In her book, `Cognitive Development: Psychological and Biological Perspectives`, Rosemary Rosser (1994) relates the view that there is an array of definitions for concepts, the subject of an "ongoing philosophical discourse" (Fodor, 1972 in Rosser, 1994). I will consider these multiple definitions for what concepts are, as well as the various types of conceptual understanding. Still, I propose each one s development involves similar cognitive processes, and so will study children s mathematical reasoning as a focused example. I will explore if and when conceptual understanding commences by analysing Spelke s (1994) term initial knowledge , and the theory of conceptual understanding as an inherent human trait rather than the result of perceptual experiences and resolve development consists of an indistinguishable relationship between the two. Finally, I will highlight the role artificial conceptual construction plays in the development of a child s understanding, and conclude there is neither a point at which it begins or ends.

Rosser (1994) defines concepts as the `core of meaning`, which is essentially the mental representation of something. This suggests conceptual understanding is the knowledge of the correct meanings behind everything. Yet she notes that this implies humans would have to learn the separate meanings to every new object they encountered, which would restrict them in novel situations and put great "strain on [their] information-processing system[s]" (1994). Jean Piaget (1954) claimed concepts are constructed through a combination of a priori and empirical deduction. His theory infers that, for humans, objects consist of a system of images and relations, conceptualised by their positions within category structures. Therefore, concepts are not merely labels to be learnt but complex and flexible identities. This is observable in how languages never translate perfectly word to word the concepts behind linguistic labels are subject to their social, cultural and historical use. I propose there are two main groups of concepts: existential (such as objects, colour, sounds) and symbolic (such as language, scientific theories, art). Traditional views would then claim that younger children s conceptual abilities are limited to the former category since their representations are instance bound where as matured humans can comprehend the symbolic concepts too because they are able to form more logical , abstract concepts (Keil, 1989). Yet, Dr. Sara Baker stated there is little distinction between perception and conception in humans (2012), and I condone opposition against a dichotomy between the perceptual and cognitive explanations for conceptual understanding and development (Scholl & Leslie, 1999). Mathematics is fundamentally made up of a structure of interrelating principles, such as the fact that fractions and decimals are a representation of division, which is a process of counting into groups and squaring is the act of multiplying a number by itself, and multiplication is the process of counting up a number of groups of a certain number.

Brian J. Scholl and Alan M. Leslie (1999) recognise the meaning of Elizabeth Spelke s (1994) phrase initial knowledge is questionable. Constructivist theorists would interpret it as the set of conceptual principles humans are born with. Children have shown to possess a foundation of numerical abilities, such as during Karen Wynn s (1992) experiments where she recorded the significantly longer looking times five-month-old infants displayed when presented with the impossible events when Mickey Mouse dolls were ostensibly removed or placed behind a flapping drawbridge they had been habituated to. This is further supported by the fact that ostensibly innate concepts can constrain an individual s overall conceptual development for instance, Bruce M. Hood s (1995) research into 2- to 4 -year olds understanding of gravity showed subjects under the age of four continued to expect a ball dropped from the opening of a chimney to come out the other end directly below, despite the fact that the tube was curved. However, this could be considered as an example of children s overextension and failure to drop perceptually obtained na ve theories. Therefore, it must be considered whether early examples of conceptual competence are actually universally innate.

Alternatively, initial knowledge could be defined as the first conceptual principles a child picks up, which as Hood s (1995) experiment displays may need to be adapted and improved. This would mean early and subsequent perceptual experiences must occur in order to establish these preliminary concepts and improve an individual s conceptual understanding. The object-tracking theory would disagree that humans are born with counting skills, arguing it is perceptual capabilities noticing the visual appeareance of an untracked token that provokes their attention, and appear to display their numerical understanding (Simon, 1997). However, six-month-old children have also exhibited the ability to enumerate changes in physical actions by showing more interest in a jumping puppet when it jumped more times than they had been habituated to it doing (Wynn, 1996). This shows that a form of enumeration beyond perceptual observation must be occuring in young children s cognition. Although theorists such as Simon (1997) would advocate that children develop the necessary understanding to count having practiced using visual or touchable objects, perceptive experiences demand a basic degree of prior conceptual abilities so that they can be interpreted and learnt from correctly. If infants were restricted to a perceptual view of their world they would only ever be reacting in the moment, and never be able to learn the concepts that are essential to accessing superior reasoning. Therefore, I contend the thoery that knowledge is only ever obtained perceptually is problematic.

First of all, Learning systems require perceptual systems that parse the world appropriately (Spelke, 1994). For example, a child could not recognise the unexpected number of dolls in the impossible events of the Mickey Mouse experiment without the ability to distingush between the drawbridge and the dolls. Secondly, perception is not always reliable. The Pulfrich double pendulum illusion is an example in which observation would provide incorrect conceptualisation regarding object solidity, where a pair of pendulums swinging on parallel planes appear to coalesce around each even though they do not touch (Leslie, 1988). The fact that perceptual systems can construct such a conceptual violation implies they cannot be responsible for establishing conceptual knowledge (Scholl & Leslie, 1999). Using the aforementioned Mickey Mouse doll experiment, this is applicable to basic numerical understanding too. Despite the fact that an object becomes no longer perceivable when it is affectively subtracted, subjects clearly still note it s movement history and adjust the value of objects they expect to be behind the drawbridge (Piaget, 1954). It is also impossible that all, particularly more complex and specific, conceptual understanding can be developed perceptually. Although simple mathematics can be grasped and solved using visual methods, certain problems like the fractional sum 17/24 + 5/18 or 5 X 4 demand a conceptual approach (Wu, 1999). It seems the way in which children perceive objects and occurences are subsequently affected by their developing conceptual systems, and the way in which concepts are constructed is subject to how they are perceived. Hence, I propose young children possess raw conceptual understanding, such as for numerical value, which is bound up in their perceptual abilities. Wider and more detailed cogintive comphrehension develops as an intricate relationship between their natural human perceptual and conceptual abilities.

Nevertheless, how then can innate conceptual understanding be distinguished from that which is developed through the complicated, interactive proccess stated above? Equally, is there then a universal point in every child s life at which this process starts to develop? From studies that compare the performance of children of different ages in conceptual tasks it would certainly appear so. For example, Xu & Carey (1996) found that 10-month-old children paid little attention when a screen was lifted to reveal objects of an unexpected identity, rather than value of objects, in contrast to 12-month-old subjects who reacted to both identity and arithmetic inconsistencies. This infers that infants experience a change in cognitive proccess between 10 to 12 months of age, which allows them to access conceptual comprehension of object identiy. Moreover, children over 4 years old, compared to the 2- to 4-year old subjects, exhibited superior gravitational conceptual understanding in Hood s (1995) aforementioned research. Several psychologists have proposed theories that children develop conceptually in stages, including Piaget (1954), Rudolf Steiner (1965), and more. However, I would argue the range of contradictory theories and evidence available shows there is no universal point of change in the processsing of every type of concept, or amongst all humans. This is also evident in variations between individuals from different contexts. Multiple studies have highlighted Asian children s significantly superior mathematical conceptual abilities in comparison to American children s, resulting from number-naming systems that make accessing some mathematical relations more difficult in English than in Chinese (Miller, Smith, Zhu, & Zhang, 1995). It is true that all mathematical reasoning beyond the elemental concepts are dependent on language, which is constructed by adult humans. So, although in Wynn s (1992) experiments children do not appear to enumerate objects verbally, by noticing the incorrect value of expected dolls, young children are operating a kind of conceptualese (Carey, 1988) which is in need of moulding to the public mode of expressing concepts that is adopted by adult humans. I argue a large quantity of conceptual systems are constructions, which can differ between societies, cultures, periods, and circumstances, just as they can differ between children and adults. Therefore, it is impossible to determine an exact point at which a change in general, or even more specifically mathematically, conceptual processing begins.

I conclude that I cannot define a point at which conceptual understanding begins in a child for the following reasons. Firstly, a certain degree of conceptual competence is an innate human quality to allow them to perceive and subsequently correctly conceptualise about their world and experiences. Although infants are not born with a complete set of operateable concepts, I propose their knowledge undergoes a complex process of development as a result of an intertwined relationship of perception and conceptual cognition. They observe and adapt to the constructed conceptual systems of the adult human world that surrounds them. The fact that this often differs slighlty depending on the culture, language, society, period, etc. of their environment is a further reason why I argue there is no point in a child s life when we can say conceptual development processes change, nevermind conceptual understanding itself begins within each of the various types of concepts.

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