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Foundations of Math Skills RTI

Interventions Jim Wright www.interventioncentral.o

rg

Mathematics is made of 50 percent formulas, 50

percent proofs, and 50 percent imagination.

Anonymous

Who is At Risk for Poor Math Performance? A

Proactive Stance

- we use the term mathematics difficulties

rather than mathematics disabilities. Children

who exhibit mathematics difficulties include

those performing in the low average range (e.g.,

at or below the 35th percentile) as well as those

performing well below averageUsing higher

percentile cutoffs increases the likelihood

thatyoung children who go on to have serious math

problems will be picked upin the screening. p.

295

Source Gersten, R., Jordan, N. C., Flojo, J.

R. (2005). Early identification and interventions

for students with mathematics difficulties.

Journal of Learning Disabilities, 38, 293-304.

Profile of Students with Math Difficulties

(Kroesbergen Van Luit, 2003)

- Although the group of students with

difficulties in learning math is very

heterogeneous, in general, these students have

memory deficits leading to difficulties in the

acquisition and remembering of math knowledge.

Moreover, they often show inadequate use of

strategies for solving math tasks, caused by

problems with the acquisition and the application

of both cognitive and metacognitive strategies.

Because of these problems, they also show

deficits in generalization and transfer of

learned knowledge to new and unknown tasks.

Source Kroesbergen, E., Van Luit, J. E. H.

(2003). Mathematics interventions for children

with special educational needs. Remedial and

Special Education, 24, 97-114..

The Elements of Mathematical Proficiency What

the Experts Say

(No Transcript)

Five Strands of Mathematical Proficiency

- 1. Understanding Comprehending mathematical

concepts, operations, and relations--knowing what

mathematical symbols, diagrams, and procedures

mean. Understanding refers to a students grasp

of fundamental mathematical ideas. Students with

understanding know more than isolated facts and

procedures. They know why a mathematical idea is

important and the contexts in which it is useful.

Furthermore, they are aware of many connections

between mathematical ideas. In fact, the degree

of students understanding is related to the

richness and extent of the connections they have

made. p. 10

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

- 2. Computing Carrying out mathematical

procedures, such as adding, subtracting,

multiplying, and dividing numbers flexibly,

accurately, efficiently, and appropriately. Compu

ting includes being fluent with procedures for

adding, subtracting, multiplying, and dividing

mentally or with paper and pencil, and knowing

when and how to use these procedures

appropriately. Although the word computing

implies an arithmetic procedure, it also refers

to being fluent with procedures from other

branches of mathematics, such as measurement

(measuring lengths), algebra (solving equations),

geometry (constructing similar figures), and

statistics (graphing data). Being fluent means

having the skill to perform the procedure

efficiently, accurately, and flexibly. p. 11

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

- 3. Applying Being able to formulate problems

mathematically and to devise strategies for

solving them using concepts and procedures

appropriately. Applying involves using ones

conceptual and procedural knowledge to solve

problems. A concept or procedure is not useful

unless students recognize when and where to use

itas well as when and whether it does not apply.

Students need to be able to pose problems,

devise solution strategies, and choose the most

useful strategy for solving problems.. p. 13

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

- 4. Reasoning Using logic to explain and

justify a solution to a problem or to extend from

something known to something less

known. Reasoning is the glue that holds

mathematics together. By thinking about the

logical relationships between concepts and

situations, students can navigate through the

elements of a problem and see how they fit

together. - One of the best ways for students to improve

their reasoning is to explain or justify their

solutions to others. Reasoning interacts

strongly with the other strands of mathematical

thought, especially when students are solving

problems. p. 14

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

- 5. Engaging Seeing mathematics as sensible,

useful, and doableif you work at itand being

willing to do the work. Engaging in mathematical

activity is the key to success. Our view of

mathematical proficiency goes beyond being able

to understand, compute, apply, and reason. It

includes engagement with mathematics. Students

should have a personal commitment to the idea

that mathematics makes sens and thatgiven

reasonable effortthey can learn it and use it

both in school and outside school.. p. 15-16

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.

Three General Levels of Math Skill Development

(Kroesbergen Van Luit, 2003)

- As students move from lower to higher grades,

they move through levels of acquisition of math

skills, to include - Number sense
- Basic math operations (i.e., addition,

subtraction, multiplication, division) - Problem-solving skills The solution of both

verbal and nonverbal problems through the

application of previously acquired information

(Kroesbergen Van Luit, 2003, p. 98)

Source Kroesbergen, E., Van Luit, J. E. H.

(2003). Mathematics interventions for children

with special educational needs. Remedial and

Special Education, 24, 97-114..

What is Number Sense? (Clarke Shinn, 2004)

- the ability to understand the meaning of

numbers and define different relationships among

numbers. Children with number sense can

recognize the relative size of numbers, use

referents for measuring objects and events, and

think and work with numbers in a flexible manner

that treats numbers as a sensible system. p. 236

Source Clarke, B., Shinn, M. (2004). A

preliminary investigation into the identification

and development of early mathematics

curriculum-based measurement. School Psychology

Review, 33, 234248.

What Are Stages of Number Sense? (Berch, 2005,

p. 336)

- Innate Number Sense. Children appear to possess

hard-wired ability (neurological foundation

structures) to acquire number sense. Childrens

innate capabilities appear also to be to

represent general amounts, not specific

quantities. This innate number sense seems to be

characterized by skills at estimation

(approximate numerical judgments) and a

counting system that can be described loosely as

1, 2, 3, 4, a lot. - Acquired Number Sense. Young students learn

through indirect and direct instruction to count

specific objects beyond four and to internalize a

number line as a mental representation of those

precise number values.

Source Berch, D. B. (2005). Making sense of

number sense Implications for children with

mathematical disabilities. Journal of Learning

Disabilities, 38, 333-339...

The Basic Number Line is as Familiar as a

Well-Known Place to People Who Have Mastered

Arithmetic Combinations

Internal Numberline

- As students internalize the numberline, they are

better able to perform mental arithmetic (the

manipulation of numbers and math operations in

their head).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

16 17 18 19 20 21 22 23 24 25 26 27 28 29

Mental Arithmetic A Demonstration

- 332 x 420 ?

Directions As you watch this video of a person

using mental arithmetic to solve a computation

problem, note the strategies and shortcuts that

he employs to make the task more manageable.

\Mental Arithmetic Demonstration What Tools Were

Used?

Math Computation Building Fluency Jim

Wright www.interventioncentral.org

"Arithmetic is being able to count up to twenty

without taking off your shoes." Anonymous

Benefits of Automaticity of Arithmetic

Combinations ( (Gersten, Jordan, Flojo, 2005)

- There is a strong correlation 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 - By internalizing numbers as mental constructs,

students can manipulate those numbers in their

head, allowing for the intuitive understanding of

arithmetic properties, such as associative

property and commutative property

Source Gersten, R., Jordan, N. C., Flojo, J.

R. (2005). Early identification and interventions

for students with mathematics difficulties.

Journal of Learning Disabilities, 38, 293-304.

Associative Property

- within an expression containing two or more of

the same associative operators in a row, the

order of operations does not matter as long as

the sequence of the operands is not changed - Example
- (23)510
- 2(35)10

Source Associativity. Wikipedia. Retrieved

September 5, 2007, from http//en.wikipedia.org/wi

ki/Associative

Commutative Property

- the ability to change the order of something

without changing the end result. - Example
- 23510
- 25310

Source Associativity. Wikipedia. Retrieved

September 5, 2007, from http//en.wikipedia.org/wi

ki/Commutative

How much is 3 8? Strategies to Solve

Source Gersten, R., Jordan, N. C., Flojo, J.

R. (2005). Early identification and interventions

for students with mathematics difficulties.

Journal of Learning Disabilities, 38, 293-304.

Math Skills Importance of Fluency in Basic Math

Operations

- A key step in math education is to learn the

four basic mathematical operations (i.e.,

addition, subtraction, multiplication, and

division). Knowledge of these operations and a

capacity to perform mental arithmetic play an

important role in the development of childrens

later math skills. Most children with math

learning difficulties are unable to master the

four basic operations before leaving elementary

school and, thus, need special attention to

acquire the skills. A category of interventions

is therefore aimed at the acquisition and

automatization of basic math skills.

Source Kroesbergen, E., Van Luit, J. E. H.

(2003). Mathematics interventions for children

with special educational needs. Remedial and

Special Education, 24, 97-114.

Big Ideas Learn Unit (Heward, 1996)

- The three essential elements of effective student

learning include - Academic Opportunity to Respond. The student is

presented with a meaningful opportunity to

respond to an academic task. A question posed by

the teacher, a math word problem, and a spelling

item on an educational computer Word Gobbler

game could all be considered academic

opportunities to respond. - Active Student Response. The student answers the

item, solves the problem presented, or completes

the academic task. Answering the teachers

question, computing the answer to a math word

problem (and showing all work), and typing in the

correct spelling of an item when playing an

educational computer game are all examples of

active student responding. - Performance Feedback. The student receives timely

feedback about whether his or her response is

correctoften with praise and encouragement. A

teacher exclaiming Right! Good job! when a

student gives an response in class, a student

using an answer key to check her answer to a math

word problem, and a computer message that says

Congratulations! You get 2 points for correctly

spelling this word! are all examples of

performance feedback.

Source Heward, W.L. (1996). Three low-tech

strategies for increasing the frequency of active

student response during group instruction. In R.

Gardner, D. M.S ainato, J. O. Cooper, T. E.

Heron, W. L. Heward, J. W. Eshleman, T. A.

Grossi (Eds.), Behavior analysis in education

Focus on measurably superior instruction

(pp.283-320). Pacific Grove, CABrooks/Cole.

Math Intervention Tier I or II Elementary

Secondary Self-Administered Arithmetic

Combination Drills With Performance

Self-Monitoring Incentives

- The student is given a math computation worksheet

of a specific problem type, along with an answer

key Academic Opportunity to Respond. - The student consults his or her performance chart

and notes previous performance. The student is

encouraged to try to beat his or her most

recent score. - The student is given a pre-selected amount of

time (e.g., 5 minutes) to complete as many

problems as possible. The student sets a timer

and works on the computation sheet until the

timer rings. Active Student Responding - The student checks his or her work, giving credit

for each correct digit (digit of correct value

appearing in the correct place-position in the

answer). Performance Feedback - The student records the days score of TOTAL

number of correct digits on his or her personal

performance chart. - The student receives praise or a reward if he or

she exceeds the most recently posted number of

correct digits.

Application of Learn Unit framework from

Heward, W.L. (1996). Three low-tech strategies

for increasing the frequency of active student

response during group instruction. In R. Gardner,

D. M.S ainato, J. O. Cooper, T. E. Heron, W. L.

Heward, J. W. Eshleman, T. A. Grossi (Eds.),

Behavior analysis in education Focus on

measurably superior instruction (pp.283-320).

Pacific Grove, CABrooks/Cole.

Self-Administered Arithmetic Combination

Drills Examples of Student Worksheet and Answer

Key

Worksheets created using Math Worksheet

Generator. Available online at http//www.interve

ntioncentral.org/htmdocs/tools/mathprobe/addsing.p

hp

Self-Administered Arithmetic Combination Drills

How to Use PPT Group Timers in the Classroom

Math Shortcuts Cognitive Energy- and Time-Savers

- Recently, some researchershave argued that

children can derive answers quickly and with

minimal cognitive effort by employing calculation

principles or shortcuts, such as using a known

number combination toderive an answer (2 2 4,

so 2 3 5), relations among operations (6 4

10, so 10 -4 6), n 1, commutativity, and so

forth. This approach to instruction is consonant

with recommendations by the National Research

Council (2001). Instruction along these linesmay

be much more productive than rote drill without

linkage to counting strategy use. p. 301

Source Gersten, R., Jordan, N. C., Flojo, J.

R. (2005). Early identification and interventions

for students with mathematics difficulties.

Journal of Learning Disabilities, 38, 293-304.

Math Multiplication Shortcut The 9 Times

Quickie

- The student uses fingers as markers to find the

product of single-digit multiplication arithmetic

combinations with 9. - Fingers to the left of the lowered finger stands

for the 10s place value. - Fingers to the right stand for the 1s place

value.

Source Russell, D. (n.d.). Math facts to learn

the facts. Retrieved November 9, 2007, from

http//math.about.com/bltricks.htm

Students Who Understand Mathematical Concepts

Can Discover Their Own Shortcuts

- Students who learn with understanding have less

to learn because they see common patterns in

superfically different sicuations. If they

understand the general principle that the order

in which two numbers are multiplied doesnt

matter3 x 5 is the same as 5 x 3, for

examplethey have about half as many number

facts to learn. p. 10

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.

Application of Math Shortcuts to Intervention

Plans

- Students who struggle with math may find

computational shortcuts to be motivating. - Teaching and modeling of shortcuts provides

students with strategies to make computation less

cognitively demanding.

Math Computation Motivate With Errorless

Learning Worksheets

- In this version of an errorless learning

approach, the student is directed to complete

math facts as quickly as possible. If the

student comes to a number problem that he or she

cannot solve, the student is encouraged to locate

the problem and its correct answer in the key at

the top of the page and write it in. - Such speed drills build computational fluency

while promoting students ability to visualize

and to use a mental number line. - TIP Consider turning this activity into a

speed drill. The student is given a kitchen

timer and instructed to set the timer for a

predetermined span of time (e.g., 2 minutes) for

each drill. The student completes as many

problems as possible before the timer rings. The

student then graphs the number of problems

correctly computed each day on a time-series

graph, attempting to better his or her previous

score.

Source Caron, T. A. (2007). Learning

multiplication the easy way. The Clearing House,

80, 278-282

Errorless Learning Worksheet Sample

Source Caron, T. A. (2007). Learning

multiplication the easy way. The Clearing House,

80, 278-282

Math Computation Two Ideas to Jump-Start Active

Academic Responding

- Here are two ideas to accomplish increased

academic responding on math tasks. - Break longer assignments into shorter assignments

with performance feedback given after each

shorter chunk (e.g., break a 20-minute math

computation worksheet task into 3 seven-minute

assignments). Breaking longer assignments into

briefer segments also allows the teacher to

praise struggling students more frequently for

work completion and effort, providing an

additional natural reinforcer. - Allow students to respond to easier practice

items orally rather than in written form to speed

up the rate of correct responses.

Source Skinner, C. H., Pappas, D. N., Davis,

K. A. (2005). Enhancing academic engagement

Providing opportunities for responding and

influencing students to choose to respond.

Psychology in the Schools, 42, 389-403.

Math Computation Problem Interspersal Technique

- The teacher first identifies the range of

challenging problem-types (number problems

appropriately matched to the students current

instructional level) that are to appear on the

worksheet. - Then the teacher creates a series of easy

problems that the students can complete very

quickly (e.g., adding or subtracting two 1-digit

numbers). The teacher next prepares a series of

student math computation worksheets with easy

computation problems interspersed at a fixed rate

among the challenging problems. - If the student is expected to complete the

worksheet independently, challenging and easy

problems should be interspersed at a 11 ratio

(that is, every challenging problem in the

worksheet is preceded and/or followed by an

easy problem). - If the student is to have the problems read aloud

and then asked to solve the problems mentally and

write down only the answer, the items should

appear on the worksheet at a ratio of 3

challenging problems for every easy one (that

is, every 3 challenging problems are preceded

and/or followed by an easy one).

Source Hawkins, J., Skinner, C. H., Oliver, R.

(2005). The effects of task demands and additive

interspersal ratios on fifth-grade students

mathematics accuracy. School Psychology Review,

34, 543-555..

How to Create an Interspersal-Problems Worksheet

Additional Math Interventions Jim

Wright www.interventioncentral.org

Math Instruction Unlock the Thoughts of

Reluctant Students Through Class Journaling

- Students can effectively clarify their knowledge

of math concepts and problem-solving strategies

through regular use of class math journals. - At the start of the year, the teacher introduces

the journaling weekly assignment in which

students respond to teacher questions. - At first, the teacher presents safe questions

that tap into the students opinions and

attitudes about mathematics (e.g., How important

do you think it is nowadays for cashiers in

fast-food restaurants to be able to calculate in

their head the amount of change to give a

customer?). As students become comfortable with

the journaling activity, the teacher starts to

pose questions about the students own

mathematical thinking relating to specific

assignments. Students are encouraged to use

numerals, mathematical symbols, and diagrams in

their journal entries to enhance their

explanations. - The teacher provides brief written comments on

individual student entries, as well as periodic

oral feedback and encouragement to the entire

class. - Teachers will find that journal entries are a

concrete method for monitoring student

understanding of more abstract math concepts. To

promote the quality of journal entries, the

teacher might also assign them an effort grade

that will be calculated into quarterly math

report card grades.

Source Baxter, J. A., Woodward, J., Olson, D.

(2005). Writing in mathematics An alternative

form of communication for academically

low-achieving students. Learning Disabilities

Research Practice, 20(2), 119135.

Math Review Incremental Rehearsal of Math Facts

Step 1 The tutor writes down on a series of

index cards the math facts that the student needs

to learn. The problems are written without the

answers.

Math Review Incremental Rehearsal of Math Facts

KNOWN Facts

UNKNOWN Facts

Step 2 The tutor reviews the math fact cards

with the student. Any card that the student can

answer within 2 seconds is sorted into the

KNOWN pile. Any card that the student cannot

answer within two secondsor answers

incorrectlyis sorted into the UNKNOWN pile.

Math Review Incremental Rehearsal of Math Facts

Math Review Incremental Rehearsal of Math Facts

Applied Math Helping Students to Make Sense of

Story Problems Jim Wright www.interventioncentra

l.org

Advanced Math Quotes from Yogi Berra

- Ninety percent of the game is half mental."
- Pair up in threes."
- You give 100 percent in the first half of the

game, and if that isn't enough in the second half

you give what's left.

Applied Math Problems Rationale

- Applied math problems (also known as story or

word problems) are traditional tools for having

students apply math concepts and operations to

real-world settings.

Sample Applied Problems

- Once upon a time, there were three little pigs -

ages 2, 4, and 6. Are their ages even or odd? - Every day this past summer, Peter rode his bike

to and from work. Each round trip was 13

kilometers. His friend Marsha rode her bike18

kilometers' each day, but just for exercise. How

much further did Marsha ride her bike than Peter

in one week? - Suzy is ten years older than Billy, and next year

she will be twice as old as Billy. How old are

they now?

Applied Math Problems Some Required Competencies

- For students to achieve success with applied

problems, they must be able to - Comprehend the text of written problems.
- Understand specialized math vocabulary (e.g.,

quotient). - Understand specialized use of common vocabulary

(e.g., product). - Be able to translate verbal cues into a numeric

equation. - Ignore irrelevant information included in the

problem. - Interpret math graphics that may accompany the

problem. - Apply a plan to problem-solving.
- Check their work.

Potential Blockers of Higher-Level Math

Problem-Solving A Sampler

- Limited reading skills
- Failure to master--or develop automaticity in

basic math operations - Lack of knowledge of specialized math vocabulary

(e.g., quotient) - Lack of familiarity with the specialized use of

known words (e.g., product) - Inability to interpret specialized math symbols

(e.g., 4 lt 2) - Difficulty extracting underlying math

operations from word/story problems or

identifying and ignoring extraneous information

included in word/story problems

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.

Math Intervention Tier I High School Peer

Guided Pause

- Students are trained to work in pairs.
- At one or more appropriate review points in a

math lecture, the instructor directs students to

pair up to work together for 4 minutes. - During each Peer Guided Pause, students are

given a worksheet that contains one or more

correctly completed word or number problems

illustrating the math concept(s) covered in the

lecture. The sheet also contains several

additional, similar problems that pairs of

students work cooperatively to complete, along

with an answer key. - Student pairs are reminded to (a) monitor their

understanding of the lesson concepts (b) review

the correctly math model problem (c) work

cooperatively on the additional problems, and (d)

check their answers. The teacher can direct

student pairs to write their names on the

practice sheets and collect them to monitor

student understanding.

Source Hawkins, J., Brady, M. P. (1994). The

effects of independent and peer guided practice

during instructional pauses on the academic

performance of students with mild handicaps.

Education Treatment of Children, 17 (1), 1-28.

Applied Problems Encourage Students to Draw

the Problem

- Making a drawing of an applied, or word,

problem is one easy heuristic tool that students

can use to help them to find the solution and

clarify misunderstandings. - The teacher hands out a worksheet containing at

least six word problems. The teacher explains to

students that making a picture of a word problem

sometimes makes that problem clearer and easier

to solve. - The teacher and students then independently

create drawings of each of the problems on the

worksheet. Next, the students show their drawings

for each problem, explaining each drawing and how

it relates to the word problem. The teacher also

participates, explaining his or her drawings to

the class or group. - Then students are directed independently to make

drawings as an intermediate problem-solving step

when they are faced with challenging word

problems. NOTE This strategy appears to be more

effective when used in later, rather than

earlier, elementary grades.

Source Hawkins, J., Skinner, C. H., Oliver, R.

(2005). The effects of task demands and additive

interspersal ratios on fifth-grade students

mathematics accuracy. School Psychology Review,

34, 543-555..

Interpreting Math Graphics

Housing Bubble Graphic New York Times 23

September 2007

Classroom Challenges in Interpreting Math Graphics

- When encountering math graphics, students may
- expect the answer to be easily accessible when in

fact the graphic may expect the reader to

interpret and draw conclusions - be inattentive to details of the graphic
- treat irrelevant data as relevant
- not pay close attention to questions before

turning to graphics to find the answer - fail to use their prior knowledge both the extend

the information on the graphic and to act as a

possible check on the information that it

presents.

Source Mesmer, H.A.E., Hutchins, E.J. (2002).

Using QARs with charts and graphs. The Reading

Teacher, 56, 2127.

Using Question-Answer Relationships (QARs) to

Interpret Information from Math Graphics

- Students can be more savvy interpreters of

graphics in applied math problems by applying the

Question-Answer Relationship (QAR) strategy. Four

Kinds of QAR Questions - RIGHT THERE questions are fact-based and can be

found in a single sentence, often accompanied by

'clue' words that also appear in the question. - THINK AND SEARCH questions can be answered by

information in the text but require the scanning

of text and making connections between different

pieces of factual information. - AUTHOR AND YOU questions require that students

take information or opinions that appear in the

text and combine them with the reader's own

experiences or opinions to formulate an answer. - ON MY OWN questions are based on the students'

own experiences and do not require knowledge of

the text to answer.

Source Mesmer, H.A.E., Hutchins, E.J. (2002).

Using QARs with charts and graphs. The Reading

Teacher, 56, 2127.

Applied Problems Individualized Self-Correction

Checklists

- Students can improve their accuracy on

particular types of word and number problems by

using an individualized self-instruction

checklist that reminds them to pay attention to

their own specific error patterns. - The teacher meets with the student. Together they

analyze common error patterns that the student

tends to commit on a particular problem type

(e.g., On addition problems that require

carrying, I dont always remember to carry the

number from the previously added column.). - For each type of error identified, the student

and teacher together describe the appropriate

step to take to prevent the error from occurring

(e.g., When adding each column, make sure to

carry numbers when needed.). - These self-check items are compiled into a single

checklist. Students are then encouraged to use

their individualized self-instruction checklist

whenever they work independently on their number

or word problems.

Source Pólya, G. (1945). How to solve it.

Princeton University Press Princeton, N.J.

Using Question-Answer Relationships (QARs) to

Interpret Information from Math Graphics 4-Step

Teaching Sequence

- DISTINGUISHING DIFFERENT KINDS OF GRAPHICS.

Students are taught to differentiate between

common types of graphics e.g., table (grid with

information contained in cells), chart (boxes

with possible connecting lines or arrows),

picture (figure with labels), line graph, bar

graph. Students note significant differences

between the various graphics, while the teacher

records those observations on a wall chart. Next

students are given examples of graphics and asked

to identify which general kind of graphic each

is. Finally, students are assigned to go on a

graphics hunt, locating graphics in magazines

and newspapers, labeling them, and bringing to

class to review.

Source Mesmer, H.A.E., Hutchins, E.J. (2002).

Using QARs with charts and graphs. The Reading

Teacher, 56, 2127.

Using Question-Answer Relationships (QARs) to

Interpret Information from Math Graphics 4-Step

Teaching Sequence

- INTERPRETING INFORMATION IN GRAPHICS. Students

are paired off, with stronger students matched

with less strong ones. The teacher spends at

least one session presenting students with

examples from each of the graphics categories.

The presentation sequence is ordered so that

students begin with examples of the most concrete

graphics and move toward the more abstract

Pictures gt tables gt bar graphs gt charts gt line

graphs. At each session, student pairs examine

graphics and discuss questions such as What

information does this graphic present? What are

strengths of this graphic for presenting data?

What are possible weaknesses?

Source Mesmer, H.A.E., Hutchins, E.J. (2002).

Using QARs with charts and graphs. The Reading

Teacher, 56, 2127.

Using Question-Answer Relationships (QARs) to

Interpret Information from Math Graphics 4-Step

Teaching Sequence

- LINKING THE USE OF QARS TO GRAPHICS. Students are

given a series of data questions and correct

answers, with each question accompanied by a

graphic that contains information needed to

formulate the answer. Students are also each

given index cards with titles and descriptions of

each of the 4 QAR questions RIGHT THERE, THINK

AND SEARCH, AUTHOR AND YOU, ON MY OWN. Working

in small groups and then individually, students

read the questions, study the matching graphics,

and verify the answers as correct. They then

identify the type question being asked using

their QAR index cards.

Source Mesmer, H.A.E., Hutchins, E.J. (2002).

Using QARs with charts and graphs. The Reading

Teacher, 56, 2127.

Using Question-Answer Relationships (QARs) to

Interpret Information from Math Graphics 4-Step

Teaching Sequence

- USING QARS WITH GRAPHICS INDEPENDENTLY. When

students are ready to use the QAR strategy

independently to read graphics, they are given a

laminated card as a reference with 6 steps to

follow - Read the question,
- Review the graphic,
- Reread the question,
- Choose a QAR,
- Answer the question, and
- Locate the answer derived from the graphic in the

answer choices offered. - Students are strongly encouraged NOT to read the

answer choices offered until they have first

derived their own answer, so that those choices

dont short-circuit their inquiry.

Source Mesmer, H.A.E., Hutchins, E.J. (2002).

Using QARs with charts and graphs. The Reading

Teacher, 56, 2127.

Mindful Math Applying a Simple Heuristic to

Applied Problems

- By following an efficient 4-step plan, students

can consistently perform better on applied math

problems. - UNDERSTAND THE PROBLEM. To fully grasp the

problem, the student may restate the problem in

his or her own words, note key information, and

identify missing information. - DEVISE A PLAN. In mapping out a strategy to solve

the problem, the student may make a table, draw a

diagram, or translate the verbal problem into an

equation. - CARRY OUT THE PLAN. The student implements the

steps in the plan, showing work and checking work

for each step. - LOOK BACK. The student checks the results. If the

answer is written as an equation, the student

puts the results in words and checks whether the

answer addresses the question posed in the

original word problem.

Source Pólya, G. (1945). How to solve it.

Princeton University Press Princeton, N.J.

Applied Problems Timed Quiz

- 4-Step Problem-Solving
- UNDERSTAND THE PROBLEM.
- DEVISE A PLAN.
- CARRY OUT THE PLAN.
- LOOK BACK.

- Suppose 6 monkeys take 6 minutes to eat 6

bananas. - How many minutes would it take 3 monkeys to eat

3 bananas? - How many monkeys would it take to eat 48

bananas in 48 minutes?

Source Puzzles Brain Teasers Monkeys

Bananas. (n.d.). Retrieved on October 22, 2007,

from http//www.syvum.com/cgi/online/serve.cgi/tea

sers/monkeys.tdf?0

Applied Problems Timed Quiz

- 4-Step Problem-Solving
- UNDERSTAND THE PROBLEM.
- DEVISE A PLAN.
- CARRY OUT THE PLAN.
- LOOK BACK.

- Mr. Brown has 12 black gloves and 6 brown gloves

in his closet. He blindly picks up some gloves

from the closet. What is the minimum number of

gloves Mr. Brown will have to pick to be certain

to find a pair of gloves of the same color?

Source Puzzles Brain Teasers Monkeys

Bananas. (n.d.). Retrieved on October 22, 2007,

from http//www.syvum.com/cgi/online/serve.cgi/tea

sers/monkeys.tdf?0

Math Computation Fluency RTI Case Study

RTI Individual Case Study Math Computation

- Jared is a fourth-grade student. His teacher,

Mrs. Rogers, became concerned because Jared is

much slower in completing math computation

problems than are his classmates.

Tier 1 Math Interventions for Jared

- Jareds school uses the Everyday Math curriculum

(McGraw Hill/University of Chicago). In addition

to the basic curriculum the series contains

intervention exercises for students who need

additional practice or remediation. The

instructor, Mrs. Rogers, works with a small group

of children in her roomincluding Jaredhaving

them complete these practice exercises to boost

their math computation fluency.

Tier 2 Standard Protocol (Group) Math

Interventions for Jared

- Jared did not make sufficient progress in his

Tier 1 intervention. So his teacher referred the

student to the RTI Intervention Team. The team

and teacher decided that Jared would be placed on

the schools educational math software, AMATH

Building Blocks, a self-paced, individualized

mathematics tutorial covering the math

traditionally taught in grades K-4. Jared

worked on the software in 20-minute daily

sessions to increase computation fluency in basic

multiplication problems.

Tier 2 Math Interventions for Jared (Cont.)

- During this group-based Tier 2 intervention,

Jared was assessed using Curriculum-Based

Measurement (CBM) Math probes. The goal was to

bring Jared up to at least 40 correct digits per

2 minutes.

Tier 2 Math Interventions for Jared (Cont.)

- Progress-monitoring worksheets were created using

the Math Computation Probe Generator on

Intervention Central (www.interventioncentral.org)

.

Example of Math Computation Probe Answer Key

Tier 2 Phase 1 Math Interventions for Jared

Progress-Monitoring

Tier 2 Individualized Plan Math Interventions

for Jared

- Progress-monitoring data showed that Jared did

not make expected progress in the first phase of

his Tier 2 intervention. So the RTI Intervention

Team met again on the student. The team and

teacher noted that Jared counted on his fingers

when completing multiplication problems. This

greatly slowed down his computation fluency. The

team decided to use a research-based strategy,

Explicit Time Drills, to increase Jareds

computation speed and eliminate his dependence on

finger-counting. During this individualized

intervention, Jared continued to be assessed

using Curriculum-Based Measurement (CBM) Math

probes. The goal was to bring Jared up to at

least 40 correct digits per 2 minutes.

Explicit Time Drills Math Computational

Fluency-Building Intervention

- Explicit time-drills are a method to boost

students rate of responding on math-fact

worksheets. - The teacher hands out the worksheet. Students

are told that they will have 3 minutes to work on

problems on the sheet. The teacher starts the

stop watch and tells the students to start work.

At the end of the first minute in the 3-minute

span, the teacher calls time, stops the

stopwatch, and tells the students to underline

the last number written and to put their pencils

in the air. Then students are told to resume work

and the teacher restarts the stopwatch. This

process is repeated at the end of minutes 2 and

3. At the conclusion of the 3 minutes, the

teacher collects the student worksheets.

Source Rhymer, K. N., Skinner, C. H., Jackson,

S., McNeill, S., Smith, T., Jackson, B. (2002).

The 1-minute explicit timing intervention The

influence of mathematics problem difficulty.

Journal of Instructional Psychology, 29(4),

305-311.

Cover-Copy-Compare Math Computational

Fluency-Building Intervention

- The student is given sheet with correctly

completed math problems in left column and index

card. For each problem, the student - studies the model
- covers the model with index card
- copies the problem from memory
- solves the problem
- uncovers the correctly completed model to check

answer

Source Skinner, C.H., Turco, T.L., Beatty, K.L.,

Rasavage, C. (1989). Cover, copy, and compare

A method for increasing multiplication

performance. School Psychology Review, 18,

412-420.

Tier 2 Phase 2 Math Interventions for Jared

Progress-Monitoring

Tier 2 Math Interventions for Jared

- Explicit Timed Drill Intervention Outcome
- The progress-monitoring data showed that Jared

was well on track to meet his computation goal.

At the RTI Team follow-up meeting, the team and

teacher agreed to continue the fluency-building

intervention for at least 3 more weeks. It was

also noted that Jared no longer relied on

finger-counting when completing number problems,

a good sign that he had overcome an obstacle to

math computation.

Group Activity Tier I Math Interventions

- Look at the math intervention ideas in your

handout. - Based on these ideas and other intervention

strategies that you may have used as an educator,

generate a list of up to FIVE Tier I

(classroom-based) interventions you would

recommend to teachers in your school. - Be prepared to share your results.