Response to Intervention Math Skills: An Introduction Jim Wright

Response to Intervention

www.interventioncentral.org

Math Skills: An IntroductionJim Wrightwww.interventioncentral.org


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“Mathematics is made of 50 percent formulas, 50 percent proofs, and 50 percent imagination.”

–Anonymous



National Mathematics Advisory Panel Report13 March 2008



Math Advisory Panel Report at:

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



2008 National Math Advisory Panel Report: Recommendations• “The areas to be studied in mathematics from pre-kindergarten through

eighth grade should be streamlined and a well-defined set of the most important topics should be emphasized in the early grades. Any approach that revisits topics year after year without bringing them to closure should be avoided.”

• “Proficiency with whole numbers, fractions, and certain aspects of geometry and measurement are the foundations for algebra. Of these, knowledge of fractions is the most important foundational skill not developed among American students.”

• “Conceptual understanding, computational and procedural fluency, and problem solving skills are equally important and mutually reinforce each other. Debates regarding the relative importance of each of these components of mathematics are misguided.”

• “Students should develop immediate recall of arithmetic facts to free the “working memory” for solving more complex problems.”

Source: National Math Panel Fact Sheet. (March 2008). Retrieved on March 14, 2008, from http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-factsheet.html



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 average…Using higher percentile cutoffs increases the likelihood that young children who go on to have serious math problems will be picked up in 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…



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.

5 Strands of Mathematical Proficiency

1. Understanding

2. Computing

3. Applying

4. Reasoning

5. Engagement

5 Big Ideas in Beginning Reading

1. Phonemic Awareness

2. Alphabetic Principle

3. Fluency with Text

4. Vocabulary

5. ComprehensionSource: Big ideas in beginning reading. University of Oregon. Retrieved September 23, 2007, from http://reading.uoregon.edu/index.php



Five Strands of Mathematical Proficiency

1. Understanding: Comprehending mathematical concepts, operations, and relations--knowing what mathematical symbols, diagrams, and procedures mean.

2. Computing: Carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately.




Five Strands of Mathematical Proficiency (Cont.)3. Applying: Being able to formulate problems

mathematically and to devise strategies for solving them using concepts and procedures appropriately.

4. Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something less known.

5. Engaging: Seeing mathematics as sensible, useful, and doable—if you work at it—and being willing to do the work.




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, 234–248.



What Are Stages of ‘Number Sense’?

(Berch, 2005, p. 336)

1. Innate Number Sense. Children appear to possess ‘hard-wired’ ability (neurological ‘foundation structures’) to acquire number sense. Children’s 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’.

2. 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

Moravia, NY Number Line: 0-144 0 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100101 102 103 104 105 106 107 108 109 110111 112 113 114 115 116 117 118 119 120121 122 123 124 125 126 127 128 129 130131 132 133 134 135 136 137 138 139 140141 142 143 144



Internal NumberlineAs 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

2 + 4 = 628 ÷ 4 = 79 – 7 = 23 X 7 = 21



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?

132,800

x 6,640

132,800 + 6000 = 138,800

132,800 + 640 = 139,440

7. Add Intermediate Products: Chunk into Smaller Computation Tasks

6,640

’66 is a famous

national road’ & ’40 is speed limit in front

of house’

6. Use Mnemonic Strategy to Remember Intermediate Product

332

x 20

332 x 10 = 3320

3320 x 2 = 6640

5. Continue with Next ‘Chunk’ of Problem: Math Shortcut

132,800

‘1=3-2’ & ‘800 is a toll-free

number’

4. Use Mnemonic Strategy to Remember Intermediate Product

1,328

x 100

132,800

3. Apply Math Shortcut: Add Zeros in One’s Place for Each Multiple of Ten332

x 4

1,328

2. Break Problem into Manageable Chunks

300

x 400

120,000

1. Estimate Answer332

X 420

Solving for…



Math Computation: Building FluencyJim Wrightwww.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




How much is 3 + 8?: Strategies to Solve…Least efficient strategy: Count out and group 3 objects; count out and group 8 objects; count all objects:

+ =11

More efficient strategy: Begin at the number 3 and ‘count up’ 8 more digits (often using fingers for counting): 3 + 8More efficient strategy: Begin at the number 8 (larger number) and ‘count up’ 3 more digits: 8 + 3Most efficient strategy: ‘3 + 8’ arithmetic combination is stored in memory and automatically retrieved: Answer = 11




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 children’s 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:1. 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.

2. Active Student Response. The student answers the item, solves the problem presented, or completes the academic task. Answering the teacher’s 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.

3. Performance Feedback. The student receives timely feedback about whether his or her response is correct—often 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, CA:Brooks/Cole.



Math Intervention: Tier I or II: Elementary & Secondary: Self-Administered Arithmetic Combination Drills With Performance

Self-Monitoring & Incentives

1. The student is given a math computation worksheet of a specific problem type, along with an answer key [Academic Opportunity to Respond].

2. 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.

3. 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]

4. 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]

5. The student records the day’s score of TOTAL number of correct digits on his or her personal performance chart.

6. 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, CA:Brooks/Cole.



Self-Administered Arithmetic Combination Drills:Examples of Student Worksheet and Answer Key

Worksheets created using Math Worksheet Generator. Available online at:http://www.interventioncentral.org/htmdocs/tools/mathprobe/addsing.php



Self-Administered Arithmetic Combination Drills…

No Reward

Reward GivenReward GivenReward Given

No RewardNo Reward

Reward Given



How to… Use PPT Group Timers in the Classroom



Math Shortcuts: Cognitive Energy- and Time-Savers

“Recently, some researchers…have 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) … and so forth. This approach to instruction is consonant with recommendations by the National Research Council (2001). Instruction along these lines may be much more productive than rote drill without linkage to counting strategy use.” p. 301




9 x 19 x 29 x 39 x 49 x 59 x 69 x 79 x 89 x 99 x 10

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 ’10’s place

value. • Fingers to the right stand for the ‘1’s 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 superficially different situations. If they understand the general principle that the order in which two numbers are multiplied doesn’t matter—3 x 5 is the same as 5 x 3, for example—they have about half as many ‘number facts’ to learn.” p. 10




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 student’s 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 1:1 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 InterventionsJim Wrightwww.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), 119–135.



Math Review: Incremental Rehearsal of ‘Math Facts’

2 x 6 =__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.

3 x 8 =__

9 x 2 =__

4 x 7 =__

7 x 6 =__

5 x 5 =__

5 x 3 =__

3 x 6 =__

8 x 4 =__

3 x 5 =__

4 x 5 =__

3 x 2 =__

6 x 5 =__

8 x 2 =__

9 x 7 =__



Math Review: Incremental Rehearsal of ‘Math Facts’

2 x 6 =__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 seconds—or answers incorrectly—is sorted into the ‘UNKNOWN’ pile.

3 x 8 =__

4 x 7 =__

7 x 6 =__

5 x 3 =__

3 x 6 =__ 8 x 4 =__

4 x 5 =__

3 x 2 =__

6 x 5 =__

9 x 7 =__

9 x 2 =__

3 x 5 =__

8 x 2 =__

5 x 5 =__

‘KNOWN’ Facts ‘UNKNOWN’ Facts



Math Review: Incremental Rehearsal of ‘Math Facts’Step 3: The tutor is now ready to follow a nine-step incremental-rehearsal sequence: First, the tutor presents the student with a single index card containing an ‘unknown’ math fact. The tutor reads the problem aloud, gives the answer, then prompts the student to read off the same unknown problem and provide the correct answer.

3 x 8 =__ 2 x 6 =__

4 x 7 =__

5 x 3 =__3 x 6 =__

8 x 4 =__

3 x 2 =__

6 x 5 =__

4 x 5 =__

Step 3: Next the tutor takes a math fact from the ‘known’ pile and pairs it with the unknown problem. When shown each of the two problems, the student is asked to read off the problem and answer it.

3 x 8 =__ 4 x 5 =__

Step 3: The tutor then repeats the sequence--adding yet another known problem to the growing deck of index cards being reviewed and each time prompting the student to answer the whole series of math facts—until the review deck contains a total of one ‘unknown’ math fact and nine ‘known’ math facts (a ratio of 90 percent ‘known’ to 10 percent ‘unknown’ material )

3 x 8 =__



Math Review: Incremental Rehearsal of ‘Math Facts’Step 4: The student is then presented with a new ‘unknown’ math fact to answer--and the review sequence is once again repeated each time until the ‘unknown’ math fact is grouped with nine ‘known’ math facts—and on and on. Daily review sessions are discontinued either when time runs out or when the student answers an ‘unknown’ math fact incorrectly three times.

2 x 6 =__

5 x 3 =__

3 x 6 =__

8 x 4 =__

3 x 2 =__

6 x 5 =__

4 x 5 =__3 x 8 =__9 x 2 =__ 2 x 6 =__

4 x 7 =__

5 x 3 =__3 x 6 =__

8 x 4 =__

3 x 2 =__

6 x 5 =__

4 x 5 =__3 x 8 =__

Step 4: At this point, the last ‘known’ math fact that had been added to the student’s review deck is discarded (placed back into the original pile of ‘known’ problems) and the previously ‘unknown’ math fact is now treated as the first ‘known’ math fact in new student review deck for future drills.



Applied Math: Helping Students to Make Sense of ‘Story Problems’Jim Wrightwww.interventioncentral.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.”





‘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

There are six bananas. Suppose 6 monkeys require 6 minutes to eat those 6 bananas. (Each monkey eats at exactly the same rate.)

• 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/teasers/monkeys.tdf?0

4-Step Problem-Solving:

• UNDERSTAND THE PROBLEM.

• DEVISE A PLAN. • CARRY OUT THE

PLAN. • LOOK BACK.



Applied Problems: Timed Quiz

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 two 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/teasers/monkeys.tdf?0

4-Step Problem-Solving:

• UNDERSTAND THE PROBLEM.

• DEVISE A PLAN. • CARRY OUT THE

PLAN. • LOOK BACK.



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 < 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 child’s 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/author/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 vocabulary—while 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/author/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 ProblemMaking 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..



Applied Problems: Individualized Self-Correction ChecklistsStudents 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 don’t 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.



Activity: Selecting the ‘Best of the Best’ Tier I Intervention

Ideas



Tier I Interventions List: Activity

• Scan the page of ‘Tier I’ (Classroom) interventions on mathematics from your packet

• Select 2-3 TOP ideas from your reading that you feel should be on teachers’ ‘Tier I’ intervention list at your school

• Be prepared to share your ideas



Interpreting Math Graphics



Housing Bubble Graphic:

New York Times23 September 2007

Housing Price Index = 100 in 1987

Housing Price Index = 171 in 2005



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 to 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, 21–27.



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.




Using Question-Answer Relationships (QARs) to Interpret Information from Math Graphics: 4-Step Teaching Sequence

1. 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.





2. 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 > tables > bar graphs > charts > 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?”





3. 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.





4. 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:

A. Read the question,

B. Review the graphic,

C. Reread the question,

D. Choose a QAR,

E. Answer the question, and

F. 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 don’t short-circuit their inquiry.




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

• Jared’s 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 room—including Jared—having 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 school’s 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 Jared’s 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.



Tier 2: Phase 2: Math Interventions for Jared: Progress-Monitoring



Tier 2: Math Interventions for JaredExplicit 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.

Documents

Response to Intervention Math Skills: An Introduction Jim Wright