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**Development of mathematical concepts as basis for an elaborated mathematical understanding**

**Annemarie Fritz ^{I}; Antje Ehlert^{I}; Lars Balzer^{II}**

^{I}Psychology Department, Faculty of Educational Sciences, Duisburg-Essen University and Centre for Education Practice Research, Faculty of Education, University of Johannesburg Soweto Campus, e-mail address: fritz-stratmann@uni-due.de

^{II}Swiss Federal Institute for Vocational Education and Training

**ABSTRACT**

A common theme of models of conceptual growth is to establish the hierarchical structures of abilities that can be interpreted along developmental lines. Integrating the literature on the development of mathematical concepts and skills in children, a comprehensive 6 level model for describing, explaining and predicting the development of key numerical concepts and arithmetic skills from age 4 to 8, is proposed. Two studies will be presented. In the first study, 1095 preschool children completed a mathematics test (MARKO-D0) based on a 5-level model. The test fitted with a one-dimensional Rasch model. The extension of the model to a sixth level was verified in a new study: 312 first-graders took part in a mathematics test based on the six levels (MARKO-D1). In order to check whether the data of both samples adhered to the principle of unidimensionality, the data of MARKO-0 and MARKO-1 were used in a common analysis for comparative purposes. The applicability of these findings for a qualitative diagnostics and an adaptive training will be discussed.

]]>**Keywords**: development of mathematical concepts, numeracy, arithmetic, assessment, adaptive training, Rasch model

**Introduction**

The acquisition of complex cognitive skills such as reading, writing and calculating depends on a variety of 'building block systems' (Spelke, 2000:1233). Studies from the past three decades indicate that the brain is equipped, from birth, with domain-specific core knowledge systems, which allow for initial representations and reasoning about particular kinds of events and entities such as objects, persons, places and numerosities (Spelke, 2000).

Based on the efficiency of the core knowledge systems for numerosities children acquire first solid mathematical concepts. When children start school, they already have a 'history' of mathematical learning and are equipped with profound mathematical knowledge. In several longitudinal studies the scope and complexity of this knowledge have been shown to be key factors for the development of mathematical knowledge, and mathematical competences, at school age. Children with sound prior knowledge have a good chance of successfully using what is offered at school for their development. Children with poor previous knowledge, however, face the risk of developing difficulties in learning math (cf. Krajewski, 2003; Aunola, Leskinen, Lerkkanen & Nurmi, 2004; Landerl & Kaufmann, 2008).

Developmental models of reading and writing have promoted the diagnostics and instruction methods for these skills, as well as the support of children who have difficulties with acquiring them (e.g., Ehri, 1995). Similarly, there is no empirically validated model for the development of mathematical competences available, and the accompanying training of mathematical knowledge, at a pre-school age, is also still in its early stages. To improve this situation, it is necessary to construct a developmental model with which the performance of kindergarten children can be described qualitatively and which forms the basis for the development of adaptive training programmes.

**Modelling mathematical development**

In the past, several models for the development of mathematical competences in early childhood up to the complete acquisition of the number concept (*cf*. Fuson, 1988; Piaget, 1965), or even to the understanding of the positional notation system (e.g. Case, 1992; Steffe, Cobb & Von Glasersfeld, 1988), have been presented. The construction of the models is based, in each case, on systematic observations and empirical studies. Yet none of these models has been empirically tested and validated as a comprehensive and rigorous model.

With this objective in mind, theoretical assumptions and empirical findings were viewed and a hypothesis on a developmental sequence was formulated. The fundamental assumption was that the key numerical concepts develop step by step, that they are based on one another, becoming slowly more elaborate and abstract as the child develops. This idea also forms the basis of the competence-level models, which are used in large-scale assessments such as the PISA study (Programme for International Student Assessment, OECD, 1999). These models assume that a complex construct is measured and that component skills, which are based on one another, are structured hierarchically.

It is assumed that competences in one field of knowledge form a continuum in which steps or levels can be distinguished. Each step is marked by a specific concept that requires certain 'cognitive processes and activities of a special quality' (Klieme, Avenarius & Blum *et al., *2003:15, translated by the authors). A hierarchical model of competence allows for a description of the performance of a child, according to a specific level in the model. If it can be asserted that the levels are arranged in a hierarchical order and are at the same time dependent on one another, then establishing which level a child has reached will amount to knowing which concepts he or she has already acquired.

**Development of non-numerical mathematical concepts**

Before children learn number-words through language and understand the meaning of numbers, they have already acquired a 'large store of non-numerical quantity knowledge' (Resnick, 1989:162). Several studies suggest that children possess primary numerical abilities. The human brain is equipped with an inborn mechanism or, at least, with very early abilities to represent quantities (Butterworth, 1999; 2005; Dehaene, 1997; Carey, 2009; Feigenson, Dehaene & Spelke, 2004; Xu, Spelke & Goddard, 2005). Recent infant studies provide evidence for these assumptions, which are further specified, both from a developmental, and a neuropsychological perspective. Feigenson *et al. *(2004), distinguish between two early core systems of representation: an analogue magnitude system of representation (referred to in Dehaene & Brannon (2011) as the approximate number system or ANS) and an object file system of representation (or in Dehaene & Brannon (2011), object tracking system, also OTS).

**Core system 1: Approximate representation of numerical magnitude**

This system allows imprecise and approximate comparisons where quantities are not represented as single, discrete units. Habituation studies demonstrate that 6-month-old infants can already distinguish illustrations of quantities, if the quantities differed markedly in a ratio of 1 : 2 (4 vs. 8 and 8 vs. 16 objects, but not 4 vs. 6 or 8 vs. 12) (Brannon, 2002: Brannon, Abbot & Lutz, 2004; Lipton & Spelke, 2003; Xu, 2003; Xu & Spelke, 2000). During their development, the 'distance effect' decreases; thus, 10-month-old infants can already distinguish quantities in a ratio of 2 : 3 (Xu *et al., *2005). The capacity to compare quantities by estimating improves as children develop. However, comparisons based on such analogue magnitude representations are only ever possible by approximation (Dehaene, 1997).

By means of the simple comparison of two quantities regarding their disparity, McCrink and Wynn's (2004) research with 9-month-old infants illustrates that the infants already have a basic understanding of simple arithmetic problems such as 5 + 5, and form expectations about the results. Non-symbolic, approximate number representations are central to human knowledge of mathematics and constitute the basis for all subsequent numerical concepts (Feigenson *et al., *2004).

**Core system 2: Precise representations of distinct entities**

build representations of objects as complete, connected, solid bodies, that persist over occlusion, and maintain their identity through time (e.g. Baillargeon, 1993; Spelke & Van de Walle, 1993) (Spelke, 2000:1233).

In Wynn's (1992b) famous, often cited experiment, a toy was presented, then occluded, then a second toy was shown and then also occluded. When the occluder was removed, the children saw either two or just one object. The fact that they looked at the unexpected result significantly longer, indicates that they kept track of each object over the time, and maintained distinct representations. This system is constrained, since it only serves to track and distinguish between no more than three individual elements.

A conceptual understanding of adding and subtracting can also be established in the handling of limited quantities: children comprehend additive or subtractive changes of quantities within this number range (Koechlin, Dehaene & Mehler, 1997; Wynn, 1992b). This means that an understanding of significant numerical concepts, such as comparing, adding and subtracting of quantities, is possible even before the acquisition of language.

**Mathematical development based on the core systems**

How important is core knowledge for the further development of mathematical competency? As Spelke states (2000:1233),

First, core systems continue to exist in older children and adults, giving rise to domain-specific, task-specific, and encapsulated representations like those found in infants. Second, core systems serve as building blocks for the development of new cognitive skills.

Two lines of development can be shown, both frequently discussed in literature: on the one hand (assuming the imprecise performance of the core concepts) the increasing elaboration of mathematical concepts in numerically imprecise contexts, and on the other hand, the development of numerically precise representations. The acquisition of number-words and digits enables the children to count precisely and to perform computations without constraints. The achievement of the verbal number-words is a prerequisite for achieving explicit knowledge about numerical concepts. However, this development does not simply evolve from the core knowledge. Learning the number-words and the counting principles requires the construction of a new representational format. This format is due to the integration of the core system-representations with the requirements of the (numerically) precise handling of numbers.

]]> Let us consider the first developmental line of numerically imprecise concepts. Before children have command over language, they are capable of tracking increasing and decreasing changes in amount. On acquiring language, they are able to evaluate changes in quantities verbally (more/less), without having to be numerically precise. Resnick (1992:403) maintains that,comparisons of quantities are made and inferences can be drawn about the effects of various changes on quantities, but no numerical quantification is involved.

Resnick speaks of proto-quantitative schemes, which develop intuitively. After the *compare *and *increase/decrease *schemes, children understand the part-part-whole scheme, that is, that quantities can be divided into parts and that these parts are equal to the original quantity, at about the age of four. In a series of studies it has been demonstrated that questions about the exchange of quantities in addition and subtraction problems, as well as questions about compensation and covariance, can be solved correctly at pre-school age, if the tasks are phrased in a numerically imprecise fashion (Irwin, 1996; Langhorst, Ehlert & Fritz, 2012; Sophian & McCorgray, 1994; Sophian & Vong, 1995). This also applies if the possible answers are limited to whether a quantity had increased, decreased, or remained the same.

At pre-school age, children can solve approximate arithmetic problems of all four basic arithmetic operations in a number range up to 60, when these problems are embedded in tasks requiring comparisons (Barth, LaMont, Lipton & Spelke; 2005; Barth, La Mont, Lipton, Dehaene *et al., *2006). For the adding tasks, for example, the children were successively shown two point sets, which were then hidden behind an occluder. The sum of both occluded point sets had to be compared to a new, visible point set, concerning the question of which point set was the larger of the two. In all these studies, the ratio of the two sets is crucial for the probability of arriving at a correct solution. The significance of these studies lies in the fact that the operations (addition, subtraction, multiplication, and division) have to be done mentally.

The examples illustrate how ever more elaborate arithmetic concepts are constructed. The acquisition of the concepts is supported by the individual's cognitive development, experiences of daily life, language, and so one. But, these representations are essentially based on the representations of the core systems.

**Development of pre-numerical mathematical concepts**

The representations of the core systems are constrained; they are imprecise, or only allow for the representation of small quantities, if these are presented in succession. Precise numerical representations, however, only occur when each number is linked to a specific quantity, which determines the relationships between numbers that can effectively be modelled. It can be assumed, in keeping with the acquisition of the number-words and the gradual grasp of the meaning of number-words, that a new representational system is constructed when each number is associated with a specific quantity. While this is related to core knowledge, it does not simply arise from the 'core knowledge systems' (this means, it needs more than core knowledge, to develop a precise numerical system).

These considerations help to understand that the meaning of numbers is acquired in succession and that this takes children many years to learn (e.g. Spelke, 2000; Goswami, 2008; Carey, 2009; Wynn, 1992a). In studies with 2½- to 4-year-old children, Wynn (1990; 1992a) and Le Corre, Van de Walle, Brannon & Carey (2006) proved that children learn the number-word line during language acquisition, yet, this does not provide them with a representation system for the precise cardinal value of the quantities. They only recite the number words like a rhyme, but are not able to count or count out a single object (cf. Fuson, 1982). It seems that the children learn the meaning of one, then that of two, then that of three in sequence, as a means of counting out objects. As soon as they understand the meaning of 4, they also know the meaning of the other numbers in 'their' number-word chain, and can use them to count, and count out quantities. It seems that at this stage they have grasped the principle of the one-to-one correspondence of number words and objects. This development takes place within a one-year period up to the age of approximately 3½ years.

]]> Simultaneously, with the sequential acquisition of the meaning of number words, arithmetic operations in the small number range become feasible (Jordan, Levine & Huttenlocher, 1994; Levine, Jordan & Huttenlocher, 1992). Here, the core knowledge seems to connect to the precise number concept, but it is not clear how the analogue magnitude and the object tracking systems integrate with the symbolic number system to determine precise quantities.In a study of our own on 4- to 8-year-old children (Langhorst, Ehlert & Fritz, 2012), we provided evidence for the separate development of both systems, the core knowledge and the symbolic number system. We were able to demonstrate that skills in the numerically imprecise domain are not simply substituted by skills from the precise domain. Understanding numerical concepts in the imprecise domain continues to precede the capability to handle relevant tasks in a mathematically precise fashion (Irwin, 1996; Resnick, 1992; Sophian & McCorgray, 1994; Sophian & Vong, 1995; Langhorst *et al., *2012).

In the following discussion, our own model of competences and its empirical validation for learning arithmetic in the age range of 4 to 8 is presented. Central questions for the modelling were: which precise mathematical concepts are fundamental for understanding mathematics from a psychological, as well as didactical, point of view, and how do these concepts build on one another? Our intention was to report on a study on the empirical adequacy of the model.

**Model for the development of numerical concepts at the age of 4 to 8 years**

**Level I: Count Number**

The first major concept in dealing with numbers is to understand number-words as count- numbers which involves the ability to count and count out small quantities. The count number concept enables children to link each number-word in succession to individual objects or their fingers - the manifestation of one-to-one correspondence. Only concretely perceivable objects can be counted by allocating one object to one number word. The counting process ends with the last number-word recited, or when the last item of the collection is reached (Steffe *et al., *1988:337). The count-number is tied to the counting process - 'an awareness of plurality drives the counting activity' (Steffe, 1992:87) - but it is not connected to the quantity. The number ceases to exist for the child once the counting episode is completed. Therefore, the children start to count again when they are asked how many objects they have counted, and the recital of the number-word sequence has to start at 'one'.

In this sense, Cobb & Wheatley (1988:3) argue that,

]]> In this way, small collections of objects are countable. The children can also count out a specific number of elements from a larger quantity. This can be ascertained with tasks such as 'Give me 5 discs!' The child, who will then give us 5 objects, does not yet understand the cardinal value of the number; the child does not understand 5 as 'fiveness'.numbers are transitory entities that have to be made and remade by actually counting and do not exist independent to the activity of counting.

With the count number concept as a basis, an understanding of the relationship between numbers develops next. The numbers are represented as a mental number line (Case, 1992; Resnick, 1983).

**Level II: Mental number line**

Counting at level I is tied to the recital of the number-word line and the successive assignment of number-words to discernible and countable objects. In this way, a qualitative representation of numbers occurs, but it is unspecific and does not give any clues as to the relationships between the numbers. A change of the representation of numbers occurs only gradually. The children construct a mental number line, in which the numbers are aligned as gradually increasing quantities. The mental number line constructed in this way is at first nothing more than an ordinal representation of the number-word line.

That is, numbers correspond to positions in a string, with the individual positions linked by a 'successor' or 'next' relationship, and a directional marker on the string specifying that later positions on the string are larger (Resnick, 1983:111).

The distances between the numbers remain as yet unreflected.

]]> Even though Resnick (1983) stresses the point that the mental number line at this level is called a 'primitive representation of numbers', Fuson (1988) suggests, (speaking rather of a mental number list here) that this representation, nonetheless, provides the children with the basis on which the relationships between numbers become precisely determinable. The construction of a linear number line enables the children to identify preceding and succeeding numbers. As the numbers on the line become progressively larger, the numbers that appear later on the line are greater. Thus, succeeding numbers are larger, while preceding numbers are smaller. With this knowledge, numbers can be compared to each other according to their position on the number-word line. Children can now correctly answer the question: 'which number is larger, 4 or 5?'

This allows children to solve a considerable number of arithmetic problems (Case & Okamoto, 1996; Siegler & Booth, 2004). The concept of adding and subtracting is familiar to the children and could already be applied to numerically imprecise quantities (Resnick, 1992; Resnick & Singer, 1993). With the precise mental number line, simple problems of adding and subtracting can be solved. Simple problems refer to tasks of the kind: a + b = ? or a - b = ?, i.e. tasks that can be solved counting forward. When adding, both partial quantities will first be counted out completely and then the sum will be determined by counting out the two combined partial quantities completely, beginning always from 1. In subtracting the initial quantity is counted out, then the partial quantity to be subtracted is determined and moved aside, and then the remaining quantity is counted out, beginning from 1. For this, the children need the aid of materials or fingers. All quantities have to be counted out individually ('counting-all strategy' or 'count from first') (Briars & Larkin, 1984, Carpenter & Moser, 1983; Riley, Greeno & Heller, 1983). Even though the problems first have to be illustrated clearly with the help of materials at this level, verbally presented number-word problems can be solved by moving forward or backward along the mental number line, while here, too, the partial quantities have to be counted out individually on the fingers (Riley, Greeno & Heller, 1983; Stern, 1992).

Findings (e.g., Case & Okamoto, 1996; Siegler & Booth, 2004) show that understanding the number line concept is crucial for the mathematical development at primary school age, as it forms the basis on which the relationships between numbers are explored further.

**Level III: Cardinality and Decomposability**

In the development of numerical concepts, the acquisition of the *cardinal number concept, *that is, the understanding that a number-word also represents a quantity with a specific number of elements, is seen as the first major step (Piaget, 1965; Piaget & Szeminska, 1975). According to Dehaene (1997), an understanding of cardinality is neither innate nor does it unfold automatically, so instruction is required.

Real cardinal understanding requires the mental integration of the counting steps, or the elements of the counted quantity to a whole. This whole is symbolised by a number-word. If, for example, 7 objects have to be counted, each object is assigned a number-word, and all objects together will be integrated into a total quantity with the number attribute seven ('sevenness'). In this way, the number becomes a composite unit (Steffe *et al., *1988), in which the 7 distinct objects are combined into one quantity. According to Piaget (1965), and Steffe *et al., *(1988), the quantity concept is achieved by means of the synthesis of a seriation feat and a classification feat: this constitutes an abstract construction feat in which concepts of ordinal sequences and cardinal quantities are integrated; the number-word line corresponds to the sequence of ascending cardinal units where the quantities follow a fixed order (quantity seriation). Accordingly, both an ordinal and a cardinal correspondence can be established simultaneously between two quantities. This acquired understanding becomes obvious when numbers and quantities can be compared with each other through the number of elements: 4 is less than 5 because the quantity 4 consists of fewer elements than the quantity 5.

Upon understanding number as quantity, attention is disengaged from the counting situation to the cardinal situation, in which the number-word stands for all elements. Therefore, a quantity with the value of 7 retains this quantitative characteristic, independently of where (irrelevance principle) and what (representation) is counted out (Piaget, 1965). Once it is understood that a number is a composite unit that consists of individual elements, it also becomes clear that numbers can be decomposed again.

Therefore, the acquisition of the cardinal number concept is the key prerequisite for the acquisition of effective calculating strategies. Addition problems now become solvable by counting the second quantity onto the first, without having to count the first quantity. Children who have acquired the cardinal number concept no longer rely on the 'counting-all strategy', they rely instead on the 'counting-on strategy' (Fuson, 1992). The first sum is understood as cardinal unit embedded in the whole, and the second sum can be counted on. The cardinal principle brings a first understanding that the first addend forms a part of the whole (Fritz & Ricken, 2008). As the concept of subtraction is understood, subtraction problems also become solvable by subtracting a partial quantity as a unit from the original quantity. This expanded understanding of numbers as cardinal entities, which can be composed and decomposed, can now also be applied to operational understanding: On an action level, the children begin to understand the connection between partial quantity - partial quantity and total quantity (2 and 3 together are 5). Thus, problems become solvable when they have to count from the second partial quantity to the total quantity, for example, 'I want 10 discs, I already have 4. How many are missing?'

**Level IV: Class inclusion and Embeddedness**

The acquisition of the part-part-whole concept is seen as the most important conceptual leap in the first years of school (*cf*. Fuson, 1988; Gerster, 2003; Resnick, 1983). According to Fuson, this is a complex conceptual feat that is acquired over a longer period of time. One part of it is that quantities slowly begin to be represented mentally by symbols (number-words), which can be decomposed and composed in *different *ways (Huttenlocher, Jordan & Levine, 1994).

In the triad 2, 5, 7, for example, 7 is always the whole; 5 and 2 are always the parts. Together, 5 and 2 satisfy the equivalence constraint for the whole: 7. The relationship among 2, 5 and 7 holds whether the problem is given as 5 + 2 = ?, 7 - 5 = ?; 7 - 2 = ?, 2 + ? = 7, or ? + 5 = 7 (Resnick, 1983:115).

There is a fixed relationship between three quantities. If two quantities are known, the third one can be deduced. With this capability, problems that require any of the three quantities become solvable.

Consequently, transformations between partial quantities are possible, without a change in total quantity. The quantity '5' can be decomposed into the partial quantities '1' and '4'. By shifting (transformation) one element from one partial quantity to the other, the partial quantities '2' and '3' are formed. Quantities can be 'composed' in different ways through decomposition and transformation. The indicator of this understanding is the children's ability to find different decompositions for numbers (Baroody, 2006; Hunting, 2003; Steffe *et al., *1988).

In understanding the part-part-whole concept, it becomes possible for the children to see addition and subtraction tasks as complementary, and to carry out solution procedures based on derived facts (compensation and covariance). In summary, the part-part-whole concept can be understood as the prerequisite for an integration of several algebraic principles, such as the commutative law, the complementarity of addition and subtraction, and the understanding that (natural) numbers are composite units (Stern, 1998).

Studies related to the acquisition of the part-part-whole (PPW) concept (Clarke, Clarke, Grüßing & Peter-Koop, 2008; Fischer, 1990; Irwin, 1996; Resnick, 1992; Sophian and McCorgray, 1994) have demonstrated that the concepts of part-part-whole, compensation and covariance, are already available in pre-school children, if problems do not require numerical precision. Children are only able to recognise the compensation principle in tasks like '5 + 2' and '4 + 3' numerically precisely, after some months of schooling (Baroody, Wilkins & Tiilikainen, 2003). It seems that experiences with a large variety of formal addition problems are necessary before children start to compare the results of different problems.

Riley & Greeno (1988), classified PPW word-problems of addition and subtraction and investigated their level of difficulty. The results revealed that problems that require the production of the final sum can be solved by nearly 100% of the first-graders correctly (this problem corresponds to Level II in our model). Problems that ask for the exchange of quantities or partial quantity, on the other hand, can only be solved by at most 50% of the pupils in first grade. Obviously the understanding of this concept develops in the first year of schooling. Children find problems that ask for the initial quantity even more difficult (approximately 30 to 39 %). In our analysis we found problems of that kind on level V. A replication of this study with German children led to similar results (Stern, 1998).

**Level V: Relationality**

The crucial concept of this level is called 'operational cardinal number concept' by Piaget (1965). For him, it reflects a deep understanding of the complex concept of the natural number and requires the grasp of cardinality, of ordinality, and of the relationship between these two concepts. The number-word line is understood as a sequence of cardinal units, in which each number in turn is an independent countable unit:

]]>Therefore, each next word presents a cardinal number that is one larger than (using the cardinal as well as the sequence meaning), the earlier word. This cardinal sequence thus comprises both class inclusion (embeddedness within the next number) and seriation. Mastery of both latter concepts was the requirement for understanding a truly operational cardinal number for Piaget (1965) (Fuson, 1992:100).

Realising that numbers do not only represent concrete quantities, but also counting acts, that can be counted themselves, is the key cognitive process at this stage. In this way, 'composite units, whose elements symbolise counting acts' emerge (Steffe *et al., *1988:338). Thus, a further level of abstraction is reached, based on which the number line concept can be further differentiated until a precise, metrical number line concept is developed, which allows the correct representation of one-, two- and later also multi-digit numbers, based on which the place-value-system concept is acquired (Campbell, 2005; Moeller, Nuerk & Willmes, 2009; Nuerk, Weger & Willmes, 2001; Nuerk & Willmes, 2005; Nuerk, Moeller, Klein, Willmes & Fischer, 2011).

Nevertheless, to reach a full understanding of the place-value-system, several further realisations are required. The number line concept states that intervals between successive numbers are of the same size (+1). Thus, numbers can also stand for a class of congruent intervals, and indicate a segment on the number line, or a relationship between two quantities (numerable-chain level; Fuson, 1992).

Based on this knowledge, the children now have a kind of scale at their disposal that enables them to compare quantities, and to determine the differences precisely.

]]>

On earlier levels, children are able to compare quantities regarding their cardinality, yet, these comparisons are limited to imprecise indications of direction (larger/smaller; more/less). Only the relational number concept provides the children with a scale on which they can count on a specific number of counting steps, from a random starting point. Thus, the number does not stand merely for a specific quantity anymore, but also for a distance or a number of counting steps on the number line.

This also means that the same number of counting steps, or equally long distances on the number line, have the same cardinality. The child now understands that the distance between 0 and 5 is equivalent to the distance between 5 and 10; and vice versa, that the number 5 indicates the distance between the numbers 0 and 5 as well as the distance between the numbers 15 and 20. Similarly, when a number is doubled or halved, identical partial quantities or sequences are created. This realisation can only be used for simple doublings and decompositions (halving), on level 5.

With the construction of a number as scale, or distance on the number line, relationships between numbers can also be modelled on the next higher level of abstraction. With this number concept, which Stern (1998) calls the relational number concept, relationships between quantities can be modelled. Comparison problems of the following kind are typically used to assess this relational number concept level: 'Maria has got 5 marbles. Hans has got 8 marbles. How many marbles has Hans got more than Maria?' (difference quantity unknown) or 'Maria has got 4 marbles. She has got 3 marbles less than Hans. How many marbles has Hans got?' (reference quantity unknown).

In order to solve comparison problems of this kind, the child has to be able, for example, to interpret '3 marbles less' as reference quantity, which first has to be found by determining the difference between the given partial quantity (4 marbles) and the difference quantity (3 marbles less). Such problems have a solution frequency among first-graders of between 16% and 28 % (Riley & Greeno, 1988; Stern, 1992). This means that the understanding of this concept is to be expected not earlier than in grade two.

]]>

**Level VI: Units in Numbers (bundling and unbundling)**

The PPW concept conveys the realisation that numbers are composite units, which can be flexibly decomposed into different partial quantities, on the one hand (8 into 1 and 7, 2 and 6, 3 and 5, 4 and 4, etc.), and into several partial quantities, on the other hand (8 into the partial quantities 1 - 3 - 4). In each case, the sum of the partial quantities is equivalent to the total quantity.

With the concept of relationality, the child realises that the distances between the numbers on the continued number-word line are always the same. Therefore, segments of the same size, or bundles, can be formed on the number line (e.g. 3 x 4). These bundles become new, abstract composite units. In the multiplication task 3 x 4, for example, 3 bundles of 4 elements each are formed, that is, 3 bundles of 4. Depending on the task, bundles of 2, 3, 4, and so on, can be formed. A bundle of 10 elements, or 10 counting steps on the number line, becomes a bundle of 10. The bundles themselves become countable units of a higher order.

Conversely, a number can obviously also be decomposed into partial quantities of the same size, or, if presented as a segment on the number line, into partial segments (bundles) of the same size (12 : 4). In keeping with the PPW understanding that quantities can be flexibly decomposed into different or several partial quantities, a quantity can be decomposed into several partial quantities of the same magnitude and a number, as a segment on the number line, also in several partial segments of the same size.

]]>

This realisation is characteristic of this developmental level: the children are able to find different decompositions of the same magnitude (bundles) for a number. Once they are able to grasp this, they are prepared for understanding division.

The fact that the number range is constructed systematically is something the children have already experienced at pre-school age, and they can recite the number line in steps of ten. By counting, even children on lower levels can count out larger quantities, and even do arithmetic operations in the number range up to 100. They count the objects one by one and write down the number determined (e.g. 35). They do not realise, however, that the 3 in the number 35 stands for 3 units of ten (Gerster & Schultz, 2004). Only after acquiring the PPW concept, and the concept of relational numbers, can the concept of bundling and unbundling of numbers be understood. This forms the basis of the place-value-system, as well as, the arithmetic operations of multiplication and division.

A profound understanding of the place-value-system, however, develops over several levels,

... the development of decimal number knowledge can be understood as the successive elaboration of the part-whole schema for numbers, so that numbers come to be interpreted by children as compositions of units and tens (and later of hundreds, thousands, etc.) and are seen as subject to special regroupings under control of the part-whole schema (Resnick & Singer, 1993:126).

On this level VI, the focus is first of all on grasping the principles of bundling and unbundling; this merely prepares the ground for understanding the place-value-system. However, the full realisation of it requires further steps.

**Method**

**Operationalisation of the mathematical levels**

MARKO-D 0 was standardised with 1 095 children aged 4 to 6 years, M = 64.6 months (SD = 7.2). The items of the one-dimensional dichotomous Rasch model show satisfactory values (weighted infit MNSQ 1±0.2 for 53 out of 55 items, weighted infit MNSQ 1 ±0.3 for 2 items), and the person reliability is at .91. The separation index of 3.26 suggests 4-5 different levels. Furthermore, the confidence intervals of the levels do not overlap, apart from a slight overlap between levels II and III. The test shows a retesting reliability of r = .89. It is also construct valid with a correlation of r = .77 with the arithmetic test OTZ. The validity was also tested with intelligence. Here, the test shows a medium correlation with the intelligence test CFT-1, of between r = .51 and r = .57, and the WPPSI-III, of between r = .41 and r = .59.

During the second step, level VI had to be verified. To do this, new items were built according to the theoretical principles for level 6, resulting in a new test-version for older children valid for all six levels. The test designed for younger children (4 - 6 years) and the one designed for the first-graders overlap in several items, the so-called anchor items. The new test version for the first grade has been tested with 312 children 86.8 months (range from 77 - 106, SD = 5.1).

Below is an overview with sample tasks for each of the six levels.

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