Compiled by James Brickwedde, Project for Elementary Mathematics

What follows a preliminary exploration of the attributes of both explicit/direct and inquiry/dialogic-based instructional models. While there are very distinct differences in the approach to learning and the belief in how students learn mathematics, there are some points of agreement that should be understood. Rather than a strict dichotomy between the two, there is some overlap in practices. Much of the practices debated fall into a matter of timing in a lesson and the intention of the instructional move; not the move itself.  At a very basic level, both research and teacher communities that ground themselves in these two outlooks on education agree that it is important that students develop a strong mathematical understanding and there is no excuse for lazy instructional habits. What constitutes understanding and what constitutes the students’ participation in the learning process is where significant differences occur.

Munter, Stein, and Smith (2015) introduced the idea of ‘adversarial collaboration’ to process these differences. This term, coined by Gilovich, Medvec, & Kahnemann (1998) “invites representatives of different perspectives to clarify and come to agreement on how they disagree.” (p. 3) The goal of such collaborations is not to come to a consensus or resolve disagreements but to surface and highlight underlying rationales, perspectives, theories, and priorities that give rise to those disagreements; to identify genuine disagreements. This is an initial review that explores the points of agreement and disagreement. While exploring the nuances of research theory tends to be a conversation stopper in many schools’ professional development settings, one needs to have enough of an understanding of the differences in what constitutes ‘evidence-based research’ to then understand why public policy decisions end up being set in legislation the way they do.

Describing each instructional approach

There is a spectrum of instructional practices and approaches to learning within both the explicit/direct instruction community as there is within the inquiry/dialogic-based instruction community. This section serves to briefly define the various instructional approaches that tend to surface in conversations/debates/arguments around how best to teach children. The following describes direct instruction, explicit instruction, inquiry/dialogic-based instruction, and discovery learning.

Direct Instruction

Direct Instruction (capital letters) and direct instruction (lowercase letters) have their roots in behaviorist, information processing and process-product approaches towards learning. Direct Instruction (capital letters) emerged in the 1960s as a scripted program where teachers follow cues and students respond largely through choral responses based on those teacher cues. More generally, direct instruction (lowercase letters) describes a set of teaching practices that consists of clearly describing an objective, articulating motivating reasons for achieving the objective and connections to previous topics, presenting requisite concepts, demonstrating how to complete the target problem type, and providing scaffolding phases of guided and independent practice, accompanied by corrective feedback (Munter, Stein, & Smith, 2015). Throughout the remainder of this article, it will be noted which version – uppercase/lowercase – that is being referenced. 

Explicit Instruction

While Explicit Instruction overlaps significantly with direct instruction practices, there are differences. Explicit Instruction is not a prescribed intervention. It is a combination of specific elements that together shape an approach towards both the design and implementation of a lesson(s) (Hughes, Morris, & Benson, 2017). Archer & Hughes (2011) outline sixteen elements that have emerged from the work of various researchers. Instruction is focused on critical content and skills that are logically sequenced. Complex skills and strategies are broken down into smaller instructional units. Lessons are organized and focused and begin with clear statements of the lesson’s goals and the teacher’s behavior expectations. Prior skills and knowledge are reviewed at the beginning of the lesson. Clear and concise language is used while providing step-by-step demonstration of the focused skill. A range of examples and non-examples are used in a process of providing guided and supported practice. Students are expected to participate in frequent responses. The teacher closely monitors students’ performance providing immediate affirmative and corrective feedback in a delivery structure that is kept at a brisk pace. There is a focus on making sure students see connections and how ideas fit. Finally, time for practice that is both distributed (multiple opportunities over time) and cumulative (integrates previous and newly acquired skills) is provided.

Differences between Explicit Instruction and formal Direct Instruction (upper case) models is that Explicit Instruction is not a script for teachers to read. It reflects an instructional design and procedures. Teachers are allowed to control the pace of the lesson and modify in-the-moment based on checking for understanding. Teachers are also given freedom to include their own student engagement strategies to manage the classroom (dataworks Educational Research website).  It is not intended to be a hard and fast ‘cookbook.’ A level of teacher responsiveness is necessary. Direct Instruction is far more grounded in curricular design. Explicit Design is not intended to reduce teaching to an act of specified set and sequence of steps.

Inquiry/Dialogic-Based Instruction

The history of inquiry-based learning has roots in the early 20th century with the work of John Dewey. Active engagement in one’s learning, learning by doing, is central to its implementation. As research into how children learn mathematics, and the launch in 1989 of the academic standards movement with math standards published by the National Council of teachers of Mathematics (NCTM), inquiry-based learning is best described as a collaborative where the teacher strategically guides and shapes student thinking based on both mathematical progressions and developmental pathways students progress through. The Standards of Mathematical Practice (what students do) (CCSSM, 2010) and the Mathematics Teaching Framework (what teachers do) (NCTM, 2017) capture the core consensus on the elements for productive and accessible mathematics classrooms.

Frequently described as problem-solving-based classrooms, teachers pose problems and elicit student thinking as students make sense of the context. Through the use of strategic questioning and prompts, the teacher assists students in visualizing solution options, thus allowing students to engage in tasks without prematurely prescribing a solution pathway. In this manner, students begin to reason, model with various tools (concrete, pictorial, and abstract) as they create representations of their solutions. Questions and prompts by the teacher while students are working provide feedback that helps students to reflect on their reasoning, clarify their decision making, and redirect their work if necessary. Public sharing is a regular component of an inquiry/dialogic-based classroom where students share strategies, probe the reasoning of their peers, and compare and contrast strategies looking for efficiencies and alternative approaches to solving a task. There is a focus on mathematical rigor where conceptual understanding, procedural skills and fluency, and connections to real-world applications are developed in tandem with each other. Formative assessment data is a rich source of evidence as students verbalize their thinking and the artifacts of their work are analyzed. Students become owners of their own thinking and become instructional resources for one another.

As with Explicit Instruction, Inquiry/Dialogic-based instruction is not a single style of lesson or instructional practice. It is based on a common set of attributes in how the role of the teacher and students are conceptualized. Much research has developed exploring which combination of elements affect student achievement gains. Further description of this research will be described in later sections.

Discovery Learning

Discovery learning is an often-maligned instructional approach within the range of inquiry/dialogic-based instructional models. Discovery learning is typically characterized as having minimal teacher guidance as students work with extensive access to hands-on materials. There are fewer teacher explanations as students are encouraged to develop multiple strategies to solve problems, look for patterns, and generalize their thinking. There is minimal repetition and memorization. Credited to the work of Jerome Bruner, the discovery process is to encourage students to use their intuition, imagination, and creativity. Projects within the classroom are often developed from students’ interests and questions. Where Direct Instruction (upper case) is criticized for being overly scripted and hyper-focused on sub-skills, Discovery Learning is criticized for being too hands off with open-ended objectives allowing students to work unfocused and potentially flounder mathematically. As with any approach to teaching and learning, a skilled teacher is necessary to implement an instructional model for the benefit of students being able to learn.

A System of Instruction, A System of Practice

A specific example of an inquiry/dialogic model of instruction, figure one captures the work of Simon (1995), Hiebert et al. (1997), the Mathematics Teaching Framework (NCTM, 2017) and the skills of teacher noticing and responsive teaching (Jacobs, et al. 2010, Richards & Robertson, 2016), among others. The term guided inquiry has been applied to this more focused inquiry/dialogic approach towards teaching and learning (Lampert, 2024) and even cognitively guided instruction (Carpenter, et al. 2015). The model is based on an unresolvable tension that teachers need to navigate. It is coined as “The Dilemma.”

How to assist students in experiencing and acquiring mathematically powerful ideas but refrain from assisting so much that students abandon their own sense-making skills in favor of following the teacher’s directions (Hiebert, et al. 1997, Ball 1993; Lampert 1991; Wheat 1941).

As described in Hiebert, et al. (1997), this system of instruction begins with teachers identifying clear mathematical learning goals for their students and, based on a deep understanding of their students’ levels of thinking, the teacher’s vision of how a lesson(s) will progress. This ‘vision’ is based on what Simon (1995) coined as the ‘hypothetical learning trajectory’ based on developmental learning frameworks identified in research. The trajectory guides the teacher during the lesson. The teacher noticing skills of attending, interpreting, and deciding responsively in-the-moment provides the feedback for the teacher  – and the students – in how to respond that shapes during the flow of the lesson(s) (Jacobs et al., 2010, Richards & Robertson, 2016). The coherency in the set of tasks moves students towards the learning goals. Feedback from students’ thinking captured both in the moment and through additional teacher assessments leads to revision in the trajectory. Tasks are purposely selected, but the sequence can be revised based on feedback of student comprehension.  Figure one provides a visual of this guided instructional approach towards teaching and learning.

 

 Teacher-Centered ←→ Student-Centered

This is a dichotomy that is often used to describe the differences between Explicit/direct and Inquiry/Dialogic instruction-based approaches. While there are distinct differences in the roles of the teacher and the students within each system, both approaches focus on aspects of student thinking and have specific roles that teachers are to serve in shaping the learning and moving students forward mathematically. If there is a distinct difference among the two sets of instructional approaches, how student thinking is accessed and for what purpose in the decision-making process of the teacher is clearly different. In Explicit/direct instruction, student thinking is probed, but it is to gain clarity of the student’s understanding of the skill that is targeted for the lesson. In Inquiry/Dialogic Instruction-based classrooms, student thinking is accessed not only for assessing one’s understanding but it central in guiding the student’s capacity to reason and generalize their decision-making process, how that work compares and contrasts with their peers’, and develop the language skills in which to communicate that thinking. Another distinct difference is in the role of communication. Talking and processing one’s thinking with peers is not central in explicit instruction/direct instruction practices. It is central to the inquiry/dialogic-based instruction models, thus the term ‘dialogic.’ (Munter, et al. 2015). 

Summary

This is an initial attempt to define terms that one hears in various discussions around ‘evidence-based’ research. In future articles, a closer look at what research guides each field and, therefore, how that research shapes instructional practices will be explored. The end goal is to provide a document that can be used within a district to decide which practices are suitable for which instructional setting and how responsive those practices are in supporting students through the learning process.