A New Hypothesis Concerning Children’s Fractional Knowledge

Abstract

The basic hypothesis that guides our work is that children’s fraction schemes can emerge as accommodations in their numerical counting schemes. This hypothesis is referred to as the reorganization hypothesis because if a new scheme is constructed by using another scheme in a novel way, the new scheme can be regarded as a reorganization of the prior scheme. There are two basic ways of understanding the reorganization of a prior scheme. The first is that the child constructs the new scheme by operating on the preceding scheme using operations that can be, but may not be, a part of the operations of that scheme. In this case, the new scheme is of the same type as the preceding scheme. But it solves problems and serves purposes that the preceding scheme did not solve or did not serve. It also solves all of the problems the preceding scheme solved, but it solves them better. It is in this sense that the new scheme supersedes the preceding, more primitive, scheme.

The basic hypothesis that guides our work is that children’s fraction schemes can emerge as accommodations in their numerical counting schemes. This hypothesis is referred to as the reorganization hypothesis because if a new scheme is constructed by using another scheme in a novel way, the new scheme can be regarded as a reorganization of the prior scheme. There are two basic ways of understanding the reorganization of a prior scheme. The first is that the child constructs the new scheme by operating on the preceding scheme using operations that can be, but may not be, a part of the operations of that scheme. In this case, the new scheme is of the same type as the preceding scheme. But it solves problems and serves purposes that the preceding scheme did not solve or did not serve. It also solves all of the problems the preceding scheme solved, but it solves them better. It is in this sense that the new scheme supersedes the preceding, more primitive, scheme.

The first type of reorganization was important in my work on children’s construction of numerical counting schemes (Steffe 1994a). For example, a child might only be able to count items in its perceptual field by coordinating the utterance of a number word in her number word sequence with pointing at each item. If the child abstracts its pointing acts from the acts of counting and uses the pointing acts as countable items, the pointing acts can stand in for perceptual items that are hidden from view. The child can still solve all the old counting problems, but now by counting pointing motions. And the child can solve new problems, such as counting the number of cookies where five cookies are showing and three more cookies are shown to the child and then hidden.

The second way of understanding the reorganization hypothesis is that the child constructs the new schemes by operating on novel material in situations that are not a part of the situations of the preceding schemes. The child uses operations of the preceding schemes in ways that are novel with respect to the situations of the schemes as well as operations that may not be a part of the operations of the preceding schemes. The new schemes that are produced solve situations, which the preceding schemes did not solve, and they also serve purposes, which the preceding schemes did not serve. But the new schemes do not supersede the preceding schemes because they do not solve all of the situations, which the preceding schemes solved. They might solve situations similar to those solved by the preceding schemes in the context of the new situations, but the preceding schemes are still needed to solve their situations. Still, the new schemes can be regarded as reorganizations of the preceding schemes because operations of the preceding schemes emerge in a new organization and serve a different purpose. It is this second way of understanding the reorganization hypothesis that is relevant in our current work.

The Interference Hypothesis

There is an alternative to our basic hypothesis that has historical roots in educational practice in the elementary school. In contrast to the reorganization hypothesis, there is a widespread and accepted belief that whole number knowledge interferes with the learning of fractions. For example, Post et al. (1993) commented that:

Children often have difficulty overcoming their whole number ideas while working with fractions or decimals…. To order fractions with the same numerator as 1/3 and 1/2, fourth graders in the RNP¹ teaching experiment often asked the clarifying question, “do you want me to order by the number of pieces or by the size of piece?”… RNP instructors thought their original lessons adequately treated the issue relating to using the size of the piece as the criterion for ordering fractions, but the children’s whole number strategies appeared to persist and temporarily interfere with the development of this new concept (p. 339).

The belief portrayed by these researchers is similar to a comment made by Streefland (1991) in a detailed report of children’s fractional knowledge: “But the only alarming ailment is the following one, namely, the temptation to deal with fractions in the same manner as with natural numbers” (p. 70). Streefland believed that we must focus on forming a powerful concept of fractions that is resistant to distractions. The distractions are of the sort that children add numerators and denominators when adding fractions. According to Streefland (1991), “we must not only focus on producing fractions, but also on grounded refutations of such misconceptions, or simply, on overcoming these misconceptions” (p. 70).

There is no question that children’s fractional knowledge involves ways and means of operating that are not available in their whole number knowledge. I essentially agree with the following comment by Kieren (1993) concerning rational number knowing: “Although intertwined with, sharing language with, and using concepts from whole numbers, rational number knowing is not a simple extension of whole number knowing” (p. 56). Even though Kieren makes a distinction between rational number knowing and fraction knowing in that the former is at a higher level than the latter, his comment concerning rational number knowing pertains as well to fraction knowing. I agree with Kieren that fraction knowing in any case is certainly not a simple extension of whole number knowing. Rather, our hypothesis is that it can emerge as a reorganization of whole number knowing in the sense that I have explained above. In a developmental analysis of the operations that produce discrete quantity and continuous quantity presented in Chap. 4, I explore whether the operations that produce each type of quantity can be regarded as unifying operations. The absence of such unifying operations would strengthen the interference hypothesis and the separation between the study of whole numbers and fractions that it seems to imply. In contrast, I find substantial similarity in the quantitative operations that produce continuous quantity and discrete quantity and provide reasons for why the quantitative operations that produce discrete quantity should be used to reconstitute (not replace) the operations that produce continuous quantity.

In a reaction to a preliminary draft of one of the chapters of this book, Kieren commented that, “while I, like you, decry the separation of whole number and fraction number knowing, I do not think that fractions as a reorganization of whole number based schemes is a necessary path or solution.” Whether or not fraction knowing is necessarily a reorganization of whole number knowing depends on whether the operations involved in fraction knowing can emerge in the continuous context² with only minimal involvement of the operations that are involved in whole number knowing. The thrust of the research in this book is not to investigate this question. Children do construct whole number knowing, and the quantitative operations that unify discrete and continuous quantity support the idea that children can and do draw on that knowing in their construction of fraction knowing.

In our work, we focus on children’s quantitative operations. Various researchers have found that these operations differ significantly from conventional ways of mathematical knowing. When presented with arithmetical problems, children use their current schemes in an attempt to solve them (Booth 1981; Erlwanger 1973; Ginsburg 1977; Hart 1983) in spite of emphasis on teaching practices that are not based on children’s methods (Brownell 1935). Using phrases like “rational number knowing,” “whole number knowing,” “decimals,” and “fractions” as if they refer to adults’ more or less conventional ways of knowing suppresses children’s quantitative operations in favor of the conventional ways. In fact, one of my primary goals is to formulate a language that can be used to refer to children’s mathematical concepts and operations without conflating them with the more sophisticated concepts referred to by standard mathematical language.

To illustrate why I regard the development of a language to refer to children’s mathematical concepts and operations as necessary, I use an example from a study by Nik Pa (1987). When interviewing nine 10- and 11-year-old children, Nik Pa found that they could not find 1/5 of ten items because “one-fifth” referred to one in five single items. The children separated a collection of ten items into two collections of five and then designated one item in a collection of five as “one-fifth.” Nik Pa’s finding is quite similar to what “sixths” meant for a 9-year-old child named Alan who thought “sixths” meant “six in each pile” (Hunting 1983). Hunting found Alan’s case to be representative, in its broad outlines, of those of the other 9-year-old children he studied. These findings indicate that children’s fractional language differs substantially from the observer’s meaning that “one-fifth,” say, can refer to making five composite units³ of indefinite but equal numerosity and then designating one of these composite units as “one-fifth.”

The children that Nik Pa and Hunting interviewed obviously gave meaning to fractional words using their numerical concepts.⁴ The constituting operations of these numerical concepts were left unspecified by Nik Pa, but whatever these operations might be, one interpretation of his findings could be that they interfered with or were detrimental to the children’s learning of fractions. This interpretation, however, does not take assimilation into account. Instead the importance of Nik Pa’s and Hunting’s findings is that the children used their numerical concepts in assimilation. With this view, the orientation to the relationship between children’s numerical concepts and fractional schemes changes. Even though the children’s numerical schemes in the main did not qualify as fractional schemes, their schemes constituted the current sense making constructs of the children. In our view, the problem is not one of trying to avoid these assimilating schemes nor of considering these schemes as misconceptions. Rather, the problem is to understand the children’s schemes and to learn how to help the children modify their current ways and means of operating.

For example, if the children in Post et al.’s (1993) experiment learned to reason in a way that one could attribute the inverse relation between the number and the size of the pieces of a whole to them, this would be a modification of their connected number⁵ concepts. To form an inverse relation between the number and the size of the parts, the children would have needed to at least compare the number and size of the parts on two different occasions, which involves the use of their whole number knowing. Of course, they would need to do more, but this is enough to establish our position that children’s whole number knowing is constitutively involved in their fractional knowing and should not be regarded as interfering with it. In this, I am in accord with McLellan and Dewey’s (1895) belief that “fractions are not to be regarded as something different from number – or at least a different kind of process” (p. 127).

The Separation Hypothesis

The interference hypothesis would imply that the study of whole numbers and fractions should be separated in the mathematics education of children. In the main, the study of whole numbers occurs in the context of discrete quantity and the study of fractions occurs in the context of continuous quantity (cf. Curcio and Bezuk 1994; Reys 1991). Some may see this separation between whole numbers and fractions as compatible with Confrey’s (1994) view of splitting and sequencing, so I will explore Confrey’s (1994) splitting conjecture and some of its implications. She has focused on splitting as a basic and primitive action and set it in opposition to sequencing.

In its most primitive form, splitting can be defined as an action of creating simultaneously multiple versions of an original, an action often represented by a tree diagram. As opposed to additive situations, where the change is determined through identifying a unit and then counting consecutively instances of that unit, the focus in splitting is on the one-to-many action. Closely related to this primitive concept are actions of sharing and dividing in half, both of which surface early in children’s activity. Counting need not be relied on to verify the correct outcome. Equal shares of a discrete set can be justified by appealing to the use of a one-to-one correspondence and in the continuous case, appeals to congruence of parts or symmetries can be made. (p. 292)

In further elaboration of a split, Confrey (1994) commented that: “A split is an action of creating equal parts or copies of an original” (p. 300). She goes on to say that the number concept derived from splitting is independent from the number concept derived from addition in its successor action (Confrey 1994, p. 324). She clearly considers splitting and sequencing to be independent in their origins.

It is important to note that splitting as Confrey defines is not restricted to the continuous case. Nevertheless, splitting is seemingly rooted in the continuous case and sequencing is seemingly rooted in the discrete case.⁶ As my own analysis will show, partitioning in continuous contexts is necessary for the construction of fraction schemes. Hence one could argue, based on Confrey’s work, that fractions are based on splitting in continuous situations, while whole numbers are based on counting in discrete situations, i.e., the fractional and whole number learning could, and perhaps should, take place separately. In addition, Confrey’s contention that splitting and counting have different experiential bases countermands the reorganization hypothesis.

In fact, Confrey (1994) herself sees the integration and coordination of the splitting and counting as essential. One reason for integration that she cites is to coordinate the number names of the quantities that arise through splitting with the number names the children construct in their (additive) number sequences. This consideration will emerge implicitly in my discussion as well. In addition, I will explain how the basic operations of splitting by n and sequencing by one are not primitive operations, but are constructed from the same basic operations. In particular, I will argue against the primitive nature of simultaneity in Confrey’s splitting conjecture, showing how the seeming sequentiality of number sequences and simultaneity of splitting are inextricably related. In my analysis of children’s fractional schemes in future chapters, I will give an account of the construction of a basic splitting operation engaged in by the children in the study using both partitioning and iteration, which is a form of sequencing. Although this operation certainly emerges as a new and powerful operation, it is not isolated from the operations involved in number sequences.

A Sense of Simultaneity and Sequentiality

In my examination of the origins of splitting and sequencing, I consider the child’s sense of simultaneity and sequentiality in the construction of two as a composite unit. I focus on simultaneity because Confrey said that the child creates simultaneously multiple versions of an original in a split, and this sense of simultaneity seems to be at odds with the sequential nature of operations when counting. My reason for choosing the construction of two as a composite unit is to demonstrate that both a sense of simultaneity and of sequentiality are involved in the construction of this most basic numerical concept. From this, I go on to argue that it follows that both a sense of simultaneity and of sequentiality are involved in number sequences as well as in splitting operations.

Confrey’s restriction that the action of splitting creates equal parts or copies of an original implies a level of cognitive functioning that is constitutively not primitive in the sense that it would not be present in very young children (Piaget et al. 1960). The splitting action in Confrey’s framework is at the level of mental operations rather than at the level of sensory-motor activity. Relaxing the restriction of creating equal parts does not diminish the value of her analysis for me because it provides a way of making a solid connection between partitioning operations and children’s number sequences. I will use the term “fragmenting” to refer to simultaneity in breaking without the restriction of there being equal parts. Of course, I include Confrey’s idea of splitting in fragmenting. Breaking off parts sequentially seemingly stands in contrast to fragmenting, and I use “segmenting” to refer to sequentiality in breaking without restriction on the size of the parts.

According to Confrey’s conjecture, segmenting would lead to counting and fragmenting to splitting. I interpret Confrey as meaning that children’s number sequences are based on segmenting rather than fragmenting and that fractional schemes are based on fragmenting rather than segmenting. In contrast, I advance the hypothesis that neither fragmenting nor segmenting is the more primitive and that both are involved in the construction of number sequences as well as in the construction of fraction schemes.

Establishing Two as Dual

If fragmenting were not involved in the construction of number sequences, sequentiality would need to be observed as the basal mechanism. Menninger (1969) claims in a historical study of the number sequence, however, that two and three did not develop in such a way that they were elements of a sequential order. In the development of two, he distinguished between two as dual and two as unity. His analysis of two is quite consistent with Brouwer’s (1913) analysis of “two-oneness.”

This neo-intuitionism considers the falling apart of moments of life into qualitatively different parts, to be reunited only while remaining separated by time as the fundamental phenomenon of the human intellect, passing by abstracting from its emotional content into the fundamental phenomenon of mathematical thinking, the intuition of the bare two-oneness, the basal intuition of mathematics. (p. 85)

This “falling apart of moments of life into qualitatively different parts” is an expression of fragmenting. It is related to Menninger’s idea of an awakening of consciousness – the isolation of self in an environment – “the I is opposed to and distinct from what is not I….” (Menninger, p. 13). The “falling apart” of moments of life has two necessary components. The first is an experiential awareness of one’s experiential self on two distinct occasions.⁷ But more is needed because the individual has to regenerate a prior experiential awareness of self in the current moment of awareness, which introduces the possibility of an awareness of precedence. Regenerating a preceding moment of awareness in the present makes possible the co-occurrence of the two moments of life. In this, there is also an awareness of one moment of life preceding the other; i.e., an awareness of sequentiality.

The second aspect is the awareness of recognizing an experiential item other than self, and then to be aware of recognizing another such experiential item while remaining aware of the first. Continued awareness of the recognition episode in the present act of recognition is again necessary for the two moments of life to be distinguished one from the other. Otherwise, the current fragment of experience would be the only item of awareness. The current awareness of the preceding fragment of experience makes possible copresent moments of life. Two moments of life are copresent if both are accessible to reflection or other possible operations.

An act of recognition is essentially an act of segmentation in that one separates what is being recognized from a background of possibilities. Recognizing an experiential item followed by recognizing another experiential item are sequential acts of segmentation. When the preceding recognition episode remains within awareness in a current recognition episode, this opens the possibility of becoming aware that the preceding moment of life comes before the current moment.

So, the separation of two fragments of experience by one preceding the other in experience does not necessarily mean that the two fragments are experienced only sequentially. By bringing the preceding recognition episode into the present through a regeneration of the preceding fragment, the individual can experience a copresence of the two fragments. It is in this conceptual sense that the breaking apart of moments of life can be regarded as simultaneous as well as sequential. Both the sense of simultaneity and the sense of sequentiality are the results of conceptual acts and both involve bringing forth a preceding fragment of experience into the present by means of a regeneration of the preceding experience. I emphasize that becoming aware that one moment of life precedes another opens the possibility of experiencing both moments of life together, and vice versa. Neither is the more primitive. That is not to say that one does not experience simultaneous or sequential events. But these experiences are the results of an active intellect that organizes the experiences in a particular way.

Establishing Two as Unity

Regenerating a preceding fragment of experience in a current recognition episode is different than uniting the two fragments into a composite unit. Regardless of whether a child experiences two items as occurring sequentially or simultaneously in a way that I have been speaking, the items would remain separate and distinct rather than be reunited into a two-oneness if the uniting operation has not emerged. What this means is that Brouwer’s basal intuition of mathematics depends on the construction of the uniting operation rather than the other way around. From a developmental perspective, Brouwer’s basal intuition of mathematics is produced by the uniting operation – that is, it is a construction and not a given intuition.

I consider two as dual to be a dyadic pattern. Such patterns have their origins in moments of life that break apart as well as in the bilateralisms of the body and in patterns such as spatial or rhythmic patterns. A dyadic pattern is not any particular spatial or rhythmic pattern nor is it any particular bilateralism. Rather, regardless of its original source, a dyadic pattern is a recognition template that can be instantiated, and further modified and generalized in its use. In this, the recognition template that constitutes the dyadic pattern can be used to recognize any pair of experiential items as co-occurring if they are experientially contiguous. When the dyadic pattern is activated, the child has a sense of simultaneity – of the copresence of a pair of perceptual unit items.

A dyadic pattern is not the numerical structure implied by the phrase “bare two-oneness.” But it is a composite structure. According to Menninger (1969), in two as a unity, “we experience the very essence of number more intensely than in other numbers, that essence being to bind many together into one, to equate plurality and unity” (p. 13). At the very core of the construction of number,⁸ then, we see the essentiality of the operation that binds many together into one – which is an experience of copresence. I call this operation the uniting operation. It is simply the unitizing operation applied to two or more items. In that I regard these items as being products of the unitizing operation, we see how two as unity is produced by the recursive use of the unitizing operation. In this, it is essential to understand that for the uniting operation to produce two as unity, the items being united are produced by a regeneration of fragments of experience and these regenerated items occur in visualized imagination.⁹ So, number is constructed through a coordination of the operations of unitizing and re-presentation (regeneration of a preceding item of experience).

Recursion and Splitting

The recursive use of unitizing in the construction of number is recapitulated in Confrey’s (1994) concept of splitting. Confrey (1994) defines the concept of unit “in any world as the invariant relationship between a successor and its predecessor; it is the repeated action” (p. 311). She goes on to elaborate, “From 1, with our first split, we create the unit of n and the first number in the sequence as n” (p. 312). This particular kind of unit produces the geometric sequence with constant multiplier n as opposed to the arithmetic sequence, with constant addend n. In either case, according to Confrey, the unit is a repeated action.

For our current purposes, I focus on how recursion is involved in Confrey’s concept of a split. In the most elementary case of a split, the breaking apart of two moments of life, I did not find it necessary to use recursion in our analysis of the copresence of the two moments. It was essential, however, for the experiencer to bring a preceding experience of self or thing into awareness in a current recognition episode. In the case of an intentional three-split, I do find it necessary to use recursion in explaining a sense of simultaneity of the results of the split. In the more general case where a unit of n is created, the splitting agent must begin with a unit and then fragment that unit item into n equal pieces. Creating n equal pieces itself involves recursive splitting actions, but Confrey goes beyond creating equal pieces and posits the creation of a unit of n.

She explicitly says that from 1 (which I take to indicate a unit item of some kind), “with our first split, we create the unit of n.” What this means to us is that the splitting agent must begin with a unit and then fragment that unit item into n equal pieces. But that is not sufficient, because “we create the unit of n.” Creating a unit of n is quite similar to the construction of the bare two-oneness as explained by Brouwer. The “falling apart of moments of life” can be thought of as analogous to starting with a unit and simultaneously breaking that unit into n parts if the necessity to make equal parts is relaxed.¹⁰ The difference is that in the construction of two-oneness, as observer I conceived of the moments sequentially as well as simultaneously. In Confrey’s analysis, there is no assumption of sequentiality. This is made possible, I believe, by the construction of splitting as an operation. If the splitting agent has already constructed what I call a partitioning structure, I indicate in a later chapter how such a structure could bring an intention of splitting forth without bringing a sense of sequentiality forth because sequentiality is symbolized. Being symbolized, however, does not eliminate sequentiality in splitting.

A similarity between the bare two-oneness and splitting is also present in the assumption that the parts produced by the split are united together into a composite unit. Further, to intentionally split the unit, 1, into n fragments, some composite unit structure has to be available to the acting agent. Early on, these composite unit structures may be dyadic, triadic, or quadriatic attentional patterns. Regardless of their nature, according to Confrey’s definition of a split, the n fragments are reunited into a composite unit of numerosity n. Thus, using the results of prior operating as input in further operating is constitutively involved in a split just as it is involved in Brouwer’s basal intuition of mathematics, albeit in a more advanced form. In our developmental analysis, the uniting operation would need to be present for Confrey’s notion of a split to make sense, and therefore, it, the split, should not be thought of as a “primitive operation.” Rather, it has to be categorized as a conceptual act.

Using the results of prior operating in further operating is even more dramatically involved in the second split. There has to be a new unit, not 1, but 1 reconstituted as a unit of n fragments which, from the observer’s perspective, could be recombined to form the original unity. In other words, a composite unit with n elements has been created as a result of the first split, which is now used as input for further operating. How this second split might occur highlights the essentiality of both fragmenting and segmenting in making a split.

Distribution and Simultaneity

We have to always operate on something, and this “something” in Confrey’s analysis of splitting is a unit of some kind. At the point of the second split, I find it necessary to introduce a new operation of distribution because it is quite unlikely that anyone can simultaneously split each of n things into n parts (Steffe 1994b, p. 21). Rather, the splitting agent would need to sequentially distribute the operation splitting by n, across the n elements produced by the first split rather than simultaneously split each of the n elements.¹¹ I argue that the distribution operation makes it possible to be aware of splitting each of n things into n fragments before the splitting action is implemented. This anticipation makes possible an awareness of simultaneously splitting all n elements. It also makes it possible to actually carry the operation out sequentially.

Splitting as a Recursive Operation

In Confrey’s analysis, I understand splitting as an operation. That is, Confrey’s definition of a unit as a repeated action involves input and output as well as mental action, which is to say that it fits well with our concept of an operation. In fact, Confrey’s idea of splitting includes our idea of a recursive operation, an operation that, in our model, yields number sequences as well as multiplicative operations. This helps to place our reorganization hypothesis on a firm conceptual foundation.

In any event, I need to reconcile Confrey’s notion of a repeated splitting action as being a unit, and our notion of a splitting action as an operation. Although the latter involves unitizing¹² in its construction, the constructive process must be distinguished from an awareness of an operation after it is constructed. von Glasersfeld (1995b) has made a similar point with respect to Confrey’s idea of a repeated splitting action as being a unit: “Operational awareness of carrying out a repetition is indispensable in generating pluralities, but it is not a requisite for the conceptual construction of units” (p. 120). In other words, children do engage in the conceptual construction of units without being aware of the involved unitizing operation. Operational awareness comes later and involves the ability to step out of the stream of direct experience, to re-present a chunk of it, and to look at it as though it were direct experience, while remaining aware of the fact that it is not (von Glasersfeld 1991, p. 47). To look at the re-presented chunk of experience as though it were direct experience involves taking what is being looked at as an experiential unit. So, operational awareness indicates that the operation has been grasped as a unit, or, in other words, the operation has become the focus of attention. In her statement, Confrey seemed reflectively aware of the operation of splitting and, in this, constituted it as a unit. But this does not mean that children cannot engage in splitting a unit into subunits without being aware of the splitting operations in which they engage.

Next Steps

The rationale for the reorganization hypothesis is still far from complete. Establishing that both simultaneity and sequentiality are involved in the construction of two as unity is a start, but it does not complete the argument for the number sequence. Following Menninger, who believed that two and three did not develop as part of a sequential order, I have argued that the initial experiences of three as trio does not include the experience of two as dual (Steffe 1988). The former experience excludes the latter in that three is not initially conceptualized as one more than two. Rather, three as trio is a triadic pattern in the same sense that two as dual is a dyadic pattern. I believe that the construction of the triadic¹³ pattern involves both a sense of simultaneity and of sequentiality and that three as unity involves a recursive use of the unitizing operation. With the possible exception of quadratic patterns, I am still to make an argument that both a sense of simultaneity and of sequentiality are involved in the construction of children’s number sequences. I am also yet to argue that children’s number sequences are relevant in the case of “continuous quantity” as well as in the case of “discrete quantity.” This is an important argument for us, because it is our goal that children, upon seeing, say, a blank stick, will regard the situation as a situation of their number sequence. That is, if fractional schemes are to be realized as reorganizations of children’s number sequences, then the latter must be used in situations that later will be regarded as fractional situations. Finally, I am yet to argue that partitioning or splitting operations and iterable units can be integrated into the same psychological structure. This argument is critical because it countermands the assumption that splitting and sequencing are built from distinctly different experiential foundations.