Today, most math educators are familiar with the concept of scaffolding instruction in mathematics. But I worry that sometimes we speak of scaffolding, without carefully considering the appropriateness of its delivery. Some speak about scaffolding as ‘help,’ ‘show,’ or ‘tell,’ and even as a trick to be deployed, when students are not getting to the ‘right’ answer. Others speak of scaffolding as removing ‘barriers’ that get in the way of underperforming learners in mathematics.
To be sure, it is tempting to think of scaffolding as ‘helping’ students get to the ‘right answer’ and is the notion which thinking of the problems faced by underperforming learners as ‘barriers’, allows us to do. If ‘barriers’ are standing in the way of the ‘right answer,’ then removing them would make everything right. Which in turn makes us feel good, as loyal warriors of social justice. But when it comes to the complex activities of learning mathematics, reasoning with quantities, noticing patterns, and connecting ideas, perhaps ‘removing things’ is not as important as adding things.
Scaffold to (asset model), involves thinking of students as having a wealth of knowledge and cognitive skill waiting to be tapped into. An appropriate scaffold does not deprive the learners from thinking, nor does it bear the weight of ‘learning’—that should always reside with the learner. On the contrary, it unleashes prior knowledge, cultural experiences, linguistic skills, and cognition onto the learning process.
Scaffold from (deficit model), is a conceptual trap for teaching. It is also a learning trap, in that it involves thinking of barriers to the ‘right answer’ as ‘gaps’ in knowledge—a gap that needs to be ‘filled’ or ‘remedied.’ It can be deceptive, too. Sometimes what we perceive as a barrier, could in fact be a learning opportunity which will quickly go to waste, if we allow ourselves to see only deficits instead of assets.
The concept of scaffolding has evolved since it was first coined in 1976 by Wood, Bruner and Ross. They set out to investigate the role of tutoring, when a child was in a position to engage in solving a problem, but which would require skills that are initially beyond them. They were fascinated by what they thought was a uniquely human ability.
“What distinguishes man as a species is not only his capacity for learning, but for teaching as well … The acquisition of skill in the human child can be fruitfully conceived as a hierarchical program in which component skills are combined into ‘higher skills’ by appropriate orchestration to meet new, more complex task requirements” (Bruner, 1973,)
Through their seminal work, they understood that the interactions between an effective tutor and the learner involve a highly complex process, which they named ‘scaffolding’.
“Intervention of a tutor is much more than modeling and imitation … It involves much more than this… it is a kind of ‘scaffolding’ process” (Wood et al., 1976, 89-100).
They named the process ‘scaffolding’ as a metaphor, for the types of scaffold we are all familiar with in the construction or repair of buildings. They could not help but see the similarities between the construction of an actual building and the construction of knowledge. The beauty of the metaphor coined by Wood, Bruner, and Ross lies in its three features, also found on an actual scaffold: access, support, and temporal intentions.
Regarding ‘access’, the metaphor is right on point. On a construction project, scaffolds provide access to the crew, allowing them to go in and out at multiple locations of the construction project when needed and to the level that is needed. So too, a scaffolding in the learning process should provide access to the learners when and where it is needed and to the degree that it is needed (i.e., no more, no less).
The aspect of ‘support’, is a bit more of a stretch, but surprisingly also communicates an important idea. Notice how the scaffold in a building under construction does not bear the weight of the building; the building must stand on its own. The crew can use the scaffold to work on reinforcing the support beams or to investigate a problem, but it will never take on the weight. Likewise, when learning mathematics, scaffolding should never subtract the thinking from the learner; on the contrary, it should add thinking to the process. When teachers intervene and ‘scaffold,’ the thinking should remain with the students, who must stand on their own feet. Support their thinking, yes—but do the thinking for them, no. That is not ultimately beneficial to the learner.
When it comes to ‘temporary,’ the metaphor is even more apt. On a construction site, scaffolds are temporary and quickly removed once they are no longer needed. Everyone is instantly happy about their removal and happy about the result. Likewise, in the learning process, scaffolds are a temporary instructional support that should be removed, to gradually transfer autonomy to the learners (i.e., agency).
In their experiment, Wood, Bruner, and Ross engineered six predetermined actions that a tutor could take. To be clear, tutors were limited to these actions: (1) recruitment or luring the learner into the activity, (2) reduction in the degrees of freedom, (3) directing and maintaining attention, (4) marking critical features, (5) frustration control, and (6) modeling. Although they were successful in demonstrating the effectiveness of these actions, their efforts were also widely misunderstood. As so often happens with research in education, the devil is not just in the details, but in the bridge to practice. The most glaring pitfall of their work was that it was interpreted (or at least implemented) by many educators as ‘simplifying’ the learning task. This was never their intention, and a careful reading of the many publications they shared would quickly put the issue to rest—they never meant to ‘simplify’ in the sense of ‘dumbing down’ a learning task. Yet, it was implemented that way by some, perhaps many.
Fortunately, the concept of scaffolding as a process to support learning has continued and has been advanced by many. Its resilience lies in the need to support all students, including those who continue to underperform in mathematics for reasons not always their fault—such as those experiencing systemic poverty. The strength of the concept of scaffolding is in its connection to Vygotsky’s Zone of Proximal Development, which is the cognitive space between what a child can do on their own and what is beyond their current level of development—the perfect place to provide scaffolds (Vygotsky & Cole, 1978)
“It is the distance between the actual developmental level as determined by independent problem solving, and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers.” (Vygotsky & Cole, 1978, 86)
In 2003, I was fortunate to have worked with, and learned from the work of Aida Walqui, a highly-regarded educator known for her work with English learners around the nation. She introduced a ‘model’ for scaffolding to districts around the country. The model prioritized six scaffolding activities: (1) modeling, (2) bridging, (3) contextualizing, (4) schema building, (5) re-presenting text and (6) developing metacognition (Walqui, 2006).
It is as if Walqui’s model took the best pieces from the research of Wood, Bruner, and Ross and integrated van Lier’s conditions for scaffolding (Lier, 2004, p.151) together with the prestigious research from the National Research Council (NRC) about how people learn. Now, findings from research are one thing, but enacting research findings into teaching practices is quite another. Walqui’s model provided a bridge to practice for three well-supported findings from the NRC:
- Students come to the classroom with preconceptions about how the world works. If their initial understanding is not engaged, they may fail to grasp new concepts.
- Organizing knowledge into conceptual frameworks is essential to learning.
- A metacognitive approach to learning empowers students to become active agents in their own learning process.
(Bransford, 2000)
I had the opportunity to work closely with math teachers to implement Walqui’s model (with a twist for mathematics) within communities of adult learners. We planned, problem-solved, taught, scaffolded, evaluated our progress, and then practiced our approach—constantly refining our facilitation to best support students, while maintaining the rigor of mathematics. And it worked; students improved. So, what’s the problem? It’s the same challenge we’ve always faced in math education: becoming an effective math teacher takes time. There are no shortcuts or magic bullets. While we’ve always understood this, we often succumb to the political forces that education is—and has always been—subject to.
Furthermore, although it was effective, we often felt like the model prioritized language development, leaving mathematics as an afterthought. While there is consensus that language mediates mathematical knowledge, it is also the case that mathematics provides valuable opportunities for language development. Neither one should be subordinate to the other; yet many math educators perceive it that way. It seems like, something essential was missing in scaffolding mathematical thinking. Learning to think mathematically is different from learning a language in that to think mathematically, the mind must be disciplined. This is the reason that math is considered a ‘discipline.’ Mathematics offers a unique perspective on the world, one which requires a more disciplined approach to thinking. Scaffolding the process of ‘disciplining’ the mind in mathematics requires that the ‘expert’ is at all times ‘doing with’ instead of ‘doing for.’ But as we have come to understand, this is easier said than done.
A ‘working’ definition! It is not surprising that today, scaffolding methodology is still widely misunderstood, often misapplied, and deeply needed. I find it more productive to think about scaffolding in terms of what it means to provide an ‘appropriate’ scaffold. Thus, drawing on the insights of many great minds who have contributed to the body of knowledge around scaffolding, while honoring Bruner’s original intention of “appropriate orchestration”, I focus on what I consider a working definition: “In the context of math education, ‘scaffolding’ refers to temporary cognitive support, that allows the learner to engage in a task that he or she would not be able to tackle independently, without degrading the rigor of the task.” This is a version I feel comfortable putting forward. With this definition in mind, we can now shift our attention to ways to provide the much-needed appropriate orchestration of scaffolds.
While it is obvious that a review of the literature gives us many aspects to consider when thinking about how to provide instructional support, I find it useful not to lose track of the ‘appropriateness’ of such support in mathematics, which has led me to identify four lenses: (1) design, (2) content, (3) language, and (4) contingency. To be clear, these lenses are not mutually exclusive; they are different perspectives from which to examine the support we provide. Nor is this list exhaustive, as there can be other ways to conceptualize instructional support.
First, design is a useful lens because math educators wrestle with the idea of considering the strength and quality of instructional materials. There are aspects of instructional materials that, by their very design, can be considered to scaffold the learning process. From the type of tasks to how they are sequenced and the expectations and use of academic language, diagrams, translations, resources, assessment, etc., instructional materials are designed to fall within such a spectrum of support. There is also an aspect of structures that can support learners on how to engage with mathematics at the edge of their abilities. Classroom rituals may fall within this category, along with visual tools, clear organization, and guided notes that help the learner focus on key points, to name a few.
Second, content is an important lens, in that often the content itself should be prioritized. Not all aspects of mathematics are equally important in learning to think mathematically. Marking those big ideas is something that requires a great deal of content knowledge on the part of the teacher. Appropriately scaffolding the process of making connections between mathematical ideas requires a great deal of expertise. Instructional routines that prioritize mathematical thinking can be examples that fall within this lens. Other examples may include finished models, diagramming, emphasizing the thought process, comparing different approaches, marking decision strategies, and highlighting the most significant features of the mathematical topic.
Third, language is an essential lens. Mathematical thinking requires language development. As mathematical knowledge develops, so too do the ways to communicate that knowledge, and make sense of it. Mathematical knowledge of the conceptual type is encoded in language. Thus, scaffolding language development is of paramount importance. Here too, instructional routines that prioritize the development of the language of mathematics can fall within this lens. Other examples may include breaking down text and content into smaller, manageable sections; engaging with mathematical texts that amplify the mathematical ideas; using graphic organizers that bridge content and language; employing concept maps that illustrate the relationships between ideas; providing sentence and paragraph frames to enrich opportunities for discourse; and incorporating prompts and questions, as well as model responses, videos, and audio, to name a few.
Finally, the lens of ‘contingency’ is probably the most difficult of all. It means providing an appropriate scaffold to a learner facing difficulty beyond his or her skills, which by definition requires moment-to-moment pedagogical actions. These situations arise spontaneously and unexpectedly, and are very challenging to prepare for. They require ‘just-right’ and ‘just-in-time’ responses and interventions but seldom—at least not in mathematics—a ‘just-in-case’ response (Lier, 2004, 149; Shannon, 2018).
Just-in-time rather than just-in-case! Providing learners with support when it is needed and/or requested requires math teachers to resist the impulse to ‘jump in’ at the first sight of struggle. However, when it is time to ‘jump in,’ the teacher’s support must be ‘appropriate.’ This is particularly essential when an assignment is given with the intention of generating a ‘productive struggle.’ The teacher should not make assumptions about what the learner does not know and should refrain from trying to make sense of things for the students by preventing errors or false starts, or by directing their steps. This requires the teacher to hold high expectations, allowing students to build their own understanding and embracing errors as a natural part of the learning process. If a student’s starting point is incorrect, the teacher should allow that student to explore where it leads, rather than rushing to provide steps that students can follow to reach the correct answer. Instead, feedback should be provided when it is needed and relevant to a particular learner. (Shannon, 2018).
References
Anghileri, J. (2006). Scaffolding Practices That Enhance Mathematics Learning. Journal of Mathematics Teacher Education, 9, 33-52. 10.1007/s10857-006-9005-9
Bransford, J. (Ed.). (2000). How People Learn: Brain, Mind, Experience, and School: Expanded Edition. National Academies Press.
Bruner, J. S. (1973). Beyond the information given: Studies in the psychology of knowing. W. W. Norton.
Harraqi, M. (2017, September 11). Review of Aida Walqui’s Scaffolding Instruction for English Language Learners: A Conceptual Framework. American Journal of Arts and Design, 2(3), 84-88. 10.11648/j.ajad.20170203.13
Lier, L. v. (2004). The Ecology and Semiotics of Language Learning: A Sociocultural Perspective (L. v. Lier, Ed.). Springer Netherlands.
Shannon, A. (2018). Scaffolding Productive Struggle: A Video Case Study (M. Cordero, Ed.). NYC Department of Education.
Shannon, A. (2018). Scaffolding Productive Struggle: A Video Case Study (M. Cordero, Ed.) [Case study with NYC teachers.]. NYC Department of Education.
Shvarts, A., & Bakker, A. (2019). The early history of the scaffolding metaphor: Bernstein, Luria, Vygotsky, and before. Mind, Culture, and Activity, 26(1), 4-23. 10.1080/10749039.2019.1574306
Vygotsky, L. S., & Cole, M. (1978). Mind in Society: Development of Higher Psychological Processes (M. Cole, Ed.; M. Cole, Trans.). Harvard University Press.
Walqui, A. (2006). Scaffolding Instruction for English Language Learners: A Conceptual Framework. The International Journal of Bilingual Education and Bilingualism, 9(2), 159-180.
Wood, D., Bruner, J. S., & Ross, G. (1976). The Role Of Tutoring In Problem SolvingJ. Child Psycho[. Psychiat.J. Child Psycho[. Psychiat. J. Child Psycho. Psychiat., 17, 89-100.versity Press.