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Math Reasoning Assisting the Struggling Middle

and High School LearnerJim Wrightwww.interventi

oncentral.org

Download PowerPoint from this workshop

athttp//www.interventioncentral.org/SSTAGE.php

Intervention Research Development A Work in

Progress

Georgia Pyramid of Intervention

Source Georgia Dept of Education

http//www.doe.k12.ga.us/ Retrieved 13 July 2007

An RTI Challenge Limited Research to Support

Evidence-Based Math Interventions

- in contrast to reading, core math programs

that are supported by research, or that have been

constructed according to clear research-based

principles, are not easy to identify. Not only

have exemplary core programs not been identified,

but also there are no tools available that we

know of that will help schools analyze core math

programs to determine their alignment with clear

research-based principles. p. 459

Source Clarke, B., Baker, S., Chard, D.

(2008). Best practices in mathematics assessment

and intervention with elementary students. In A.

Thomas J. Grimes (Eds.), Best practices in

school psychology V (pp. 453-463).

Tier 1 What Are the Recommended Elements of

Core Curriculum? More Research Needed

- In essence, we now have a good beginning on the

evaluation of Tier 2 and 3 interventions, but no

idea about what it will take to get the core

curriculum to work at Tier 1. A complicating

issue with this potential line of research is

that many schools use multiple materials as their

core program. p. 640

Source Kovaleski, J. F. (2007). Response to

intervention Considerations for research and

systems change. School Psychology Review, 36,

638-646.

Limitations of Intervention Research

- the list of evidence-based interventions is

quite small relative to the need of RTI. Thus,

limited dissemination of interventions is likely

to be a practical problem as individuals move

forward in the application of RTI models in

applied settings. p. 33

Source Kratochwill, T. R., Clements, M. A.,

Kalymon, K. M. (2007). Response to intervention

Conceptual and methodological issues in

implementation. In Jimerson, S. R., Burns, M. K.,

VanDerHeyden, A. M. (Eds.), Handbook of

response to intervention The science and

practice of assessment and intervention. New

York Springer.

Schools Need to Review Tier 1 (Classroom)

Interventions to Ensure That They Are Supported

By Research

- There is a lack of agreement about what is meant

by scientifically validated classroom (Tier I)

interventions. Districts should establish a

vetting processcriteria for judging whether a

particular instructional or intervention approach

should be considered empirically based.

Source Fuchs, D., Deshler, D. D. (2007). What

we need to know about responsiveness to

intervention (and shouldnt be afraid to ask)..

Learning Disabilities Research Practice,

22(2),129136.

What Are Appropriate Content-Area Tier 1

Universal Interventions for Secondary Schools?

- High schools need to determine what constitutes

high-quality universal instruction across content

areas. In addition, high school teachers need

professional development in, for example,

differentiated instructional techniques that will

help ensure student access to instruction

interventions that are effectively implemented.

Source Duffy, H. (August 2007). Meeting the

needs of significantly struggling learners in

high school. Washington, DC National High School

Center. Retrieved from http//www.betterhighschool

s.org/pubs/ p. 9

Intervention Key Concepts

Big Ideas The Four Stages of Learning Can Be

Summed Up in the Instructional Hierarchy

(Haring et al., 1978)

- Student learning can be thought of as a

multi-stage process. The universal stages of

learning include - Acquisition The student is just acquiring the

skill. - Fluency The student can perform the skill but

must make that skill automatic. - Generalization The student must perform the

skill across situations or settings. - Adaptation The student confronts novel task

demands that require that the student adapt a

current skill to meet new requirements.

Source Haring, N.G., Lovitt, T.C., Eaton, M.D.,

Hansen, C.L. (1978). The fourth R Research in

the classroom. Columbus, OH Charles E. Merrill

Publishing Co.

Scripting Interventions to Promote Better

Compliance

- Interventions should be written up in a

scripted format to ensure that - Teachers have sufficient information about the

intervention to implement it correctly and - External observers can view the teacher

implementing the intervention strategy andusing

the script as a checklistverify that each step

of the intervention was implemented correctly

(treatment integrity).

Source Burns, M. K., Gibbons, K. A. (2008).

Implementing response-to-intervention in

elementary and secondary schools. Routledge New

York.

Intervention Script Builder Form

Increasing the Intensity of an Intervention Key

Dimensions

- Interventions can move up the RTI Tiers through

being intensified across several dimensions,

including - Type of intervention strategy or materials used
- Student-teacher ratio
- Length of intervention sessions
- Frequency of intervention sessions
- Duration of the intervention period (e.g.,

extending an intervention from 5 weeks to 10

weeks) - Motivation strategies

Source Burns, M. K., Gibbons, K. A. (2008).

Implementing response-to-intervention in

elementary and secondary schools. Routledge New

York. Kratochwill, T. R., Clements, M. A.,

Kalymon, K. M. (2007). Response to intervention

Conceptual and methodological issues in

implementation. In Jimerson, S. R., Burns, M. K.,

VanDerHeyden, A. M. (Eds.), Handbook of

response to intervention The science and

practice of assessment and intervention. New

York Springer.

Research-Based Elements of Effective Academic

Interventions

- Correctly targeted The intervention is

appropriately matched to the students academic

or behavioral needs. - Explicit instruction Student skills have been

broken down into manageable and deliberately

sequenced steps and providing overt strategies

for students to learn and practice new skills

p.1153 - Appropriate level of challenge The student

experiences adequate success with the

instructional task. - High opportunity to respond The student

actively responds at a rate frequent enough to

promote effective learning. - Feedback The student receives prompt

performance feedback about the work completed.

Source Burns, M. K., VanDerHeyden, A. M.,

Boice, C. H. (2008). Best practices in intensive

academic interventions. In A. Thomas J. Grimes

(Eds.), Best practices in school psychology V

(pp.1151-1162). Bethesda, MD National

Association of School Psychologists.

Core Instruction, Interventions, Accommodations

Modifications Sorting Them Out

- Core Instruction. Those instructional strategies

that are used routinely with all students in a

general-education setting are considered core

instruction. High-quality instruction is

essential and forms the foundation of RTI

academic support. NOTE While it is important to

verify that good core instructional practices are

in place for a struggling student, those routine

practices do not count as individual student

interventions.

Core Instruction, Interventions, Accommodations

Modifications Sorting Them Out

- Intervention. An academic intervention is a

strategy used to teach a new skill, build fluency

in a skill, or encourage a child to apply an

existing skill to new situations or settings. An

intervention can be thought of as a set of

actions that, when taken, have demonstrated

ability to change a fixed educational trajectory

(Methe Riley-Tillman, 2008 p. 37).

Core Instruction, Interventions, Accommodations

Modifications Sorting Them Out

- Accommodation. An accommodation is intended to

help the student to fully access and participate

in the general-education curriculum without

changing the instructional content and without

reducing the students rate of learning (Skinner,

Pappas Davis, 2005). An accommodation is

intended to remove barriers to learning while

still expecting that students will master the

same instructional content as their typical

peers. - Accommodation example 1 Students are allowed to

supplement silent reading of a novel by listening

to the book on tape. - Accommodation example 2 For unmotivated

students, the instructor breaks larger

assignments into smaller chunks and providing

students with performance feedback and praise for

each completed chunk of assigned work (Skinner,

Pappas Davis, 2005).

Core Instruction, Interventions, Accommodations

Modifications Sorting Them Out

- Modification. A modification changes the

expectations of what a student is expected to

know or dotypically by lowering the academic

standards against which the student is to be

evaluated. Examples of modifications - Giving a student five math computation problems

for practice instead of the 20 problems assigned

to the rest of the class - Letting the student consult course notes during a

test when peers are not permitted to do so - Allowing a student to select a much easier book

for a book report than would be allowed to his or

her classmates.

Intervention Footprint 7-Step Lifecycle of an

Intervention Plan

- Information about the students academic or

behavioral concerns is collected. - The intervention plan is developed to match

student presenting concerns. - Preparations are made to implement the plan.
- The plan begins.
- The integrity of the plans implementation is

measured. - Formative data is collected to evaluate the

plans effectiveness. - The plan is discontinued, modified, or replaced.

Interventions Potential Fatal Flaws

- Any intervention must include 4 essential

elements. The absence of any one of the elements

would be considered a fatal flaw (Witt,

VanDerHeyden Gilbertson, 2004) that blocks the

school from drawing meaningful conclusions from

the students response to the intervention - Clearly defined problem. The students target

concern is stated in specific, observable,

measureable terms. This problem identification

statement is the most important step of the

problem-solving model (Bergan, 1995), as a

clearly defined problem allows the teacher or RTI

Team to select a well-matched intervention to

address it. - Baseline data. The teacher or RTI Team measures

the students academic skills in the target

concern (e.g., reading fluency, math computation)

prior to beginning the intervention. Baseline

data becomes the point of comparison throughout

the intervention to help the school to determine

whether that intervention is effective. - Performance goal. The teacher or RTI Team sets a

specific, data-based goal for student improvement

during the intervention and a checkpoint date by

which the goal should be attained. - Progress-monitoring plan. The teacher or RTI Team

collects student data regularly to determine

whether the student is on-track to reach the

performance goal.

Source Witt, J. C., VanDerHeyden, A. M.,

Gilbertson, D. (2004). Troubleshooting behavioral

interventions. A systematic process for finding

and eliminating problems. School Psychology

Review, 33, 363-383.

How Do We Reach Low-Performing Math Students?

Instructional Recommendations

- Important elements of math instruction for

low-performing students - Providing teachers and students with data on

student performance - Using peers as tutors or instructional guides
- Providing clear, specific feedback to parents on

their childrens mathematics success - Using principles of explicit instruction in

teaching math concepts and procedures. p. 51

Source Baker, S., Gersten, R., Lee, D.

(2002).A synthesis of empirical research on

teaching mathematics to low-achieving students.

The Elementary School Journal, 103(1), 51-73..

Profile of Students With Significant Math

Difficulties

- Spatial organization. The student commits errors

such as misaligning numbers in columns in a

multiplication problem or confusing

directionality in a subtraction problem (and

subtracting the original numberminuendfrom the

figure to be subtracted (subtrahend). - Visual detail. The student misreads a

mathematical sign or leaves out a decimal or

dollar sign in the answer. - Procedural errors. The student skips or adds a

step in a computation sequence. Or the student

misapplies a learned rule from one arithmetic

procedure when completing another, different

arithmetic procedure. - Inability to shift psychological set. The

student does not shift from one operation type

(e.g., addition) to another (e.g.,

multiplication) when warranted. - Graphomotor. The students poor handwriting can

cause him or her to misread handwritten numbers,

leading to errors in computation. - Memory. The student fails to remember a specific

math fact needed to solve a problem. (The student

may KNOW the math fact but not be able to recall

it at point of performance.) - Judgment and reasoning. The student comes up with

solutions to problems that are clearly

unreasonable. However, the student is not able

adequately to evaluate those responses to gauge

whether they actually make sense in context.

Source Rourke, B. P. (1993). Arithmetic

disabilities, specific otherwise A

neuropsychological perspective. Journal of

Learning Disabilities, 26, 214-226.

Team Activity Define Math Reasoning

- At your table
- Appoint a recorder/spokesperson.
- Discuss the term math reasoning at the

secondary level. Task-analyze the term and break

it down into the essential subskills. - Be prepared to report out on your work.
- What is the role of the Student Support Team in

assisting teachers to promote math reasoning?

Assisting Students in Accessing Contextual,

Conceptual, Procedural Knowledge When Solving

Math Problems

- Well-structured, organized knowledge allows

people to solve novel problems and to remember

more information than do memorized facts or

procedures... Such well-structured knowledge

requires that people integrate their contextual,

conceptual and procedural knowledge in a domain.

Unfortunately, U.S. students rarely have such

integrated and robust knowledge in mathematics or

science. Designing learning environments that

support integrated knowledge is a key challenge

for the field, especially given the low number of

established tools for guiding this design

process. p. 313

Source Rittle-Johnson, B., Koedinger, K. R.

(2005). Designing knowledge scaffolds to support

mathematical problem-solving. Cognition and

Instruction, 23(3), 313349.

Types of Knowledge Definitions

- Conceptual Knowledge integrated knowledge of

important principles (e.g., knowledge of number

magnitudes) that can be flexibly applied to new

tasks. Conceptual knowledge can be used to guide

comprehension of problems and to generate new

problem-solving strategies or to adapt existing

strategies to solve novel problems. p. 317

Source Rittle-Johnson, B., Koedinger, K. R.

(2005). Designing knowledge scaffolds to support

mathematical problem-solving. Cognition and

Instruction, 23(3), 313349.

Types of Knowledge Definitions

- Procedural Knowledge knowledge of

subcomponents of a correct procedure. Procedures

are a type of strategy that involve step-by-step

actions for solving problems, and most procedures

require integration of multiple skills. For

example, the conventional procedure for adding

fractions with unlike denominators requires

knowing how to find a common denominator, how to

find equivalent fractions, and how to add

fractions with like denominators. 318

Source Rittle-Johnson, B., Koedinger, K. R.

(2005). Designing knowledge scaffolds to support

mathematical problem-solving. Cognition and

Instruction, 23(3), 313349.

Types of Knowledge Definitions

- Contextual Knowledge our knowledge of how

things work in specific, real-world situations,

which develops from our everyday, informal

interactions with the world. Students contextual

knowledge can be elicited by situating problems

in story contexts. p. 316

Source Rittle-Johnson, B., Koedinger, K. R.

(2005). Designing knowledge scaffolds to support

mathematical problem-solving. Cognition and

Instruction, 23(3), 313349.

Math Problem Scaffolding Examples (Modeled after

Rittle-Johnson Koedinger, 2005)

Source Rittle-Johnson, B., Koedinger, K. R.

(2005). Designing knowledge scaffolds to support

mathematical problem-solving. Cognition and

Instruction, 23(3), 313349.

Math Problem Scaffolding Examples (Modeled after

Rittle-Johnson Koedinger, 2005)

Source Rittle-Johnson, B., Koedinger, K. R.

(2005). Designing knowledge scaffolds to support

mathematical problem-solving. Cognition and

Instruction, 23(3), 313349.

Leveraging the Power of Contextual Knowledge in

Story Problems Use Familiar Student Contexts

- Past research on fraction learning indicates

that food contexts are particularly meaningful

contexts for students (Mack, 1990, 1993). p. 319

Source Rittle-Johnson, B., Koedinger, K. R.

(2005). Designing knowledge scaffolds to support

mathematical problem-solving. Cognition and

Instruction, 23(3), 313349.

Math Problem Scaffolding Examples (Modeled after

Rittle-Johnson Koedinger, 2005)

Source Rittle-Johnson, B., Koedinger, K. R.

(2005). Designing knowledge scaffolds to support

mathematical problem-solving. Cognition and

Instruction, 23(3), 313349.

Math Problem Scaffolding Examples (Modeled after

Rittle-Johnson Koedinger, 2005)

Source Rittle-Johnson, B., Koedinger, K. R.

(2005). Designing knowledge scaffolds to support

mathematical problem-solving. Cognition and

Instruction, 23(3), 313349.

Research is Unclear Whether Math Problems in

Story or Symbolic Format Are More Difficult

- The reason for contradictory findings about the

relative difficulty of math problems in story or

symbolic format may be explained by

grade-specific challenges in math. First,

young children have pervasive exposure to

single-digit numerals, but some words and

syntactic forms are still unknown or unfamiliar

explaining why younger students may find story

problems more challenging. In comparison, older

children have less exposure to large, multidigit

numerals and algebraic symbols and have much

better reading and comprehension skills

explaining why older students may find symbolic

problems more challenging. p. 317

Source Rittle-Johnson, B., Koedinger, K. R.

(2005). Designing knowledge scaffolds to support

mathematical problem-solving. Cognition and

Instruction, 23(3), 313349.

Strands of Math Proficiency

- 5 Strands of Mathematical Proficiency
- Understanding
- Computing
- Applying
- Reasoning
- Engagement

Source National Research Council. (2002).

Helping children learn mathematics. Mathematics

Learning Study Committee, J. Kilpatrick J.

Swafford, Editors, Center for Education, Division

of Behavioral and Social Sciences and Education.

Washington, DC National Academy Press.

Five Strands of Mathematical Proficiency

- Understanding Comprehending mathematical

concepts, operations, and relations--knowing what

mathematical symbols, diagrams, and procedures

mean. - Computing Carrying out mathematical procedures,

such as adding, subtracting, multiplying, and

dividing numbers flexibly, accurately,

efficiently, and appropriately. - Applying Being able to formulate problems

mathematically and to devise strategies for

solving them using concepts and procedures

appropriately.

Source National Research Council. (2002).

Helping children learn mathematics. Mathematics

Learning Study Committee, J. Kilpatrick J.

Swafford, Editors, Center for Education, Division

of Behavioral and Social Sciences and Education.

Washington, DC National Academy Press.

Five Strands of Mathematical Proficiency (Cont.)

- Reasoning Using logic to explain and justify a

solution to a problem or to extend from something

known to something less known. - Engaging Seeing mathematics as sensible, useful,

and doableif you work at itand being willing to

do the work.

Source National Research Council. (2002).

Helping children learn mathematics. Mathematics

Learning Study Committee, J. Kilpatrick J.

Swafford, Editors, Center for Education, Division

of Behavioral and Social Sciences and Education.

Washington, DC National Academy Press.

Five Strands of Mathematical Proficiency (NRC,

2002)

- Understanding Comprehending mathematical

concepts, operations, and relations--knowing what

mathematical symbols, diagrams, and procedures

mean. - Computing Carrying out mathematical procedures,

such as adding, subtracting, multiplying, and

dividing numbers flexibly, accurately,

efficiently, and appropriately. - Applying Being able to formulate problems

mathematically and to devise strategies for

solving them using concepts and procedures

appropriately. - Reasoning Using logic to explain and justify a

solution to a problem or to extend from something

known to something less known. - Engaging Seeing mathematics as sensible, useful,

and doableif you work at itand being willing to

do the work.

Motivation

Five Strands of Mathematical Proficiency (NRC,

2002)

- Table Activity Evaluate Your Schools Math

Proficiency - As a group, review the National Research Council

Strands of Math Proficiency. - Which strand do you feel that your school /

curriculum does the best job of helping students

to attain proficiency? - Which strand do you feel that your school /

curriculum should put the greatest effort to

figure out how to help students to attain

proficiency? - Be prepared to share your results.

- Understanding Comprehending mathematical

concepts, operations, and relations--knowing what

mathematical symbols, diagrams, and procedures

mean. - Computing Carrying out mathematical procedures,

such as adding, subtracting, multiplying, and

dividing numbers flexibly, accurately,

efficiently, and appropriately. - Applying Being able to formulate problems

mathematically and to devise strategies for

solving them using concepts and procedures

appropriately. - Reasoning Using logic to explain and justify a

solution to a problem or to extend from something

known to something less known. - Engaging Seeing mathematics as sensible, useful,

and doableif you work at itand being willing to

do the work.

RTI Secondary LiteracyExplicit Vocabulary

Instruction

Comprehending Math Vocabulary The Barrier of

Abstraction

- when it comes to abstract

mathematical concepts, words describe activities

or relationships that often lack a visual

counterpart. Yet studies show that children grasp

the idea of quantity, as well as other relational

concepts, from a very early age. As children

develop their capacity for understanding,

language, and its vocabulary, becomes a vital

cognitive link between a childs natural sense of

number and order and conceptual learning. - -Chard, D. (n.d.)

Source Chard, D. (n.d.. Vocabulary strategies

for the mathematics classroom. Retrieved November

23, 2007, from http//www.eduplace.com/state/pdf/a

uthor/chard_hmm05.pdf.

Math Vocabulary Classroom (Tier I)

Recommendations

- Preteach math vocabulary. Math vocabulary

provides students with the language tools to

grasp abstract mathematical concepts and to

explain their own reasoning. Therefore, do not

wait to teach that vocabulary only at point of

use. Instead, preview relevant math vocabulary

as a regular a part of the background

information that students receive in preparation

to learn new math concepts or operations. - Model the relevant vocabulary when new concepts

are taught. Strengthen students grasp of new

vocabulary by reviewing a number of math problems

with the class, each time consistently and

explicitly modeling the use of appropriate

vocabulary to describe the concepts being taught.

Then have students engage in cooperative learning

or individual practice activities in which they

too must successfully use the new

vocabularywhile the teacher provides targeted

support to students as needed. - Ensure that students learn standard, widely

accepted labels for common math terms and

operations and that they use them consistently to

describe their math problem-solving efforts.

Source Chard, D. (n.d.. Vocabulary strategies

for the mathematics classroom. Retrieved November

23, 2007, from http//www.eduplace.com/state/pdf/a

uthor/chard_hmm05.pdf.

Promoting Math Vocabulary Other Guidelines

- Create a standard list of math vocabulary for

each grade level (elementary) or course/subject

area (for example, geometry). - Periodically check students mastery of math

vocabulary (e.g., through quizzes, math journals,

guided discussion, etc.). - Assist students in learning new math vocabulary

by first assessing their previous knowledge of

vocabulary terms (e.g., protractor product) and

then using that past knowledge to build an

understanding of the term. - For particular assignments, have students

identify math vocabulary that they dont

understand. In a cooperative learning activity,

have students discuss the terms. Then review any

remaining vocabulary questions with the entire

class. - Encourage students to use a math dictionary in

their vocabulary work. - Make vocabulary a central part of instruction,

curriculum, and assessmentrather than treating

as an afterthought.

Source Adams, T. L. (2003). Reading mathematics

More than words can say. The Reading Teacher,

56(8), 786-795.

Vocabulary Why This Instructional Goal is

Important

- As vocabulary terms become more specialized in

content area courses, students are less able to

derive the meaning of unfamiliar words from

context alone. - Students must instead learn vocabulary through

more direct means, including having opportunities

to explicitly memorize words and their

definitions. - Students may require 12 to 17 meaningful

exposures to a word to learn it.

Enhance Vocabulary Instruction Through Use of

Graphic Organizers or Displays A Sampling

- Teachers can use graphic displays to structure

their vocabulary discussions and activities

(Boardman et al., 2008 Fisher, 2007 Texas

Reading Initiative, 2002).

4-Square Graphic Display

- The student divides a page into four quadrants.

In the upper left section, the student writes the

target word. In the lower left section, the

student writes the word definition. In the upper

right section, the student generates a list of

examples that illustrate the term, and in the

lower right section, the student writes

non-examples (e.g., terms that are the opposite

of the target vocabulary word).

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Semantic Word Definition Map

- The graphic display contains sections in which

the student writes the word, its definition

(what is this?), additional details that extend

its meaning (What is it like?), as well as a

listing of examples and non-examples (e.g.,

terms that are the opposite of the target

vocabulary word).

Word Definition Map Example

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Semantic Feature Analysis

- A target vocabulary term is selected for

analysis in this grid-like graphic display.

Possible features or properties of the term

appear along the top margin, while examples of

the term are listed ion the left margin. The

student considers the vocabulary term and its

definition. Then the student evaluates each

example of the term to determine whether it does

or does not match each possible term property or

element.

Semantic Feature Analysis Example

- VOCABULARY TERM TRANSPORTATION

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Comparison/Contrast (Venn) Diagram

- Two terms are listed and defined. For each term,

the student brainstorms qualities or properties

or examples that illustrate the terms meaning.

Then the student groups those qualities,

properties, and examples into 3 sections - items unique to Term 1
- items unique to Term 2
- items shared by both terms

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Provide Regular In-Class Instruction and Review

of Vocabulary Terms, Definitions

- Present important new vocabulary terms in class,

along with student-friendly definitions. Provide

example sentences/contextual sentences to

illustrate the use of the term. Assign students

to write example sentences employing new

vocabulary to illustrate their mastery of the

terms.

Generate Possible Sentences

- The teacher selects 6 to 8 challenging new

vocabulary terms and 4 to 6 easier, more familiar

vocabulary items relevant to the lesson.

Introduce the vocabulary terms to the class. Have

students write sentences that contain at least

two words from the posted vocabulary list. Then

write examples of student sentences on the board

until all words from the list have been used.

After the assigned reading, review the possible

sentences that were previously generated.

Evaluate as a group whether, based on the

passage, the sentence is possible (true) in its

current form. If needed, have the group recommend

how to change the sentence to make it possible.

Provide Dictionary Training

- The student is trained to use an Internet lookup

strategy to better understand dictionary or

glossary definitions of key vocabulary items. - The student first looks up the word and its

meaning(s) in the dictionary/glossary. - If necessary, the student isolates the specific

word meaning that appears to be the appropriate

match for the term as it appears in course texts

and discussion. - The student goes to an Internet search engine

(e.g., Google) and locates at least five text

samples in which the term is used in context and

appears to match the selected dictionary

definition.

RTI Secondary LiteracyExtended Discussion

Extended Discussions Why This Instructional Goal

is Important

- Extended, guided group discussion is a powerful

means to help students to learn vocabulary and

advanced concepts. Discussion can also model for

students various thinking processes and

cognitive strategies (Kamil et al. 2008, p. 22).

To be effective, guided discussion should go

beyond students answering a series of factual

questions posed by the teacher Quality

discussions are typically open-ended and

exploratory in nature, allowing for multiple

points of view (Kamil et al., 2008). - When group discussion is used regularly and

well in instruction, students show increased

growth in literacy skills. Content-area teachers

can use it to demonstrate the habits of mind

and patterns of thinking of experts in various

their discipline e.g., historians,

mathematicians, chemists, engineers, literacy

critics, etc.

Use a Standard Protocol to Structure Extended

Discussions

- Good extended classwide discussions elicit a

wide range of student opinions, subject

individual viewpoints to critical scrutiny in a

supportive manner, put forth alternative views,

and bring closure by summarizing the main points

of the discussion. Teachers can use a simple

structure to effectively and reliably organize

their discussions

Standard Protocol Discussion Format

- Pose questions to the class that require students

to explain their positions and their reasoning . - When needed, think aloud as the discussion

leader to model good reasoning practices (e.g.,

taking a clear stand on a topic). - Supportively challenge student views by offering

possible counter arguments. - Single out and mention examples of effective

student reasoning. - Avoid being overly directive the purpose of

extended discussions is to more fully investigate

and think about complex topics. - Sum up the general ground covered in the

discussion and highlight the main ideas covered.

RTI Secondary LiteracyReading Comprehension

Reading Comprehension Why This Instructional

Goal is Important

- Students require strong reading comprehension

skills to succeed in challenging content-area

classes.At present, there is no clear evidence

that any one reading comprehension instructional

technique is clearly superior to others. In fact,

it appears that students benefit from being

taught any self-directed practice that prompts

them to engage more actively in understanding the

meaning of text (Kamil et al., 2008).

Assist Students in Setting Content Goals for

Reading

- Students are more likely to be motivated to

read--and to read more closelyif they have

specific content-related reading goals in mind.

At the start of a reading assignment, for

example, the instructor has students state what

questions they might seek to answer or what

topics they would like to learn more about in

their reading. The student or teacher writes down

these questions. After students have completed

the assignee reading, they review their original

questions and share what they have learned (e.g.,

through discussion in large group or cooperative

learning group, or even as a written assignment).

Teach Students to Monitor Their Own Comprehension

and Apply Fix-Up Skills

- Teachers can teach students specific strategies

to monitor their understanding of text and

independently use fix-up skills as needed.

Examples of student monitoring and repair skills

for reading comprehension include encouraging

them to - Stop after every paragraph to summarize its main

idea - Reread the sentence or paragraph again if

necessary - Generate and write down questions that arise

during reading - Restate challenging or confusing ideas or

concepts from the text in the students own words

Teach Students to Identify Underlying Structures

of Math Problems

Algebra Word Problems Predictable Elements

- Most algebra problems contain predictable

elements - Assignment statements Assign a particular

numerical value to some variable. - Relational statements Specify a single

relationship between two variables. - Questions Involve the requested solution

(e.g., What is X?). - Relevant facts Any other type of information

that might be useful for solving the problem. - Problem translation The process of taking each

of these forms of information and using them to

develop corresponding algebraic equations. - The most common problems were noted
- Students may fail to discriminate relevant from

irrelevant information. - Students may commit translation errors when

processing relational statements.

Source National Mathematics Advisory Panel.

(2008). Foundations for success The final Report

of the National Mathematics Advisory Panel

Chapter 4 Report of the Task Group on Learning

Processes. U.S. Department of Education

Washington, DC. Retrieved from http//www.ed.gov/a

bout/bdscomm/list/mathpanel/reports.html

Developing Student Metacognitive Abilities

Importance of Metacognitive Strategy Use

- Metacognitive processes focus on self-awareness

of cognitive knowledge that is presumed to be

necessary for effective problem solving, and they

direct and regulate cognitive processes and

strategies during problem solvingThat is,

successful problem solvers, consciously or

unconsciously (depending on task demands), use

self-instruction, self-questioning, and

self-monitoring to gain access to strategic

knowledge, guide execution of strategies, and

regulate use of strategies and problem-solving

performance. p. 231

Source Montague, M. (1992). The effects of

cognitive and metacognitive strategy instruction

on the mathematical problem solving of middle

school students with learning disabilities.

Journal of Learning Disabilities, 25, 230-248.

Elements of Metacognitive Processes

- Self-instruction helps students to identify and

direct the problem-solving strategies prior to

execution. Self-questioning promotes internal

dialogue for systematically analyzing problem

information and regulating execution of cognitive

strategies. Self-monitoring promotes appropriate

use of specific strategies and encourages

students to monitor general performance.

Emphasis added. p. 231

Source Montague, M. (1992). The effects of

cognitive and metacognitive strategy instruction

on the mathematical problem solving of middle

school students with learning disabilities.

Journal of Learning Disabilities, 25, 230-248.

Combining Cognitive Metacognitive Strategies to

Assist Students With Mathematical Problem Solving

- Solving an advanced math problem independently

requires the coordination of a number of complex

skills. The following strategies combine both

cognitive and metacognitive elements (Montague,

1992 Montague Dietz, 2009). First, the student

is taught a 7-step process for attacking a math

word problem (cognitive strategy). Second, the

instructor trains the student to use a three-part

self-coaching routine for each of the seven

problem-solving steps (metacognitive strategy).

Cognitive Portion of Combined Problem Solving

Approach

- In the cognitive part of this multi-strategy

intervention, the student learns an explicit

series of steps to analyze and solve a math

problem. Those steps include - Reading the problem. The student reads the

problem carefully, noting and attempting to clear

up any areas of uncertainly or confusion (e.g.,

unknown vocabulary terms). - Paraphrasing the problem. The student restates

the problem in his or her own words. - Drawing the problem. The student creates a

drawing of the problem, creating a visual

representation of the word problem. - Creating a plan to solve the problem. The student

decides on the best way to solve the problem and

develops a plan to do so. - Predicting/Estimating the answer. The student

estimates or predicts what the answer to the

problem will be. The student may compute a quick

approximation of the answer, using rounding or

other shortcuts. - Computing the answer. The student follows the

plan developed earlier to compute the answer to

the problem. - Checking the answer. The student methodically

checks the calculations for each step of the

problem. The student also compares the actual

answer to the estimated answer calculated in a

previous step to ensure that there is general

agreement between the two values.

Metacognitive Portion of Combined Problem Solving

Approach

- The metacognitive component of the intervention

is a three-part routine that follows a sequence

of Say, Ask, Check. For each of the 7

problem-solving steps reviewed above - The student first self-instructs by stating, or

saying, the purpose of the step (Say). - The student next self-questions by asking what

he or she intends to do to complete the step

(Ask). - The student concludes the step by

self-monitoring, or checking, the successful

completion of the step (Check).

Combined Cognitive Metacognitive Elements of

Strategy

Combined Cognitive Metacognitive Elements of

Strategy

Combined Cognitive Metacognitive Elements of

Strategy

Combined Cognitive Metacognitive Elements of

Strategy

Combined Cognitive Metacognitive Elements of

Strategy

Combined Cognitive Metacognitive Elements of

Strategy

Combined Cognitive Metacognitive Elements of

Strategy

Applied Problems Pop Quiz

- Q To move their armies, the Romans built over

50,000 miles of roads. Imagine driving all those

miles! Now imagine driving those miles in the

first gasoline-driven car that has only three

wheels and could reach a top speed of about 10

miles per hour. - For safety's sake, let's bring along a spare

tire. As you drive the 50,000 miles, you rotate

the spare with the other tires so that all four

tires get the same amount of wear. Can you figure

out how many miles of wear each tire accumulates?

Directions As a team, read the following

problem. At your tables, apply the 7-step

problem-solving (cognitive) strategy to complete

the problem. As you complete each step of the

problem, apply the Say-Ask-Check metacognitive

sequence. Try to complete the entire 7 steps

within the time allocated for this exercise.

- 7-Step Problem-SolvingProcess
- Reading the problem.
- Paraphrasing the problem.
- Drawing the problem.
- Creating a plan to solve the problem.
- Predicting/Estimat-ing the answer.
- Computing the answer.
- Checking the answer.

A Since the four wheels of the three-wheeled

car share the journey equally, simply take

three-fourths of the total distance (50,000

miles) and you'll get 37,500 miles for each

tire.

Source The Math Forum _at_ Drexel Critical

Thinking Puzzles/Spare My Brain. Retrieved from

http//mathforum.org/k12/k12puzzles/critical.think

ing/puzz2.html

RTI Math Reasoning Key Next Steps

At your tables, Discuss these key next steps

for moving forward with RTI Math Reasoning in

your school or district. Begin to draft an

action plan to implement each of these steps. Be

prepared to report out on your work.

- Define math reasoning and task-analyze to

create a checklist of subskills that make up that

term. (This checklist can be framed as student

look-for behaviors and adjusted to each grade

level). - Develop school-wide screening measures to

identify students at-risk for math computation

and math reasoning skills. Also develop the

capacity to complete diagnostic math assessments

for students with more severe math deficits. - Set up knowledge brokers in your school who

will monitor math instructional and intervention

programs and research findings by attending

workshops, visiting websites, reading

professional journals, etc.and give them

opportunities to share these updates with school

staff.

Team Activity Favorite Math Websites

- At your table
- Discuss math websites that you have used and have

found to be helpful. - Be prepared to report out on your favorite math

websites.

Secondary Group-Based Math Intervention Example

Math Mentors Training Students to Independently

Use On-Line Math-Help Resources

- Math mentors are recruited (school personnel,

adult volunteers, student teachers, peer tutors)

who have a good working knowledge of algebra. - The school meets with each math mentor to verify

mentors algebra knowledge. - The school trains math mentors in 30-minute

tutoring protocol, to include - Requiring that students keep a math journal

detailing questions from notes and homework. - Holding the student accountable to bring journal,

questions to tutoring session. - Ensuring that a minimum of 25 minutes of 30

minute session are spent on tutoring. - Mentors are introduced to online algebra

resources (e.g., www.algebrahelp.com,

www.math.com) and encouraged to browse them and

become familiar with the site content and

navigation.

Math Mentors Training Students to Independently

Use On-Line Math-Help Resources

- Mentors are trained during math mentor sessions

to - Examine student math journal
- Answer student algebra questions
- Direct the student to go online to algebra

tutorial websites while mentor supervises.

Student is to find the section(s) of the websites

that answer their questions. - As the student shows increased confidence with

algebra and with navigation of the math-help

websites, the mentor directs the student to - Note math homework questions in the math journal
- Attempt to find answers independently on

math-help websites - Note in the journal any successful or

unsuccessful attempts to independently get

answers online - Bring journal and remaining questions to next

mentoring meeting.

(No Transcript)

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