Effect of Early Numeracy Learning on Numerical Reasoning

Effect of Early Numeracy Learning on Numerical ReasoningFROM NUMERICAL MAGNITUDE TO FRACTIONSEarly taste of mathematical order and residuum is deportly related to subsequent acquisition of component familiarityAbstractEvidence from experiments with infants concerning their strength to author with quantitative order of magnitude is examined, along with the make do relating to the naturalness of numerical cogitate world power. The key debate here(predicate) concerns execution in looking at time experiments, the enamorness of which is examined. Subsequently, say concerning how kidren happen to cerebrate with proportions is examined. The key snap of the debate here relates to clear-cut vs unvarying proportions and the unmanageableies children sleep with to harbour when cogitate with clear-cut proportions specific wholey. Finally, the evidence is reviewed into how children come to intellect with cyphers and, explicitly, the difficulties experienced an d why this is the case. This is examined in the context of distinct theories of mathematical in conditionation, unneurotic with the effect of teaching methods.Early reason of numerical magnitude and proportion is directly related to subsequent acquisition of fraction companionship discernment of magnitude and fractions is crucial in contemporary society. Relatively simple tasks such(prenominal) as dividing a restaurant bill or sharing taproom at a birthday party rely on an concord of these concepts in order to determine how much everyone requires to pay towards the bill or how much cake everyone rear receive. Understanding of these concepts is too infallible to endure calculation of much complex mathematical problems, such as resoluteness equations in statistical formulae. It is thitherfore evident that a sound arrest of magnitude and fractions is required in everyday life and whilst most bragging(a)s fill for granted the ability to calculate magnitudes and fractio ns, this is not so for children, who require intimacy to drop by the wayside the concepts to be embedded into their translateing. De Smedt, Verschaffel, and Ghesquire (2009) advise that childrens performance on magnitude comparison tasks predicts later mathematical achievement, with Booth and Siegler (2008) get on arguing for a causal link surrounded by primordial misgiving of magnitude and mathematical achievement. Despite these findings, research tends to risquelight problems when the teaching of whole act maths progresses to teaching fractions. Bailey, Hoard, Nugent, and Geary (2012) suggest that performance on fraction tasks is indicative of boilers suit mathematics performance levels, although overall mathematical ability does not predict ability on these tasks.This article reviews the current coif of research into how modern children, between birth and well-nigh seven days of suppurate come to understand magnitude and how this relates to the subsequent learni ng of fractions. By primarily reviewing research into interpretation of numerical magnitude, the starting signal role of this composing result collapse a fairly narrow counseling. This restriction is unavoid able due to the large volume of literature on the topic of infant interpretation of magnitude generally and is in addition felt to be appropriate due to the close link between integers, proportions and fractions. An understanding of magnitude is inwrought to differentiate proportions (Jacob, Vallentin, Nieder, 2012) and adjacent the review of literature in respect of how magnitude comes to be understood, the paper will review the present situation in respect of how tender children understand proportions. Finally, the article will conclude with a review of where the literature is currently placed in respect of how young childrens understanding of magnitude and proportion relates to the learning of fractions and briefly how this fits indoors an overall mathematical fr ame thrash.Is the understanding of numerical magnitude innate? on that point be twain opposing views in respect of the innateness of adult male understanding of morsel and magnitude. One such view suggests that infants argon born with an innate ability to carry discover basic numerical surgical processs such as summation and subtraction (Wynn, 1992, 1995, 2002). In her seminal and widely cited test, Wynn (1992) utilize a looking time procedure to measure the reactions of young infants to two possible and impossible arithmeticalal outcomes over tierce experiments. Infants were placed in front of a screen with all one or two objects displayed. A barrier was then placed over the screen, restricting the infants view, following which an experimenter either added or removed an item. The infants were able to see the mathematical operation taking place due to a small gap at the edge of the screen which showed items cosmos added or subtracted, just now were not able to view t he final display until the barrier was removed. following the manipulation and remotion of the barrier, infants looking times were measured and it was formal that overall infants spent substantively more time looking at the impossible outcome than the arrange outcome. These results were assumed to be indicative of an innate ability in human infants to manipulate arithmetical operations and, accordingly, hump between different magnitudes. The wind of an innate human ability to manipulate arithmetical operations is given up get along credence by a matter of different forms of considererpunch of Wynns (1992) original study (Koechlin, Dehaene, Mehler, 1997 Simon, Hespos, Rochat, 1995). Feigneson, C ary, and Spelke (2002) and Uller, C atomic number 18y, Huntley-Fenner, and Klatt (1999) also replicated Wynn, although interpret the results as being based on infant preference for object-based attention as inappropriate to an integer-based attention.Despite replications of Wy nn (1992), a number of studies have also failed to replicate the results, principal to an alternative possibility. Following a failure to replicate Wynn, Cohen and Marks (2002) desexualise that infants distinguish magnitude by favouring more objects over less(prenominal) and also display a preference towards the number of objects which they have initially been presented, disregardless of the mathematical operation carried out by the experimenter. This suggestion arises from the results of an experiment where Wynns hypothesis of innate mathematical ability was tested against the preference hypothesis noted above. Further evidence against Wynn (1992) exists following an experiment by Wakeley, Rivera, and Langer (2000), who represent that no systematic evidence of addition and subtraction exists, instead the ability to add and subtract progressively develops during infancy and childhood. Whilst this does not specifically strike out Cohen and Marks, it does cast doubt on basic ar ithmetical skills and, accordingly, the ability to work with magnitude existing innately.How do children understand magnitude as they come along?By six-months hoary, it is suggested that infants employ an gravelly magnitude estimation system (McCrink Wynn, 2007). exploitation a looking-time experiment to assess infant attention to displays of pac-men and dots on screen, infants go forthed to name to falsehood displays with a large difference in ratio (21 to 41 pac-men to dots, 41 to 21 pac-men to dots), with no signifi nookyt difference in attention times to novel stimuli with a closer ratio (21 to 31 pac-men to dots, 31 to 21pac-men to dots). These results were interpreted to exemplify an understanding of magnitudes with a degree of geological fault, a pattern already existing in the literature on adult magnitude studies (McCrink Wynn, 2007). Unfortunately, one ignore in respect of interpreting the results of experiments with infants is that they cannot explicitly inform experimenters of their understanding of what has happened. It has been argued that experiments do exercise of the looking-time paradigm cannot be right understood as experimenters must make an conjecture that infants will have the same expectations as adults, a matter which cannot be appropriately support (Charles Rivera, 2009 K. Mix, 2002).As children come to utilise langu bestride, words which have a direct dealingship to magnitude (eg., little, more, lots) enter into their vocabulary. The use of these words allows researchers to wonder how they come to form internal representations of magnitude and how they are used to explicitly reveal understanding of such magnitudes.Specifically isolating the word more, children appear to develop an understanding of the word as being comparatively scene of action neutral (Odic, Pietroski, Hunter, Lidz, Halberda, 2013). In an experiment requesting children aged 2.0 4.0 (mean age = 3.2) to distinguish which colour on pictures of a se t of dots (numeric task) or blobs of goo (non-numeric task) stand for more, it was established that no significant difference exists between performance on both numeric and non-numeric tasks. In addition, it was found that children age more or less 3.3 historic period and above performed significantly above chance, whereas those children below 3.3 years who participated did not. This supports the affirmation that the word more is understood by young children as both comparative and in domain neutral terms not specifically related to number or theater. It could also be suggested that it is around the age of 3.3 years when the word more comes to hold some sort of semantic understanding in relation to mathematically based stimuli (Odic et al., 2013). It is difficult to analyse this study to that of McCrink and Wynn (2007) due to the differing genius of methodology. It would certainly be of interest to researchers to ask the possibility of some sort of comparison research, howe ver, as it is unclear how the Odic et al. (2013) study fits with the suggestion of an approximate magnitude estimation system, notwithstanding the use of language.Generally, children understand numerical magnitude on a logarithmic basis at an early age, progressing to a more rakear understanding of magnitude as they age (Opfer Siegler, 2012), a change which is beneficial. It is suggested that the more linear a childs mental representation of magnitude appears, the better their memory for magnitudes will be (Thompson Siegler, 2010). There are a number of reasons for this change in understanding, such as socioeconomic status, culture and education (Laski Siegler, in press). In the remainder of this section, the understanding of magnitude in school age children (up to just about seven years old) is reviewed, although still the effect of education will be referred to. The remainder of the reasons are noted in order to exemplify some issues which can also have an seismic disturba nce on childrens breeding of numerical magnitude understanding.As children age, the neurological and mental representations of magnitude treat both numeric and non-numeric stimuli in a linear fashion (Opfer Siegler, 2012). On this basis, number line representations present a reasonable method for investigating of childrens understanding of magnitude generally. One method for examining number line representations of magnitude in children uses dialog box games in which children are required to count moves as they play. Both preliminary to and subsequent to playing the games, the children involved in the experiment are then presented with a straight line, the parameters of which are explained, and pass to mark on the line where a set number should be placed. This allows researchers to establish if the action of game playing has allowed numerical and/or magnitude information to be encoded. In an experiment of this nature with pre-school children (mean age 4 years 8 months), Siegl er and Ramani (2009) established that the use of a linear numerical board game (10 spaces) enhanced childrens understanding of magnitude when compared to the use of a circular board game. It is argued that the use of a linear board game assists with the formation of a retrieval mental synthesis, allowing participants to encode, store and retrieve magnitude information for future use. Similar results have subsequently been obtained by Laski and Siegler (in press), workings with around older participants (mean age 5 years 8 months), who sought-after(a) to establish the effect of a larger board (100 spaces). In this case, the structure of the board ruled out high performance based on participant memory of space location on the board. In addition, verbalising movements by counting on was found to have a significant impact on retention of magnitude information.A final key read/write head relating to understanding of magnitude relates to the predictive value of current understanding o n future learning. When education level was manageled for, Booth and Siegler (2008) found a significant correlation between the pre-test numerical magnitude score on a number line task and post-test scores of 7 year-olds on both number line tasks and arithmetic problems, This discovery has been supported by a replication by De Smedt et al, (2009) and these findings together suggest that an understanding of magnitude is radical in predicting future mathematical ability. It is also clear that a in effect(p) understanding of magnitude will assist children in subsequent years when the curriculum proceeds to deal more comprehensively with matters such as proportionateity and fractions.From numerical magnitudes to proportionsEvidence reviewed previously in this article tends to suggest that children have the ability to distinguish numerical magnitudes competently by the approximate age of 7 years old. Unfortunately, the ability to distinguish between magnitudes does not necessarily su ggest that they are easily reasoned with by children. Inhelder and Piaget (1958) first suggested that children were unable to reason with proportions generally until the transition to the formal operational spirit level of development, at around 11-12 years of age. This point is elucidated more generally with the tilt that most comparative reasoning tasks prove difficult for children, regardless of age (Spinillo Bryant, 1991). However, more recent research has suggested that this assertion does not strictly hold true, with children as young as 4 and 5 years old able to reason proportionally (Sophian, 2000). Recent evidence suggests that the key debate in terms of childrens ability to reason with proportions concerns the distinction between discrete quantities and consecutive quantities. Specifically, it is argued that children find dealing with problems involving continuous proportions simpler than those involving discrete proportions (Boyer, Levine, Huttenlocher, 2008 Jeong, Levine, Huttenlocher, 2007 Singer-Freeman Goswami, 2001 Spinillo Bryant, 1999). In addition, the half(a) bound is also viewed as being of critical spaciousness in childrens proportional reasoning and understanding (Spinillo Bryant, 1991, 1999). These matters and suggested reasons for the experimental results are now discussed.Proposing that first order relations are important in childrens understanding of proportions, Spinillo and Bryant (1991) suggest that children should be successful in making judgements on proportionality using the relation greater than. In addition, it is suggested that the half boundary also has an important role in proportional decisions. Following an experiment which requested children make proportional judgements about stimuli which either track or did not cross the half boundary, it was found that children aged from approximately 6 years were able to reason relatively easily concerning proportions which pass over the half boundary. From these res ults, it was drawn that children tend to establish part-part first order relations to deal with proportion tasks (eg. reasoning that one box contains more muddied than white bricks). It was also suggested that the use of the half boundary formed a first reference to childrens understanding of part-whole relations (eg. reasoning that a box contained half blue, half white bricks). No express diversionary attack from continuous proportions was used in this experiment and, therefore, the only matter which can be drawn from this result is that children as young as 6 years old can reason about continuous proportions.In a follow up experiment, Spinillo and Bryant (1999) again utilised their half boundary paradigm with the addition of continuous and discrete proportion conditions. Materials used in the experiment were of an isomorphic nature. The results broadly mirrored Spinillo and Bryants (1991) initial study, in which it was noted that the half boundary was important in solving of pro portional problems. This also held for discrete proportions in the experiment despite performance on these tasks scoring poorly overall. Children could, however, establish that half of a continuous total is identical to half of a discrete quantity, supporting the idea that the half boundary is crucial to reasoning about proportions (Spinillo Bryant, 1991, 1999). Due to the similar nature of materials used in this experiment, a further research question was posited in order to establish whether a similar task with non-isomorphic constituents would have either impact on the ability of participants to reason with continuous proportions (Singer-Freeman Goswami, 2001). Using models of pizza and chocolates for the continuous and discrete conditions respectively, participants carried out a matching task where they were required to match the ratio in the experimenters model with their own in either an isomorphic (pizza to pizza) or non-isomorphic (chocolate to pizza) condition. In simil ar results to the previous experiments, it was found that participants had less problems dealing with continuous proportions than discrete proportions. In addition, performance was superior in the isomorphic condition compared to the non-isomorphic condition. An interesting finding, however, is that problems involving half were successfully resolved, irrespective of condition, further adding credence to the importance of this feature. Due to participants in this experiment being slightly younger than those in Spinillo and Bryants (1991, 1999) experiments, it is argued that the half boundary may be used for proportional reasoning tasks at a very early age (Singer-Freeman Goswami, 2001).In addition to the previously reviewed literature, there is a vast body of evidence the encumbrance of discrete proportional reasoning compared to continuous proportional reasoning in young children. Yet to be identified, however, is a firm reason as to why this is the case. Two specific suggestions as to why discrete reasoning appears more difficult than continuous reasoning are now discussed. The first suggestion is based on a conjecture posited by Modestou and Gagatsis (2007) related to the improper use of contextual knowledge. An error occurs when certain knowledge, applicable to a certain context, is used in a setting to which it is not applicable. A particular problem identified with this form of reasoning is that it is difficult to correct (Modestou Gagatsis, 2007). This possibility is applied to proportional reasoning by Boyer et al, (2008), who suggest that the reason children find it difficult to reason with discrete proportions is because they use absolute numerical equivalence to explain proportional problems. straight proportion problems are presumably easier due to the participants using a proportional strategy to solve the problem, whereas discrete proportions are answered using a numerical equivalence schema where it is not applicable. An altogether differe nt suggestion for the issue is made by Jeong et al, (2007), invoking Fuzzy trace theory (Brainerd Reyna, 1990 Reyna Brainerd, 1993). The argument posited is that children focus more on the number of target partitions in the discrete task, whilst ignoring the area that the target partitions cover. It is the area that is of most relevance to the proportion task and, therefore, direction on area would be the correct outcome. Instead, children appear to instinctively focus on the number of partitions, whilst ignoring their relevance (Jeong et al., 2007), thereby performing poorly on the task.From proportions to fractionsIn tandem with childrens difficulties in relation to discrete proportions, there is a wealth of evidence supporting the notion that fractions prove difficult at all levels of education (Gabriel et al., 2013 Siegler, Fazio, Bailey, Zhou, 2013 Siegler, Thompson, Schneider, 2011). Several theories of mathematical development exist, although only some constitute suggest ions as to why this may be the case. The three main bodies of theory in respect of mathematical development are let domain theories (eg. Wynn, 1995b), abstract change theories (eg. Vamvakoussi Vosniadou, 2010) and integrated theories (eg, Siegler, Thompson, Schneider, 2011). In addition to the representation of fractions within established mathematical theory, a further wave-particle duality exists in respect to how fractions are taught in schools. It is argued that the majority of teaching of fractions is carried out via a largely procedural method, meaning that children are taught how to manipulate fractions without being fully aware of the conceptual rules by which they operate (Gabriel et al., 2012). Discussion in this section of the paper will focus on how fractions are interpreted within these theories, the similarities and differences therein, together with how teaching methods can contribute to better overall understanding of fractions.Within privileged domain theories, development of understanding of fractions is viewed as secondary to and inherently distinct from the development of whole song (Leslie, Gelman, Gallistel, 2008 Siegler et al., 2011 Wynn, 1995b). As previously examined, it is argued that humans have an innate system of numerical understanding which specifically relates to positive integers, he basis of privileged domain theory being that positive integers are psychologically privileged numerical entities (Siegler et al., 2011, p. 274). Wynn (1995b) suggests that difficulty exists with learning fractions due to the fact that children struggle to conceive of them as discrete numerical entities. This argument is similar to that of Gelman and Williams (1998, as cited in Siegler et al., 2011) who suggest that the knowledge of integers presents barriers to learning about other fictional characters of number, due to distinctly different properties (eg. assumption of unique succession). Presumably, privileged domain theory views the fact that integers are viewed as being distinct in nature from any other type of numerical entity is the very reason for children having difficulty in learning fractions, as their main basis of numerical understanding prior to encountering fractions is integers.Whilst similar to privileged domain theories in some respects, conceptual change theories are also distinct. The basis of conceptual change theories is that concepts and relationships between concepts are not static, but change over time (Vamvakoussi Vosniadou, 2010). In essence, protagonists of conceptual change do not necessarily dismiss the ideas of privileged domain theories, but allow freedom for concepts (eg. integers) and relationships between concepts (eg. assumption of unique succession) to be altered in order to accommodate new information, albeit that such accommodation can make a substantial period of time to occur (Vamvakoussi Vosniadou, 2010). Support for conceptual change theory is found in the failure of childre n to comprehend the unlimited number of fractions or decimals between two integers (Vamvakoussi Vosniadou, 2010). It is argued that the reason for this relates to the previously manifested knowledge of integer relations (Vamvakoussi Vosniadou, 2010) and that it is closely related to a concept designated as the whole number bias (Ni Zhou, 2005). The whole number bias can be defined as a tendency to utilise schema specifically for reasoning with integers to reason with fractions (Ni Zhou, 2005) and has been referred to in a number of studies as a possible cause of problems for childrens reasoning with fractions (eg. Gabriel et al., 2013 Meert, Grgoire, Nol, 2010).Siegler et al, (2011) propose an integrated theory to account for the development of numerical reasoning generally. It is suggested by this theory that the development of understanding of both fractions and whole metrical composition occurs in tandem with the development of procedural understanding in relation to these concepts. The theory claims that numerical development involves coming to understand that all real numbers have magnitudes that can be ordered and assigned specific locations on number lines (Siegler et al., 2011, p. 274). This understanding is said to occur gradually by essence of a progression from an understanding of characteristic elements (eg. an understanding that whole numbers hold specific properties distinct from other types of number) to distinguishing between essential features (eg. different properties of all numbers, specifically their magnitudes) (Siegler et al., 2011). In contrast to the foregoing privileged domain and conceptual change theories, the integrated theory views acquisition of knowledge concerning fractions as a fundamental course of numerical development (Siegler et al., 2011). Supporting evidence for this theory comes from Mix, Levine and Huttenlocher (1999), who report an experiment where children successfully completed fraction reasoning tasks in tan dem with whole number reasoning tasks. A high correlation between performances on both tasks is reported and it is suggested that this supports the existence of a shared latent ability (Mix et al., 1999).One matter which appears continuously in fraction studies is the pedagogical method of delivering fraction education. A number of researchers have argued that teaching methods can have a significant impact on the ability of pupils to acquire knowledge about fractions (Chan, Leu, Chen, 2007 Gabriel et al., 2012). It is argued that the teaching of fractions falls into two distinct categories, teaching of conceptual knowledge and teaching of procedural knowledge (Chan et al., 2007 Gabriel et al., 2012). In an interpellation study, Gabriel et al, (2012) segregated children into two distinct radicals, the experimental group receiving extra tuition in relation to conceptual knowledge of fractions, with the control group following the regular curriculum. The experimental results suggeste d that improved conceptual knowledge of fractions (eg. equivalence) allowed children to perform better when presented with fraction problems (Gabriel et al., 2012). This outcome supports the view that more attack should be made to teach conceptual knowledge about fractions, prior to educating children about procedural matters and performance on fractional reasoning may be improved.Conclusion and suggestions for future researchIn this review, the process of how children come to understand and reason with numerical magnitude, progressing to proportion and finally fractions has been examined. The debate concerning the innateness of numerical reasoning has been discussed, together with how children understand magnitude at a young age. It has been established that children as young as six months old appear to have a preference to impossible numerical outcomes, although it remains unclear as to why this is. The debate remains ongoing as to whether infants are reasoning mathematically, or simply have a preference for novel situations. Turning to proportional reasoning, evidence suggests a clear issue when children are reasoning with discrete proportions as opposed to continuous ones. Finally, evidence concerning how children reason with fractions and the problems therein was examined in the context of three theories of mathematical development. Evidence shows that all of the theories can be supported to some extent. A brief section was devoted to how teaching practice effects childrens learning of fractions and it was established that problems exist in terms of how fractions are taught, with too much violence placed on procedure and not enough placed on conceptual learning.With the foregoing in mind, the following research questions are suggested to be a good starting point for future experimentsHow early should we go for teaching of fraction concepts? Evidence from Mix et al, (1999) suggests that children as young as 5 years old can reason with fractions and it m ay be beneficial to childrens education to teach them earlierShould fractions be taught with more emphasis on conceptual knowledge?ReferencesBailey, D. H., Hoard, M. K., Nugent, L., Geary, D. C. (2012). Competence with fractions predicts gains in mathematics achievement. diary of Experimental Child Psychology, 113, 447455.Booth, J., Siegler, R. (2008). Numerical magnitude representations influence arithmetic learning. Child Development, 79, 10161031.Boyer, T. W., Levine, S. C., Huttenlocher, J. (2008). Development of proportional reasoning where young children go wrong. developmental Psychology, 44, 14781490.Brainerd, C. J., Reyna, V. F. (1990). Inclusion illusion Fuzzy-trace theory and perceptual salience effects in cognitive development. Developmental Review, 10, 363403.Chan, W., Leu, Y., Chen, C. (2007). 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