Explicit, systematic instruction in mathematics requires educators to clearly teach the steps involved in solving mathematical problems using a logical progression of skills (Hudson, Miller, & Butler, 2006; Montague & Dietz, 2009). 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 (Gersten et al., 2009; Kroesbergen & Van Luit, 2003; Maccini, Mulcahy, & Wilson, 2007). Regardless of the concept or skill being taught, explicit, systematic instruction should include the following components (Archer & Hughes, 2011; Hudson et al., 2006):
1. Advance Organizer: Providing students with an advance organizer allows them to know the specific objective of the lesson and its relevance to everyday life.
2. 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.
For example, if students have previously learned regrouping strategies in addition and subtraction, are they able to generalize these concepts to regrouping in multiplication and division? In addition, instructors should ask students questions about the new concept to assess their knowledge of the concept.
3. 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.
4. 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.
5. 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.
Instructors often want to know how much time they should spend on each phase. Although there are no specific guidelines concerning how much time should be devoted to each phase, the bulk of the instruction should occur within the guided practice phase (National Center on Intensive Intervention, 2013)