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# The Hierarchical Progression of Mathematical Learning

Updated: Apr 7

Competency in Mathematics is gained through a strong foundation, and being able to apply these foundations onto newer concepts learnt. Unlike some subject areas, Mathematics can only be learnt through a bottom-up process, and not top-down. Without a solid foundation, learning new concepts usually prove to be a challenge, thus resulting in incomplete understanding. Without patching up these gaps in the foundations first, it is tricky to build strong competency thereafter. There are a variety of pre-requisite and support skills required to learn Mathematics, even before beginning on numbers itself, such as classification of information and objects, matching, sequencing, following multi-step directions, and visual clustering. These skills make up only part of the number of skills required before formal Mathematical learning takes place. Mathematics is also an abstract subject, as we are unable to see numbers or the numerical relationship between objects. Manipulatives are thus important in concretising the abstraction of Mathematics, to foster understanding in the child. Number sense is usually the first skill to be developed among kindergarteners. Number sense involves the knowledge of what numbers mean, understanding their relationship to other numbers and understanding the symbolic representations of these numbers. This involves counting and sequencing of numbers. It is with this knowledge that children are then able to move on to the next step of the four operations (addition, subtraction, multiplication and division). Even within the four operations, there is a hierarchy for children to learn about them. Children first learn about addition, before going on the subtraction. Addition and subtraction then becomes repeated addition (multiplication) and repeated subtraction (division). By understanding repeated addition, children are able to learn how it can also apply to understanding time. By understanding division, children are then able to learn how fractions are a form of division. It is important that children also understand the links between each concept, allowing them to see the relationship between the concepts and how foundational concepts can be applied in different situations. At the same time, this enhances their foundational skills. Our world is filled with numbers and we are constantly exposed to numbers in our daily lives. However, being effectively able to make sense of these numbers requires a lot of sequential work and understanding. One cannot expect a child to excel in fractions, when he or she is struggling to recount his or her multiplication tables. When a child thus struggles in Mathematics, it is crucial to check for his or her foundations, and whether the child is able to fully understand them. For example, if a child is unable to perform division, check if the child is able to do multiplication. If the child is able to do multiplication, then the gap is understanding how multiplication and division are related. However, if the child is unable to do multiplication, then it is important to check if the child is able to do addition or subtraction, place values, or even sequencing of numbers. Spending more time in solidifying the foundations is crucial, as it will aid in future Mathematics learning. 