Academic Quarterly

Marvelous Mathematician Spring 2019

Applying Explicit Instructional Techniques to both Direct Instruction and Inquiry-based Learning


Teachers are often concerned about choosing between direct instruction or inquiry-based lessons. But, what is it about teaching that influences learning in math? In a meta-analysis of research on this question Hiebert and Grouws (2007) concluded that two things make the biggest overall difference in students understanding mathematical concepts.

  • Teaching attends explicitly to concepts – to connections among mathematical facts, procedures, and ideas (p. 383).
  • [Students are regularly engaged in] struggling or wrestling with important mathematical ideas . . . that students expend effort to make sense of mathematics, to figure something out that is not immediately apparent . . . solving problems that are within reach (p. 387).

Hiebert and Grouws further explain that both student-centered (inquiry-based learning) and teacher-centered (direct instruction) lessons can provide targeted and structured activities in which students, with teacher guidance, grapple with challenging concepts and problems and mathematical connections are made clear. The National Mathematics Advisory Panel (2008) stated in its Final Report that, “all-encompassing recommendations that instruction should be entirely ‘student centered’ or ‘teacher directed’ are not supported by research” (p. xxii). Hattie, Fisher, and Frey (2017) found that both dialogic (inquiry-based learning) and direct instruction have high effect sizes when done well (0.82 and 0.59, respectively). Both methods can provide a high degree of success with learning math. However, the best way to ensure that both direct instruction and inquiry-based lessons succeed with the two elements that Hiebert and Grouws found essential for effective mathematics instruction is to incorporate evidence-based, explicit instructional techniques.

Explicit Instruction

math instruction design and deliveryArcher and Hughes (2011) explain that explicit instruction is a “structured, systematic, and effective methodology for teaching academic skills” (p. 1). It is an instructional approach that includes specific delivery and design elements that provide students with the best chance to succeed. The box to the right shows a list of these elements. The elements of explicit instruction dictate neither a direct instruction nor inquiry-based model for lessons. However, incorporating these elements is instrumental in making either type of lesson a success for students. 

Models of Direct Instruction and Inquiry-based Instruction with Explicit Instruction

areas of a triangleThe diagram to the right is part of a lesson on how to determine the areas of triangles. Diagrams such as this provide an opportunity for students to see the connections between the areas of triangles and the areas of rectangles that circumscribe the triangles (fit exactly around the triangles). The rest of this article describes examples of good direct instruction and good inquiry-based instruction for this lesson and make clear how explicit instruction plays a critical role in both types of lessons.

Model of Direct Instruction

Below are directions for a direct instruction lesson utilizing explicit instructional techniques. The lesson focuses on learning how to determine the areas of triangles and recognizing the connections between the areas of triangles and the areas of rectangles. The explicit instructional techniques are called out with bold italic text throughout the lesson plan.

Direct Instruction Math Lesson Plan

The direct instruction lesson plan begins with a strategic warm-up problem and a clear lesson objective. The lesson is sequenced step-by-step both for careful building of the concept and to facilitate a brisk pace. It is designed to keep students engaged and involved. The plan calls for key mathematical connections to be made explicit and to guide students to reason about important ideas. Student work and understanding are monitored frequently with feedback provided as needed. At the end of the lesson, judicious practice is provided, and an exit ticket used as a final means to assess learning. This lesson plan is one example of good direct instruction and exemplifies how explicit instructional techniques provide an excellent opportunity for student success.

Model of Inquiry-based Instruction

Below are directions for an inquiry-based lesson focused on the same math concepts as above and utilizes elements of explicit instruction.

Inquiry Based Math Lesson Model

The inquiry-based lesson plan began with reviewing key prior knowledge and stating the lesson goal. The beginning triangle activity is intended to make explicit what students are going to be doing during the inquiry portion of the lesson. The inquiry portion of the lesson is designed for step-by-step progress with students focused on a sequential set of ideas. This step-by-step process allows for both building understanding and keeping a sense of forward momentum to the lesson. Students are required to grapple with important mathematical ideas on their own and in groups. At strategic points work is shared, clarified, corrected, if needed, and key mathematical connections are made explicit. After completing the inquiry part of the lesson, the teacher provides clear models applying the new formula, checks for understanding, and assigns judicious practice. Organizing an inquiry-based lesson with explicit instructional techniques provides the lesson with an excellent chance for success.


The inquiry-based lesson plan and the direct instruction lesson plan have much in common – they both make effective use of explicit instructional techniques. Using these techniques creates a great lesson flow in both models. Students are active, engaged, and reason about important mathematical ideas, while key mathematical connections are made clear and explicit. Hattie, Fisher, and Frey (2017) state that “It should never be an either/or situation. . . Both dialogic [inquiry-based] and direct instruction have a role to play throughout the learning process, but in different ways” (p. 24). Applying explicit instructional techniques provides the best chance for success with both models.


Archer, A. L., & Hughes, C. A. (2011) Explicit Instruction: Effective and efficient teaching. New York, NY: Guilford.

Hattie, J., Fisher, D., & Frey, N. (2017). Visible Learning for Mathematics. Thousand Oaks, CA: Corwin

Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371-404). Reston, VA: National Council of Teachers of Mathematics.

National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. Retrieved from


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