3.1. Conceptual content

Two logically independent and empirically distinct properties of core cognition representations lead me to attribute them conceptual content. First, contrary to empiricist proposals, they cannot be reduced to spatiotemporal or sensory vocabulary. One cannot capture concepts such as goal, agent, object, or approximately 10 in terms of primitives such as locations, paths of motion, shapes, and colors. Second, they have a rich, central, conceptual role. The representations in core cognition are centrally accessible, represented in working memory models, and support decision and control mechanisms leading to action such as reaching. For example, infants make a working memory model of the individual crackers in each of two buckets and guide their choice of which bucket to crawl to from quantitative computations using those models. A variety of quantitative computations are defined over working memory representations of sets of objects. Preverbal infants can sum continuous quantities or compare models on the basis of one-to-one correspondence and categorically distinguish singletons from sets of multiple individuals. Moreover, young infants represent objects relative to the goals of agents, and infants' representations of physical causality are constrained by their conceptualization of the participants in a given interaction (as agents capable of self-generated motion or as inert objects). Thus, the conceptual status of the output of a given core cognition system is confirmed by its conceptual interrelations with the output of other core cognition systems.

In other respects, the representations in core cognition resemble perceptual representations. Like representations of depth, the representations of objects, agents, and number are the output of evolutionarily ancient, innate, modular input analyzers. Like the perceptual processes that compute depth, those that create representations of objects, agents, and number continue to function continuously throughout the life span. And like representations of depth, their format is most likely iconic.

3.2. Dedicated input analyzers

A dedicated input analyzer computes representations of one kind of entity, and only of that kind. All perceptual input analyzers are dedicated in this sense: The mechanism that computes depth from stereopsis does not compute color, pitch, or causality.

Characterizing the mechanisms that identify the entities in systems of core cognition is important for several reasons, including that knowing how an input analyzer works bears on what aspects of the world it represents. Analysis of the input analyzers underlines the ways in which core cognition is perception-like, and provides one source of evidence for the continuity of core cognition throughout development. For example, the primacy of spatiotemporal information in creating object representations and tracing object identity through time is one of the signatures that identifies infant object representations with those of mid-level object-based attention and working memory in adults. Characterizations of the input analyzers bear on other theoretically important issues as well. For example, whether analog magnitude symbols are computed through a serial accumulation mechanism or by a parallel computation bears on whether they straightforwardly implement a counting algorithm, thus being likely to underlie learning to count (TOOC, Ch. 4 and 7). The evidence favors a parallel process. Or for another example, Chapter 5 (TOOC) presents evidence that infants use spatiotemporal descriptions of the motions and interactions among objects to assign agency, goals, and attentional states to them, and also that the static appearance of the entities in an event (e.g., presence of eyes and hands) also plays a role in creating representations of agency. Such results raise the question of whether one of these sources of information is primary. For example, infants may initially identify agents through patterns of interaction, and may then learn what these agents look like. Alternatively, the innate face detectors infants have may serve the purpose of identifying agents, allowing them then to learn how agents typically interact. A third possibility is that agency detection is like mother recognition in chicks, such an important problem for human infants that evolution hedged its bets with two dedicated input analyzers to do the trick.

3.3. Innateness

What I mean when I say that a representation is innate, is that the input analyzers that identify its referents are not the product of learning. That is, a representational capacity is innate, not necessarily any particular instantiated symbol for some entity in the world. This characterization is empty, of course, without a characterization of learning mechanisms. Broadly, these are mechanisms for which environmental input has the status of evidence. Obviously, theories of learning and theories of mental representations are mutually constraining. The first place we would currently look to explain the existence of innate input analyzers would be evolution; either they are adaptations under some direct selection pressure or byproducts of representational mechanisms that arose under selective pressure.

For the most part, the evidence reviewed in TOOC for core cognition does not derive from experiments with neonates. Rather, the evidence for the object representations of core cognition comes from studies of infants 2 months of age or older, and that for core representations of intentional agency comes from infants 5 months of age or older. Five months, and even two months, is a lot of time for learning. Why believe that the representations tapped in these experiments are the output of innate input analyzers, and why believe that the demonstrated inferential role that provides evidence for the content of the representations is unlearned? I discuss this question in each case study, appealing to four types of arguments.

First, that a given representational capacity may be innate in humans is suggested by evidence that it is manifest in neonates of other species. Examples offered were depth perception, which emerges without opportunities for learning in neonate goats and neonate rats, and object representations, which are observed in neonate chicks. This line of evidence is obviously indirect, providing only an existence proof that evolution can build input analyzers that create representations with the content in question.

Second, the simultaneous emergence of different aspects of a whole system also provides indirect evidence for the innateness of the input analyzers and computational machinery that constitute core cognition. As soon as infants can be shown to form representations of complete objects, only parts of which had been visible behind barriers, they also can be shown to use evidence of spatiotemporal discontinuity to infer that two numerically distinct objects are involved in an event, and also to represent object motion as constrained by solidity (TOOC, Ch. 2 and 3). Similarly, different aspects of intentional attribution emerge together. For example, an infant's representing an entity as capable of attention increases the likelihood she will represent its action as goal directed, and vice versa (TOOC, Ch. 5). If the generalizations that underlie infants' behavior are learned from statistical analyses of the input (represented in terms of spatiotemporal and perceptual primitives), it is a mystery why all of the interrelated constraints implicated in the core cognition proposals emerge at once. Infants have vastly different amounts of input relevant to different statistical generalizations over perceptual primitives. Relative to the thousands of times they have seen objects disappear behind barriers, 2-month-old infants have probably never seen rods placed into cylinders, and have rarely seen solid objects placed into containers. Yet the interpretation of both types of events in terms of the constraints on object motion that are part of core cognition emerge together, at 2 months of age. Statistical learning of perceptual regularities would be expected to be piecemeal, not integrated.

Third, learnability considerations also argue that the representations in core cognition are the output of innate input analyzers. If the capacity to represent individuated objects, numbers, and agents are learned, built out of perceptual and spatiotemporal primitives, then there must be some learning mechanism capable of creating representations with conceptual content that transcend the perceptual vocabulary. In the second half of TOOC, I offer Quinian bootstrapping as a mechanism that could, in principle, do the trick, but this type of learning process requires explicit external symbols (e.g., words or mathematical symbols), and these are not available to young babies. Associative learning mechanisms could certainly come to represent regularities in the input. For example, a baby could form the generalization that if a bounded stimulus disappeared through deletion of the forward boundary behind another bounded stimulus, there is a high probability that a bounded stimulus resembling the one that disappeared will appear by accretion of the rear boundary from the other side of the constantly visable bounded surface. But such generalizations would not be formulated in terms of the concept object. There is no proposal I know for a learning mechanism available to nonlinguistic creatures that can create representations of objects, number, agency, or causality from perceptual primitives.

Fourth, success at some task provides support for some target representational capacity needed to perform it, whereas failure is not necessarily good evidence that the target capacity is lacking. Some other capacity, independent of the target one, may be needed for the task and may not yet be available (not yet learned or not yet matured). TOOC provides several worked-out examples of successful appeals to performance limitations masking putatively innate competences. For example, the A/not B error made by infants between ages 7 and 12 months is at least in part explained by appeal to immature executive function. For another example, infants' failure until 2 months of age to create representations of a complete rod partially hidden behind a barrier when they are shown the protruding ends undergoing common motion is at least partly explained by their failure to notice the common motion across the barrier. Thus, they lack the critical input to the putatively innate computation.

3.4. Iconic format

A full characterization of any mental representation must specify its format as well as its content and conceptual role. How are the mental symbols instantiated in the brain: are they language-like, diagram-like, picture-like, or something else? I intend the distinction between iconic and noniconic formats to be the same distinction that was at stake in the historical debates on the format of representation underlying mental imagery. Iconic representations are analog; roughly, the parts of the representation represent parts of the entities that are represented by the whole representation.

We know little about the format of most mental representations. Of the core cognition systems discussed in TOOC, the question of format is clearest for number representations, so my discussion of format was concentrated there (Ch. 4). The very name of “analog magnitude representations” makes a claim for their format. Analog representations of number represent as would a number line: the representation of 2 is a quantity that is smaller than and is contained in the representation of 3. We do not know how these analog representations are actually instantiated in the brain. Larger quantities could be represented by more neurons firing or by faster firing of a fixed population of neurons, for example. Many plausible models have been proposed (see TOOC, Ch. 4). That discrimination satisfies Weber's law (is a function of the ratio of set sizes) suggests that number representations work like representations of length, time, area, brightness, and loudness. All proposals for how all of these continuous dimensions are represented also deploy analog magnitudes.

TOOC speculates that all of core cognition is likely to be represented in iconic format. Consider the working memory models that constitute the parallel individuation system of object representations. The fact that these representations are subject to the set-size limit of parallel individuation implicates a representational schema in which each individual in the world is represented by a symbol in working memory. This fact does not constrain the format of these symbols. A working memory model for two boxes of different front surface areas, for instance, could consist of image-like representations of the objects (), or they could be symbolic (object[3 square inches], object[4 square inches]). These models must include some representation of size, bound to each symbol for each object, because the total volume or total surface area of the objects in a small set is computable from the working memory representations of the sets. The most plausible model for how this is done implicates iconic representations of the objects, with size iconically represented, as well as shape, color, and other perceptual properties bound to the symbols iconically. The iconic alternative laid out in TOOC Chapter 4 explains the set size limits on performance even when continuous variables are driving the response.

I have several other reasons for suspecting that the representations in core cognition are iconic. Iconic format is consistent with (though not required by) the ways in which the representations in core cognition are perception-like, assuming, as I believe to be the case (contrary to Pylyshyn Reference Pylyshyn2002), that perceptual representations are iconic. Second, just as static images may be iconic or symbolic, so also may representations of whole events. If infants represent events in iconic format, like a movie that can be replayed, this could help make sense of the apparently retrospective nature of the representations that underlie many violation-of-expectancy looking-time experiments (Ch. 2–6). Finally, that core cognition may be represented in terms of iconic symbols, with some of its content captured in encapsulated computations defined over these symbols, may help to make sense of the extreme lags between understanding manifest in infant looking-time studies, and that manifest only much later in tasks that require explicit linguistic representations (TOOC Ch. 3, 5, 8–12). The guess that the format of all core cognition is iconic is just that: a guess. But the considerations just reviewed lead me to favor this hypothesis.

3.5. Constant through the life span

Continuity through the life span is an important property of core cognition for several reasons. We seek an account of cognitive architecture that carves the mind into meaningful sub-systems; and most conceptual representations are not continuous throughout development. Core cognition is one very distinctive part of the human mind: no other systems of conceptual representations share its suite of characteristics.

If innate input analyzers are the product of natural selection, one might think that they must be useful to adults as well as children. Expectation of continuity might seem to be the default. However, this first thought is not necessarily correct. Some innate representational systems serve only to get development started. The innate learning processes (there are two) that support chicks' recognizing their mother, for example, operate only in the first days of life, and their neural substrate actually atrophies when the work is done. Also, given that some of the constraints built into core knowledge representations are overturned in the course of explicit theory building, it is at least possible that core cognition systems themselves might be overridden in the course of development.

Thus, it is most definitely an empirical question whether core cognition is constant throughout the life span. TOOC argues that the answer is “yes” for the core cognition systems described therein. Evidence for continuity includes the same signatures of processing in adulthood and infancy. Under conditions where core cognition is isolated from other conceptual resources, adults display the same limits on computations, and the same modular input analyzers, as do infants. For example, for both populations, the input analyzers that create object representations privilege spatiotemporal information over other perceptual features in the service of individuation and tracking numerical identity.

3.6. A dual factor theory of the concepts within core cognition

Dual factor theory straightforwardly applies to the representations that articulate core cognition. There are aspects of innate conceptual role that remain constant throughout development. These specify the narrow content of representations within core cognition. Furthermore, the representations within core cognition are the output of innate perceptual input analyzers. These input analyzers most probably come into being through natural selection, a process that, in this case, explains how the extension of the concepts within core cognition may be determined by causal connections between entities in their domains (objects, agents, goals, and cardinal values of sets) and the mental symbols that represent them.

4.1. Representations of cause

The existence of innate conceptual representations embedded within systems of core cognition does not preclude other innate conceptual representations as well, including non–domain-specific central ones. Chapter 6 takes the concept cause as a case study. Michotte (Reference Michotte1946/1963) proposed that innate causal representations are the output of innate perceptual analyzers that take object representations and spatiotemporal relations among their motions as input. This is tantamount to the claim that causal representations are part of core object cognition, as Spelke (2002) suggested. Chapter 6 contrasts Michotte's hypothesis with the proposal that there may be innate central representations of causation. According to Michotte's proposal, the earliest causal representations should be sensitive only to spatiotemporal relations among events, and should be limited to reasoning about causes of object motion. An impressive body of empirical data establishes that by 6 months of age, infants represent Michottian motion events (launching, entraining, and expulsion) causally. Nonetheless, TOOC rejects Michotte's proposal on the grounds that causal cognition integrates across different domains of core cognition (object representations and agent representations), encompassing state changes as well as motion events, from as early in development as we have evidence for causal representations at all.

Innate central causal representations could come in either of two quite different varieties. There may be innate central processes that compute causal relations from patterns of statistical dependence among events, with no constraints on the kinds of events. Or there may be specific aspects of causality that are part of distinct core cognition systems (e.g., Michottian contact causality within the domain of core object cognition, and intentional causality within the domain of agent cognition) and these may be centrally integrated innately. These possibilities are not mutually exclusive; both types of central integration of causal representations could be part of infants' innate endowment.

4.2. Public symbols, logical and linguistic capacity

TOOC says little about two important aspects of conceptual development. First, I assume that domain-specific learning mechanisms, jointly comprising a language acquisition device, make possible language acquisition, but I have made no effort to summarize the current state of the art in characterizing the language acquisition device. Whatever its nature, it is another way innate cognitive architecture goes beyond core cognition, for the symbols in language are not iconic, and language, like causality, integrates representations across separate core domains. Second, I have said almost nothing about the logical capacities humans and other animals are endowed with, although these are independent of core knowledge and I make use of various of them in my bootstrapping proposals. These are topics for other books.

Language acquisition and conceptual development are intimately related. The representations in core cognition support language learning, providing some of the meanings that languages express. TOOC Chapter 7 considers how pre-linguistic set-based quantification supports the learning of natural language quantifiers, and how pre-linguistic representations of individuals support the learning of terms that express sortals. But because my concern is the origin of concepts, I focused mainly on the complementary influence of language learning on conceptual development. Language learning makes representations more salient or efficiently deployed (Ch. 7; so-called weak effects of language learning on thought), and plays a role in conceptual discontinuities (strong effects of language learning on thought).

Chapter 7 reviews two cases in which very early language learning affects nonlinguistic representations. First, learning, or even just hearing, labels for objects influences the establishing/deploying of sortal concepts. Second, mastery of explicit linguistic singular/plural morphology plays a role in deploying this quantificational distinction in nonlinguistic representations of sets. Although TOOC argues that these are most likely weak effects of language learning on thought, “weak” does not entail “uninteresting” or “unimportant.” Creating representations whose format is noniconic paves the way for integrating the concepts in core cognition with the rest of language.

Furthermore, most of the second half of the book concerns how language learning also shapes thought in a much stronger way. Language learning plays a role in creating new representational resources that include concepts that previously could not be entertained.

6.1. Natural number

Core cognition contains two systems of representation with numerical content: parallel individuation of small sets of entities in working memory models, and analog magnitude representations of number. Within the language acquisition device, a third innate system of representation with numerical content supports the learning of natural language quantifiers. These are the CS1s. CS2, the first explicit representational system that represents the positive integers, is the verbal numeral list embedded in a count routine. Deployed in accordance with the counting principles articulated by Gelman and Gallistel (Reference Gelman and Gallistel1978), the verbal numerals implement the successor function, at least with respect to the child's finite count list. For any numeral that represents cardinal value n, the next numeral in the list represents n+1.

CS2 is qualitatively different from each of the CS1s because none of the CS1s has the capacity to represent the integers. Parallel individuation includes no symbols for number at all, and has an upper limit of 3 or 4 on the size of sets it represents. The set-based quantificational machinery of natural language includes symbols for quantity (plural, some, all), and importantly contains a symbol with content that overlaps considerably with that of the verbal numeral “one” (namely, the singular determiner, “a”), but the singular determiner is not embedded within a system of arithmetical computations. Also, natural language set-based quantification has an upper limit on the sets' sizes that are quantified with respect to exact cardinal values (singular, dual, trial). Analog magnitude representations include symbols for quantity that are embedded within a system of arithmetical computations, but they represent only approximate cardinal values; there is no representation of exactly 1, and therefore no representation of +1. Analog magnitude representations cannot even resolve the distinction between 10 and 11 (or any two successive integers beyond its discrimination capacity), and so cannot express the successor function. Therefore, none of the CS1s can represent 10, let alone 342,689,455.

This analysis makes precise the senses in which the verbal numeral list (CS2) is qualitatively different from those representations that precede it: it has a totally different format (verbal numerals embedded in a count routine) and more expressive power than any of the CS1s that are its developmental sources.

As suggested by CS2's being qualitatively different from each of the CS1s that contain symbols with numerical content, it is indeed difficult to learn. American middle-class children learn to recite the count list and to carry out the count routine in response to the probe “how many,” shortly after their second birthday. They do not learn how counting represents number for another 1½ or 2 years. Young 2-year-olds first assign a cardinal meaning to “one,” treating other numerals as equivalent plural markers that contrast in meaning with “one.” Some 7 to 9 months later they assign cardinal meaning to “two,” but still take all other numerals to mean essential “some,” contrasting only with “one” and “two.” They then work out the cardinal meaning of “three” and then of “four.” This protracted period of development is called the “subset”-knower stage, for children have worked out cardinal meanings for only a subset of the numerals in their count list.

Many different tasks that make totally different information processing demands on the child confirm that subset-knowers differ qualitatively from children who have worked out how counting represents number. Subset-knowers cannot create sets of sizes specified by their unknown numerals, cannot estimate the cardinal values of sets outside their known numeral range, do not know what set size is reached if one individual is added to a set labeled with a numeral outside their known numeral range, and so on. Children who succeed at one of these tasks succeed at all of them. Furthermore, a child diagnosed as a “one”-knower on one task is also a “one”-knower on all of the others, likewise for “two”-knowers, “three”-knowers, and “four”-knowers. The patterns of judgments across all of these tasks show that parallel individuation and the set-based quantification of natural language underlie the numerical meanings subset-knowers construct for numeral words.

In sum, the construction of the numeral list representation is a paradigm example of developmental discontinuity. How CS2 transcends CS1 is precisely characterized, CS2 is difficult to learn, adult language expressing CS2 is represented by the child in terms of the conceptual resources of CS1, and children's performance on a wide variety of tasks consistently reflects either CS1 or CS2.

6.2. Rational number

TOOC Chapter 9 presents another parade case of developmental discontinuity within mathematical representations. In this case CS1 is the count list representation of the positive integers, enriched with an explicit understanding that there is no highest number, and so number means natural number. Early arithmetic instruction depends upon and further entrenches this representational system, representing addition and subtraction as counting up and counting down, and modeling multiplication as repeated addition. In CS2, “number” means any point on a number line that can be expressed x/y, where x and y are integers. In CS2, rather than it being the case that integers are the only numbers, there is an infinity of numbers between any two integers. The question of the next number after n (where n might be an integer or not) no longer has an answer. Therefore, CS2 has more expressive power than CS1 (CS1 cannot represent one-half as a number, nor any of the infinite noninteger rational numbers), and numbers are related to each other differently in the two systems. The new relation in CS2 is division. Division cannot be represented in terms of the resources of CS1, which model only addition, subtraction, and multiplication of integers. CS2's division cannot be represented as repeated subtraction of integers.

CS2 is extremely difficult for children to learn. One-half of college-bound high school students taking the SAT exams do not understand fractions and decimals. Furthermore, explicit instruction concerning rational number is initially assimilated to CS1, and children are consistent over a wide range of probes as to how they conceptualize number. Whether children can properly order fractions and decimals, how they justify their ordering, how they explain the role of each numeral in a fraction expressed x/y, whether they agree there are numbers between 0 and 1 and whether they believe that repeated division by 2 will ever yield 0 are all interrelated. What the child does on one of these tasks predicts what he or she will do on all of the others. CS1 and CS2 are each coherent conceptual systems, qualitatively different from each other.

8.1. Bootstrapping representations of natural number

TOOC draws on Quinian bootstrapping to explain all the developmental discontinuities described in the previous section. In the case of the construction of the numeral list representation of the integers, the memorized count list is the placeholder structure. Its initial meaning is exhausted by the relation among the external symbols: They are stably ordered. “One, two, three, four…” initially has no more meaning for the child than “a, b, c, d….” The details of the subset-knower period suggest that the resources of parallel individuation, enriched by the machinery of linguistic set-based quantification, provide the partial meanings children assign to the placeholder structures that get the bootstrapping process started. The meaning of the word “one” could be subserved by a mental model of a set of a single individual (i), along with a procedure that determines that the word “one” can be applied to any set that can be put in one-to-one correspondence with this model. Similarly “two” is mapped onto a long term memory model of a set of two individuals (j k), along with a procedure that determines that the word “two” can be applied to any set that can be put in one-to-one correspondence with this model. And so on for “three” and “four.” This proposal requires no mental machinery not shown to be in the repertoire of infants: parallel individuation, the capacity to compare models on the basis of one-to-one correspondence, and the set-based quantificational machinery that underlies the singular/plural distinction and makes possible the representation of dual and trial markers. The work of the subset-knower period of numeral learning, which extends in English-learners between ages 2.0 and 3.6 or thereabouts, is the creation of the long-term memory models and computations for applying them that constitute the meanings of the first numerals the child assigns numerical meaning to.

Once these meanings are in place, and the child has independently memorized the placeholder count list and the counting routine, the bootstrapping proceeds as follows: The child notices the identity between the singular, dual, trial, and quadral markers and the first four words in the count list. The child must try to align these two independent structures. The critical analogy is between order on the list and order in a series of sets related by additional individual. This analogy supports the induction that any two successive numerals will refer to sets such that the numeral farther in the list picks out a set that is 1 greater than that earlier in the list.

This proposal illustrates all of the components of bootstrapping processes: placeholder structures whose meaning is provided by relations among external symbols, partial interpretations in terms of available conceptual structures, modeling processes (in this case analogy), and an inductive leap. The greater representational power of the numeral list than that of any of the systems of core cognition from which it is built derives from combining distinct representational resources: a serially ordered list; set-based quantification (which gives the child singular, dual, trial, and quadral markers, as well as other quantifiers); and the numerical content of parallel individuation (which is largely embodied in the computations performed over sets represented in memory models with one symbol for each individual in the set). The child creates symbols that express information that previously existed only as constraints on computations. Numerical content does not come from nowhere, but the process does not consist of defining “seven” in terms of mental symbols available to infants.

8.2. Bootstrapping in the construction of explicit scientific theories

Historians and philosophers of science, as well as cognitive scientists, working with daily records of scientists' work, have characterized the role of Quinian bootstrapping in scientific innovation. Chapter 11 draws out some of the lessons from case studies of Kepler (Gentner Reference Gentner, Magnani and Nersessian2002), Darwin (Gruber & Barrett Reference Gruber and Barrett1974), and Maxwell (Nersessian Reference Nersessian and Giere1992).

In all three of these historical cases, the bootstrapping process was initiated by the discovery of a new domain of phenomena that became the target of explanatory theorizing. Necessarily, the phenomena were initially represented in terms of the theories available at the outset of the process, often with concepts that were neutral between those theories and those that replaced them. Incommensurability is always local; much remains constant across episodes of conceptual change. For Kepler, the phenomena were the laws of planetary motion; for Darwin, they were the variability of closely related species and the exquisite adaptation to local environmental constraints; for Maxwell, they were the electromagnetic effects discovered by Faraday and others.

In all three of these cases the scientists created an explanatory structure that was incommensurable with any available at the outset. The process of construction involved positing placeholder structures and involved modeling processes which aligned the placeholders with the new phenomena. In all three cases, this process took years. For Kepler, the hypothesis that the sun was somehow causing the motion of the planets involved the coining of the placeholder concept vis motrix, a term that designated the force/energy emitted by the sun responsible for the motion of the planets. Analogies with light and magnetism allowed him to fill in many details concerning the nature of vis motrix, as well as to confirm, for him, its existence. It became the ancestor concept to Newton's gravity, although Newton obviously changed the basic explanatory structure. Gravity due to the sun doesn't explain the planets' motion, but rather their deviations from constant velocity.

For Darwin, the source analogies were artificial selection and Malthus' analysis of the implications of a population explosion for the earth's capacity to sustain human beings. For Maxwell, a much more elaborate placeholder structure was given by the mathematics of Newtonian forces in a fluid medium. These placeholders were formulated in external symbols – natural language, mathematical language, and diagrams.

Of course, the source of these placeholder structures in children's bootstrapping is importantly different from that of scientists. The scientists posited them as tentative ideas worth exploring, whereas children acquire them from adults, in the context of language learning or science education. This difference is one reason metaconceptually aware hypothesis formation and testing is likely to be important in historical cases of conceptual change. Still, many aspects of the bootstrapping process are the same whether the learner is a child or a sophisticated adult scientist. Both scientists and children draw on explicit symbolic representations to formulate placeholder structures and on modeling devices such as analogy, thought experiments, limiting case analyses, and inductive inference to infuse the placeholder structures with meaning.

8.3. Bootstrapping processes underlying conceptual change in childhood

Historically, mappings between mathematical structures and physical ones have repeatedly driven both mathematical development and theory change. In the course of creating the theory of electromagnetic fields, Maxwell invented the mathematics of quantum mechanics and relativity theory. Another salient example is Newton's dual advances in calculus and physics.

In childhood, as well, constructing mappings between mathematical and physical representations plays an essential role in conceptual change both within mathematics and within representations of the physical world. Chapter 11 illustrates this with case studies of the creation of concepts of rational number and the creation of theory of the physical world in which weight is differentiated from density. These two conceptual changes constrain each other. The child's progress in conceptualizing the physical world exquisitely predicts understanding of rational number and vice versa. Children whose concept of number is restricted to positive integers have not yet constructed a continuous theory of matter nor a concept of weight as an extensive variable, whereas children who understand that number is infinitely divisible have done both.

Smith's bootstrapping curriculum (Smith Reference Smith2007) provides insight into the processes through which material becomes differentiated from physically real and weight from density. Although developed independently. Smith's curriculum draws on all of the components of the bootstrapping process that Nersessian (Reference Nersessian and Giere1992) details in her analysis of Maxwell. First, Smith engages students in explaining new phenomena, ones that can be represented as empirical generalizations stated in terms of concepts they already have. These include the proportionality of scale readings to overall size (given constant substance), explaining how different-size entities can weigh the same, predicting which entities will float in which liquids, and sorting entities on the basis of whether they are material or immaterial, focusing particularly on the ontological status of gases. She then engages students in several cycles of analogical mappings between the physical world and the mathematics of extensive and intensive variables, ratios, and fractions. She begins with modeling the extensive quantities of weight and volume with the additive and multiplicative structures underlying integer representations. The curriculum then moves to calculating the weight and volume of very small entities, using division. These activities are supported by thought experiments (which are themselves modeling devices) that challenge the child's initial concept weight as felt weight/density, leading them into a contradiction between their claim that a single grain of rice weighs 0 grams, and the obvious fact that 50 grains of rice have a measurable weight. Measuring the weight of a fingerprint and a signature with an analytical balance makes salient the limits of sensitivity of a given measurement device, and further supports conceptualizing weight as an extensive variable that is a function of amount of matter.

To complete the differentiation of weight from density Smith makes use of visual models that represent the mathematics of extensive and intensive quantities. The visual models consist of boxes of a constant size, and numbers of dots distributed equally throughout the boxes. Numbers of dots and numbers of boxes are the extensive variables, numbers of dots per box the intensive variable. Students first explore the properties of these objects in themselves, discovering that one can derive the value of any one of these variables knowing the values of the other two, and exploring the mathematical expression of these relations: dots per box equal number of dots divided by number of boxes. The curriculum then moves to using these visual objects to model physical entities, with number of boxes representing volume and number of dots representing weight. Density (in the sense of weight/volume) is visually represented in this model as dots/box, and the models make clear how it is that two objects of the same size might weigh different amounts, for example because they are made of materials with different densities; why weight is proportional to volume given a single material, and so on. The models are also used to represent liquids, and students discover the relevant variables for predicting when one object will float in a given liquid and what proportion of the object will be submerged. This activity is particularly satisfying for students, because at the outset of the curriculum, with their undifferentiated weight/density concept, they cannot formulate a generalization about which things will sink and which will float. Differentiating weight from density in the context of these modeling activities completes the construction of an extensive concept weight begun in the first part of the curriculum.

The formula D=W/V (density equals weight divided by volume) is a placeholder structure at the beginning of the bootstrapping process. The child has no distinct concepts weight and density to interpret the proposition. As in all cases of bootstrapping, the representation of placeholder structures makes use of the combinatorial properties of language. Density here is a straightforward complex concept, defined in terms of a relation between weight and volume (division), and the child (if division is understood) can understand this sentence as: something equals something else divided by something (most children also have no concept of volume at this point in development). The dots per box model is also a placeholder structure at the beginning of the bootstrapping process, a way of visualizing the relations between an intensive variable and two extensive variables related by division, and therefore provides a model that allows the child to think with, just as Maxwell's visual models allowed him to think with the mathematics of Newtonian mechanics as he tried to model Faraday's phenomena. At the outset of the process the child has no distinct concepts weight and density to map to number of dots and number of dots/box, respectively.

Although straightforward conceptual combination plays a role in these learning episodes (in the formulation of the placeholder structures), the heart of Quinian bootstrapping is the process of providing meaning for the placeholder structures. At the outset weight is interpreted in terms of the child's undifferentiated concept degree of heaviness (see Chapter 10) and density has no meaning whatsoever for the child. It is in terms of the undifferentiated concept degree of heaviness that the child represents the empirical generalizations that constitute the phenomena he or she is attempting to model. The placeholder structure introduces new mental symbols (weight and density). The modeling processes, the thought experiments and analogical mapping processes, provide content for them. The modeling process, using multiple iterations of mappings between the mathematical structures and the physical phenomena, makes explicit in a common representation what was only implicit in one or the other representational systems being adjusted to each other during the mapping.

That bootstrapping processes with the same structure play a role in conceptual change both among adult scientists and young children is another outgrowth of cognitive historical analysis. As I have repeatedly mentioned, of course there are important differences between young children and adult scientists engaged with metaconceptual awareness in explicit theory construction. Without denying these differences, Chapters 8–12 illustrate what the theory–theory of conceptual development buys us. By isolating questions that receive the same answers in each case, we can study conceptual discontinuities and the learning mechanisms that underlie them, bringing hard-won lessons from each literature to bear on the other.