Direct Instruction and Inquiry in Mathematics

In 2010, my first year of teaching, I was coached on the benefits of the “I Do, We Do, You Do” approach to teaching mathematics, a type of direct instruction. Each lesson began with a hook to capture the student’s interest and then almost immediately transitioned into guided notes and worked examples, followed by guided and independent practice. If students could progress far enough in independent practice, they could tackle some practice problems or more interesting tasks, but these were mostly reserved for the fastest workers or students who were motivated to complete at least a few pages of memorization exercises. .

Even as a first-grade teacher, I knew there was a problem with this approach. My students’ work all looked exactly the same and looked exactly like the examples I presented. And when they encountered the problem in a slightly different format, they quickly gave up when repeating the same steps didn’t work. I wasn’t teaching them the tools to engage in deeper problem solving and apply math skills to new situations.

This all changed with the adoption of the new Common Core State Standards and aligned assessments later that year. It was clear that students needed to be able to apply mathematical thinking as outlined in the chapter. Standards of Mathematical Practice will be evaluated in many different contexts and will be evaluated in terms of problem solving in addition to procedures.

Over the next school year, I attended numerous professional development sessions on how adopting inquiry-based learning techniques and incorporating rich math tasks into the classroom would be key to implementing the Common Core State Standards and Standards for Mathematical Practice. These inquiry-based tasks differed from the procedure problems I studied because they often confirmed more than one solution path, incorporated multiple problem-solving skills, and allowed students to build a conceptual understanding of mathematics.

During the 2011–12 school year, I added a research assignment to each course that challenged students to investigate and solve original problems. In the beginning, I didn’t have much of a system to support this inquiry-based work; I would present students with a task and give them some time to try the task with a partner, turning and talking, and then we would come back together to discuss the topic. I noticed that it was often the same students who dealt with this problem every day, while other students preferred to take a more passive role.

after reading Creating Thinking Classrooms In 2020, for the research-based portion of the class, I decided to try creating random groups of students to work on vertical whiteboards hung on the walls around the classroom. I immediately noticed how this format changed the way students participated in research. Working together on whiteboards produced more group accountability that encouraged students to participate, and it was easier to ensure that each student had a role. Additionally, vertical whiteboards allow for information sharing across multiple rounds during the briefing of the inquiry-based task.

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

We inform students after they are given a certain amount of time to complete the problem. I usually start with a gallery walk to observe other groups’ work, or with two groups swapping boards. After completing the task, students return to their normal table pairings to discuss the gist of the task based on what they observed from their own group work or from other groups.

Through both group work on the whiteboard and structured debriefing discussion, tasks become much more accessible to all students and their confidence in the concepts covered in the task increases.

While I have spent years developing the inquiry-based part of my course structure, I have never changed the lecture notes and guided practice portion of the course. I have found that after dedicating a significant amount of time (usually 20-30 minutes) to an inquiry-based task and learning about the key takeaways, it is critical for students to mark that learning by summarizing the main ideas in notes and practicing them explicitly. procedures and concepts are in a guided format that allows for collective and individual feedback. An inquiry-based task provides students with a critical “aha!” Create the “need” for the content introduced in the lesson.

Note-taking and explicit instruction are an opportunity to come to a common understanding of key vocabulary, concepts, and procedures. Additionally, by practicing a variety of different problem types in a guided format, students can experiment with mathematics increasingly independently in different contexts as they gain confidence. This gives them the opportunity to make mistakes, get feedback, and experience how different contexts can change the way they solve.

There is a growing debate pitting inquiry-based learning against open education in mathematics education. Dividing them is a false dichotomy that ultimately harms both educators and students. Proponents of inquiry-based learning will argue that open instruction prevents students from constructing their own meaning of mathematics, and advocates of open instruction will argue that inquiry-based learning is time-consuming and less effective and leads to confusion or a lack of knowledge. learning. Both sides of the debate sometimes act as if there is no room for the other in the same classroom, but this is simply not true.

There is clear scope for these two approaches to co-exist, and students actually benefit from both approaches for different reasons. Inquiry-based learning gives students experience in building problem-solving habits and encourages student solutions, student thinking, and meaningful “aha!” understands.

Open education provides clarity and common language and formalizes concepts and procedures so they can be applied to new contexts. The inquiry opens questions which it answers following clear instruction. Students benefit most when the two go hand in hand.