Response to Intervention for Early Mathematics

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ematics in two primary ways. First, we will describe skills and professional stan- ... State Standards Initiative, 2010; Ginsburg, Lee, & Boyd, 2008; National Research. Council, 2009). .... (Haring, Lovitt, Eaton, & Hansen, 1978). Each day, children ...
11 Response to Intervention for Early Mathematics Scott A. Methe and Amanda M. VanDerHeyden

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esponse to intervention (RTI) is a system of decision making that uses child performance data to allocate instructional supports and resources to advance child learning and growth toward important outcomes (VanDerHeyden & Snyder, 2006). The use of RTI is particularly well suited to early intervening systems because RTI emphasizes prevention through provision of high-­quality instruction to all children, active universal screening to detect children who may need additional support, and data-­driven progress monitoring to evaluate instruction and system improvement over time. Through the use of progress monitoring, RTI avoids the high error rate associated with single-­point-­in-­time decisions that are particularly problematic for young children whose skills are rapidly emerging in response to both intervention and development. Applying principles of RTI to early mathematics can be straightforward despite the novelty of this topic in the research literature. Over the past decade, professional organizations have worked to reduce the complexity and breadth of the early mathematics curriculum to focus on fewer skills related to number sense. As a result, research into formative assessment has advanced considerably and many useful instruments are available to support the types of data-­based decisions necessary to build and sustain an RTI model. This chapter is intended to assist implementers of an RTI model in early mathematics in two primary ways. First, we will describe skills and professional standards that define early mathematics to best align instruction and assessment with these skills and standards. Second, we will detail how the structures and steps of RTI (including screening, progress monitoring, and monitoring intervention effectiveness) can facilitate learning of early mathematical skills.

EARLY MATHEMATICS AS THE DEVELOPMENT OF NUMBER SENSE The most focused set of recommendations for young children to date is the National Research Council’s Committee on Early Childhood Mathematics (National Research Council, 2009). These recommendations informed the Common Core State Standards (CCSS, 2010), which are rapidly being adopted as the de facto set of learning standards that will drive the courses of study in districts across the country. In early childhood, the CCSS initiative helped to focus what the NCTM (Fennell, 2006) identified as five separate strands of knowledge (number 169

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and operations; algebra; geometry; measurement; and data analysis and probability) and emphasizes learning and teaching of number sense. Number sense can be thought of as the key that opens the door to deeper understanding of mathematical reasoning and problem solving. Griffin (2004) defines number sense as a child knowing that numbers represent quantity and therefore have magnitude, that one number may be bigger or more than another number (or quantity), and that numbers occupy fixed positions in a counting sequence and therefore have a fixed order, with numbers appearing later in the sequence representing more (greater quantities) than numbers appearing earlier in the sequence. With regard to specific skills necessitating focus, number sense provides the core for counting in sequence, counting objects, ordering, classifying, and adding and subtracting numbers (Sarama & Clements, 2009). Number sense has been seen as the foundation for the entirety of the mathematics curriculum because it establishes the basis for understanding the properties of number such as the associative property, the commutative property, and the distributive property (Harniss, Stein, & Carnine, 2002). Very young children who use a sum strategy for counting to five, for example, may sometimes use two fingers on the left hand and three on the right hand but at other times use two fingers on the right hand and three on the left hand, thus demonstrating rudimentary knowledge of commutativity. Developing number sense through early experience is critical to support a preschooler’s transition to kindergarten, which is characterized by an introduction to increasingly abstract depictions of number. The preschool years are a critical developmental period for mathematics learning because knowledge and curricular content change as children move toward kindergarten. Experts in mathematical cognition and curriculum design have identified the completion of preschool and the transition into kindergarten as the interface of informal and formal mathematical knowledge and curricular content (Clements, Sarama, & DiBiase, 2004; Ginsburg, 1997; Ginsburg & Seo, 1999; National Research Council, 2009). The terms “informal” and “formal” specifically refer to the content of knowledge and curriculum rather than instructional rigor. Additionally, these terms were coined to both 1) indicate the presence of abstract numerals in learning standards and 2) depict the contrast between learning that does and does not involve numerals as abstract entities (Ginsburg, 1997). Development from informal to formal mathematics learning takes children from a language-­and object-­based understanding of number to an understanding of number that can be represented through the use of numerals, the latter of which is notable in kindergarten curriculum standards (Common Core State Standards Initiative, 2010; Ginsburg, Lee, & Boyd, 2008; National Research Council, 2009). Without robust early learning experiences that focus on building informal knowledge prior to kindergarten, problems may arise in kindergarten because the ability to understand and use numerals is predicated upon experience with language and number concepts like quantities and magnitude. In research and policy documents, preschool learning standards emphasize “connecting and communicating” (p. 43, National Research Council, 2009), suggesting that developing a language for mathematics (e.g., less, more, fewer, seven, three, etc.) is of critical importance. Additionally, research into preschool learning standards recommends rich representational environments where mathematical talk, pictures, and objects can be frequently used to represent number and numerals (Clements, Sarama, &



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DiBiase, 2004; National Research Council, 2009). Overall, flexibility in representing number using words, objects, and then more abstract means (i.e., numerals) should be emphasized as children ready themselves for the challenges of the more numeral-­heavy kindergarten curriculum. A facile number sense is achieved when a child learns to use concrete, representational (i.e., pictorial), and abstract (i.e., numerals) depictions of number (Griffin & Clements, 2007), and replaces less-­efficient and lower-­order knowledge with more efficient higher-­order skill (i.e., replacing counting on fingers to solve addition to quickly retrieving the fact when presented with a written problem such as 5 + 7). Instruction that changes in form from concrete to representational to abstract is popularly known as the concrete-­ representational-­ abstract (CRA) sequence. Indeed, this relatively linear CRA sequence not only is an effective organizer for the development of basic to advanced number sense but also characterizes the overall system of prompts to be used as a child gains knowledge and experiences success. A critical step in development occurs when young children no longer need to depict a number with their fingers, for example, and instead use a word, drawings, or a numeral to characterize the number. From the perspective of RTI, interventionists must use content-­valid assessments to detect these transitions (Baroody, 2004) as they provide interventions to support movement toward higher levels of abstraction, especially for students who do not respond to core classroom instruction provided to all learners. Much like the CRA sequence, but more rigorous in terms of its bases in neuropsychology, the number sense access view indicates that success is predicated upon a seamless transition from and interconnections among 1) perceptual knowledge of magnitudes, 2) language or verbal knowledge, and 3) visual knowledge in the form of a numeral (Dehaene, Piazza, Pinel, & Cohen, 2005). Young children who are beginning to count and represent numerals must rely on a concrete representation of number, and manipulatives are useful for this purpose. At this stage, a wide array of concrete representations can be used to build knowledge of the base-­ 10 system, such as counters or base-­ten blocks. During the process of counting and representing number with concrete objects, young children learn to associate number words (i.e., seven, three, twenty-­two) with these quantities. Educational psychologists who are examining and building early mathematics intervention curricula, such as Number Worlds (Griffin & Clements, 2007), use the CRA sequence (Griffin, Case, & Siegler, 1994; Griffin, 2004; Griffin & Clements, 2007) to build number sense access. When intervention is sequenced correctly, concrete representations of number are faded as children become accurate and fluent with identifying and combining number symbols. Effective interventions teach children that concrete representations are essentially impractical when children are faced with and given opportunities to use representations like pictures and eventually number symbols. Over time, numerals become the primary vehicle for learning more advanced mathematics content.

NUMBER SENSE LEARNING STANDARDS The CCSS represent the most recent effort to articulate and sequence expected learning outcomes across grade levels. This effort is notable because it attempts to ensure that all states build their instructional programs to accomplish the same end goals

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(i.e., mastery of the learning standards). Although the CCSS were influenced heavily by the many research committees and policy guidelines that preceded them, they are distinguished from past recommendations because they are streamlined rather than exhaustive of every possible learning task that a child might encounter during the school year. Although the CCSS may be considered the best-­organized set of recommendations for learning standards to date, it should be noted that prekindergarten standards are not included in the CCSS. However, a logical extrapolation to preschool expectations can be made by emphasizing the kindergarten domain of counting and cardinality and the development of mathematical language (National Research Council, 2009; Sarama & Clements, 2009). Many comprehensive resources have been developed to refine and clarify skills within the number sense and operations strand for preschool and beyond. Clements (2004) arranged the broad number sense and operations standard into six areas: counting, comparing and ordering, equal partitioning, composing and decomposing, and adding to and taking away. This work was subsequently refined into a learning trajectories approach that emphasized five areas (Sarama & Clements, 2009). The five-­faceted learning trajectories approach informed much of the work of the Committee on Early Childhood Mathematics that focused on number (as opposed to geometry): 1) quantity, number, and subitizing, 2) verbal and object counting, 3) comparing, ordering, and estimating, 4) early addition, subtraction, and counting strategies, and 5) composition of number, place value, and multi-­digit addition and subtraction. Although many references for prekindergarten and kindergarten mathematics include counting, the CCSS emphasizes both counting and cardinality as major curriculum topics for kindergarten. Cardinality is a fundamentally important early transition in knowledge because it follows one-­to-­one correspondence and indicates that the last number counted is the number that makes up a set (Baroody, 2004; National Research Council, 2009). With cardinality comes an advance in logic; for example, children learn to see a counted set of six items as equal to one group. In early mathematics, three critically important interrelated ideas that follow cardinality are ordinality, composition, and decomposition. These ideas make up the core of the NCTM Focal Points for prekindergarten and kindergarten that focus on “representing, comparing, and ordering whole numbers and joining and separating sets.” Ordinality introduces the idea of movement along a number line, thus facilitating the knowledge that number can be compared in terms of magnitude and position. Contrasting cardinality with ordinality, the cardinal language of “one, two, three” becomes first (relative to nothing and second), second (relative to first and third), and third (relative to second and fourth). Knowing that a counted set equals one complete unit is important because sets are frequently composed to make up another set. However, inflexibility can develop if a cardinal number is not understood as a number that can be decomposed. Therefore, understanding decomposition (Clements, 2004; Ma, 1999) can assist children in moving beyond cardinality and toward partitioning and rationing. Not only is a counted set one group of that number (e.g., six), but it is also two groups of three, for example, or one group of four and one group of two. The CCSS also includes number and operations in base ten. In kindergarten, knowledge of base ten facilitates skill in place value and grouping (Clements, 2004). Place value and grouping are treated as higher-­order concepts that are informed by counting, comparing and ordering, composing and decomposing, and partitioning.



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Harniss, Stein, and Carnine (2002) emphasized the importance of decomposition in the transition from number sense to operating with numerals for addition and subtraction. Furthermore, their work supports the recommendations of the National Mathematics Advisory Panel (2008) which emphasized algebraic thinking across all levels of the curriculum. Because a fundamental task in algebra is solving for an unknown quantity, it becomes important to emphasize how basic number sense can facilitate algebraic thinking. Related to identifying a missing quantity, Baroody (2004) discusses part-­whole relations where, for example, knowledge of three and “how many more” might result in five, and the “how many more” (two) comprises the unknown quantity. Additionally, children learning to recompose around a base-­ten unit can benefit from knowledge of part-­whole relations. Knowing that the number 5 is made of 2 and 3 can help a child when adding 8 and 5 because this problem can be treated as (8 + 2) + 3. Therefore, part-­whole knowledge can support understanding of the associative property (Harniss, Stein, & Carnine, 2002), place value, and proportions.

Standards as Instructional Goals Specifying the skills that are most essential for mastery in pre-­K and kindergarten provides teachers with meaningful instructional targets (Slentz & Hyatt, 2008). However, specifying a sequence of skills that represent critical outcomes of instruction can be a point of confusion for some teachers. Some teachers may view such a list as representing the full range of all instruction and may also assume that skills can best be taught through rote memorization methods. Consistent with policy and research recommendations, we recommend balanced mathematics instruction that is sequenced logically, paced to permit children to reach mastery for essential skills, matched to student instructional level, and connected to real-­world problem solving (Ginsburg et al., 2008). We have argued that before teachers can decide how to teach, they must decide what to teach. In the next section, we will describe a framework for ensuring child mastery of essential early learning skills.

IMPORTANT FEATURES OF CORE INSTRUCTION IN EARLY MATHEMATICS The goal of early mathematics instruction is to connect children’s informal knowledge to number representations and operations in base-­ten language. Children have a natural curiosity and deep understanding of pivotal math concepts (Ginsburg et al., 2008) that can and should be met by the teacher and facilitated with intentional instructional techniques. The first step toward promoting early mathematical understanding is to focus content and specify which skills should be targeted for instruction. The second step is to provide well-­sequenced instruction that facilitates conceptual understanding and facility using numbers and numerals to solve problems. Teachers and interventionists must understand the instructional hierarchy (IH), and use this model to facilitate mastery of learning standards. The IH is a fundamental and progressive set of four levels, or benchmarks, related to learning: acquiring skill and knowledge, developing fluency with the skill, generalizing the skill to other skills and settings, and adapting the skill to solve real-­world problems (Haring, Lovitt, Eaton, & Hansen, 1978). Each day, children should participate in

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mathematics learning opportunities that provide acquisition-­level support for new or emerging skills. New skills are skills that a child cannot yet perform accurately. During acquisition instruction, teachers should emphasize strategies like modeling correct and incorrect responding, providing immediate corrective feedback that is more detailed and concludes by asking a child or the whole class to give or show the correct response, and guided practice completing the task accurately. Once children can respond accurately when presented with a task, and have attained the next step above acquisition in the IH, fluency-­building strategies can be used to facilitate growth toward skill mastery. Fluency-­building strategies include using short intervals of timed practice (sometimes referred to as timed trials), embedding multiple opportunities to respond into a math game or applied activity, setting goals for more fluent performance, and rewarding fluent performance. At this stage of instruction, the teacher can vary the task presentation as long as support is available to ensure that correct responding continues. Once children have become accurate and fluent, the teacher can and should introduce opportunities to move further up the instructional hierarchy and use the mastered skill to solve applied or more complex problems. For example, a task of counting coins can be used to teach counting object correspondence and counting by fives and tens. As children become fluent in counting coins, concepts related to conservation of number, equivalence, place value, and associative and distributive property can be introduced. Teachers should guide discussion around these concepts and provide opportunities for students to construct hypotheses, estimate solutions, test solutions, and discuss findings related to math concepts. Although core instruction should be guided by standards and assessment of how children are learning these standards, the curriculum and its related assessment do not prescribe teaching. Rather, a sequence of skills based upon early mathematics standards provides teachers with a series of measurable outcomes that can be tracked to ensure that children are mastering essential skills over time. Using instructional standards in number sense as essential outcomes allows teachers to avoid blind adherence to a specific set of curriculum materials that may include sub-­optimal content. When teachers use research-­based criteria for selecting skills, they are free to select the instructional tools and resources that will best advance students to mastery on these essential skills. In addition to focusing on content, teachers must have a system for knowing the children’s development on a given skill. A common mistake in mathematics instruction is to advance content or task difficulty when prerequisite skills have not actually been mastered. This mistake is costly because children will be prone to errors and misunderstanding as learning tasks become more complex. Using the steps described next, we have identified a number of techniques and models (i.e., the instructional hierarchy) to ensure that instruction is sensitive to the developmental levels and learning needs of students.

THE PROCESS OF RESPONSE TO INTERVENTION The purpose of RTI is to ensure that the core instructional environment is adequate for most children and equipped with resources to provide systematic support to children who do not readily respond to instruction. Core classroom instruction (also referred to as Tier 1) in early mathematics is the most fundamental structure of an RTI model and should be guided by a sequence of instructional standards,



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include formative assessments, and be carefully balanced to provide the type of instruction that is needed by children at various stages of the learning or instructional hierarchy (acquisition, fluency-­building, and generalization). Rigorous scientific standards for all materials adopted in core classroom instruction should be used to choose materials that are most likely to result in student learning. When children struggle to learn expected skills, supplemental small-­group (referred to as Tier 2) and/or individualized (referred to as Tier 3) intervention is provided. To accomplish RTI, student data must be collected and used by decision makers. These data can be obtained from universal screening and allow decision makers to reach conclusions about the adequacy of Tier 1 instruction and further pinpoint instructional needs for students.

Step 1: Conduct Universal Screening Teachers should select a skill for screening that 1) has been taught within the current curriculum, and 2) most children are expected to be able to do to benefit from the instruction. With the knowledge of key early mathematics content discussed in the first part of this chapter, readers are urged to use the screening measures identified in Table 11.1. Users of these measures should briefly assess the extent to which a learning standard is exemplified by the measure, and choose the correct screener based on this assessment. If a selected measure such as identifying which object set has more or less (i.e., quantity discrimination; Floyd, Hojnoski, & Key, 2006)

Table 11.1.  Curriculum-­based measures for screening and progress monitoring in preschool Quantity, number, and subitizing

• Number Selection (VanDerHeyden et al., 2004; http://www.isteep.com) • Number Naming (Floyd, Hojnoski, & Key, 2006; Methe, Hintze, & Floyd, 2008; VanDerHeyden et al., 2004; http://www.isteep.com) • Count Objects, Say Number (VanDerHeyden et al., 2004; http://www .isteep.com) • Count Objects, Select Number (Methe, Hintze, & Floyd, 2008; VanDerHeyden, Witt, Naquin, & Noell, 2001; http://www.isteep.com) • Count Objects, Write Number (VanDerHeyden et al., 2001; www.isteep .com). • Quantity Array (Lembke & Foegen, 2009)

Verbal and object counting

• Oral Counting (Clarke & Shinn, 2004; www.aimsweb.com; VanDerHeyden et al., 2004; www.isteep.com)

Comparing, ordering, and estimating

• Quantity Discrimination (Floyd, Hojnoski, & Key, 2006; VanDerHeyden et al., 2011) • Relative Size (Methe, Begeny, & Leary, 2011) • Ordinal Position Fluency to Five (Methe, Hintze, & Floyd, 2008; Methe, Begeny, & Leary, 2011) • Identify/Discriminate the Object That Is Different (VanDerHeyden et al., 2001, VanDerHeyden et al., 2004; http:// www.isteep.com) • Equal Partitioning Fluency (Methe, Begeny, & Leary, 2011)

Composition and place value

• Math Numbers and Operations and Algebra (http://www.easycbm.com) • Group Five (Methe, Begeny, & Leary, 2011) • Math Concepts and Operations (http://www.aimsweb.com)

Note. For more detailed reports of these and other screening and progress monitoring measures, see http://www .rti4success.org

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indicates low performance for a given student, teachers can target one or more of the prioritized skills that are prerequisite to quantity discrimination and use this measure to monitor progress. Once a screening measure has been selected, teachers should follow standard or scripted instructions to administer the screening measures to all children in the classroom. Following scripted instructions allows the teacher to compare students and to track individual student progress over time. Figure 11.1 depicts 11 separate classrooms that used the count and circle number measure (VanDerHeyden, Witt, Naquin, & Noell, 2001). Student performance on this measure was aggregated by classroom to look at the percent of students who had attained an important instructional benchmark. Ideally, all classes within a program at the same grade or level will administer the same measure so that the progress of all children across classes can be compared and tracked over time. All children should participate in screening in the fall, winter, and spring, and progress monitoring should occur more frequently for classes and students when a learning problem is detected. Adequate screening measures are well aligned with learning expectations in the classroom, yield sufficiently reliable scores that correlate well with more comprehensive measures of mathematics proficiency, forecast mathematics proficiency over time, and require minimal time to administer and score. All of the early numeracy measures listed in Table 11.1 meet these criteria.

9

16

30

43

78

45

91

6 94

50

64

48

27

84 73

70 57

55

52

50 36

Teachers Figure 11.1.  Classroom performance on the Count and Circle Number probe.

Class 11

Class 10

Class 9

Class 8

Class 7

Class 6

Class 5

Class 4

Class 3

22

Class 2

110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

Class 1

Percentage

Instructural Effects Grade K, Date 7/1/2010–10/30/2010 Assessment: KEHALA, Numeracy, Count and Circle Number



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Step 2: Identify and Address Class, Grade, and Schoolwide Problems Once screening data are collected, they can be examined to identify the extent to which problems in student learning can be differentiated from problems in classroom, grade-­level, or schoolwide instruction. Common problems in the instructional ecology include poor teacher agreement on instructional goals, failure to use research-­based curricula and intervention programs, poor professional development, lack of common planning time, and other related issues (Woodward & Kaufman, 1998). Using an assessment instrument that broadly represents a set of curriculum-­relevant skills for the grade and age level (see Table 11.1), the aggregate score obtained for any classroom can be compared to a benchmark criterion or goal that the instructional leadership deems ambitious. Examining Figure 11.1 closely, it is notable that some classrooms (i.e., classroom 4, 5, 6, 8, 9, and 10) are not attaining the same level of skill development as the others, indicating a grade-­level problem. Comparing the obtained performance with a more desirable goal state (i.e., average classroom performance exceeds 80–­90% accuracy, as in classrooms 1 and 2 from Figure 11.1) helps determine whether most of the children in the class are learning effectively and also helps to uncover effective classroom strategies. Where a classroom aggregated score is well below the criterion, a problem that should be addressed through classwide intervention is indicated (VanDerHeyden, Witt, & Naquin, 2003). This same procedure can be used to identify grade-­ and school-­ level problems. Where more than half of the classes at any given age or grade-­level exhibit scores below the benchmark, a larger systems-­based intervention may be necessary. When this pattern is detected, the first step taken by the intervention team must be to evaluate the instructional “basics” in the classroom, including consensus for a sequence of learning outcomes for early mathematics linked to a calendar of instruction, availability of and correct use of curriculum materials to facilitate child learning of essential skills, use of routine child assessment to guide instruction, sufficient instructional time allocated to early mathematics instruction, and more frequent progress monitoring to track whether or not corrective actions reduce the numbers of students scoring in the risk range on subsequent screenings. Where classwide learning problems are detected, classwide intervention can be planned and delivered using a research-­supported program like Peer Assisted Learning Strategies (PALS; http://kc.vanderbilt.edu/pals/).

Step 3: Identify Individual Child Learning Problems for Tier 2 or Tier 3 Intervention If a classwide problem has been addressed through intervention, the scores of all children in the class can be evaluated to identify individual children in need of additional intervention. Figure 11.2 depicts the different levels of skill demonstrated by students within a classroom when they are assessed with the count and circle number probe. After depicting student scores in this type of graphical array, the next step is to differentiate among two types of skill deficits in a classroom. This differentiation (or diagnosis) is critical because homogenous grouping is still frequently deployed in schools and it cannot be assumed that every child shares the same root cause of poor performance. To further differentiate within an instructional grouping, one decision rule that can be applied is to use the bottom

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Score Can’t Do/Won’t DO School Teacher Grade K Mean Score: 9.05 Median Score: 8.5 Assessment: 9/30/2010—KENALA, Numeracy, Count and Circle Number

Some Risk Low Risk

18 15

16 13

14 12 Percentage

16

10 8

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10 10 10 10

11 11

6 4

3

2 0

Teachers Figure 11.2.  Ranked performance by classroom on the Count and Circle Number probe.

16th percentile of children in a classroom as a cutoff point to indicate the need for further assessment (VanDerHeyden et al., 2003). After these children are identified, interventions should be judiciously selected as a means to pinpoint the cause of poor performance. The intervention provider is therefore linking assessment and intervention by examining which intervention, implemented in brief trials, leads to the most robust student response. Therefore, this step requires a followup assessment and/or progress monitoring. The functional assessment literature (Martens & Gertz, 2009) provides an empirical basis for the idea that this procedure will indeed differentiate students and identify the intervention that will best indicate the root cause of poor performance. Additionally, the indicated intervention can be selected and adapted for more extended use. Common interventions to use at this step include maximizing motivation (tested by offering incentives for improved performance), reducing task difficulty (tested by administering successively easier tasks at skill level and at the prerequisite skill level), and providing guided support to correctly respond (tested by



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conducting a brief instructional session followed by a timed trial; see the appendix following this chapter). The cause has been identified when students respond more readily to one or more interventions than to others. In addition to a functional assessment, the student’s screening data should be used to differentiate between accuracy and fluency deficits. This type of problem analysis uses the instructional hierarchy to aid in intervention selection. For example, if student scores on a number naming assessment indicate that they incorrectly named 10 numbers out of 30 they attempted in one minute, this low (66%) accuracy indicates a need for acquisition-­level support. Acquisition-­level support would not be used for a student who, for example, says 15 of 15 numeral names in one minute. Instead, intervention for the “accurate but not fluent” student would be targeted at fluency-­ building strategies. Figure 11.3 indicates how two students, Danae and Katelyn, responded to the chosen intervention and in turn demonstrated two very different types of skill deficits. Katelyn’s response to the intervention indicated that she is in great need of acquisition-­level support, as she has not likely attained the skill of counting a set and circling the number that corresponded to the set. Therefore, a fluency-­level intervention strategy like the one detailed in the appendix is not likely to be appropriate for Katelyn. Danae’s response to the intervention indicates that she has learned this skill but is either not motivated to perform the skill or has not practiced the skill enough to have attained fluency. The intervention in the appendix may be more useful for Danae. The cases depicted in Figure 11.3 have been chosen to exemplify step 3 in an RTI model but also to warn educators that placing children with similar screening scores in homogenous groups without a problem analysis may be inappropriate.

Step 4: Effectively Deploy Interventions

Correctly circled numbers per minute

One of the most prevalent causes of intervention failure is poor intervention implementation (McIntyre, Gresham, DiGennaro, & Reed, 2007). Research suggests that intervention failures should be rare events (Torgesen et al., 2001; Vaughn, Linan-­ Thompson, & Hickman, 2003; VanDerHeyden, Witt, & Gilbertson, 2007), and that intervention failure should be examined prior to concluding that the student has not learned the skill. When interventions fail to produce the desired change in learning, the first step should be to verify correct intervention use. A variety of excellent

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Intervention

10

Not at Risk

Screening

8

Incentives

6

At Risk

4 2 0 Danae

Kaitlyn

Class Median

Figure 11.3.  Response to a fluency-­building intervention for two students.

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resources are available at www.interventioncentral.org that can help implementers identify the critical components of an intervention and observe the intervention as it is implemented. Witt, VanDerHeyden, and Gilbertson (2004) indicate that using data to evaluate intervention integrity is a critical aspect of ensuring intervention success and that these data should be used in concert with progress monitoring data. By identifying the specific aspects of the intervention tasks and using checklists to evaluate the extent to which they have been accomplished, schools not only facilitate exemplary intervention adherence, but also help teachers, students, and staff to collaborate and maximize learning outcomes. After implementers have decided which intervention is needed and ensured the effective deployment of the intervention, they must decide how and when to monitor progress of those interventions, and how to train and support high-quality intervention implementation.

Step 5: Monitor Intervention Effects and Evaluate Outcomes Critical components of this step include 1) establishing schedules for monitoring all students, 2) selecting a progress monitoring measure or measures that are sensitive to growth, 3) ensuring that data are displayed visually (Riley-­Tillman & Burns, 2009), and 4) ensuring that implementers agree on rules that allow ineffective interventions to be modified as needed. With regard to monitoring schedules, screening results can be used to indicate the frequency of monitoring. Using a three-­tiered model of risk (i.e., risk for not meeting the end-­of-­year learning standards) schools can use criteria, normative data, or teacher nominations of risk level to prescribe monitoring schedules. Students at low risk (Tier 1) are likely to respond to classroom instruction and can be monitored infrequently in the fall, winter, and spring. Students at some risk (Tier 2) should be monitored every two weeks, and students who are at high risk (Tier 3) should be monitored weekly. With regard to the choice of a progress monitoring measure, in some cases the adopted measure may be too difficult for a student. Although it is important to continue monitoring in this skill, implementers should feel free to select an easier skill to monitor concurrently. A visual display of data allows implementers to focus on one key goal, which is to reduce the discrepancy between actual and expected child performance (i.e., the aimline). When growth is not demonstrated over a period of two to three weeks, assuming that a student whose screening data indicated the need for weekly monitoring and intensive intervention, implementers should verify intervention integrity and then make systematic changes to the intervention and reevaluate after another two to three weeks. The most difficult part of monitoring intervention effects is ensuring that there is enough time for the intervention to work, but not waiting too long and concluding that the student is not responding to the intervention. If the intervention implementation has been monitored, however, and measures have been selected that are sensitive to growth, implementers can be more confident that the graphed data reflect reality. Because similar data are collected across all classrooms and evaluated relative to a benchmark criterion, effects of intervention efforts should also be evaluated over time (Shapiro & Clemens, 2009) to evaluate the extent to which groups of children experiencing unexpected rates of failure at baseline close the performance gap between themselves and not-­at-­risk peers. Examining proportions of children in each group at the beginning, middle, and end of the year, as well as over time, allows school



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personnel to evaluate the effectiveness of the overall program of instruction in early mathematics.

SUMMARY This chapter was written to assist in implementing an RTI model for early mathematics. We focused content on prekindergarten skills with an emphasis on effective transition into kindergarten. Furthermore, we gave attention to both informal and formal skills that can be used as assessment and intervention targets. It is critical for teachers choosing assessment and intervention methods in an RTI model to understand that number is a code-­based system and that many children experiencing problems with combining numerals, an earmark of mathematics disability (Gersten & Jordan, 2005), need instruction that helps them understand number language and concepts as well as numerals and basic numeral combinations. Thankfully, broad-­based recommendations from policy-­making bodies indicate that substantial momentum is building in the area of early mathematics. The process of RTI discussed in this chapter allows teachers to understand how the content of early mathematics can be effectively combined with the process of screening, progress monitoring, intervention integrity, and ongoing program evaluation. Although this chapter represents an updated collection of research in the foundations of effective RTI models in early mathematics, such as early learning standards, assessment, and intervention, it is likely that more advances will be made in the short term. Readers are urged to stay updated on research and policy changes and to advocate for a more focused curriculum that can be evaluated with innovative and precise instructional tools designed to maximize learning for all students.

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