Elementary Math Academic Interventions

Three-Tiered RtI Mathematics Model Characteristics

Tier 1 is the mathematics instruction that all students in a classroom receive. It entails universal screening of all students, regardless of mathematics proficiency, using valid measures to identify students at risk for future academic failure—so that they can receive early intervention. There is no clear consensus on the characteristics of instruction other than that it is “high quality.”

Tier 2 interventions include schools provide additional assistance to students who demonstrate difficulties on screening measures or who demonstrate weak progress. Tier 2 students receive supplemental small group mathematics instruction aimed at building targeted mathematics proficiencies. These interventions are typically provided for 20 to 40 minutes, four to five times each week. Student progress is monitored throughout the intervention.

Tier 3 interventions are provided to students who are not benefiting from tier 2 and require more intensive assistance. Tier 3 usually entails one-on-one tutoring along with an appropriate mix of instructional interventions. In some cases, special education services are included in tier 3, and in others special education is considered an additional tier. Ongoing analysis of student performance data is critical in this tier. Students often receive relevant mathematics interventions from a wide array of school personnel, including their classroom teacher.

Student and teacher talking

Laying the Foundation for Mathematical Reasoning

Map for Laying the foundation for Math Reasoning

Practices for Intensive Intervention in Mathematics

Explicit, Systematic Instruction

Explicit instruction may take the form of teaching students how to use manipulatives, teaching specific algorithms for solving computational problems, or teaching strategies for solving more advanced mathematical concepts.

Systematic instruction considers the scope and mathematical trajectories, such as the types of examples used for developing the foundational skills prior to introduction/re-teaching of grade-level material.

Advance Organizer: Providing students with an advance organizer allows them to know the specific objective of the lesson and its relevance to everyday life.
Assessing Background Knowledge: In assessing background knowledge, instructors determine whether students have mastered the prerequisite skills for successful problem solving in the new concept area. If the prerequisite skills were recently covered, assessment of background knowledge should be conducted quickly. If, however, those skills were taught several weeks ago, more time may be needed to refresh students’ memories. Instructors can also determine whether students are able to generalize previously learned concepts to the new concept.
Modeling: During the modeling phase, instructors “think aloud” as they model the process of working through a computation problem; read, set up, and solve a word problem; use a strategy; or demonstrate a concept. During modeling, instructors should be clear and direct in their presentation; they also should be precise and mindful in using general and mathematical vocabulary as well as in selecting numbers or examples for use during instruction. During modeling, instructors should involve students in reading the problems and should ask questions to keep students engaged in the lesson.
Guided Practice: During guided practice, instructors engage all students by asking questions to guide learning and understanding as students actively participate in solving problems. During this phase, instructors prompt and scaffold student learning as necessary. Scaffolding is gradually eliminated as students demonstrate accuracy in using the material being taught. Positive and corrective feedback is provided during this phase, and instruction is adjusted to match student needs. Students should reach a high level of mastery (typically 85 percent accuracy or higher) before moving out of the guided practice phase.
Independent Practice: After achieving a high level of mastery, students move to the independent practice phase where they autonomously demonstrate their new knowledge and skills. During independent practice, the instructor closely monitors students and provides immediate feedback as necessary. Countless independent practice activities can be used with students, and the primary focus of the independent practice activity should be related to the content of the modeling and guided practice. If students demonstrate difficulty at this stage, instructors evaluate and adjust their instruction to re-teach concepts as needed. 6. Maintenance: Students with disabilities often have a difficult time maintaining what they have learned when the knowledge is not used on a regular basis. Students are given opportunities to independently practice these skills during the maintenance phase. During this phase, instructors use distributed practice to assess student maintenance at regularly scheduled intervals. Distributed practice is focused practice on a specific skill, strategy, or concept. The frequency of these practice assessments is determined by the difficulty level of the skill and according to individual student needs. Maintenance may also include cumulative practice.

Concrete, Representational/Visual/Pictorial, and Abstract/Symbolic Models

Using multiple representations to teach mathematics allows students to understand mathematics conceptually, often as a result of developing or “seeing” an algorithm or strategy on their own. By building strong conceptual understanding, students are able to better generalize skills and understand algorithms.

Concrete: In this phase, students use three-dimensional manipulatives to solve problems and to gain a better conceptual understanding of a concept. Examples of manipulatives include counting bears, snap cubes, base-10 blocks, real or plastic money, clocks, fraction tiles, geoboards, Algeblocks, algebra tiles, and others. It is helpful to use a variety of manipulatives (if possible) to teach concepts so that students can generalize the concept being taught. Using an assortment of manipulatives is not always possible, however; some concepts can only be taught using a specific manipulative. It is important to note that although students may demonstrate proper use of a manipulative, this does not mean that they understand the concepts behind use of the manipulative. Explicit instruction and student verbalizations, such as explaining the concept or demonstrating use of the manipulative while they verbally describe the mathematical procedure, should accompany all manipulative use.
Representational/Visual/Pictorial: Students use two-dimensional pictures, drawings, or diagrams to solve problems. These pictures, drawings, or diagrams may be given to the students, or they may draw them when presented with a problem. These representations should be used to connect and solve the same concepts previously taught using concrete objects. Representational models also may be presented virtually through websites or tablet applications. With a virtual representation, students move the image with a mouse or with their hands.
Abstract/Symbolic: During this phase, students are expected to solve problems through the use of numbers and symbols rather than with the use of concrete objects or visual representations. Students are often expected to memorize facts and algorithms as well as to build fluency.

Example: Solving the Problem 4 + 5: