Abstract
Students who fail to develop proficiency in math skills in the primary grades are more likely to experience difficulties in the math curriculum later on. These students may be in need of a more targeted intervention, or Tier 2 supports, in mathematic instruction. Although the instructional principles of an effective math intervention are becoming better understood, the practice of designing and implementing an effective Tier 2 math intervention system challenges many schools. This article documents a case study of one school’s experience in designing a Tier 2 math intervention system for 10 third graders to provide a model of effective Tier 2 systems.
Within a response to intervention (RTI) framework, there are three tiers of instruction. The first tier (Tier 1) is the core instruction that all students are provided in the general curriculum. The second tier (Tier 2) of instruction is supplemental instruction provided to those students who are struggling, at risk, or otherwise not meeting grade-level benchmarks in Tier 1. The third tier (Tier 3) of instruction includes intensive interventions for those students with significant needs or disabilities. A foundational component of RTI is the use of data as the basis for making decisions about the intensity of instruction that students require to meet their learning needs (National Center on Response to Intervention [NCRTI], 2010).
When RTI is properly understood and used, its focus is on improving student response to instruction and student outcomes. Currently, much of the literature and research regarding RTI has focused on the area of reading instruction. However, RTI is not limited to just reading; rather, it can be applied to all academic areas. When generalizing from reading to math, though, it can be deduced that without early identification, intervention, and progress monitoring to determine students’ response to instruction and intervention, many young students with difficulties and deficits in math skills may not develop a level of automaticity that is necessary for becoming proficient in math (Bryant, Bryant, Gersten, Scammacca, & Chavez, 2008; Fuchs & Fuchs, 2005; VanDerHeyden, 2005).
There is a growing body of research on young children’s mathematics cognition that has contributed to our knowledge of early numeracy skills that prove problematic for young children at risk for math disabilities (Bryant, Bryant, Gersten, Scammacca, & Chavez, 2008; Bryant, Bryant, Gersten, Scammacca, Funk, et al., 2008; Fuchs & Fuchs, 2005; Fuchs, Fuchs, & Hollenbeck, 2007). Students who fail to develop proficiency and automaticity in computational skills (e.g., addition, subtraction, multiplication, and division) and problem solving in the primary grades are more likely to experience difficulties in math curriculum later (Miller, Stringfellow, Kaffar, Ferreira, & Mancl, 2011). Thus, it is critical to provide quality math computational skill instruction and intervention to ensure early mastery of these foundational skills.
Students who are not making progress and are struggling with the general education math curriculum may be in need of more targeted intervention, or Tier 2 supports. The goal of Tier 2 is to reduce current cases of academic failure to prevent student problems from escalating to an intensity that requires individualized intervention and support (NCRTI, 2010). In other words, Tier 2 intervention is supplemental to Tier 1 instruction, with the goal of alleviating difficulties in the general education curriculum. Hallmark characteristics of Tier 2 are (a) use of screening data and criteria for placement, (b) interventions based on strong instructional design principles, and (c) frequent progress monitoring (NCRTI, 2010).
This article describes how one school approached Tier 2 for math for third graders, including (a) the instructional principles of a Tier 2 math intervention, (b) the process and implementation of a math intervention program that shows promise for providing targeted instruction in computational skills and problem solving (i.e., VMath), (c) progress monitoring and data-based decision making within the program, and (d) the effectiveness of this intervention. Pseudonyms are used for the names of the school and students for confidentiality (see Note 1).
Mathematics Interventions at Tier 2
Mathematics interventions at the Tier 2 level should incorporate the following instructional principles: explicit instruction, an instructional design that alleviates the learning challenge, an emphasis on drill and practice, a cumulative review as part of the drill and practice, and motivators to help students regulate their attention and behavior (Fuchs, 2009; Fuchs et al., 2008).
Explicit instruction
Research provides consistent support for using an explicit approach to teaching math concepts at the secondary prevention level (Baker, Gersten, & Lee, 2002; Bryant, Bryant, Gersten, Scammacca, & Chavez, 2008; Fuchs et al., 2008; Fuchs & Fuchs, 2001; Kroesbergen & Van Luit, 2003). The math curriculum at Tier 1 (e.g., core math curriculum) generally consists, at least in part, of a constructivist instructional style (Fuchs, 2009; Fuchs & Fuchs, 2001). Students who are at risk for or have math skill deficits are often unsuccessful or fail to profit from this style of instruction and may require additional direct instruction to fully profit from the core math curriculum (Fuchs, 2009; Fuchs & Fuchs, 2001; Kroesbergen & Van Luit, 2003). Bottge (2001) stated that by continuing to provide students with explicit instruction, teachers can foster competence in basic math skills. An analysis of 34 intervention studies found explicit instruction to be more successful in 32 of the 34 studies reviewed (Adams & Engelmann, 1996). Kroesbergen and Van Luit (2003) reported in a meta-analysis of 58 research reports on math instruction that direct instruction was found to be more effective than discovery-oriented instruction. Another meta-analysis of 15 studies conducted by Baker et al. (2002) revealed moderate to strong effects for the use of explicit instruction for teaching mathematics to high-risk students.
This research provides evidence that an explicit, or direct instruction, pedagogy facilitates the learning process in math. “Explicit instruction can be viewed as providing a series of instructional supports or scaffolds—first through the logical selection and sequencing of content, and then by breaking down that content into manageable instructional units” (Archer & Hughes, 2011, p. 3). Delivery of explicit instruction is characterized by the use of clear descriptions or instructions, demonstration of the skill, supported practice, and timely feedback. It is called explicit because it is a direct approach to teaching in design and delivery.
Instructional design
However, explicit instruction is not enough to ensure student understanding and success. Careful instructional design is the process of analyzing learning needs and goals of students and developing a system of delivery to meet those needs. It includes development of instructional materials and activities and evaluation of the effectiveness of those materials and activities. Instructional design that alleviates the learning challenge is an often-overlooked principle of Tier 2 math interventions (Fuchs et al., 2008). Reducing the learning challenge may mean, for example, that the material used when teaching is processed and packaged to facilitate easy learning. Another example would be using a script for providing instruction. When teachers anticipate and reduce misunderstandings and use specific explanations and sequenced instruction, they are alleviating the learning challenge for students. This principle is particularly important for math instruction. Math is highly proceduralized, continually building on previous knowledge for successful learning, and involves distinct skill sets—each with its own conceptual and procedural requirements. Many students with math skill difficulties struggle to master computation skills, and those who fail to develop proficiency related to computational operations are likely to struggle through the math curriculum (Bryant, Bryant, Gersten, Scammacca, & Chavez, 2008; Miller et al., 2011). Strategic instructional design should begin with the instruction of foundational skills that can be applied across the math program. For example, counting up, two-digit calculations, solving algebraic equations, and checking work are all skills that can be applied to most aspects of the math curriculum.
Drill and practice with cumulative review
Drill and practice is another essential instructional principle (Fuchs et al., 2008). The goal of drill and practice is for students to rehearse and practice math skills to improve fluency and memorization (i.e., automatic retrieval). Students at the secondary intervention level benefit from drill and practice opportunities. The provision of multiple opportunities to practice skills supports the automaticity of recall and proficiency, which leads to mastery. These opportunities may include practice in sorting problems into problem types, the mixing of problem types (once more than one problem type has been introduced), and daily review (Fuchs et al., 2008). Other examples include the use of flash cards, peer tutoring, and computerized practice. Lessons should incorporate a cumulative review, another principle of instruction (Fuchs et al., 2008; VanDerHeyden, 2005). This principle is reflected in the review of (a) skills taught initially in the lesson and (b) previously taught problem types. Review can be in various forms: warm-up activities, computerized practice, and paper-and-pencil review (Fuchs et al., 2008).
Motivators
Last, a motivational environment promotes high levels of learning among all students (Fuchs & Fuchs, 2001; VanDerHeyden, 2005). Unfortunately, students who are receiving Tier 2 interventions have often experienced at least some failure and difficulty, which may cause some to avoid participation in the intervention. For this reason, incorporating tangible (e.g., stickers, prizes) and intangible (e.g., motivating statements, positive feedback) motivators and reinforcers is key to successful Tier 2 intervention. As students become more capable and feel better about their math abilities, their confidence increases and they are more motivated to persevere, which increases their chances of success.
Mountain View Elementary: Math Intervention
Although the principles of effective math intervention are becoming increasingly well understood, the practice of designing and implementing an effective Tier 2 math intervention system often eludes many schools (Shores & Chester, 2009). The complexity of orchestrating a system comprising several moving parts can be challenging for schools. In the remainder of this article, we share a case study of one school’s experience in designing a Tier 2 math intervention system. The purpose of the case study is to provide a model of effective Tier 2 systems.
Mountain View Elementary, a K–5 school located in the U.S. Northwest, has implemented a comprehensive tiered service delivery model incorporating RTI for academics and positive behavioral interventions and supports for behavior. This was their second year of implementation for this comprehensive approach (see Note 1).
This case study reports effective Tier 2 systems with a focus on a Tier 2 math intervention for third graders that follows the instructional principles described previously, as well as the process for determining the math learning needs of the students and implementation and effectiveness of the intervention at this particular school.
Determining Math Learning Needs
Screening
A central component of RTI is early screening to identify those at risk for difficulties. All students were screened at the beginning, middle, and end of the year to determine which students were most likely to be at risk for academic failure and behavioral problems. Mountain View used a “direct route” to intervention, in which students whose screening performance fell below a specific cut point were targeted to immediately receive a Tier 2 intervention, in this case, math.
From the three third-grade classrooms with a total of 60 students, 10 students were identified as needing secondary intervention by the math screening test (i.e., EasyCBM and Math Navigator). Of this group of 10 students, there were 9 boys and 1 girl. Students identified by the math screening were placed in a small pull-out group where they were to receive supplemental instruction four times a week, for 30 minutes a day. Mountain View used VMath as its Tier 2 intervention program. A description of what a typical day/week in this group looks like and how the program was implemented with these students follows after a description of the specific intervention. It is worth noting that although VMath includes it own progress monitoring tool, Mountain View chose to monitor progress using the Monitoring Basic Skills Program (MBSP) Concepts & Applications (Fuchs, Hamlett, & Fuchs, 1998; http://www.proedinc.com) to remain consistent with district guidelines.
VMath
VMath (Voyager Learning, n.d.), produced by Voyager (http://www.voyagerlearning.com), is a secondary intervention for struggling students that incorporates the instructional principles that have been found to be effective for these students (National Mathematics Advisory Panel, 2008). The VMath curriculum focuses on computational fluency and problem-solving skills. This intervention uses direct, systematic instruction with a carefully sequenced, specific teacher dialogue for every lesson. Daily lessons are explicit and highly structured using a direct instruction model. VMath instruction is characterized by a four-step lesson format: getting started, guided practice, independent practice, and test preparation and error analysis. A description of the components the four-step lesson format can be seen in Figure 1.
Components of VMath lesson.
Implementation of the Tier 2 Math Program at Mountain View
As previously mentioned, once the fall math screening data were collected and evaluated for the third graders at Mountain View, struggling students (i.e., those below the cut point on the screener) were identified for placement in a Tier 2 math intervention group that supported their specific needs. After the grouping(s) were determined, the small-group instructor took baseline data by giving an initial MBSP. The MBSP provides grade-level benchmarks, and these benchmarks are used to determine the expected rate of growth for each student. As stated earlier, the MBSP was chosen to follow district guidelines. (More on progress monitoring using the MBSP follows after a description of the intervention.)
After baseline and a target benchmark were determined, the instructor began to provide the VMath intervention. A typical lesson for the intervention group included the following:
Warm-up activities: When students arrived in the small-group classroom, several warm-up problems were on the board for them to complete. These warm-up problems are generally problems from previous lessons. They provided the students with recall and additional practice. After everyone finished their warm-up problems, the answers were shared as a group. By reviewing as a group, the teacher was able to provide corrective feedback to the group as needed. (This step ensured cumulative review from the previous lesson.)
Getting started activities: The teacher used the script provided in the instructor manual to introduce and model the new skill or strategy needed for the lesson objective. This script is explicit and walks the teacher and the student through each task. At Mountain View, the teacher realized that the script was, at times, “too wordy” to hold the students’ attention. Therefore, the teacher reviewed the script before the lesson and noted times when to veer from the script, provide more or less information, or break the task into smaller chunks. The script was still used; it was just modified for the needs of this particular group. (This step incorporated explicit instruction and instructional design that alleviated the learning challenge by scaffolding.)
Guided practice: For guided practice, the teacher followed a script that guides and scaffolds the students in acquiring the new skill or lesson objective. During guided practice, the students did the problems in their workbook together and then reviewed the answers. The teacher used prompts and questions to scaffold the instruction and transition the students from initial learning to independent practice. If issues arose, the teacher provided more guided practice. (This step used explicit instruction, instructional design that alleviated the learning challenge, and cumulative review.)
Independent practice: In the next step of the lesson, students worked independently in their workbooks, practicing the skills taught and practiced in guided practice. The teacher monitored and checked their work at the end of the lesson to see if any reteaching needed to occur in the next lesson. (This step provided drill and practice.)
Progress monitoring: Once a week the students were given the MBSP. This is a timed test. Before the warm-up activities, students were given an MBSP form and 6 minutes to complete as much as they could. Their scores on the MBSP were graphed for progress monitoring purposes.
Implementation of the intervention and the progress monitoring was systematic and consistent. Motivators and reinforcers were provided throughout the lesson. The instructor reviewed student work daily and monitored progress weekly. Students were informed of their progress. Data from the MBSP were input in a data collection file immediately.
The Tier 2 math intervention for these third graders met recommendations for best practice (Miller et al., 2011). Instruction was explicit and clear and included teacher demonstrations with verbal and pictorial descriptions of the process, guided practice (i.e., high levels of support initially with a systematic decrease in support as student proficiency increased), and independent practice. Instruction was highly interactive (e.g., choral response), briskly paced, and clearly presented, fostering student engagement and response.
Data Collection and Interpretation
Progress monitoring is an efficient tool for gauging the effectiveness of instruction on a regular basis and can assist school teams in making decisions about appropriate instruction and intervention for students. The main purpose of progress monitoring at Tier 2 is to determine whether the intervention provided is effective and successful (Johnson, Mellard, Fuchs, & McKnight, 2006). The Tier 2 instructor at Mountain View collected daily and weekly progress monitoring data on the students receiving this supplemental instruction.
Student workbooks were reviewed daily for work completed in independent practice. The teacher looked for skills or areas in need of remediation or additional practice. If students completed their work satisfactorily (i.e., did not need reteaching or remediation), then the group moved on to the next lesson the next day. Weekly progress monitoring data were collected using the MBSP. The teacher scored the MBSP and graphed results to see if students were making adequate progress.
The MBSP data were analyzed collectively, in that all student responses were compared as well as analyzed individually. The purpose of looking at the data collectively was to determine how well the intervention was working for most students. The students in this intervention continuously showed an upward trend that followed the goal line, indicating that the intervention was generally successful for the majority of the students.
Individual student data were also closely monitored and used to guide instructional decisions. Figure 2 presents an individual student’s graph. In this graph, there are no decision points because the student consistently responded to the intervention and required no instructional changes to the program. This particular student was above the target benchmark after 13 weeks of intervention. The Tier 2 problem solving team made the decision to exit the student from Tier 2 intervention.
Progress monitoring data for an individual third-grader in the intervention group.
At Mountain View, the teachers used a shared drive on the network to store student data so that all faculty and staff had access to assessment and progress monitoring data. This system allowed for easy input of data and instant access at team meetings. How data were used in the decision-making process follows.
Decision-Making Process
At Mountain View, there was a Tier 2 Intervention Support Team that met every other week. The team consisted of the school principal, the school psychologist, the special education teacher, and general education grade-level representatives. During this meeting, data were reviewed for all students receiving Tier 2 supports (see Pool, Carter, & Johnson, 2013 for more information on the Tier 2 team process). The third grade Tier 2 instructor provided the team with the MBSP data as well as any anecdotal information about specific students. The team then reviewed these data to determine whether individual students (a) had made successful progress in Tier 2 and should return to Tier 1 instruction, (b) were making progress but had not progressed enough to warrant removal from Tier 2 instruction, or (c) needed adaptations to the Tier 2 intervention to improve rate/growth of progress, assessment for intensive intervention (i.e., Tier 3 intervention), and/or referral for special education services.
The intervention instructor informed the general education third grade teachers of their students’ progress in Tier 2 on a weekly basis. In addition, when the Tier 2 Intervention Support Team met about a particular teacher’s student to make instructional decisions, that teacher was notified. For example, Mrs. Green had two students in the intervention group, Lisa and Mike. When the support team met to determine if Mike should be exited from the intervention group, Mrs. Green was notified and invited to the meeting. Mrs. Green could then also provide information to the team about Mike’s performance in the Tier 1 math curriculum.
Strategies for Motivation
Difficulties with attention, motivation, and self-regulation may play a critical role in learning and academic outcomes (Fuchs et al., 2006). To mediate such difficulties, the Tier 2 instructor implemented strategies to promote motivation and engagement. Since the school had implemented schoolwide positive behavior support (SWPBS) and used “Stallion Pride” tickets as incentives, these tickets were also used in the intervention group. When a student was “caught” being on-task, following directions the first time, participating in the activity, and/or working hard, the Tier 2 instructor gave the student a ticket. At the end of the intervention period, the students either cashed in their tickets for a reward or saved them up for a bigger reward at the end of the week.
Another strategy for motivation was immediate verbal and nonverbal feedback to the group or individual. Students were provided with praise, redirects, or corrective feedback as soon as a behavior was seen. Nonverbal positive feedback included giving a thumbs up, high five, or pat on the back. Nonverbal corrective feedback could have included gestures such as pointing to the expectations chart as a reminder, finger to the lips to indicate silence, and proximity to the student. The instructor made sure feedback was specific, consistent, and immediate. Oftentimes this verbal feedback was accompanied by a Stallion Pride ticket. To ensure that the students were successful, the instructor reviewed expectations briefly at the beginning of every lesson and had the expectations listed on the board. These expectations were consistent with the SWPBS expectations. When a behavior occurred, the student was reminded of the expectations.
The practice of charting student progress as a motivational strategy was employed as well. The visual representation of progress can be very motivational for students. The scores from the MBSP were shared and graphed with the student. In this way, students could see their progress and try to improve on their previous score. Students who may not have reached their goal had a one-on-one meeting with the instructor where they were praised for their effort and discussed strategies for success the next week.
Effectiveness of the Targeted Math Intervention
Response to the VMath intervention was generally positive for the 10 third graders. As students exited the group, new students took their place. Some examples of success and decision making follow.
Tommy
Tommy was exited from the group after receiving 3 weeks of the intervention. His baseline MBSP score was near but below the cutoff. However, Tommy’s scores were above the third grade benchmark on the MBSP the following 3 weeks. At the Tier 2 support meeting, it was determined that perhaps his baseline score had been inaccurate and he should return to the Tier 1 math curriculum. After 12 weeks of struggling in general education math instruction, his teacher provided evidence of his struggles and requested he return to the Tier 2 group. His MBSP score on second entry to the intervention group was similar to his initial baseline score. He was making slow, steady progress by the end of the school year.
Marcus and Anthony
Marcus and Anthony were exited from the intervention group after 12 weeks of instruction. Their MBSP scores consistently rose each week. It was determined in the 12th week that they could return to their regular third grade math instruction. They remained in the general education math curriculum for the remainder of the school year.
Adam
Adam received intervention for both math and behavior. His math data showed improvement each week and made him a good candidate for exiting the intervention group. At the Tier 2 support team meeting, it was determined that he could move back into the general education class for math, based on his progress monitoring data, but would remain in the CICO behavioral intervention because his progress indicated he still required support in this area. The following week, while back in general education math instruction, Adam’s inappropriate behavior increased dramatically and he spent a good part of the week in the principal’s office. When discussing the issue with Adam, he indicated that VMath was his favorite part of the day. The math intervention instructor, the general education teacher, and the Tier 2 support team discussed the matter and determined his participation in the intervention seemed to support his positive behavior and that Adam would return to the intervention group despite the fact that his math data indicated he did not need the intervention anymore. The Tier 2 team also decided to increase the intervention group size by one and place another struggling student in the group. Adam would be promoted to peer tutor in an effort to increase his positive behavior and academic experiences and to support other students in the group. Moreover, the team and the general education teacher began work on determining the ways to restructure the environment to help better support Adam’s needs in the general education classroom. This is an unusual situation, but one that worked at the time.
Conclusion
This article described evidence-based practices that assisted in the provision of a high-quality Tier 2 math intervention. In summary, Mountain View successfully implemented a secondary intervention program that encompassed the principles of instruction: explicit instruction, an instructional design that alleviates the learning challenge, an emphasis on drill and practice, and a cumulative review as part of the drill and practice. In addition, multiple motivational strategies were used to help students regulate their attention and behavior while in the intervention group. Finally, Mountain View had a system in place for consistent progress monitoring, data collection and interpretation, and data-based decision making.
Footnotes
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project was funded by a grant from the Idaho State Department of Education.