PSSM From a Constructivist Perspective

Leslie P. Steffe

Department of Mathematics Education

University of Georgia

Athens, GA 30602

This paper was written for the Research Symposium Linking Research and the New Early Childhood Mathematics Standards held at the 2000 NCTM Research Precession, Chicago, Illinois.

This paper was written as part of the Activities of the project Interdisciplinary Research on Number (IRON) of which NSF Project REC 9814853 is a part. All opinions are those of the author.

Reaffirmation that mathematics instructional programs should promote the learning of mathematics by all students is not only necessary, but it is critical in the mathematics education of children. It has widespread and pervasive implications for school mathematics that can be enacted only through what is considered as mathematics and mathematics learning. The disposition toward mathematics in PSSM is clearly stated in the discussion of The Mathematics Curriculum Principle.

It is customary for mathematics educators to speak in this way of mathematics as content for students to learn quite apart from the students who are to learn it. There are passages in the PSSM document, however, that recognize that young children think in ways that are different from that of an adult.

I find the first of three sentences in the above passage compatible with my concept of the mathematics of children. However, the last sentence seems to be added to reassure the reader that, although a child’s ways of knowing and communicating mathematically is different from that of an adult, these ways will be replaced by more conventional adult ways. In fact, it is recommended in the discussion of the mathematics curriculum principle that "students should understand and be fluent with whole-number computation by the end of grade 5" (p. 30). Apparently, permitting children at the pre-K-2 level to "develop strategies for computing that make sense to them and that are efficient and accurate." (p. 115) is done so the students will understand and be fluent with whole-number computation by the end of grade 5.

The Mathematics of Children

Recognizing that a child’s ways of knowing and communicating is different from that of an adult but yet holding fast to the proposition that whole number computation is a major goal of mathematics education in the elementary school raises the issue of what is to be regarded as the rational bridgehead of school mathematics. Shouldn’t we consider children as rational beings whose mathematical knowledge is every bit as legitimate as the mathematical knowledge of the adults who teach them? I don’t see how we can take the equity principle seriously unless we adults are willing to attribute mathematical concepts and operations to children that are as rational and as coherent as our own mathematical concepts and operations. A fundamental reason why "school mathematics has been viewed as a sorting machine, in which many students are considered unlikely to study higher mathematics and in which a few students are identified as capable of succeeding in the discipline of mathematics or in mathematically related fields of study." (PSSM, p. 24) is that mathematics has been attributed an objective, ontological existence apart from human beings. The belief is that "Mathematics is there. It’s beautiful. It’s this jewel we uncover" (Hoffman, 1987, p. 66). Surely, this realists’ view of mathematics has led to the unfortunate result of mathematics being only for the elite and mathematics being a classical, elitist subject. The concept that I would like to develop is mathematics of children because these are the concepts of mathematics that permit an equity stance. Above all, I want to avoid the trap of promoting an equity stance while at the same time maintaining a view of mathematics that has led to mathematics serving as a sorting machine.

I find the proposition compelling that children indeed have mathematical concepts and operations that are distinct from my own even though an attribution of such concepts and operations to a child is a conceptual construct. I find the proposition compelling for at least two reasons. Were I to deny that the mathematical concepts and operations of another person are distinct from my own, then the other person, not being unlike myself, could also claim that my mathematical concepts and operations were his or her own concoction. This claim is unacceptable to me because I am reluctant to admit that my mathematical concepts and operations are simply a concoction of another not unlike myself. In principle, my only choice is to attribute mathematical concepts and operations to the other that are indeed distinct from my own. This attribution orients us in mathematics education appropriately because, rather than being oriented toward teaching students objective, ontological mathematical concepts and operations, we are oriented toward exploring the mathematical concepts and operations of students. Simply put, as mathematics teachers, we are oriented toward learning the mathematical concepts and operations of our students and making them the rational bridgehead of school mathematics.

I use the phase "children’s mathematics" to refer to the mathematical concepts and operations I attribute to children. Such an attribution, however, is not helpful to a mathematics teacher without understanding those concepts and operations. Consequently, I introduce the phrase "mathematics of children" to refer to those models an adult makes to explain his or her observations of children’s mathematical language and actions. Mathematics of children is a construction of the observer, and it is a kind of mathematics that only children can teach us. That children have indeed taught me their version of mathematics through their language and actions, and that I have been able to formulate rational models of their language and actions constitutes the second reason that I find the proposition compelling.

Is the mathematics of children important mathematics? The three criteria given in PSSM for mathematics that is important is the place of the topic within mathematics itself, the role of the topic outside of mathematics, and the relevance of the topic for the learner (p. 28). While the mathematics of children certainly seems relevant for the learner, the two other criteria emanate from the way mathematics is viewed in PSSM. Should the aim of the teacher be seem as assisting students to learn the mathematical culture of adult mathematicians as specified in a mathematics curriculum? This view of the aim of the mathematics teacher emphasizes the mathematical knowledge of the teacher, but it makes no mention of the mathematical knowledge that the teacher can learn by interacting with children. The phrases "everyday knowledge" or "informal knowledge" do not respect children’s mathematics as legitimate and rational mathematical knowledge in the way that the mathematical knowledge of the teacher is viewed as legitimate and rational. Because it is different from that of an adult, I often see it referred to using the terms "misconception" or "misunderstanding". These terms, of course, arise in the context of adults interpreting children’s mathematics in terms of adult mathematical knowledge. The use of the terms do indicate that the adults who use them experience children’s mathematics as different from their own. The problem seems to be to learn how to take advantage of children’s mathematics and to make it the rational bridgehead of school mathematics.

Constructing a mathematics of children is especially crucial in that case where promoting the learning of mathematics by all students is taken seriously. Perhaps an example will best illustrate what I mean. In a course on children’s mathematical learning I offered in the Fall Semester of 1999, the students in the course spent one hour each of 10 weeks teaching a first grade child. My request to the Assistant Principal of the school that the students be permitted to teach first graders resulted in 31 of the 32 students being assigned a developmentally delayed child. The 31 students amounted to approximately 40% of the first-grade population of the school. Although this percentage of students might seem greater than one would expect, it is not at all out of the ordinary in the Southeastern region of the United States.

There are a host of questions that immediately arise concerning these developmentally delayed children. Among them are the following:

1. Is there a model to explain these children’s mathematics other than the phase "developmentally delayed" that would useful to a mathematics teacher?
1. What does it mean for teachers to have high expectations for these students (PSSM, p. 24)?

These 31 children clearly were at risk and they are precisely those students mentioned in PSSM for whom teachers have low expectations. But, are low expectations on the part of teachers wholly unwarranted? There are excellent reasons why the first-grade teachers of the cooperating school considered these children developmentally delayed and they were quite eager for the students in the course on children’s mathematical learning to teach these children one-on-one. Of course, I was on site coaching my students as they learned to communicate mathematically with these children, so I do have first-hand experience concerning the constraints these children presented.

In the four weeks prior to teaching the children, we discussed three basic counting schemes of young children, we watched and discussed video-tapes of children who exemplified these schemes, and we developed lateral and vertical goals along with situations of learning that the students might find useful when actually teaching the children. My goal here is rather modest in that I present these basic counting schemes as mathematics of children.

Counting as activity is a coordination of two productive activities; the production of a number word sequence and the production of a sequence of countable items. A counting scheme includes counting as activity. But it also includes a situation of counting, a goal relative to the situation of counting, and the results of counting.

Perceptual Collections. A situation of counting is construed to be the child’s situation and the results of counting is construed to be the child’s results. Appreciating situations of counting entails understanding the composite wholes children use to recognize an observer’s situation as a situation of their counting scheme. The composite wholes of immediate concern are perceptual collections, figurative collections, and numerical composites. Perceptual items like those that constitute a perceptual collection are constructed during the first two years of a child’s life. Piaget (1955/1934) demonstrated that a child can recognize a perceptual item long before it is aware of the item after it leaves the child’s visual field. The construction of a recognition template is essential in the construction of object concepts, but to say the child has constructed an object concept, the child must be able to use the recognition template in a regeneration of the item without otherwise experiencing the item. That is, the child must be able to produce a visualized image of the item, an ability which von Glasersfeld (1991) called a re-presentation of the item.

The houses on the hill or the boats in the harbor are both examples of perceptual collections. These examples imply that the child has already learned to categorize the items together to form collections. Classifying items together is based on experiencing a current item as having been experienced before. In such an experience, the involved recognition template is used not only in recognition, but also in regenerating an incipient awareness of an item that the child compares with the currently experienced item. This ability to compare an item experienced in the past with a currently experienced item is the basis of an awareness of more than one item and is fundamental in establishing co-occurring items. The co-occurrence of a pair of items may be sufficient for the child to unite the two items together into a pair of items, which is an act of classification. It is also the fundamental act which produces the material which will be used to construct the concept of two.

Uniting a pair of co-occurring items together is a special act of classifying the items together. The realization that a pair of items may co-occur without being united together leads to differentiating classification from the construction of number. We should not say that classifying items together is what produces numerical structure because the operation of uniting is involved as well as the operations of re-presentation and reprocessing. Nevertheless, classification produces operations which are essential in the construction of number. When comparing a pair of experientially co-occurring items, the child may abstract one or more properties of the items which the child then attributes to both of the items as well as to each item. These properties serve in forming a new recognition template that supersedes but not necessarily replaces the templates used in recognition. The new recognition template can then be used in scanning the items, which is another way of saying that the child runs through the items focusing attention on each item using the abstracted properties. These operations produce a pair of perceptual items that have been categorized together using a single template. But they produce more in that repetition is introduced into the recognition template. Repetition of the recognition template produces an awareness of more than one perceptual item, which is an awareness of perceptual plurality (von Glasersfeld, 1981).

An Awareness of Plurality. The houses on the hill or the boats in the harbor both afford the child an opportunity to construct a single recognition template that can be used in classifying the items together into perceptual collections. In the classification process, the child establishes an awareness of more than one perceptual item, or an awareness of plurality. An awareness of perceptual plurality is a quantitative property introduced into the perceptual collection by the acting child. In fact, it is one of the first quantitative properties and it is essential in the construction of a counting scheme as it serves the child in establishing a goal of counting. However, an awareness of perceptual plurality is to be distinguished from an awareness of numerosity, which is a quantitative property of a more abstracted structure than a perceptual plurality.

A perceptual collection is an experiential structure that exists for a child in the immediate here and now. How it might come to be established as a structure that is considered to exist beyond the immediate here and now involves the ability of the child to regenerate its experience of the collection in the absence of the items of the collection in the perceptual field of the child. We know that during the first two years of a child’s life, the child can use a recognition template in re-presenting a single item, which is essential in establishing the item as a permanent item (Piaget, 1955/34). However, using a recognition template more than once in re-presentation, which I consider as essential in the construction of numerical structure, does not follow immediately from the ability to use a recognition template in re-presentation (Steffe, 1994). In fact, I consider the production of a plurality of figurative items as a major step in the construction of numerical structure.

Unitizing Operation. When a child produces an image of a perceptual item, I call the image a figurative item to distinguish it from the perceptual item. There is a further distinction that I introduce between a perceptual item and the perceptual unit items that can be constructed in the context of counting. Perceptual items are produced by compounding sensory material from various sensory channels together, and a recognition template is simply the records of this operation. Compounding sensory material together is the beginning of the unitizing operation, an operation which continues to function throughout our lifetimes. In that case when a child uses a recognition template (which is a unitizing operation) to reprocess the perceptual items of a collection, the child may focus, not on particular sensory features of the perceptual items, but rather on the perceptual items as unitary things, thus essentially ignoring the sensory features of the perceptual items that were used in classifying them together. The criterion of classification can then change from particular recurrent sensory features to recurrent unit items. A result of this change is that classification based on recurrent sensory features is not a basic element in the construction of number. Rather, when a child can establish perceptual unit items, any particular perceptual items can be taken together as unit items to be counted. An example might be a small red triangular logic block, a ball, a pipe cleaner, a marble, a doll, and a bracelet. There still needs to be some perceptual items to use in making unit items, but what those perceptual items might be is not relevant for the classification. They are simply things that can be put together to form a collection of things.

Perceptual unit items certainly are established outside of the counting context. In the counting context, the child initially establishes a collection of items that it intends to count. Counting is an intentional activity whose goal is to make an awareness of more than one countable item definite. In that case where a child initially establishes a collection of perceptual items as countable, in an act of counting the child establishes the perceptual item being counted as a unit item and coordinates it with the utterance of a number word. The collection of counted perceptual items along with an awareness of definite plurality constitutes the results of counting.

The Perceptual and Figurative Stages of the Counting Scheme

Children who can count collections of perceptual items but who also require such collections to be in their perceptual field in order to count are called counters of perceptual unit items. Of the 31 children I mentioned above, at least 25 of them were counters of perceptual unit items. These children were in the perceptual stage of their counting scheme and the others were in the figurative stage. It is easy to distinguish between children in these two stages because the children in the figurative stage can count items that are not in their immediate perceptual field. That is, these children can establish collections of figurative items prior to engaging in the activity of counting them. When counting a collection of perceptual items hidden from view, some children just entering the figurative stage point to or otherwise indicate places on the screening device behind or under which they think there are perceptual items, and coordinate the pointing act with the utterances of number words. They in fact create figurative unit items in the acts of counting when they indeed intend to count the items of the hidden perceptual collection. The items they intend to count and the items they actually count do differ. I refer to these children as counters of figurative unit items.

My sense is that counters of figurative unit items use their recognition template to produce a pattern of two or three figurative items prior to counting. So if they are told that there are, say, eight items hidden prior to counting, they can begin to count the items of the pattern. In this case, they are indeed aware that there are items hidden other than the items of the figurative pattern if for no other reason than it is their goal to count to "eight". Counting propels them forward and they continue to create figurative unit items in the activity of counting. They are usually consumed by the counting activity and lose track of their goal in counting. This is indicated by their stopping fortuitously, usually when reaching a boundary of the screening device.

The re-presentation of a perceptual collection is a dynamic operation and the child is considered as in the activity. There is a result of the activity, such as a pattern of figurative items, but one should not assume that the child can take the result as input for further operating. That is, one should not assume that the child sets the result at a distance and treat it as if it were the perceptual collection while remaining aware of the fact that it is not. To do this requires an additional operation. If the child reprocesses the results of re-presenting a perceptual collection by unitizing the figurative items using its recognition template, this creates a sequence of figurative unit items prior to counting. This frees the child from creating figurative unit items in the activity of counting. The results are easy to observe because the child creates substitute countable items in the activity of counting such as the motoric acts of repeatedly pointing with a finger or sequentially putting up fingers. These children always start to count from "one", and they still need to count to establish a collection of counted items. Counting is still a sensory-motor activity that children cannot take as a given. They necessarily count to establish meaning for number words beyond the range of their figurative patterns.

I consider the perceptual and figurative stages of the children’s counting schemes as prenumerical stages. These children do produce unit items, but the unit items are sensory-motor or figurative unit items rather than abstract or arithmetical unit items. The children are yet to construct a number sequence although they do produce sequences of counted items as the results of using their counting schemes. However, these sequences are experiential sequences that exist for the child in the immediate here and now and have no permanence. The child by necessity has to produce an experiential sequence or there is literally no sequence of counted items that exists for the child. The situation is similar to the earlier time in the child’s life when the child had established recognition templates but was yet to use those templates in regenerating experiences of the items in their immediate absence. That is, the child can use its counting scheme to actually count just as it could earlier use its recognition template to establish experiential items. Using the counting scheme in a regeneration of counting is yet to be established by the child in the way it has already established using recognition templates in re-presenting images of perceptual items.

Numerical Counting Schemes

The major advance the child makes in constructing the initial number sequence is that the child can use its counting scheme to generate (as well as regenerate) an experience of counting without actually counting. That is, the items of the sequence of countable items the child establishes prior to counting already contain records of counting acts, and, thus, the sequence of countable items is also a sequence of counted items. This may seem puzzling at first sight if one imagines the countable items as a collection of perceptual items. However, if one imagines the collection of perceptual items as screened from view and another such screened collection, the situation differs. Imagine also that the child is told how many items are behind each screen and then asked to find how many items are behind both screens. The child who has constructed the initial number sequence can generate an experience of counting the items behind one of the two screens without actually counting, and then extend counting beyond this imagined experience when counting the items behind both screens. There is still a certain restriction in how the child keeps track of the counting acts in the extension of counting in that the child stops when recognizing a numerical pattern. The child is yet to learn to explicitly count the counting acts of the extension. Nevertheless, a major advancement has been made in the child’s counting scheme in that the scheme has been constituted as a number sequence.

One might feel compelled to ask how the child learns to regenerate an experience of counting without actually counting because, prior to this event, an experiential sequence of counted items that was produced by the sensory-motor activity of counting constituted the child’s "number sequence". When considering that, in principle, the child can regenerate the experience of counting up to any number word, it is not at all easy to explain such learning. But monitoring counting is the key. Monitoring counting is the accommodation which engenders the reorganization of the figurative counting scheme, but it is insufficient to explain the reorganization (Steffe, 1991). Monitoring counting--which can be thought of as keeping track of the counting acts in an extension of counting--has to be learned, and learning it involves the coordination of two operations. First, the child regenerates the counting acts of an extension of counting, and, second, the child unitizes each generated counting act, thus producing an arithmetical unit item. The result of these two operations is a sequence of arithmetical unit items that contain records of the re-presented counting acts. Because they contain records of counting acts, I have referred to them as arithmetical units.

Monitoring counting involves only a small number of counting acts. For example, I observed a child, Tyrone, learn to monitor his counting acts "8, 9, 10, 11, 12" (Steffe, 1991). However, it wasn’t until some time later that he reorganized his counting scheme into an initial number sequence. He had learned to regulate his counting acts by monitoring counting in specific uses of his counting scheme, but it was autoregulation of the monitoring activity that produced the reorganization.

Only one child of the 32 six-year-old children that my students worked with could indeed count-on and he could count-on before the student to whom he was assigned started to work with him. The 31 other children did not learn to count on over the ten week period that the students worked with them, and it was not at all apparent to me that they would learn to count-on in the near future. So, although I am in great sympathy with the equity principle that "Mathematics instructional programs should promote the learning of mathematics by all students" (PSSM, p. 23), it is also obvious to me that major modifications in our conception of what constitutes mathematics and mathematics learning are necessary.

When regarding school mathematics as consisting of important mathematics like place value, function, scaling and similarity, and structure of the number system, we suppress important mathematics-of-children concepts such as the perceptual counting scheme, the figurative counting scheme, and the initial number sequence. Place value, for example, is regarded in PSSM as a teachable mathematical concept much like van Oers (1996) regards the cultural practice of mathematics. According to van Oers (1996), the cultural practice of mathematics "can be transformed into curriculum content and, as such, it can be taught" (p. 94). Personal meanings that students may attach to the "actions, rules, methods, and values as provided by a school subject" are a constitutive part of van Oers’ basis for instructional practice in mathematics education, but the personal meanings are left unspecified because they are relative to the cultural practices of mathematics.

Counting-On as a Non-Teachable Scheme

Unlike van Oers concept of the cultural practice of mathematics, the mathematics of children emerges from within children and it cannot be transferred from the heads of teachers to the heads of children by means of the words of the language. A very experienced mathematics educator for whom I have great respect once asked me if I thought counting-on could be taught. Should it be considered as a part of the mathematics curriculum? I was apprehensive then and I still am that if we agree that counting-on is an important kind of counting scheme (the initial number sequence), teachers will consider it as teachable in the sense that van Oers considers the cultural practice of mathematics as teachable. For children who have constructed the initial number sequence, counting-on does not need to be taught, and for children in the stage of the figurative or perceptual counting scheme, it should not be a goal of the teacher to teach the children to count-on. In other words, it should never be a goal of a teacher to teach children to count-on unless there is good reason for the teacher to believe that her or his students can easily curtail counting. Even in this case, I would prefer that the teacher not demonstrate to the children how to count-on. Rather, I would prefer that the teacher design a sequence of situations of learning that affords the children an opportunity to curtail counting on their own.

The concept of excellence that is promoted in PSSM should be interpreted relative to the mathematics of children. What constitutes excellence for children in the perceptual stage of the counting scheme differs from what constitutes excellence for children in the figurative stage, and both differ from what constitutes excellence for children in the stage of the initial number sequence. The common practice of creating one curriculum for all children in mathematics education should be abandoned just as we should abandon the concept of tracking in school mathematics. Both of these concepts of curriculum are based on conventional mathematical concepts and if the mathematics of children is given any consideration at all, it is regarded as those personal meanings that students may attach to the "actions, rules, methods, and values as provided by a school subject".

The mathematics of children is a legitimate mathematics that should be taken very seriously by mathematics educators. It certainly does not stop with counting-on (the initial number sequence). In fact, the construction of the initial number sequence is a starting point in children’s mathematical constructions rather than an end point. Nevertheless, it opens possibilities which are not available to children in the figurative stage of the counting scheme. The most basic advance is that children no longer need to count to establish meaning for number words. Rather, a number word refers to a numerical composite which is a sequence of arithmetical units. For example, a number word such as "eight" refers to a sequence of arithmetical units that contain records of counting acts from "one" up to and including "eight". The number word is yet to refer to a composite unit containing the sequence--just to the sequence. To understand why, it is sufficient to consider the process that produced the sequence--the monitoring of an extension of counting. In this operation, the child focuses on the re-presented counting acts and takes each one as a unit item in reprocessing them using the unitizing operations that its recognition template constitutes.

Early Childhood Mathematics Scope and Sequence

I now turn to a more intensive investigation of the implications of considering the mathematics of children as the rational bridgehead of school mathematics by comparing and contrasting what I have said with an early version of the Early Childhood Mathematics Scope and Sequence (Clements & Sarma, Electronic Copy). I am engaging in this activity for a purpose other than critiquing the particular mathematics scope and sequence, as I understand that this scope and sequence is still in preparation. My purpose is to illustrate how our language as well as our thinking changes when making the mathematics of children of central concern in school mathematics. In making the comparison and contrast, the reader need not be familiar with the particulars in the scope and sequence as I will try to say enough to make my comments comprehensible.

I start with Number and Operation, the first major heading of what constitutes important mathematics. I hasten to say that within my understanding of children’s mathematics, my interpretation of this topic indeed constitutes "important mathematics" in the mathematics of children. The first subheading under this main heading has to do with Number Meanings and Systems, and under that heading the overall learning objective is for children to count with understanding and recognize "how many" in sets of objects. It might seem pedantic, but I do not use the phrase "sets of objects" when speaking using the language of the mathematics of children, because "sets of objects" can be interpreted to imply that a Boolean Algebra is operative in the mind of the knowing agent. Rather, I use a more differentiated way of speaking that respects what children are able to establish as "sets of objects"--perceptual collections, figurative collections, or numerical composites. Moreover, "counting with understanding" differs dramatically across the three stages of the counting scheme. For children in the Perceptual Stage, an awareness of perceptual plurality is fundamental in establishing the goal of counting perceptual unit items. The child is aware of more than one perceptual item, and its goal for counting is to make the indefiniteness of more than one perceptual item definite by counting. So, in this case, I interpret "counting with understanding" as first establishing this goal for counting and then count to satisfy the goal.

For children in the Figurative Stage of the counting scheme, the need to make an awareness of more than one figurative item definite is what drives counting. Although it is possible to differentiate an awareness of figurative plurality even further according to whether a child is a counter of figurative unit items or a counter of motor unit items, I refrain from doing this as the distinction between an awareness of more than one perceptual item and of more than one figurative item is sufficient for most purposes. So, "counting with understanding" here means to first establish the goal of making an awareness of figurative plurality definite and then count to satisfy the goal.

For children in the stage of the Initial Number Sequence (which is a stage of the counting scheme), the goal of the child in counting can be similar to the goals of the children in the two preceding stages. A child who has constructed the initial number sequence never loses his or her ability to count perceptual or figurative unit items, but these items are not constituted in the same way as they are constituted by children in the preceding two stages. The reason is that the object concepts of the children in the stage of the initial number sequence have been (or can be) constituted at the same level as the child’s arithmetical units. I call these object concepts abstract units to differentiate them from perceptual unit items and figurative unit items. Abstract units are constructed by the same process that the child uses to construct arithmetical units--he or she reprocesses re-presented figurative unit items using the recognition template that was used to make the figurative unit items. When the child’s object concept, dog, for example, is constituted as an abstract unit, the child can willfully use it to produce an image of more than one dog. In fact, the child can produce an image of many dogs. Although these images are figurative items, they can be very minimal images. It is important to note here that I do not consider an abstract unit (or concept) to be used without producing some perhaps minimal image to make the abstract unit tangible. Although it may seem confusing to insist that the child needs to make an abstract concept tangible by producing an image, I maintain that mathematical thought always involves images of some kind. The essential difference between perceptual unit items, figurative unit items, and abstract unit items consists in the level of interiorization of the items.

Borrowing a term from von Glasersfeld (1981), I call a sequence of abstract unit items an arithmetical lot to distinguish the composite structure these children can produce from the composite structures already designated as perceptual collections and figurative collections. Arithmetical lots are established at the same level as numerical composites and are essentially uncounted numerical composites. One way to make a distinction between arithmetical lots and numerical composites in terms of conventional mathematics is to say that arithmetical lots are the beginnings of set theory and numerical composites are the beginnings of number theory.

In the case of an arithmetical lot, as I have already indicated, the child can form an image of a plurality of figurative unit items that are countable, and the child is aware of more than one of them. This differs from an awareness of figurative plurality in that the figurative unit items are produced by means of activated abstract units and thus can consist of the most minimal of figurative material. Any figurative material will suffice for the establishment of countable items. I call the awareness of the results of counting these countable items an awareness of figurative numerosity to emphasize an awareness of the figurative items as countable items as well as the awareness of the results of counting them, which is a numerosity. So, "counting with understanding" in the present case means to make an awareness of figurative numerosity definite.

"Verbal Counting"

Under the main objective of to count with understanding and recognize "how many" in sets of objects, there are 25 subobjectives. Five of these concern what is called "verbal counting". For children in the perceptual and figurative stages of counting, I interpret this phrase to mean saying number words in sequence, which is not counting at all. It is possible for children who are in the most advanced subperiod of the figurative stage to count by saying a sequence of number words. These children are counters of verbal unit items. They always start to count from "one", but saying a sequence of number words can constitute counting because they have abstracted the sequence of number words from a counting sequence where they established motoric unit items as countable items in the activity of counting. In the case of the initial number sequence, children can and do say sequences of number words when it is their goal to make their awareness of figurative numerosity definite.

It is definitely important for children to learn to extend their standard number word sequences to one thousand. It always appalls me when I encounter children who have constructed the initial number sequence but who cannot experience the fascination of what to them are large numbers because they do not know the appropriate language. But for children in the perceptual and figurative stages of their counting scheme, it is not a realistic to expect them to establish a number word sequence to "one thousand". For these children, I advocate that they learn to extend their number word sequences as far as possible by means of acoustic counting, a concept introduced by van den Brink (1991). Acoustic counting is not counting at all, but it is saying number words in rhythmic auditory patterns consisting of two or three number words in each pattern. It is far easier for children to string patterns of three number words together than it is individual number words.

For children in the stage of the initial number sequence, I advocate that these children learn to extend their verbal number sequence in the context of purposeful counting. Estimating the numerosity of collections of perceptual items and then counting the items to check the estimate is one context. Another context that should be highlighted is where the children use their counting schemes to solve what to an adult is an addition situation; for example, "How many is seven more than ninety-eight?" Counting "ninety-nine, one hundred, one hundred-one, one hundred-two, one hundred-three, one hundred-four, one hundred-five" can be a breakthrough for a child if the child abstracts juxtaposing saying "one hundred" with saying the verbal number sequence from "one" through "99".

Up to this point, I haven’t explained the production of the verbal number sequence. When children who have constructed the initial number sequence generate an experience of counting by re-presenting the auditory records in their arithmetical units, I refer to this image as a verbal number sequence. It is an important sequence for many reasons and it is what I mean be verbal counting when children use it to count. It is this number sequence which should be extended to "one thousand". But it also serves children in many other ways some of which I discuss later in the paper.

In this same set of 25 objectives, there are six which deal with extensions or declensions of counting. One of these is simply worded "count on; just before or just after". I have already expressed my apprehension that counting-on could be regarded as teachable in the sense that van Oers considers the cultural practice of mathematics teachable. Let me repeat again that for children who have constructed the initial number sequence, counting-on does not need to be taught, and for children in the stage of the figurative or perceptual counting scheme, it should not be a goal of the teacher to teach the children to count-on. With this admonition, I did insert the caveat that it should never be a goal of a teacher to teach children to count-on unless there is good reason for the teacher to believe that her or his students can easily curtail counting.

The relations "just before" and "just after" are, for children in the perceptual and figurative stages, relations that pertain to their number word sequences or counting sequences and these children can learn to say the number word just after or just before a given number word. But this learning should be differentiated from the construction of the initial number sequence, and it should not be used as a means to train children to count on. Counting on is learned by a process of autoregulation of monitoring and it should not be regarded simply as a procedural aspect of counting that can be taught like extending a number word sequence can be taught.

Counting backwards from 10 to 1 is also cited as a learning objective. This can be learned as a modification of counting forward from "one" to "ten" by children in the figurative stage of the counting scheme. When the a number word just before a given number word is not known, if the child spontaneously drops back to "one" and counts forward to generate the desired number word, then it is well worth encouraging such children to count rows of perceptual items with two through ten items per row both forward and backward. It is worth it because it provides the child with a sense of accomplishment and because counting backward is important in the construction of subtracting schemes after the child becomes numerical. However, it is not ethical to attempt to train a child to count backwards from 10 to 1 just as it is not ethical to train them to count on. They must be able to generate the backward counting sequence on their own, and, above all, they must want to do it. Seeing other children count backwards may be sufficient to generate such a need.

Basing school mathematics on the mathematics of children definitely changes what is traditionally considered as a mathematics scope and sequence. In the six objectives concerned with extensions and declensions of counting, there are two which concern keeping track of the counting acts of an extension or a declension. Children in the stage of the initial number sequence can indeed keep track of the counting acts of an extension or a declension of counting, but I have already pointed out their limitations. These children cannot yet double count, or count their counts either forward or backward. But they can monitor their counting acts forward or backward using numerical patterns. A numerical pattern can be a spatial, auditory, or rhythmic pattern consisting of up to as many as seven elements in some cases. In the case of finger patterns, I have witnessed children make numerical patterns of up to twenty items (four open hands). What I call sophisticated finger patterns are also made by some children in that, say, three fingers can refer to either three or thirteen. If the child arranges their spatial patterns linearly in an extension or declension of counting, this is a good indication that these patterns are indeed numerical patterns. Rhythmic patterns are usually produced as a rhythmic motion of some kind, and auditory patterns are usually produced as a sequence of number words. Numerical patterns have their origins in monitoring counting, and through the process of autoregulation of monitoring counting, numerical composites emerge beyond the range of the patterns used in monitoring.

The key in understanding the difference in the initial number sequence and the tacitly nested number sequence, that next number sequence in the vertical learning of the child, is that the initial number sequence cannot be taken as its own input for making countable items. In the case of the tacitly nested number sequence, the child can take a sequence of counting acts as unit items to count. The tacitly nested number sequence is a recursive counting scheme. It is as if the child has two number sequences, one to use as material to make countable items, and the other as the counting operations to count these countable items, which is what I refer to as double counting.

Just as counting on should not be regarded as teachable, neither should double counting. Children who have already constructed the tacitly nested number sequence can double count, and it should not be a goal of the teacher to teach children who have constructed only the initial number sequence to double count. Double counting is not a procedural aspect of counting even though one observes children double count. But again, I introduce the caveat that a teacher may have good reason to believe that a child is on the verge of double counting and thereby may create a sequence of learning situations which may afford the child the opportunity to construct double counting on his or her own.

One reason a teacher might believe that a child is on the verge of constructing double counting is it may be apparent that the child’s method of keeping track of counting carries the force of double counting. This may be especially apparent if the child can find how many of a collection of perceptual items are hidden from view. As an example, say that a child counts the items of a perceptual collection and finds that there are sixteen items in the collection. If some of the items are then hidden and if the child can find how many are hidden by first counting the visible items and then continuing to count up to "sixteen," where a counting act consists of uttering a number word and putting up a finger, this is an indication that keeping track of counting in the extension of counting the visible items carries the force of double counting. Keeping track of counting in this way carries the force of double counting because the child generates an experience of continuing to count beyond counting the visible items before actually making the extension of counting. In the extension of counting, the child does indeed take the counting scheme as its own input because what the child counts are not images of perceptual items, but rather images of counting acts. Had the child been simply told that there are sixteen items in the collection rather than first count the items, it would be more compelling that the child’s counting acts carried the force of double counting. The reason is that there would be no recent results of counting that the child could re-present. The child would need to generate rather than regenerate the counting acts referred to by "sixteen".

Adding and Subtracting Schemes

I now turn to the learning objective Operation Concepts: Addition & Subtraction. The first objective under this main objective is to understand the meaning of these operations and how they relate to one another. So, rather than discuss each of the 22 objectives following, I will interpret this first objective with respect to the mathematics of children.

In the perceptual and figurative stages of the counting scheme, we have to consider the whole of these counting schemes in order to formulate a model of children’s addition in these two stages. In the perceptual stage, it is important for the involved collections of perceptual items to have category names, like marbles or toys, so that an adding language may be developed along with the particular actions. The situations that can be imagined are inexhaustible, so I will present only example situations and leave it to the reader to contextualize them. At the most basic level, I involve children in the perceptual stage in sensory-motor action. For example, I might ask one child count out four toys and another three toys. I might then ask each child how many toys the other child counted out, and then ask each to find how many toys would be in a box if they put all of the toys into the box.

Asking the children to find how many toys the other child counted out is done to encourage decentering and to take another child’s results of counting into consideration. Asking how many toys would be in a box if they put all of the toys into the box is done to encourage the children to reprocess the toys in the two physically separated collections using their recognition template, toy, and take them as perceptual unit items to be counted. Reprocessing the toys is essentially a process of reclassification of the toys into a new collection of perceptual unit items. Reprocessing the perceptual items of two separate collections to form a single collection of perceptual unit items is the basic meaning of an addition situation in this stage. It is a conceptual joining action that in my opinion is far more important than generating an imagined action of physically placing the toys together.

The way in which the questions of the sequence of questions are asked does encourage counting at each step. As each child counts out four or three toys, as the case may be, counted collections of perceptual unit items are produced assuming that the children do indeed establish perceptual unit items as countable items in counting. Then, asking each child to find how many toys the other child counted out again encourages establishing a second counted collection of perceptual unit items. Finally, asking the last question of the sequence of three questions encourages the children to reprocess collections of counted perceptual unit items. This is an important point because the children conceptually join four counted perceptual items and three counted perceptual unit items.

It is not unusual to find children in the perceptual stage of the counting scheme who simply count their own collection of perceptual items when asked to find how many toys there would be in a box if both children put all of their toys into the box. In this case, I encourage these children to work alone and actually enact putting the two collections of toys they count out into a box and then count the toys in the box. These children usually establish no relationship between the counted collection of toys in the box and the two counted collections of toys prior to placing them into the box. No claim of an adding scheme on the part of these children should be made.

In the figurative stage of the counting scheme, the goal is to encourage children to reprocess the items of two collections of counted figurative unit items in categorizing the items of the two collections together into a single collection, and then count the items of the single collection. An example might be where a child counts out seven discs and places them into an opaque container, counts out five more discs and places them into the container, and then counts all of the discs in the container without looking at them. The child might touch the container seven times at specific places in synchrony with uttering the number words "one, two, ..., seven", and then continue on counting, touching the container five more times in synchrony with uttering the number words "eight, nine, ten, eleven, twelve". The child knows to stop when recognizing a spatial pattern for "five" formed by the locations of the points of contact with the container.

The adding scheme of children in the stage of the Initial Number Sequence involves numerical composites or arithmetical lots rather than collections of perceptual or figurative unit items. The difference is relatively easy to observe. Given, for example, the following situation:

If the child says there are seven marbles in the cup and then proceeds to count the additional marbles, "8, 9, 10, 11--eleven!", it suggests that in uttering "seven" the child knows that the number word, in the given context, stands for a specific collection of individual perceptual unit items that satisfy the template "marble" and that, if counted, they could be coordinated with utterances of the number words from "one" to "seven". The child knows this and therefore does not have to run through the counting activity that would actually implement it.

But there is more going on in the child’s adding scheme than its meaning of "seven". First, the child’s meaning of "four" is quite similar to his or her meaning of "seven". Moreover, the template "marble" is an interiorized template, so the particular images of marbles that the child may establish are instantiations of this abstract unit item. This is important because the child made and maintained a separation between the seven marbles that were originally in the container and the four that were placed with them as indicated by how the child counted. Children in the figurative stage can also make and maintain such a separation, but they also start counting from "one" whereas the child in the current case started counting from "seven". To explain the difference, I assume the child established two numerical composites corresponding to "seven" and "four" prior to counting, and then recategorized them into a numerical composite of unknown numerosity by reprocessing them using its "marble" template. I consider the structure created using the two sequences of units, seven and four, as a sequence of unit items that satisfies the template "marble," with a separation between the last unit of the first sequence and the first unit of the last sequence. Another way of saying this is that the two sequences, seven and four, were juxtaposed into one sequence of unknown numerosity as a result of reprocessing the two sequences. The reader might wonder why I didn’t simply model this as the child taking the two sequences as unitary wholes and moving them together in juxtaposition, as is more conventionally assumed for addition. The reason is that the child is yet to construct the sequences a unitary items--as composite units. In the current stage, "seven" refers to seven individual unit items in sequence but not to a composite unit containing this sequence.

The goal of the child in counting "8, 9, 10, 11--eleven!" is to specify the numerosity of the numerical composite established using the two numerical composites seven and four. It is in this sense that this counting scheme is to be regarded a child-generated addition algorithm that I refer to as counting-on with numerical extension.. A similar claim concerning the "child generatedness" of the adding schemes of children in the two preceding two stages can be also made. In the case of the perceptual stage, I refer to the adding scheme as counting-all with simple extension, and the adding scheme in the figurative stage as counting-all with intuitive extension. I use the "counting-all" to refer to the child counting all of the unit items in the joined collection from one. A simple extension is an extension of counting where the child counts perceptual unit items, and an intuitive extension is an extension of counting where the child uses the items of a figurative pattern to keep track of counting.

Adding Schemes and the Learning Objectives.

Only three of the 21 learning objectives stated for addition and subtraction are appropriate for children who are restricted to counting-all with simple or intuitive extension and these three would need to be modified to remove reference to subtraction and any reference to numbers. Clearly, much more attention needs to be given to developing learning objectives for these children if we are to promote equity and excellence in their mathematics education.

For those children who are restricted to counting-on with numerical extension, there are only two objectives which are clearly appropriate and pieces of two others. This may come as a shock, but these children have not yet constructed the necessary operations to establish subtracting as the inversion of adding nor can they investigate the commutativity of addition or examine the associativity of addition. These children would be put at a clear disadvantage given the stated learning objectives, and mathematics would continue to be a sorting machine for these children very early on in their school life. Appropriate learning objectives need to be written for these children as well as for the children in the preceding two stages that can be placed in their zones of potential construction.

Before moving on to analyze some of the other objectives, it is worth noting that the learning objective of adding and subtracting numbers to 10 with objects may not be appropriate for children who can count-on with numerical extension, because they may revert back to counting-all with simple or intuitive extension rather than use their most advanced adding scheme. It is not unusual for children to revert back to a more primitive way of proceeding if they believe that the teacher prefers that particular way. A great deal of sensitivity must be exercised by the teacher in bringing forth and sustaining children’s most sophisticated ways and means of operating. I do advocate that children who can count-on with numerical extension should engage in "reality checks" by counting the perceptual items to which their number words refer, but to encourage children to count perceptual unit items as a method of solution when they can operate in a more abstract way may exclude the more abstract way of operating and the productive learning it engenders.

Subtracting Scheme in the Stage of the Initial Number Sequence: It’s Purposes.

There are no compelling arguments that I know for attempting to bring forth a counting all scheme for subtracting in the case of children in the perceptual and figurative stages of their counting schemes. For those children who can count-on with numerical extension, however, there are good reasons for developing very particular learning situations involving what is commonly known as "take away" subtraction. An example situation is to ask the child to place, say, fifteen blocks into an opaque container and then to take out three of them and put them into another opaque container so they can’t be seen. The child is to find how many blocks remain in the first container. In presenting this situation, I assume that the child can count backward from at least "twenty".

The goals in presenting the situation are two-fold. The first goal is for the child to re-present the forward counting acts to "fifteen" and use these counting acts starting from "fifteen" as material for making countable items in counting backward: "fifteen, fourteen, thirteen". This is done to encourage the child to take the initial number sequence as its own input material, which is what is necessary for the children to construct the next number sequence after the initial number sequence--the tacitly nested number sequence. The second goal is to encourage the children to unite the three counting acts together into a composite unit after they reach "thirteen is three". If the child has the goal to find how many blocks are left in the container, the child could stop at "thirteen," "step out" of the sensory experience of counting, and "look at" what he or she has just done. By "looking at" what he or she has just done, the child may very well set the three counting acts at a distance and take them as one thing. The uniting operation has to be constructed just as any other recognition template has to be constructed. The process is quite similar, but instead of compounding sensory material from different sensory channels into a unitary thing, the child compounds the trio of re-presented counting acts into a unitary thing.

If the child does in fact unite a re-presentation of "fifteen, fourteen, thirteen" together into a unit, this moves the child to a plane above counting. The child is now "at" thirteen looking backward toward one. There are three possibilities now for how the child operates and I have observed all of them. First, the child may re-present the counting acts backward from "twelve’ to "one" and then change their direction from backward to forward counting acts. In this case, the child simply says "twelve" to indicate how many blocks are left. This bidirectionality of counting indicates that the child takes the counting acts as material of the uniting operation and unites them together into a composite unit because then the child would be "above" the counting acts and operating on them. The child knows that the counting acts from "twelve" down to "one" are the self same counting acts as counting from "one" up to "twelve," but in a different direction.

Second, the child may continue to count from "twelve" down to "one". Here, there are two possibilities. The first is where the child may simply utter the number words without keeping track. This is important, because the child will not have reached its goal after counting. Consequently, the child may independently reinitialize counting, this time keeping track by sequentially putting up fingers in synchrony with uttering the number word sequence. The second is where the child does keep track of its counting acts by putting up fingers upon initializing counting starting from "twelve’. In both cases the child monitors counting and thereby takes each counting act as a unit item. This is what I call reinteriorizing counting acts and it leads to the next number sequence. After the child completes its counting activity, there is a distinct possibility that the child will unite the records of counting into a composite unit.

Third, the child may introduce a novelty--double counting. That is, the child might count "twelve is one, eleven is two, ..., one is twelve. Twelve!" By saying "twelve", the child indicates that it unites the counting acts into a composite unit. So, introducing "take away" subtraction to children in this stage in the way I have explained is done not to teach subtraction, but rather for the purpose of the children making vertical progress in the construction of their number sequence.

Additive situations where the numerosity of the numerical extension is beyond the range of the child’s numerical patterns also can be used to encourage the construction of the next number sequence. In sum, the main goal of the teacher for children whose adding scheme is counting-on with numerical extension should be for the children to make vertical progress to the next number sequence. Of course, there are many subsidiary goals that I refer to as lateral learning goals that can be elaborated in a mathematical scope and sequence which do entail important mathematics for these children to learn which is within their learning level.

I turn now to the very important objectives of children constructing subtracting as the inversion of adding. There is an objective, however, which precedes these objectives and it reads as follows: "Construct and solve open sentences that have variables (e.g., c + 7 = 10; a + b = 10)". At the outset, this objective should be reformulated to place it in contexts other than open sentences so that the situations of a child’s adding scheme become as broad as possible. In any event, I have already commented on the fact that children who have constructed the tacitly nested number sequence can indeed find how many items of a collection of items of known numerosity are hidden from view because they can take their number sequence as its own input. In other words, when these children establish a situation like that in Figure 1, they do not simply generate images of the hidden items.

Figure 1: Seven of sixteen items hidden.

Rather, they generate an image of their verbal number sequence from "nine" up to "sixteen" and take the items of this verbal number sequence as countable items in the activity of counting. They either explicitly double count or their counting acts carry the force of double counting. At this point in their mathematical development, I can find no compelling reason to ask these children to symbolize the situation using conventional notation like "9 + ___ = 16" and then to find what goes in the blank by counting from "nine" up to "sixteen." Their natural language serves as their primary symbol system and it is the means whereby children maintain the spontaneity of their spontaneous development.

The scheme that I discussed with respect to Figure 1 is the counting-up-to scheme. Like counting-on with numerical extension, it is indeed a child-generated algorithm. I do not refer to it as a subtraction algorithm because there is no sense on the part of the child of finding the difference of sixteen and nine, nor is there any sense of taking nine away from sixteen. Rather, after counting the nine visible items, the goal of the child is to find the numerosity of the counting acts from "nine" up to "sixteen" by counting them. I also refrain from calling it an addition algorithm because the child separates the first nine of the sixteen counting acts from those remaining.

Counting-up-to does not necessarily indicate that the child disembeds the remainder of the counting acts past "nine" from the whole of the sequence of counting acts from one to sixteen. And this is why I referred to the number sequence of the child who solves the task of Figure 1 as a novelty as tacitly nested. The child can make a distinction between the two parts of the number sequence up to sixteen, but it leaves them embedded in sixteen. That is, the child can make two composite units, one containing the first nine elements of the number sequence and the other containing the remainder of these elements in the number sequence to sixteen. Further, the child can use "nine" to symbolize the elements of the first part. This is an important achievement because the child considers the first nine elements as counted items and the remainder as countable items. In fact, the elements in the extension of counting beyond "nine" serve a dual role. First, each serves as the next countable item and thus belongs to the second part of sixteen. And, second, each counting act serves as a counted item and thus belongs to the first part of the counting acts to sixteen. That is, the child establishes an expanding sequence of counted items and a corresponding contracting sequence of countable items.

But the child is yet to disembed the composite unit containing the remainder of the first nine elements of the number sequence to sixteen from the whole number sequence without destroying the sequence. When this operation emerges, I refer to the child’s number sequence as explicitly nested. An analogy to the disembedding operation in conventional mathematics is the relation of a subset to a set which includes it. A subset may be disembedded from a set without destroying the set. The subset is in fact a set and it is a subset only with respect to a set in which it is included.

There is a second feature of the explicitly nested number sequence which, when coupled with the disembedding operation, produces great economy in the child’s reasoning. In the case of the tacitly nested number sequence, a number word like "seven" refers to a composite unit containing a sequence of seven unit items, which I symbolize using conventional notation for the purposes of illustration: {1, 2, 3, 4, 5, 6, 7}. In the case of the explicitly nested number sequence, "seven" refers to a composite unit containing a singleton unit that can be iterated seven times to fill out the composite unit. This structure can be illustrated as: {w }. These two advancements permit the child to collapse a number up to, say, sixteen into two unitary items, nine and the remainder of nine in sixteen. The child can then disembed both nine and the remainder of nine in sixteen from sixteen and consider them as two component parts of sixteen apart from sixteen while leaving them in sixteen. So, the child can produce three numbers, nine, the remainder of nine in sixteen, and sixteen and establish relationships among them.

If the child establishes the numerosity, seven, of the remainder of nine in sixteen by counting from nine up to sixteen, the child will also understand that if seven is taken away from sixteen, the result will be nine. These operations permit the child to construct subtracting as the inversion of adding because the child has only three unitary items to deal with--9, 7, and 16. It also permits the child to establish subtraction as the difference of two numbers rather than as the more primitive concept of take away. To understand what is meant by the difference of sixteen and nine, a child maintains nine as embedded in sixteen and also as a number apart from sixteen. The difference of the two numbers can then be conceptualized as the gap between them, how far it is from the lesser to the greater. The essential scheme which permits children to find the difference between two numbers is a more advanced version of counting-up-to or counting-down-to that I call part-to-whole counting-up-to (or down-to).

After children have constructed the explicitly nested number sequence, the tendency might be to head as quickly as possible toward such learning objectives as "adding or subtracting numbers through 999 + 999". If this is interpreted as meaning the children should learn standard computational algorithms, I would consider it as a major disaster in their mathematical education. But if it is interpreted to mean that the children produce child-generated algorithms, then there is a chance for the children to maintain their insight and creativity in mathematics while learning what all too often is considered as procedures. The children certainly have constructed some very powerful ways and means of operating and it should be the teacher’s goal to bring forth accommodations of these ways and means of operating so that the children’s methods blossom into ever more powerful and spontaneous methods.

The children’s explicitly nested number sequence is a number sequence involving the unit of one and the children who have constructed this number sequence essentially live in a "one’s world". "One hundred", for example, refers to one iterated one hundred times, but not to fifty iterated twice, to twenty-five iterated four times, to ten iterated ten times, to five iterated twenty times, etc. A major goal for children who have constructed the explicitly nested number sequence is for them to construct composite units as iterable in the way that their unit of one is iterable. An illustrative task is to ask the children a question like "If you count up to twelve by threes, how many threes would you count?" The typical first attempt is "1, 2, 3, that is one; 4, 5, 6, that is two; 7, 8, 9, that is three; 10, 11, 12, that is four. Four." This may be one of the first times the child repeats a composite unit more than once. Nevertheless, the child has already constructed its composite units at the same level as the abstract unit of one. So, it would seem that the construction of three as an iterable unit would be within their zone of potential construction.

In any case, a child who has recently constructed the explicitly nested number sequence is yet to construct three or any other composite unit as an iterable unit. This realization is staggering, because in order to reorganize, say, sixty into a unit of ten units of ten, ten must be available to the child as an iterable unit. Bringing forth the construction of the operations that produce a unit of units of units is the major goal for the children in the stage of the explicitly nested number sequence. This goal, when elaborated within the mathematics of children, supersedes all conventional learning objectives concerned with multiplication and division as well as those concerned with place value.

Criteria for Judging when a Composite Unit is Iterable. For a composite unit to be judged as iterable, the child must be aware of such composite units prior to operating. That is, the child must have constructed a composite unit containing another composite unit that can be iterated so many times, a structure which is strictly analogous to the numerical structure the child constructs in the case of the iterable unit of one. Rather than speak of seven ones, for example, the child can also speak of seven fours because the child is aware of a composite unit containing a composite unit of four that can be iterated seven times. The ability to solve the following missing composite units task is an excellent indication that the child has constructed the composite unit of four as iterable if the child solves it as I explain.

If the child proceeds to find how many toys originally were under the cover by counting by four six times, and then continues on making units of four while keeping track of how many more units of four could be made, this is a strong indication that the child has constructed four as an iterable composite unit. For example, the child might proceed as follows: "Four, that is one; eight, that is two; twelve, that is three; sixteen, that is four; twenty, that is five; twenty-four, that is six. Twenty-eight, that is one; thirty-two, that is two; thirty-six, that is three; forty, that is four; forty-four, that is five. Five!". In the solution, I would infer that the child could imagine making a continuation of counting by four six times prior to counting. In this continuation, the child would be aware of an unknown numerosity of fours that could be made from the toys added.

So, in situations designed to bring forth the construction of composite units as iterable, rather than simply ask children to count-by-four up to a certain number, I ask them, for example, how many stacks of four they could make out of, say, 24 blocks. I ask this question because I want to encourage the children to imagine a sequence of stacks of four blocks each, and I might encourage them to describe what they intend to do as they solve the task before they engage in solving activity. In this way, I encourage them to visualize the results of making stacks of four and, in doing so, I also encourage them to use their numerical concept of four several times in producing visualized images of stacks of four blocks. If I am successful, the child usually refers to these visualized images as they engage in keeping track of counting by four.

Once I establish with the children that they can, in fact, find how many stacks of two, three, four, or five blocks can be made from given numbers of blocks without actually making the stacks, several options open up. First, I can ask the children to regenerate parts of their solutions. For example, in the case of the 24 blocks, I might ask the children how many blocks are in six of the stacks they made. This orients the children to take the results of their solution as input for further operating, which should be regarded as the primary means of children making progress. It is also the very beginnings of recursion, a process that the authors of PSSM emphasize along with iteration and the comparison of algorithms "because of their utility in a technological world (PSSM, p. 28)". Throughout this paper, I have emphasized all three of these processes as basic in the mathematics of children, basic not only as a product of the mental operations that constitute this mathematics, but also as basic in the construction of these mental operations.

There are other retrospective questions that can be asked, such as "How many blocks are in four of the stacks?" and "How many blocks are in two more than three of the stacks?" These questions, other than emphasizing reprocessing sequences of units of four in order that an iterable unit might be abstracted that can be used in review, also emphasize establishing strong numerical connections among the involved numbers. In this emphasis, the children’s basic facts emerge as a product of their operating rather than as something to be memorized.

In corroboration of how a child might construct a composite unit as iterable, consider how Johanna, a child who had constructed the explicitly nested number sequence, found how many blocks were in five rows of blocks she had made with four blocks per row after the blocks were hidden from view (Steffe, 1992, pp. 292 ff.). After the blocks were hidden, Johanna was first asked how many blocks were in the first three rows.

After sitting silently for about 25 seconds with her hands resting in her lap in deep concentration, she said "twelve." So, she was asked how many more rows she had, and she said, "two" and that there were eight blocks in the two rows. In reply to the question, "How many blocks in all five rows?" Johanna again sat quietly for about 15 seconds and replied, "twenty!" In explanation, she said, "Because I added up. Twelve plus four is 16, and 16 plus 4 is 20!"

Whatever figurative material Johanna generated as meaning for "four," I believe that she produced five units of four in visualized imagination, which constitutes a re-presentation of the five rows of blocks with four blocks per row. Although she may have simply produced a pattern of five rows of four in visualized imagination, she appeared quite capable of uniting four visualized blocks of a row into a unit that she could operate with as a singleton unit as well as a composite unit, and then unite these five composite units into another composite unit containing them as a sequence. This capability is indicated by her reply "twelve" to the question of how many blocks were in the first three rows and the 25 seconds she sat quietly. This is a rather longish period of time to count by fours to twelve. Indeed, there was sufficient time for her to separate the five units of four into three units and two units, a separation that is clearly indicated by her knowing almost immediately that there were two remaining rows of four after she said "twelve".

Johanna’s initial goal was to find the number of blocks in the first three rows. Such a goal is essential in taking a sequence of composite units just produced as input material for uniting. It is quite plausible that she transformed a composite unit of four into a singleton unit and then made a composite unit containing the first three of these singleton units and another composite unit containing the last two of these singleton units. She could then unpack the singleton units in the sense of producing their elements (blocks), and count the blocks in the first three units of four and then in the two units of four that remained. That she was aware of a composite unit of three units of four is indicated by her starting from "twelve" and continuing to count the blocks in two more units of four beyond "twelve".

Establishing Ten as an Iterable Unit. Because of its importance in the decimal system of numeration, one of the first composite unit I begin with is the unit of ten. Unfortunately, I have found that children in the stage of the explicitly nested number sequence can produce the number word sequence "ten, twenty, thirty, ..., one hundred" before they have constructed the unit of ten as iterable. This introduces a practical difficulty because such children often circumvent constructing an iterable unit of ten simply because they can solve most of the tasks they encounter in school mathematics without constructing this type of unit. For example, when asked to find how many dimes could be traded for one hundred pennies, I have observed children simply utter "10, 20, ..., 100" in synchrony with putting up fingers and answer "ten", but yet have no idea that they could make ten stacks of pennies with ten per stack out of one hundred pennies. Because they treat a dime as an abstract unit of one rather than a composite unit containing ten individual unit items, I refer to their apparent counting-by-ten activity as pseudo counting-by-ten. I have observed a similar phenomenon in the case of counting-by-two.

Because of pseudo counting-by-ten, it might seem easier for children to find how many stacks of ten can be made from, say, 70 blocks than how many stacks of three can be made from, say, 15 blocks. But I have not found this to be the case at all because pseudo counting-by-ten is not activated by this conceptual problem. Rather, it appears to the child to be a novel problem that has to be solved. One of my goals in presenting this problem is for the children to abstract counting-by-ten in their solution and reconstitute pseudo counting-by-ten as a part of their emerging multiplying and dividing schemes. It is also my goal for the children to abstract certain regularities in the results of their solution, especially those regularities that will permit them to curtail counting by ten and simply answer "twelve" when presented with the question of how many stacks of ten could be made from one hundred-twenty blocks.

Once children have constructed ten as an iterable unit, they can take counting-by-ten as a given in a way that is quite analogous to how they can take counting-by-one as a given in the case of units of one. This opens the possibility for the children to make major accommodations in their previous schemes involving units of one in the further construction of child-generated algorithms. For example, when meeting a situation involving 73 blocks in a container and 47 more placed into the container, the child who can take counting-by-ten and-one (a slight modification of counting-by-ten) as a given can count-on by ten and-one to find how many blocks in the container. For example, the child might count "73; 83, 93, 103, 113--114, 15, 116, 117, 118, 119, 120. One hundred-twenty." Such child-generated algorithms permit the child to maintain ownership of their mathematics and to build great confidence in their own ways and means of operating. Are they to be considered as a step in the construction of standard paper and pencil algorithms? I considerWord Work File D 2046"€è’"`TEXTMSWD
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þÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿP>Making the mathematics of children the rational bridgehead of school mathematics does involve a change of paradigm, but it involves far more. It also involves a willingness to open oneself to the world of children and to assume that within that world, children’s mathematical thinking is every bit as coherent and rational as the thinking of the adult. How could it be otherwise if we consider that children are not unlike ourselves?

It is sometimes difficult to imagine what the mathematical concepts and operations of children might be like and how they might be constructed in establishing a coherent view of children’s mathematics under the assumption that children are indeed rational beings. Starting with the object concepts that human beings construct during the first two years of life, I traced a constructive itinerary that begins with perceptual unit items, then proceeds to figurative unit items, abstract or arithmetical unit items, the iterable unit of one, and finally to iterable composite units. Further, I identified the operations which produce these six different unit types as re-presenting, reprocessing, unitizing, and uniting. By reprocessing the items of a perceptual collection, a child constructs perceptual unit items. Because the child learns to focus on the unitary aspect of the perceptual items rather than on some specific sensory feature, the child can classify what would otherwise be quite disparate perceptual items together on the basis of their unitariness. This is the first abstraction in the construction of the numerical units I have called abstract or arithmetical units. In this itinerary, classifying does not produce numerical structure. Rather, it produces perceptual collections. A regeneration of perceptual collections produce what I referred to as figurative collections, but neither of perceptual nor figurative collections can be said constitute numerical structure. Numerical structure is produced by reprocessing re-presented perceptual structures. Reprocessing in the context of re-presenting is the operation that produces the numerical structure that has been called arithmetical lot.

An arithmetical lot is a sequence of unit items that have been abstracted from perceptual unit items. These unit items, called abstract units, can be used in producing countable units, but they do not contain records of the experience of counting acts. Consequently, the items of an arithmetical lot are not regarded as the items of a number sequence. The concept of scheme opens the way to explain the construction of children’s number sequences. The most advanced number sequence that I mentioned, the explicitly nested sequence, is nothing but an interiorized counting scheme. This understanding of children’s number sequences provides constructivism with a new insight into children’s construction of number and extends it into the province of their mathematics education. The explanation offered by Piaget & Szeminska (1952) for children’s construction of number culminated with a structure similar to the arithmetical lot. This is a crucial structure in children’s conceptual development, but it pertains more to their class concepts than it does to their number sequences.

I traced the construction of the number sequence through essentially four stages: the perceptual stage, the figurative stage, the stage of the initial number sequence, and the stage of the explicitly nested number sequence. The tacitly nested number sequence is a more or less transitional stage between the preceding and succeeding number sequences. At every stage, I stressed counting as a scheme rather than as an activity. As a scheme, counting is goal directed and purposeful. It serves in the construction of adding and subtracting schemes, child generated algorithms, the structure of a unit of units of units, multiplying and dividing schemes, and the establishment of units for measuring numbers. Children’s number sequences are fundamental in their mathematical education, but they are not to be regarded as being constituted in the same way as the conventional concept of a number sequence in mathematics.

References