1 Introduction

The previous chapter titled Whole number thinking, learning and development embraces many theoretical issues that enrich our understanding of the early attainment of arithmetic skills, followed by a description of concrete applications that teachers can implement in the classroom. I would like to focus this commentary on two issues: (a) cardinal and ordinal numbers and (b) patterns and structure. Additionally, I will add some comments that go beyond the scope of the chapter, but are relevant to the study of whole numbers and their operations at the early stages of arithmetic learning.

2 Cardinal and Ordinal Numbers

2.1 Philosophical Musings

Sinclair and Coles (2015) challenge current emphases on cardinal awareness in learning number and suggest focusing on the development of ordinality. Their hypothesis states that what is significant in the learning of number (and more generally in mathematics) is not being able to link symbols to objects but being able to link symbols to other symbols. They also challenge the emphasis that is put on linking number symbols to collections of objects (i.e. on cardinality) in the first years of schooling. I will discuss these points from two perspectives: the philosophical point of view and the child development point of view.

From a philosophical point of view, cardinality is based on sets and the basic concept involved is the ‘one-to-one correspondence’ (Fraenkel 1942). The limitation of this approach is that it signifies that the sets are equal in number but does not specify the number of items in a set (Russell 1919). The exact number of a set is based on counting, which is based on order (this will be elaborated on later).

The basis for numbers from an ordinal point of view was made by Peano (in Russell 1919) who suggested three primitive notions: ‘0’, ‘number’ and ‘successor’ along with five axioms:

  • 0 is a number.

  • The successor of any number is a number.

  • No two numbers have the same successor.

  • 0 is not the successor of any number.

  • Any property which belongs to 0, and also to the successor of every number which has the property, belongs to all numbers. (Russell 1919/1971, p. 5)

However, as Russell (1919) wrote, these axioms can serve for any progression and not rather for the series of natural numbers. This leads me to elaborate on another important point made by Sinclair and Coles (2015):

There is an intriguing parallel, however, between our hypothesis and the (perhaps neglected) work of Gattegno (1961) and Davydov (1975) both of whose curriculum for early number were based on developing awareness of relations between lengths (Dougherty 2008), where what are symbolized are relations between objects (greater than, less than, double, half), rather than, say, using numerals to label ‘how many’ objects are in a collection. (Sinclair and Coles 2015, p. 253)

Bearing in mind Russell’s comment, we can notice that Gattegno (1961) and his use of Cuisenaire rods (this volume Sects. 9.3.1.1 and 10.3.3) provides an analogy to Peano’s axioms (Nesher 1972). Cuisenaire rods are a didactic tool consisting of coloured rods in which the difference in lengths of two consecutive rods is exactly the length of the white rod (the unit rod). The children recognise these rods by colour. If we let ‘the white’, ‘a rod’ and the ‘follower in length’ be the primitive concepts, then the analogous reading of Peano’s axioms will be as follows:

  1. 1.

    ‘The white’ is ‘a rod’.

  2. 2.

    The ‘follower in length’ of any rod is ‘a rod‘.

  3. 3.

    No two rods have the same ‘follower in length’.

  4. 4.

    ‘The white’ is not a ‘follower in length’ of any rod.

  5. 5.

    Any property that belongs to ‘the white’ and also to the ‘follower in length’ of every rod that has the property belongs to all the rods.

Of course, one does not teach these axioms to children, yet noticing the isomorphism between the Cuisenaire rods and the axioms of natural numbers guarantees that all properties of natural numbers can be demonstrated accurately with tangible objects.

Cuisenaire rods have an additional important property. Going back to philosophy, it was Frege (1884/1980) who elaborated on the definition of ‘number’ and came to the conclusion that ‘number’ in general is a concept, and individual numbers (such as ‘4’, ‘9’, etc.) are singular objects falling under this concept. It is hard to visualise that the meaning of the number ‘4’ is an object and that it does not represent a set four objects. However, as a mathematical object, the number ‘4’ represents the set of all sets of four objects (Russell 1919).

It might be interesting to note that in natural language (in English, as well as in many other languages, including Hebrew) one says ‘4 is an even number’ and not ‘4 are an even number’, which conveys the notion of a singular object. We often hear in natural language, as well as in word problems in mathematics, expressions such as ‘4 apples are on the table’. However, here the number 4 is employed as a quantifier to the subject of the sentence which is the apples, while in the sentence ‘4 is an even number’, ‘4’ is in the nominative position and refers to a mathematical object.

2.2 A Note About Number and Counting

The development of counting is inherently dependent on the advancement through various levels, for example, having a stable list of number names and the ability to tag objects of a set by the ‘one-to-one correspondence’ abilities that are culminated by the process of encapsulation at the notion of a number as an object of mathematics. By fully completing this process, the child is ready to perform symbolically mathematical operations and master more complex mathematical concepts.

Thus, the difficulties of the early stages of arithmetic learning lie not in teaching too many cardinal aspects of number, but rather in the fact that we sometimes confuse counting with cardinal number. Mathematics literature is full of examples in which the child counts (‘all’ the entities in two collections or ‘on’, starting from the number of entities in the first collection and going on with the second collection) and describes it as if it was an addition operation, while it is actually a continuation of counting. The use of the ‘+’ sign among numbers (i.e. 5 + 3) symbolises the mathematical operation of addition with numbers which are themselves objects of mathematics. We frequently observe children who are given a symbolic exercise such as ‘5 + 3’ and solve it by counting. Some explain: ‘they have not yet mastered ‘number facts’’. I would like to suggest that they have not yet made the encapsulation of number as a mathematical object and have not learned the real mathematical interpretation of the symbols in the ‘additive structure’ as being something different in principle from counting, to which I will refer in the next section.

Learning mathematics is a long process and is achieved in an individual manner. I do not suggest forcing those who have not yet progressed beyond counting to jump ahead. However, I think that teachers should be aware of the difference between counting and the cardinal number as an object of mathematics. Saying that the last number in counting is the cardinal number is half of the story; splitting the cardinal number from counting and encapsulating it into a mathematical object is the one-step jump into mathematics. This is, of course, resonant with Piaget’s (1941/1965) notion of number and addition becoming operational:

It is operations that are the essence of thought, and it is of the nature of operations continuously to construct something new. Thus, if 1 + 1 + 1 = 3, the three units that are added are identical with three in the sense that the total three can again give, by enumeration, the three units identical with the original three, but the additive operation has created a new entity, the totality three [italics added]. (p. 202)

Without intending to trouble kids or people who are not interested in the philosophical grounds of number (or linguistics), it is good to know that each of the Cuisenaire rods is a tangible continuous object, one that masks counting (though enables counting by measuring the length of each rod by units), and can help in this transition. Since young children use concrete materials for exemplification, it is advantageous that there are such materials, as recommended by Sinclair and Coles (2015), which enable the child who already counts reliably to relate to numbers as mathematical objects.

2.3 Psychological Considerations

Mounds of psychological and neuroscience research in the last 20 years were devoted to the question of whether the count-based representation of the natural numbers is the work of evolution or that of human culture (Butterworth 2005, 2015; Dehaene 1997; Feigenson et al. 2004). While all agree on the core capacity for numerical processing (e.g. subitising – this volume Sect. 7.2.1., representing nonsymbolic numerical magnitudes, etc.), there are theoretical disagreements as to whether these core endowments with which young children (babies and toddlers) are equipped are analogue in nature (Dehaene 1997) or characterised by a distinction of object files as evident by subitising (Carey 2004; Le Corre and Carey 2007).

While Dehaene, Piazza, Pinel and Cohen (2003) in their neuroscience studies map regions in the brain to three distinct numerical capabilities (i.e. a visual Arabic mode, an analogical magnitude code and a verbal code), Le Corre and Carey (2007) examine young children’s first experiences in counting and trace its progression. Though their experiments examined mainly cardinality, they point to the ordinal development of number concepts.

Before attending to Carey’s (2004) theory, let us recall Piaget’s (1941/1965) seminal work, The child’s conception of number, in which after delving into detailed levels of seriation and cardinality, and after elaborating on the nature of symmetrical relations that form classes (hence, cardinality) and the asymmetric relation of order that forms ordinality, he writes:

There is then no doubt as to the explanation of the coordination between ordinal and cardinal numbers…Finite numbers are therefore necessarily at the same time cardinal and ordinal, since it is the nature of number to be both a system of classes and of asymmetrical relations blended into one operational whole. (p. 157, Piaget 1965 edition)

Returning to Carey (2004), most researchers agree upon the range of subitising (quantities of one to three or four) in which the comprehension of a set’s quantity is fast and seems to be performed perceptually without counting. Carey, who studied the emergence of cardinal numbers, describes the process of very young children who when asked to give one item give the one object but when asked to give two will give any bunch of objects. She names them ‘one-knowers’. Six months later, they can distinguish ‘one’ and ‘two’ (these she calls ‘two-knowers’; they give exactly two objects, but fail for the exact amounts of other number names). Carey suggests that the within this range, children learn the numbers the way they learn the meaning (the extension) of other quantifiers such as ‘many’ and ‘all’ in natural language, and numbers beyond this range are taken as ‘many’. This, according to her theory, changes for greater cardinals such as 7 or 12 (Carey 2004; Nesher 1988).

Carey (2004) suggests that in the meantime the child learns the list of counting words that initially has no meaning to it and is recited as a ritual. Fuson and Hall (1983) have described in detail the process of learning how to count. She describes how the child progresses in mastering the order of the numerals. At each stage, the child acquires some knowledge about numbers that comprises a stable ordered list, followed by numbers recited in the right order, but with skips. Then, from not knowing the right order or missing larger number names, the child starts to repeat the previous known number names. These stages are of course dynamic and the ranges of stable lists grow with age and experience. Similarly, Gelman and Gallistel (1978) have described the principles underlying counting including a stable ordered list of number names, the one-to-one principle of attaching the number words as tags to every counted object (without repeating or omitting objects), that the order of counting objects is not important and comprehending that the last counted number name is the cardinal number of the counted set.

However, Carey suggests that the knowledge of the stable ordered list of number words in natural language is the key for learning the concept of a successor. The successor principle, operationalised as the knowledge that adding one object to a set (i.e. n) results in an increase of exactly one unit on the count list (i.e. n + 1). It is this knowledge that enables coordination between the ordered list of words and the sets to be enumerated, in order to establish their cardinal number.

It should be noted that the stable string of number words holds an asymmetrical relation between the words and necessarily fosters ordinality in counting the sets. This occurs according to researchers such as Fuson and Hall (1983) and Gelman and Gallistel (1978) before the child acquires cardinality. As soon as the child succeeds in tagging the sets correctly (i.e. mastering one-to-one correspondence between the ordered words and the objects), he is already in the mode of ordinality, which is embedded in the notion of cardinality.

In sum, though Sinclair and Coles (2015) are saying ‘it is important to balance ordinal and cardinal aspects of number sense development in the primary grades’, it seems that they call more attention to the aspect of patterns in mathematics and overlook the aspect of structure and as a result also missed a central aspect of Gattegno’s (1962) work with the Cuisenaire rods.

3 Structure

3.1 Structure in Mathematics

Structure refers to the way in which the various elements are organised and related, and to the essence of operations between numbers. In starting the section on structure (this volume Sect. 7.3.3), the author mentions additive principles such as commutativity (a + b = b + a) and the addition-subtraction inverse (a + b – b = a). These principles and others are derived from the structure of additive relations as stressed by many researchers (e.g. Mulligan and Mitchelmore 2009; Nesher 1989; Roberts 2015; Schmittau 2011; Vergnaud 1982). The notion of structures is central in mathematics and is stressed in higher education (e.g. in learning about groups, fields, rings). The interest in structure in primary grades has had its revival recently with the attempts to teach pre-algebra as early as possible (see Cai and Knuth 2005; see also this volume Chaps. 13 and 14).

This brought attention back to Davydov’s (1975) work. Davydov’s (1975) approach starts with quantities symbolised in lengths and derives the notion of number from a unit of measurement. Children use letters to express the relationships between quantities, learn to express part-whole relationships between quantities, transform inequalities into equalities and find missing wholes and parts using addition and subtraction. The relation of the whole-part additive structure is presented before numbers.

A similar approach is taken by Gattegno (1962) who used the coloured rods to probe the additive structure. Like Davydov (1975), who uses letters before numbers, Gattegno employs colours to distinguish between lengths. And since the set of rods up to ten are constructed under the idea of ‘successor in length’ (which forms monotonic steps for the child), the rods serve as a concrete apparatus that can be easily manipulated and is isomorphic to the numbers yet emphasises first structures and relations.

Let us consider the following exemplification suggested by Gattegno (Gattegno 1971 and Fig. 8.1): Putting rods ‘end to end’ will exemplify addition and the symbol ‘+’ (see A in the drawing), putting rods ‘side by side’ (B) will exemplify subtraction and will be symbolised by the ‘−’ sign, and finally, completing the structure to two equal lengths of rows (C) will justify the ‘=’ sign.

Fig. 8.1
figure 1

Example elaborated by the author: exemplification according to Gattegno’s approach to the additive relation (See Gattegno 1971, p. 24.) (Note: This is a simplified version of the original figure.)

Full size image

Thus, the sentence ‘A + B = C’ has a full concrete analogy, and the child can absorb that symbols such as ‘+’, ‘−’ and ‘=’ have a distinct, though currently limited, meaning within a structure. This structure will be elaborated on and enriched in the future. The alternation between instructions in natural language, well understood by the child (e.g. ‘end to end’, ‘equal lengths’, etc.), exemplification by concrete materials and introducing the symbolic signs of arithmetic as analogy to these structures supports the learning of the language of arithmetic. In a way, the above exemplification of the additive structure of the rods represents the semantics of the signs ‘+’, ‘−’ and ‘=’ (to be discussed in the next section).

3.2 School Practice

This approach is entirely different from the practice in many schools. Recently, Bruun, Diaz, and Dykes (2015) suggested teaching the language of mathematics and the meaning of the arithmetic operations in a detached manner. The children in their classes learn to define mathematical words and give an example and a non-example. So, for instance, the following definition of addition is given: ‘A mathematical operation in which the sum of two numbers or more is calculated, usually a plus sign (+)’. A non-example is given: ‘9 – 3’ (p. 532). Then, the following definition of subtraction is given: ‘The operation or process of finding the difference between two numbers using the (-) minus sign’. A non-example is given: ‘2 + 2’. (p. 533).

Instead of teaching the full structure of the additive relations, the ‘+’ and the ‘–’ operations are not connected and even stand as negative examples. What then might a student do with an open sentence such as: ‘3 + = 9?’ Is it ‘addition’ or ‘subtraction?’ In fact, answering this question by adding 3 + 9 and replying 12 is a most common error performed by children – clear evidence of the misunderstanding of the additive structure and the full meaning of ‘+’ and ‘−’. Many children also interpret the ‘=’ sign as a non-symmetric command: ‘do it’ rather than a symmetric sign of equivalence.

The Cuisenaire rods are by no means the only tool that can be used to acquire the semantics of the mathematical signs. One can develop manipulative materials with discrete models of numbers such as grids of rectangles or sets of circles or a number line as well.

For example, let’s consider the approach proposed by Carraher and Schliemann (2015). They suggest, and even experimented with, learning pre-algebra in third to fifth grade. They consider the four operations learned as functions and suggest approaching ‘+ 3’ as a function that can be employed in open sentences such as ‘n + 3’, where ‘n’ can receive any number. Carraher and Schliemann claim that this approach enables children to integrate arithmetic with algebra and geometry. The major concrete exemplification they employ is the number line, though they’ve tried their ideas in other contexts such as a box of candies or heights. This approach, too, departs from counting and relates, like Davydov (1975), to numbers as units of measurements. Their interpretation of the ‘+’ sign is as a one-argument function of ‘adding’ or ‘advancing’, and the ‘=’ sign is the comparison of two functions (Carraher and Schliemann 2015).

3.3 Concrete Materials

I would like to emphasise that it is not sufficient to introduce concrete materials that represent numbers, but rather, a sound pedagogy needs to support the learning of the semantics of the mathematical signs such as ‘+’, ‘−’ and ‘=’, because these signs have semantics and knowing them means understanding the relevant structure. By the semantics in early arithmetic learning I mean the following.

In addition to the concrete materials, numbers on the two sides of the ‘+’ sign refer to the parts (named ‘addends’), and the number after the ‘=’ sign refers to the equivalent whole amount (the ‘sum’). In subtraction, the role of the numbers differs. The number to the left of the ‘−’ sign refers to the sum, and the number to the right of the ‘−’ sign is one of the addends. The number to right of the ‘=’ sign in subtraction refers to the second addend. However, both ‘+’ and ‘−’ refer to the same underlying structure.

The rods’ additive construction with its language game offers even young children a microworld (or a model) that implements relations such as parts and whole similar to the semantics of the mathematical additive structure of natural numbers. A child who works according to the rules of the exemplification can realise a temporary meaning of the symbols ‘+’, ‘−’ and ‘=’, and for him or for her, a string of symbols such as ‘3 = 4 + =’ is meaningless. All the relations mentioned in Chap. 7 (this volume), such as commutativity (a + b = b + a) and the addition-subtraction inverse (a + b – b = a), also (c > a) and (b < a) from Davydov (1975), etc. are visible in the model and can be understood.

4 Final Remarks

To sum up, one should be aware that shifting from counting, which is learned within natural language, and acting within the arithmetical discipline is an enormous step for the child and its difficulty is not fully appreciated.

Sometimes the child adds or subtracts correctly by non-arithmetical means such as counting, and we mistakenly regard it as if he or she understands the ‘+’ operation of arithmetic. The ‘+’ operation is between numbers, and as long as the child did not grasp the cardinal numbers, he or she essentially did not learn the meaning of the ‘+’ sign, which gets its true meaning within the context of additive structure.

The nature of the shift from counting to arithmetical operations is semantic as well as ontological, though it is not yet fully understood. However, we can and should smooth the way for this shift by engaging young children with tangible objects as substitutes for the true reference of the abstract objects of mathematics. However, these exemplifications should represent the true nature of the objects and relations of arithmetic.

Mathematical symbolism has developed as a need to express ideas that were ambiguous in natural language or to symbolise new ideas that were advanced. The uniqueness of mathematical symbolism is in its being a condensed language – a rigorous sign language that has a strict interpretation. Moreover, mathematical language makes more subtle distinctions that the natural language cannot make (unless intonation is involved). Let us take the following written phrase: ‘A fifth of a number decreased by 4’. Is the intention 1/5x – 4 or 1/5(x – 4)? Mathematical symbolism clearly makes the distinction between the two interpretations (meanings).

Another example is the interpretation of the word ‘is’. In natural language, its interpretation is derived from the given context. However, this word receives further distinctions in the formal language of mathematics and logic (Ayer, 1936/1972):

‘Is’ in the case of equality:

A = B

 

‘Is’ in the case of class membership

a ε B

 

‘Is’ in the case of class inclusion

A ⊃ B

 

‘Is’ in the case of existence

∃X

(p. 63)

This is the power of formal language vs natural language that made it so powerful in all sciences and many applications. It is also true of arithmetic and its simple operations, and we should not underestimate the precision of these expressions.

Admittedly, it is a problematical task to convey the meaning of the rigorous signs of arithmetic by translation to natural language. Neither ‘put end to end’, ‘put together’ or ‘go forward’ for the ‘+’ sign nor ‘take away’, ‘put side to side’ or ‘go back’ or ‘descend’ for the ‘–’ sign stands for the semantics of the mathematical additive relationships. It is important that teachers understand that the symbols of whole numbers, their operations and relations in arithmetic are not merely a new syntax for concepts learned in the past in the everyday environment, but rather a difficult jump into a new symbolic domination. Mastering the symbolic language can be afterwards enriched and applied in everyday contexts via word problems given in natural language or other inquiry projects.

It is my conviction that acknowledging the big step the child must accomplish and devising new learning environments to assist in bridging the gap will avoid the failure of so many children who already feel alienated from mathematics in the early primary grades.