Keywords

Lev Semyonovich Vygotsky (1896–1934) was born in present-day Belarus. He was, by all accounts, a man of great personal charm. In 1913 he qualified as part of a three percent Jewish student quota for enrolment at the Moscow State University. There he studied law and developed an early additional interest in the arts, including the study of philosophy and history . His own experiences as a Russian intellectual living in tumultuous times during the profound social upheaval of the 1917 Socialist Revolution contributed to his wish to create a psychology that engaged both the laws of science and of society. The policy work he was charged to carry out for the education of marginalized (and often homeless) children of the time led him to support the Marxist principle that people change history and, in the process, people are themselves changed, or as he put it, ‘human nature has changed in the course of history ’ (Luria & Vygotsky, 1992, p. 41). His interest in the psychology of change was formalized when he attended the Institute of Psychology in Moscow and completed his dissertation on ‘The Psychology of Art’ in 1925. That interest became focused on developmental psychology and, in particular, on the ways that we shape who we are through others, by appropriating cultural resources, which are, in turn, historically inflected. He came to the conclusion that the mind is not intrapsychic; rather it is the emergent outcome of cultural-historical processes.

Considered a controversial figure within the Soviet Union, during his own lifetime, Vygotsky failed to gain the same international stature that the Swiss developmental psychologist, Jean Piaget , achieved for his work on human development. Circumstances of the time in the Soviet Union contributed to the suppression of much of his creative work, and ‘selective editing’ and dubious translations of the work that was published did not help matters. It is only during the past three decades that his ideas have become influential in Russia, thanks principally to members of the Vygotskian circle, an informal network of scholars from a range of disciplines, who sought to preserve the legacy of Vygotsky’s ideas. Although Vygotsky’s ideas were initially introduced to North America during the late 1920s, they did not become widely known in the West until around the 1970s in the form of the book Mind in Society, presenting new (yet not always accepted) approaches to developmental and educational psychology . In time, the ideas became increasingly influential not only to developmental psychology but also to other disciplines as diverse as anthropology, philosophy, business, sociology , communication and systems design.

As with many of the other theorists discussed in this book, Vygotsky was involved in a wide range of interests and the development of a number of theories over the course of his academic life. Disciplinary fields and areas of interest as diverse as the philosophy of science, education, developmental psychology , methodology, the psychology of art , the relation between language and thought, and also between learning and human development, learning disabilities and abnormal development, play, and the construction of concepts served to capture his attention. Accompanying the changes in interest were changes in relation to his own thinking. Minick (1987) has recorded that Vygotsky moved “from ‘the instrumental act’ and the ‘higher mental functions’…to the emergence of ‘psychological systems’” (p. 24) and then to his third and final argument that ‘the analysis of the development of word meaning must be carried out in connection with the analysis of word in communication’ (p. 26). We cannot claim to know ‘one true’ Vygotsky. Indeed, he did not wish to be known as such, arguing that ‘he wished his ideas to be used, transcended, and even refuted’ (Daniels, Cole, & Wertsch, 2007, p. 9).

His interests and the changes in thinking that emerged become apparent in his prolific writing over a short course of time, including Pedology of the School Age (1928), Pedology of the Juvenile (1934), Educational Psychology (1926) and Outlines of the Development of Behaviour (1930), to name a few. However, it was the relationship between speaking and thinking that became an enduring interest. In his most-read text, Thinking and Speech (1934), he introduced the idea of the mediational role of speech, analysing how speech as an example of a cultural artefact shapes possibilities for thought and action, while simultaneously showing how speech itself is shaped by those who use it. He wrote over 250 scientific articles and six of his most significant works were written over a period of only 10 years. In their translation into English, however, the scholarly integrity of the texts was sometimes lost with a resulting distortion and misinterpretation of the original intent of the ideas.

In his short 38 years, Vygotsky became a pioneering psychologist, claiming that the discipline of psychology , through its two divisions—naturalistic psychology and idealistic psychology—and as offered at universities during the early 1900s, failed to fully account for the complexity that characterizes human personality. As he argued, ‘The tragedy of all modern psychology consists in the fact that it cannot find a way to understand the real sensible tie between our thoughts and feelings on the one hand, and the activity of the body on the other hand’ (Vygotsky, 1933a, pp. 196–197). In his view, thinking can be explained from a developmental perspective by linking higher mental functions to their origin. For him, at the heart of a child’s psychological development was the child’s social development. At that time, at least in the United States, the study of children was a relatively small low-status enterprise. The study of developmental change focused on adaptations to the environment and often using procedures drawn from Pavlov initially developed to study small animals. In contrast, while he also took an evolutionary approach to knowledge and human development, Vygotsky emphasized the importance of social interaction in human development and undertook laboratory work that explored how, in interactions with others, a child develops new ways of thinking and being and, in the process, new social mechanisms are developed. Put another way, he sought to find out how, through new complex mental function interrelations, new patterned ways of the child’s behaviour are formed.

Further to this proposal and with his group of students including Leont’ev (1978, 1981), Luria (1979) and Luria and Vygotsky (1992), Vygotsky (see Roth & Lee, 2007) created a research programme to analyse the origin of complex mental functions, including the constructs of selective attention, memory , language understanding and cognitive functioning development. This programme of work was marked by an attempt to understand complex mental functions from the perspective of the way in which people use signs and symbols which are necessarily nested within cultural practices, including language (whose meaning necessarily changes over time), to shape memory and reasoning processes. From there he was able to propose his theory of the significant integration of human consciousness. The theory was an explanation of the ways in which cultural and social interaction patterns contribute to the forms of mediation and developmental trajectories.

His experimental studies were undertaken not simply for providing answers to theoretical problems. They have made significant contributions to a range of areas. For example, in psychiatry they have moved the study of schizophrenia forward. In cognitive science, the influence is apparent in the move from intelligent tutoring systems to interactive computer programs. Applied to education, the ideas lie at the core of contemporary interpretations of social constructivism, sociocultural theory, societal-historical theory, cultural-historical theory and situated cognition. In educational psychology the ideas form the basis of pedagogical practices that emphasize interactions within the classroom. In general and special education, the ideas are used to predict child development. They also reveal how children take up new knowledge, the influences of that take-up and the consequences of that experience in terms of development, knowledge, preferences and the mental tools learned. Put simply, Vygotsky’s studies revealed how children’s development is influenced by their interactions with other people and the wider social environment which encompasses cultural and historical artefacts and practices. In this formulation, interactions do not merely play a part as a social modifier in the process; they are deeply intertwined with human development. As Vygotsky (1978, p. 58) has explained:

Every function in the child’s cultural development appears twice: first, on the social level, and later, on the individual level; first, between people (interpsychological) and then inside the child (intrapsychological). This applies equally to voluntary attention, to logical memory , and to the formation of concepts . All the higher functions originate as actual relationships between individuals.

This is not to suggest a one-way process in which children mirror their everyday world. Rather than being passive or acted upon, children actively construct knowledge through these interactions . By way of example, in the classroom, teachers stand at the front, side or back of the room. They write on the board, show PowerPoint slides, assign mathematical tasks, offer explanations and feedback, listen and notice, facilitate discussions, set homework, evaluate students’ book work and check attendance. Students sit at desks, work, watch and listen to the teacher and other students and work from textbooks and worksheets. In all these activities, in Vygotskian understanding, the content and contours of what happens constantly change from one moment to another and from one lesson to another and in relation to their cultural and material reality. The mind is not intrapsychic; rather it is the emergent outcome of cultural-historical processes. Since thought and mind are social, the question of the student’s thinking becomes a question of how the student acts upon the activities and transforms them. As Luria (1979, p. 23) has noted, building on Vygotsky’s ideas:

We should not look for the explanation of behavior in the depths of the brain or the soul but in the eternal living conditions of persons and most of all in the external conditions of their societal life, in their social-historical forms of existence.

Like all ideas, Vygotsky’s did not develop in a vacuum. With inspiration from de Spinoza’s (1989/1677) seventeenth-century work and building on ideas provided by a number of others such as Marx and Engels (1968/1890, 1978/1924), he proposed an explanation from a developmental point of view, of the ways in which higher cognitive functions develop in children . He was interested in Freud’s work but soon came to the realization that Freud’s theory of unconscious intention, as grounded in the individual, set up a dualist relation with the social. Similarly, he found that Marxist ideas, centred on an individual psychology , did not adequately account for the part that social factors play in thinking and being. Piaget’s work, he believed, was simplistic in its understanding that thinking first emerges as pleasure seeking, with no relation to social reality. Reacting to the individualistic and ahistorical tendencies that dominated psychology, Vygotsky undertook to conceptualize a social and cultural theory of mind. He proposed that reasoning emerges through practical activity in the social environment, and it is this proposal that provided a significant contribution to the genesis of the psychological functions of the child.

Vygotsky’s explanations of cognitive development, known as Social Development Theory or Cultural-Historical Theory, made a significant contribution to developmental psychology . They mark a specific tradition of thought, namely, the dialectical tradition, whose key players are often named as Marx, Hegel, Vygotsky and Il’enkov. Dialectical thought expresses the idea that every life is constantly in motion and changing in a structured way (Hegelian lineage). It proposes inner contradictions that are always in a process of resolving and understands nature and the mind as interdependent. The dialectical tradition offers a rich intellectual tradition, formed, as it is, from a complex mix of ideas about the nature and development of human life (both individually and collectively), embodied in an interrelated network of concepts . Drawn from a monistic approach to psychology, a modern materialist view focusing on the conditions for human development, and historical materialism, the composition marks the ‘specific scientific tradition to which [Vygotsky] was objectively related and to which he subjectively attached himself’ (Davydov & Radzikhovskii, 1985, p. 39). The framing of Vygotsky’s work within a dialectical tradition is important since it allows us to capture the full intent, rather than a restricted version of the intent, of those ideas.

In Vygotsky’s view, the idea of dialectic as applied in psychology represents the study of consequences of material change, focusing on the whole of human practice. Conceiving of Vygotsky’s thought as grounded within the dialectical tradition does not overlook the point that it is possible to detect traces of his work that reveal an allegiance with other traditions. The argument made to the effect that his oeuvre is consistent with dialectics is intended to challenge those who restrict their analyses to piecemeal pickings of his work, neglecting the way that Vygotsky taps into the ideas of Hegel. Dialectics provides a way of conceiving human interaction , learning and development, different from theories that fail to address the interaction of mind and society (those being intrapsychic theories of cognition). Hegelian dialectics was meant to challenge the dualistic views of thinking and being and the closure of the spectrum divided by binary pairs. The dialectical tradition has been critiqued more recently for not adequately addressing how cognition and material reality are enmeshed. A dialectic is not quite a dichotomy, but it is a theory that rests on contradiction, negation and resolution of contesting forces. And yet dichotomies, as fundamental as these are, and fixed oppositions such as micro and macro, internal and external, mental and material, individual and social, thought and action, quantitative and qualitative, observation and intervention, agency and structure all conceal the extent to which they are in fact interdependent (see Derrida, 1978). In other words, binary pairs derive their meaning from an established contrast where one term is prior to or dominant over the other. Scott (1988) and Lloyd (1984), among others, have taken this point further to argue against the western philosophical tradition itself (and indeed against dialectics as itself a play of pairings), drawing attention to its foundations as resting on binary oppositions such as unity/diversity, identity /difference and presence/absence (see the chapter on Deleuze and Barad for a similar approach to subverting dichotomies and dialectics of contradiction). In this re-evaluation of dualistic views of thinking and being, it is important to note that one term is not able to be reduced to the other. We cannot separate one from the other.

Post-Hegelian theorists have critiqued the dialectical tradition for still clinging to a sense of negation and contradiction between members of a paired couple (master-slave, for instance). Central to the tradition are a number of premises, some of which have an ontological basis and some of which are epistemologically derived. Not all theories of the dialectic are the same, and it’s important to read originary sources (such as Hegel or Marx) to ensure that your particular approach is grounded in the philosophical tradition. For example, for some dialecticians, the environment is not conceived of as a cluster of objectively specifiable states or conditions. Rather, people actively change their material conditions. The potential for change is influenced historically through a range of experiences and practices, situated, for example, within the physical environment, the material environment within the immediate context such as the workplace, home, leisure and local or within the mediated context such as social networks or communities within national and international contexts, all of which may require action such as problem solving and meeting new demands. In interacting with these experiences and practices and as captured through language use, people themselves change. The individual and the environment are mutually constitutive, although it is evident that dialecticians have historically tended to emphasize the human power in this dialectical relationship, while neglecting the force of matter and the material (see chapter on Barad in this book). For dominant interpretations of Vygotsky, thinking is embodied and situated and distributed across material and social settings (see Latour for how he takes this approach further). Thinking for Vygotsky happens from experience, and it is possible to suggest that cognition develops in and for the purpose of action, and so it’s interesting to consider what he might have made of current neurocognitive research on the plasticity of the brain.

Vygotsky’s analyses reveal a strong commitment to studying the human mind in the process of becoming. Like Piaget , he was interested in an analysis of the historical conditions of human life and how this required an understanding that people are constantly changing. Vygotsky said he was influenced by Spinoza, although it is important to mention that philosophers have shown how Hegel misconstrued and misrepresented Spinoza who was not a dialectical thinker. Whether Vygotsky was drawing on Hegel’s version of Spinoza or his own is not yet completely clear. Setting aside such issues for now, Vygotsky said, ‘The individual becomes for himself what he is in himself through what he manifests for others’ (Vygotsky, 1931b, p. 105). Whereas the way in which we express ourselves may be the result of a range of influences, nevertheless, the words we use for the expression do not speak themselves. The corollary may be stated in this way: the individual has the status of agency, yet that agency derives from history , culture and society. This kind of argument provided Vygotsky with a frame for understanding not only for how people might engage in the creation or transformation of conditions but also for understanding how particular material and social conditions might contribute to or hinder possibilities for people to achieve full developmental potential.

The Key Theme in Vygotsky’s Work

For Vygotsky, all learning is social. It takes place primarily through cultural and psychological tools. Cultural tools derive from human cultural and historical activity . They represent what human beings within groups, communities and societies have developed over time in order to assist people in thinking about, reflecting on and representing their values, ideas, feelings, principles and practices. Such tools embody a social intelligence, to the effect that members of the society share an understanding in relation to the symbolic meaning of and purpose for the tools. For Vygotsky, the development of higher mental functions is associated with the mastery of social practices: ‘…social relations, real relations of people, stand behind all the higher functions and their relations…[T]he mental nature of man represents the totality of social relations internalized’ (Vygotsky, 1931, p. 106). It is through using cultural tools such as language , symbols, road signs, technology , music, art , writing, painting, music and dance, among others, that people become aware of their own thoughts. From there it is a small step to critical reflection and self and social transformation.

Concepts Fundamental to Vygotsky’s Work

Internalization is a process in which an individual begins to be able to gain control over external processes. It signals that the individual has ‘picked up’, informally in many instances, through her engagement in a range of practices, certain cultural tools and is able to use them. Together, these tools which include basic linguistic and conceptual structures, and the use of fundamental psychological tools and techniques, form a set of cultural concepts and forms of thought and reasoning. By way of example, in a Western classroom, a student is asked to order a set of numbers . She is asked to use specific cultural tools of her society that, as a much younger student, would have been beyond her capabilities. When the student has learned the number system and is familiar with how to use it for ordering numbers, she can engage in the ordering activity , selecting those numbers that are smaller or larger, and if she applies the relevant numerical rules correctly, she is drawing on higher mental concepts to engage in the activity as a member of her mathematical learning community.

The process of internalization can be understood to some extent as ‘knowing how’: the student has both learned the cultural tools and how to use them. She has developed a higher-order mental function, in this case, the ability to compare and order. In another context that higher mental function could include the ability to analyse, to remember, to generalize, to make deliberate intentional movements, to consciously pay attention to something, to put categorize, to reason logically and so forth. Once the student has developed a repertoire of concepts and forms of numerical thought and reasoning, the concepts and numerical operations are internalized when she is able to ‘make them her own’. The basic numeracy concepts and operations may have been picked up as cultural tools, but it is only when the student is able to use them as a vehicle for her own activity and actively deploy them that we can speak of her agency in the process. It is important to note that the cultural tools relating to numeracy are typically considered to be inert. Semiosis challenges this understanding, invoking the dialectical tradition to suggest that students and others who use numeracy concepts exert limited or no control over their use. ‘Signs are not mere instruments. They exert an agency of their own’ (Colapietro, 1993, p. 178). Even as we might understand that we think, speak and act for ourselves, our capacity to think and act is produced by mutually reinforcing collaborative activities and practices.

Internalization , as Vygotsky has explained, is not a simple matter of transplanting a social activity onto an inner plane, precisely because in the process of internalization, the internalized practice is transfigured. To that end, the student’s development in mathematics, as it is in any other realm, is a process of individualizing the social. However, as Roth (2012) pointed out, in the Vygotskian understanding, internalization is also a process of socializing the individual. The processes of individualizing and socializing should be understood as representing the same developmental process.

The student may, of course, take up the cultural tool of ordering numbers , in her own unique way. The concept of appropriation is used to describe that process. She may, for example, list, in order, all the even numbers separately from her ordered list of odd numbers. Rather than providing a single list of ordered numbers, she takes up the tool and makes it her own for her own ends. The student has learned a tradition of thought, but has offered a critical reflection of that tradition. Vygotsky argued that the goal of education should not be focused on students’ assimilation of received wisdom but rather, that it should be aimed at enhancing students’ independent critical appreciation and interrogation of mathematical concepts (and other concepts emanating from other disciplines) that they encounter. He points out :

For present-day education it is not so important to teach a certain quantity of knowledge as it is to inculcate the ability to acquire such knowledge and to make use of it….Where he [the teacher] acts like a simple pump, filling up students with knowledge, there he can be replaced with no trouble at all by a textbook, by a dictionary, by a map by a nature walk….Where he is simply setting forth ready-prepared bits and pieces of knowledge, there he has ceased to be a teacher. (Vygotsky, 1997a, b/1926, p. 339)

Vygotsky believed that mental tools extend mental abilities. In formulating his concept ‘tools of the mind’, he expressed the view that such tools are necessary in order to find creative solutions to both small- and large-scale problems. Developing in students ‘tools of the mind’ such as independence and critical appreciation and interrogation tools, rather than focusing on transmitting facts, will contribute not only to cognitive development but also to their physical, social and emotional development. Ultimately it will enable them to make worthwhile contributions to economic, political and social life .

Mediation

Students develop higher mental functions, moving from everyday mental functions and concrete thinking to more abstract thinking, through mediated, social and collaborative activity . Mediation brings about qualitative changes in thinking with the use of cultural tools and signs such as ‘language; various systems for counting; mnemonic techniques; algebraic symbols; works of art ; writing; schemes, diagrams, maps, and mechanical drawings; all sorts of conventional signs’ (Vygotsky, 1981, p. 137). As students engage in various practices, they ‘pick up’ a cultural toolkit which is, in the first instance, initiated or scaffolded by others. Through mediation and the use of cultural tools, students come to internalize the social expression of preferences, feelings and so forth, learn strategies for everyday living and come to reflect critically on their own wants and needs. Through the process they move their dependency of explicit forms of mediation to more implicit forms such an inner speech, shifting their dependency on others towards an independence associated with remembering, internalizing and using the cultural tools.

It is through the mediation of others, through the mediation of the adult that the child undertakes activities. Absolutely everything in the behavior of the child is merged and rooted in social relations. Thus, the child’s relations with reality are from the start social relations, so that the newborn baby could be said to be in the highest degree a social being. (Vygotsky, 1932)

From birth a child’s responses to the world are shaped by constant intervention of adults and significant others. For example, the child’s attention might be drawn to the rain falling outside, or she hears the same words repeated frequently, or she might be read a story. In Vygotskian understanding, the significant others in the child’s life are mediating the child’s contact with the world and with the people and objects in it. The role as external mediating agents will in time be minimized as the child begins to initiate the processes herself. The processes that were initially inter-psychological, shared between child and adult, over time become intra-psychological, marked by the child’s own similar responses to the world. In this framework, the cultural tool is the ‘subjective reality of an inner voice, born of its externalization for the Other, and thus also for oneself as for the Other within oneself’ (Vygotsky, 1929, p. 17, original emphasis).

The Interrelationship Between Thought and Language

Vygotsky’s formulation of the relationship between thought and language development is significant in any discussion of key concepts . In Thinking and Speaking, working on the assumption of different developmental roots for thought and speech, he argued that a critical point in development is reached when the two pathways converge. Thought, in Vygotskian understanding, represents the development of mental concepts and cognitive awareness and is, for example, manifest in basic problem solving activities, whereas speech represents both inner speech and oral language and is manifest in primitive communicative utterances. It is important to note that the use of ‘self-talk’ or ‘thinking out loud’ begins with the primary purpose as a tool for social interaction and later becomes a tool for self-regulated behaviour, taking the form of inner speech.

According to Vygotsky, the moment of significance occurs when thought becomes linguistic and speech becomes rational (see a critique of this emphasis on the linguistic in the chapters on Barad and Deleuze). That moment arises when the child utterances are deemed meaningful to her and are used by her as a form of communication in relation to that meaning. Thinking, of course, can occur without language. However, when thinking is mediated by language, it develops to a more sophisticated level. Thought, at that moment, then begins to encroach consciousness or the system of all higher mental functions, since, as Vygotsky (1932) has argued, meaning has become their common currency. Her perception of the world begins to take on a meaning for her. However, since meaning is derived from shared understandings, her perception of reality is highly normative. As Vygotsky has noted, ‘[C]onsciousness as a whole has a semantic structure ’ (Vygotsky, 1933b, p. 137).

The Psychology of Play

Play, Vygotsky proposes, is an important step in the development of a child. It marks the beginning of the decontextualization of meaning. When a child engages in play, she signals that she is able to think about something, despite its lack of presence, and, importantly, that she has grasped the meaning of that entity. She demonstrates that she is able to move beyond the visual field to the field of sense or meaning. Since very young children are not able to imagine objects, the decontextualization process, as Vygotsky has expressed it, is a specifically human form of conscious activity . Thus the development of abstract meaning of a concrete object is a critical feature in the development of higher mental functions. In other words, the point at which meaning rather than physical object comes to predominate is a mark of fundamental significance in relation to cognitive development.

In imaginary games and pretend play situations, the child is developing social rules of behaviour and speech. Walkerdine (1989) has provided compelling evidence of how the roles and talk of mothers, fathers, sons and daughters within families are acted out in play settings in nursery education, as young children adopted a role of a family member different to their own. In imaginary games and pretend play situations, the child may also use an object to stand for another. She is using a sign or a symbol. As she matures, that process will be internalized as she relies less and less on using an object for another. She is using a sign or a symbol.

For the young child, play represents for her a transitional stage in distinguishing meaning from an object (Smagorinsky, 2001). The play involves the application of the child’s own ‘rules’ that relate to the meaning she has attributed to the situation or object. By way of example, a block in the manipulative box is picked up by a child who uses it as a mobile phone. Applying the rules of mobile phones, the child holds the block to her ear, speaks into it and then holds it up and uses it to take a photograph. In representing a mobile phone, the block of wood becomes a ‘pivot’ for separating the meaning of phone from a real phone. Indeed, in the child’s mind, the block of wood is the mobile phone. Using the example of the use of a stick for a horse, Vygotsky explained:

In a critical moment when for a child a stick is a horse, i.e., when an object (a stick) constitutes a prop for separating the meaning of a horse from a real horse, the fraction becomes reversed and the sense: sense/object becomes predominant. (Vygotsky, 1978, p. 80)

Zone of Proximal Development

The Zone of Proximal Development (ZPD) is generally taken to be Vygotsky’s most far-reaching concept . It encapsulates the key objects of attention specific to the discipline of psychology and offers credible explanations for relationships between those objects of attention. According to del Río and Álvarez (2007), the ZPD is a ‘frontier territory [encompassing the] situated-embodied mind and the cognitive mind, the individual mind and the social mind, the development already attained and the development to be attained’ (p. 277). The concept offered a credible explanation for the relation between a student’s learning and her cognitive development. It offered an explanation that was at odds with a number of leading positions at the time. For example, it contrasted with the following understandings: that development always precedes learning, that learning and development occur simultaneously and that learning and development are separate but interactive processes. In his prospective view, development always follows the student’s potential to learn.

While the ZPD is Vygotsky’s best known concept , at least in the Western world, it is the least understood. For example, Lave and Wenger (1991) have argued that the operational definition of ZPD has been interpreted in many ways. They note interpretations ranging from a ‘scaffolding’ , a ‘cultural’ and a ‘collectivist/societal’ interpretation of the original formulation of ZPD. For example, Engeström (1993), taking a broad perspective focused on social transformation beyond formal pedagogy, has defined the ZPD as the ‘distance between the everyday actions of individuals and the historically new form of the societal activity that can be collectively generated’ (p. 174). Within formal education, the ZPD is sometimes cited as a justification for pedagogical practices that are in fact incompatible with Vygotsky’s intention. As a result, the discussion of the ZPD in educational circles often misses the key point that instruction leads to development (see Chaiklin, 2003). To clarify the intended meaning of the ZPD, in Vygotsky’s own words, the ZPD is the distance between a child’s ‘actual developmental level as determined by independent problem solving’ and their higher level of ‘potential development as determined through problem solving under adult guidance or in collaboration with more capable peers’ (Vygotsky, 1978, p. 86).

The concept first arose as a consequence of attempting to reconcile a number of paradoxical results found from students’ intelligence test results. His response to the dilemma was to postulate the initial and terminal thresholds within which development could take place, as a way of finding out, as Van der Veer and Valsiner (1991) have pointed out, the critical periods associated with ‘mental age’ to accomplish certain educational goals. In other words, he was interested in finding out the difference between what an individual can do independently and what can be accomplished with assistance, or what the individual can do when ‘stretched’. In Vygotsky’s formulation, ‘actual developmental level characterizes mental development retrospectively, while the zone of proximal development characterizes mental development prospectively’ (1978, pp. 86–87). The ZPD was not conceptualized as a permanent state but, rather, as a stage towards independent knowing or acting. It was Vygotsky’s methodological approach for dealing with the need to anticipate the course of development. Vygotsky (1978, pp. 85–86) explained:

When it was first shown that the capability of children with equal levels of mental development to learn under a teacher’s guidance varied to a high degree, it became apparent that those children were not mentally the same and that the subsequent course of their learning would obviously be different. This difference …is what we call the zone of proximal development. It is the difference between the actual development level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaborations with more capable peers.

The initial threshold of ZPD represents the level of development for the child working independently, whereas the terminal threshold represents the level of potential development that the child might reach with assistance. It is important to observe that the ZPD opens development to diverse possible trajectories. It not only considers development in terms of an individual’s psychological growth. It also considers development as set within a cultural, social and political setting. Of fundamental significance to the ZPD is the notion of potential or, more correctly, ‘proximal’. Learning is most effective when the student is introduced to new concepts that are, for the student, on the cusp of emergence.

Scaffolding

Scaffolding is a concept often used in association with the concept of the ZPD. It was first used by Wood, Bruner, and Ross (1976) as an instructional metaphor to explain the process whereby adult assistance enables a student to solve a problem, carry out a task or achieve a goal beyond her unassisted efforts at that point in time. For them, scaffolding involved the control of task elements by an adult to enable the student to perform the task within her current capabilities. However, Wood, Bruner and Ross did not make explicit reference to Vygotsky’s work. Cazden (1979) was the first author to draw such links. Interestingly, Vygotsky himself never used the term. The way in which scaffolding is interpreted in relation to Vygotsky’s work distinguishes between support for the initial performance of tasks and subsequent performance without assistance. It requires a sensitivity to the level of support or the incremental change in information required in order to support the learner up to the highest level they can achieve with support, that is, in order to meet the cognitive potential of the child. Or, more simply, applying Vygotsky’s (1978) own words, ‘[T]he distance between problem solving abilities exhibited by a learner working alone and that learner’s problem solving abilities when assisted by or collaborating with more-experienced people’ (p. 86).

In some interpretations, the concept of scaffolding is perceived as an offer of a preordained ‘climbing frame’ in the form of hints and supports that contribute to the transfer of skills from the more or less capable partner. In many ways this kind of perception reinforces a view of a rigid scaffold aligned with behaviourist teaching principles. Newman, Griffin, and Cole (1989) argue against this view, maintaining that the ZPD is created through negotiation between the more capable partner and the student. Moll (1990) agrees :

Vygotsky never specified the forms of social assistance to learners that constitute a ZPD… he wrote about collaboration and direction and about assisting children ‘through demonstration, leading questions, and by introducing the initial elements of the task’s solution’; but did not specify beyond these general prescriptions. (p. 11)

Applications

Sociocultural theories, once seen as on the fringe of a mainly cognitive field, now take their place squarely within mainstream mathematics education journals….Concepts such as ‘communities of practice’, ‘learning as participation/belonging’, and ‘out-of-school math’ are being used by researchers. The shift toward social issues has allowed us to uncover the importance of students and teachers needing to belong to something larger and for changes in one’s identity to serve as evidence of learning. As such, it has opened doors for researchers to study classroom culture, participation structures, socialization processes, and teacher professional development in whole new ways. (Gutiérrez, 2013, p. 2)

Since the late 1980s, the mathematics education literature has experienced what Tsatsaroni, Lerman, and Xu (2003) have named as a ‘social turn’. Researchers draw primarily from the disciplines of cultural and social psychology , anthropology and cultural sociology , and each of these offers broader theoretical tools for interpreting the social origins of knowledge and thought. By the ‘social turn’, Lerman (2000) intends to convey the ‘emergence into the mathematics education research community of theories that see meaning, thinking, and reasoning as products of social activity’ (p. 23). These theories have enabled the exploration of a broader range of research questions and issues that theoretical traditions such as cognitivism and behaviourism, grounded in a positivist paradigm, would permit. Vygotsky’s theory of cognitive development (alternatively known as Social Development Theory and as Cultural-Historical Theory) introduced the possibility of accounting for individual cognition and difference as informed by social activity and as a consequence, new perspectives, topics, questions and methodologies soon became apparent, providing social and cultural dimensions to mathematics education.

Reference to Vygotsky’s work, as Lerman (2000) has noted, first appeared in a review of Wertsch (1981) by Crawford (1985), published in the journal of Educational Studies in Mathematics (ESM). Crawford also referenced Vygotsky in the 1988 PME proceedings, and in that same year, Bishop (1988) published an article, drawing on Vygotsky, in ESM. The following year, Cobb (1989) published an article, referencing Vygotsky, in For the Learning of Mathematics. In employing Vygotsky’s work, the intent of these researchers was to understand mathematics teaching and learning rather than to predict it. The trend soon gained momentum. Acknowledged by Lerman (2000), Gutiérrez (2013) and Morgan (2014), among others, the use of Vygotskian psychology by researchers has received a clear expression within the proceedings of the conferences of the International Group for the Psychology of Mathematics Education (PME). Jablonka, Wagner, and Walshaw (2013), in their analysis of PME proceedings 2007–2012, found that the sociocultural frameworks of Vygotskian and neo-Vygotskian theories were more highly cited than any other theoretical framework.

Goos (2004) has drawn on Vygotsky’s theoretical tradition to analyse the development of a classroom community over time. She made use of the concept of ZPD as a framework for exploring teaching and learning practices within a secondary school classroom, in general, and for investigating, in particular, the way in which participants were ‘pulled forward into mathematical inquiry’ (p. 262). Fundamental to the exploration, in keeping with the dialectical tradition, was the agency of the learner. In the analysis the ZPD was utilized from three perspectives: (1) as scaffolding, (2) student-student collaboration and (3) interweaving. The scaffolding perspective begins in the understanding that teachers support students’ mathematical activity by establishing classroom communities in which there is access to social, discursive, visual and technological resources for mathematical understanding. Peer collaboration takes as its starting point the idea that collaborative activity within a small supportive environment allows students not only to exchange ideas but also to test those ideas critically. Interweaving begins in the understanding that teachers or expert others socialize students from their intuitive worlds into a larger mathematical world that honours standards of reasoning and rules of practice.

Scaffolding practices that enhance mathematical learning are the key focus of a study by Anghileri (2006). Drawing on evidence of support strategies provided in the literature, a hierarchy of pedagogical interactions was established between teacher and students that was found to promote new learning. The three levels, beginning with the most basic scaffolding strategies, were offered as environmental provision, explaining, reviewing and restructuring and finally developing conceptual thinking. The levels were illustrated with examples drawn from data from young children learning geometry and from older students’ learning numeracy.

Vygotskian ideas will be utilized as a means to explore the way in which one teacher (Ms B) pulled her students forward in a unit on algebra. However, the analysis deviates from Anghileri’s by proposing that every social activity performed by the teacher, from environmental provision to explanations, and so forth, as part of a larger matrix of practice, has the potential to foster conceptual thinking in students. Thus, all classroom activities are taken as the unit of analysis.

Mercer (2000) has argued that teachers tend to use specific linguistic strategies to strengthen the connection between students’ motivations, knowledge and competencies and the curriculum-based goals of the activity , in an ongoing way, to allow students to enhance their present existing knowledge and to consolidate their new knowledge as a shared understanding. In using linguistic strategies as tools for guiding, monitoring and assessing the activities that they organize for their students and in the context of national pronouncements of effective mathematics teaching, teachers build student confidence, establish norms of participation, shape students’ mathematical language , elaborate, clarify, ask questions, summarize previous knowledge and relate that knowledge to new knowledge, press for understanding, revoice students’ thinking, provide cognitive structure and fine-tune mathematical thinking and make connections between students’ contributions. In all these activities, language is used as a tool for describing and consolidating a shared experience and understanding within the class. That is to say, the development of shared understanding is a joint activity between and mutual achievement of teacher and student. It is an outcome of what is made possible and what is ‘taken up’ within the classroom environment (Holzman & Karliner, 2005). If the teacher’s talk fails to keep the student’s mind attuned to the teacher’s, scaffolding loses its impact and the development of shared understanding is minimized. It is not simply an issue of whether or not specific language techniques are in use. Rather, in the study at hand, it is an issue of how those techniques are used to create and maintain shared knowledge.

At the time of the study, Ms B had been teaching for around 20 years and had been head of the Mathematics department for the past 12 years at the single-sex girls’ school in which the research took place. In the following analysis and through a Vygotskian framework, we unpack her teaching practice and, in particular, her teaching of algebra to an accelerated class of average age 13 years. Grounding the analysis is the idea that students’ mathematical development is influenced by their interactions with others and practices within the classroom environment which necessarily encompasses cultural and historical artefacts and practices. Interactions do not merely play a part as a social modifier in the process; rather, they are deeply intertwined with the development of modes of thinking.

In creating an understanding of the teacher’s practice in relation to the development of students’ mathematical thinking, a three-step process of identification and categorization was carried out in relation to the strategies she used to move students’ thinking forward. First, from observations of and field notes made of the unit of work spread over two weeks, a number of preliminary categories were developed. In the second phase, evidence of further categories was sought from viewing of the video transcripts and these were added to the categories already established. In the third phase of the process, the categories were matched against data from an interview with the teacher and an interview with four of the classroom students in order to create a connection between the teacher’s activities and the students’ activities in the classroom.

Step 1: Preliminary categories of effective mathematics teaching, as identified from observations and field notes, were listed as follows:

  • Establishing norms of participation

  • Linking tasks and new knowledge to students’ prior knowledge and existing proficiencies

  • Initiating and connecting ideas

  • Eliciting information to determine students’ understanding of a new idea

  • Arranging for peer learning

  • Arranging for individual thinking time

  • Providing constructive feedback

  • Building student confidence

  • Valuing students’ contributions

  • Revoicing students’ responses

  • Encouraging mathematical argumentation

  • Providing opportunities for students to explain and justify thinking and solutions

After viewing the videos and reading the transcripts, during the second phase of the process, evidence was found for an additional four categories. These were:

  • Providing challenge

  • Using artefacts to assist in the development of knowledge

  • Providing opportunities for students to monitor progress and understanding

  • Pressing for understanding

  • Providing cognitive structure and fine-tuning mathematical thinking

  • Making connections between ideas

During this second phase of the process, the category ‘revoicing students’ responses’ was merged with ‘providing constructive feedback’, and the category ‘building student confidence’ was merged with ‘valuing students’ contributions’. Similarly, the category ‘linking tasks and new knowledge to students’ prior knowledge and existing proficiencies’ was joined with ‘initiating and connecting ideas’ to form a new category ‘providing tasks and introducing and consolidating new knowledge in ways that align with students’ current knowledge and existing proficiencies’.

A third phase offered a further means for developing theoretical insights about the activities in this class. In this phases the category ‘familiarizing students with and modelling the use of mathematical conventions and language ’ was added, and the categories ‘pressing for understanding’ and ‘providing cognitive structure and fine-tuning mathematical thinking’ merged to become ‘fine-tuning mathematical thinking by intervening or providing intermediate steps or by ‘pressing’ students in order to move understanding forward’. In addition, the category ‘encouraging mathematical explanation and argumentation in whole-class discussion’ was formed from the amalgam of ‘encouraging mathematical argumentation’ and ‘providing opportunities for students to explain and justify thinking and solutions’. The interview data, along with close inspection of the video data relating to the teacher’s and the students’ actions in the classroom, expanded the categories and provided a validation point for the categories already developed. The entire process was iterative in that it involved continuous searching, both forward and back, for evidence of activities related to the development of mathematical thinking in the classroom.

From the evidence it was possible to draw up a list of teacher activities and associated student activities, relevant to three specific domains that had emerged: situational activities, pedagogical activities and mathematizing activities. In comparison, Goos (2004) identified three perspectives—scaffolding , student-student collaboration and interweaving—as central to the students’ development of mathematical inquiry. The domains that emerged in the current study also contrasted with Anghileri’s (2006) hierarchy of pedagogical interactions , consisting of the level of environmental provision, the level of explaining, reviewing and restructuring and, finally, the level of developing conceptual thinking.

The resulting framework developed from the data is illustrated in Table 2.1. A more detailed explanation is offered, along with classroom evidence, following. In viewing the framework, it is important not to conceive of the classroom activity as atomistic, but rather, as a complex interrelated unit in which social, cultural, historical and cognitive elements come into play. Thus, the classroom activity, as the unit of analysis, is dynamic; it is able to grow and change. Vygotsky (1994) clarifies further, ‘It is not just the child who changes, for the relationship between him and his environment also changes, and the same environment now begins to have a different influence on the child’ (p. 346)

Table 2.1 Classroom activity related to the development of mathematical thinking
Full size table

Different Kinds of Activity

Explanation of Notation Used for the Following Sections

  • T: teacher

  • S: student

  • L4 (2.38): lesson 4, 2 min 38 s into the lesson

  • Ms B: the teacher

Situational Activities

The daily practices and rituals of this classroom provided students with ‘insider’ knowledge of what to do and say, mathematically, from the norms associated with those daily practices. This knowledge evolved as students took part in the ‘socially developed and patterned ways’ (Scribner & Cole, 1981, p. 236) of the classroom. By scaffolding the development of those patterned ways, the teacher regulated the mathematical opportunities available in the classroom.

L1 (2.59):

T: Okay, quickly. (Teacher does two or three heel raisers to indicate she’s waiting). So you’re not writing anything, you need to look this way. Let’s just go over a few things.

L1 (3.25):

T: Just a couple of reminders about what you need to have in class and you need to make sure you bring your textbook every lesson and I expect that to be out on the desk as soon as you come in, so these are the books that are out on your desk. Your exercise books and your notes.

L3 (22.33):

T: I want you to stop and listen. I know some of you want to jump into it, but there’s a couple of things I want to remind you about from the start of the year. I want you to only choose a few, and I’m not interested in you doing the first five, okay? Beginning, middle, end. [Then] I want you to mark. You might do a couple of questions and then you mark it. If you’re getting something incorrect, you’ve got to find out there and then what to do to correct them.

L3 (23.11):

T: So this is an opportunity for you to get a little bit of extra practice on this and push yourself, right?

If, as in Vygotskian understanding, all learning is social, then mathematical thinking begins with a taken-as-shared sense of the expectations and obligations of mathematical participation. In this classroom the teacher worked at creating social norms surrounding behaviour and participation in mathematical discussion.

L1 (17.02):

T: Okay, so I want you to compare your own answers with the person next to you, have you got the same answers or have you got different ones, okay? There are lots of different questions I’ve asked you, so I’d like you to discuss this with the person next to you and you might even get into a discussion with more of you, okay, because you might have different stuff. Go.

The way in which students view their relationship with mathematics is influenced, as Whitenack, Knipping, and Kim (2001) have argued, by the value that is given to students’ thinking and their contributions. By validating contributions and asking further questions with the intent of allowing other students to access knowledge, the teacher used students’ ideas to shape instruction and to occasion particular mathematical understanding in the classroom.

L1 (31.13)

T: Factorize, so when I say factorize, what is it that you do? What is it that you do? Suata?

 

 

S: Um, so you just do the opposite of what you did with the expanding.

 

T: Okay. So with the expanding, I multiply it out, so what am I going to do to go backwards? Do you want to talk me through?

 

S: Um, divide it?

 

T: Yeah, what kind of, yep, you are sort of dividing. How do I know what to take?…Keep going, Suata?

L1 (32.41)

S: Um, you find the common factor from both of the things.

Claire: Question 2, there’s another way of doing it too. 2, 8× and 2 again.

 

T: Okay good, I’m actually, so 2, in brackets 8x minus 2. Okay? I’m actually really pleased that Claire’s brought this up because it highlights something quite important.

Social nurturing and confidence building within this classroom were related to an overall goal structure that included consistent affective support. This support conveyed the message that student ideas were valued. In turn, the positive support from the teacher encouraged further student effort—an effort that was consistent with their own demonstrated proficiencies. Ella explained:

Ella: …with Miss B there’s not many things that anyone dislikes about maths, like I wasn’t such a fan of maths last year and this year it’s one of my favourite subjects.

In return for the support from the teacher, students were expected to monitor their own progress and understanding. This was a gradual process for the students, as the teacher explained:

These are bright girls, you know. They’re pretty sharp, but to encourage them to actually do some thinking and take some responsibility for their learning was a real challenge. In the sense that many of them had been used to been given a worksheet and they just do that, you know they’d do the same kind of concept 50 times. They weren’t able to apply their knowledge to think, and, and to make those connections themselves. And also to learn for understanding as opposed to a set rule that you just regurgitate and you just keep doing that 100 times. So the start of the year was exceptionally challenging.

The classroom was organized to include peer group work, providing a rich forum for students to develop their mathematical thinking and understanding. As Ms B pointed out:

One of the things that they do really, really well is the way that they interact and work in groups and they discuss things with each other and they help each other…They’re really keen to share with someone else and to share their understanding.

The students volunteered that the peer groups served as an important resource for developing their mathematical thinking. They asked their peers about the nature of task demands and how those demands could be met. In the course of working through problems with another student, students extended their own framework for thinking. Benefits accrued as they listened to what their peers were saying and tried to make sense of it and coordinate it with their own thoughts on the situation. Ella explained that she ‘really enjoy[ed] talking to other people and discussing the problems and finding the answer’. However, she also noted a limitation of such strategies and the need for the teacher to arbitrate between and simplify competing conjectures. ‘Sometimes I think that, I think it…makes it harder to understand so that’s why it’s good to have Ms B to sort of just simplify it and explain it for me’.

Pedagogical activities

The teacher in this classroom purposefully provided information and asked questions of her students. The approach, as Lobato, Clarke, and Ellis (2005) have proposed, is directed at developing students’ conceptual knowledge rather than their memory skills. This form of telling does not take away from students the agency for making sense of mathematics (Hiebert & Wearne, 1993). More specifically, she negotiated meaning through ‘telling’, tailored to students’ current understandings. She appeared to dilute her own knowledge into a less polished, less final form, working backwards from a mature understanding of the content, as a means of understanding students’ current thinking. She shared and then transferred responsibility so that her students could attain greater agency. In this classroom, telling was followed by a pedagogical action that had the express intent of finding out students’ understandings and interpretations of the given information.

L1 (33.28):

T: Have I got the highest factor here? [referring to problem as noted at 32.41]. If you look at the 8 and the 2, there’s still something common, agree? Okay? So this is actually a really important point, that you’ve got to factorize it fully.…A lot of students can make the mistake of writing that as the answer but it’s not fully factorized.

Mathematical conventions and language were important in the teacher’s lesson. She focused on shaping the development of her students to speak the precise language of mathematics. In endeavouring to do this, she made connections between ideas, distinguishing between terms, sensitizing students to the particular nuances between them. By reframing student talk in mathematically acceptable language , she provided students with an opportunity to enhance connections between language and conceptual understanding.

L1(16.15)

T: Okay, so on this side, up the top, they’ve called them expressions. Why have they called them expressions? What’s an expression as opposed to an equation? What’s the difference?

S: Is that because there’s like the variables instead of…

T: There’s variables instead of?

S: Numbers.

T: Numbers. So can you have variables instead of numbers for equations and expressions? Could you? What’s the difference ? Rebecca?

Rebecca: Is it because there’s no equal sign?

T: There’s no equal sign to an expression. Do you notice that all of these don’t have an equal sign? Whereas an equation will have an equal sign.

L5(10.45)

T: What happens to my 8?

S: It’s cancelled.

T: Yeah, I usually like to use the word ‘simplify’ rather than ‘cancel’. Okay, so why do they simplify? Because I know as things go, I can cross that off and I can cross that one off. Yes, Rebecca?

In particular, she drew out the specific mathematical ideas embedded within students’ methods, shared other methods, clarifying understanding of appropriate mathematical conventions. By reframing student talk in mathematically acceptable language , she provided students with an opportunity to enhance connections between language and conceptual understanding. The students, for their part, believed that their teacher ‘was really good at explaining things and really clear’ [Ella]. As was further explained:

 

Ella: [Our previous teacher earlier in the year] didn’t cater for our needs as much as Miss B did, like if you didn’t understand something, she wouldn’t explain it as well as Miss B could.

Maddy: [Ms B] is always a bit structured in the way she does it and it kind of fitted us.

Later she pointed out:

 

There’s a lot of simplifying the equations, so that it’s more easy to figure out and also expanding and factorizing as well. And we’ve spent quite a lot of time on those and also finding, like a balance on either side so there will be like an equal sign in between that we had to figure out what was on the side of the equation.

Interviewer:

Yes, so that’s a different idea about the equal sign from what you perhaps in primary [elementary] school were used to.

Michelle:

I really, I found it strange. Like there was a different meaning about equal signs. A balance, I found that quite hard to get used to.

Ella and Maddy had their own views of their particular class:

Ella:

Basically we’ve been put in our classroom because we are all accelerant, and being in the mathematics classroom, we often get very challenged in our problems. So, all the time we’re doing things that extend us beyond our capabilities to try to get us to try new things.

Maddy:

It’s like at, at we get more of a challenge and because, I guess, we enjoy that most. Well, I do enjoy the challenge.

An element of challenge was embedded in the lessons. Alton-Lee (2003) has argued that teachers who provide moderate challenges for their students signal high expectations. Their students, in turn, report higher self-regulation and self-efficacy together with a greater inclination to seek help. In Lesson 3, after Ms B had carried out a ‘what’s my number’ exercise, she explained to the class:

T:

…here’s your challenge. What I would like you to do is write the algebra with that. I want you to use algebra to prove what happens. Okay, so I’ll just write down the instructions up on the board and I’d like you to use algebra to prove it.

Mathematizing Activities

Ms B’s presentations modelled the use of multiple representations, meaningful exploration and appropriate mathematical justification, often in the format of a class presentation of a solution. Successive presentations would sometimes illustrate multiple ways of approaching a problem. She invited students to offer explanations, and her questions, comments and feedback revealed that she was seeking justifications and meaning. Morrone, Harkness, D’Ambrosio, and Caulfield (2004) have argued that when a teacher ‘presses a student to elaborate on an idea, attempts to encourage students to make their reasoning explicit, or follows up on a student’s answer or question with encouragement to think more deeply’ (p. 29), the teacher is able to provide an incentive for the student to enrich that knowledge.

L3 (34.55)

T: First of all what is it that you’re trying to do when you’re solving equations? What is it that you’re trying to do?

S: Find the answer.

T: Find the answer. What do you mean, find the answer? The answer’s four. What…?

S: Find the missing thing.

T: Find the missing thing. What’s the missing thing?

S: The unknown.

T: Find the unknown. Good, okay. I’ve got one question for you, and I might have to leave that hanging until you get up to Year 11 or 12 [age 15/16 and 16/17]. Could you have more than one answer?

From whole-class discussions and particularly from the students’ physical response to her teaching, Ms B could gauge the student’s thinking and the way in which the knowledge had been internalized. She explained:

[What is particularly rewarding about teaching this class] is their development and their, you know, their light switch goes on when they’re like, oh I get that, oh. And then they’re really excited about something that they understand…and you know, when you’re sort of working with the class and they sit there nodding at you and you can see that the light, you know, that things make, they make those connections.

Ms B used ‘revoicing’ to fine-tune her students’ mathematical thinking. She repeated the students’ talk in order to clarify or highlight content, extend reasoning, include new ideas or move discussion in another direction. Through careful questioning and purposeful interventions, she resolved competing student claims and addressed misunderstanding and confusion. For their part, the students took increasing responsibility for the important activity of making conjectures, for providing justifications and for practices of generalization (see Whitenack et al., 2001).

In whole-class discussions, students were invited to contribute answers to questions and expand on the teacher’s explanations that were offered in a measured pace. These answers were put under interrogation within whole-class discussion. For example, the class had been working on solving 3x + 2 = −7.

L3 (44.37)

S: Now that’s 3x plus 2 equals negative 7. It doesn’t make sense because what the, what the answer, the x is…

T: Shall we go through it and solve the rest of it?

Ss: Yes.

S: Because I know what the answer is to 4x plus 2 equals x minus 7, but now I don’t know.

T: Okay, let’s go through and finish it off. Right, so from here what are we going to do? Yep, Grace?

S: 3x equals negative 9.

T: Can you just talk us through that, how you got to the negative 9?

S: I take 2 away from both sides.

T: Yep, okay. And if you notice the working that I’m using, this is what I quite like, okay? For some of you it might work, for some of you it may not. So I go negative 7 take away 2 is going to give me negative 9. You’ve got to be really careful with your negatives, yep, happy?

T: Okay, last thing. X is equal to what?

1 (45.40)

Students: Negative 3.

2 (45.43)

Teacher: Yeah. You can put this line of working in, or you can go straight to the answer, it’s up to you.

T: Okay, now I just want to finish, there’s one other actual point that I want to make. How can I check my answer? How can I check my answer? How do I know whether I have this correct or not? How can I check that, what can I do? What can I do, Emma?

Ms B was not content to stop at students’ explanations. She pressed students forward and in the process, a conceptual shift was made by the student who had originally been confused. In whole-class discussion, the teacher sustained the discussion, by nudging students’ contributions in mathematically enriching ways. Her feedback on the strategies chosen to solve the problem became a rich resource by which students were able to gauge the quality of their own individual performances. The feedback signalled to them where they might move their thinking forward.

Ella summed up the mathematizing element of the classroom from her perspective:

[Regarding the class’s teacher earlier in the year] If you didn't understand something she wouldn’t explain it as well as Ms B could and if you, it was almost as though the way, it’s like, the activities that she was putting us through, one, they weren’t very challenging and two, they weren’t, we weren't getting the same benefit out of them as Ms B’s activities. Like the ones Ms B puts us through, they’re very good and they teach you like a lot of things and you actually learn a lot from them.

In Vygotskian understanding, higher psychological functions, such as mathematical thinking, emerge from societal relations and activity. In this framework, thinking, acting and being have their basis in the individual’s relation with history and culture. A Vygotskian approach to classroom activity that focuses on societal relations and activity can enrich our understanding of the development of mathematical thinking. In this approach, mathematical thinking is not a construct that can be understood in simply cognitive terms. Like all other development, it is, rather, ‘an aspect of a socio-historically specific institutionally defined setting’ (Wertsch, 1985, p. 212). It is constituted by past, present and potential relationships within constantly changing circumstances and conditions.

A Vygotskian approach is able to unearth what it is precisely that occasions new thinking in the mathematics classroom. It is able to analyse the dialectic nature of mathematical thinking, revealing that while it might appear we could contribute cognitive agency to the individual student, that agency derives from history , culture and society. Foregrounding societal relations allows us to shift our attention away from proposing mathematical thinking as an inner individual resource towards a proposal that ‘our capacities to act [and] think…in formal mathematics situations are produced by mutually reinforcing societal activity’ (Roth & Walshaw, 2015, p. 228). More specifically, the situational, pedagogical and mathematizing characteristics of the classroom become more than mediators of students’ cognitive development; they can then be named as its origins.

Summary

The key theme that underpins all of Vygotsky’s work is the cultural context. In Vygotsky’s understanding, we are constituted by our social experiences and our interactions with people, as well as by the ideas and the cultural tools we encounter and with which we engage throughout our lives. Contextualizing an individual’s development is a society’s organization of people, tasks and ideas. That is to say, the organization of the society within which we live and the people, ideas, beliefs, tools and value systems of the people within that society provide us with socially structured patterned ways for attending to tasks such as work, education and everyday matters and making available physical and mental tools to accomplish the many tasks we encounter. These all play a critical part in our constitution.

Cultural-historical approaches amplify the contingent and, in doing so, inspire a reshaping of the teaching imaginary . In these understandings, mathematics teaching revolves around the potential of the student, rather than demonstrated achievements, as the focus of teaching. Teaching occasions the development of students, through active participation that is characterized by negotiation and collaboration and transference of ownership of learning to the student.