FROM NUMERICAL MAGNITUDE TO FRACTIONS
Early understanding of numerical magnitude and proportion is directly related to subsequent acquisition of fraction knowledge
Evidence from experiments with infants concerning their ability to reason with numerical magnitude is examined, along with the debate relating to the innateness of numerical reasoning ability. The key debate here concerns performance in looking time experiments, the appropriateness of which is examined. Subsequently, evidence concerning how children progress to reasoning with proportions is examined. The key focus of the debate here relates to discrete vs continuous proportions and the difficulties children come to have when reasoning with discrete proportions specifically. Finally, the evidence is reviewed into how children come to reason with fractions and, explicitly, the difficulties experienced and why this is the case. This is examined in the context of different theories of mathematical development, together with the effect of teaching methods.
Early understanding of numerical magnitude and proportion is directly related to subsequent acquisition of fraction knowledge
Understanding of magnitude and fractions is crucial in contemporary society. Relatively simple tasks such as dividing a restaurant bill or sharing cake at a birthday party rely on an understanding of these concepts in order to determine how much everyone requires to pay towards the bill or how much cake everyone can receive. Understanding of these concepts is also required to allow calculation of more complex mathematical problems, such as solving equations in statistical formulae. It is therefore evident that a sound understanding of magnitude and fractions is required in everyday life and whilst most adults take for granted the ability to calculate magnitudes and fractions, this is not so for children, who require education to allow the concepts to be embedded into their understanding. De Smedt, Verschaffel, and Ghesquière (2009) suggest that children’s performance on magnitude comparison tasks predicts later mathematical achievement, with Booth and Siegler (2008) further arguing for a causal link between early understanding of magnitude and mathematical achievement. Despite these findings, research tends to highlight problems when the teaching of whole number mathematics progresses to teaching fractions. Bailey, Hoard, Nugent, and Geary (2012) suggest that performance on fraction tasks is indicative of overall mathematics performance levels, although overall mathematical ability does not predict ability on these tasks.
This article reviews the current position of research into how young children, between birth and approximately seven years of age come to understand magnitude and how this relates to the subsequent learning of fractions. By primarily reviewing research into interpretation of numerical magnitude, the first section of this paper will have a fairly narrow focus. This restriction is necessary due to the large volume of literature on the topic of infant interpretation of magnitude generally and is also felt to be appropriate due to the close link between integers, proportions and fractions. An understanding of magnitude is essential to differentiate proportions (Jacob, Vallentin, & Nieder, 2012) and following the review of literature in respect of how magnitude comes to be understood, the paper will review the present situation in respect of how young children understand proportions. Finally, the article will conclude with a review of where the literature is currently placed in respect of how young children’s understanding of magnitude and proportion relates to the learning of fractions and briefly how this fits within an overall mathematical framework.
Is the understanding of numerical magnitude innate?
There are two opposing views in respect of the innateness of human understanding of number and magnitude. One such view suggests that infants are born with an innate ability to carry out basic numerical operations such as addition and subtraction (Wynn, 1992, 1995, 2002). In her seminal and widely cited study, Wynn (1992) used a looking time procedure to measure the reactions of young infants to both possible and impossible arithmetical outcomes over three experiments. Infants were placed in front of a screen with either 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 being added or subtracted, but were not able to view the final display until the barrier was removed. Following the manipulation and removal of the barrier, infants’ looking times were measured and it was established that overall infants spent significantly more time looking at the impossible outcome than the correct outcome. These results were assumed to be indicative of an innate ability in human infants to manipulate arithmetical operations and, accordingly, distinguish between different magnitudes. The suggestion of an innate human ability to manipulate arithmetical operations is given further credence by a number of different forms of replication of Wynn’s (1992) original study (Koechlin, Dehaene, & Mehler, 1997; Simon, Hespos, & Rochat, 1995). Feigneson, Carey, and Spelke (2002) and Uller, Carey, Huntley-Fenner, and Klatt (1999) also replicated Wynn, although interpreted the results as being based on infant preference for object-based attention as opposed to an integer-based attention.
Despite replications of Wynn (1992), a number of studies have also failed to replicate the results, leading to an alternative hypothesis. Following a failure to replicate Wynn, Cohen and Marks (2002) posit that infants distinguish magnitude by favouring more objects over less and also display a preference towards the number of objects which they have initially been presented, regardless of the mathematical operation carried out by the experimenter. This suggestion arises from the results of an experiment where Wynn’s 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 argue 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 support Cohen and Marks, it does cast doubt on basic arithmetical skills and, accordingly, the ability to work with magnitude existing innately.
How do children understand magnitude as they age?
By six-months old, it is suggested that infants employ an approximate magnitude estimation system (McCrink & Wynn, 2007). Using a looking-time experiment to assess infant attention to displays of pac-men and dots on screen, infants appeared to attend to novel displays with a large difference in ratio (2:1 to 4:1 pac-men to dots, 4:1 to 2:1 pac-men to dots), with no significant difference in attention times to novel stimuli with a closer ratio (2:1 to 3:1 pac-men to dots, 3:1 to 2:1pac-men to dots). These results were interpreted to exemplify an understanding of magnitudes with a degree of error, a pattern already existing in the literature on adult magnitude studies (McCrink & Wynn, 2007). Unfortunately, one issue 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 making use of the looking-time paradigm cannot be properly understood as experimenters must make an assumption that infants will have the same expectations as adults, a matter which cannot be appropriately verified (Charles & Rivera, 2009; K. Mix, 2002).
As children come to utilise language, words which have a direct relationship to magnitude (eg., “little,” “more,” “lots”) enter into their vocabulary. The use of these words allows researchers to investigate 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 domain 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 set of dots (numeric task) or blobs of “goo” (non-numeric task) represented “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 approximately 3.3 years and above performed significantly above chance, whereas those children below 3.3 years who participated did not. This supports the assertion that the word “more” is understood by young children as both comparative and in domain neutral terms not specifically related to number or area. 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 compare this study to that of McCrink and Wynn (2007) due to the differing nature of methodology. It would certainly be of interest to researchers to investigate the possibility of some sort of comparison research, however, 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 linear understanding of magnitude as they age (Opfer & Siegler, 2012), a change which is beneficial. It is suggested that the more linear a child’s 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 approximately seven years old) is reviewed, although only 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 impact on children’s development of numerical magnitude understanding.
As children age, the neurological and mental representations of magnitude encompass both numeric and non-numeric stimuli in a linear fashion (Opfer & Siegler, 2012). On this basis, number line representations present a reasonable method for investigation of children’s’ understanding of magnitude generally. One method for examining number line representations of magnitude in children uses board games in which children are required to count moves as they play. Both prior 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 requested 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), Siegler and Ramani (2009) established that the use of a linear numerical board game (10 spaces) enhanced children’s 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 structure, allowing participants to encode, store and retrieve magnitude information for future use. Similar results have subsequently been obtained by Laski and Siegler (in press), working with slightly older participants (mean age 5 years 8 months), who sought 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 question relating to understanding of magnitude relates to the predictive value of current understanding on future learning. When education level was controlled 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 fundamental in predicting future mathematical ability. It is also clear that a good understanding of magnitude will assist children in subsequent years when the curriculum proceeds to deal more comprehensively with matters such as proportionality and fractions.
From numerical magnitudes to proportions
Evidence 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 suggest 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 stage of development, at around 11-12 years of age. This point is elucidated more generally with the argument that most proportional 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 children’s ability to reason with proportions concerns the distinction between discrete quantities and continuous 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” boundary is also viewed as being of critical importance in children’s 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 children’s 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 crossed 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 crossed the “half” boundary. From these results, 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 blue than white” bricks). It was also suggested that the use of the “half” boundary formed a first reference to children’s understanding of part-whole relations (eg. reasoning that a box contained “half blue, half white” bricks). No express deviation 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 Bryant’s (1991) initial study, in which it was noted that the “half” boundary was important in solving of proportional 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 quantity 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 any 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 similar 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 Bryant’s (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 difficulty 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 theory 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 theory 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. Continuous proportion problems are presumably easier due to the participants using a proportional schema to solve the problem, whereas discrete proportions are answered using a numerical equivalence schema where it is not applicable. An altogether different 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, focussing 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 fractions
In tandem with children’s 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 propose suggestions as to why this may be the case. The three main bodies of theory in respect of mathematical development are privileged domain theories (eg. Wynn, 1995b), conceptual 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 dichotomy 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 numbers (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 types 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 take a substantial period of time to occur (Vamvakoussi & Vosniadou, 2010). Support for conceptual change theory is found in the failure of children to comprehend the infinite 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 children’s reasoning with fractions (eg. Gabriel et al., 2013; Meert, Grégoire, & Noël, 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 numbers 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 means 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 tandem 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 intervention study, Gabriel et al, (2012) segregated children into two distinct groups, the experimental group receiving extra tuition in relation to conceptual knowledge of fractions, with the control group following the regular curriculum. The experimental results suggested 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 effort 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 research
In 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 children’s learning of fractions and it was established that problems exist in terms of how fractions are taught, with too much emphasis 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 experiments:
- How early should we implement 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 may be beneficial to children’s education to teach them earlier;
- Should fractions be taught with more emphasis on conceptual knowledge?
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