For the past several years, Building Thinking Classrooms (BTC) has been one of the most visible movements in mathematics education. Teachers have experimented with randomized groups, vertical non-permanent surfaces, rich problems, student discussion, and classroom routines intended to get more students thinking.

At the same time, interest in the Science of Math and the broader Science of Learning has grown. Educators are asking whether mathematics instruction should place greater emphasis on explicit teaching, worked examples, retrieval practice, carefully sequenced content, foundational fluency, and the limits of working memory.

The resulting conversation is often framed as a choice: Should teachers use Building Thinking Classrooms or move toward Science of Math? That framing is understandable, but it oversimplifies the issue. Building Thinking Classrooms is an instructional model. Science of Math is better understood as an evidence-informed way of evaluating instructional decisions through research from cognitive science, educational psychology, and mathematics education.

The more useful question is not which label wins. It is this: What do students need at this point in their learning, and which teaching practices are most likely to help them learn accurately, efficiently, and durably?

Why This Conversation Is Growing

Teachers are rarely debating these ideas in the abstract. They are responding to practical classroom realities: short periods, unfinished learning, large classes, behavior challenges, wide ranges of prior knowledge, demanding pacing guides, and pressure to improve achievement.

In a lengthy educator discussion that inspired this article, teachers described very different experiences with BTC. Some reported strong engagement and valuable mathematical discussion. Others said the approach became difficult to sustain when group management consumed limited instructional time or when students lacked the prerequisite knowledge needed to engage productively. Several educators described moving toward more explicit instruction, while many others advocated a deliberate blend rather than an all-or-nothing shift.

Those comments are practitioner experiences rather than controlled research findings, but they reveal the questions teachers are trying to solve: How do we preserve student thinking without leaving novices to discover too much too soon? How do we build fluency without reducing mathematics to imitation? How do we make collaboration purposeful rather than merely active?

What Is the Science of Math?

Science of Math is not a single curriculum, program, book, or mandated lesson format. The term generally refers to the application of research about learning, memory, instruction, and mathematics achievement to classroom practice.

A Science of Math perspective commonly emphasizes:

• Explicit explanation and modeling when students are learning unfamiliar content.
• Carefully sequenced examples that break complex ideas into manageable components.
• Guided practice with frequent feedback before fully independent work.
• Retrieval practice that requires students to recall and use previously learned material.
• Spaced and cumulative review so knowledge is revisited over time.
• Attention to working-memory limits and unnecessary cognitive load.
• Systematic development of conceptual understanding, procedural knowledge, and fluency.
• Purposeful problem solving once students possess enough background knowledge to participate productively.

This does not mean every Science of Math classroom looks identical. It means instructional decisions are evaluated against what is known about how novices acquire knowledge, how practice strengthens memory, and how teachers can gradually transfer responsibility to learners.

What Is Building Thinking Classrooms?

Building Thinking Classrooms is an instructional framework developed by mathematics educator Peter Liljedahl. It is designed to increase student thinking by changing the tasks students receive, how groups are formed, where students work, how teachers answer questions, and how lessons are consolidated.

Common BTC practices include:

• Rich or thinking-oriented tasks.
• Frequently randomized student groups.
• Vertical non-permanent surfaces, such as whiteboards.
• Student discussion and visible problem-solving processes.
• Teacher moves intended to maintain thinking rather than immediately provide answers.
• Consolidation that connects student strategies to important mathematical ideas.

BTC has resonated with educators because it addresses a genuine problem: students can appear compliant while doing very little mathematical thinking. Its routines are intended to make participation visible and create more opportunities for students to reason, communicate, and learn from one another.

Science of Math vs. Building Thinking Classrooms: Comparison Table

Area	Science of Math	Building Thinking Classrooms
Primary purpose	Align instruction with research about learning, memory, and mathematics achievement.	Create classroom conditions in which more students visibly think, discuss, and solve problems.
Type	Evidence-informed framework or lens for selecting instructional practices.	Instructional model with specific classroom practices and routines.
New content	Often begins with clear explanation, modeling, worked examples, and guided practice.	Often begins with a problem or task that students explore collaboratively.
Role of the teacher	Actively models, checks understanding, provides feedback, and gradually removes support.	Selects tasks, forms groups, monitors thinking, prompts students, and consolidates learning.
Role of prior knowledge	Prior knowledge is central because novices and experts learn differently.	Tasks are designed to elicit thinking, though student readiness can affect productive participation.
Cognitive load	Explicitly considers working-memory limits and seeks to reduce avoidable overload.	May create productive challenge, but task, group, and novelty demands must be carefully managed.
Collaboration	Used strategically when collaboration supports the learning goal.	A central feature of many lessons and routines.
Practice	Emphasizes guided practice, retrieval, spacing, cumulative review, and feedback.	Emphasizes thinking tasks, discussion, strategy development, and consolidation.
Common tools	Worked examples, mini whiteboards, questioning, retrieval quizzes, examples and non-examples.	Vertical whiteboards, random groups, thinking tasks, hints, and student strategy sharing.
Best combined use	Build knowledge and fluency, then apply it through discussion and problem solving.	Use rich collaborative tasks when students have sufficient knowledge and the task advances the intended objective.

Two Approaches, One Instructional Goal

The two approaches can be summarized as different starting points that can lead toward the same goal: students who possess strong mathematical knowledge and can use that knowledge flexibly.

SCIENCE OF MATH Activate prerequisite knowledge Explain and model clearly Use worked examples Guide practice and check understanding 5.  Strengthen memory through retrieval Remove support gradually Apply knowledge to richer problems

BUILDING THINKING CLASSROOMS Launch a thinking task Place students in collaborative groups Make thinking visible Encourage strategy development Prompt without immediately rescuing Compare student approaches Consolidate the mathematics

Shared destination: accurate knowledge + flexible problem solving + mathematical confidence

The Central Issue: Novices Do Not Learn Like Experts

A central idea in the Science of Learning is that prior knowledge changes how a learner experiences a task. Experts have extensive knowledge organized into schemas. Those schemas allow them to recognize patterns, choose efficient strategies, and manage complex information without overwhelming working memory.

Novices do not yet have those mental structures. A task that feels stimulating and open-ended to an expert may feel confusing to a student who is still trying to remember basic facts, vocabulary, symbols, or procedures. This is why a strategy that works beautifully with one class, topic, or group of students may be ineffective with another.

The expert reversal effect describes how instructional support that benefits novices can become unnecessary or even counterproductive as expertise increases. The practical implication is not that teachers must always lecture. It is that the level of guidance should change as students gain knowledge.

Cognitive Load Theory and Mathematics Instruction

Cognitive Load Theory begins with a simple constraint: working memory can process only a limited amount of new information at one time, while long-term memory can store a vast amount of organized knowledge.

In mathematics, a student may need to hold numbers, symbols, vocabulary, procedures, intermediate results, and the overall goal of a problem in mind at the same time. Add unfamiliar group routines, competing strategies, movement around the room, or uncertainty about what to do next, and the total demand can exceed the student's available working-memory capacity.

Effective instruction manages this load by:

• Breaking complex procedures into smaller, teachable steps.
• Removing distracting or irrelevant information.
• Using clear examples and visual representations.
• Connecting new content to previously learned knowledge.
• Providing guided practice before independent performance.
• Automating foundational skills through appropriately designed practice.

The NSW Centre for Education Statistics and Evaluation provides an accessible research review and classroom guide explaining cognitive load and the value of explicit guidance, practice, and feedback. Read the CESE cognitive load resources

Explicit Instruction Is More Than “I Do, We Do, You Do”

Explicit instruction is sometimes reduced to a slogan or confused with uninterrupted teacher talk. Strong explicit instruction is much more responsive. It includes clear explanations, purposeful examples, frequent student responses, checks for understanding, immediate feedback, guided practice, and gradual release.

In an effective explicit lesson, students are not passive. They may solve on mini whiteboards, explain a step, identify an error, compare examples, answer carefully sequenced questions, retrieve prior learning, and practice until the new idea becomes secure.

The teacher's role is to make important thinking visible before asking students to perform that thinking independently. This can be especially valuable when students lack background knowledge or when errors are likely to become entrenched.

Worked Examples and Example Selection

Worked examples show students how a problem is solved while directing attention to the decisions and relationships that matter. They reduce the need for novices to search blindly for a method and free working memory to focus on understanding the structure of the solution.

Worked examples are strongest when teachers do more than display completed steps. Students should be asked to explain why a step is valid, identify what changed, compare two methods, complete a partially worked solution, or diagnose an incorrect example.

Example selection also matters. A carefully designed sequence can vary one feature at a time, highlight boundaries between concepts, and help students notice patterns that random practice may hide.

Retrieval Practice, Spacing, and Interleaving

Learning a method during today's lesson does not guarantee that students will remember it next week. Durable learning requires opportunities to retrieve knowledge after some forgetting has occurred.

Retrieval practice asks students to recall and use information without simply rereading it. In mathematics, this may include short cumulative quizzes, mini-whiteboard questions, exit tickets, flashcards for foundational facts, or warm-ups that revisit prior content.

Spacing distributes practice over time. Interleaving mixes related problem types so students must decide which strategy applies. Together, these practices strengthen retention and discrimination, although they should be introduced thoughtfully so practice remains challenging without becoming chaotic.

Where Building Thinking Classrooms Can Be Especially Valuable

A Science of Math perspective does not require teachers to abandon rich tasks, collaboration, vertical whiteboards, or mathematical discussion. Those practices can be highly valuable when they serve a clear learning purpose and students possess enough knowledge to engage successfully.

BTC-inspired routines may be especially useful for:

• Applying previously taught concepts in unfamiliar situations.
• Comparing multiple valid solution strategies.
• Developing mathematical communication and justification.
• Surfacing misconceptions that are difficult to see in independent written work.
• Encouraging students to connect representations and generalize patterns.
• Building classroom norms in which mathematical thinking is shared rather than hidden.

The question is not whether students should think. Every strong mathematics lesson should involve thinking. The question is what kind of thinking the task requires and whether students have the knowledge needed to perform it productively.

A Practical Blended Lesson Sequence

Consider a middle school lesson on solving two-step equations. A blended sequence might look like this:

15. Retrieve prerequisite knowledge about inverse operations, equality, and one-step equations.
16. Model a carefully chosen two-step equation while explaining the reasoning behind each transformation.
17. Use a second worked example and ask students to anticipate the next step before it is revealed.
18. Have every student solve short examples on personal whiteboards so the teacher can check understanding in real time.
19. Provide guided and then independent practice, including common error patterns.
20. Move students into small groups for a richer task that requires comparing equations, creating examples, or explaining why two methods are equivalent.
21. Consolidate the lesson by connecting student strategies to the underlying structure and revisiting the learning goal.

This sequence preserves explicit teaching, accountability, retrieval, collaboration, and rich problem solving. More importantly, each practice is used at the point where it is most likely to support learning.

What Teachers Are Reporting

The educator comments that prompted this article do not produce a single verdict. They show that context matters.

• Some teachers reported that BTC worked well with smaller classes, longer periods, strong classroom norms, or students with sufficient prior knowledge.
• Others described difficulty sustaining group work in 50-minute periods, large classes, or settings with significant behavior and attendance challenges.
• Several teachers said personal mini whiteboards produced broad participation with less group-management demand.
• Some educators reported improved student confidence, behavior, or test performance after increasing explicit instruction.
• Many experienced teachers rejected an all-or-nothing choice and described using collaborative problem solving after foundational instruction.
• Teachers with multiple preparations or rigid pacing guides raised concerns about the planning time required to build complete task sequences.

These experiences should not be treated as proof that one approach always succeeds or fails. They do, however, reinforce the need to evaluate instructional methods by more than visible engagement. Teachers also need evidence of understanding, retention, transfer, and achievement.

Common Misconceptions

“Science of Math means students only mimic procedures.”

Poorly implemented instruction can produce imitation without understanding, but that is not the goal of evidence-informed teaching. Strong instruction connects procedures to concepts, representations, language, and reasoning. Students are taught enough to participate in deeper thinking rather than being expected to reason without a foundation.

“Building Thinking Classrooms means teachers never teach explicitly.”

BTC includes teacher facilitation and consolidation, and individual teachers implement the model in different ways. The tension usually concerns the timing and amount of guidance, not whether teachers ever explain mathematics.

“Engagement proves learning.”

Engagement creates opportunities to learn, but activity alone does not establish achievement. Teachers need checks for understanding during the lesson and opportunities to determine what students retain and can use later.

“Balance means using every strategy equally.”

Balanced instruction is not a fixed percentage of direct teaching and group work. It means selecting the method that best fits the learning goal, student knowledge, classroom context, and evidence of progress.

Questions Teachers Can Use to Choose an Approach

• What prerequisite knowledge does this task require?
• Are students learning a new procedure or applying an established one?
• What information must students hold in working memory?
• Could I reduce unnecessary complexity without reducing the mathematical goal?
• How will I check every student's understanding?
• Does collaboration improve the mathematics, or merely change the seating arrangement?
• When should I model, and when should I withhold support?
• How will students retrieve this learning in future lessons?
• What evidence will show that learning lasted beyond today's activity?

Where to Start: Recommended Resources

How I Wish I’d Taught Maths by Craig Barton

A practical entry point into worked examples, retrieval, diagnostic questions, example selection, and lessons drawn from classroom experience.

https://mrbartonmaths.com/books/

The Mr Barton Maths Podcast

Long-form conversations with teachers and researchers about mathematics instruction, assessment, cognitive science, and classroom practice.

https://podcast.mrbartonmaths.com/

Chalk & Talk with Anna Stokke

Interviews and discussions focused on effective mathematics teaching, educational research, and common claims in math education.

https://www.annastokke.com/podcast

The Science of Math

A resource hub focused on evidence-based mathematics instruction and related research.

https://www.thescienceofmath.com/

CESE Cognitive Load Theory

An accessible research review and practical guide for teachers on working memory, explicit guidance, practice, and feedback.

https://education.nsw.gov.au/about-us/education-data-and-research/cese/publications/literature-reviews/cognitive-load-theory

The Learning Scientists

Classroom-friendly explanations of retrieval practice, spacing, interleaving, elaboration, concrete examples, and dual coding.

https://www.learningscientists.org/

Frequently Asked Questions

What is the Science of Math?

Science of Math is an evidence-informed approach to mathematics instruction that draws on cognitive science, educational psychology, and mathematics education. It commonly emphasizes explicit instruction, carefully sequenced examples, guided practice, retrieval, feedback, and cognitive load management.

Is Science of Math a curriculum?

No. It is not one curriculum or commercial program. It is a framework for evaluating teaching practices based on evidence about learning and mathematics instruction.

Is Science of Math the same as direct instruction?

Explicit instruction is an important component, but Science of Math is broader. It also includes retrieval practice, spacing, formative assessment, feedback, curriculum sequencing, conceptual development, and purposeful problem solving.

What is Building Thinking Classrooms?

Building Thinking Classrooms is an instructional model developed by Peter Liljedahl. It uses practices such as thinking tasks, randomized groups, vertical non-permanent surfaces, student discussion, and lesson consolidation to increase student thinking.

Is Science of Math replacing Building Thinking Classrooms?

Not necessarily. The two are not equivalent categories. Teachers can evaluate BTC practices through a Science of Learning lens and use them when they support the intended learning goal.

Can teachers use Science of Math and BTC together?

Yes. A teacher might explicitly teach and practice a new concept, then use a BTC-style collaborative task to apply, compare, or extend that knowledge.

Does Science of Math eliminate inquiry and problem solving?

No. It suggests that the timing and level of guidance should reflect student expertise. Novices often need more instruction and scaffolding before open-ended problem solving becomes productive.

Does BTC work for every classroom?

No instructional model works equally well in every setting. Class size, student behavior, prior knowledge, lesson length, teacher experience, curriculum, and implementation quality can all affect results.

What is Cognitive Load Theory?

Cognitive Load Theory explains that working memory has limited capacity. Instruction can support learning by reducing unnecessary demands, organizing information clearly, and helping students build knowledge in long-term memory.

What is the best starting resource for teachers?

Many educators recommend Craig Barton’s How I Wish I’d Taught Maths and Anna Stokke’s Chalk & Talk podcast because both connect research to practical questions about mathematics teaching.

Final Thoughts

The debate between Science of Math and Building Thinking Classrooms can become unproductive when either side is reduced to a caricature. Explicit instruction is not synonymous with passive learning. Collaborative problem solving is not automatically productive thinking. Both depend on design, timing, teacher expertise, and student readiness.

Building Thinking Classrooms has helped many educators rethink participation, group work, task design, and the visibility of student reasoning. Science of Math asks teachers to place those choices within a broader understanding of memory, prior knowledge, cognitive load, practice, and instructional guidance.

The strongest path forward may be neither complete rejection nor unquestioning fidelity. It may be disciplined professional judgment: build knowledge carefully, check learning frequently, provide enough guidance for success, and create meaningful opportunities for students to use mathematics independently and collaboratively.

Students deserve classrooms where they are not merely busy, compliant, or entertained. They deserve instruction that helps them understand mathematics, remember it, and use it with confidence.