The Development of Mathematical Skills
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The Development of Mathematical Skills

Chris Donlan, Chris Donlan

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eBook - ePub

The Development of Mathematical Skills

Chris Donlan, Chris Donlan

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About This Book

Current research into the psychology of children's mathematics is extremely diverse. The present volume reflects this diversity; it is unique in its breadth, bringing together accounts of cutting-edge research from widely differing, sometimes opposing viewpoints. The reader with a grounding in developmental psychology but no knowledge of mathematical development will enjoy a wide ranging and challenging summary of current trends. Those already familiar with some of the work may take the opportunity to broaden their knowledge and to evaluate new methodologies and the insights they offer.
The book is an invitation to explore a complex set of phenomena for which no unitary explanation can be offered. It aims to show that apparently disparate research perspectives may be complementary to each other; and to suggest that progress towards a comprehensive account of mathematical skills may require a broad-based understanding of research from more than one viewpoint.

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Year
2022
ISBN
9781317715450

SECTION ONE:

Pre-school mathematical understanding

INTRODUCTION

Chapters 1 and 2 offer the reader lucid and authoritative, but also contrastive accounts of early mathematical development. When Karen Wynn published a paper in Nature in 1992 under the provocative title “Addition and subtraction by human infants” she presented evidence that five-month old babies show suprise when their numerical expectations concerning the outcomes of events are violated. The innovative design of the 1992 study appeared to exclude explanations based on pattern recognition. Participants’ reactions demonstrated their attention to cumulative sequences of events; Wynn claimed to have evidence that, before any sort of numerical language is acquired, infants are capable of encoding ordinal (numerical) information.
In our opening chapter Wynn presents the 1992 study and subsequent work within a thoughtful and broad-ranging literature review. The reader will welcome the clarity with which the methodologies as well as the findings of infant research are presented, as well as the detailed interpretation of findings.
Wynn takes a strong nativist position, arguing that human infants are innately endowed with arithmetical abilities. She makes it clear towards the end of Chapter 1 that she does not seek to explain all subsequent mathematical or even arithmetical development within the narrow confines of her model. However, it seems fair to conclude that Wynn’s account entails a continuity whereby the enumeration system present in infancy supports the development of counting in early childhood, and therefore that innate structures are central to mathematical development in the pre-school years.
In Chapter 2 Catherine Sophian offers an equally thoughtful alternative account. Addressing the infant research, she proposes a perceptually based interpretation of findings, in opposition to Wynn, and suggests that definitive evidence for one or other position has yet to emerge. Sophian goes on to examine in detail the findings of her own and others’ research into the development of counting. She shows that the refinements of counting skill that take place from 3 to 5 years of age are such as to test quite severely the constraints of an innate enumeration mechanism.
Sophian proposes that the developmental process is best characterised as a “virtuous circle” of interaction between the child’s goal-directed numerical activities and the conceptual advances that these bring about. The goals or uses of counting clearly change as the child grow older, and a part at least of this change is, according to Sophian, the result of social interaction. Thus her account of pre-school mathematical development is a dynamic one within which the achievement of socially mediated goals is a driving force.
The final chapter in this opening section builds, to some extent, on the social orientation of Chapter 2. Penny Munn presents new research exploring children’s acquisition of symbolic skills, in particular their use of written numerals, and the influence this new skill may have on the understanding of number in general. She presents intriguing evidence to support the claim that children’s understanding of the function of numerical symbols is crucial in the development of a cognitive model of number. For Munn this cognitive model is essentially a social one, it is the product of negotiation and forms the basis of shared understanding of numerical communication. There is considerable force behind this argument, and it has powerful educational implications, especially within the United Kingdom where formal schooling starts at age five. Munn draws an important distinction between the socially mediated learning that takes place in the pre-school and the formal methods of teaching in school, noting that a quarter of children in her sample entered school before they had become aware of the function of conventional number symbols.
At a theoretical level Munn’s proposal is far-reaching. She presents, in prototype at least, a theory of symbolic development within which emergent numeracy and literacy skills are grounded in social context. An important contribution of this chapter is its requirement that the reader consider at some length the complex meanings that attach to the simple written numeral, the function of which is taken for granted in much of the research that is to follow.

CHAPTER ONE

Numerical competence in infants

Karen Wynn
University of Arizona, Tucson, USA

INTRODUCTION

The abstract body of mathematical knowledge that has been developed over the last several thousand years is one of the most impressive of human achievements. What makes the human mind capable of grasping number? Philosophers and psychologists have long speculated about the origins of numerical knowledge and concepts. The empiricist account of how we possess such knowledge is that we acquire even the simplest understanding of numerical relationships from our observations of the world. The alternative is a nativist account in which some understanding of number is inherent in the structure of the mind.
Number is different from perceptual properties such as colour and size. As the 19th century philosopher and logician Gottlob Frege (1893) observed, number is not an inherent property of any portion of the physical world. That is, there is no one number that describes a particular portion of matter. To give an example, there may be three dogs—but that same group of dogs is also an example of some different number (probably twelve) of paws, of yet a different number of hairs, of an astronomical number of molecules, and so on. In short, there is no “right” number for describing that particular portion of the material world. Number is not a property of the physical world itself, but rather is determined as a result of how we choose to carve up the physical world into individual elements—as dogs, as hairs, as paws, and so on.
The basic empiricist story of the origins of numerical knowledge, found in both philosophy and psychology, is that we learn numerical facts by induction over observations of the world. The philosopher John Stuart Mill (1843) held that we can identify the number of a group of things by identifying the individual elements and arranging them, in our minds or in the actual world, into a recognizable pattern. We can recognise that there are three flowers, for example, by perceiving the individual flowers and realizing we can always arrange them in the same pattern, for example,
Image
We then go on to learn mathematical truths, such as that 1+2 = 3, by observing it to be true over and over again, for many different objects. We observe that one flower (∘) and two flowers (∘∘) make three flowers
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that one cookie plus two more cookies make three cookies; that one kitten and two kittens make three kittens; and that in general, we can arrange our idea of one thing (∘) and our idea of two things (∘∘) so as to make up our idea of three things
Image
. Finally, we induce, from observing many such examples, that it is always true that one and two make three.
Philip Kitcher (1984) is a contemporary philosopher who has proposed an alternative empiricist theory of the origins of mathematical knowledge. Kitcher’s main thesis is that we learn the simplest of numerical knowledge by observing the results of our own actions, and we learn the rest from authorities in mathematics—parents and teachers, and ultimately mathematicians. The body of mathematical knowledge that has been developed through history was itself built (and is currently being expanded further) by experts in mathematics, who first learn from and then build on the knowledge of their predecessors, who in turn built on the knowledge of their predecessors, and so on. The whole chain of knowledge, both for individual people and over the course of history, is grounded in the activities of young children, who learn about the mathematical structure of reality through their actions and interactions with the physical world: “Mathematical knowledge arises from rudimentary knowledge acquired by perception. Several millennia ago, our ancestors set the enterprise in motion by learning through practical experience some elementary truths of arithmetic and geometry” (Kitcher, 1984, p. 5). Kitcher gives an example of the way in which we perceptually acquire a particular arithmetical truth through practical experience—namely, that 2 + 3 = 5: “We recognize, for example, that if one performs the collective operation called ‘making two’, then performs on different objects the collective operation called ‘making three’, then performs the collective operation of combining, the total operation is an operation of ‘making five’.” (Kitcher, 1984, p. 108).
But for as long as there have been empiricist theories of how we attain numerical knowledge, there has also been a nativist alternative to the empiricist view: philosophers, psychologists and cognitive scientists have long argued that some understanding of number is inherent in the structure of the mind. For example, Kant (1781/1965) has argued that “Mathematical propositions are always judgements a priori, not empirical; because they carry with them necessity, which cannot be derived from experience.” More recently, Chomsky (1980) has speculated that “It seems reasonable to suppose that this [number] faculty is an intrinsic component of the human mind… the capacity to deal with the number system … is surely unlearned in its essentials.”
The debate over how mathematical understanding originates has been going on for many hundreds of years, but for most of this time has been limited to a priori and philosophical arguments. Within the past 20 years, however, experimental methodologies have been developed in psychology that allow us to investigate the minds of young infants, and so to obtain some understanding of the inherent structure of the mind prior to the influences of language and culture. In this chapter, I will review evidence showing that humans come into the world already equipped with the concept of number. They are able to distinguish different numbers of entities, and to recognize numerical equivalence across different instances. This ability to represent number applies to a wide range of different kinds of entities, and to entities perceived through different sensory modalities. Moreover, in addition to being able to represent number, infants are able to engage in processes of numerical reasoning—they can determine the numerical relationships that hold between different numbers. But this numerical competence is not unique to humans—I will next review evidence that many different animal species are also able to represent and reason about number. I will conclude by describing a mechanism that has been proposed to account for this numerical competence in human infants and nonhuman animals—a hardwired mental mechanism dedicated to enumerating and performing numerical operations—and discuss some limitations to this psychological foundation of numerical knowledge.

NUMERICAL COMPETENCE IN HUMAN INFANTS

Infants are sensitive to number

Findings over the past 20 years have shown that infants are sensitive to number. These studies use a habituation methodology. Infants tend to look longer at things that are new or unexpected to them. By accustoming, or habituating, infants to displays that have a given property, we can determine if infants are sensitive to this property by then presenting them with displays that do not have that property and seeing whether they look longer at these new displays.
In one study, 5-month-old infants were divided into two groups. The infants in one group were habituated to displays of 2 circular spots of light arranged in a line, whereas those in the other group were habituated to displays of 3 spots of light. Each baby was repeatedly presented with displays of 2 (or 3) spots, and the infant’s looking time to each display was measured. (As an infant becomes accustomed to a display, he or she begins to lose interest in it and will look at it for shorter and shorter periods of time, indicating he or she is becoming habituated.) Once an infant was habituated, as determined by the decrease in looking time, the test phase began: the infant was presented with new displays, some containing 2 spots, some containing 3 spots, and his or her looking time to these displays was measured. The experimenters found that infants looked longer when they were shown a new number of spots than when shown the number they had been habituated to, indicating that they could discriminate 2 spots from 3 spots (Starkey & Cooper, 1980). In a second experiment conducted in a similar fashion, infants did not discriminate 4 spots from 6. Because the ratio of 2:3 is the same as that of 4:6, infants were apparently not distinguishing the ratio of dark-to-light in the displays or using a general “more-numerous/less-numerous” response, both of which would distinguish 4 from 6 just as strongly as 2 from 3. Rather, they were responding to the number of spots per se. In a separate study, similar results were obtained with infants of 1 to 3 days of age (Antell & Keating, 1983).
In another study, 7-month-old infants were habituated to visual displays of either 2 or 3 randomly arranged objects. The displays were constructed of photographs of various household objects that were different in each picture—for example, one picture might consist of an orange and a glove, the next might include a key chain and a banana, the next a bell pepper and a sponge, and so on (Starkey, Spelke, & Gelman, 1990). Moreover, the photographs of the different objects varied considerably in size, and the items were placed in a different spatial arrangement in each picture, guaranteeing that the overall configuration of the items changed from picture to picture. Following habituation, infants were shown test pictures of 2 items and of and 3 items, containing new objects not seen in the habituation pictures. Infants looked significantly longer at test pictures containing the number of items that differed from what they had been habituated to, indicating that they discriminated between the two numbers.
In an intriguing series of experiments (van Loosbroek & Smitsman, 1990), infants of 5, 8, and 13 months were habituated to displays of 2, 3, or 4 moving items displayed on a video screen, and then tested on the habituated number and its immediate neighbours. Each item was a random “checkerboard” pattern, constructed by randomly filling in a proportion of the rectangles defined by a 16 × 16 grid. Each pattern had its own path of motion on the computer screen. The different paths of motion occasionally intersected so that checkerboards would overlap each other on occasion. Thus, a static view of the display would not guarantee correct number information—the number of items could only be determined by watching the display in motion over time. Nonetheless, infants of all ages tested discriminated 2 from 3 and even 3 from 4, and the older two groups of infants also distinguished 4 from 5.
Infants are able to determine numbers of many different kinds of entities. Not only can they identify the number of items in perceptually very different displays of visual objects, from spots of light, to photographs of household objects, to computer-generated random “checkerboard” patterns, they can also determine numbers of sounds, as shown in a series of experiments by Starkey et al. (1990). In these experiments, 6- to 9-month-old infants were habituated to photographs of 2 items or of 3 items as described earlier. On habituation, they were presented with a black disk on a display stage. On some test trials, the disk would emit 2 drumbeats, on other trials 3 drumbeats, and infants’ looking time to the disk was measured. It was found that infants looked longer at the disk when it emitted the same number of drumbeats as the number of objects they had been habituated to. That is, infants habituated to pictures of 2 objects looke...

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