On Number Language: A Commentary on Chapter 3

Abstract

This chapter primarily offers a commentary on Chap. 3, before moving off at the end into some wider issues. I have organized my comments under four broad headings – links among what is said, written and gestured vis-à-vis number; place value; the (dissolving) distinction and its pertinence between count nouns and mass nouns (and the place of English-language number words in relation to this distinction); and some similarities and differences among the systems of cardinal, ordinal and fractional number words – before concluding with a (very) few summary remarks.

1 Introduction

I start with a policy statement, pretty unrelated to the previous chapter. I am always a little taken aback to see numbers or other mathematical symbols (e.g. ‘7’ rather than ‘seven’, ‘+’ rather than ‘plus’) presented inside classroom transcripts, which supposedly provide a written account of what was said. Everything that is said is said by someone in some natural language (or natural language mix – cf. code-switching, e.g. Setati 1998 – such as where a somewhat bilingual speaker may know how to say the higher number words in one language only). Non-verbal numerals (of whatever sort) are not part of any natural language,¹ so they require ways to be read aloud into such a language. Because of this, I believe it is important to be very, very precise about marking such distinctions. In Pimm (1987), for instance, I distinguished between what I termed a ‘spelling’ reading and an ‘interpretative’ reading of written mathematics: for example, is the Biblical ‘number of the beast’ (666) to be said as ‘six six six’ or is it ‘six hundred and sixty-six’ (in British English) or ‘six hundred sixty-six’ (in the North American version)? What it is not, however, is ‘six hundreds (and) sixty-six’, something I will come back to later on.²

Is ‘six six six’ even a spoken number or simply a time-ordered string of digits being listed in turn (one no different from a reading into English of ‘6, 6, 6’ rather than ‘666’), one that ignores the positional structure? In French, certain numbers (such as phone numbers) whose cardinal value is seldom of interest are frequently read (and written) as sequences of two-digit numbers: 02 65 47 23 46. I will come back to this later too when querying whether the number-word system of any language reflects place value (or better put perhaps, in relation to speech, ‘temporal value’). My broader point is that there are significant differences between speaking and writing in relation to numbers, most particularly when it comes to engaging with the written symbolism of mathematics (not least of number), differences that are forgotten at our peril.

As numerical anthropologist Stephen Chrisomalis claims, ‘The linkages between number words, computational technologies, and number symbols are complex, and understanding the functions each serves (and does not serve) will help illustrate the range of variability among the cognitive and social systems underlying all mathematics’ (Chrisomalis 2009, p. 496).

2 What Is Written and What Is Said

I start by echoing the claim from early on in Chap. 3 that ‘whole number arithmetic is not culture-free, but deeply rooted in local languages and cultures with the inherent difficulty of transposition and culture perspective’ (this volume, Sect. 3.1.2). As mathematician René Thom once observed:

when learning to speak, a baby babbles in all the phonemes of all the languages of all the world, but after listening to its mother’s replies soon learns to babble in only the phonemes of its mother’s language. (cited in Ziman 1978, p. 18)

Also, from Chap. 3’s opening page, the expression ‘cultures of speaking’ brought to mind the fact that there are ‘cultures of writing’ too (e.g. the order of writing of the two numerals within a single fraction – see Bartolini Bussi et al. 2014) and that these two may not perfectly align within a single ‘culture’ (see later for a further example involving grouping of digits within a large number in relation to how they are read). And these both influence and are influenced by the physical actions and gestures implicated in counting and computation (a fact worthy of the historical and geographical term ‘cultures of gesture’, such as varied forms of finger counting and finger calculation³ – for many examples and a classification scheme, see Bender and Beller 2012). It is important to remember that, in many times and places, these two mathematical actions (counting and computation) were barely connected at all – e.g. the combined but unrelated use of Roman numerals to hold numbers and counting boards with which to calculate (see Tahta 1991; Chrisomalis 2010).

In response to the piece about fraction writing order by Bartolini Bussi et al. (2014), I wrote:

With fractions written by hand, the composite symbol is produced in a given, temporal order. How might that gestural order relate either to what is said or to how what is said is conventionally written? In English, the first word spoken in time is the numerator: is this so for any language? When a fraction word is written down in English, left to right, the numerator is again the first word to be written. (Ditto the question about other languages.) But when the composite symbol for the fraction is produced, there are variations possible, as their terrific vignettes from China and Burma attest. But both examples point to the arbitrary nature of manual symbol formation (in Hewitt’s 1999 use of that word) and to the fact that, once made, the symbol retains (almost) no trace of its making [not least its order]. (Pimm 2014, p. 15)

The many cultures of number are fascinating and intricate, and the particularities of language vis-à-vis time and place, in interaction with computational technological devices (which have existed for at least 5000 years), offer a most worthwhile focus for profound attention. In relation to very recent work concerned with what might be termed ‘tangible technological gestures’ (see Sinclair and de Freitas 2014, not least in regard to Jackiw and Sinclair 2014), languages themselves at times encode forms of gesturing that have their own transparencies and opacities, their own generalities and idiosyncrasies, all of which form part of the complex symbolic world into which all children are born.

For Wittgenstein, language is, initially but fundamentally, reactive, the word not being the origin:

The origin and the primitive form of the language game is a reaction; only from this can more complicated forms develop. Language – I want to say – is a refinement, ‘im Anfang war die Tat’ [in the beginning was the deed]. (Wittgenstein 1937/1976, p. 420)⁴

In relation to the deed of counting, the specific pedagogic language of computational practice (e.g. the English arithmetic metaphor in addition of ‘borrowing’ and ‘paying back’) brings with it the possibility that it was at one point literal. One potential example taken from Chap. 3 relates to the suggested link between the medieval Latin expression reservare in manibus (‘to keep in the hands’) and the more contemporary French term à retenir (‘to keep in mind’). It crossed my mind that the former, in relation to abaci and counting boards, might literally refer to what the hands had to do. Elsewhere, Wittgenstein also commented, ‘Remember the impression one gets from good architecture, that it expresses a thought. It makes one want to respond with a gesture’ (Wittgenstein 1932, p. 22e). This observation reminded me of the Egyptian hieroglyph for million (), plausibly a human whole-body gesture at the large size of the number.

Language is not separable from culture nor from gesture (especially not in the context of counting). Gestures perhaps have evolved over a longer period and perhaps have left their trace on the language.⁵ There is also the possibility of temporal slippage of one system in relation to a development within the other, not dissimilar to those identified by Lakoff and Núñez (2000) with regard to the calculus, whereby the (static) talk is mid-nineteenth century, while associated (dynamic) gestures are more seventeenth century in nature (more fitting to the notion and language of a moving variable, a language that is returning with dynamic geometry environments).

In particular, Raphael Núñez examined the co-production of gestures and speech of Guershon Harel proving a result from real analysis. Núñez observes:

The study of gesture production and its temporal dynamics is particularly interesting because it reveals aspects of thinking and meaning that are effortless, extremely fast, and lying beyond conscious awareness (therefore not available for introspection). (2009, p. 319)

But it is also true that the gestures are co-produced when counting (and that in certain circumstances constitute counting), phenomena that are equally worthy of study as their higher mathematical counterparts. Nevertheless, the focus of this chapter as well as its predecessor is on number language and not number gestures, even though I do not wish to dismiss the latter as epiphenomenal, the way labelling them ‘paralinguistic’ does.

3 On Place Value

With regard to place value, one of Chap. 3’s central themes, I have three main observations to make.

First, I would like to consider whether the phenomenon of place value exists solely in relation to written numerals (i.e. written marks, nowadays usually, but not always, employing what are termed Hindu-Arabic numerals⁶) and not in respect of written words or characters from a natural language and whether it also could describe aspects of spoken number words in a natural language as well (or even gestural language – query: what is the structure of number signs in British, American or Chinese sign language?). This question reflects my increasing uncertainty as to what place value actually is, as well as echoing Tahta’s (1991) informed assertion that place value is appreciably overemphasised in Western mathematics teaching – and the discussion of Chinese numeration in Chap. 3 relates to this, when the authors claim, ‘The transparency of Chinese [number] names is likely to foster students’ understanding for place value’ (this volume, Sect. 3.2.2). Though if, as I argue below, place value is simply a convention, then there is a strong question as to whether it is something that is amenable to being ‘understood’, rather than simply complied with (see Hewitt 1999).

My questioning arose from reading Chap. 3. The authors claim that traces (which they nicely term ‘memories’) exist within many spoken numeration systems within natural languages. But these are, at best, ordinal traces, with regard to how number names are said in a conventional order (in English, in decreasing powers of ten, although exceptions like four-and-twenty still exist; in German, the decades are systematically said after the units, e.g. 54 is vier-und-fünfzig, ‘four-and-fifty’). This raised two sub-questions: does it make sense even to ask whether written (or spoken) natural language numeration systems are or are not place value and, in regard to written numeration systems that were not place value (e.g. the ancient Egyptian one), what were their spoken counting systems like?

In regard to the first question, my (admittedly strong, potentially over-strong) conjecture is no spoken language-based numeration system is place value (not even Asian ones, which would be the most likely contenders). This is because the structure of how number words are formed ensures that their decimal value is encoded as part of the string, thus changing either the written order (of language-specific symbols on the page) or (temporally) the spoken order in which the various parts of the numeral are said aloud does not alter the combined total. It may go against convention (as ‘four-and-twenty’ does), but it does not produce a different number. (Of course it is true that simply interchanging the ‘six’ and the ‘seven’ in ‘sixty-seven’ and ‘seventy-six’ changes the value but that ignores the fact that ‘six’ is part of ‘sixty’.) So possibly place value is solely a phenomenon of written, non-language-based numeration systems, and whichever natural language is used cannot help with this.

My second place value observation, which relates to the first, has particular force because of the particularities and peculiarities of manipulatives such as Dienes blocks (also known as multibase arithmetic blocks – see this volume, Sect. 9.3.1.2), which are regularly promulgated as a means to assist with acquiring the concept of place value. See Fig. 4.1.

Fig. 4.1

Dienes blocks (1000, 100, 10 and 1)

It is a commonplace pedagogic move in English-language primary schools to use large sheets of paper and columns labelled (from left to right), thousands (or Th), hundreds (or H), tens (or T) and units (or U).⁷ The blocks are collected and placed in the respective columns and then ‘Hindu-Arabic’ digits are used to record the number of them in each column, hugely finessing the fact that it is actually the paper columns and not the blocks themselves that are both ‘holding’ the places and, consequently, carrying ‘place value’.

However, as with the instance mentioned in Chap. 3 where a 7-year-old writes 10013 instead of 113, an accurate (as opposed to conventionally correct) notating from a corresponding Dienes block paper configuration would be 1H, 1T, 3Us = 100, 10, 3 = 100103. The question of quantity versus place is an intricate and arbitrary one (again in Hewitt’s 1999 sense), and there is no necessary reason why there cannot be any number of blocks of any size in a single column (something an abacus masks by each spike having a set, uniform height relative to the diameter of the beads). Indeed, a higher (and linear-algebra-influenced) mathematical perspective has any whole number generated by the basis consisting of powers of 10 (and includes decimal fractions, if negative powers of 10 are permitted) with the coefficients 0–9. It is partly for this reason that I mentioned the six, six, six reading of ‘666’ in the opening paragraphs of this commentary (as well as linking to David Fowler’s 1987 historical reconstruction, via the arithmetic process of anthyphairesis, of a pre-Euclidean functioning definition of ratio).

Before raising further difficulties, there are three more observations I wish to make about Dienes blocks themselves. The first is that they can actually be modified to display any whole number or decimal fraction. To have ten thousand, for instance, one simply needs to stick together ten of the large cube size; for hundred thousand, a square array of a hundred of the large cube size; and for million, a cube array of a thousand the large cube size. For decimal fractions, merely rename one of the larger blocks (e.g. the larger cube or the ‘long’ or the ‘flat’ as they are sometime called) as ‘one’.⁸ Secondly, this visuo-geometric repetition every 10³ exactly fits the SI (Système Internationale) emphasis on grouping whole number digits into triples, as well as reflecting the standard metric naming structure of measures (though if we wanted to use this to refer to a kilo-something, a milli-something or a micro-something, we would need a word for a standard counting unit – other than ‘unit’). My third observation is straightforward: there is a ‘natural’ and directly observable sense of decimal equivalence between each power of ten and the next one.

However, it strikes me that Dienes blocks are at least as good a fit to the base-ten system of Egyptian hieroglyphic numerals (which uses the repetition of vertical lines, hoops, scrolls, lotus flowers, etc. where there are no links whatsoever among the symbols for 1, 10, 100, 1000 and so on, either to record numbers or to calculate with them).⁹ (See Fig. 4.2.) And this numeration system is decidedly not a place value one.

Fig. 4.2

A depiction of Dienes blocks (with ancient Egyptian hieroglyphs for 1000, 100, 10 and 1 appended)

In general, there are two alternate principles for generating words or other symbols for numbers: repetition (related to tallying) and cypherisation, namely, the use of distinct and independent symbols for each number. Many older symbolic systems use repetition as the primary principle. For example, below (see Fig. 4.3) depicts an example using the ancient Egyptian numeration system: on the left is the conventional order and on the right in a scrambled order. (Whole-number adding is totally unproblematic as the symbols themselves are simply combined, and any excess over nine of one power of ten is converted into one of the next power up.)

Fig. 4.3

An ancient Egyptian numeral (in conventional and scrambled order)

‘Full’ cypherisation means every initial numeral up to one less than the base has a different symbol (think 1, 2, 3, 4, 5, 6, 7, 8, 9). Chinese rod numerals (discussed in Chap. 3) reflect enormous flexibility with very limited cypherisation (the vertical rod and the horizontal rod), akin to Roman numerals I and V, only without the latter’s subtractive principle¹⁰ and much repetition, likely because they became traces of actual piles of rods used on counting boards, where an attribute of the board (lines, positions) took care of the place value. Likewise, ancient Babylonian numeration consists in its entirety of only two distinct stylus edge marks (one the same as the other only rotated through 90°), together with a mixed-base system (ten and sixty), repetition, a form of place value and contextual ‘floating point’.

One pragmatic test for any written system (presuming it makes repeated use of the same set of symbols or objects) with regard to its being place value or not is whether one can generically scramble the order of the marks and not affect the numerical value represented: this is true with Dienes blocks and also with ancient Egyptian numerals. In passing, it is also true of early Greek (Ionian) numerals (and are still used today for depicting ordinals), where the numeral for 1 bears no relation to that of 10 or 100, being different letters of the alphabet. The Egyptian numerals (as do the Greek) retain their specific decimal value, even when rearranged – see Pimm (1995) for more on these complexities of symbol/object manipulation. Consequently, Dienes blocks cannot ‘contain’ place value. So, if they do ‘work’, how do they ‘work’? The ‘value’ is there, but the ‘place’ is not.

Caleb Gattegno (e.g. Gattegno 1974) repeatedly proposed systematising language-based counting systems in elementary schools in different European languages, in order to make them easier to learn by being a far closer fit to the standard Western written numeration system. In particular, in English he wanted ten to be said as ‘one-ty’, eleven as ‘one-ty-one’ and twelve as ‘one-ty-two’¹¹ and then twenty as ‘two-ty’, thirty as ‘three-ty’ and so on. With the later decades (sixty, seventy, eighty, ninety), the changes merge with the actual empirical system. There is a samizdat-style community within Anglophone mathematics education (especially within the UK, but in North America and Western Europe too), which works extensively with Gattegno’s ideas (and the teaching aid of the Gattegno number chart, among others), in order to support the acquisition of structured fluency in number naming (for one recent instance, see Coles 2014).¹² The Chinese numeration system detailed in Chap. 3 (which presents in several other Asian languages too) has these properties already.

My third place value observation relates to the notion of linguistic/conceptual transparency as employed in Chap. 3, the generation of number terms for powers of ten and how they relate to the standard (SI) means of writing large whole numbers using ‘Hindu-Arabic’ numerals. In Chinese, 千 (qiān ¹³) is the character for ‘thousand’ although none is needed, as it is not in English either, based on the principle that a new power-of-ten name is only needed when the same two terms would otherwise be next to each other. ‘Ten tens’ gives rise to ‘hundred’, but ten hundreds (‘thousand’) should cause no difficulty (and does not in naming centuries, e.g. ‘the seventeen hundreds’) and need not exist, while ‘hundred hundreds’ is the next one that should generate a new term. In Chinese, that character is 万 (wàn), while English speakers just say ‘ten thousand’. (To belabour the point, notice it is not said as ‘ten thousands’, as conventional pluralisation rules of English would demand – though see the next section on the distinction between mass and count nouns.¹⁴) It is this same principled issue that causes divergent interpretations of ‘billion’ (‘hundred million’ in North America, ‘million million’ in the UK, at least historically): likewise with ‘trillion’. But the generation of these new words for certain powers of ten allows the use of the same number words from one to nine to be combined to name every whole number.

The SI number convention declares:

The digits of numerical values having more than four digits on either side of the decimal marker are separated into groups of three using a thin, fixed space counting from both the left and right of the decimal marker. Commas are not used to separate digits into groups of three. (http://physics.nist.gov/cuu/Units/cheklist.html)

Thus, for example, 213 154 163 is how this number should be written. However, this convention makes a (false, universal) presumption in relation to every natural language on the planet with regard to the structure of number words within each language, because, as I mentioned at the outset, written numerals are not part of any natural language.

So in relation to transparency and Chap. 3’s claimed ‘perfect’ match of Chinese numeration and ‘the mathematician’s arithmetic’, this is one place where the Chinese language numeration system does not match SI at least, namely, with regard to delimiting (whether by means of commas, full stops or spaces) numerals with more than four digits. For instance, twelve thousand is written in Chinese as 一万两千 (yīwàn liǎngqiān; in other words, one wàn two qiān – one ten-thousand and two thousand), which does not match 12 000. The written symbolic form of numbers can thus aid or interfere with generating the corresponding correct, language-specific spoken form.

In an article about this particular mismatch, Arthur Powell opens his account as follows:

In May and June of 1984, while conducting a series of mathematics teacher education workshops in Beijing, capital of the People’s Republic of China, I was introduced to some pedagogical problems in Chinese numeration. They involve the teaching and learning of how to speak numerals with fluency in Chinese, using Hindu-Arabic written numerals. A salient feature of these problems manifests itself when Chinese students attempt to read numerals longer than four digits. For example, even graduates of senior middle schools find it necessary to read 6,721,394 by first pointing at and naming from right to left the place value of each digit before knowing how to read the “6” in the millionth place and the rest of the numeral. (Powell 1986, p. 20)

Powell’s proposals with regard to a way of ameliorating this difficulty in this article relate to a suggestion generated by Gattegno’s ideas of the power of pedagogic modification of certain elements of number-naming systems in order to emphasise structure:

[this proposed alternative approach] allows learners to become aware of the regularity of Chinese numeration. It also helps learners to develop strategies for by-passing reading difficulties caused by the employment of a convention of delimiting digits which is contradictory to the linguistic structure of Chinese. (p. 20)

So, by putting a space or comma after every four digits (rather than three) and reading the delimiter as wàn, correctly spoken Mandarin Chinese numeration follows, rather than it having to be memorised.¹⁵ But this does raise the question of where, in regard to mathematics, does a specific language ‘stop’.

Finally, I was led to wonder, if place value can be so transparent in some systems, whether it becomes hard to think about change of base. But I cannot go into this here. The next core element of this commentary relates to the complex issue of numerical units.

4 Count Nouns and Mass Nouns: The Question of Units

The motivation for exploring the issues of this section arose in part from the interesting and important discussion of number classifiers in Chinese provided in Chap. 3, but also from my simple curiosity wondering why in lots of settings English number words have features of nouns that reflect both singular and plural forms: for instance, in the everyday expression ‘hundreds and hundreds’ compared with ‘two hundred (and) fifty-three’. Or, the spoken number following ‘ninety-nine’ is ‘one hundred’ or ‘a hundred’, yet the number following one hundred ninety-nine is ‘two hundred’). Why is it that ‘hundred’, when used as a power-of-ten unit, is singular (e.g. two hundred and forty-two, rather than two-hundreds and forty-two)? Why do number words put pressure on the straightforward singular/plural distinction in English? How does this play out in the units for the countable noun and are numbers themselves such units? As Wittgenstein observed, ‘Grammar tells what kind of object anything is’ (Wittgenstein 1953, p. 116); hence, this uncertainty (present also in singular or plural verbs) potentially reflects an ontological instability at the heart of (English-language) number.

The earlier discussion of Dienes blocks and the paper tabular presentation also had column headings that were called ‘thousands’, ‘hundreds’, ‘tens’ and ‘units’. Yet once numerals were used to replace the (multiple) blocks, the differentially marked plural forms vanished in both the corresponding spoken and written English. (With regard to fractions words in English, not least the question of whether ‘three-fifths’ is a singular or plural noun and how it differs syntactically from ‘three fifths’, see the next section.)

In order to pursue some of the challenges that were mentioned in the previous section, I wish to examine certain morphosyntactic aspects of number words in English. One broad distinction in English grammar (which has commented upon in the literature since at least the early 1900s¹⁶) is the distinction between count nouns and mass nouns (the latter is sometimes referred to as ‘non-count’ nouns, though the two categories are not the same – see Laycock 2010), albeit one currently eroding (as is the case with round numbers) in interesting ways.

Edward Wisniewski begins his chapter on the potential cognitive basis for such a distinction as follows:

English and other languages make a grammatical distinction between count nouns and mass nouns. For example, “dog” is primarily used as a count noun, and “mud” is primarily used as a mass noun. Count nouns but not mass nouns can be pluralized and preceded by numerals (as in “three dogs” but not “three muds”). Count nouns but not mass nouns can appear with the indefinite determiner “a” (as in “A dog ate the chicken” but not “A mud covered the chicken”). On the other hand, mass nouns can appear with indefinite quantifiers, such as “much” or “little” (as in “much mud” but not “much dog”), whereas count nouns can appear with indefinite quantifiers such as “many” and “few” (as in “many dogs” but not “many muds”).¹⁷ (2010, p. 166)

There are many things to be said about this distinction. One key observation concerns the potential that any English noun, in certain circumstances, can come to take on both count and mass aspects, rather than, as suggested above, that these are two disjoint noun categories. This is also marked by the failing distinction between fewer (count) or less (mass), likewise many and much. In a mathematics context, this flexibility can be seen in the nineteenth century with the terms ‘algebra’ or ‘geometry’, where mathematical developments (non-Euclidean geometry, Boolean algebra) subsequently enabled ‘an algebra’ or ‘two geometries’ to be spoken of (and related to highly significant shifts in the perception of the underlying mathematics being referred to). In the late twentieth century, ‘technology’ has morphed to allow ‘a technology’ or ‘digital technologies’. All instances of conventionally mass nouns admit count possibilities and characteristics, and vice versa.¹⁸

A second observation has to do with the way that mass nouns are quantified: traditionally, this was by using various instances of the ‘a unit of’ construction (e.g. ‘a slice of’ or ‘a loaf of’ bread, ‘a grain of’ rice), where the unit could always be quantified, i.e. was itself a count noun. (Though, contrariwise, seeing four as ‘a quartet of’¹⁹ permits all count nouns to be seen as mass nouns – albeit in the plural form – where the number words themselves can be quantified: two quartets of, three septets of, and so on.²⁰)

However, the most educationally significant thing by far that this distinction relates to in regard to this chapter and the previous one is the extent to which number words themselves (in English or other languages), when functioning as nouns (as they do in arithmetic), do so as mass or count nouns. This is centrally related to the passing comments I have repeatedly made so far about whether it should be ‘hundred’ or ‘hundreds’.

In the specific context of this chapter, however, my interest lies with English number words themselves in their nominal form, one, two, three, etc., and the corresponding ordinals, first, second, third (which may or may not function as nouns), and the somewhat bewildering connection in some languages between ordinals and fraction terms (see Pimm and Sinclair 2015, as well as the next section). With all of these sets of number words, the question is: mass or count?

Consider the English word count sequence ‘one, two, three, etc.’. One of the ambiguities in English in respect of multiplication has to do with whether it ‘should be’ four twos is or four twos are eight.²¹ Notice the distinct pluralisation of ‘two’ marks it as a count noun, as does the ‘count’ word ‘four’, as does the verb agreement of ‘are’ with the pluralised noun form ‘twos’. The presence of a count noun permits the question ‘How many?’ to be asked in relation to it (for much more on this, see Sinclair and Pimm 2015a). Yet in the count sequence one, two, three, etc., the number words act more like mass nouns. And, as always, what cognitive shifts or chasms underlie such linguistic uncertainties?

Look at the ordinal terms: first, second, third, fourth, etc. While it is possible to imagine scenes where a count noun perspective is possible (e.g. in an athletic meeting, asking a runner: how many firsts, and how many seconds?, meaning first places and second places), these act more like mass than count nouns. But notice what happens when we shift to the related fraction forms: again, we get two sevenths and three tenths, which are plural and count nouns (and would encourage this syntax for statements like ‘two sevenths are bigger than three tenths’). But then the unifying hyphen may show up (two-sevenths, three-tenths), with the effect of singularising these composite forms.²²

Once again, this section simply contains some brief comments and observations about number language in certain contexts. In the next section, I move towards aspects of a range of number word systems.

5 Cardinal, Ordinal and Fractional: Three Interlocking Linguistic Subsystems

Whole numbers are not the only game in town. Languages also have systematic ways of naming (of summoning, of calling into being) ordinals and fractions (decimals or otherwise) as well. Most of what follows specifically concerns the English language, though a more diverse discussion (concerning some twenty different languages) can be found in Pimm and Sinclair (2015), which explored variations among these three sets of number words across a variety of languages and language groups. The motivation to do so arose from the two papers in For the Learning of Mathematics (Bartolini Bussi et al. 2014; Pimm 2014), most specifically in relation to close links (in some languages near identical) between how ordinal words and fraction words are formed (and why this might be).

In an attempt to summarise some of what was found, here are four diagrams that reflect different relationships among cardinal (C), ordinal (O) and fraction (F) words within specific languages from my dataset. The arrows indicate ‘adding’ a suffix to the previous sets of words to form the new set. Figure 4.4(a) captures, e.g. Norwegian, while (b) exemplifies one common relationship (e.g. German): (c) is the ‘degenerate’ case of (b) that fits some Western European languages (e.g. English, French, Italian and Spanish), while (d) reflects Hungarian.

Fig. 4.4

(a–d) Various relationships among sets of number words within a single natural language

As I mentioned at the outset, one question I was led to examine was how these naming systems relate one to another within a given language, as well as how they relate to gestures on the one hand and to trans-linguistic written numerals on the other. Ordinality primarily concerns the sequential aspect of whole numbers, as opposed to their quantitative (cardinal) one. And there is a key and fundamental question about which came first, cardinal or ordinal. (For much more on this, see Seidenberg 1962 and Sinclair and Pimm 2015a.) But spoken ordinal terms carry a significant difference from cardinal terms in that the core issue becomes which one comes before or is said before or after another, rather than which one is bigger or smaller. Thus, ordinality is strongly related to temporality rather than magnitude.

Here are two minute observations. First, there is a commonly employed, hybrid written notation that seems both to pull ‘Hindu-Arabic’ numerals into a specific language and to privilege the cardinal over the ordinal: 1st, 2nd, 3rd, 4th, 5th, etc. (even though in French it is 1^er, 2^ième, 3^ième, 4^ième, 5^ième, etc.). The second one is specific to English and relates to the supposed cardinal counting decade words: both ‘thirty’ and ‘fifty’ show explicit ordinal over cardinal traces – ‘thir-ty’ as the third ‘-ty’, ‘fif-ty’ as the fifth ‘-ty’ – a visible (and audible) trace, not least because of the distinction between the English words ‘three’ and ‘third’ and ‘five’ and ‘fifth’ (from the two closely related English language systems of cardinal and ordinal words), whereas neither ‘four’ and ‘fourth’ exactly ‘fit’ ‘forty’.

There is an ordinal regularity in English after five, both of forming ordinals ‘from’ cardinals and the presumed economy (and greater ease of pronunciation) of potentially dropping the ‘-th’ suffix from a possible, historical sixthty, seventhty, eighthty and ninethty. But in regard to my discussion in Pimm (2014) of ‘the fifth part’ (in regard to the singularity of unit fractions in Ancient Egyptian arithmetic), there is some appreciable specificity implied, in that ‘the sixthty’ (seen as ‘the sixth ‘-ty’’) would have to be unique and ‘a sixthty’ or ‘two sixthtys’ would not be feasible.

6 A Few Concluding Remarks

The main focus of this commentary piece has been to draw attention to certain features of number language, both language-specific ones (mostly in regard to English) and across certain classes of language (in terms of the presence or absence of certain distinctions, such as mass/count or classifiers) that may have some pertinence or significance in learning to number and to count. But underneath it has been an attempt to keep an ear and an eye out for ‘traces’ (‘memories’ in the terminology of Chap. 3) of what has passed before or en route (both within individuals and within cultures) to our present-day set of practices and forms with regard to number.

In particular, in my attempt to localise place value away from natural language and primarily into written symbolic notation systems (though it is important not to forget physical manifestations of the same, such as with khipu – see Chrisomalis 2009), I have endeavoured to make distinctions among the interlinked systems of language, notation and the world. In regard to mathematics education, far more generally, the potential overvaluing of cardinal number as the pedagogically presumed dominant form with regard to arithmetic and mathematics has some serious consequences, as has the consequent downplaying of ordinality and its significant role in learning how to count (see Tahta 1991, 1998; Sinclair and Pimm 2015a, b; Coles 2017).

In Fig. 4.5, there is my first attempt at trying to depict this (even though I already can see problems, oversimplifications, omissions and errors). It draws on the distinction between metaphoric and metonymic relations, as outlined in Tahta (1991, 1998), which he links with the abacist and the algorist, respectively: the use of physical objects (which become metaphors for number) versus the ‘manipulation’ of numerals.

Fig. 4.5

Metaphor and metonymy in relation to the interlinked systems of natural language, notation and the world

Tahta writes:

Metaphor and metonymy are not necessarily distinct polarities, but more like aspects that can be stressed or ignored as desired. One of our problems in teaching arithmetic is the move from the stress on metaphor to the stress on metonymy. We offer children counters and rods and so on, in order to mimic processes which we eventually want them to transfer to written or spoken numerals. (1998, p. 6)

As an individual becomes more and more numerically fluent, the separation between number words and numerals becomes less and less: but this does not mean that those distinctions and separations cease to leave their traces.

One final observation: Chrisomalis’s (2010) fascinating book on the history of numerical notation is over five hundred pages long. The world and its (linguistic) history in regard to whole number is a very complex and sophisticated mix. But also an engaging and, at times, fascinating one.