Why Building Thinking Classrooms Works: Using Evidence from Cognitive Science Research to Explain the Effectiveness of the BTC Framework

Peter Liljedahl’s Building Thinking Classrooms (BTC) framework has become central in discussions about K-12 math education since the publication of his book, Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning, in 2020. Its 14 teaching practices have gained significant traction across Canada, the U.S., and beyond, with BTC sessions—often featuring Liljedahl himself—appearing at nearly every major mathematics education conference.

I started incorporating some of the BTC practices into my high school math classes in 2019 and have supported a number of teachers with them for a few years prior as a math coach. Through my own experiences with the framework, I observed significant improvements in students meeting learning objectives, their problem-solving enjoyment, and the quality and quantity of math discussions amongst students. For me, this experience highlighted the framework’s potential for fostering productive and enjoyable math learning.

As a math leader for my school board, I have collaborated with Peter along with a team of math coaches to train thousands of K-12 educators in BTC practices through a lens of equity and inclusion. Over two years, the response from teachers has been overwhelmingly positive, with many teachers in almost every school in my district experimenting with the practices in their classrooms.

However, the widespread adoption of BTC across North America has invited scrutiny from parents, educators, and others interested in public education. Critics question the scientific evidence supporting BTC’s effectiveness, especially since cognitive science research often highlights explicit instruction as the most effective math teaching method.

I want to be clear that I welcome this scrutiny and value the debate about math teaching methods. Engaging in respectful discussions is crucial for professional growth and improving the field of math education. I also believe that the BTC framework has room for improvement and will continue to evolve. Given the apparent conflict between cognitive science research and the positive experiences of BTC users—including my own—I began exploring this body of research and its related theories to try and understand BTC’s effectiveness from a cognitive science perspective.

My exploration is revealing that cognitive science research—and specifically cognitive load theory—offer valuable insights into why BTC practices can enhance student learning. Instead of opposing viewpoints, I found that many BTC principles align with established cognitive science research.

While this post will not fully address all criticisms of BTC or inquiry-based instruction in general, it aims to offer insights into why BTC practices work (and sometimes don’t) and how their use is actually supported by the same research base that is often used to argue against them. I hope this piece contributes to a more nuanced discussion about math teaching, fosters ongoing constructive dialogue and helps to integrate diverse strategies to meet the complexities of inclusive classroom instruction.

Part I: Background

Since this post is aimed at classroom teachers, parents, researchers, and other education stakeholders who may be unfamiliar with cognitive science or Building Thinking Classrooms beyond the basics, I will provide a brief overview to help illustrate the connections I will be making between these two areas.

What is Explicit (or Direct) Instruction?

Explicit instruction, also referred to as direct instruction, is a teaching approach that introduces new concepts through clear, step-by-step explanations, teacher modelling, and structured guidance. Students are actively engaged through answering questions, practicing, and receiving feedback to achieve understanding and mastery.

Barak Rosenshine (1976) suggested that direct instruction was an optimal form of teaching primary school students from low socioeconomic backgrounds. In 2012, Rosenshine shared 10 of the most salient strategies in Principles of Instruction: Research-Based Strategies That All Teachers Should Know. These practices are supported by evidence from cognitive science research and classroom studies and assist students with developing strong content knowledge, understanding, and fundamental skills in many subjects, including mathematics.

It is important to make a distinction between the terms direct instruction and Direct Instruction (with capital letters). The latter refers to a specific model of teaching developed by Siegfried Engelmann in the 1960s that involves carefully sequenced and scripted lessons (National Institute for Direct Instruction, 2024). I will use the term explicit instruction instead of direct instruction to avoid any conflation with Engelmann’s Direct Instruction approach.

A main distinction between explicit instruction and inquiry-based instruction, such as BTC, lies in the sequence of explicit teaching and problem-solving. Advocates of explicit instruction follow an instructional hierarchy that emphasizes clear teaching and mastery of mathematical procedures prior to engaging in problem-solving experiences. In contrast, inquiry-based instruction uses problem solving as a means to learn concepts and procedures, with explicit instruction as one way to summarize the learning at or towards the end of lessons.

What is Cognitive Load Theory?

Much of the research supporting explicit instruction in K-12 math classrooms is based on cognitive load theory, developed by John Sweller in 1988. This theory proposes that the human brain has a limited working memory capacity for processing new information. Generally, a person can hold about 7 chunks of information at a time, plus or minus 2 (Miller, 1956), but it can be as low as 3 to 5 chunks for young adults (Cowan, 2010). When this capacity is exceeded, learning is hindered. However, retrieving information from long-term memory into working memory appears to have no limits (Kirschner, Sweller, & Clark, 2006).

When learning new information, there are three sources of cognitive load: intrinsic cognitive load, extraneous cognitive load, and germane cognitive load. Intrinsic load is the inherent complexity of the information that a learner has to process. This complexity is often measured in terms of element interactivity (Kirschner et al., 2017, Ashman, Kalyuga, & Sweller, 2019). 

A task with low element interactivity involves processing bits of information that are independent of one another, such as learning the names of three-dimensional solids (e.g. sphere, cylinder, cone). Since these names are unrelated, the intrinsic cognitive load of this task is low.

Conversely, a task with high element interactivity requires processing multiple pieces of information simultaneously. For example, determining the relationship between the volumes of spheres, cylinders, and cones as the radius increases involves calculations and comparisons, thereby increasing element interactivity and intrinsic cognitive load. However, if a student is skilled in calculating volume, then the intrinsic load is decreased. Therefore, element interactivity depends on both the task and the level of expertise of the learner (Kirschner et al., 2017).

Extraneous cognitive load, in contrast, refers to the mental effort caused by factors unrelated to the learning task (Kirschner et al., 2017). Examples of extraneous load are the manner in which the task information is presented, the clarity of the instructions, or the environmental stimulus of the classroom during a task. To reduce extraneous load, teachers should design tasks and environments that minimize distractions, use clear language, and focus on the learning goals.

Germane cognitive load refers to the mental effort involved in integrating new learning into long-term memory through schemas—mental representations of concepts that help to make connections to different ideas and improve efficiency. Unlike intrinsic and extraneous cognitive load, germane load benefits learning and should be promoted.

In summary, cognitive load theory implies that teachers should design instruction to:

• minimize intrinsic and extraneous cognitive load to avoid overwhelming learners 
• promote drawing on students’ long-term memory to support new learning 
• maximize germane cognitive load to promote information processing into long-term memory for later use

It is mainly from this perspective that advocates of explicit instruction argue that inquiry-based learning with minimal teacher guidance is ineffective (Kirschner, Sweller, & Clark, 2006). “Discovery learning” is seen as imposing an unnecessarily high cognitive load on novice learners by leaving students to aimlessly figure out problems without direction and fail to develop useful schemas for understanding math concepts. Explicit instruction proponents, therefore, advocate for teachers to lead the learning of math concepts and skills through clear explanations, modelling solutions, scaffolding learning for novice learners, asking lots of questions, checking for student understanding, and providing guided practice (Rosenshine, 2012).

What is Building Thinking Classrooms?

The Building Thinking Classrooms framework is a collection of 14 teaching practices developed by Peter Liljedahl over 15 years of research. When implemented in concert, these practices are designed to increase student thinking and improve math learning. These practices are described in detail in his book, Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning, published in 2020. Updates to the framework are available in his second book, Mathematics Tasks for the Thinking Classroom, Grades K-5, co-authored with Maegan Giroux.

The BTC framework was developed to address what Liljedahl identified as a lack of meaningful and deep thinking on the part of students in mathematics classrooms, which he attributes to institutional norms that either inhibit student thinking or fail to encourage it. These norms include ineffective teacher-led lessons that prioritize mimicry over understanding, seating students independently in rows rather than in clusters to promote collaboration, and over-scaffolding activities that foster learned helplessness. The BTC framework is meant to disrupt these norms by elevating students’ expectations while developing a supportive classroom environment with teacher guidance.

The most recognized aspect of the BTC framework is having students solve problems on vertical non-permanent surfaces (e.g., whiteboards, chalkboards, and windows). However, the framework is much more than that one strategy, encompassing a variety of teacher moves, task designs, and assessment practices that guide students to solve math problems effectively and to learn skills and concepts efficiently. For a more detailed overview of each of the 14 BTC practices, you can read the executive summary.

Part II: Connections Between Cognitive Science and Building Thinking Classrooms

In this section, I link cognitive science research to three BTC practices that are often criticized for lacking scientific evidence:

1. Constructing students’ understanding of math skills or topics using “thin-sliced” tasks with hints and extensions.
2. Having students work collaboratively in random groups of three.
3. Encouraging students to make meaningful notes for their “future forgetful self.”

As I delved deeper into explicit instruction, cognitive load theory, and the myriad instructional effects demonstrated by cognitive science research, I began to see why these three BTC practices often succeed in math classrooms. My analysis also provides insight into the conditions that are necessary for these practices to be successfully implemented.

1. Constructing students’ understanding of math skills or topics using “thin-sliced” tasks with hints and extensions

Overview

A thin-sliced task is a series of questions strategically designed to build a math skill or concept in small, progressive steps. The questions gradually increase in complexity and difficulty, but begin by drawing on students’ prior knowledge. The initial questions should be simple enough for anyone in the class to answer. The next set of questions go deeper into a new math topic, and the final set reaches the lesson’s learning goals with higher complexity. Within each group, questions incrementally increase in difficulty, with only one aspect of the question changing at a time. This prompts students to focus on specific aspects of the mathematics, allowing them to gradually and sequentially build their skills and understanding.

Here is an example of a thin-sliced task that my colleague, Michelle Cavarretta, and I developed to introduce and develop skills for simplifying algebraic expressions. We cut the stages into strips of paper so students would only focus on one stage at a time. Once a group of students felt that they had successfully answered all three questions, they would call me over to check, and if they were correct, the group moved onto the next stage of questions.

Stage	Tasks
A	Simplify these statements: 5 MOOSE + 4 SHEEP + 3 MOOSE – 2 SHEEP 8 MOOSE + 5 SHEEP – MOOSE – 4 SHEEP 6 MOOSE + 2 SHEEP – 5 MOOSE + 3 SHEEP + 8 MOOSE – SHEEP **CHECK WITH MR. TO!**
B	Simplify these algebraic statements: 8M + 5S – 2M + 3S 6M + 8S – M + 2S 9x + 3y + 2x + 4y **CHECK WITH MR. TO!**
C	Simplify: 9x + 3 + 2x + 4 7x + 6 – 3x + 5 9x – 4 + 3x + 6x – 3 + 5x **CHECK WITH MR. TO!**
D	Simplify: 6x2 + 5x + 8 + 3x2 – 2x – 1 8x2 – 2x + 4 – 6x2 – 3x – 7 9x2 – 5x – 6 – 2x2 + 2x – 3 **CHECK WITH MR. TO!**
E	Simplify: 2a2 – 6 + 4a + 6 – 5a2 – 3a 9x2 – 5x – 6 – 2x2 + 2x – 3 6x2y + 2xy2 – 3x2y + 6xy2  **CHECK WITH MR. TO!**

An example of a thin-sliced task.

The use of thin-sliced tasks can be described as a form of guided discovery (Mayer, 2004) because it incorporates regular feedback and just-in-time guidance to students. The feedback can be intrinsic to the mathematics being learned (e.g., students checking their answers when solving equations by substituting the value of the variable into the original equation) or built into the flow of the task and provided externally (e.g., by a teacher or peers). Students need to confirm that they are on the right track and grasping the intended concepts. The teacher plays a crucial role in constantly circulating and monitoring progress.

Guidance during thin-sliced tasks is also given through hints and extensions when groups are stuck or ready for greater challenge. According to Liljedahl, hints can either increase skill (e.g., by providing a strategy) or decrease challenge (e.g., by giving part of the answer). The purpose of providing hints is to keep groups productive and working in a state of flow, a concept made popular by psychologist Mihály Csíkszentmihályi (Liljedahl, 2020).

Supporting evidence from cognitive science research

The strategy of developing a mathematical concept or skill in small, incremental steps is supported by cognitive science as an effective way to introduce new learning. Rosenshine (2012) highlights that effective teachers present small amounts of new material at a time to avoid overloading students’ working memory. To ensure mastery before moving on, teachers regularly check for understanding, provide additional explanations, and ensure students have the requisite information to work independently (Rosenshine, 2012).

Guidance throughout the learning process, such as the support throughout thin-sliced tasks, is also crucial for successful teaching. The main issue with a pure discovery-based approach (i.e. with minimal to no guidance) is that when students are given a problem to solve without hints or prompts, they may simply stumble upon a solution that may not actually lead to the development of a useful schema, leading to minimal or no learning (Mayer, 2004; Clark, Kirschner, & Sweller, 2012). To avoid this, explicit guidance in the form of teacher monitoring, hints, prompts, and regular feedback is essential for learning to take place.

In addition to teacher guidance, the intrinsic guidance of thin-sliced tasks can help students focus on the learning objective. Mayer (2004) highlights that when students are told what learning to attend to with guided discovery methods, they learn more effectively and retain information better compared to students who learned using pure discovery methods. Thin-sliced tasks are intentionally designed using variation theory (Kullberg, Runesson Kempe, & Marton, 2017) to direct students’ attention to the key elements of the question.

Mayer (2004) also notes that prior studies demonstrate the superiority of guided discovery in comparison to pure discovery and expository methods (i.e. explicit instruction) in mathematics when it comes to tests on immediate retention, delayed retention, and transfer problems. Guided discovery helps students construct appropriate knowledge to make sense of new information and integrate it with their existing knowledge. Thin-sliced tasks as a form of guided discovery, therefore, have the potential to be the most effective method of mathematics instruction.

There is additional evidence in the cognitive science literature suggesting that problem solving followed by explicit instruction can be effective for tasks with low complexity or low element interactivity (Ashman, Kalyuga, & Sweller, 2020). If a thin-sliced task is well-designed to build students’ expertise gradually, then element interactivity stays relatively low, allowing problem solving to be beneficial for learning.

Summary

Thin-sliced tasks with hints and extensions minimize cognitive load by guiding students through the acquisition of mathematical skills in small, incremental steps with support. By using variation theory and strategically changing only one facet of each question at a time, these tasks help focus students’ attention on the most important aspects, providing the guidance that pure discovery methods lack. Additionally, feedback—whether embedded in the questions or provided by the teacher—ensures that students stay on track and learn the intended material.

2. Having students work collaboratively in random groups of three.

Overview

According to Liljedahl, random groups of three students strike an optimal balance between knowledge redundancy (e.g., shared vocabulary, math understanding, and learning experiences) and diversity of ideas, approaches, and perspectives, enabling effective collaboration to engage in problem solving, including thin-sliced tasks. By probability, students will inevitably work with each of their peers at least once during the school year or semester.

For these groups to be productive, students need to learn how to collaborate effectively. Liljedahl suggests starting the school year with 3 to 5 non-curricular tasks. These engaging activities, while not necessarily tied to curriculum standards, help stimulate discussions and establish positive norms for collaboration.

To further develop collaboration skills, Liljedahl recommends co-creating simple rubrics with students that outline desired behaviors during group work. These rubrics serve as an accountability tool to ensure effective teamwork. As collaboration skills improve, social barriers break down, knowledge sharing increases, and student thinking flourishes.

Supporting evidence from cognitive science research

Cognitive load theory, traditionally used to study individual learning, has been applied by Kirschner et al. (2009) to explain the benefits of collaborative learning. As task complexity increases, individual learning becomes less effective compared to group learning. In difficult tasks, cognitive load is high, but it can be distributed among the working memory capacity of individual group members, thus reducing the burden on any single working memory (Zambrano R., Kirschner, & Kirschner, 2020). This concept is known as the collective working-memory effect (Kirschner, Paas, & Kirschner, 2011; Sweller, van Merriënboer, & Paas, 2019).

Essentially, the collective working-memory effect describes how individuals can pool together their working memory resources to achieve a collective working memory that eases the strain on any one person. This effect is particularly beneficial in groups of learners with low prior knowledge (Kirschner et al., 2018), which supports the use of collaborative learning among novices. Group work, therefore, can act as a scaffold for students to attempt more difficult tasks—a hallmark of effective teaching (Rosenshine, 2012).

However, collaborative learning can also introduce extraneous cognitive load. To minimize this, Zambrano R. et al. (2020) suggest:

• Keeping group sizes small to limit the demands of group interactions and reduce the risk of “social loafing” (Kirschner et al., 2018).
• Distributing information as homogeneously as possible amongst group members.
• Ensuring group members are familiar with each other and have prior experience working together on tasks with similar structure.
• Clarifying expectations for group work.

It is worth reiterating the importance of developing students’ collaboration skills for reducing the extraneous cognitive load associated with group dynamics. Kirschner et al. (2018) caution that “if learners have not acquired these skills prior to beginning on the collaborative task, the load induced here could be so high as to hinder collaborative learning.” Providing non-curricular tasks and co-creating rubrics to establish norms for collaboration, therefore, are the concrete steps that teachers can take to explicitly support students’ acquisition of the requisite collaboration skills needed to unlock the benefits of group work. In addition, creating random groups facilitates students gaining experience working with each of their peers and ensures that collaborative norms are consistent for all students.

Additionally, tasks must be sufficiently challenging to justify the collaborative effort. If every group member can complete the task independently, then group work becomes unnecessary and even counterproductive (Kirschner et al., 2018). Despite these caveats, the benefits of collaborative learning far outweigh the costs (Kirschner, Paas, & Kirschner, 2011).

Summary

Organizing students into random groups of three helps to reduce the extraneous cognitive load associated with collaboration while maximizing the benefits of diverse perspectives and the formation of a collective working memory needed to tackle complex math tasks that could not be done by any one individual alone. Investing in the development of collaboration skills using non-curricular tasks and rubrics further minimizes the cognitive load costs of group work. The collective working-memory effect enables students “to process information elements deeply and construct higher quality schemas in their [long-term memories] than learners working individually” (Kirschner, Paas, & Kirschner, 2011).

3. Encouraging students to make meaningful notes for their “future forgetful self.”

Overview

Liljedahl argues that traditional note-taking, where students simply copy down notes written by the teacher, is a cognitively passive activity that hinders learning. Students can usually either listen to what the teacher is saying or they can focus on copying down what the teacher is writing; however, doing both is extremely difficult. Liljedahl advocates for a shift from passive note-taking to active note-making.

After a lesson consolidation in a thinking classroom, students are encouraged to work collaboratively to synthesize the main ideas and create notes that will help them remember what they have learned. Liljedahl refers to this as making notes to your future forgetful self.

For some students, creating personalized notes is a very open experience and can feel daunting, so initial or ongoing support may be needed. Peer collaboration can help ensure that notes are comprehensive, and a graphic organizer, such as the one below, can guide students in capturing key information and details.

A template for a meaningful note (Liljedahl and Giroux, 2024).

An essential component of effective notes is the inclusion of worked examples—questions with full solutions and, if necessary, diagrams for context. Worked examples can include annotations with self-explanations to clarify the reasoning behind key steps. Students are encouraged to include multiple worked examples to cover various problem types.

Supporting evidence from cognitive science research

Cognitive science research strongly supports the use of worked examples in mathematics learning. Well-designed worked examples, particularly those integrating visuals, can facilitate knowledge construction more effectively than solving equivalent problems without guidance (Kirschner, Sweller, & Clark, 2006; Rosenshine, 2012; Sweller, van Merriënboer, & Paas, 2019). This is known as the worked example effect. As students practice and develop fluency, referencing worked examples from their notes can provide significant benefits.

However, simply viewing worked examples is not enough. Chi et al. (1994) found that prompting students to provide self-explanations while studying worked examples greatly enhances their effectiveness. Annotating steps and explaining the reasoning behind them encourages deeper thinking and maximizes the benefits of worked examples—a phenomenon known as the self-explanation effect (Sweller, van Merriënboer, & Paas, 2019).

Collaborative note-taking also has some evidence in terms of enhancing information processing. Costley and Fanguy (2021) found that when students collaborated on note-taking in an online setting that involved explaining ideas to others and providing feedback, they experienced higher levels of germane cognitive load (i.e., effective information processing) and greater understanding compared to those who took notes individually.

Summary

The process of students collaboratively writing their own notes can improve information processing by increasing germane cognitive load. These student-generated notes can take advantage of the worked-example and self-explanation effects by incorporating a number of fully worked-out solutions to various problems that also include students’ own annotations to explain the underlying reasoning behind the steps.

Part III: Conclusion

Peter Liljedahl’s Building Thinking Classrooms framework, perhaps coincidentally, leverages instructional effects and principles from cognitive science to create powerful mathematics learning environments. As a form of guided inquiry-based instruction, BTC practices foster conditions for students to solve problems as a means of developing understanding and procedural skills. Unlike the stereotype of discovery-based learning where students are left to discover math concepts on their own, the BTC framework provides extensive support through task design (e.g. thin-sliced tasks that build skills incrementally), collaborative learning (e.g., forming random groups of three to create a collective working memory capacity and foster collaboration skills), and teacher guidance (e.g., monitoring progress, offerings hints and extensions, providing feedback, encouraging meaningful note-making). These elements provide the substantial guidance that all students, but particularly novice learners, need to achieve the desired learning outcomes in K-12 math classrooms.

This piece is not intended to argue that BTC is superior to explicit instruction in teaching mathematics. While I agree with de Jong et al. (2023) that a combination of inquiry-based instruction and explicit instruction is beneficial, my focus here is to show that BTC, like explicit instruction, aligns with instructional principles based on cognitive load theory and cognitive science research. Specifically, I argue that beginning with problem-solving experiences in the form of thin-sliced tasks, supported with hints and group work, is an effective instructional approach rooted in established cognitive science research. Contrary to Sweller et al.’s (2024) assertion that novice learners require concepts to be fully explained and procedures fully modelled before they apply those procedures, I hope I have illustrated that collaborative problem solving is a valid initial approach, especially for low element interactivity material and when collaborative cognitive load principles are applied. In those cases, the evidence suggests that problem solving is actually the more effective approach.

Given BTC’s increasing popularity among K-12 teachers, I encourage colleagues and educational researchers to further explore how cognitive science can enhance its effectiveness. A deeper understanding of cognitive load theory and cognitive science research can only refine and optimize the implementation of the Building Thinking Classrooms framework for the benefit of student learning.

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