

ry

Data Analysis and ProbabilityAlg

ebra

NCTM Strands With Number Sense

Number &

Operation

s

Measurement

Geomet

ry

Data Analysis and Probability

Algebr

a

Fluency (Automaticity)

Strong correlation (relationship) between poor retrieval of arithmetic combinations (math facts) and global math delays

Automatic recall of arithmetic combinations frees up student ‘cognitive capacity’ to allow for understanding of higher-level problem-solving

Working memory!!!!

How to Increase Fluency PRACTICE!!!!!!!!

However, practice alone may not be sufficientArea of concern Interventions

Speed and Fluency 1. Immediate feedback2. Contingent free time3. Interspersal of easy and

difficult material on worksheets (folding-in technique)

Speech, Fluency, AND accuracy

1. Using simple reminders2. Reminding students of

steps3. Visual representations of

problem

31 1=Language in Mathematics


Are these the same?


Break During the break, read the article:“ Early Identification and Interventions for Students With Mathematics Difficulties” (Gersten, Jordan & Flojo, 2005)

Discussion upon returning to group

Early Identification

1. Describe the nature of math difficulties.2. How are mathematical difficulties and

reading difficulties related?3. What is number sense as operationalized by

Kalchman, Moss and Case in this article?4. What are the two distinct factors in

mathematics proficiency in kindergartners?5. What is the role of socioeconomic status as

stated by Griffin, Case and Sigler in 1994?6. What are the instructional implications for

the findings? NCDPI RtI Foundations Training

Tier I Consultation

Between Teachers-Parents

Tier II Consultation With Other Resources

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INTENSITY OF NEEDS

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Tier IVIEP

Consideration

Tier IIIStudent Study Team

Intensive Interventions 1-7%

Strategic Interventions 5-15%

Core Curriculum 80-90%

How Does Math Fit into the RtI Model?


What Works Clearinghouse

Intervention Evidence

Extent of Evidence

Odyssey Math +

Small

Everyday Math +

Medium to Large

Progress in Math 0

Small

Scott Foresman-Addison Wesley Elementary Math (enVision)

0

Small

Houghton Mifflin Math

0

Medium to Large

Saxon Elementary Math

0

Small

Tier I - Math

Characteristics:1. Classroom

teacher2. Core Math

instructional time (60 minutes recommended)

3. Benchmarks - Fall, Winter, Spring

Instruction:1. Quality Lesson

Design2. Research-Based

Strategies3. Differentiated

Instruction4. Explicit

Instruction5. Questioning6. Connections

Assessment At Tier I Universal Screening/Benchmarking

Fall, Winter, Spring Investigating Further. . . .

Number Knowledge test (handout) Money test (handout)

Use Data to: Develop groups for differentiation Groups for supplemental intervention/instruction

Groups for enriched instruction

http://clarku.edu/numberworlds

Differentiated Instruction in Math



What is Differentiated Instruction?

A teaching theory based on the premise that instructional approaches….

…should vary and be adapted…

…in relation to individual and diverse students in classrooms.

Ways to Differentiate in Math: Manipulate Content/Topic Manipulate Process/Activities Manipulate Product Manipulate Environment

How have you been successful using these techniques in your classroom?

Manipulation of Content

Multiple Options for Taking in Information

Classroom Objective: All students will subtract using renaming.

Manipulation: Some students may learn to subtract two-digit numbers, while others learn to subtract larger numbers in the context of word problems.

(Tomlinson, 1999)

Manipulation of Process

Multiple Options in which a Student Accesses Material

One student may explore a learning center, while another student collects information from the web.

(Tomlinson, 1999)

Manipulation of Project

Multiple Options for Expressing What They Know

To demonstrate understanding of a geometric concept, one student may solve a problem set, while another builds a model.

Tomlinson, 1999

Manipulation of Learning Environment

Multiple Ways to Create an Environment to Stimulate Learning

Create places in the room to work quietly and without distraction

Provide materials that reflect a variety of cultures and home settings

Set out clear guidelines for independent work that matches individual needs

Develop routines that allow students to get help when teachers are busy with other students and cannot help immediately

Allow learners to move around if it increases learning

(Tomlinson, 1999)

Strong Tier I Math Instruction Tiered Assignments Adjusting Questions Compacting Curriculum Learning Centers/Groups Choice Boards Flexible Grouping Acceleration/Deceleration Peer Teaching Independent Study Projects Anchoring Projects

Tiered Assignments Focus of Differentiation: Readiness

Example:In a unit on measurement, some students are taught basic measurement skills, including using a ruler to measure length. Other students can apply measurement skills to problems involving perimeter.

Adjusting Questions Focus of Differentiation: ReadinessExample:During large group discussion activities, teachers direct higher level questions to students who can handle them and adjust questions accordingly for students with greater needs.

With written quizzes, teacher may assign specific questions for each group of students. They all answer the same number of questions, but complexity required varies from group to group.

Compacting Curriculum Focus of Differentiation: ReadinessExample:A 3rd grade class is learning to identify the parts of fractions. Diagnostics indicate that two students already know the parts of fractions. These students are excused from completing activities and are taught to add and subtract fractions.

Interest Centers/Groups Focus of Differentiation: Readiness, Interest

Example:Interest Centers - centers can focus on specific math skills, such as addition, and provide activities that are high interest such as counting jelly beans or adding eyes on aliens.

Interest Groups - students work in small groups to research a math topic of interest such as how geometry applies to architecture or how math is used in art.

Flexible Grouping Focus of Differentiation: Readiness, Interest, Learning Profiles

Example:The teacher may assign groups based on readiness for direct instruction on algebraic concepts and allow students to choose their own groups for projects that investigate famous mathematicians.

Learning Contracts Focus of Differentiation: Readiness, Learning Profile

Example:A student decides to follow a football team over a 2-month period and make inferences about players’ performances based on scoring patterns. The student develops a plan for collecting and analyzing data and present to class.

Choice Boards Focus of Differentiation: Readiness, Interest, Learning Profile

Example:Students can choose to complete an inquiry lesson where they measure volume using various containers, use a textbook to read about measuring volume, or watch a video where steps are explained. The activities are based on visual, auditory, kinesthetic, and tactile learning. Students must complete two activities from the board and choose these activities from two different learning styles.

Acceleration/Deceleration Focus of Differentiation: ReadinessExample:Students demonstrating a high level of competence can work through the curriculum at a faster pace. Students experiencing difficulties may need adjusted activities that allow for a slower pace in order to experience success.

Independent Study Projects Focus of Differentiation: Readiness, Interest, and Learning Profiles

Example:Assign a research project where students learn how to develop the skills for independent learning. The degree of help and structure will vary between students and depend on their ability to manage ideas, time, and productivity.

Peer Teaching Focus of Differentiation: Readiness, Interest, Learning Profiles

Example:Occasionally, a student may have personal needs that require one-on-one instruction that go beyond the needs of peers. After receiving this instruction, the student could be designated as the “resident expert” for that concept or skill and get practice by teaching others. Both students would benefit.

Anchoring Activities Focus of Differentiation: ReadinessExample:List of activities that a student can do at any time when they have completed present assignments

Short assignments at the beginning of each class as students organize themselves and prepare for work.

**These activities must be worth of a student’s time and appropriate to learning needs.**

Tier II/III - Math Instruction Direct/Explicit Instruction Components of Number Sense Use of appropriate LANGUAGE Concrete-Representational-Abstract Approach (CRA)

Small group/individualized instruction

Frequent monitoring of progress

IES Recommendations - Tier II, III

Interventions should: Include instruction on how to solve word problems that is based on common underlying structures

Intervention materials need to include opportunities for students to work with visual representations of math ideas

Interventions at all grade levels should devote about 10 minutes to building fluent retrieval of basic arithmetic facts!!!!!

IES Recommendations, cont. Instructional materials should:

Focus on in-depth treatment of:•Whole numbers in grades K-5 and•Rational numbers in grades 4-8

Monitor progress of students receiving supplemental instruction

Include motivational strategies in Tier II and Tier III

Tier II/III Intervention Instruction http://dww.ed.gov/practice/?T_ID=28&P_ID=71

The multimedia overview discusses characteristics of effective Tier 2 and Tier 3 instruction, offering examples of systematic and explicit instruction, guided practice, and feedback. It also details the use of concrete materials and visual representations to develop abstract concepts as well as the role of motivation in learning math.

RtI: Tier II: Intervention (Secondary Prevention)

Focus

Instruction

Interventionist

Grouping

Time

Assessment

For students identified with mathematics difficulties

Personnel determined by the school (e.g., classroom teacher, mathematics interventionist)

Small group instruction (no more than 1:6)

30 minutes per day/2-3 days per week in addition to 60 minutes of mathematics instruction

2 times per month using AIMSweb Early Numeracy,MCOMP, or MCAP

Standard research-validated intervention protocol

Fidelity Observations conducted on fidelity of implementation

RtI: Tier III: Intervention (Tertiary Prevention)

Focus

Instruction

Interventionist

Grouping

Time

Assessment

For students identified with mathematics difficulties whoHave NOT responded to Tier II instruction based on data.

Personnel determined by the school (e.g., classroom teacher, mathematics interventionist)

Small group instruction (no more than 1:3)

30 minutes per day; 3 days per week in addition to 60 minutes of mathematics instruction

2 times per week using AIMSweb Early Numeracy,MCOMP, or MCAP

Standard research-validated intervention protocol

Fidelity Observations conducted on fidelity of implementation

Concrete-Representational-Abstract (C-R-A)



Research-based Tier II instruction for small groups

C-R-A Approach

CONCRETE REPRESENTATIONAL ABSTRACT Hands-on physical modelsManipulatives to represent numbers

Semi-concreteDraws or uses pictures of the models

Numbers as abstract symbols of picture displays

(Witzell, Smith, and Brownell, 2001)

Concrete Phase Students provided with manipulatives and other materials that will provide them opportunity to explore a math concept or process by doing it with the tools.

This is the stage of “getting their hands dirty” with the intent that having an actual experience will enable the construction of the knowledge being targeted.

Concrete Modeling Examples







Representational Phase Students begin to develop mental images of the manipulatives by drawing on other means for understanding the target knowledge.

Students are encouraged at this time to step back from the manipulatives and other concrete tools and focus on the concept involved in performing actions with the tools.

Representational Example



Abstract Phase Students could manipulate concepts or processes in the absence of the tools that were important in the early phase of learning.

Abstract Example



Moving from Concrete to Representational Students work on computer to use virtual manipulatives to solve problems

Use self-dialogue or think-aloud strategies

Provide songs, rhymes, or rhythms to help remember facts

Chunk material (i.e. fact families)

Moving from Representational to Abstract Work with student individually or in small group to ensure understanding

Fade use of guided worksheets to prepare for abstract stage

Use cooperative groups and encourage self-talk and talking to others

Provide opportunity for practice

What Does C-R-A look like?Title: Sharing Pizzas (grades 1-6)1. Gather appropriate materials2. Divide students into groups of 43. Ask students to show how they would

share one pizza using manipulatives4. Encourage students to describe and

draw their thinking to create equal shares

5. Ask students to show how they would share two pizzas with the group

6. Repeat for three, four, and five pizzas

(Compiled by Bryan, Horn, Jones, 2008)

Direct/Explicit Instruction Teacher-centered instructional approach

Focuses on “what” to teach and “how”

Can be a scripted program Number Worlds

Very systematic Step-by-step format Fast-paced instruction

(National Council of Teachers of Mathematics)



Example of Direct Instruction





http://www.k8accesscenter.org/training_resources/DirectExplicitInstruction_Mathematics.asp

What does the research say? Students receive immediate feedback Continuous modeling by teachers which fades as mastery is acquired

Often used in small groups (appropriate for Tier II and III interventions)

Effective for low achieving students and Exceptional Children students

Cover, Copy, Compare Practice visualizing and computing through a sequence of easy-to-remember steps

Allows for immediate, corrective feedback

Modeling Videohttp://www.ecu.edu/cs-cas/psyc/rileytillmant/Cover-Copy-Compare-Modeling-Video-1.cfm

Measuring Progress Curriculum Based Measurement

AIMSweb•MCOMP (grades 1-5)•MCAP (grades 2-5)•Early Numeracy (K-1)

• Number Identification• Oral Counting• Quantity Discrimination• Missing Number

AIMSweb MCAP Domains & NCTM Content Areas



DISCLAIMER!!!! We do NOT have the same correlations as we have with reading curriculum based assessments

We need to be CAUTIOUS about applying what we know about Reading to the world of Mathematics

In Math, we must think about what we are measuring. We are not in the same place (with regard to research) as we are in Reading.

Collecting Strong DataMULTIPLICATION: 4-digit number times 2-digit number: with regrouping

7083x 5749581354150403,731

6 correct digits = 6 points

Tier III Positive Response





Tier III Negative Response

Making “sense” of Math. . .

Ma and Pa Kettle Math video



Comments/Questions? ? ? ?

responsiveness to instruction: mathematical understanding rowan-salisbury schools rti foundations...

Documents

math content slide

9a7ef76e53a5e67c slide

south africa slide

program specialist slide

grade students

high level content slide

math concepts

nc math results