Unveiling the Power of Singapore Math and Math Heuristics

This is a Theoretical Dive into Bruner, Skemp, and Dienes. In case you do not know or first time hearing this, read on.

In every parent’ quest to provide the best education for one’s child, one may have come across the term “Singapore Math.” This teaching approach, which has gained international recognition for its effectiveness, is rooted in strong theoretical underpinnings. Today, we’ll take a closer look at the theories of Jerome Bruner, Richard Skemp, and Zoltan Dienes that have heavily influenced Singapore Math with its heuristics.

Understanding these theories can help you appreciate why this approach is so successful and how it can benefit your child’s mathematical journey.

Jerome Bruner’s Constructivist Theory

Jerome Bruner, a prominent American psychologist, laid the foundation for constructivism in education. His ideas form a core component of Singapore Math’s approach to teaching mathematics. Constructivism propounds that learning is an active process, where individuals construct their knowledge through their experiences.

Here’s how Singapore Math incorporates Bruner’s constructivist theory:

Concrete-Pictorial-Abstract (CPA) Approach
This is a central concept in Singapore Math. It aligns perfectly with Bruner’s theory of learning by doing. Students begin with concrete materials, move on to pictorial representations, and finally, transition to abstract symbols. This progression allows children to build a deep understanding of mathematical concepts and establishes a strong mathematical foundation for him.

Spiral Curriculum
Bruner advocated for a curriculum that revisits and deepens concepts over time. Singapore Math follows this principle, ensuring that students encounter mathematical ideas repeatedly but at increasing levels of complexity as they advance through grades.

Problem-Solving Emphasis:
Bruner believed that meaningful problem-solving is key to learning. Singapore Math emphasizes problem-solving from an early age, encouraging students to explore, make mistakes, and learn from them.

Richard Skemp’s Relational Understanding

Richard Skemp, a British mathematician and educator, contributed significantly to education in the field of mathematics. His work on the concept of “relational understanding” has profoundly impacted the philosophy behind Singapore Math:

Instrumental and Relational Understanding
Skemp distinguishes between instrumental understanding (knowing how to do something) and relational understanding (understanding why something works). Singapore Math prioritizes relational understanding, ensuring that students don’t just memorize procedures but comprehend the underlying mathematical concepts. Conceptual understanding of topics and questions is far more superior than mere procedural fluency.

Concrete Manipulatives
Singapore Math provides students with hands-on experiences using concrete manipulatives, aligning with Skemp’s belief that students need tangible experiences to develop a relational understanding of mathematics.

Visual Representations
Skemp emphasized the importance of visual representations to aid understanding. In Singapore Math, especially with math heuristics to solve problems, students use diagrams and models extensively to visualize and conceptualize before arriving at solutions, thus, fostering a deeper connection with mathematical concepts.

Zoltan Dienes and Manipulatives

Zoltan Dienes, a Hungarian mathematician, championed the use of manipulatives in mathematics education. His work on the “Dienes blocks” has had a profound impact on how mathematics is taught, and Singapore Math incorporates his ideas in the following ways:

Concrete Learning
Dienes believed that students should begin learning mathematics concretely, through hands-on experiences with manipulatives. Singapore Math adheres to this principle by using physical objects and models in the early stages of mathematical education.

Progression from Concrete to Abstract
Dienes’ theory suggests that students should gradually transition from concrete to abstract thinking. Singapore Math’s CPA approach ensures that this progression is a fundamental part of the curriculum.

Developing Number Sense
Dienes’ work emphasized the importance of developing number sense through manipulation and exploration. Singapore Math encourages students to develop a strong foundation in number sense, which is essential for advanced mathematical thinking.

The Role of Heuristics in Singapore Math

In addition to the theoretical foundations mentioned above, heuristics play a crucial role in Singapore Math. Math heuristics are problem-solving strategies or techniques that guide students in finding solutions.

Here is a list of some of the strategies students can make use of to tackle complex problems with confidence:

Model Drawing
One of the most renowned heuristics in Singapore Math is model drawing. This technique involves representing complex problems with simple visual models. It helps students break down intricate problems into manageable parts, making problem-solving more accessible.

Guess and Check
Singapore Math encourages students to make educated guesses and check their solutions. This heuristic promotes critical thinking and problem-solving skills, as children learn to eliminate incorrect options through systematic checking.

Working Backwards
In certain situations, it’s easier to work from the solution backward to find the answer. This heuristic encourages students to consider possible solutions and backtrack to identify the correct one.

Act It Out
For problems involving real-world scenarios, acting out the situation can help students visualize and solve the problem. This heuristic encourages creativity, visual and kinesthetic learning.

Make Suppositions
Here it involves making an educated guess or supposition to simplify a problem. By assuming a value or condition, students can explore the problem and derive a solution more easily.

Before and After Concept
This math heuristic method is applied where the questions show there is a change resulting in a ‘before’ situation and an ‘after’ situation. One will need compare the two situations in order to understand the question fully and find a way to solve it. At best, the method is combined with the model drawing method.


Incorporating the theories of Jerome Bruner, Richard Skemp, and the principles of Zoltan Dienes, Singapore Math has evolved into a highly effective approach to teaching mathematics. Its emphasis on constructivism, relational understanding, and the use of manipulatives aligns with research-backed pedagogical principles. The inclusion of problem-solving math heuristics further can empower your child to become a confident and an independent mathematical thinker.

As parents, understanding the theoretical foundations and heuristics behind Singapore Math can help you appreciate the value of this approach in your child’s education. By nurturing a deep understanding of mathematics, Singapore Math equips your child with the tools he needs to excel not only in math but in problem-solving and critical thinking throughout his academic journey and beyond.

More resources: Explore SAM Hougang Program

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