Overcoming the Challenges of Providing Additional Instructional Support While Fostering Productive Struggle
Fostering productive struggle among math students is a fundamental and now widely-accepted instructional strategy, intended to give students the space to make sense of math concepts and build their own understanding (Hiebert & Grouws, 2007). And since it is desirable to let students wrestle with unfamiliar ideas in order to eventually master them, isn’t it inherently contradictory for teachers to provide ‘additional’ instructional support (e.g., scaffolding) beyond what is available in the existing curriculum? This is the first question we will consider (i.e., the nature of such support). The second: What counts as too much, too little, or just enough instructional support in this context?
Is it even necessary?
First, consider the wide range of academic and emotional needs students bring to the math classroom: from stubborn misconceptions and knowledge gaps to varied trajectories of cognitive experiences, individual readiness for learning, academic language development, and more. Taken together, these factors clearly show that students are a far from monolithic group; they come to the math classroom with diverse needs. (Ausubel et al., 1978; Vygotsky & Cole, 1978; Tomlinson, 2014; DWECK, 2017 (Martin et al., 2024)
Second, consider our own experiences as math educators. For instance, I find that there are times when additional instructional support (i.e., scaffolds) is indeed necessary. Here’s why: when a mathematics teacher sets out to engage students in an investigation with an open-ended mathematics task—even with strong tasks and effective teaching—it is likely that, at some point, some students will encounter a mathematical idea that demands skills that lie beyond their current understanding. Simply put, they may stall. How teachers respond to this moment will determine whether productive struggle is fostered or undermined.
What is One Common Way That Productive Struggle is Undermined?
In my many interactions with math teachers, I have observed the following: The teacher sets up the math task wonderfully and follows best practices. Students engage with the task in pairs or triads while the teacher circulates the room, conducting moment-to-moment formative assessment and managing students’ engagement with both the math task and each other. Then, all of a sudden, some students encounter a difficulty beyond their current math skills. The teacher intervenes with effective teaching moves such as prompting, guiding questions, and more. Sometimes, students respond well to these interventions, and nothing further is needed. Other times, students do not respond well enough and begin to disengage from the math task.
You might think that this is precisely the moment to encourage perseverance—and of course, that would be ideal—but perseverance is a skill that must be developed and supported. Unfortunately, what tends to happen in many math classrooms is that when a few students disengage, teachers take over. Once a teacher takes over and walks students through the math, the lesson often slides down a slippery slope toward “show-and-tell.” (Takahashi, 2008).
As it turns out, perseverance is just as important for teachers—to not give in and take over—as it is for students—to not give up on reasoning, sense-making, and problem-solving. I argue therefore, that the ‘scaffolds’ are there not only to support student perseverance but also to prevent teachers from taking over.
Some professionals might object that if the teacher’s interventions were truly effective, students would remain engaged, and perhaps more professional learning around effective teaching moves is all that is needed. This is a valid point, and probably true in many cases. However, I would respond that this can also become a semantic debate: “effective teaching moves,” “instructional supports,” and “scaffolds” are intersecting ideas—different descriptions of the same phenomenon from different angles. Yet their intentions, design, and execution can be quite different.
More About Productive Struggle
Today, among mathematics educators, it is widely recognized that productive struggle is essential for developing deep conceptual understanding. Although the term itself was formally articulated as an instructional principle in NCTM’s Principles to Actions (2014), the ideas underlying productive struggle have roots that extend much further back in the history of mathematics education.
In 1945, George Polya normalized the idea that difficulty is inherent and necessary for problem solving in mathematics. Soon after (in 1954), Jean Piaget’s theories of cognitive conflict and disequilibrium laid the foundation for struggle as necessary for conceptual understanding. Then in 1961, Jerome Bruner emphasized that students must grapple with ideas to construct understanding. In 1976, Wood, Bruner, and Ross coined the term “scaffolding,” connecting struggle and support as impactful to the teaching and learning process. In the 1980s and 1990s, Ernst von Glasersfeld provided the epistemological foundation through his seminal work on ‘Radical Constructivism,’ where he argued that knowledge is not passively received; rather, it is built up by the thinking subject.
As research in mathematics education continued to advance, researchers and publications flooded the field: Stein and Smith (1998) introduced the framework of ‘cognitively demanding tasks’; Hiebert and Grouws (2007) made the first research-backed argument that struggle is essential for learning mathematics. Today, emerging literature continues to highlight productive struggle as central in the era of Equity and Mathematics identity, while juggling intersecting ideas such as frustration, over-scaffolding, or withholding support.
More About Instructional Support
In my experience, there is always a tension between providing too much instructional support, too little, or just enough. Three factors (among others) contribute to the complexity of this tension. First, it is important to recognize that effective teachers and high-quality curriculum materials already incorporate instructional support into daily lessons for all students. From visual cues, annotated examples, guiding questions, graphic organizers, multiple entry points, instructional and language routines, and student-to-student collaboration, many activities are already designed to support all learners. As a result, discussions about adding additional instructional support (i.e.,scaffolds) can understandably engender skepticism among professionals.
Second, while some professionals question whether students truly need additional instructional support beyond what is already built into the lesson, others worry that providing additional support may compromise the rigor of the math tasks and, in doing so, violate a highly-valued principle of equity and access to grade-level content. In other words, students who receive support beyond what is embedded in the lesson may risk having a learning experience that is diminished. Unfortunately, many well-intentioned scaffolds can slip into “dumbing down” the math, turning tools meant to empower students into practices that inadvertently create a less robust learning experience than that of peers receiving less support. To be sure, this is a legitimate concern—one that requires careful consideration.
Third, this tension becomes even more relevant and pronounced when curriculum materials are designed around a learning trajectory that relies on investigations and open-ended math tasks (i.e., a problem-based curriculum). This is because a fundamental feature of problem-based approaches is active student engagement in productive struggle. Thanks to a substantial body of research, we now know that “effective mathematics teaching uses students’ struggles as valuable opportunities to deepen their understanding of mathematics. Students come to realize that they are capable of doing well in mathematics with effort and perseverance in reasoning, sense making, and problem solving.” (National Council of Teachers of Mathematics, 2014, p. 52). So, not surprisingly, the idea of “over-scaffolding”—which can jeopardize students’ learning experiences and undermine productive struggle—warrants caution.
More on Scaffolding: Aye or Nay?
When it comes to providing additional instructional support (i.e., scaffolds) beyond what is already embedded in the lesson and curriculum materials, we find a variety of perspectives and approaches among educators—ranging from offering very little additional support to offering far too much, and from focusing support on completing the task to focusing it on the thinking process, with an emphasis on language, or with no emphasis at all.
Compounding this multiplicity of approaches is a reality most math teachers face: not all math tasks are equally demanding, and not all students require the same level of support. If we acknowledge that the learning experience can vary from task to task and from student to student, it follows that the idea of a single instructional support for all is not without challenges. A small but important distinction about “scaffolds”: in my view, they are not intended for every student.
The concept of providing ‘additional’ instructional support is very complex, and it is precisely this challenge that Wood, Bruner, and Ross investigated in depth and reported on in their seminal paper, The Role of Tutoring in Problem Solving (Wood et al., 1976). They coined the term “scaffolding” in their effort to describe the process of providing instructional support. They wrote:
“More often than not, it involves a kind of ‘scaffolding’ process that enables a child or novice to solve a problem, carry out a task or achieve a goal which would be beyond his unassisted efforts…, But, we would contend the learner cannot benefit from such assistance unless one paramount condition is fulfilled. In the terminology of linguistics, comprehension of the solution must precede production. ” (Wood et al., 1976, 90).
In this quote, it has always struck me as significant that the researchers felt the need to clarify that “comprehension of the solution must precede production.” This resonates with my own professional experience, which underscores the important notion that without conceptual understanding, the benefit of scaffolding is undeniably limited.
According to Wood, Bruner, and Ross (1976), scaffolding is a process in which a teacher provides successive levels of temporary support that help students achieve higher levels of understanding and skill. This support is intended to be gradually withdrawn as students gain competence and independence.
While many existing teaching moves and curriculum materials may fall within this definition, I think school communities could benefit from taking stock of what is already in place and determining what else, if anything, is needed.
Too Much?
When is additional instructional support that is beyond what is already built into the lesson for all students too much? I propose that it is too much when it takes the thinking and sense making away from the learner. I argue that after a teacher does whatever it is he or she calls ‘scaffolding,’ it begs the question: Who is doing the thinking? If the thinking has been sabotaged or subtracted from the learning experience, then the scaffolding is simply too much. This is especially relevant in the context of productive struggle and problem-based curricula.
Some obvious categories of ‘too much scaffolding’ are step-by-step instructions, “let me show you how to do it” examples, and “do not look at that, look at this” suggestions. Other less obvious categories of ‘too much scaffolding’ are versions of breaking a task into smaller ‘nuggets’ or sub-tasks, which could be very controversial given it could easily go wrong. There is a lot of nuance under the umbrella of ‘chunking’. To be clear, it is my experience that ‘chunking’ can be done correctly, but a lot of thought must be put into it in order to preserve the rigor of the task.
Too Little?
When is instructional support too little? In the context of productive struggle, this is very difficult to answer. We cannot simply use a student’s initial difficulty as evidence that they are unable to do the task—they may just need more time for the embedded supports to work. Perhaps a stronger enactment of an instructional routine is all that is needed. We could continue down this line of reasoning, examining every element that strengthens instruction, but reviewing every element of instruction is a formidable undertaking.
I propose that support becomes “too little” or “not enough” when a student disengages despite the presence of effective teaching practices and a rich learning environment. I would settle on a scaffold that keeps the student engaged in active thinking and sense-making. And by “engaged,” I mean truly engaged—the kind of engagement we experience when we cannot put down a puzzle even when we haven’t yet solved it.
Just Enough?
When is instructional support just enough? I propose that a scaffold provides enough support when it successfully re-engages the student, who is beginning to disengage, in actively thinking and sense making.
I suggest that in the context of productive struggle, an effective scaffold is not one that helps a student complete the task or get to the ‘right answer’. Far from it: an appropriate scaffold is an instructional support that maintains students’ engagement with the task and their thinking processes when they encounter a situation that demands mathematical skill that is beyond what they can do unassisted. It is important to note that if we shift the purpose of the scaffold from one that supports ‘task completion’ or the ‘right answer’ to maintaining engagement in ‘their thinking processes’ and making sense, then such a scaffold is distinctive in purpose (and perhaps in form) from what many practitioners may consider ‘scaffolding’.
A Framework
What does a scaffold intended to support thinking and sense making look like? It looks like more thinking, not less (i.e., amplification). And I argue that it is here where the language that encodes thinking becomes essential and indispensable to appropriate scaffolding efforts. This is one place where academic language, beyond just communicating complex ideas, becomes a tool for thinking and making sense. (Nagy & Townsend, 2012).
You might have noticed that today, most mathematics curriculum materials provide a list of the vocabulary targeted in each unit, and that such vocabulary falls within two categories: (1) discipline specific (e.g., angle, hypotenuse, function); and (2) vocabulary that encodes ‘thinking’ (i.e., compare, contrast, examine, analyze). Uccelli & Galloway (2016) referred to this second category as ‘metalinguistic vocabulary’. I propose that it is precisely this metalinguistic vocabulary that is paramount for building scaffolds that are appropriate to support and promote productive struggle.
Furthermore, given that many curriculum materials list vocabulary by units, it follows that such metalinguistic vocabulary tends to cluster together by units of study. Let us consider one example of a ‘cluster’ of metalinguistic vocabulary: make sense, identify, describe, infer, and interpret. Notice that metalinguistic vocabulary does not only cluster together, but also tends to happen in sequences. That metalinguistic vocabulary ‘clusters and sequences’ is an important observation because it offers a framework to provide scaffolds directed to the precise location where the thinking process of a student may be stalling.
Consider this example:
A dining hall had a total of 25 tables—some long rectangular tables and some round ones. Long tables can seat 8 people. Round tables can seat 6 people. On a busy evening, all 190 seats at the tables are occupied. How many long tables, x, and how many round tables, y, are there? (IM, Alg1, U2, L12, A12.3)
In this example, the metalinguistic vocabulary would present a demand on student thinking that follows a particular path:
- Making sense: In this context, it simply means the students have seen the structure of this word problem before.
- Identifying: This means the student can name it. Naming the structure as a system of equations can activate all kinds of prior knowledge and prior experiences related to it.
- Inferring: Encoded in each sentence of the word problem are quantities and their relationships, some explicit, some implied.
- Writing algebraic descriptions in the form of a system of equations, using the information that was just identified and inferred.
- Solving the system of equations, which requires both procedural and conceptual knowledge.
- Interpreting the result in the context of the original problem.
This analysis of the thinking process matters only because it can expose the precise location where a scaffold might be appropriate. And if such a scaffold were to be provided, it needs to meet one very important criterion: it should add more thinking to the process, not less (i.e., amplification).
If you consider such analysis demanding, then note that in recent work with a group of teachers, we found that students across the district were having difficulty with one particular process: inferences. This discovery made doable the design of scaffolds specifically designed to amplify and facilitate the cognitive activity around inferences.
How does it look? Like this:
- If students are having difficulties with inference and begin to disengage with struggling productively, then amplify their learning experience by having them identify incorrect inferences and explaining what makes them incorrect, then cycle back to inferring.
- Likewise, if after making sure that the enactment of teaching moves is the best that it can be, students still stall with making sense of a task, then amplify their learning experience by having them identify elements of the task and sorting them in order of familiarity. Then go back to making sense of the task.
- If a student needs additional support building an explanation, then amplify their learning experience by having them describe different aspects and then sequence the descriptions in ways that make sense to them. Perhaps just critique the sequence done by peers.
We could go on with infinite examples, but the overarching principle is this: Difficulties with thinking that threaten to stall a student’s engagement can benefit from additional thinking that offers a different perspective. In his seminal work, Edward de Bono (1961; 1970; 1985; 1991; 1999; 2009) referred to this principle as ‘perceptual thinking’. In simpler terms, amplification of the thinking process can be achieved by offering another point of view. We could call it a different but comparably demanding ‘entry point’ to their thinking process.
References
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