Abstract
Data-based individualization (DBI) is a framework that allows educators to make timely and informed decisions about student progress in academics or behavior. In this article, we focus on the DBI framework as applied to math intervention within a tiered support model for students experiencing math difficulty. We review how DBI starts with an instructional platform paired with progress monitoring. Then, we discuss decision making within a DBI framework and highlight how diagnostic data informs instructional adaptations. Finally, we emphasize the on-going use of DBI to determine the best set of instructional practices in math for each student.
Ms. Dolston is a special educator at Hill Ridge Middle School who teaches math intervention for Grades 6, 7, and 8. Students participating in her intervention classes met eligibility criteria as students with a specific learning disability, and all have at least one individualized education program (IEP) goal in math. One afternoon while planning for the upcoming unit for her seventh-grade intervention class, Ms. Dolston decided to review data from the recently administered district-wide benchmark assessment. Ms. Dolston noted that three of her students (Jasmin, Kristen, and Lucas) scored much lower than the rest of the students in the intervention group. Puzzled, she compared their assessment scores to informal data she routinely collected during lessons and from homework assignments. She found that Jasmin, Kristen, and Lucas’s scores showed minimal improvement over the past several weeks despite consistent math intervention support. With the evidence in front of her, she recognized it was necessary to adjust her instruction, but like many educators, she felt unsure about where to start or what resources to utilize to effectively address these students’ needs.
To help educators like Ms. Dolston support students like Jasmin, Kristen, and Lucas in math, a data-based individualization (DBI) framework can be beneficial. The DBI framework allows for organized implementation of math instruction for students who experience difficulty with math, and use of DBI can lead to timely and guided decision making about changes to instruction (Powell, Lembke et al., 2021). Time is of the essence for students like Jasmin, Kristen, and Lucas so their math difficulties do not continue to compound. Such difficulties with math in the earlier grades may lead to later math difficulties (Siegler et al., 2012), which could result in fewer options for college and limited career pathways (Lee, 2012). DBI can help Ms. Dolston quickly know when and how to make changes to her math instruction. In this article, we focus on the DBI framework and its application in math intervention.
The DBI Framework
Intensive and individualized intervention supports are necessary for students with disabilities who have persistent academic or behavioral challenges (D. Fuchs & Fuchs, 2015; Lemons et al., 2019). In response to the growing need for guidance on intervention intensification, the National Center for Intensive Intervention (NCII; 2013) expanded an approach to intensive intervention rooted in the concept of DBI. Generally speaking, DBI involves the ongoing, systematic evaluation of student data to determine how to appropriately intensify academic or behavior intervention. When implemented by educators, DBI can lead to improvements in academic outcomes for students (Jung et al., 2018). As noted, we focus on DBI in the academic area of math for students who experience math difficulty and describe DBI as researched by our team (Powell, Lembke, et al., 2021) and others (Connor et al., 2018; Schumacher et al., 2017; van der Scheer & Visscher, 2018; van Geel et al., 2016).
“Utilizing the DBI framework enables educators to make real-time adjustments to intervention components when student performance falls below an adequate level.
Successful implementation of DBI in math occurs in the context of an existing multitiered, validated academic program and involves the collection of both progress monitoring data and diagnostic assessment data. Throughout the DBI framework, decisions regarding intervention length, intensity, duration, and instructional delivery are objective and based on an individual student’s response to the intervention (Lemons et al., 2018; Powell & Stecker, 2014). The DBI framework relies on educator expertise and knowledge of their students’ present levels of performance in combination with frequent data collection that teachers examine and are responsive to. Thus, utilizing the DBI framework enables educators to make real-time adjustments to intervention components when student performance falls below an adequate level (i.e., before a long-term goal that has been set).
Figure 1 depicts the DBI framework as highlighted by the NCII (2013). This framework involves a series of steps for implementation and is intended to address the needs of students with persistent academic difficulties (i.e., students who are not making adequate progress despite intervention support). Therefore, this framework is typically used for students at Tier 2 (i.e., targeted small-group intervention) or Tier 3 (i.e., intensive individualized instruction) in a multitiered system of supports model. In some districts, Tier 3 instruction may also include students receiving special education services and supports. That said, the DBI framework is not typically used with students in Tier 1 because instruction at Tier 1 consists of universal, core instruction provided to all students.
Data-based individualization framework from the National Center on Intensive Intervention
The first step of DBI in math requires educators to start with a validated instructional platform that aligns to core instruction and targets students’ academic needs (see Step 1 in Figure 1). The instructional platform must include an evidence-based intervention or (if unavailable) evidence-based practices that have been validated through high-quality experimental research and result in improvements to student learning outcomes in math (Hughes et al., 2016). To determine the instructional platform, educators should (a) collect diagnostic and formative data to identify students’ strengths and weaknesses, (b) use data to develop the scope and sequence for instruction, and (c) identify the evidence-based intervention or evidence-based practices that target students’ identified area(s) of weakness (NCII, 2013). Remember, when implementing DBI, instruction should be delivered at a greater level of intensity than instruction the student received previously.
Second, educators should select a validated progress-monitoring measure for the ongoing, weekly assessment of the student’s response to the instructional platform (see Step 2 in Figure 1). The term “progress monitoring” here refers to a method of standardized (i.e., formal) formative assessment using Curriculum-Based Measures (CBMs). Unlike other progress-monitoring approaches (e.g., Mastery Measures), CBMs are brief measures that assess students’ overall growth on the curriculum over time and include a mix of problem types. Such measures enable educators to check student understanding of not only skills learned that week but also skills learned in months prior and provide important insight into a student’s ability to generalize learning. Data collection concurrent with instruction is necessary to evaluate if the student is making adequate progress toward the predetermined goal (see Step 3 in Figure 1). If the student is making growth, an educator continues to provide the instructional platform with continued math progress monitoring.
If the student is not showing adequate response, the fourth step is to collect diagnostic data on the student’s individual math strengths and weaknesses (see Step 4 in Figure 1). This data collection process can incorporate both formal diagnostic data and informal data from key stakeholders (educators, parents, and related service providers; NCII, 2013). At Step 5, adaptations to the intervention are selected based on student performance with the original instructional platform and performance on diagnostic assessments (see Step 5 in Figure 1). Adaptations can include quantitative adaptations (e.g., changes to the intervention setting, duration, personnel), qualitative adaptations (e.g., changes to how instruction is delivered), or a combination of both (NCII, 2013). The sixth and seventh steps (see Figure 1) involve the continuation of ongoing progress monitoring to evaluate the efficacy of the intervention on the student’s academic growth. If the student continues to make progress, educators continue to implement the intervention as is. However, if the student does not make adequate progress, educators should return to Step 4 for additional diagnostic data collection. It is important to note that this systematic approach to integrating instruction and assessment does not supplant educators’ daily use of classroom-level data to guide in-the-moment decision making; educators are continually observing and noticing students’ understanding through formal and informal means (e.g., checks for understanding, exit tickets) and using these data to provide an alternate explanation, work another example, provide corrective feedback, and so on. These practices work in tandem with DBI.
Mathematics and DBI
Instructional Platform
The foundation of the DBI framework is an instructional platform. An instructional platform consisting of evidence-based practices has been shown to improve the math outcomes for students experiencing math difficulty (L. S. Fuchs et al., 2021; Jitendra et al., 2018; Powell, Mason, et al., 2021; Stevens et al., 2018). This instructional platform may consist of a validated intervention tested by researchers and supported by data. For some grades or students, however, validated packaged interventions may not be available. In these instances, educators can develop their own instructional platform by using a combination of some of the following evidence-based practices: (a) systematic and explicit instruction, (b) multiple representations, (c) precise math language, (d) word-problem instruction, (e) mnemonics, (f) graphic organizers, and (g) fluency-building activities (see Figure 2). Each of these practices has a strong evidence base in the area of math intervention (L. S. Fuchs et al., 2021; Powell, Mason, et al., 2021). We describe each evidence-based practice in greater detail in the following sections. When implementing a validated intervention or a set of evidence-based practices, educators should ensure they implement the intervention or practices with high levels of fidelity.
Evidence-based practices for the instructional platform
Systematic and explicit instruction
Systematic and explicit instruction forms the bedrock of high-quality math intervention (L. S. Fuchs et al., 2021), and it has been shown to effectively increase math outcomes across a variety of skills and grades (Bouck et al., 2020; Doabler et al., 2015; Morgan et al., 2015). Consequently, the majority of existing math interventions include systematic and explicit instruction as a key component (Jitendra et al., 2018; Stevens et al., 2018). Systematic instruction includes the planning of sequential instruction with learning outcomes in mind (L. S. Fuchs et al., 2021). Explicit instruction includes a step-by-step explanation followed by highly scaffolded and purposeful practice opportunities (Archer & Hughes, 2011). To ensure that all students understand the math concepts, it is important for educators to model the math topic, include purposeful examples, consistently engage students in the math content, and provide clear and specific academic feedback. During both the initial modeling and subsequent practice opportunities, educators should include a variety of opportunities for students to respond, including low-level and high-level questions followed by corrective feedback (Doabler et al., 2017).
Multiple representations
A second cornerstone of many evidence-based math interventions is the use of multiple representations (Bouck et al., 2018). The three primary forms of multiple representations typically include concrete, pictorial, and abstract representations. Across all grades in math, the use of hands-on, or concrete, manipulatives has been shown to increase students’ math performance by building conceptual understanding of math concepts (Namkung & Bricko, 2021).
Pictorial representations, which can be on paper or virtual, are often two-dimensional representations of the three-dimensional concrete manipulatives. Sometimes, pictorial representations may be referred to as semiconcrete representations (L. S. Fuchs et al., 2021). Common pictorial models include 10 frames, area models, arrays, and number lines. Pictorial representations provide students with opportunities to create their own representations of learned content in unobtrusive ways. When practiced systematically, students can develop the habit of portraying the math content on their own via pictorial representations (Strickland, 2017).
Abstract representations of math include numbers, symbols, and words, and educators should display abstract representations of math when they are using concrete and pictorial representations to help students make the connections among the different forms. Although there is no hierarchy between the three forms of representations, students benefit when educators integrate the use of multiple representations into a systematic and explicitly taught lesson (Strickland, 2017).
Precise math language
One barrier to math content many educators may not consider is the math language students must master to be able to understand and apply math concepts. Many students struggle with acquiring the math language needed to participate in class (Powell et al., 2017). It is important for students to have multiple opportunities to read, write, hear, and speak precise math terms on a regular basis (Powell et al., 2019). To help students master formal math terms, educators themselves need to use precise and concise math language. For example, instead of using terms like “number on the bottom” or “number on the top” when discussing fractions, refer to the “denominator” and the “numerator.” Educators should also explicitly teach math terms just as they would teach math concepts to students. Tools, such as Frayer models, math journals, vocabulary practice sheets, and interactive word walls, can help students learn the language of math.
Solving word problems
Successful interpretation of word problems relies on a student’s ability to access many different skills and understandings of math, including math language, computation, and semantic skills. Many students, therefore, require explicit instruction in applying strategies to dissect and solve word problems (Powell & Fuchs, 2018). Accordingly, an attack strategy provides students with a step-by-step process to tackle any word problem. Often, an attack strategy includes at least four steps (see Figure 3).
Sample word-problem attack strategy
Beyond teaching an attack strategy, students benefit from schema instruction (Jitendra et al., 2015). Schemas refer to the structure of the word problem, like a change problem where an amount increases or decreases or an equal groups problem where there are groups with an equal number in each group. When students identify the schema, or structure, of the word problem, it then becomes easier to determine and use the key information needed to solve for the missing information. It is also important for educators to not focus on teaching key words alone; the strategy of using key words to dissect a word problem falls apart when students are in older grades and need to solve multistep problems (Karp et al., 2019; Powell et al., 2022).
Mnemonics
Many word-problem attack strategies employ a mnemonic device or mnemonic strategy and graphic organizers to help students remember necessary steps. Mnemonics are an evidence-based strategy that can help students remember a cognitive or metacognitive strategy (Cuenca-Carlino et al., 2016). Mnemonics are often composed of a series of letters in which each letter prompts a step or strategy. For example, the SOLVE mnemonic helps students remember to Study the problem, Organize the facts, Line up a plan, Verify the plan with action, and Evaluate the answer (Freeman-Green et al., 2015). Mnemonics should be taught to students explicitly, with adequate time allotted for students to learn both the mnemonic and the strategy the mnemonic is designed to prompt or assist in recalling.
Graphic organizers
Often used as a tool alongside mnemonics, graphic organizers provide a visual prompt for students to recall specific concepts or strategies. Graphic organizers on their own or as part of a comprehensive math intervention have been shown to improve students’ math outcomes (van Garderen, 2007). Moreover, when prompting students to recall strategies or steps, graphic organizers can help students make connections or review math content to develop deeper conceptual understanding (Shin & Bryant, 2017).
Fluency-building activities
Finally, as we consider high-quality math strategies for students, fluency-building activities are an important evidence-based practice to allow students to develop mastery of math facts (and later, computation), leaving more energy available to solve complex math problems. Many students with math difficulty have challenges with short-term memory and may spend so much time and energy recalling addition or multiplication facts that they become mentally exhausted before they attempt more complex tasks (Peng & Fuchs, 2016). Therefore, it is recommended that educators include brief (e.g., 1–3 minutes) fluency activities every day in math class to build automaticity of math fact recall (Burns et al., 2010). Students may require explicit instruction in addition, subtraction, multiplication, and division strategies if they are not developing efficient means of fluently responding to these activities. Examples of fluency practice include cover-copy-compare, dice or domino games, prepackaged fluency cards or tools, or online games that provide corrective feedback.
Checking in with Ms. Dolston
Before she recognized that Jasmin, Kristen, and Lucas needed focused math intervention, Ms. Dolston realized she used many of these evidence-based practices already. She decided to formalize her instructional platform and name the practices she used with Jasmin, Kristen, and Lucas. Her instructional platform included systematic and explicit instruction with the use of multiple representations, especially concrete manipulatives, to help the students understand rational number concepts. Ms. Dolston also decided to focus on math vocabulary through use of a math glossary created by each student. When students created their own math glossary, each entry included the target vocabulary word, a contextually relevant definition, and an associated visual (e.g., pictures, drawing, symbols) to support their understanding of the term. As she considered additional layers of support she could provide, she thought about how her students struggled to organize and structure their ideas when learning new content. She determined that students would also benefit from the use of graphic organizers when appropriate. Lastly, to support rational number understanding, Ms. Dolston incorporated opportunities for students to build fact and computational fluency. She did this by restructuring her lesson plans to include 2 to 3 minutes of fluency practice during each intervention session. When she planned the fluency activities (e.g., flashcards, games), she intentionally integrated both new and previously learned math facts.
Ms. Dolston documented her instructional platform by creating a document about the instructional platform with a brief description of each practice and a checklist for implementation of each practice. This documentation would allow Ms. Dolston to review the platform, share with others, and understand the essential components of each practice to implement with high levels of fidelity. Once Ms. Dolston started using her instructional platform, it became time for her to also collect progress-monitoring data (see Figure 1, Step 2). In the next section, we describe progress monitoring.
Progress Monitoring
The DBI framework cannot be applied without progress monitoring. In math, the content of progress-monitoring measures varies based on grade level. In the early elementary grades, students may be assessed using measures that contain content about number identification, identifying a missing number in a sequence, or comparing quantities (Clarke et al., 2008; Lembke et al., 2008). As students develop foundational addition and subtraction knowledge, educators may administer computation or concepts and application progress-monitoring measures. In the elementary and middle school grades, these are the two most common types of math progress-monitoring measures. On a computation measure, students answer grade-level prompts with addition, subtraction, multiplication, and division of whole and rational numbers. On an application measure, students answer math questions in which they apply their math knowledge in context. Such questions may ask about measurement, geometry, fractions, word problems, place value, and so on. In middle or high school, the content of progress-monitoring measures may involve solving problems related to prealgebra or algebra content (Foegen, 2008).
“The DBI framework cannot be applied without progress monitoring.
Resources are available (e.g., the NCII) to help educators decide which measures are best suited for their situation. Understanding different progress-monitoring measures may help narrow the options. When determining which progress-monitoring measure to use, educators should consider several factors. First, how much time do they have to administer and score the measure? Some progress-monitoring measures take 2 minutes to administer, whereas others take 10 minutes or more. This may also impact the sensitivity of the measure for capturing small changes in students’ understanding; measures in which students respond only to a few items may not be as sensitive to student growth as measures that elicit more responses. Also, some measures are quick to score, whereas others take more time, especially if scoring by the number of correct digits in a student’s response. Some measures can be administered to a small group of students, but some need to be administered individually by a trained assessor.
Second, educators should consider if they should use technology to administer and score progress-monitoring measures. Technology-based options are available for some measures in which students complete their measure on a computer, and then the program automatically scores the student’s work. Before opting to use a technology-based option, consider: Does the school have the technology to support regular data collection? Do all students have regular access to technology and know how to use it?
Third, what information needs to be collected? Do educators want to understand student computation performance, or do they want to understand real-life application of math knowledge? The content of the progress-monitoring measures should relate to the instructional math content. Once an educator determines the progress-monitoring measure, they should use grade-level measures unless the student is working on math content at a different grade level. For example, if a fifth-grade student is learning third-grade content, an educator should use progress-monitoring measures focused on third-grade content.
In DBI, progress-monitoring measures should be administered every 1 or 2 weeks to make timely decisions about whether the instructional platform is helping the student. Most often, educators administer one measure each week. Determining the frequency of administration is based on several factors, including the intensity of support the student is receiving (e.g., more intensive support is monitored more frequently) and the sensitivity of the measures for capturing growth (e.g., more sensitive measures are administered more frequently). Schedule a regular time to administer the measure (e.g., the first 3 minutes of the last day of each week’s small-group tutoring session) and score the measure (e.g., immediately following the small-group tutoring session). Once data have been collected, it is important to graph the data (or access the data via a technology-based platform) to make decisions about student progress. We discuss decision making in the next section.
Decision Making
To determine progress, an initial step is to formulate a goal for each student. This should be established prior to delivery of the instructional platform. In the next section, we discuss goal setting.
Goal setting
There are several methods for determining a goal. One method involves using a prescribed benchmark. If a publisher (e.g., testing company or university) produced a progress-monitoring measure, it will typically provide benchmarks by grade level or by fall, winter, and spring in a grade level. If using a benchmark, plot the benchmark on a graph in the final week of planned intervention. Then, draw a goal line from the initial data points to the goal.
Another method involves using a prescribed rate of improvement (i.e., slope). Similar to benchmarks, the publisher should provide slope expectations by grade level. If using a slope, identify the slope at the appropriate grade level (e.g., a publisher provided a slope of 0.8 for a student using sixth-grade measures) and multiply this slope by the number of weeks remaining in intervention (e.g., 0.8 × 12 weeks = 9.6). Then, add this product (e.g., 9.6) to the baseline of scores. Plot the sum on a graph in the final week of intervention and draw a goal line from the initial data points to the goal. If a slope is not provided by a publisher, an educator can calculate the student’s own slope. Software with graphing tools (e.g., Microsoft Excel) can provide slope calculations. To calculate slope without software, divide all the data points into three fairly equal groups. Then, subtract the third median data point minus the first median data point and divide the difference by the total number of data points minus 1. To set an ambitious slope for a student, multiply the student’s slope by 1.5 to expect 50% more than the student’s current pattern of performance.
In Figure 4, we show a graph with a goal and goal line. In this graph, the educator set goals by using a benchmark of 20. This 20 is plotted with a green “X,” and the green goal line shows the project path to meeting this goal. Having multiple options for setting a goal (e.g., benchmark, publisher-provided slope, or student’s own slope) ensures each student’s goal is sufficiently ambitious for them.
Sample progress-monitoring graph
Making decisions
After several weeks of implementing the instructional platform, it is time to examine the data to determine whether a student’s response to the instruction is sufficient. After an educator has collected enough data to determine a trend, usually six to eight data points, the educator should draw a trend line on the graph. In Figure 4, this trend line is in blue.
An examination of the trend line compared to the goal line provides information about whether a student is on track, needs adjustments to their goal, or needs adjustments to their instruction. If the trend line and goal line are similar to one another, the decision should be to continue with the current instructional platform. If the trend line is steeper than the goal line, it is time to increase the student’s goal. When the trend line is less steep than the goal line, it is necessary to make instructional adaptations.
Checking in with Ms. Dolston
Ms. Dolston decided to start using prealgebraic progress-monitoring measures, similar to those of Ketterlin-Geller et al. (2015). She selected the Quantity Discrimination measure in which students compared two numbers, either whole or rational. She administered different forms of this measure each week for the suggested length of time, and she placed the data on a graph. Each student had their own graph. Figure 4 shows Kristen’s graph. After 8 weeks of using the instructional platform and collecting data, Ms. Dolston looked at Kristen’s graph. She quickly saw that Kristen’s trend line (in blue) was well below her goal line (in green). The current instructional platform was not boosting Kristen’s math performance in the way Ms. Dolston planned. She decided to make adaptations to the platform but wanted to make these adaptations based on data. Ms. Dolston planned to collect diagnostic data to inform the adaptations. We focus on diagnostic data in the next section.
Diagnostic Data
When an educator has determined the initial instructional platform does not provide sufficient support that a student needs, it is helpful to collect diagnostic data to inform instructional adaptations. Educators can collect diagnostic data through formal assessments, like the Test of Early Mathematics Ability (Ginsburg & Baroody, 2003), KeyMath3 (Connolly, 2007), Group Mathematics Assessment and Diagnostic Evaluation (Pearson, n.d.), or the Diagnostic Online Math Assessment (Seton Testing, n.d.). Educators can also collect diagnostic data from unit or chapter tests aligned with a math curriculum or by analyzing student work by using error analysis techniques (e.g., Ketterlin-Geller & Yovanoff, 2009; Kingsdorf & Krawec, 2014). The ultimate goal of the diagnostic is to have a detailed view of student strengths and weaknesses in math. Instructional adaptations, discussed in the next section, will focus on playing into student strengths and alleviating student weaknesses.
Instructional Adaptations
When students do not make desired progress, educators need to adapt their instruction to more effectively meet their students’ needs. There are a number of potential adaptations educators can make to intensify their instruction (L. S. Fuchs et al., 2017; NCII, 2019). Several factors will impact how educators choose to adapt their instruction. School resources, classroom and school logistics, and the structure of the intervention will all impact which adaptation is the best fit for the student. It is recommended educators implement only a few adaptations at a time—using diagnostic data to select the adaptations that have the greatest likelihood of improving outcomes—and continue to collect regular progress-monitoring data before deciding whether to alter the adaptation or choose a different adaptation to intensify the math instruction. Recommended adaptations include (a) implement with greater fidelity, (b) embed behavioral supports, (c) increase dosage, (d) adapt math content, (e) utilize systematic and explicit instruction, and (f) teach for transfer (see Figure 5). We describe each instructional adaptation in more detail in the following section.
“It is recommended educators implement only a few adaptations at a time.
Adaptations to the instructional platform
Implement with greater fidelity
If a student is not progressing as expected or desired, before making any major changes to the student’s instruction, it is important for educators to first assess their own implementation of instruction. Evidence-based programs and strategies contain key components educators must include and teach as designed for the instruction to be effective. If educators are not carrying out the instruction as intended, then the intervention may not yield the same benefits. Some interventions, especially packaged interventions available from companies or universities, may contain secondary components educators may customize or personalize; examples of secondary components include the rewards students receive for completing tasks or the names or context used in word problems (but often the structure of the word problem is important). To evaluate if they are implementing with high fidelity, educators can self-assess or ask a colleague to observe and assess them on the key components of instruction. Many packaged interventions include a checklist to ensure high fidelity. If a checklist is not provided, the research team can often be contacted, or the educator, with administrative support or support from an instructional coach, can identify the key components of instruction that need to be included to see student growth.
Embed behavioral supports
Some students experiencing difficulty with math may also demonstrate nonproductive behavior during instruction, which decreases the effectiveness of the instruction. For these students, it may be beneficial to embed behavioral supports to increase their productive behavior. For students who simply require additional reminders or incentives to stay engaged, it may be enough to include activities to encourage students to stay on task. For example, students can earn tally marks for staying on task and then receive a reward (e.g., a small prize) after a period of time based on the number of earned tally marks. If a student requires more targeted behavioral supports, it will be important to operationally define the desired behavior and model this behavior or explicitly teach the expected behavior. Educators may need to find a reinforcer that the student is willing to work for and then set a plan for rewarding the student when he or she demonstrates the desired behavior.
Increase dosage
Dosage often refers to the number of minutes (i.e., duration) and the frequency of an intervention, but a broader definition can also refer to the number of opportunities students have to engage with a math topic. For educators or schools that have greater flexibility, increasing dosage may look like increasing the duration or frequency of the intervention. For example, educators may be able to increase dosage by pulling the student for an additional 5 minutes per day or meeting for one additional tutoring session per week. Alternatively, educators can reduce the number of students in a small group, increasing the number of opportunities each student has to directly engage with the educator. If educators are not able to adjust the duration, frequency, or size of the groups, they can potentially increase the number of opportunities students have to respond by including more class-wide responses or creating more pair or small-group responses so students have more opportunities to engage with the math.
Adapt math content and use systematic and explicit instruction
Perhaps one of the broadest options to intensify instruction is adapting the math content. For whole-class instruction, educators can adapt their own instruction to include more evidence-based practices, described previously, such as using precise math language or teaching word-problem schemas. Educators should see an increase in most, if not all, student outcomes if the educators themselves include evidence-based practices regularly. Additional examples of ways educators can adapt math content include showing worked examples, chunking problems into smaller steps, and teaching alternate algorithms (L. S. Fuchs et al., 2021; Star et al., 2015). For some students, these adaptations may not be enough, and they may require more intensive individualized adjustments. In these instances, it may be appropriate to alter the scope and sequence and modify the student’s curriculum. However, if changing the scope and sequence or the curriculum, educators need to work with the student’s IEP team prior to making modifications because these changes may alter the student’s opportunity to learn grade-level content. In addition to adapting math content, including or increasing the systematic and explicit instruction a student receives (described previously) is an evidence-based method for increasing the intensification of an intervention.
Teach for transfer
Some students may seem to be able to grasp a specific math concept, but when asked to apply that understanding to a new but similar problem, they may struggle to apply the same procedures or concepts they just mastered. These students may have difficulty with transfer and will require explicit instruction in how to transfer their skills to a new problem (L. S. Fuchs et al., 2017). To explicitly teach for transfer, educators must first identify similar characteristics between old problems and novel problems and then ask students to identify the relationship between the two problems. It is key to have students focus on the ways problems are similar. To help students extend their transfer, educators can ask them to represent both problems in the abstract. Finally, it is important to continue to draw students’ attention to similarity between problems whenever teaching a new skill set.
Checking in with Ms. Dolston
Based on her instructional time with Kristen and the diagnostic data she collected, Ms. Dolston decided to implement three adaptations to the instructional platform. First, she realized Kristen was often off task and had difficulty focusing when working independently. Ms. Dolston decided to embed a self-monitoring system for Kristen where Kristen checked her focus every 2 to 5 minutes. Second, Ms. Dolston decided to adapt the math content. Kristen needed a stronger understanding of fractions and decimals, so Ms. Dolston used the chunking strategy and broke down fraction and decimal concepts and procedures into smaller, more manageable steps. Third, Ms. Dolston focused on specifically teaching for transfer. For example, when she taught a lesson about fractions, she explicitly linked fractions to decimals. Ms. Dolston documented each of these adaptations on the same document in which she outlined her instructional platform.
Ms. Dolston decided this suite of adaptations might be helpful to Kristen, but she would continue to monitor progress to learn if that was true. She marked the week in which the instructional adaptations began (from Weeks 8 to 9; see Figure 6) and continued graphing Kristen’s Quantity Discrimination scores on Kristen’s graph.
Kristen’s progress-monitoring graph
Progress Monitoring, Decision Making, and . . .
Once educators have determined which adaptations to implement, frequent progress monitoring and timely decision making help educators determine whether the suite of adaptations is adequate for students. Progress monitoring should occur regularly, which most often means weekly. Decision making should occur every six to eight data points.
“The DBI framework continues with every evaluation of the data.
Even though the DBI model (see Figure 1) visually ends after adaptations and progress monitoring, the DBI framework continues with every evaluation of the data. If a student demonstrates adequate progress and meets goals, math intervention may not be necessary. If a student does not show adequate progress, educators collect more diagnostic data, make different instructional adaptations, and continue with progress monitoring to gauge the impact of such adaptations. This DBI framework can be used for as long as necessary.
Conclusion
After 7 weeks, Ms. Dolston looked at Kristen’s data (see Figure 6 in orange) and saw an increase in Kristen’s trend line. Kristen was still not reaching the planned goal, so Ms. Dolston collected more diagnostic data and reviewed the adaptations to the instructional platform. She would continue using the DBI framework to support Kristen and others.
The DBI framework allows educators to have structure to their instruction and assessment for students participating in math intervention. The instructional platform gives every educator a great jumping-off point for designing math intervention, and knowledge of adaptations allows for efficient and wise changes to the instructional platform. Knowledge about assessment in DBI helps educators be mindful about the importance of regular collection of assessment data, the need to graph data, and the regular review of data to make instructional changes when changes are necessary. DBI has led to improvements in the math outcomes for many students (Powell, Lembke, et al., 2021) and has helped many educators formalize the framework for providing math support to students.
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 work was supported by Grant H326M170006 from the Office of Special Education Programs, U.S. Department of Education.
