By David Allsopp and David Hoppey
The effective implementation of response to intervention (RTI) at the secondary level is proving to be especially challenging for school leaders. The structure of secondary schools and the complexity and depth of the secondary curriculum can complicate the integration of RTI across specific content areas. The purpose of this article is to give secondary school leaders guidance as they think about the implementation of mathematics RTI. The following five questions can stimulate critical thinking about the implementation of mathematics RTI.
Does my school have a systematic framework for effectively implementing mathematics RTI?
Schools make a mistake when implementing mathematics RTI if they fail to do it in a systematic way that emphasizes important aspects of mathematics success for students: the mathematics curriculum; the ways that students learn, understand, and do mathematics; the processes for screening and progress monitoring to make effective instructional decisions; and the structure of learning experiences that target foundational mathematical concepts and skills. (See Five questions can guide secondary schools in adopting mathematics RTI at the secondary level. figure 1 for a basic framework for the systematic implementation of mathematics RTI at the secondary level.)
A primary goal of RTI is to prevent failure among students by getting effective instruction to them early. A good place to start is to focus on mathematics courses that are gatekeepers for earning a standard high school diploma (i.e., Algebra 1 and 2 and Geometry). Each of those courses connects to and builds upon foundational ideas that are addressed in the K–8 curriculum. When students’ mathematical learning needs are analyzed in relation to targeted gatekeeping courses, then scheduling can be done in ways that truly address the needs of those students.
At the secondary level, mathematics RTI should purposely target the big ideas that are essential to success in those courses. When screening and progress monitoring measures incorporate the specific big ideas that are aligned with gatekeeping courses, they become more effective and the focus of tiered instructional intervention can be more precise in nature. That allows informal mathematics assessments to be used to pinpoint a student’s mathematical learning needs that are specific to a course or a mathematical content area, thereby informing instructional and intervention decisions. (See figure 2 for a summary of the components of an informal mathematics assessment process that are suitable for making tiered intervention decisions.)
To what extent does my school critically evaluate the mathematics content that is emphasized in our day-to-day instruction?
When we discuss mathematics RTI with school leaders and teachers, someone often asks us, “Can you tell us what evidence-based effective mathematics interventions to use for tier two and tier three?” Certainly, this is valid question. First, however, schools should evaluate the mathematics content that is being emphasized by teachers across tiered instruction. As we alluded earlier, an effective mathematics RTI framework entails making purposeful decisions about the particular mathematics content—the big ideas—that are being emphasized for screening, progress monitoring, and instruction and intervention.
Let’s examine algebra, for example. At the high school level, algebra is typically conceptualized as two courses, Algebra 1 and Algebra 2. The content in these courses is, for the most part, an extension of the broader areas of number, operations, and algebraic thinking. When secondary schools target key foundational concepts and skills that relate to those areas and frame their screenings, progress monitoring, and instruction and interventions accordingly, then struggling learners are better positioned to develop the type of mathematical “understandings” necessary to achieve success in Algebra 1 and Algebra 2. (See figure 3 for the big ideas and related concepts and skills that are foundational to success in high school algebra that could be areas of focus.)
Does my school use researchsupported effective mathematics instruction across the tiers?
Although the research about effective mathematics RTI instruction is limited, particularly at the secondary level, there is plenty of research about effective mathematics instruction. It is important to evaluate whether the actual teaching practices across tiers reflect this research. School leaders can use six questions to evaluate both mathematics teaching practices and the mathematics curricula in their schools:
- Are students doing math in a variety of ways?
- Is math instruction explicit?
- Is math being taught using realworld examples?
- Do students have multiple opportunities to apply newly learned concepts and skills?
- Is students’ progress being monitored continually?
- Are teachers using maintenance activities so that students retain prior learning?
Evaluating the answers to those six general questions can help school leaders identify areas of strength and weakness. (See figure 4 for more details about the six areas of researchsupported mathematics instruction.) For example, one effective practice is to ensure that students have multiple opportunities to apply newly learned mathematical concepts and skills. A school might discover that students are not really getting multiple opportunities to practice new mathematical understandings, particularly in Algebra 1 courses.
After examining the textbook, the evaluation team discovers that the practice section for each lesson actually provides as many practice items for previously introduced concepts as it does for the new concept. Further, the district’s pacing guide encourages teachers to cover a lesson each day, and the teachers who teach Algebra 1 feel that they need to hurry through lessons, so they reduce the amount of questioning they incorporate in their daily instruction, thereby limiting response opportunities for students during teacher-directed instruction. The evaluation team begins to understand why students, particularly those most at risk for math failure, seem to either understand a concept one day but forget it the next day or have only partial understanding of previously taught mathematics concepts and skills.
Through systematic self-evaluation, the school team is able to pinpoint an area of weakness in their mathematics instruction and then judiciously plan how to turn it into a strength. For example, teachers might incorporate more practice items that address new concepts into each lesson, identify lessons that address the most foundational concepts and skills and plan to spend more instructional days covering that material, or use peer coaching to increase the level of questioning during teacher-directed instruction.
How does scheduling affect my school’s implementation of mathematics RTI?
A vital aspect of RTI implementation at the secondary level is the master schedule and how it accommodates multitiered instruction. Certainly, integrating tier two and three support in a specific content area can be a scheduling nightmare. Typically, decisions about the master schedule are made in isolation using a computer and paying little attention to the specific content-related learning needs of individual students. Supplemental instruction for struggling students is usually scheduled around the already-established master schedule.
We advocate for scheduling strategies that align varying levels of support for individual students who are identified through targeted assessments. To ensure that students in need of tier two and three interventions get the instructional support they require, we recommend that the interventions be designed and scheduled before tier one classes. Of course, this requires a more flexible approach to scheduling.
Besides accounting for individual students’ needs first, a strength of such a flexible scheduling model is that once the master schedule is designed, targeted professional development can be planned to support teachers as they implement the more individualized tiered mathematics delivery system called for in RTI. For example, coteaching involving a mathematics teacher and a special education teacher or intervention specialist is one possible structure for providing more intensive tier one mathematics instruction. The mathematics faculty collectively might decide, on the basis of targeted screening data, to move toward a coteaching model for Algebra 1, but teachers might have limited knowledge of how to effectively plan for that type of mathematics instruction. Therefore, professional development for coteaching that simultaneously embeds designing effective research-supported mathematics teaching practices can be planned.
This approach to scheduling is best done collaboratively with an interdisciplinary team (e.g., administrators, guidance counselors, content specialists, and special educators). Collaboratively, team members determine how to provide support for the students who are most in need. For example, in many schools, instruction for tiers two and three uses a small group pull-out model in addition to daily tier one instruction. This model ideally requires specialists or math coaches to communicate and collaborate effectively with math teachers to design and supplement core mathematics instruction.
On the other hand, in some schools interventions for tiers two and three take place in small groups or one-to-one within a general education mathematics classroom, which often requires collaboration between two or more teachers who can effectively differentiate whole class and small group instruction. The effective implementation of mathematics RTI requires that schools find ways to use the collective talents of teachers and staff members to collaboratively scaffold instruction and provide support across multiple instructional tiers.
How can my school work collaboratively to effectively implement mathematics RTI?
Effective mathematics RTI requires collaboration among mathematics teachers, special educators, and other specialists who have expertise in addressing the needs of students at risk for school failure (e.g., teachers of English language learners, speech and language professionals, and counselors). One effective method for enacting purposeful collaboration with respect to mathematics RTI is through professional learning communities (PLCs).
A PLC is a space where school faculty members come together with a shared sense of purpose and collaboratively engage in inquiry. A PLC should have a clear focus on continuous improvement using data-based decision making and evidence-based practices coupled with the current realities of teaching (Dufour, DuFour, & Eaker, 2008). PLCs can be structured in a variety of ways that meet the needs of a school community (e.g., face-to-face or online book study groups, coteaching, classroom observations of peers, and data-based decision making). PLCs have the potential to deepen both breadth and depth of content, pedagogical, and curricular knowledge. The focus of a PLC can be general or specific, depending on the needs of the school, teachers, and students.
More importantly, PLCs can be implemented for different purposes. For example, PLCs might be formed to support faculty and staff members who are working toward implementing mathematics RTI and its associated responsibilities (e.g., identifying foundational mathematics concepts and skills in core courses; implementing informal assessments, such as the mathematics dynamic assessment, to inform instruction; or collecting and analyzing progress monitoring data). The key to success for a PLC that has been formed to address mathematics RTI is its flexible approach to professional learning that can address the complexities of teaching mathematics in a multitiered system.
PLCs that are focused on mathematics RTI should guide all instruction associated with implementing it, including the enhancement and understanding of mathematics content, how adolescents learn mathematics, and effective mathematics assessment and pedagogy. In particular, PLC participants should focus on understanding how effective mathematics pedagogy and the mathematics content intersect. As teachers blend their knowledge of content and pedagogy into an understanding of how instruction can be transformed, organized, and adapted to meet the diverse needs of students, they are better able to determine the content understandings that students lack and select specific research-supported instructional practices that address these deficits.
PLCs can be used to address the immediate and broader mathematics RTI needs in a school. PLCs can assist teachers and administrators as they wrestle with the shifting roles and responsibilities required to effectively implement mathematics RTI. In addition, PLCs can be used to support the inevitable changes in curriculum and instruction that need to be addressed within a school (e.g., helping administrators, teachers, and support staff members understand how data-based decisions influence mathematics instruction and intervention so that they align with the big ideas of the mathematics curriculum). The implementation of research-supported mathematics practices within RTI is a logical topic for PLCs and, in turn, can deepen content knowledge and understanding of more effective mathematics instructional practices. This outcome has the potential to positively affect students’ mathematical outcomes.
The effective implementation of mathematics RTI can be a daunting notion. It takes time and energy to make it happen. A pragmatic place to start is to identify essential components of the effective implementation of mathematics RTI at the secondary level and to think critically about where your school is in the process using the five questions to help guide this process.
A systematic approach to the implementation of mathematics RTI includes making judicious decisions about the mathematics curriculum, including identifying foundational big ideas that are related to Algebra 1, Algebra 2, and Geometry; developing targeted screening and progress monitoring measures; evaluating current mathematics teaching practices to pinpoint areas for improvement; scheduling on the basis of the mathematics learning needs of individual students; and emphasizing productive collaboration among faculty and staff members through PLCs. PL
- DuFour, R., DuFour, R., & Eaker, R. (2008). Revisiting professional learning communities at work: New insights for improving schools. Bloomington, IN: Solution Tree.
David Allsopp (firstname.lastname@example.org) is a professor and David Hoppey is an assistant professor in the Department of Special Education at the University of South Florida.
|Big Ideas in High School Algebra
- Sorting, identifying, and creating numerical patterns
- Understanding and working with numerical equalities and inequalities
- Understanding and working with variables and expressions
- Using words, tables, graphs, and rules to describe numerical relationships
- Working with and solving equations and word problems
Related Concepts and Skills for Algebra 1 Screening and Tiers Two and Three Interventions
Evaluating expressions for given values of variables and writing algebraic expressions for word statements
- Determine whether a math statement is true or false
- Recognize the relationship of multiplication to numbers and variables (i.e., 6y)
- Identify word statements and one-variable algebra equations that have one variable
- Identify mathematical expressions containing one variable
Working with order of operations and properties of rational numbers
- Simplify numerical and variable expressions by using rules for order of operation
- Simplify and rewrite expressions using the commutative and associative properties of addition and multiplication, the distributive property, the multiplication identity of 1, the multiplication property
- Additive identity of 0
- Simplify variable expressions by combining like terms
Working with absolute value and integers
- State the coordinate of a point on a number line and graph integers on a number line
- Add integers on a number line
- Identify math symbols and their meanings (<, >, =, <, >, =) and use symbols to compare integers representing equalities and inequalities (e.g., 3 < 10)
- Demonstrate understanding of the comparison property and the transitive property
- Graph solution sets on a number line
- Define absolute value, identify the symbol | |, and name the rules for adding positive and negative integers
- Demonstrate understanding of the additive inverse property and name the rules for subtracting positive and negative integers
- Adding and subtracting positive and negative integers
Working with one-variable algebraic equations, word problems, and graphs
- Solve one-variable one- and two-step algebraic equations
- Solve word problems that can be solved using one-variable one and two-step algebraic equations
- Given a linear equation and its graph, describe how changes in the equation (i.e., the slope and the intercept) affect the graph of the line
- Given a set of ordered pairs that represents a relation, represent the relation in graph and table form
- Given different representations (table of values, graphs, and equations) of the same function, compare their representations for similarities and differences
- Given a table of values or a graph, write an algebraic equation that represents the relation represented in the table of values