# Scaffolding in Mathematics Education

**DOI:**https://doi.org/10.1007/978-3-319-77487-9_136-2

## Keywords

Support system Help Zone of proximal development Educative strategy## Definition

Scaffolding is generally conceived as an interactional process between a person with educational intentions and a learner, aiming to support this learner’s learning process by giving appropriate and temporary help. Scaffolding in mathematics education is the enactment of this purposive interaction for the learning of mathematical actions and problem solving strategies.

Scaffolding is an intentional support system based on purposive interactions with more competent others, which can be adults or peers; the support can be individualized (one teacher scaffolding one student) or collective (a group scaffolding its members in a distributed way).

The support consists of employing instructional means that are supposed to help learners with the accomplishment of a new (mathematical) task by assisting him/her to carry out the required activity through providing help at parts of the activity that aren’t yet independently mastered by the learner; this is to be distinguished from just simplifying the task by cutting it down into a collection of isolated elementary tasks.

Scaffolding aims at providing learners help that is contingent on the learner’s prior qualities and contributes to the development of knowledge, skills, and confidence to cope with the full complexity of the task; as such scaffolding is to be distinguished from straightforward instruction in correct task performance.

As a support system scaffolding is essentially a temporary construction of external help that is supposed to fade away in due time.

## Characteristics

### Tutoring Learning

The notion of educational support systems for the appropriation of complex activities was first introduced by Bruner in the 1950s in his studies of language development in young children. In opposition to the Chomskyan explanation of language development resulting from an inherent Language Acquisition Device (LAD), Bruner advocated a theory of language development that holds that parent–infant interactions constitute a support system for children in their attempts to accomplish communicative intentions. In Bruner’s view it is this Language Acquisition Support System (LASS) that “scaffolds” children’s language development.

- 1.
*Recruitment*: Scaffolding should get learners actively involved in relevant problem solving activity. - 2.
*Reduction in*degrees of freedom, i.e., keeping students focused on those constituent acts that are required to reach a solution and that they can manage while preventing them from being distracted by acts that are beyond their actual competence level; these latter actions are supposed to be under the control of the scaffolding tutor. - 3.
*Direction maintenance*: The tutor has the role of keeping students in pursuit of a particular objective and keeps them motivated to be self-responsible for the task execution.

Without explicitly mentioning the Vygotskian notion of the zone of proximal development, the formulations used by Bruner and his colleagues (see quote above) unequivocally refer to one of Vygotsky’s operationalizations of this notion (see Vygotsky 1978, p. 86) as the discrepancy between what a learner can do independently and the learner’s performance with help (support) from more knowledgeable others. In later explanations and elaborations of scaffolding, most authors have taken this notion of the zone of proximal development as a point of reference.

Using a Vygotskian theoretical framework, the work of Stone and Wertsch has contributed significantly to the understanding of scaffolding. Stone and Wertsch (1984) have examined scaffolding in a one-to-one remedial setting with a learning-disabled child. They could show how adult language directs the child to strategically monitor actions. Their analyses articulated the temporary nature of the scaffold provided by the adult. Close observation of communicative patterns in the adult–child interactions showed a transition and progression in the source of strategic responsibility from adult (or other-regulated) actions to child (self-regulated) actions. The gradual reduction of the scaffolding (“fading”) is possible through the child’s interiorization of the external support system (transforming it into “self-help”).

Stone (1993) made a critical analysis of the use of the scaffolding concept as a purely instrumental teaching strategy. He pointed out that until the early 1990s most conceptions of scaffolding were missing an important Vygotskian dimension that has to do with the finality of scaffolding for the learner. Especially the learner’s understanding of how the scaffolding and learning make sense beyond the narrow achievement of a specific goal adds personal sense to the cultural meaning of the actions to be learned through scaffolding. Stone refers to this dimension with the linguistic notion of “prolepsis” which can be seen here as an understanding in the learner of the value of the scaffolded actions in a future activity context. Until today many applications of the scaffolding strategy are still missing this proleptic dimension and neglect the process of personal sense attachment to the scaffolded actions.

The use of scaffolding in various contexts has led to different educative strategies for implementing scaffolding in classrooms with varying levels of explicitness as to the help given (for an excellent, recent, and very informative overview and empirical testing of scaffolding strategies, see van de Pol 2012). The most used scaffolding strategies with increasingly specific help are *modeling* (showing the task performance), *giving advice* (providing learners with suggestions that might help them to improve their performance), and providing *coaching* in the accomplishment of specific actions (giving tailored instructions for correct performance). Following Stone’s critique on current scaffolding conceptions, however, it is reasonable to add, as a useful educative strategy, *embedding*, which entails luring the learner in familiar sociocultural practices in which the new knowledge, actions, operations, and strategies to be learned are functional components for a full participation in that practice. This embedding in familiar sociocultural practices helps students to discover the sense of both these learning goals and the teacher’s scaffolding.

Attempts at employing scaffolding strategies in mathematics education can be generated from the above summarized general theory of scaffolding, provided that it is clear what kind of mathematical learning educators try to promote. If the formation of mathematical proficiency is reduced to learning to perform mathematical operations rapidly and correctly, then scaffolding should include *embedding* to make clear how the mastery of these operations may help students to participate autonomously in future practices. The choice for *coaching* on these specific actions in order to take care that they are mastered in correct form may be an important way of scaffolding the learning by repetition and practicing. If, however, the focus is on learning mathematics for understanding and hence on developing the ability of concept-based communication and problem solving with mathematical tools, a broader range of scaffolding strategies is needed. First of all the strategy of *embedding* is important: helping students to connect the actions to be learned with a sociocultural practice that is recognizable and accessible for them. One may think of practices like being a member of the mathematical community, but most of the time this scaffolding strategy consists in embedding the mathematical problem solving process in cultural practices like industrial design (e.g., designing a tricycle for toddlers), or practicing a third-world shop in the upper grade of primary school, or enacting everyday life practices (going to the supermarket or calculating your taxes). In a process of collaborative problem solving (and exploratory talk, see Mercer 2000) under guidance of the teacher, the teacher has to take care of the contingency of the actions and solutions on all participants’ prior understandings but also of tailoring the scaffolding to the varying needs of the students: *modeling* general solutions (if necessary, when the students have problems to find the direction of where to find the solution of the mathematical problem at hand), giving hints (i.e., *giving advice*, if necessary, when the group’s problem solving seems to go astray), or even stepwise *coaching* the execution of complex new actions when these actions are important for the resolution of the problem but go beyond the actual level of the participants’ competences. In this latter case it is important that the teacher sensitively monitors the contingency of the steps in the learning process in the students.

How to reconcile dialogical agreements in a group of students with the extensive body of proofs and understandings in the wider professional mathematical community? How can a teacher scaffold the students’ processes of becoming a valid and reliable mathematics user in a variety of cultural contexts?

How to scaffold the emergence of mathematical thinking in young children that opens a broad and reliable basis for the development of rich and valid mathematical thinking? How can we meaningfully scaffold the process of learning to talk, informed by mathematical concepts? Although practical and theoretical know-how is currently being expanded (see van Oers 2010; Fijma 2012), further ecologically valid empirical studies are needed.

How can teachers scaffold the process of mastery of automatization in mathematics while maintaining the foundations of this process in understanding and meaningful learning?

How can teachers support the gradual fading of the teacher’s scaffolding and turn this external (interpersonal) scaffolding into a personal quality of self-scaffolding? For this it is necessary to encourage the students to make and discuss their own personal verbalizations of the shared concepts and solutions. More study is needed into this formation of personalized regulatory abilities on the basis of accepted mathematical understandings, using a combination of dialogue (interpersonal exploratory talk) and polylogue (critical discourse with the wider mathematical community).

## Cross-References

## References

- Fijma N (2012) Learning to communicate about number. In: van Oers B (ed) Developmental education for young children. Springer, Dordrecht, pp 253–270CrossRefGoogle Scholar
- Mercer N (2000) Words and minds. Routledge, LondonGoogle Scholar
- Pimm D (1995) Symbols and meanings in school mathematics. Routledge, LondonGoogle Scholar
- Sfard A (2008) Thinking as communicating. Human development. The growth of discourses, and mathematizing. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Stone CA (1993) What is missing in the metaphor of scaffolding? In: Forman EA, Minick M, Stone CA (eds) Contexts for learning. Sociocultural dynamics in children’s development. Oxford University Press, New York, pp 169–183Google Scholar
- Stone CA, Wertsch J (1984) A social interactional analysis of learning disabilities remediation. J Learn Disabil 17:194–199CrossRefGoogle Scholar
- van de Pol J (2012) Scaffolding in teacher- student interaction. Exploring, measuring, promoting and evaluating scaffolding. Dissertation University of Amsterdam. Faculteit der Maatschappij- en Gedragswetenschappen, AmsterdamGoogle Scholar
- van Oers B (2010) The emergence of mathematical thinking in the context of play. Educ Stud Math 74(1):23–37CrossRefGoogle Scholar
- Vygotsky LS (1978) Mind in society. Harvard University Press, CambridgeGoogle Scholar
- Wood D, Bruner JS, Ross G (1976) The role of tutoring in problem solving. J Child Psychol Psychiatry 17:89–10CrossRefGoogle Scholar