Although preschool students have already known about numbers and counting, they still have a problem in understanding the amount/quantity. The primary reason of this problem is that the students have not yet construct a big mathematical idea about counting in their mind, their mind only construct about numbers. Some students can compare two, three and four objects and understand the magnitude of each group of objects; they know which group is the biggest without counting. They know it because the group which has more objects seems bigger than the others. We can conclude that young children can perceive amount without counting, in this case student use magnitude. Some students even seem to understand about one to one correspondence when they match some objects with number in dice but they don’t understand that the number in the dice represent the amount of objects corresponds with. To understand about amount or quantity of objects, the student need to understand about cardinality – the idea that number means amount. Cardinality is one of the big ideas in a young child’s mathematical development. Once students have constructed the concept of cardinality, they naturally double check themselves and recount more carefully when they get different answers.
Subitizing, seeing small amount as group, is more perceptual than mathematical therefore the idea of magnitude is much easier for children to construct than cardinality. If students only can do subitizing and magnituding, they will only know about ‘which one is more and which one is less’. They will have a problem in addition and subtraction because they don’t know the meaning of ‘amount’, they just seeing and comparing the size. To solve this problem, teacher can use many strategies in the learning of counting such as using games, using daily/routine activities and/or designing and using investigations. Counting activity will bring students to the idea of cardinality because by counting activity students will understand about the sense of numbers , the students will really understand that a number represents a particular amount of ‘object’. Another consequence of counting activity is students will know that ‘four’ is included in ‘five’ or ‘four’ is a part of ‘five’, because they can realize that before reaching ‘five’ they have to have ‘four’ first.
After understanding about cardinality, it is easier for student to understand about hierarchical inclusion, understanding that whole numbers grow by once each time and therefore that numbers nest inside each other : six is inside seven (by removing one), etc. It means that student can understand that the smaller number is a part of the bigger number. Here the students can say that four is less than five, they know it not by comparing four and five but they know it because they know that four is a part of five (‘a part’ of ‘something’ means that ‘a part’ is smaller than ‘something’). This mathematical idea requires a logical interference, an operation on the whole. Once student construct an understanding of hierarchical inclusion, they can begin to consider how if 6 + 2 = 8, then necessarily 5 + 3 = 8 as well, because while 1 more has been removed from 4 it has been added to 2 in order to get same amount, it is an example of compensation. Finally the student will realize and understand that 6+2 = 5+3 = 8. We can conclude that when a student has really understood about cardinality, hierarchical inclusion, and compensation, it will be easier for he/she to really understand about the meaning amount/quantity of objects (understand about ‘how many’). Compensation also bring the students to new stage, that is addition-subtraction.
v Big idea :
A mathematical idea that often requires puzzlement, reflection and discourse. It is important in mathematics, but also it is a developmental leap, a new perspective or way of organizing, for children
Understanding that the amount one ends on when counting tell how many objects there are in set.
v Hierarchical inclusion
Understanding that whole numbers grow by once each time and therefore that numbers nest inside each other: six is inside seven (by removing one), etc.
Comparing amounts and knowing which is more
Seeing small amount as a group, without having to count or do an operation
Written by de King
(Summarized from Dolk and Fosnot: Young Mathematician at Work)