Response to Intervention Foundations of Math Skills & RTI Interventions Jim Wright .

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Foundations of Math Skills& RTI InterventionsJim Wrightwww.interventioncentral.org


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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..



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...



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:

–(2+3)+5=10– 2+(3+5)=10

Source: Associativity. Wikipedia. Retrieved September 5, 2007, from http://en.wikipedia.org/wiki/Associative



Commutative Property

• “the ability to change the order of something without changing the end result.”

• Example:

– 2+3+5=10– 2+5+3=10

Source: Associativity. Wikipedia. Retrieved September 5, 2007, from http://en.wikipedia.org/wiki/Commutative



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

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 Intervention: Tier I or II: Elementary & Middle School: Cover Copy Compare

• The student is given a math worksheet with 10 number problems and answers on the left side of the page.

• For each problem, the student:– Studies the correctly completed problem on the left side of the page.– Covers the problem with an index card.– Copies the problem from memory on the right side of the page.– Solves the problem.– Uncovers the correct model problem to check his or her work.– If the student’s problem was done incorrectly, the student repeats the process

until correct.

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.



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.



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: Math Interventions for Jared

• Jared did not make sufficient progress in his Tier 1 intervention. So his teacher consulted with other teachers at her grade level and the principal. It was 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 the 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: Math Interventions for Jared: Progress-Monitoring



Tier 3: Math Interventions for Jared• Progress-monitoring data showed that Jared did not make

expected progress in his Tier 2 intervention. So he was referred to the RTI Intervention Team. 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, taped problems, to increase Jared’s computation speed and eliminate his dependence on finger-counting.

During the Tier 3 intervention, Jared contiued 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.



Tier 3: Math Interventions for JaredTime-Delay Math Computation Intervention: • A series of math worksheets were created (2 digit times 2

digit; no regrouping). • For each worksheet, a digital recording (MP3 file) was

made in which the computation problems and their answers were read aloud several times. – In the first reading, there was no delay between the problem and

the answer.– In the second reading, there was a four-second delay between

the problem and its answer. – In the third reading, there was a two-second delay between the

problem and its answer.– In the final reading, there was a two-second delay between the

problem and its answer. Source: McCalum, E., Skinner, C. H., Turner, H., & Saecker, L. (2006). The taped-problem intervention: Increasing multiplication fact fluency using a low-tech, classwide, time-delay intervention. School Psychology Review, 35, 419-434.



Tier 3: Math Interventions for JaredTime-Delay Math Computation Intervention (cont.): • Jared was given a math worksheet 3 times per week. With

the worksheet, he was given an IPod with the digital file of the problems and answers being read.

• In each session, Jared listened to the tape while he completed the math worksheet. He was encouraged to ‘beat the tape’ by answering the problem before the taped reader read the answer.

• At the end of the session, Jared’s worksheet was collected.• Jared’s math computation speed was independently

assessed using CBM math computation probes.

Source: McCalum, E., Skinner, C. H., Turner, H., & Saecker, L. (2006). The taped-problem intervention: Increasing multiplication fact fluency using a low-tech, classwide, time-delay intervention. School Psychology Review, 35, 419-434.



Tier 3: Math Interventions for JaredTime-Delay Math Computation Intervention: Building

Intervention Capacity: • High school students were recruited to make the digital

files of problems and answers read aloud.• The school saved the worksheets and related digital files

for the taped-problems intervention to use with other students.




Tier 3: Math Interventions for Jared: Progress-Monitoring



Tier 3: Math Interventions for JaredTime-Delay Math Computation Intervention: Outcome• The progress-monitoring data showed that Jared was well

on track to meet his computation goal. At the SST 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.


Response to Intervention Foundations of Math Skills & RTI Interventions Jim Wright .

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