Building Thinking Classrooms in Mathematics

Educator and researcher Peter Liljedahl’s 2020 book, Building Thinking Classrooms in Mathematics: 14 Practices for Enhancing Learning, has grown in popularity in classrooms across the US in the years since it was published. Driven by concerns over a decade plus drop in math scores and a widening achievement gap, educators are seeking new approaches to improve student learning and engagement in math. Given the growing interest in the Building Thinking Classrooms movement, let’s unpack exactly what a thinking classroom is and how you can get started using this approach in your class.

What are thinking classrooms in math?

Thinking classrooms are designed to get students to more actively think when they engage in math. This may leave you wondering: What are my students doing in math class now if they aren’t thinking? 

Both the book and the methods within it were born out of observations Liljedahl made in K–12 math classrooms across 40 different schools. What Liljedahl noticed was that regardless of the student makeup or factors like the school type or location, most students were engaging in four behaviors instead of thinking:

• Slacking:  Not attempting the given task at all.
• Stalling:  Not attempting the given task but masking their slacking by doing things like sharpening pencils or getting a drink of water.
• Faking:  Masking not doing the task in a more academic way, like looking at a textbook without producing any significant work.
• Mimicking:  Attempting the task but only following rote steps provided by the teacher rather than developing real understanding. 

Liljedahl found that only about 20% of students actually tried to engage in the math in a way that built their understanding of the concepts being taught. This instructive percentage led him to identify practices that support deeper student engagement and thinking. 

The goal of a thinking classroom is to move students from being passive receivers of mathematical knowledge to being mathematical thinkers. In a thinking classroom, the focus shifts from getting the right answer to exploring the process and understanding the concepts that underlie the problem. Instead of asking “am I right,” students explore, discuss, and test approaches to progressively more challenging problems. Meanwhile, teachers spend their time and energy thinking about the sequence of problems and presenting open-ended questions that will help students uncover key mathematical understandings. 

What does a thinking classroom look like?

Thinking classrooms look and sound different than non-thinking classrooms. Most classrooms are oriented around the teacher at the front of the room. Students are at their desks, or sometimes on a rug for a group lesson in elementary classes. Lessons typically follow the “I do, we do, you do” model consisting of group instruction, group practice, and individual practice.

In a thinking classroom students stand in groups, working together on what Liljedahl terms vertical, non-permanent surfaces, which are essentially chart paper-sized dry erase boards. They work through a series of carefully chosen tasks that start at a low enough entry point that all students can engage and then build to help students develop understanding of the mathematical concepts that are the focus of the current learning. Students are talking and writing and considering each other’s work. The teacher is moving around asking questions to help clarify or stretch student thinking and directing them to consider the work of other groups that might challenge or push their thinking.

Importantly, teachers answer “keep thinking” questions, which students ask to help them continue engaging with the task, while avoiding answering “stop thinking” questions like “Will this be on the test?” or “Is this right?” These are a student’s way of shutting down learning.

Practices for building a thinking classroom in math

Ready to get started making your classroom a thinking classroom? The book itself is structured to support you on this journey. In each chapter, Liljedahl lays out one of 14 teaching practices, which his observations show support student thinking. Each chapter concludes with FAQs about the practice and a list of “micro” and “macro” moves you can do to implement the practice. 

The 14 practices are organized into four clusters called toolkits. Liljedhal suggests how to implement the practices within each toolkit, sometimes implementing multiple practices at once, and sometimes moving to the next practice only after the previous one is well-established. The choice of the word “building” in the book’s title is deliberate, as you need to invest the time to build these new practices and habits in both yourself and your students.

Let’s dive into the first three practices, which are the best way to get started with this new way of engaging with math

Practice 1: Choose the right tasks to encourage thinking

When students are only presented with routine, cookie-cutter problems, they fall back into mimicking behaviors. When transitioning to Building Thinking Classrooms, start with highly engaging, non-curricular tasks, and then once students are comfortable with the practices, transition to more scripted curricular tasks. Liljedahl found that the engagement and thinking that comes out of the non-curricular tasks carries over into the curricular-based tasks.

Teacher move: When you first start establishing a thinking classroom, the first four to six tasks you present should be non-curricular. This immerses students in this approach, allowing them to build new thinking habits. Starting with non-curricular tasks also allows students to feel freer to take risks since this material “won’t be on the test.” 

Practice 2: Form collaborative groups

A critical component of a thinking classroom is active student collaboration, yet adults know that true collaboration can be challenging. Liljedahl found that the most effective way to form true collaborative groups was to transparently randomize the groups and to make groups of no more than three students for Grades 3 and up.

Teacher move: The groups must also be refreshed frequently—think every hour. Small, randomized, short-term groups decrease social stress and increase knowledge sharing, enthusiasm, and engagement.

Practice 3: Work on vertical, non-permanent surfaces

In an NCTM podcast, Liljedahl talks about classrooms as a system. Systems have cultures, rhythms, and normative structures. Systems are also resistant to change. Because of this, Liljedahl suggests you need to overwhelm the system to help students unbuild old habits and build new ones. This is why he suggests you implement all three strategies of the first toolkit all at once and start with a series of non-curricular tasks.

Teacher move: In a thinking classroom students primarily work standing with their group around vertical dry erase boards. Liljedahl tested multiple variations of where students work and found the vertical non-permanent surface strategy nearly eliminates passivity, task avoidance, and mimicking. It also allows teachers to more easily see each group’s thinking and orchestrate discussions across groups.

What makes a good thinking task in the math classroom?

Ok, so you are ready to jump in, but you need some good thinking tasks to get started. When looking for rich thinking tasks, look for problems that require students to do the following:

• Get stuck and think
• Test out and revise solution paths
• Apply knowledge in novel ways
• Step outside of routine thinking
• Engage with a variety of mathematical ideas and concepts

For non-curricular tasks, Liljedahl provides examples on the Building Thinking Classrooms website, or see our article on Open Middle math problems as another possible place to look. For curricular tasks, consider how they can meet the criteria above. When considering a task, instead of asking “Is this a good task?” ask, “What is this task good for?” This reframing helps you think about the mathematical concepts that a task can support for students. 

Thinking classroom math activities

You can implement the thinking classrooms approach regardless of what curriculum you use. Start by examining how the various aspects of the curriculum relate to the 14 practices, tweaking them as needed. Here are some examples of Into Math program components that can support a thinking classroom:

• Open Middle problems: These problems, created by Robert Kaplinsky, are great examples of thinking tasks. Their low floor, high ceiling approach offers both entry and challenge for all students. They are designed to push students beyond rote thinking to use what they know about math to solve the problem flexibly, without mimicking.
• Math language routines: These research-based routines help students hone their ability to communicate mathematically, which is critical to collaborating with others. “Compare and Connect” can be used to help students in different groups compare their approaches to solving the same problems. “Stronger and Clearer Each Time” can help students improve how they interact at their vertical white boards both in terms of explaining their ideas and in terms of asking questions of one another. Along with additional math language routines, Into Math includes Peer Coach videos, which show grade-level students modeling a language routine as a clear step-by-step demonstration for the teacher.
• Teacher prompts and support: Teaching strategies and supports embedded within Into Math can help support thinking classrooms:
  • Open-ended questions designed to spur student thinking
  • Universal Design for Learning supports that include helping teachers identify critical precursor content for their launch activities to activate prior knowledge relevant to a given task
  • Common error explanations that can help teachers think in advance about questions meant to move student thinking forward, without actually naming the student’s error or prescribing how to proceed

Don’t be afraid to start

Implementing a thinking classroom involves shifting well-worn routines and practices, so getting started may seem daunting. Last year, I interviewed Beverly Broedlow and Rachel Stalford, two teachers from California who implemented the practices. When I asked the best way to get started, Broedlow said, “It doesn’t have to be perfect; you just have to start it. If you start somewhere, it gets better over time. You get sharper with the tools by just trying one thing, and then trying another thing, and then continuing on and on until you have the full thinking classroom.”

***

HMH’s core math solution Into Math for Grades K–Algebra 1 includes a variety of routines, real-world connections, and problem-solving strategies that deepen students’ mathematical understanding.

We want to hear from you! For questions, comments, or feedback, please contact us at Shaped@hmhco.com or follow us on Instagram, LinkedIn, and YouTube.