In Singapore, mathematics education is guided by a framework covering the development of five domains — concepts, skills, processes, attitudes and metacognition, where mathematical problem-solving is the central focus, and this framework sets the direction for teaching, learning and assessment in all schools in Singapore from primary to the ‘A’ levels. The different levels of syllabuses are classified according to topics and subtopics in arithmetic, algebra, geometry, measurement, trigonometry, statistics, probability and calculus. Students are exposed to the different topics and subtopics in all levels (although advanced topics in calculus are usually covered in the upper levels), and hence the level of abstraction and generalisation becomes deeper as the students progress in their years in school.

At the heart of the framework is the ability to solve real-life mathematical problems. To support this, students are expected to be equipped with not only the mathematical knowledge of concepts, processes and skills but also to possess the right attitudes such as perseverance and appreciation of the discipline, as well as meta- cognition—the ability to reflect about one’s own thinking and learning.

While the framework provides the ideal scenario for mathematics teaching and learning, the actual delivery in schools does not always follow as such. With a tight timeframe for curriculum delivery and a content-heavy syllabus, much focus is placed on the skills and procedural knowledge as these are deemed critical for students to know how to solve problems because of the way they are being assessed in the school and national exams. It is also easier to determine whether a child has understood a particular concept by having him evaluate a problem and see if he is able to attain the correct solution. It is then assumed that students will internalise the concepts through demonstration of how concepts are applied. For example, the topic of indices seeks to address how numbers can be represented and compared using the laws of indices (concepts). Students then apply these laws on new problems, and their understanding is assessed by how well they are able to solve questions requiring the application of those concepts.

Very often, mathematics educators do not distinctively separate between concepts and facts when delivering lessons in the classroom. In many cases, facts and concepts are taught simultaneously, because facts are examples of how mathematical concepts work, while concepts are actually the generalisations of those facts. For example, in the teaching of differentiation in Calculus, the first derivative of a function comes from the concept of limits:

This concept transcends across all types of functions. It also involves other subconcepts such as slope and, of course, functions. Using this concept, the first derivative of many functions can be found, such as:

A mathematical concept such as first derivative is hence used to generate a nonexhaustive list of mathematical facts.

It is always a challenge to teach mathematical concepts in the classroom because very often it involves deep thinking and time for the learner to understand how the concept works. Without the deep understanding, the learner often resorts to memorisation of facts, so that when a new situation arises, whereby the problem is not in the same ‘form’ as one of those facts that he memorised, he is not able to apply the concept properly and will have difficulties following up with the solution.

With the varying nature of problems in mathematics and a whole repertoire of heuristics to problem-solving, true conceptual understanding may not be attained even if the child is able to obtain the correct answer. For example, the application of the null factor law (principle/concept) in solving quadratic equations has become a procedural skill in secondary school students, and students, if given sufficient practice, would have little problem applying the skill. However, do they really know the concept behind how the principle works? One can extend the problem to cubic equations or polynomial equations of higher order, and it is not surprising that even when presented in the factorised form, some students may not know that the same principle (null factor law) can be applied to solve a similar type of question. So, while students can rely on hard work put into the many hours of drill and practice, they may stumble when faced with extensions of problems and may not know how to respond accordingly, even though the principle and concepts that underlie the questions are similar.

To address this lack of ability in transferring of knowledge to new situations, educators should devise lesson activities which help students to discover the concepts through inquiry learning or inductively exploring a variety of examples where the concepts would be applicable. Instead of spending time working on similar types of questions, students can be exposed to a wider variety of problem-solving situations, thus broadening the scope where the concept can be applied. Through regular reference to the use of concepts in problem-solving, rather than specific facts or formulae, students can better appreciate what they are learning and are intellectually stimulated in the classroom.