Over the past fifteen years, the Internet has triggered a boom in research on human behavior. As growing numbers of people interact on a regular basis in chat rooms, web forums, listservs, email, instant messaging environments and the like, social scientists, marketers, and educators look to their behavior in an effort to understand the nature of computer-mediated communication and how it can be optimized in specific contexts of use. This effort is facilitated by the fact that people engage in socially meaningful activities online in a way that typically leaves a textual trace, making the interactions more accessible to scrutiny and reflection than is the case in ephemeral spoken communication, and enabling researchers to employ empirical, micro-level methods to shed light on macro-level phenomena. Despite this potential, much research on online behavior is anecdotal and speculative, rather than empirically grounded. Moreover, Internet research often suffers from a premature impulse to label online phenomena in broad terms, for example, all groups of people interacting online are “communities”; the language of the Internet is a single style or “genre.” Notions such as community and genre are familiar and evocative, yet notoriously slippery, and unhelpful (or worse) if applied indiscriminately. An important challenge facing Internet researchers is thus how to identify and describe online phenomena in culturally meaningful terms, while at the same time grounding their distinctions in empirically observable behavior.

This series for Cambridge University Press is becoming widely known as an international forum for studies of situated learning and cognition.

Innovative contributions are being made by anthropology; by cognitive, developmental, and cultural psychology; by computer science; by education; and by social theory. These contributions are providing the basis for new ways of understanding the social, historical, and contextual nature of learning, thinking, and practice that emerges from human activity. The empirical settings of these research inquiries range from the classroom to the workplace, to the high-technology office, and to learning in the streets and in other communities of practice.

The situated nature of learning and remembering through activity is a central fact. It may appear obvious that human minds develop in social situations and extend their spheres of activity and communicative competencies. But cognitive theories of knowledge representation and learning alone have not provided sufficient insight into these relationships.

This series was born of the conviction that new and exciting interdisciplinary syntheses are underway as scholars and practitioners from diverse fields seek to develop theory and empirical investigations adequate for characterizing the complex relations of social and mental life and for understanding successful learning wherever it occurs. The series invites contributions that advance our understanding of these seminal issues.

The term community is widely and often uncritically used to characterize two kinds of groups that are central to this book. First, there are groups that come together to learn, in classes, workshops, and professional associations. Most professionals would refer to these as classes, workshops, and associations. Some educators like to refer to all of these kinds of groups as learning communities. The second kind of group is one that participates in an electronic forum (e-forum), such as an Internet Relay Chatroom, a professional listserv, a distance education course, or an online auction. It has become equally commonplace to refer to such groups as communities – virtual communities. This book examines issues for the developers and participants of electronic forums that could facilitate learning.

We believe that the casual use of the term community to characterize groups that are engaged in learning, or groups that participate in e-forums, is seriously misguided. As we shall see, developing a group into a community is a major accomplishment that requires special processes and practices, and the experience is often both frustrating and satisfying for many of the participants. The extent to which a group develops certain desirable community-like characteristics should be based on empirical observation rather than on assumptions or aspirations.

WIRING SCHOLARLY NETWORKS

Rapid developments in computer-mediated communication are associated with a paradigm shift in the ways in which institutions and people are connected. This is a shift from being bound up in small groups to surfing life through diffuse, variegated social networks. Although the transformation began in the pre–Internet 1960s, the proliferation of the Internet both reflects and facilitates the shift.

Much social organization no longer fits a group-centric model of society. Work, community, and domesticity have moved from hierarchically arranged, densely knit, bounded groups to social networks. In networked societies, boundaries are more permeable, interactions are with diverse others, linkages switch between multiple networks, and hierarchies are flatter and more recursive. People maneuver through multiple communities, no longer bounded by locality. They form complex networks of alliances and exchanges, often in transient virtual or networked organizations (Bar & Simard, 2001). Workers – especially professionals, technical workers, and managers – report to multiple peers and superiors. Work relations spill over their nominal work group's boundaries and may even connect them to outside organizations. In virtual and networked organizations, management by network has people reporting to shifting sets of supervisors, peers, and even nominal subordinates (Wellman, 2001).

How people learn is becoming part of this paradigm shift. There has been some movement away from traditional classroom-based, location-specific instruction to online, virtual classrooms. There has also been some movement away from teacher-centered models of learning to student-centered models and flatter hierarchical relations. Physically dispersed learning is part of this shift.

This volume brings together a series of chapters focused on the theoretical, design, learning, and methodological questions with respect to designing for and researching virtual communities to support learning. We are at an interesting time in education and technology, with terms such as communities of learners, discourse communities, learning communities, knowledge-building communities, school communities, and communities of practice being the zeitgeists of education and the Internet serving as a much touted medium to support their emergence. More generally, any time a new technology is introduced, it suggests the promise of the revolution of education. Thomas Edison was convinced that film would transform education and make the teacher obsolete. Although the Internet offers much promise and the potential to support new environments for learning, we are just beginning to understand the educational potential of community models for learning and whether community can be designed online or face-to-face. In fact, we know very little about whether something such as community can be designed and, if so, whether this can be done online. We are witnessing instructional designers employing usability strategies effective for understanding human-computer interactions, but we have little appreciation of how to design to facilitate sociability – that is, supporting human–human interactions as mediated by technology.

In the previous chapter, we analysed the relationship between numbers and objects. We encountered numbers as efficient tools in different kinds of ‘measurements’, that is, tools that we assign to objects in a meaningful way in different contexts. Our discussion has identified the different purposes numbers fulfil in these contexts, and has shown us how we manage to employ them to tell us something about empirical properties. Now that we have an understanding of what we use numbers for, and how we do that, let us explore the philosophical background of our discussion and see what would be a good characterisation of numbers, in order to carve out the area in which our investigation will take place.

What does it mean to be a number? One of the earliest definitions of numbers that have been passed on to us is an analysis that the Greek mathematician Euclid gave in his ‘Elements’:

So, are numbers multitudes? What must a definition of numbers account for? Which characteristics distinguish numbers from other entities? At the end of the nineteenth and beginning of the twentieth centuries, these questions were discussed intensely in the philosophy of mathematics.

The criteria-based approach characterises numbers as elements of an infinite progression that is used as a tool in cardinal, ordinal, and nominal assignments. This account gives rise to a range of possible number sequences. We are not looking for the number sequence anymore, but for sequences that are employed as numbers, sequences that fulfil the function of numbers in our daily lives. In the present chapter we explore the consequences that can be derived from this view. The criteria-based view carves out the principal area of numbers, it gives us the criteria to recognise possible number sequences; with this characterisation in hand, we can now ask which sequences it is that we use for numerical purposes: what are the ‘numbers’ we employ in number assignments, the entities that constitute the core of our cognitive number domain?

So, what we are looking for now are sequences that fulfil the three ‘number’ requirements and that are actually used in the different types of number assignments: we want a designated number sequence that can serve as the basis for the cardinal, ordinal, and nominal number concepts we will discuss in chapters 5 and 6. Let us have a look at how we come to grasp numerical tools first: which sequence is it that children initially encounter when they are exposed to numbers which sequence do they employ in numerical routines like counting?

A striking feature of numbers is their enormous flexibility. A quality like colour, for instance, can only be conceived for visual objects, so that we have the notion of a red flower, but not the notion of a red thought. In contrast to that, there seem to be no restrictions on the objects numbers can apply to. In 1690 John Locke put it this way, in his ‘Essay Concerning Human Understanding’:

This refers to our usage of numbers as in ‘four men’ or ‘four angels’, where we identify a cardinality. This number assignment works for any objects, imagined or existent, no matter what qualities they might have otherwise; the only criterion here is that the objects must be distinct in order to be counted. In a seminal work on numbers from the nineteenth century, the mathematician and logician Gottlob Frege took this as an indication for the intimate relationship between numbers and thought, a relationship that will be a recurring topic throughout this book:

And this is only one respect in which numbers are flexible.

CARDINAL, ORDINAL, AND ‘#’-CONSTRUCTIONS

Cardinal, ordinal, and ‘#’-constructions express concepts of cardinality, numerical rank, and numerical label. Cardinal measure constructions (as opposed to cardinal counting constructions) include concepts of measure functions as part of their meaning. The different numerical concepts are integrated in the semantic representation of complex number word constructions. In the following representations, a is an empirical object (in the case of cardinal constructions the empirical object a is a set), p is a progression, s is a set, and /θriː/ is an element of the English counting sequence C (that is, /θriː/ is an element of a set of numerical tools).

Semantic representations for the different classes of number word constructions

• Cardinal counting construction: three pens: ɛa[PEN⊕(a) ∧ NQ(a, /θriː/)]

• Cardinal measure construction: a pumpkin of 3 kg: ɛa[PUMPKIN1(a) ∧ NQ(KG(a), /θriː/)]

• Ordinal construction: the third player: ιa ∃p[PLAYER1(a) ∧ NR(a, p, /θriː/)]

• ‘#’-construction (ordinal): player #3: ιa ∃p[PLAYER1(a) ∧ NR(a, p, /θriː/)]

• ‘#’-construction (nominal): player #3: ιa ∃s[PLAYER1(a) ∧ NL(a, s, /θriː/)]

Explanations of the elements used in the formulae

1. ɛ: The ‘ɛ’-operator is used to model nominal terms: F(ɛx(G(x))) =df. ∃x(G(x) ∧ F(x)).

2. ι: The ‘ι’-operator is used to model definite terms: ‘ιx(F(a))’ stands for the most salient F, that is, the most salient realisation of a concept F.

3. F1: ‘ɛa(F1(a))’ stands for ‘one F’, that is, F1(a) is true iff a is a singleton of realisations of F; a singleton of realisations of F is defined as a set of Fs with the numerical quantity ‘one’;

4. hence, employing our definition of NQ and of English words as an instance of numerical tools, we can state that F1(a) is true iff NQ(a, /w∧n/) is true.

5. […]

What is the status of non-verbal numerals like ‘5’ or ‘V’? Obviously they are not linked to a particular language, but used cross-linguistically; for instance, both ‘5’ and ‘V’ are used in English as well as in French contexts (although they are not used universally). And they are also not part of a particular alphabet: roman numerals are not restricted to contexts where we use, say, the Latin alphabet, but can be used together with the Cyrillic alphabet as well as with Chinese script, and the same is true for arabic or Chinese numerals. Which numeral system is used in the context of a particular script is a cultural phenomenon. The usage is not based on an inherent link between numeral systems and particular alphabets, but rather is due to historical coincidences or to the relative advantages of different numeral systems. And as the case of arabic and roman numerals illustrates, it is also quite common that two (or more) numeral systems are used within the same culture and together with the same script.

The independence of non-verbal numerals from script is particular obvious in Arabic. In contexts where Arabic script is used, one normally finds the East-Arabic numerals, which, like the West-Arabic ones that are used in Europe, originated in the Indian Brahmi-script. This script goes from left to right (hence like Latin, but unlike Arabic script).

As a result of our investigation thus far, we have developed a view of numbers as a set of species-specific mental tools, and we have characterised counting words as verbal manifestations of this toolkit and as the first numbers our species might have grasped. In the previous chapter, identified language as that human faculty which gave us the mental background to develop a systematic number concept. I suggested a route to number in human evolution which started from iconic representations of cardinality, went via finger tallies and iconic sequences of words, and led to cardinal number assignments as the first context in which counting words might have been used as genuine numerical tools. Let us now have a look at the emergence of number not in human history, but in the individual development of children. What does the picture look like here? What route do children take when they acquire a systematic concept of number?

The present chapter is dedicated to these questions. In particular I will discuss the acquisition of cardinal number assignments, relating our view of counting words and number contexts to the psychological evidence for their representation in individual development. I show what role the numerical status of counting words plays in their acquisition, and how children learn to apply them to empirical objects for the first time. Again, we will find initial stages of iconic cardinality representations, and will see how the development of the language faculty supports the emergence of genuine number assignments.

A note on the definitions of the functions NQ, NR, and NL, which account for our concepts of numerical quantity, numerical rank, and numerical label, respectively:

In what follows, I introduce an ontology for our conceptualisation of number contexts that is grounded in the representation of numerical tools. In order to keep things simple, I focus on counting sequences as our primary numerical tools. However, according to our criteria-based approach there are also other progressions that can serve as numerical tools. When we acquire these progressions, their representation is integrated into our concept of ‘number sequence’.

The definitions are hence generalised in order to cover all sequences that are introduced as number sequences: instead of ‘counting sequence’ I use the general term ‘numerical sequence’ for any sequence that fulfils the number criteria and is conventionally used as a set of numerical tools (where the above-defined sequence C is one instance for such a sequence).

As shown in chapter 7 non-verbal numerals can also play this role. In this case, the final argument of NQ, NR, and NL, respectively (the argument identified as ‘ɑ’ in the definitions) is not an element of a counting sequence C, but for instance an element of the sequence A of arabic numerals whose initial element in these mappings is then the element corresponding to ‘one’, namely ‘1’.