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Something that will break down need by skill/concept

Star Math?

Dibbles Next - math*

Aims Web**

Delta Math

Easy CBM - Paid version

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RTI Toolkit/Guidelines *P*

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RTINetwork.org guides educators and families in the effective implementation of Response to Intervention (RTI) as a means to improve educational outcomes for all students.

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Opportunity to register for this CoP is now closed. Additional opportunities for local leaders will occur in the future. Are you an local education agency leader or district level personnel interested in learning more about how to support and scale data-based individualization (DBI) implementation?

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Intervention Center

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Educators and classroom leaders look to academic interventions to help promote response to intervention and rti programs for classroom management

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Big names:
Russell Gersten
Mark Shinn
Fuchs and Fuchs

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RtI is now referred to as MTSS (multi-tiered systems of supports)

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Research Based Programs to implement:

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Teaching strategies for interventions:

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How to build number sense and conceptual understanding?

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Do all effective math intervention programs have a fact fluency element?

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PROMPT Intervention - Proactive approach to Interventions

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*P*

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Best Practices for RTI Math Interventions *P*

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Overview of RTI/MTSS

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Pre-teaching/Frontloading?  Is this approach more effective?

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Intensive Intervention for Students with Mathematics Disabilities: Seven Principles of Effective Practice *P*

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Lynn S. Fuchs, Ph.D., Douglas Fuchs, Ph.D., Sarah R. Powell, M.S., Pamela M. Seethaler, M.S., Paul T. Cirino, Ph.D., and Jack M. Fletcher, Ph.D.

Abstract

The focus of this article is intervention for third - grade students with serious mathematics deficits at third grade. In third grade, such deficits are clearly established, and identification of mathematics disabilities typically begins. We provide background information on two aspects of mathematical cognition that present major challenges for students in the primary grades: number combinations and story problems. We then focus on seven principles of effective intervention. First, we describe a validated, intensive remedial intervention for number combinations and another for story problems. Then, we use these interventions to illustrate the first six principles for designing intensive tutoring protocols for students with mathematics disabilities. Next, using the same validated interventions, we report the percentage of students whose learning outcomes were inadequate despite the overall efficacy of the interventions and explain how ongoing progress monitoring represents a seventh, and perhaps the most essential, principle of intensive intervention. We conclude by identifying issues and directions for future research in the primary and later grades.

Approximately 5-9% of the school-age population may be identified with mathematics disability (e.g., Badian, 1983; Gross-Tsur, Manor, & Shalev, 1996). Although this estimate is similar to the prevalence of reading disability, less systematic study has been directed at mathematics disability ( Rasanen & Ahonen, 1995), even though poor mathematics skills are associated with life-long difficulties in school and in the workplace. For example, mathematics competence accounts for variance in employment, income, and work productivity even after intelligence and reading have been explained ( Rivera-Batiz, 1992).

Some research illustrates how prevention activities at preschool (e.g., Clements & Sarama, 2007), kindergarten (e.g., Griffin, Case, & Siegler, 1994), or first grade (e.g., Fuchs, Fuchs, Yazdian, & Powell, 2002) can substantially improve math performance. For example, at the beginning of first grade, Fuchs and colleagues (2005) identified 169 students in 41 classrooms as being at risk for math difficulties based on their low initial performance. These children were randomly assigned to a control group or to receive small-group tutoring three times per week for 16 weeks. Results showed that math development across first grade was significantly and substantially superior for the tutored group than for the control group on computation, concepts, applications, and story problems. In addition, the incidence of students with mathematics disability was substantially reduced at the end of first grade, and this reduction in mathematics disability remained in the spring of second grade, one year after tutoring ended ( Compton, Fuchs, & Fuchs, 2006).

Nevertheless, despite the efficacy of tutoring, students were not universally responsive. A subset of the tutored students, approximately 3-6% of the school population (depending on the measure and cut-point for the severity of mathematics performance used), manifested severe mathematics deficits. As this example illustrates, the need for intensive remedial intervention persists even when prevention services are generally effective.

In this article, we focus on serious mathematics deficits at third grade. In third grade, serious mathematics deficits are clearly established, and identification of mathematics disabilities typically begins (Fletcher, Lyon, Fuchs, & Barnes, 2007). We provide background information on two aspects of mathematical cognition that present major challenges for students in the primary grades: number combinations and story problems. We describe validated, intensive interventions for number combinations and another for story problems and use these interventions to illustrate six principles for designing interventions for students with mathematics disabilities. Next, using the same validated interventions, we report the percentage of students whose learning outcomes were inadequate despite the overall efficacy of the interventions and we how ongoing progress monitoring therefore represents a seventh and perhaps the most essential principle for intensive intervention. We conclude by identifying issues and directions for future research in the primary and later grades.

Number Combinations and Story Problems

Number combinations refer to problems with single-digit operands, which can be solved by counting or committing to long-term memory for automatic retrieval. Widespread agreement exists that number combination skill is essential for all children to acquire. The National Research Council ( Kilpatrick, Swafford, & Findell, 2001), for example, identified number combination skill as a key component of math proficiency. Moreover, research shows that number combination skill is a significant path to procedural computation and story-problem performance ( Fuchs et al., 2006).

To answer number combination problems, typical children gradually develop procedural efficiency in counting. First they count the two sets (e.g., 2+3) in their entirety (i.e., 1, 2, 3, 4, 5); then they count from the first number (i.e., 2, 3, 4, 5); and eventually they count from the larger number (i.e., 3, 4, 5). Also, as their conceptual knowledge about number becomes more sophisticated, children develop additional back-up strategies for deriving answers (e.g., decomposition: [2+2] = 4 + 1 = 5). As increasingly efficient counting and back-up strategies help children consistently and quickly pair problems with correct answers in working memory, associations become established in long-term memory, and children gradually favor memory-based retrieval of answers, although most continue to use a mix of strategies over time.

Students with mathematics disabilities, on the other hand, manifest greater difficulty with counting (e.g., Geary, Bow-Thomas, & Yao, 1992; Geary, Hoard, Byrd-Craven, Nugent, & Numtee, in press). They also persist with immature back-up strategies. Not surprisingly, therefore, they fail to make the shift to memory-based retrieval of answers ( Fleishner, Garnett, & Shepherd, l982; Geary, Widaman, Little, & Cormier, l987; Goldman, Pellegrino, & Mertz, l988). When children with mathematics disabilities do retrieve answers from memory, they commit more errors and manifest unsystematic retrieval speeds compared to younger, typically developing counterparts (e.g., Geary et al., in press; Geary, Brown, & Samaranayake, l991; Gross-Tsur et al., l996; Ostad, l997). In fact, number combinations constitute a signature deficit of students with mathematics disabilities (e.g., Fleishner et al., 1982; Geary et al., 1987; Goldman et al., 1988).

Conventionally, number combinations are incorporated into the curriculum at kindergarten through second grade, although most general education programs do not allocate systematic attention to developing strategies for fluent number combination performance. Nevertheless, most typically developing students are well on their way toward automatic retrieval of number combinations as their major solution strategy by the beginning of third grade. Therefore, when students still manifest deficiencies involving number combinations at the beginning of third grade, a pressing need exists for remediation.

In contrast to number combinations and other aspects of calculation skill, story problems incorporate linguistic information that requires children to construct a problem model. Whereas a calculation problem is already set up for solution, a story problem requires students to use text to identify missing information, construct the number sentence, and derive the calculation problem for finding the missing information. This transparent difference would seem to alter the nature of the task and, in fact, some research suggests that computation and math problem solving may be distinct aspects of mathematical cognition ( Fuchs et al., in press).

Four large-scale studies have addressed indirectly the distinction between computation and problem solving by examining the cognitive correlates of these domains of mathematical cognition in large representative samples.

In a study of 353 first through third graders, Swanson and Beebe-Frankenberger (2004) identified working memory as an ability that contributed to strong performance across both areas of mathematical cognition, but some unique cognitive abilities were also important, including phonological processing for computation and fluid intelligence as well as short-term memory for story problems.

In the second study, Swanson (2006) followed these students' development of calculation and problem-solving skill over one year. He identified predictors of computation (inhibition or controlled attention, vocabulary knowledge, visual-spatial working memory) that differed from problem solving (working memory's executive system; i.e., listening span, backward digit span, and digit/sentence span).

The third study involved a sample of 312 third graders. Here Fuchs et al. (2006) examined the concurrent cognitive correlates of computation versus story problems, this time controlling for the role of arithmetic skill within story problems. Teacher ratings of inattentive behavior emerged as a correlate common to both domains of math, but the remaining abilities differed: for computation, phonological decoding and processing speed; and for story problems, nonverbal problem solving, concept formation, sight word efficiency, and language.

Finally, the fourth study ( Fuchs et al., 2005) used beginning-of-the-year cognitive abilities to predict the development of skill across the year among 272 first graders. Results again suggested some common and some unique patterns of cognitive abilities. The common predictors were working memory and ratings of attentive behavior. The unique predictors were phonological processing for computation and nonverbal problem solving for simple word problems.

Across these studies, some findings recur; others are idiosyncratic. But together, results indicate that some abilities underlying these aspects of mathematical cognition are unique. This provides the basis for hypothesizing that the cognitive dimensions underlying difficulty in each of these domains may also be distinct.

More recently, Fuchs et al. (in press) assessed this question directly. Using a sample of 924 third graders from 89 classrooms, we (a) assessed the students on computation and problem solving; (b) classified them as having difficulty with computation, problem solving, both domains, or neither domain; and (c) measured them on nine cognitive dimensions. Difficulty occurred across the two math domains with the same prevalence as difficulty with a single domain, and specific difficulty with calculation occurred as frequently as specific difficulty with story problems. Moreover, specific computational difficulty was associated with a strength in language and weaknesses in attentive behavior and processing speed. By contrast, problem-solving difficulty was associated with poverty and deficient language.

These findings lend support to the hypothesis that computation and problem solving represent distinct domains of mathematical cognition in students at the lower ranges of performance, who might be identified with mathematics learning disabilities in the schools. Together, therefore, these lines of research suggest that practitioners need to consider computational skill and problem-solving skill separately in diagnosing and instructing students with learning disabilities.

Number Combination Deficits in Third Graders: A Validated Intervention

Math Flash

The number combinations intervention we researched is referred to as Math Flash because number combinations "flash" during the computerized practice activity. The Math Flash protocol relies on scripts (a) to clarify for tutors how to frame precise, effective explanations; and (b) to provide tutors a concrete model for how to implement the lessons. Tutors study scripts; they do not read them. Each lesson lasts 20-25 minutes, and the Math Flash standard protocol runs 16 weeks, with three sessions per week.

Math Flash addresses the 200 number combinations with addends and subtrahends from 0 to 9. Math facts are introduced in a deliberate order. For the first two lessons, tutors address facts of + 1 and - 1, using manipulatives and the number line, teaching the commutative property of addition, and emphasizing that this property does not apply to subtraction. In the next two lessons, facts of + 0 and - 0 are introduced, again using manipulatives and the number line. In lessons five and six, + 1, - 1, + 0, and - 0 math facts are reviewed.

In lesson seven, students begin learning doubles that run from 0 through 6 (i.e., 0 + 0, 1 + 1, 2 + 2, 3 + 3, 4 + 4, 5 + 5, 6 + 6, 0 - 0, 2 - 1, 4 - 2, 6 - 3, 8 - 4, 10 - 5, 12 - 6). Students work on these doubles using manipulatives and rehearsing doubles chants. At this point in Math Flash, mastery criteria are introduced, and students spend a minimum of one day on each lesson topic (so they do not waste time on facts they already know) and a maximum of four days on each lesson topic (to avoid students getting "stuck" on a topic and losing content coverage). Mastery is assessed in each lesson during computerized practice (see below). After doubles, students learn facts with + 2 and - 2. Manipulatives and the number line practice are used again.

Next, students are taught to use two strategies for answering a math fact. They are taught that if they "just know" the math fact, they "pull it out of your head." However, if they do not know an answer immediately, they count up. Counting-up strategies for addition and subtraction are taught using the number line and their fingers. To count up addition problems, students start with the bigger number and count up the smaller number on their fingers. The answer is the last number spoken.

For subtraction counting-up, new math vocabulary is introduced. The minus number is the number directly after the minus sign. The number you start with is the first number in the equation. To count up subtraction problems, students start with the minus number and count up to the number you start with. The answer is the number of fingers used to count up. From this point on, during every subsequent lesson, students are reminded to "Know It or Count Up."

Because students are now equipped with two strategies for answering math facts, the tutor introduces additional sets of number combinations, beginning with the 5 set. This includes all addition problems equaling 5 and all subtraction problems with 5 as the minuend: 0 + 5, 1 + 4, 2 + 3, 3 + 2, 4 + 1, 5 + 0, 5 - 0, 5 - 1, 5 - 2, 5 - 3, 5 - 4, 5 - 5. After mastery of the 5 set, students progress to the 6 set, then the 7 set, and so on: 8, 9, 10, 11, 12, 13, 14, 15, 16, and 17-18. The tutor works with the student on each set for a maximum of four days. Between the 12 set and the 13 set, students work on doubles of 7 through 10 (i.e., 7 + 7, 8 + 8, 9 + 9, 10 + 10, 14 - 7, 16 - 8, 18 - 9, 20 -10). If a student masters all sets before session 48, he/she reviews for the remaining Math Flash sessions.

Each of the 48 Math Flash daily lessons comprises five activities: flash card warm-up, conceptual and strategic instruction, lesson-specific flash card practice, computerized practice with mastery assessment, and paper-pencil review. In addition, throughout every lesson, a systematic reinforcement program is used to motivate good attention, hard work, and accurate work.

With flash card warm-up, tutors show flash cards, one at a time, for 2 minutes. The cards are a representative sample of the pool of the 200 number combinations addressed in Math Flash. Flash cards answered correctly are placed in a "correct" pile. When students answer incorrectly, the tutor instructs them to "count up." Students count up to produce the correct answer, but the card is placed in the "incorrect" pile. At the end of 2 minutes, the flash cards answered correctly are counted, and the student graphs this number on his/her flash card graph.

During conceptual and strategic instruction, tutors introduce or review concepts and strategies. Throughout, tutors emphasize the two strategies for deriving answers -"Know It or Count Up"- provide practice in counting up, and require students to explain how to count up addition and subtraction problems. Tutors then work with students on the day's number combinations set (e.g., +1 and -1, doubles 0-6, combinations of 12) using the number line and manipulatives.

After the tutor-led lesson, tutors conduct lesson-specific flash card practice for 1 minute. Lesson-specific flash cards consist of the math facts that are the focus of the day's lesson. (For example, if a lesson focuses on the 5 set, lesson-specific flash cards are facts with addends that sum to 5 and with minuends of 5.) Correctly answered flash cards are placed in the "correct" pile. When students answer incorrectly, tutors require them to "count up." The cards for which an incorrect answer is given are placed in the "incorrect" pile. After 1 minute, the lesson-specific flash cards answered correctly are counted, but the score is not graphed. On the second, third, and fourth days of a given lesson topic, students get a chance to beat their lesson-specific flash card score. Tutors remind students what their score on the first minute was and encourage them to answer more cards during the upcoming minute. Scoring and feedback are the same as in the first minute. Tutors praise students when they beat their score.

For the next 7.5 minutes, students complete computerized practice to build fluency with number combinations and to assess their mastery of the day's number combination set. Math facts presented on the computer game include 10 lesson-specific facts and 5 review facts.

The game goes like this. A math fact flashes on the screen for 1.3 seconds. Students rehearse the fact (e.g., 3 + 2 = 5) while it briefly appears; when the fact disappears, students retype the entire fact (e.g., addends and answer). If the answer is correct, the student hears applause and earns a point. If it is incorrect, the student has another chance to enter the problem correctly. The computer game ends after the student has answered each of the 10 lesson-specific facts correctly twice or after 7.5 minutes. The student then receives feedback on his/her mastery of correctly answered number combinations for that day's session.

Mastery on the lesson-specific number combinations set is assessed automatically as the student completes the computer game. Immediately after the game ends, the computer provides a report to the tutor. If the student answers each of the 10 lesson-specific facts correctly twice before 7.5 minutes elapses, mastered appears on the screen. If not, review appears on the screen. The tutor takes note of the student's mastery status, moving students onto the next number combinations set when mastery occurs, or after four days on a given set.

Finally, students complete a paper-and-pencil review. The student has 1 minute to complete 15 lesson-specific facts on one side of a paper and then has another minute to complete 15 review facts on the other side of the paper. At the end of 2 minutes, tutors circle correct answers and write the score at the top of the paper. Students take home these papers each day.

A systematic reinforcement program is also incorporated. Tutors award gold stars following each component of the tutoring session, with the option to withhold stars for inattention or poor effort. At the end of a session, each gold star is placed on a "Star Chart." Sixteen stars lead to a picture of a treasure box and, when this goal is reached, the student chooses a small prize from a real treasure box. The student keeps the Star Chart, and a new chart is provided during the next lesson.

Evaluation of Math Flash

During the 2006-2007 school year, Fuchs et al. (2007) randomly assigned 133 students in the Nashville Metropolitan Public Schools and in the Houston Independent School District who experienced substantial difficulty with computation and story problems (a) to receive the Math Flash standard intervention protocol, as just described; (b) to receive the Pirate Math standard intervention protocol addressing story problems, as described next; or (c) to continue in school without any additional math tutoring.

Half of the students in each intervention condition had math difficulties without reading difficulties; the other half experienced concurrent difficulty with math and reading. Each session was audiotaped. A representative sample of lessons was coded for fidelity against the tutoring scripts: fidelity was strong for both interventions. Students were pre- and posttested on measures of fluency with number combinations and on measures of problem solving. Math Flash and Pirate Math students (which incorporated instruction and practice on counting-up strategies, but less drill and practice and less conceptual instruction on number combinations) improved on number combinations fluency significantly more than students in the control group. The effect size comparing Math Flash to the control group was large: 0.85 standard deviation units. The effect size comparing Pirate Math to the control group was somewhat smaller but also substantial: 0.72. The difference in effect sizes may reflect less practice and less conceptual instruction on number combinations in Pirate Math.

Problem-Solving Deficits in Third Graders: A Validated Remediation

Pirate Math

The story-problem intervention is referred to as Pirate Math because posters and materials rely on a pirate theme. Again, scripts are studied, not read. Each lesson lasts 25-30 minutes, and the Pirate Math standard protocol runs 16 weeks, with three sessions per week. The 48 lessons are divided into four units. An introductory unit addresses skills foundational to story-problem instruction. Specifically, tutors (a) teach students the counting-up strategy for solving addition and subtraction number combinations; (b) review double-digit addition and subtraction; (c) teach students to solve for "X" in any position in simple algebraic equations (i.e., a + b = c; x - y = z); and (d) teach students to check their story-problem work.

The remaining three units focus on solving story problems while incorporating and reviewing the foundational skills taught in the introductory unit. Each unit introduces one story-problem type and, after the first problem-type unit, subsequent units provide systematic, mixed cumulative review that includes previously taught problem types. The problem types are Total (two or more amounts being combined), Difference (two amounts being compared), and Change (initial amounts that increase or decrease). In the Total unit, the first of the three problem types taught, tutors teach students to RUN through a problem: a three-step strategy prompting students to Read the problem, Underline the question, and Name the problem type. Students use the RUN strategy across all three problem types.

Next, for each problem type, students are taught to identify and circle relevant information. For example, for Total problems, students circle the item being combined and the numerical values representing that item, and then label the circled numerical values as "P1" (i.e., for part one), "P2" (i.e., for part two), and "T" (i.e., for the combined total). Students mark the missing information with an "X" and construct an algebraic equation representing the underlying mathematical structure of the problem type. For Total problems, the algebraic equation takes the form of "P1 + P2 = T." The "X" can appear in any of the three variable positions. Students are taught to solve for X, to provide a word label for the answer, and to check the reasonableness and accuracy of work. The strategy for Difference problems and Change problems follows similar steps but uses variables and equations specific to those problem types. For Difference problems, students are taught to look for the bigger amount (labeled "B"), the smaller amount (labeled "s"), and the difference between amounts (labeled "D"), and to use the algebraic equation "B - s = D." For Change problems, students are taught to locate the starting amount (labeled "St"), the changed amount (labeled "C"), and the ending amount (labeled "E"); the algebraic equation for Change problems is "St +/- C = E" (+/- depends on whether the change is an increase or decrease in amount).

Across problem types, explicit instruction to identify transfer features occurs in four ways. First, students are taught that because not all numerical values in story problems are relevant for finding solutions, they should identify and cross out irrelevant information. Second, students are taught to recognize and solve problems with the missing information in the first or second position. Third, students learn to apply the problem-solving strategies to problems that involve addition and subtraction with double-digit numbers, including those requiring regrouping. Finally, students are taught to find relevant information for solving problems in pictographs, bar charts, and pictures. Across the three problem-type units, previously taught problem types are included for review and practice.

Following the introductory unit (six lessons), each Pirate Math daily lesson comprises four activities: flash card warm-up, conceptual and strategic instruction, problem-type flash card practice, and paper-pencil review. Also, throughout every lesson, a systematic reinforcement program is used to motivate good attention, hard work, and accurate work.

The first activity, flash card warm-up, is identical to the flash card warm-up used for Math Flash. The second activity, conceptual and strategic instruction, lasts 15-20 minutes. Tutors provide scaffolded instruction in solving the three types of story problems, along with instruction on identifying and integrating transfer features, using role-playing, manipulatives, instructional posters, modeling, and guided practice. In each lesson, students solve three problems, with decreasing amounts of support from the tutor.

The third activity is sorting word problems. Tutors read aloud flash cards; one word problem is printed on each card. The student listens to the problem, identifies the word-problem type, and places the card on a mat with four boxes labeled "Total," "Difference," "Change," or "?." Students do not solve problems. They simply sort the problems into problem type. To discourage students from associating a particular cover story with a problem type, the cards use similar cover stories but with varied numbers, actions, and placement of missing information. After 2 minutes, the tutor notes the number of correctly sorted cards and provides corrective feedback for up to three errors.

For the last activity, paper-and-pencil review, students work independently. They have 2 minutes to complete 10 addition and subtraction number combinations and four addition and subtraction double-digit computation items, two of which require regrouping. Then, students have 2 minutes to complete one story problem on the back of the paper. Tutors provide corrective feedback and note the number of correct problems on the top of sheet. Students take home the paper-and-pencil review sheets. The Pirate Math reinforcement program is similar to the one used for Math Flash.

Evaluation of Pirate Math

In the efficacy study described earlier for Math Flash ( Fuchs et al., 2007), one of the three conditions was Pirate Math. In that study, story-problem performance improved significantly more for Pirate Math than for both Math Flash and the control group. The effect size comparing Pirate Math to Math Flash was large (0.72). The effect size comparing Pirate Math to the control group was somewhat larger but in the same range (0.89).

The First Six Principles of Effective Intervention for Students with Mathematics Disabilities

The Math Flash and Pirate Math interventions illustrate six principles of effective intervention for students with mathematics disabilities (see ). The first is instructional explicitness. Typically developing students profit from the general education mathematics program, which relies, at least in part, on a constructivist, inductive instructional style. Students who accrue serious mathematics deficits, however, fail to profit from those programs in a way that produces understanding of the structure, meaning, and operational requirements of mathematics.

Table 1

Instructional explicitness

Instructional design to minimize the learning challenge

Strong conceptual basis

Drill and practice

Cumulative review

Motivators to help students regulate their attention and behavior to work hard

Ongoing progress monitoring

A meta-analysis of 58 math studies ( Kroesbergen & Van Luit, 2003) revealed that students with math disability benefit more from explicit instruction than from discovery-oriented methods. Therefore, effective intervention for students with math disability requires an explicit, didactic form of instruction, in which the teacher directly shares the information the child needs to learn.

Explicitness is not sufficient, however. A second and often overlooked principle of effective intensive mathematics intervention is instructional design to minimize the learning challenge. The goal is to anticipate and eliminate misunderstandings with precise explanations and the use of carefully sequenced and integrated instruction so that the achievement gap can be closed as early as possible. This may be especially important for mathematics, which involves many branches and strands that may be distinct, each with its own conceptual and procedural demands. So, given the ever-changing and multiple demands of the mathematics curriculum, instructional efficiency is critical, creating the need for the tutor, or the program on which the tutor relies, to minimize the learning challenges for the student.

Careful instructional design is illustrated in Math Flash in terms of sequencing, where the introduction of number combination sets is ordered to capitalize on the knowledge the student possesses, while maximizing the student's rate of acquisition and sense of accomplishment. So, for example, we introduce + 1/ - 1 first for three reasons: It corresponds well to a basic number line concept that is easy to teach, if not already mastered; it eases counting demands; and it results in the student mastering 29 of the 200 number combinations in the first week of the program. We introduce + 0/- 0 next for three reasons: Although conceptually demanding, it builds well on + 1/- 1; it presents no counting demands; and it efficiently creates mastery of another large group of number combinations. In a related way, foundational concepts are presented only as they become instrumental to learning, and strategies are taught only as they become broadly applicable. So, for example, "Know It or Count Up" is introduced (a) after enough "Know It" combinations (+ 1/- 1, + 0,- 0, easy doubles) have been addressed to make the "Know It" strategy salient for the student and (b) after enough number combinations for which counting-up is an efficient strategy have been introduced.

Instructional design that minimizes the learning challenge is also illustrated in Pirate Math. Take three examples. Pirate Math begins by teaching a set of foundational skills the student can apply across the entire program: counting up for number combinations, two-digit calculations, solving algebraic equations, and checking work. Once story-problem instruction begins, these foundational skills can be taught as intact instructional targets and then applied efficiently across subsequent units.

A second example is that Pirate Math purposefully conceptualizes, organizes, and teaches students to recognize problem types that pertain broadly to the kinds of problems found in the general education curriculum and in high-stakes tests. That way, novel story problems are not random events, each requiring the student to create a solution strategy. Rather, the student recognizes novel problems as familiar, using schemas for problem types the program teaches, and thereby deciphers when to apply which set of solution rules they have learned.

A third example is that Pirate Math conceptualizes transfer within the same problem-type structure, so that irrelevant information, finding missing information in any of the three slots of an equation, and finding relevant information within charts/graphs recur predictably and efficiently across problem-type instructional units.

The third principle of effective intensive mathematics intervention is the requirement that instruction provide a strong conceptual basis for the procedures that are taught. Special education is already strong in emphasizing drill and practice, a critical and fourth principle of effective practice. However, it has sometimes neglected the conceptual foundation of mathematics, and such neglect can result in confusion, learning gaps, and a failure to maintain and integrate previously mastered content. Math Flash illustrates the need for conceptual instruction as it relies strongly on manipulatives and the number line to introduce and review each number combination set. Pirate Math illustrates the need for conceptual foundation with its use of role-playing for each problem type.

Drill and practice, the fourth principle of effective intensive mathematics intervention, is also evident in our sample interventions: in Math Flash, with the use of flash cards, computerized practice, and daily review; in Pirate Math, with practice in sorting problems into problem types, the mixing of problem types within the daily lesson (once at least two problem types have been introduced), and daily review.

We note that this practice is rich in cumulative review, a fifth principle of effective intensive mathematics intervention. This is reflected in Math Flash's flash card warm-up activity, computerized practice, and paper-pencil review. It is also incorporated in Pirate Math's continual reliance on the foundational skills taught in the introductory unit, the use of mixed problem types within conceptual instruction, sorting practice, and paper-pencil review.

Finally, intensive mathematics interventions need to incorporate motivators to help students regulate their attention and behavior and to work hard. Students with learning disabilities often display attention, motivation, and self-regulation difficulties, which may adversely affect their behavior and learning ( Fuchs et al., 2005, 2006; Montague, 2007). By the time students enter intensive intervention, they have experienced repeated failure, causing many to avoid the emotional stress associated with mathematics. As a result, they no longer try to learn for fear of failing. For this reason, intensive intervention must incorporate systematic self-regulation and motivators, and for many students, tangible reinforcers are required. Math Flash illustrates this principle (a) using the "beat your score" flash card activity, where goal-directed behavior is required; (b) when students graph scores they achieve; and (c) in the use of stars as rewards. Pirate Math incorporates similar activities to help students regulate attention and behavior and to work hard.

Valid Is Not the Same as Universally Effective: Progress Monitoring as the Most Essential Principle of Intensive Intervention

As summarized in this article (see Fuchs et al., 2007, for a full report), Math Flash and Pirate Math are demonstrably efficacious, resulting in statistically significant and practically important effects on number combinations fluency (i.e., Math Flash and Pirate Math, each compared to a control group) and on problem solving (i.e., Pirate Math, compared both to a control group and to the Math Flash group). Yet, no instructional method, even those validated using randomized control studies, works for all students.

We use Math Flash and Pirate Math to illustrate this point. Although Math Flash resulted in statistically significantly better improvement in number combinations fluency compared to the control group, with an effect size of 0.88, 3 of 21 Math Flash students failed to realize improvement that exceeded the 30th percentile among the 66 study participants. Moreover, although the effects for Pirate Math were statistically significant (effect sizes of 0.82 and 0.97), the improvement/outcome scores of 8 of 22 students did not exceed the 30th percentile - this time, using a comparison group of 159 students whose math performance was typically developing and who received conventional classroom problem-solving instruction over the course of third grade.

When a standard protocol of validated instruction proves ineffective for a given student, this is unexpected because research has shown that the intervention works for the majority of students. Therefore, when unexpected unresponsiveness occurs, we assume the student has individual needs that are unusual or special, such that he or she requires an individually tailored instructional program.

Because schools must assume that validated intervention protocols will work for most, but not all, students, schools need to monitor the effects of interventions on individual children's learning. That way, children who do not respond adequately can be identified promptly, and the teacher can adjust the intervention to develop an individually tailored instructional program that does work for the student. This leads us to propose a seventh and most essential element of intensive remedial programming: ongoing progress monitoring. Progress monitoring is used to determine whether a validated treatment protocol is, in fact, effective for a given student. When progress monitoring reveals that a student is failing to respond as expected to a validated intervention protocol, progress monitoring is then used for a second purpose: to formulate an individually tailored instructional program that is in fact effective for that student.

Curriculum-Based Measurement

In this section, we describe curriculum-based measurement (CBM), the form of progress monitoring for which the preponderance of research has been conducted. CBM differs from most forms of classroom assessment in two major ways ( Fuchs & Deno, 1991). First, it is standardized so that the behaviors to be measured and the procedures for measuring them are prescribed, with documented reliability and validity. Second, CBM's testing methods and content remain constant, with equivalent weekly tests that span much, if not all, of the school year. (The primary reason for long-term consistency is so that progress can be monitored systematically and coherently over time.)

To illustrate how CBM is used, let's say Roberto, a hypothetical student, developed sizeable math deficits over the course of first and second grade, despite strong general education programming and despite small-group tutoring implemented during the spring semester of second grade. At the beginning of third grade, Roberto is identified for remedial intervention. Mrs. Hayes, the special education teacher, sets Roberto's mathematics goal for year-end performance as competent second-grade performance (which is transparently connected to the third-grade mathematics curriculum, but includes some easier problem types that create the platform for learning the harder, third-grade problems).

Relying on established methods, Mrs. Hayes creates or identifies enough CBM tests to assess Roberto's performance each week across the school year. Each of these tests systematically samples the second-grade mathematics curriculum in the same way. Each week, Mrs. Hayes administers one computation test in exactly the same way; Roberto has 3 minutes to complete as many problems as he can. The score is the number of correct digits written in answers. Each week, Mrs. Hayes also administers one concepts/applications test in exactly the same way. Roberto has 6 minutes to complete as many problems as he can; the score is the number of correct points written in answers.

Each progress-monitoring test collected across the school year is of equivalent difficulty, and the score on each week's CBM test is an indicator of mathematics competence at second grade. At the beginning of the year, we expect Roberto's performance to be low. As Mrs. Hayes teaches and Roberto learns the second-grade curriculum, we expect Roberto's score to gradually increase. Because each progress-monitoring test collected across the school year is of equivalent difficulty, each week's computation scores can be graphed and directly compared to each other. Also, each week's concepts/application scores can be graphed and compared to each other. Moreover, on each graph (computation; concepts/applications), a slope can be calculated on the series of scores. This slope quantifies Roberto's rate of mathematics improvement in terms of the weekly increase in score. In addition, because each week's assessment samples the annual curriculum in the same way, Mrs. Hayes can derive a systematic analysis of which skills Roberto has and has not mastered at any point in time, and Mrs. Hayes can look across time at a given skill to determine whether Roberto is retaining mastery.

A large body of work indicates that CBM progress monitoring enhances teachers' capacity to plan mathematics programs and to effect stronger mathematics achievement among students with serious learning problems ( Fuchs & Fuchs, 1998). To inform instructional planning, teachers rely on the CBM graphed scores. Once the teacher sets the year-end goal, she draws the desired score on the graph at the date corresponding to the end of the year. The teacher then draws a straight line connecting the student's beginning-of-the-year score with the year-end goal. This line is called the goal line. It represents the approximate rate of weekly improvement (or slope) we hope a student will achieve. When a student's trend line (i.e., the slope through the student's actual scores) is steeper than the goal line, the teacher increases the goal for the student's year-end performance. On the other hand, when a student's trend line is flatter than the goal line, the teacher relies on her knowledge about the student along with her CBM analysis of the student's skills, derived from the CBM data, to revise the instructional program in an attempt to boost the weekly rate of student learning.

Research shows that with CBM decision rules, teachers design more varied instructional programs that are more responsive to individual needs ( Fuchs, Fuchs, & Hamlett, 1989b), that incorporate more ambitious student goals ( Fuchs, Fuchs, & Hamlett, 1989a), and that result in stronger end-of-year scores on commercial, standardized reading tests (e.g., Fuchs, Fuchs, Hamlett, & Stecker, 1991).

Let's return to Roberto and Mrs. Hayes to illustrate how a teacher uses CBM in these ways to monitor the effectiveness of a validated treatment protocol and, when that protocol proves ineffective for an individual student, how a teacher uses CBM to inductively develop an individually tailored instructional program. When Mrs. Hayes assumed responsibility for Roberto's math remediation program, she decided to use the validated protocol Pirate Math. This entailed tutoring for 30 minutes per session three times per week. As Mrs. Hayes began to implement this validated protocol, she also began to administer second-grade CBM tests once each week for computation and once each week for concepts/applications.

After the first three weeks, Mrs. Hayes calculated the median of the first three computation test scores (5) and the median of the first three concepts/application test scores (3). These represented Roberto's baseline or beginning-of-the-year scores. Using CBM guidelines for goal setting, Mrs. Hayes decided that her year-end goal for Roberto would seek a weekly increase of .5 digits for computation and a weekly increase of .6 points for concepts/applications. So 25 weeks later, at the end of the school year, Roberto would score 18 digits correct on second-grade CBM computation and 18 points correct on second-grade CBM concepts/applications.

Mrs. Hayes drew these scores onto Roberto's graphs, one for computation and the other for concepts/applications, and then connected Roberto's baseline scores with the year-end goal to show the goal lines, or desired weekly rates of improvement (see for computation; see for concepts/applications). Ten weeks later, Mrs. Hayes then drew lines of best fit through Roberto's actual CBM scores and compared these trend lines to the goal lines. The CBM data showed that Pirate Math, with its focus on the counting-up strategy for number combinations and with its foundational focus and ongoing review of double-digit addition and subtraction, was producing strong growth for Roberto. As shown in , Roberto's actual rate of improvement (solid diagonal line) was steeper than the goal line (broken diagonal line).

By contrast, Roberto was proving insufficiently unresponsive to Pirate Math's word-problem instruction. As shown in , Roberto's actual rate of improvement was dramatically less steep than the goal line (broken diagonal line), indicating that he was growing slower than hoped and was unlikely to achieve his year-end goal. Therefore, Mrs. Hayes modified the Pirate Math standard protocol. She considered Roberto's performance during tutor sessions and reviewed Roberto's performance on the CBM concepts/applications story problems. She determined that Roberto was having difficulty differentiating problem types when irrelevant information was included in problems and when the missing information in story problems occurred anywhere but in the final position in the number sentence. Based on this analysis, Mrs. Hayes decided (a) to add instruction on mixed problem types, (b) to lengthen the problem-type sorting activity, and (c) to add instructional time on irrelevant information and deriving number sentences when the missing information is in the first or second slot of the equation. This revision in the intervention protocol is signified on Roberto's CBM concepts/applications graph with the dotted vertical line. As she implemented this revision in the intervention protocol, Mrs. Hayes continued to monitor Roberto's responsiveness using weekly CBM concepts/applications assessments. As indicated in , Roberto's learning, as shown in the new trend line, improved and was now steeper than the goal line Mrs. Hayes had set for Roberto. It is in this formative, inductive, and recursive way that teachers use CBM to derive individual instructional programs that are effective for individual students and increase the probability of improved student outcomes.

Implications for Practice

In this article, we have argued for the importance of using seven principles in designing effective intensive interventions for students with mathematics disabilities: instructional explicitness, instructional design to minimize learning challenges, conceptual foundation, drill and practice, cumulative review, systematic motivation to promote self-regulation and encourage students to work hard, and ongoing progress monitoring to quantify response and formulate individually tailored programs as needed. We noted that two of these principles, instructional design to minimize learning challenges and attention to the conceptual foundation of the mathematics, are often overlooked. We also emphasized that the last principle, ongoing progress monitoring to quantify response and formulate individually tailored programs, may be the most essential principle of intensive intervention.

Issues and Directions for Future Research in the Primary and Later Grades

Although we focused on third grade here, specifically with respect to deficits with number combinations and story problems, similar instructional principles recur in other programs of research (e.g., Allsop, 1997; Jitendra, DiPipi, & Perron-Jones, 2002; Macinni, Mulcahy, & Wilson, 2007; Miller & Hudson, 2007; Montague, 2007) conducted at higher grades and on other aspects of the mathematics curriculum. However, additional work is required. Major aspects of the mathematics curriculum have received inadequate attention. For example, as school systems ratchet up high-school graduation requirements, interventions (see, e.g., Allsop, 1997) need to be expanded to ensure mastery of the algebra curriculum. In a related way, as the demands of a technology-rich environment continue to grow, additional attention is needed to address everyday mathematics problem solving, which requires flexible use of computational skills combined with knowledge about how to find relevant information within real-life situations.

Given the scope of intervention development still needed to ensure intensive intervention in mathematics, a second major issue that warrants the field's attention is professional development. The question is how to prepare new and practicing special educators to master and integrate the principles of effective practice so they (a) can design their own intensive interventions to address the skill deficits their students manifest and (b) have the knowledge to formatively develop effective program revisions when students do not respond as expected to intensive intervention protocols.

Unfortunately, special education training programs in universities and special education professional development opportunities within school districts focus disproportionately on reading intervention, with less attention to mathematics. Given the scope of the mathematics curriculum, along with many teachers' discomfort with their own mathematics skills, coursework and additional texts must be developed to assist university professors and school districts in preparing their trainees and teachers to understand and effectively employ principles of effective intensive mathematics intervention.

Finally, and in a related way, although a sense of urgency for improving student outcomes has been effected for reading, the same is not true for mathematics. The majority of students with learning disabilities suffer large deficits in both domains, but time is allocated disproportionately to reading intervention ( Rasanen & Ahonen, 1995). If mathematics difficulties are to be addressed effectively, a similar sense of urgency needs to be mustered to (a) permit adequate opportunity for intensive intervention in mathematics and (b) open doors for researchers to develop and validate intensive tutoring protocols.

Acknowledgments

The research described in this paper was supported in part by Grant #1 P01046261 and Core Grant #HD15052 from the National Institute of Child Health and Human Development to Vanderbilt University. Statements do not reflect the position or policy of these agencies, and no official endorsement by them should be inferred.

Contributor Information

Lynn S. Fuchs, Vanderbilt University.

Douglas Fuchs, Vanderbilt University.

Sarah R. Powell, Vanderbilt University.

Pamela M. Seethaler, Vanderbilt University.

Paul T. Cirino, University of Houston.

Jack M. Fletcher, University of Houston.

References

Badian NA. Dyscalculia and nonverbal disorders of learning. In: Myklebust HR, editor. Progress. New York: Grune & Stratton; 1983. pp. 235-264.

Compton DL, Fuchs LS, Fuchs D. The course of reading and mathematics disability in first grade: Identifying latent class trajectories and early predictors. 2006 Manuscript submitted for publication.

Fletcher JM, Lyon GR, Fuchs LS, Barnes MA. Learning disabilities: From identification to intervention. New York: Guilford; 2006.

Fuchs LS, Fuchs D, Stuebing K, Fletcher JM, Hamlett CL, Lambert WE. Problem solving and calculation skill: Shared or distinct aspects of mathematical cognition? Journal of Educational Psychology in press. [PMC free article]

Fuchs LS, Powell SR, Seethaler PM, Fuchs D, Hamlett CL, Cirino P, Fletcher JM. Intensive intervention on number combination and story problem deficits in third graders with math difficulties, with and without concurrent reading difficulties. 2007 Manuscript in preparation.

Geary DC, Bow-Thomas CC, Yao Y. Counting knowledge and skill in cognitive addition: A comparison of normal and mathematically disabled children. Journal of Experimental Child Psychology. 1992;54:372-391.

Geary DC, Hoard MK, Byrd-Craven J, Nugent L, Numtee C. Cognitive mechanisms underlying achievement deficits in children with mathematics learning disability. Child Development in press. [PMC free article]

Griffin SA, Case R, Siegler RS. Rightstart: Providing the central conceptual prerequisite for first formal learning of arithmetic to students at risk for school failure. In: McGilly K, editor. Classroom lessons: Integrating cognitive theory and classroom practice. Cambridge, MA: MIT Press; l994. pp. 25-50.

Gross-Tsur V, Manor O, Shalev RS. Developmental dyscalculia: Prevalence and demographic features. Developmental Medicine and Child Neurology. 1996;37:906-914.

Kilpatrick J, Swafford J, Findell B, editors. Adding it up: Helping children learn mathematics. Washington, DC: National Academic Press; 2001.

Ostad SA. Developmental differences in addition strategies: A comparison of mathematically disabled and mathematically normal children. British Journal of Educational Psychology. 1997;67:345-357.

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Intensive Intervention for Students with Mathematics Disabilities: Seven Principles of Effective Practice

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Lynn S. Fuchs, Ph.D., Douglas Fuchs, Ph.D., Sarah R. Powell, M.S., Pamela M. Seethaler, M.S., Paul T. Cirino, Ph.D., and Jack M. Fletcher, Ph.D. The focus of this article is intervention for third - grade students with serious mathematics deficits at third grade.

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School-Wide Strategies for Math Interventions *P*

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School-Wide Strategies for Managing... MATHEMATICS | Intervention Central

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Mathematics instruction is a lengthy, incremental process that spans all grade levels. As children begin formal schooling in kindergarten, they develop 'number sense', an intuitive understanding of foundation number concepts and relationships among numbers. A central part of number sense is the student's ability to internalize the number line as a precursor to performing mental arithmetic.

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"Do The Math"

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Help Kids Learn Math with Do The Math by Marilyn Burns

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Help Kids Learn Math with Math Instruction from America's Most Trusted Educator using Scholastic's Do The Math program by Marilyn Burns

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Effective Math Instruction *P*

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RTI and Math Instruction *P*

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Clements, D. H., & Sarama, J. (2007). Effects of a preschool mathematics curriculum: Summative research on the Building Blocks project.

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Using RTI to Improve Learning in Mathematics

Response to Intervention (RTI) has become a vehicle for system reform because it provides a framework in which data can be relied on as the basis for making relative judgments (e.g., determining who needs help the most and how much they need) and for distributing instructional resources to promote the greatest good for the greatest number of students.
Much of the writing and research on RTI has occurred in the area of reading, but RTI is not limited to reading. Rather, it is a science of decision making that can be applied to a variety of "problem" behaviors. RTI, properly understood and used, is focused on improving student learning. Ensuring the development of mathematics competence during the primary grades is essential to later learning success. Key findings in the literature highlight the need to focus on early mathematics instruction:

1. Children who have had less experience or exposure to mathematical concepts and numeracy are at high risk for mathematics failure (Griffin & Case, 1997).

2. Most students fail to meet minimal mathematics proficiency standards by the end of their formal schooling (U.S. Department of Education, 2003).

3. Students identified with specific learning disabilities perform lower and grow at a slower pace relative to their peers in learning mathematics.

4. Existing instructional tools and textbooks often do a poor job of adhering to important instructional principles for learning in mathematics (National Mathematics Advisory Panel, 2008).

5. Math is highly proceduralized and continually builds on previous knowledge for successful learning. Hence, early deficits have enduring and devastating effects on later learning, as indicated in The Head Start Path to Positive Child Outcomes (U.S. Department of Health and Human Services, 2001) and elsewhere (e.g., National Mathematics Advisory Panel, 2008; National Council of Teachers of Mathematics [NCTM], 2000; U.S. Department of Education, 2003).

6. Early mathematics intervention can repair deficits and prevent future deficits (Clements & Sarama, 2007; Fuchs, Fuchs, & Karns, 2001; Fuchs, Fuchs, Yazdian, & Powell, 2002; Griffin & Case, 1997; Sophian, 2004).

In mathematics, a reform process similar to what occurred in reading in the 1990s appears to be underway. Whereas math has been underresearched relative to reading, research findings are available to guide the application of RTI in mathematics. Specifically, research is available to guide the selection of adequate screening measures, selection of adequate progress-monitoring measures, development of decision criteria, and development of intervention protocols appropriate for use at all tiers of instruction. To use RTI in mathematics, a district or school must first select a model of RTI, identify adequate screening and progress-monitoring measures, and plan for effective delivery of intervention at Tiers 1, 2, and 3.

Useful Screening and Progress Monitoring Measures in Mathematics

Data are required for decision making in RTI. Generally, three decisions must be made: Who needs intervention? What type of intervention is needed? And is the intervention working?

Who Needs Mathematics Intervention?

To identify who needs intervention, educators need sensitive screening tools. The screening task should be a task that is closely aligned with expectations for learning in the classroom at that point in the instructional program. Published performance standards available in every state are an excellent basis for selecting a screening task. The screening task is used to make both relative (how one student's performance compares to another's in the same class or grade) and absolute judgments (how one student's or all students' performance compares to expectations for performance at that time in the year). Both judgments are necessary to correctly define a problem and determine what type of intervention or interventions are needed. Some screening models make only one judgment (relative or absolute), and this causes decision errors.

Curriculum-based measurement (CBM) probes of basic (e.g., sums to 12, subtraction 0-20, fact families) and advanced computation skills (e.g., finding least common denominator, multidigit multiplication with regrouping, converting numbers to percentages, solving equations) are empirically supported for screening (VanDerHeyden, Witt, & Naquin, 2003; VanDerHeyden & Witt, 2005). These measures have been found to yield reliable scores over time that correlate moderately with other more comprehensive measures of mathematics performance. Research indicates that the use of computation-only assessment and intervention has demonstrated value for early identification of children who are likely to struggle with advanced problem solving in mathematics. Because these probes can be administered to an entire class at one time and require only two minutes of the student's time, they are currently the measures of choice for screening in mathematics. To identify the screening task, the RTI consultant should print out the state standards for mathematics, review the computation-oriented objectives in sequence, and consult with teachers at a grade-level meeting to determine where students are in the instructional program (i.e., what students are expected to know how to do at that time of year to benefit from continued mathematics instruction). CBM probes can be purchased from a variety of sources (e.g., Sopris West Educational Services) or built for free using online tools (e.g., Intervention Central) or using inexpensive software (such as those on the Schoolhouse Technologies Web site).

What Type of Intervention Is Needed?

As noted previously, areas where many children (e.g., 25%-50% of those screened) perform poorly indicate the need for system-wide interventions in which all children receive the intervention (i.e., Tier 1). Where small numbers of children perform below the criterion (e.g., 2-4 students per class), small- or large-group intervention is indicated (i.e., Tier 2). Where only a few children perform poorly (e.g., fewer than 1 student per class on average), individualized intervention (i.e., Tier 3) may be immediately planned and implemented for those students. Alternatively, students who fail to respond at Tier 1 or Tier 2 may be provided with a higher level intervention-Tier 2 or Tier 3, respectively.

Following the collection of screening data, the decision team must determine whether a systemic problem exists. Where systemic learning problems are identified, the core program of instruction should be evaluated to ensure that a research-supported curriculum is being used, that instruction is being delivered for sufficient duration and with sufficient quality, and that adequate resources are available to support effective instruction. The adequacy of the core instructional program in mathematics can be evaluated by comparing existing instructional procedures to elements of known effective instructional programs. Several panels have identified the use of routine assessment to continuously guide and refine instruction efforts (and effects) as a hallmark of effective instruction in mathematics (e.g., NCTM, 2000; U.S. Department of Education, 2003).

Effective mathematics instruction should include a system for monitoring student learning and adjusting instructional efforts to ensure adequate learning or accelerate it where needed. Other variables of effective instruction that are relevant include a well-sequenced program of instruction that logically builds on existing skills and periodically returns to previously mastered skills to ensure maintenance, demonstration of correct and incorrect responses, and substantial opportunity to practice performing newly learned skills with direct support (especially immediate corrective feedback) followed by more independent practice once the probability of errors is very low (or once accurate responding is a relatively sure thing).

Once a systemic problem is ruled out (or resolved through intervention), two groups of students might remain. First, there might be a subset of students who are performing below their classmates and in the risk range. These students' performances may be similar to each other. These students may be targeted for Tier 2 intervention programming. Because their performances are similar, intervention materials and procedures can be geared toward the needs of the group (e.g., introduction of a skill to be learned, fluency building of an already acquired skill, teaching a prerequisite skill to fluency, guided practice to apply the skill under novel conditions). Tier 2 program features should include similar characteristics to those of effective Tier 1 programs (e.g., well-sequenced, ensuring mastery of skills as instruction progresses, adequate corrective feedback matched to student level of competence).

Effective Tier 2 programs for mathematics will emphasize matching the task difficulty to the capability of the students in the group, providing high numbers of opportunities to practice the skill and receive performance first under tightly controlled and stable conditions and later as the skill improves under variable conditions (e.g., with different materials, different problem presentations). Supplemental programs can be purchased to assist with intervention at Tier 2 (e.g., FASTT Math, Accelerated Math), and existing resources in the school can often be retooled to better serve the needs of children in need of Tier 2 intervention (e.g., the resource teacher or intervention coach could provide small-group intervention to Tier 2 students, with daily progress monitoring).

The second group likely to remain once Tier 2 students have been identified would comprise those students whose performances were below that of their classmates and in the risk range at screening, and for whom subsequent assessment shows extensive skill gaps. For these students, Tier 3 intervention should be implemented. Functional academic assessment will be necessary to build an intervention that adequately addresses weak skills for this group of students. Tier 3 programs should include a data-based process for identifying specific causes of poor performance in mathematics, and individual interventions should be developed to target those specific deficits while monitoring both intervention-specific and generalized improvements in mathematics. Children who receive Tier 3 intervention may require specific training to learn how to apply learned skills under conditions that are required in the regular classroom.

Intervention procedures and materials at Tiers 2 and 3 might come from published resources (see the What Works Clearinghouse site for reviews of intervention programs), but up-front assessment will be needed to match the student with the right intervention and to obtain formative data to alter the intervention as needed to maximize intervention effects. Movement between the tiers can be bidirectional and children can go directly to the most intensive intervention level (Tier 3) if needed. Students who are already identified and receiving special education services could participate in RTI and receive instruction at Tiers 1, 2, or 3. Finally, assessment data collected at Tier 3 may be most useful for eligibility determination and individualized education program planning if a student is found eligible.

Is the Mathematics Intervention Working?

To evaluate intervention effects, two judgments must be made. First, was the intervention provided as planned for a sufficient period of time? Second, did student performance improve "enough?" The first question requires the collection of evidence of intervention implementation. The second question involves the collection of student performance data before intervention and with intervention relative to some criterion. To decide if intervention growth was sufficient, implementers might use a student performance criterion that reflects the average performance of students who are not at risk at screening at that school or that reflects the level of performance that has been associated with "passing" the accountability test in the spring, or the average performance of peers in the same class on that task. This second question poses greater measurement challenges in RTI (although research is making steady progress toward improving the adequacy of measurement tools and procedures for this purpose).

The most rigorous analysis of whether the intervention has successfully solved a learning problem is to consider post-intervention performance on both the original and an updated screening task and evidence that the learning improvements caused by the intervention generalize to improved classroom performance and learning. Importantly, some researchers have demonstrated that direct measures of conceptual understanding in mathematics have a value-added effect to understanding a student's capacity in mathematics. In other words, computational skill is a meaningful and important precursor to successful mathematics learning, including applying learned skills to solve novel and more complex problems, but some students may demonstrate high levels of computational fluency with basic facts and operations yet fail to gain conceptual understanding. One logical and empirically supported approach is to address computation fluency problems where they are detected with intervention, and to directly assess conceptual understanding on related tasks (e.g., counting vs. more/less judgments; multiplication/division facts vs. solving a word problem that involves building equal sets).

Attending to Implementation Fidelity and System Change to Ensure Desired and Sustainable Outcomes for Mathematical Learning

It is important to note that whatever program of instruction is identified and implemented, research data tell us that deliberate planning and monitoring of implementation fidelity will be necessary to ensure desired outcomes. To enhance the sustainability of the RTI effort in mathematics, implementers should make every attempt to integrate the RTI math effort with ongoing system reform efforts. For example, RTI progress-monitoring data may have benefits in the following areas:

1. Personnel review and professional development;

2. Standards-based instruction efforts;

3. Efforts to promote effective instruction schoolwide, and

4. Monitoring outcomes of change efforts

Implementation fidelity should be directly measured at all tiers. The most efficient way to monitor implementation fidelity is to track student performance (e.g., use the progress-monitoring data) and where student performance is not adequate, conduct a direct observation of instruction in the classroom to determine the percentage of intervention steps that are being completed as planned. This observation provides an occasion to provide direct coaching on how to effectively implement the intervention.

RTI is a logical system of data-based decision making that can permit districts, schools, and teachers to evaluate the adequacy of ongoing mathematics instruction and to systematically chart a plan to accelerate learning in mathematics for all students and for those who are at risk for failure without intervention.

References

Journal for Research in Mathematics Education, 38, 136-163.

Fuchs, L. S., Fuchs, D., & Karns, K. (2001). Enhancing kindergartners' mathematical development: Effects of peer-assisted learning strategies. The Elementary School Journal, 101, 495-510.

Fuchs, L. S., Fuchs, D., Yazdian, L., & Powell, S. R. (2002). Enhancing first grade children's mathematical development with peer-assisted strategies. School Psychology Review, 31, 569-583.

Griffin, S., & Case, R. (1997). Re-thinking the primary school math curriculum: An approach based on cognitive science. Issues in Education, 3, 1-49.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Mathematics Advisory Panel (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel, U.S. Department of Education: Washington, DC.

Sophian, C. (2004). Mathematics for the future: Developing a Head Start curriculum to support mathematics learning. Early Childhood Research Quarterly, 19, 59-81.

U.S. Department of Education. (2003). Mathematics and science initiative concept paper. Retrieved June 1, 2004, from http://www.ed.gov/rschstat/research/progs/mathscience/concept_paper.pdf.

U.S. Department of Health and Human Services, Administration on Children, Youth and Families/Head Start Bureau. (2001). The Head Start path to positive child outcomes. Washington, DC: Author.
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RTI can ensure the development of mathematics competence during the primary grades which is a key to later learning success.

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Mathematics Intervention at the Secondary Prevention Level of a Multi-Tier Prevention System: Six Key Principles *P*

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Mathematics interventions at the Tier 2 (secondary prevention) level of a multi-tier prevention system must incorporate six instructional principles:

1. Instructional explicitness

2. Instructional design that eases the learning challenge

3. A strong conceptual basis for procedures that are taught

4. An emphasis on drill and practice

5. Cumulative review as part of drill and practice

6. Motivators to help students regulate their attention and behavior and to work hard

This article describes each of these principles and some of their research underpinnings, with consideration given to how the principles can be implemented in real-world teaching contexts.

The first principle of effective intervention in mathematics at the secondary prevention level is instructional explicitness. Typically developing students profit from the general education mathematics program even though it relies, at least in part, on a constructivist, inductive instructional style. Students who are at risk for serious mathematics deficits, however, fail to profit from those programs in a way that produces understanding of the structure, meaning, and operational requirements of mathematics. A meta-analysis of 58 math studies (Kroesbergen & Van Luit, 2003) revealed that students with math disability benefit more from explicit instruction than from discovery-oriented methods. Therefore, effective intervention in Tier 2 requires an explicit, didactic form of instruction in which the teacher directly shares the information the child needs to learn.

Explicitness is not, however, sufficient. A second and often overlooked principle of effective secondary mathematics intervention is instructional design that eases the learning challenge. The goal is to anticipate and eliminate misunderstandings by means of precise explanations and with the use of carefully sequenced and integrated instruction. The purpose is to close the achievement gap as quickly as possible. This may be especially important for mathematics, which involves many branches and strands that may be distinct, each with its own conceptual and procedural demands. So, given the ever-changing and multiple demands of the mathematics curriculum, instructional efficiency is critical, creating the need for the tutor or the program on which the tutor relies to minimize the learning challenges for the student.

Careful instructional design begins by teaching a set of foundational skills the student can apply across the entire program: counting up for number combinations, two-digit calculations, solving algebraic equations, and checking work. These foundational skills can be taught as intact instructional targets and then applied efficiently across subsequent units once word problem instruction begins. Then instruction purposefully conceptualizes, organizes, and teaches students to recognize problem types that pertain broadly to the kinds of problems found in the general education curriculum and in high-stakes tests. That way, novel word problems are not random events for students, each of which requires the creation of a solution strategy. Rather, the student recognizes novel problems as familiar, using schemas for problem types that the program teaches, and thereby deciphers when to apply which set of solution rules he or she has learned. Finally, instruction conceptualizes transfer within the same problem-type structure, so that irrelevant information, finding missing information in any of the three slots of an equation, and finding relevant information within charts or graphs recurs predictably and efficiently across problem-type instructional units.

The third principle of effective secondary mathematics intervention is the requirement that instruction provide a strong conceptual basis for procedures that are taught. Special education is already strong in emphasizing drill and practice, a critical and fourth principle of effective practice. Special education has, however, sometimes neglected the conceptual foundation of mathematics, and such neglect can result in confusion, learning gaps, and a failure to maintain and integrate previously mastered content. Drill and practice, the fourth principle of effective secondary mathematics intervention, includes practice in sorting problems into problem types, the mixing of problem types within the daily lesson (once at least two problem types have been introduced), and daily review. We note that this practice should be rich in cumulative review, a fifth principle of effective secondary mathematics intervention. This is reflected in continual reliance on the foundational skills taught in the introductory unit and the use of mixed problem types within conceptual instruction, sorting practice, and paper-and-pencil review.

Finally, secondary mathematics interventions need to incorporate motivators to help students regulate their attention and behavior and to work hard. Students at risk for poor academic outcomes often display attention, motivation, and self-regulation difficulties that may adversely affect their behavior and learning (Fuchs et al., 2006; Montague, 2007). By the time students enter secondary intervention, they often have experienced at least some failure, which may cause some to avoid the emotional stress associated with mathematics: They no longer try to learn for fear of failing. For this reason, secondary intervention must incorporate systematic self-regulation and motivators, and for many students, tangible reinforcers are required.

Although mathematics preventative tutoring at the Tier 2 level has received less attention than in reading, several validated protocols do exist in the primary grades. Further information is available from the RTI Action Network or can be obtained by contacting Flora Murray at Vanderbilt University.

References

Fuchs, L.S., Fuchs, D., Compton, D.L., Powell, S.R., Seethaler, P.M., Capizzi, A.M., Schatschneider, C., & Fletcher, J.M. (2006). The cognitive correlates of third-grade skill in arithmetic, algorithmic computation, and arithmetic word problems. Journal of Educational Psychology, 98, 29-43.

Fuchs, L.S., Fuchs, D., Prentice, K., Burch, M., Hamlett, C.L., Owen, R., & Schroeter, K. (2003). Enhancing third-grade students' mathematical problem solving with self-regulated learning strategies. Journal of Educational Psychology, 95(2), 306-315.

Fuchs, L.S., Fuchs, D., Karns, K., Hamlett, C.L., Katzaroff, M., & Dutka, S. (1997). Effects of task-focused goals on low-achieving students with and without learning disabilities. American Educational Research Journal, 34(3), 513-544.

Kroesbergen, E. H., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special needs: A meta-analysis. Remedial and Special Education, 24, 97-114.

Montague, M. (2007). Self-regulation and mathematics instruction. Learning Disabilities Research and Practice, 22, 75-83.

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Mathematics Intervention at the Secondary Prevention Level of a Multi-Tier Prevention System

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Mathematics interventions at the Tier 2 (secondary prevention) level of a multi-tier prevention system must incorporate six instructional principles: Instructional explicitness Instructional design that eases the learning challenge A strong conceptual basis for procedures that are taught An emphasis on drill and practice Cumulative review as part of drill and practice Motivators to help students regulate their attention and behavior and to work hard This article describes each of these principles and some of their research underpinnings, with consideration given to how the principles can be implemented in real-world teaching contexts.

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Effective Intervention Programs

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Explore/Evaluate DreamBox

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Research-based math intervention works

Effective support at each tier

Response to Intervention (RtI), a key component of multi-tiered system of supports (MTSS), is an approach for designing and establishing research-based teaching and learning environments that are effective for students, families, and educators. RtI is not just for students with special needs or a learning disability; it is a preventative measure for all students. Educators will use interventions, or a set of teaching practices, with any student struggling with a skill or topic to help them succeed in the classroom.

There are three Tiers in RTI:

RTI Tier 1 is general education. Students in Tier 1 are performing on or above grade level. Instruction offered is research-based and is aligned to state or regional standards.

RTI Tier 2 is targeted instruction given to students who are one level below grade level. Instruction is often delivered within the classroom in small groups.

RTI Tier 3 is more intensive, often individualized, intervention for students who are generally more than one grade level behind. Students who do not respond to Tier 2 interventions are assigned to Tier 3.

Read a blog about the Four Principals for effective math intervention, plus how technology gives RtI a boost.

Read about effective, evidence-based recommendations by the Institute for Education Sciences "What Works" report for RtI Interventions for Math.

Progress Monitoring and RtI

Within the RtI framework, progress monitoring is used to determine if students are benefitting from the instruction, if they are making adequate progress in their learning, and if adjustments need to be made to the intervention program. Progress monitoring in RtI has the following benefits:

Student learning is accelerated because they a receive instruction appropriate for their individual needs.

Teachers have access to actionable data that enables them to make more informed instructional decisions.

Communication between educators and families about student learning improves.

Watch the webinar Tracking Progress for Tier 2 Students in Response to Intervention (RtI).

Assessment in the RtI Framework

RtI includes a comprehensive system that integrates assessment and instruction, so that educators can identify the level of intervention needed by a student, diagnose skill, ability, and knowledge gaps, and measure progress. They also support educators determining if the intervention is successful and when instructional adjustments need to be made.

Watch the webinar Prevention to Intervention - Formative Assessment Reimagined.

Dreambox Learning and Response to Intervention

DreamBox Learning® Math offers a research-based math intervention program for students identified as struggling with math or who require additional academic support. Use DreamBox Learning independently, in small groups, or with instructors to overcome math challenges at each of the three Tiers.

Differentiated instruction by design

Intelligent Adaptive Learning™ means that each student receives the right next lesson, at the right level of difficulty, with scaffolding tailored to his or her level of comprehension. By integrating assessment and instruction, DreamBox Learning Math provides adaptation based not just on answers, but on strategies. As a result, struggling students receive individualized instruction focused on the basic foundational concepts they need, along with appropriate practice for mastery. Current reports and alerts keep teachers informed of student progress and provide actionable data for one-to-one instruction. Dashboard and detailed reports show each student's progress and comprehension level on curriculum units based on the Common Core and other standards.

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Research-based math intervention works Effective support at each tier Response to Intervention (RtI), a key component of multi-tiered system of supports (MTSS), is an approach for designing and establishing research-based teaching and learning environments that are effective for students, families,...Read more ›

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Mathematics Interventions: What Strategies Work for Struggling Learners or Students With Learning Disabilities? *P*

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In a search of evidence-based materials from research organizations, we found several resources that are written for teachers and other practitioners. These resources have the following strategies in common: providing systematic and explicit instruction; teaching visual representation of functions and relationships, such as manipulatives, pictures, and graphs; providing peer-assisted instruction; and using ongoing, formative assessment.

These are just a few of the resources available that are in the public domain.

Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools [PDF]
This 2009 practice guide from the What Works Clearinghouse provides eight recommendations to help teachers, principals, and school administrators use RtI to identify students who need assistance in mathematics and to address the needs of these students through focused interventions. Each recommendation has practical suggestions for implementation.

Best Evidence Encyclopedia
The Best Evidence Encyclopedia reviews mathematics programs and rates them according to the overall strength of the evidence supporting their effects on student achievement. The reviews summarize evidence on three types of programs: mathematics curricula, computer-assisted instruction, and instructional process programs.

Effective Strategies for Teaching Students with Difficulties in Mathematics
Based on meta-analyses of more than 50 studies, this research brief from the National Council of Teachers of Mathematics reviews six aspects of instruction that have been found to be consistently effective in teaching students who have difficulties with mathematics: systematic and explicit instruction, think-alouds, peer-assisted learning, and formative assessment data.

Math Education Practices for Students with Disabilities and Other Struggling Learners: Case Studies of Six Schools in Two Northeast and Islands Region States
This report from the Regional Educational Laboratory Northeast and Islands describes in-depth practices at six schools that are making targeted efforts to improve math education for students with disabilities and other struggling learners. It examines each school's practices for improving the math learning of all students as well as specific supports for students with disabilities and other struggling learners and identifies the challenges that schools face to serve students with diverse needs.

Mathematics Instruction for Students with Learning Disabilities or Difficulty Learning Mathematics: A Guide for Teachers
This guide from the Center on Instruction describes seven effective instructional practices for teaching mathematics to K-12 students with learning disabilities that were identified in the Center's synthesis of intervention research, and also incorporates recommendations from "The Final Report of The National Mathematics Advisory Panel."

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Mathematics Interventions: What Strategies Work for Struggling Learners or Students With Learning Disabilities?

# data[].attributes.attachment_link.description
In a search of evidence-based materials from research organizations, we found several resources that are written for teachers and other practitioners. These resources have the following strategies in common: providing systematic and explicit instruction; teaching visual representation of functions and relationships, such as manipulatives, pictures, and graphs; providing peer-assisted instruction; and using ongoing, formative assessment.

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Guidelines for teachers/Implementation guide for math interventions

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Explore/Evaluate TenMarks more

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Incorporate ideas into School Improvement Plan

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What research based math intervention programs are a viable/affordable option for WH?

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How does reading interventions interfere with math progress?  How could WH reorganize/reconfigure intervention blocks to target both reading and math?  How should students be prioritized between reading and math interventions?

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Specific math skills that impact achievement: