**2. Young children's mathematical intuition**

Jung [5] and Kammii and DeClark [6] describe young children's mathematical intuitions as internal abilities possessed by young children. In discussing the origins of such abilities [5] employs Piaget's three kinds of knowledge: "physical; logicomathematical and social" (p. 7) knowledge. The difference amongst these knowledge is derived from their sources and modes of structuring. Physical knowledge is knowledge of "objects in external reality." This knowledge can be observed, touched or felt using senses. This knowledge cannot develop without external influence or experience. On the other hand, logico-mathematical knowledge is formed internally through connections that are mentally made. For example, [5]'s story of a 6-yearold girl trying to understand the concept of Santa Claus. This girl started noticing some patterns about Santa Claus that made her ask some questions such as: "How come Santa Claus uses the same wrapping paper as we do?"; "How come Santa Claus has the same writing as Daddy?" (p. 45). Kammii's story of Santa Claus indicates that a young child believes all what her parents tells her. However, as she develops she begins to make connections in her brain. The girl in Kammii's story believe in Santa Claus/Father Christmas in South African language but one day he made some observations: (1) she observed that Santa Claus's writing on his presents is the same as her father, and (2) the way the presents are wrapped is the same as the way her mother wrap up presents at home. This pushed the girl to make connections that, she has never met Santa but Santa always knows what she wants. She therefore realized that there is no Santa as Santa is her father assisted by her mother in wrapping up the present. This indicates that the external or physical knowledge observed was then internalized by this child and became internal knowledge and connections were made and a logico-mathematical knowledge is formed.

Furthermore, [6–8] affirm that children demonstrate spontaneous quantitative recognition at a young age. This spontaneity is a natural ability that is identified as intuitions by researchers [9]. For example, at 6 months old children are able to discriminate small numbers [6]. Clements [7] suggest that this skill demonstrated by the children at this infant stage is subitizing. Hyde and Spelke [10] extend this discovery by suggesting that babies under 9 months own two systems of nonverbal numerical cognition: one that is retorting to small quantities of individual objects and the other to approximately larger quantities. These systems contribute to the development of counting skills [11]. In addition, [12] brings forth the spatial intuitions of children's crawling which allows them to navigate and understand space. The above literature clearly indicates children's mathematical intuitions that need one to tap on in developing them further and deeper. For this chapter the big

**85**

*Self-Regulation in Early Years of Learning Mathematics: Grade R Observed Self-Efficacy Skills…*

question is on how does this intuition assist in developing independent learning of mathematics to young children? Kammii presents Piaget's concept of autonomy as a vehicle to mediate mathematics to young children capturing their intuitions and

The autonomy concept comes from Piaget's theory of construction of knowledge that originates from children's experiences of their world. The intuitions discussed above resonates well with the children's experiences. These experiences happen through senses physically. Kammii and DeClark [5] defines autonomy as being directed by yourself. In other words, being self-driven. In the case of mathematics learning the concept of autonomy will refer self-dependency and independent learning. There are two types of autonomy from the original theory: moral auton-

Moral autonomy is an ability to choose between right and wrong without pleasing others taking own responsibility for doing the right thing without expecting any rewards [13]. This ability is one of the important characteristic needed for leadership and citizenry. Kammii and DeClark [5] advocate for nurturing of this ability to children at an early age to develop responsible, accountable, adults with integrity.

Intellectual autonomy is important for successful learning because it challenges the mind. Feza [13] describes intellectual autonomy as "the connection made by children within the physical world that leads them to question things they observe that do not connect" (p. 63). The curiosity demonstrated by children in knowing more and relationships made between patterns. For example, a young child when s/he gets a toy plays with it for a moment and start dismantling it until it is broken, once it is in pieces the child will try hard to rebuild the toy. This indicates that when the child was breaking the toy into pieces the purpose was not to destroy it but to figure out something about its composition. Therefore, after seeing all the components of it the child wants to be able to deconstruct it, but unfortunately cannot and

Another example of figuring out things come from [14] as a unique difference between animals and humans. A story of a crawling baby and a dog fetching a cloth stuck on the tree explain Vygotsky's meaning of this difference. In this story the baby and the dog are playing with a cloth amongst the two. A strong wind came and snatched the cloth and threw it on the branch of the tree. The dog started barking and jumping in attempt to reach the cloth the baby on the other hand sat and looked up. After sometime the baby crawled towards a stick lying on the ground. The baby took the stick and crawled back to the tree with it he tried to reach the cloth until on the third attempt the stick hooked the cloth and the cloth fell on the ground. This story again supports Kammii's example of Santa Claus. The baby observed the physical space and distance and realised that, nor matter how high the dog jumps, it's strides are small and the distance is longer. Therefore, the baby looked around

for something that can cover the observable distance and found a stick.

The stories give account to children's intellectual autonomy that needs to be nurtured by educators as it arises. A number of opportunities come in classrooms

*DOI: http://dx.doi.org/10.5772/intechopen.88497*

interests to nurture independent development.

**3. Autonomy**

omy and intellectual autonomy.

**3.1 Moral autonomy**

**3.2 Intellectual autonomy**

they cry with frustration.

question is on how does this intuition assist in developing independent learning of mathematics to young children? Kammii presents Piaget's concept of autonomy as a vehicle to mediate mathematics to young children capturing their intuitions and interests to nurture independent development.
