Mathematicsforall

Centre for Teaching and Learning of Mathematics

The Centre for Teaching-Learning of Mathematics (CTLM) is based in Framingham, Massachusetts, with offices in England and India. It was founded by Professor Mahesh Sharma with a vision to develop and promote efficient, elegant, and effective methods of teaching and learning mathematics, from early childhood through high school and teacher training.

At CTLM, we are committed to bringing the best practices in mathematics teaching—drawn from research, classroom experience, and observations of teachers around the world—into the classroom. We use these teaching techniques in our courses, workshops, webinars, classroom demonstrations, and private tutoring. We have helped thousands of educators, from pre-service teachers to veteran teachers, improve their instruction so that children can develop mastery of mathematical ideas, including concepts, procedures, and skills.

We at CTLM believe that children should develop mastery of mathematical concepts with a clear emphasis on language, concepts, and procedures. Ultimately, these concepts and procedures should become skills, tools, and ideas through their application in various settings—intra-mathematical, interdisciplinary, and extracurricular.

We believe that children can achieve mastery only when lessons are delivered in ways that allow students to experience concepts at six levels of knowing: beginning with the intuitive level of knowing, progressing through the concrete, pictorial, and abstract/symbolic levels, and then applying these ideas in problem-solving to reach the communication level of knowing. This approach is called CPVA/S.

CPVA/S stands for Concrete–Pictorial–Visualization–Abstract/Scripts and represents a coherent progression in the teaching and learning of mathematical concepts.

C – Concrete: The learner experiences and explores a mathematical concept through concrete materials, manipulatives, models, and real-life experiences. These experiences help the child build a strong conceptual foundation and form a concrete conceptual schema.

P – Pictorial: The learner represents the concrete experience through pictures, diagrams, drawings, and other iconic and non-iconic representations. The pictorial level helps the child move beyond physical materials while retaining a meaningful representation of the mathematical concept.

V – Visualization: The learner forms, holds, and manipulates an image of the mathematical concept in the mind’s eye. Visualization serves as the critical bridge between pictorial representation and abstract thinking. With the support of an appropriate script, the child learns to mentally manipulate the image without depending on concrete materials or external pictures.

A – Abstract: The learner expresses and works with the mathematical concept using numbers, symbols, mathematical notation, algorithms, and other abstract representations. Because the abstract representation has developed from meaningful concrete, pictorial, and visualization experiences, the symbols carry conceptual meaning rather than being treated merely as rules to memorize.

S – Scripts: Scripts are linguistic containers of mathematical concepts and procedures. They provide precise, coherent, and connected mathematical language that accompanies the learner through the concrete, pictorial, visualization, and abstract levels. The script binds the different representations of a concept together, enabling the learner to experience them not as isolated stages but as one coherent conceptual whole.

The CPVA/S approach therefore differs from the conventional CPA approach in two important ways: it explicitly recognizes Visualization (V) as a crucial bridge between pictorial and abstract thinking, and it uses Scripts (S) as the linguistic thread that connects and binds the concept and its procedures across all levels. In this way, CPVA/S helps students develop conceptual understanding, procedural fluency, mathematical language, visualization, and ultimately the ability to apply and communicate mathematical ideas with understanding and confidence.

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