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Methods in Math

Building Thinking Classrooms (BTC) Approach

The Building Thinking Classrooms (BTC) approach, developed by Dr. Peter Liljedahl, is a research-backed framework designed to help students think deeply about math rather than rely on memorization. BTC has been shown to increase student engagement, deepen understanding, and improve problem-solving skills (Liljedahl, 2020).

Thinking Tasks

Students tackle open-ended, rich math problems that encourage exploration and multiple solution paths.

- Use rich, open-ended problems that promote thinking rather than rote application of procedures.
- Problems are designed to encourage multiple approaches and solutions.

Vertical Non-Permanent Surfaces (VNPS)

Students solve problems on erasable surfaces (whiteboards, chalkboards, or digital boards), making thinking visible and dynamic.

- Students work on whiteboards encouraging visibility and collaboration.
- The vertical setup allows teachers to observe student thinking and promotes active engagement.
- Erasable surfaces reduce the fear of making mistakes, fostering a growth mindset.

Building Perseverance

Building Perseverance – Students are encouraged to struggle productively, learning that mistakes are a natural part of problem-solving (Boaler, 2016).

- We act as facilitators, providing minimal direct instruction and instead prompting students to discover solutions independently or through group collaboration.
- Students take ownership of their learning, with the teacher intervening only when absolutely necessary to guide thinking.

Collaborative Problem Solving

Instead of passive learning, students discuss, defend, and refine their reasoning through guided questioning.

Intentional Notes

Students do not take notes while solving problems. Instead, they record key learnings at the end of the lesson, ensuring that notes are meaningful and purposeful.

Strategic Questioning

We ask thought-provoking questions instead of directly giving answers, helping students develop critical thinking.

Kagan Cooperative Learning Structures

Kagan Cooperative Learning is a set of structured engagement techniques that promote active participation, communication, and peer learning. Developed by Dr. Spencer Kagan, these structures ensure that every student is actively involved in the learning process (Kagan, 1994).

Kagan Structures We Use:

Think-Pair-Share – Students first think independently, then discuss with a partner, and finally share ideas with the group. This reinforces mathematical reasoning and communication skills.
Rally Coach – One student solves a problem, while the partner coaches and checks their work. This builds confidence and reinforces step-by-step thinking.
RoundRobin – Students take turns sharing ideas or solving steps in a problem, ensuring equal participation and multiple perspectives.

One-on-One Tutoring Approach

Active Thinking Tasks – Instead of traditional worksheets, students engage in hands-on problem-solving, using visual models and whiteboards to explore math concepts.
Guided Discovery – We ask strategic questions to lead students to understanding rather than giving direct answers (Liljedahl, 2020).
Kagan Partner Strategies – Even in a one-on-one setting, we use Rally-Coach Style discussions, where students explain their reasoning and how to solve problems to the tutor.
Building Math Confidence – Struggling students often fear making mistakes. We create a safe learning environment that encourages productive struggle and growth mindset thinking (Dweck, 2006).

Small Group Instruction Approach

Collaboration and Communication – Using Kagan structures like RoundRobin and Rally Coach, students explain their thinking, which strengthens retention.
Problem-Solving Together – Students work on BTC-inspired thinking tasks at vertical whiteboards, discussing multiple solution strategies.
Peer Teaching – Explaining math to peers reinforces understanding and builds confidence (Slavin, 2014).
Individualized Support – Even in small groups, we tailor instruction to meet each student's needs, ensuring no one falls behind.

Why These Approaches Work for Struggling Math Learners

Increases Engagement – Students actively participate rather than passively listen.
Develops Deeper Understanding – Students discover concepts rather than memorize steps.
Improves Problem-Solving Skills – Encourages logical thinking and persistence.
Boosts Confidence – By normalizing mistakes, students develop a growth mindset (Dweck, 2006).

These combined strategies create a math learning experience that is interactive, supportive, and effective for struggling learners.

References

Boaler, J. (2016). Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages, and Innovative Teaching. Jossey-Bass.
Dweck, C. (2006). Mindset: The New Psychology of Success. Random House.
Kagan, S. (1994). Cooperative Learning. Kagan Publishing.
Liljedahl, P. (2020). Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning. Corwin.
Slavin, R. (2014). Cooperative Learning and Academic Achievement: Why Does Groupwork Work? Anales de Psicología, 30(3), 785-791.

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