An Instructional Approach That Boosts Student Confidence in Math

In my first full year of teaching in 2010, I was coached on the merits of the “I Do, We Do, You Do” approach to math instruction, a form of direct instruction. Each lesson started with a hook to pique student interest and then almost immediately pivoted into guided notes and worked examples followed by guided and independent practice. If students got far enough into the independent practice, they might get to tackle some application problems or more interesting tasks, but those were mostly reserved for the fastest workers or at least the students who were motivated to wade through a couple of pages of rote practice.

Even as a first-year teacher, I knew there was something wrong with this approach. My students’ work all looked exactly the same and exactly like the examples I had presented. And when they faced a problem with a slightly different format, they quickly gave up when replicating the exact same steps didn’t work. I wasn’t teaching them the tools necessary to engage in deeper problem-solving and apply their mathematical skills to novel situations. 

All of this changed with the adoption of the new Common Core State Standards and aligned assessments later that year. It was clear that students would need to be able to apply mathematical thinking as outlined in the Standards for Mathematical Practice to many different contexts and would be assessed on problem-solving in addition to procedures. 

The following school year, I attended numerous professional development sessions about how adopting inquiry-based learning techniques and incorporating rich mathematical tasks in the classroom would be key to implementing the Common Core State Standards and the Standards for Mathematical Practice. These inquiry-based tasks were different from the procedural problems I had been working with because they often validated multiple solution paths, incorporated multiple problem-solving skills, and allowed students to construct a conceptual understanding of the mathematics. 

Incorporating Inquiry-Based Learning

Throughout the 2011–12 school year, I added an investigation task to each lesson that invited students to engage in inquiry and authentic problem-solving. At first, I didn’t have many systems to support this inquiry-based work; I would present students with a task and give them some time to attempt it with a partner turn-and-talk, and then we would come back together to discuss it. I noticed that it was often the same students engaging in this problem each day while other students preferred to take a more passive role.

After reading Building Thinking Classrooms in 2020, I decided to try using randomized groupings of students who would work at vertical whiteboards posted on the walls around the classroom for the inquiry-based portion of class. I noticed immediately how this format changed the way that students engaged in the investigation. Working together at the whiteboards produced more group accountability that encouraged students to participate, and it was easier to ensure that every student had a role. Additionally, vertical whiteboards allow for multiple rounds of knowledge sharing during the debrief of the inquiry-based task.

It did take some time to develop strong norms that supported students in productive collaboration during this time, and success was not instantaneous. As with any new routine, I went through different iterations before finding the norms that worked best in my classroom.

After students have been given a set amount of time to complete the problem, we engage in a debrief. I often start with either a gallery walk or two groups swapping boards to observe the work of other groups. After completing the task, students return to their regular desk pairings to discuss the main idea of the task, based on what they observed from their own group work or other groups. 

Through both the group work at the whiteboards and the structured debrief discussion, the tasks become far more accessible for all students, building their confidence with the concepts embedded in the task.

Why Blending is Best

Even as I spent years developing the inquiry-based portion of my lesson structure, I never changed the class notes and guided practice portion of the lesson. I have found that after investing a significant chunk of time (usually 20–30 minutes) working through an inquiry-based task and debriefing the key takeaways, it is critical for students to stamp that learning by summarizing the main ideas in notes and explicitly practicing the procedures and concepts in a guided format that allows for batch and individual feedback. An inquiry-based task can give students a critical “aha!” moment and create the “need” for the content that is introduced in the lesson. 

The note-taking and explicit instruction are an opportunity to norm on a common understanding of key vocabulary, concepts, and procedures. Additionally, by practicing a variety of different problem types in a guided format, students are able to attempt the mathematics in different contexts with increasingly more independence as they build confidence. This allows them the opportunity to make errors and receive feedback and to experience how different contexts might alter the solution path.

Increasingly, there has been a debate in math education that pits inquiry-based learning against explicit instruction. Dividing them is a false dichotomy that is ultimately harmful to educators and students alike. The inquiry-based learning advocates will argue that explicit instruction takes away from students’ constructing their own meaning of mathematics, and the proponents of explicit instruction will argue that inquiry-based learning is time-consuming and less effective and leads to confusion or lack of learning. Both sides of the argument sometimes act as though there is no room for the other one in the same classroom, but this is simply not true. 

There is clearly space for both of these approaches to coexist, and students do in fact benefit from both for different reasons. Inquiry-based learning gives students experience with building problem-solving habits and centers student solution paths in a way that draws out student thinking and meaningful “aha!” moments. 

Explicit instruction provides clarity and common language and formalizes concepts and procedures so that they can be applied to new contexts. Inquiry opens up the questions that explicit instruction then answers. Students benefit most when the two go hand in hand.