Encyclopedia of Mathematics Education

Living Edition
| Editors: Steve Lerman

Activity Theory in Mathematics Education

  • Wolff-Michael RothEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-77487-9_4-13


Vygotsky Leont’ev Dialectics Consciousness Personality Change 


Activity theory is the result of an attempt to construct a psychology that draws on and concretely implements epistemological principles of materialist dialectics as K. Marx presented them (Leont’ev 1978; Vygotsky 1997). Like Marx’s Das Kapital, activity theory is intended to explain change, learning, and development as an immanent feature of a system rather than in terms of externally produced cause-effect relations.

History of Activity Theory

L. S. Vygotsky generally is recognized as the founding father of activity theory because he introduced the idea of tool-mediated activity as a way of overcoming on-going psychological ideas consistent with stimulus–response or disembodied thinking approaches to cognition. Responding to the crisis of psychology, he explicitly stated the need for developing a Marxist psychology. Expanding on Vygotsky’s work, A. N. Leont’ev articulated what is now known as second-generation cultural-historical activity theory in his Activity, Consciousness, and Personality (Leont’ev 1978), in which the first two chapters are devoted to establish the Marxist foundation of the theory. The third-generation activity theory was formulated in two different lineages. The Helsinki version originally established by Y. Engeström (1987) focuses on the structural-systemic aspects of activity, whereas the Berlin version, developed by K. Holzkamp (1993) and his colleagues, is a subject-oriented psychology that focuses on the person and consciousness. Fourth-generation cultural-historical activity theory builds on both third-generation versions and also includes emotions (affect) and ethics as irreducible fundamental moments of human activity (Roth and Lee 2007).

Minimum Unit

In all other psychologies, individuals and objects are the minimal units of analysis. Thus, for example, the transformation of a square into a parallelogram by means of shearing would involve a human agent, who, by acting on the square, would turn it into the result (Fig. 1a). The action is external to the object and thus external to the unit. The human subject and his/her actions are the causes for the transformation. Activity theory, on the other hand, conceives the situation in a radically different way. In this theory, the entire production of some outcome from the beginning conception to its material realization is the minimum unit (Fig. 1b). This unit bears an inner contradiction, because depending on how and when we look at it, we would see a square, a person, a parallelogram, none of which exists independent of the entire unit. Because of this inner contradiction, the unit is referred to as a dialectical unit; it sublates – simultaneously integrates and overcomes – what manifests itself in irreconcilable differences. If we were to look at school mathematics, then prior knowledge, post-unit knowledge, grades, teacher, and students would all be part of this minimal unit and could not be understood independent of it. By using this unit, change is immanent to the minimum category and does not require external agents. To understand the key principles, consider the following two scenarios.
  • Scenario 1: Connor and his peers in a second-grade mathematics class sort objects into groups, which become constitutive of geometrical relations within and between the objects.

  • Scenario 2: Erica, a fish culturist, talks about the production of coho salmon smolt to be released into the river to increase natural stocks; she monitors and controls the production process using a spreadsheet-based database and mathematical functions such as graphs, histograms, and mathematical functions (e.g., to calculate amount of food).

Fig. 1

(a) In traditional theories, people and objects are the minimum units; change is the result of outside actions on objects (elements). (b) In activity theory, the minimum unit encompasses the entire change process; it is impossible to speak of causes and effects

In the first scenario, the minimal unit would be schooling; as part of doing schooling, the second-grade students complete tasks. That is, not their grouping task, where a collection of objects is sorted is the activity, but the before and after in the context of schooling belongs into the minimal unit as well (Fig. 2). This is so because the ultimate productions that really count are grades and grade reports. Activity theory explains learning as a by-product in the production of grades. It does not account for mathematical activity as if it could occur outside and independent of the schooling context. In the second scenario, the ultimate product is a population of young salmon released into the river. The computer and the mathematics that Erica draws on are means employed in the production. What she can be observed to do is subsumed into the one category of salmon production – which contributes to increased opportunities for commercial fishing (generalized dietary needs), native sustenance fishing (specific dietary needs), and tourism focusing fishing (leisure) (Roth et al. 2008).
Fig. 2

The entire process by means of which a collection of objects into a groups of like-objects, together with institution, tools, and people constitutes the minimum unit

Structure of Activity

The structural approach, as embodied in the mediational triangle, is perhaps the most well-known and used version of cultural-historical activity. As Fig. 3 shows, it makes thematic seven moments that constitute the parts of the irreducible unit of productive activity: subject, object, means (of production), product, rules, society, and division of labor. All production is oriented towards ultimate consumption, which meets some generalized basic (e.g., food, shelter) or extended need (e.g., leisure). Schooling, in the course of which the second-graders complete the sorting, involves teachers and students who have different roles (division of labor), school buildings, (school) rules of engagement, and society. It is society that ultimately comes to be reproduced in the activity of schooling, both in terms of certain practices as in the hierarchical relationships between those who go to university and those who end up doing menial labor or drop out and never finish school. Society also benefits from the activity in which Erica is a part, because the salmon she contributes to producing ultimately lead to the generalized satisfaction of needs. Thus, dietary needs may be satisfied directly or in exchange of a salary for working on a fishing vessel or in the tourism industry (as fishing guide or maid in a hotel or lodge). What Erica does with the mathematical tools and with mathematics, for example, graphs, numbers, equations, and histograms, cannot be understood outside of the system as a whole. This is important for capturing the changing nature of the subject and its changing relation to the tool and object. Activity theorists are not interested in understanding the structure at a given point; rather, the entire transformation of goods into products is an integral part of the same unit. A better representation would be similar to Fig. 1, with two different triangles within the same unit. Nothing within the unit makes sense on its own. That is why activity theorists speak of the mediation of actions by the activity as a whole.
Fig. 3

Productive activity may be analyzed in terms of the seven moments that constitute a system; the products are exchanged, coming to be distributed in society, and ultimately are consumed (or used up)

Central to activity theory is the transformation of the object into a product, which initially only consists ideally. The intended transformation is the motive of activity. Activity theorists therefore speak of the object/motive; this makes thematic the in-order-to nature of all human activity. By definition, this category includes both material (the materials started with) and ideal dimensions (e.g., the envisioned product). For Erica, the intended outcome is clear. She wants to end the work cycle with a healthy brood of about one million coho salmon, with an average weight per specimen of about 20 g. In contrast, the second-grade students do not know the intended outcome of their task; and they are not likely aware of the ultimate motive of schooling (Roth and Radford 2011). As a result, they have to engage in the activity constituting task without knowing its object/motive. With respect to the task, they can become conscious of the reasons for doing what they do – that is, they can become aware of the goal – only when they have completed their task. It is when they realize the grouping in Fig. 2 that they can come to understand why the teacher, for example, asked Connor to rethink his actions when he placed one of the squares with the rectangles rather than with the other squares.

Subjectification and Personality

Cultural-historical activity theory allows us to understand two developmental processes. On the one hand, when a person participates in an activity, such as Connor in schooling or Erica in producing young coho salmon, they undergo subjectification. This concept names the process by means the person, together with everything else that makes the activity system, undergoes change. This change can be noted as the emergence of new capacities for actions of a body together with new forms of talk, neither of which has been identifiable previously. Together, these changes in objects, bodies, and forms of talk reconfigure the field of experience. Thus, for example, as Erica uses the spreadsheet to track and model the coho salmon population, she becomes more proficient with spreadsheets, mathematical models, calculating feed needs, graphs, histograms, and calculations. With these changes, her entire field of experience is reconfigured. Most importantly, activity theory does not allow us to speak of her development independent of everything else at her worksite; her transformation also means transformation of the entire field.

For Erica, working in the hatchery is only part of her everyday life, just as for Connor going to school is only part of his everyday life. Both participate in many other activities: as family members, shoppers, participants in leisure activities, or as members in urban traffic systems. That is, in the course of their everyday lives, both contribute to realizing other object/motives other than producing a population of young coho salmon and doing schooling. Leont’ev introduced the category personality to integrate all these object/motives that an individual takes on when participating in the correlated activities. Thus, personality is understood as a network of societal object/motives. That is, personality is made up of an ensemble of collective object/motives. However, each network is highly individual. Personality, therefore, is utterly singular while being entirely constituted by societal/collective moments.

A Holistic Psychology

Cultural-historical activity theory is a holistic approach to psychology. It does not reduce the individual to its thoughts (mental constructions). It in fact integrates body and mind, on the one hand, and individual and collective, on the other hand (Vygotsky 1989). It focuses on change as inherent in life and society and, therefore, inherent in individual life and understanding. That is, everything we do has to be understood in terms of this intersection of dimensions, which manifest themselves in mutually excluding ways: mind versus body, individual versus society, and the natural world versus the social world. The developments at the different timescales also are irreducible and therefore mutually constitute each other. The moment-to-moment changes – e.g., the transformation of a collection of objects in the second-grade mathematics classroom or the entering of fish size and weight into the database – are related to the development of system and people (i.e., subjectification) over time, and because people and systems are part of society, the cultural-historical changes of the human life form.



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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Applied Cognitive ScienceUniversity of VictoriaVictoriaCanada

Section editors and affiliations

  • Yoshinori Shimizu
    • 1
  1. 1.University of TsukubaGraduate School of Comprehensive Human ScienceTsukuba-shiJapan