The Core of the Matter

The Core of the Matter Issue #20

Thomas Nobili — Thu, 27 Aug 2026 19:53:26 GMT

A new school year is upon us, which brings with it many new things, not the least of which is a class, or classes, of fresh faced students. And as teachers are preparing for the start of the year, they are undoubtedly thinking hard about those first days of school. And one of the mantras that runs on loop in many educators’ minds during this planning is the importance of building positive relationships with their students. So much so, that it is not at all uncommon to hear a teacher explain that they will be using the first week of school or more on relationship building.

On its face this makes sense: teachers should be investing time and resources into cultivating positive relationships with their students, as we have ample evidence that they are correlated with things like motivation, self-efficacy, engagement, and achievement (Hamre & Pianta, 2001; Li & Julien, 2012; Roorda, Koomen, & Oort, 2011). Although the idea that relationships matter in schools is indisputable, what is meant by relationship is the subject of much more debate. In many schools, teacher-student relationships are characterized by whether the teacher knows the name of a student’s dog, what sports they play, or that they have two siblings. Although there is certainly nothing wrong with taking interest in these things, they are far from sufficient in cultivating the type of relationship that leads to the outcomes mentioned previously. According to the National Scientific Council on the Developing Child (2004). “.... Relationships engage children in the human community in ways that help them define who they are, what they can become, and how and why they are important to other people (p.1).” Bronfenbrenner (1979) goes even further:

“Learning and development are facilitated by the participation of the developing person in progressively more complex patterns of reciprocal activity with someone with whom that person has developed a strong and enduring emotional attachment and when the balance of power gradually shifts in favor of the developing person” (p. 60).

Therefore, it would seem that the type of relationship between a teacher and a student goes well beyond superficial knowledge and social pleasantries. Rather, it is a multfaceted construct rooted in development. The Search Institute has done a brilliant job in concretizing the complexity of teacher–student relationships in the form of The Developmental Relationships Framework (DRF). The DRF frames relationships as having five dimensions: express care, challenge growth, provide support, expand possibilities, and share power.

The important takeaway here is that while expressing care is part of a strong developmental relationship, it is not the sole contributor, even though relationships are sometimes defined in this arrested way. This limited conception of relationship often means that educators conceive of building relationships as something that occurs outside of instruction. In fact, in many instances teachers see relationship building as a prerequisite to ambitious instruction. However, thinking of relationships in a more dynamic sense challenges this notion. Instruction is one of the primary places where relationships are actually built. Instruction is where we challenge student growth through cognitively demanding tasks, provide support without diluting demand, expand possibilities by introducing students to new people, places, concepts, and ways of thinking, and share power through democratizing thinking, distributing cognitive authority, and nurturing agency. Although a student might feel a sense of connection because their teacher knows that Minecraft is their favorite video game, that pales in comparison to the connection they experience when they are stuck on a cognitively demanding task and know that their teacher believes they can work through it—and that they are safe to struggle along the way. In our work, my colleagues and I refer to the conditions that make this possible as the classroom holding environment: an environment in which students experience both the safety and support necessary to take intellectual risks and the press to engage in meaningful, challenging work. My colleague Isobel has written more about the idea of a holding environment : CL 178 and CL 179.

Yet the kinds of interactions required to build these deeper, more developmental relationships are surprisingly rare in classroom practice. In a study of more than 2,500 classrooms, the NICHD (2005) found that more than 85% of students’ opportunities for academic activity and learning occurred through teacher-directed whole-group instruction or individual seatwork, rather than in settings that might allow teacher–student relationships to play a more meaningful role. The typical student interacted individually or in a small group with their teacher fewer than four times per hour, and even those interactions were largely compliance focused. Perhaps more importantly, instructional exchanges overwhelmingly centered on the performance of basic skills, rather than creating opportunities for students to analyze, reason, wrestle with uncertainty, or work through more complex problems with the support of their teacher.

To be clear, the choice here is not between expressing care and the other dimensions of the DRF, dimensions, which primarily fall under the umbrella of instructional press. Rather, it is the combination of support and press which leads to the most ideal outcomes. In fact, the combination of instructional support and positive climate has been identified as a powerful predictor of subsequent student achievement (Hamre & Pianta, 2001). Further, an analysis of citywide middle school data in Chicago found that both high levels of social support and strong academic press were positively associated with achievement in reading and mathematics. More importantly, the strongest outcomes occurred when students experienced both high academic press and high social support. Regardless of students’ backgrounds or the demographic characteristics of their schools, when one was strong and the other weak, students learned less (Lee et al., 1999).

This matters because it pushes us toward a different conception of what a strong teacher–student relationship actually looks like. It is not simply a teacher who is warm, caring, and knows a great deal about a student. Nor is it simply a teacher who maintains high expectations and pushes students academically. It is a teacher who communicates, through their interactions with students, some version of: I care about you, I believe you are capable of doing important and difficult things, and I am going to support you in doing them (Nobili et al., 2026). Relationships, therefore, cannot be thought of as separate from instruction, because it is only through instruction that teachers can get that message across. It’s where they walk the talk.

There is one more problem with the way we commonly talk about relationship building at the beginning of the school year: it tends to treat relationships as relatively fixed. When educators talk about taking the first week or two of school to “build relationships,” there is an implicit theory underneath that language—that relationships are something we establish early and then carry with us through the rest of the year. But relationships in schools are highly dynamic and malleable. Hunter et al. (2012), for instance, found that students’ and teachers’ perceptions of their relationships at the end of the year frequently differed from their initial impressions. From the students’ perspective, these relationships also tended to become less positive over the course of the year. Importantly, changes in the quality of those relationships were associated with changes in students’ homework submission, self-efficacy, and effort.

In other words, relationships are not something we build in August or September and then draw upon for the next nine months. They are continually being built, reinforced, weakened, and renegotiated through the thousands of interactions that occur between teachers and students across the school year.

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The question, then, may be less, What am I going to do during the first week to build relationships with my students? and more, What will students experience in my classroom, day after day, that communicates care, belief, support, challenge, and agency? The answer to which will not be found in a beginning-of-the-year icebreaker or an All About Me poster, but in the work of teaching and learning itself.

And there is no reason why this work cannot start from the get-go. Giving students an opportunity to wrestle with a cognitively demanding task on day one communicates: Thinking is what we do here. Providing support without diluting the challenge communicates: You belong here, and we are in this together. Asking students to explain and defend their thinking communicates: Your ideas are worth listening to. Treating errors and uncertainty as material for learning communicates: You are safe taking intellectual risks here.

Perhaps most importantly, the way a teacher responds when a student becomes stuck sends a powerful relational message. A teacher can tell students that they believe in them during an advisory activity, but that belief becomes much more consequential when a student experiences it in the exact moment they are uncertain and looking to the teacher for rescue. Pressing a student to continue thinking, rather than immediately stepping in to remove the difficulty, communicates: I believe you are capable. Supporting the student while preserving the challenge simultaneously communicates both care and confidence.

None of these moves requires waiting until a relationship has been established. In fact, they are among the ways the relationship gets established in the first place. The first days of school certainly provide opportunities to learn about students, express care, and begin building familiarity, but they also provide an opportunity to establish a much deeper relational message. That is relationship building too. And as we have examined in this post, in many ways, it may be the kind that matters most.

The Core of the Matter Issue #19

Thomas Nobili — Wed, 22 Jul 2026 13:57:43 GMT

Schools spend an enormous amount of time, money, and personnel on intervention. Students are screened, progress monitored, moved between tiers, and placed into additional programs or blocks of instruction. Yet, despite all of this investment, intervention is not consistently producing the outcomes schools expect (Balu et al., 2015; Tennessee Department of Education, 2016; SCORE, 2023; TNTP, 2026).

TNTP has recently taken up this problem through a series of reports and resources on instructional coherence. In its 2024 analysis of nearly 28,000 elementary and middle schools where students began below grade level, they found that only 5 percent helped the average student catch up by the time they left the school. More recently, its Coherence by Design report (2026) argued that a big reason students fail to catch up is that core instruction, intervention, tutoring, and other academic supports are often disconnected from one another. Students may encounter different materials, routines, representations, and goals as they move through their day, leaving them responsible for making connections the system has failed to make for them.

TNTP’s major recommendation is to align these experiences more closely. They argue students should not receive one approach to a concept during core instruction, another in intervention, and still another during tutoring.

However, alignment alone is insufficient and overly simplistic. Boiling the failures of intervention down to an alignment problem alone, fails to account for several other plausible causes.

The major assumption in TNTP’s logic is that the core instruction students receive is itself effective. However, it is well documented, including by TNTP in their Opportunity Myth report, that many students are not receiving access to quality grade-level instruction That raises an obvious problem: if core instruction is inadequate, aligning intervention to it may simply provide students with more of the same instruction that did not work the first time.

The original logic of Response to Intervention was supposed to help schools make this distinction. RTI was intended, in part, to prevent schools from treating low achievement as automatic evidence of a disability when inadequate instruction could reasonably explain the difficulty. Students would receive research-based instruction and increasingly intensive intervention, while their progress was monitored to determine how they responded.

Importantly, early accounts of RTI made clear that the number of students struggling mattered. If a large share of students failed to make progress, the instructional program itself was supposed to be examined and modified. Targeted intervention made sense when strong general education instruction was working for most students but remained insufficient for a much smaller group (Batsche et al., 2005; Fuchs & Fuchs, 2006). The problem is that this logic only works if schools know something about the quality of the instruction students received before they were placed in intervention. More specifically, it is not enough to say that Tier 1 occurred without having a shared definition of what Tier 1 instruction is supposed to entail.

Imagine a patient with a serious infection who is prescribed antibiotics. After several days, the patient has not improved. Perhaps the infection is resistant to the medication. Perhaps the diagnosis was wrong. Perhaps the patient needs a more intensive treatment. Before reaching any of those conclusions, however, the doctor would want to know whether the patient actually received the prescribed treatment. Was it the correct medication? Was the dosage appropriate? Was it taken consistently? Was the full course completed?

Now imagine that some patients received the full dosage, others received half, some missed several days, and others were given a different medication altogether. All of these experiences were nevertheless recorded as “antibiotic treatment.” It would make little sense to conclude that every patient who remained sick had failed to respond to the same treatment.

The same issue exists with Tier 1 instruction. A student’s schedule may show that they were present during a Tier 1 mathematics or literacy block, but that tells us very little about the quality of instruction that occurred during that time. In one classroom, students may spend a majority of their time working with important grade-level content. They may be doing sustained thinking, be expected to make that thinking visible, and experience teaching that responds to their ideas while preserving the cognitive demand of the work. In another classroom, students may spend the same amount of time with the same grade-level content while passively watching the teacher explain ideas or demonstrate procedures, with very little cognitive activity taking place on the part of students.

Both are considered Tier 1, but they are not the same instructional experience. Further, this variation should not simply be treated as a matter of individual teacher quality or competence. Tier 1 is also what a school or district has defined, resourced, supported, and made possible.

This is the central weakness in TNTP’s coherence argument. A district can align intervention to Tier 1 without improving either setting. If Tier 1 responds to difficulty by reducing cognitive demand, intervention may simply intensify that low-demand work. If Tier 1 relies on excessive prompting and teacher rescuing, intervention may reproduce those same practices with fewer students and more adult attention.

In some districts, upwards of 40 percent of the student body receives special education, 504, or RTI services. When this is the case, the solution cannot be greater alignment of the intervention system. Rather, the system needs to determine what students are consistently failing to receive during core instruction and what must change so that far more students learn without requiring additional support.

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In our book, Maximizing Opportunity to Learn (Nobili et al., 2026), we define opportunity to learn as the interaction of three factors: the cognitive demand of the tasks students receive, the amount of time they spend working with grade-level content, and the quality of the teaching they experience. Our model for translating this idea into instruction is the HEAT, the Heuristic for Equitable and Adaptive Teaching. It focuses on students having access to grade-level content, engaging cognitively in worthwhile tasks, making their thinking visible, experiencing teaching that responds to their thinking while preserving cognitive demand, and developing both academic and social belonging.

This is our model. Schools may use it or choose another one. But they need something more concrete than general language about rigor, engagement, differentiation, or research-based instruction. The model also has to shape curriculum, professional learning, coaching, teacher collaboration, and leadership structures and decisions. The work involves making that kind of instruction common enough that schools can actually say students received it.

Until this kind of instruction is experienced with reasonable consistency across classrooms, schools cannot confidently use intervention in the way RTI originally envisioned because they cannot know whether a student’s difficulty reflects inadequate core instruction, or a genuine need for more intensive or specialized support.

TNTP is right that intervention should not operate as a fragmented parallel system, but coherence is only valuable when we are coherent around something worth reproducing. Without a clear and shared understanding of equitable and ambitious instruction, schools risk becoming more efficient at providing students with more of what did not work the first time.

The Core of the Matter Issue #18

Thomas Nobili — Fri, 19 Jun 2026 13:16:11 GMT

Small group instruction has become the multi-purpose solution to any ailment--a bit like “take two aspirin and call me in the morning.”

Students are struggling? Use small groups.

The class has a wide range of needs? Use small groups.

Teachers need to differentiate more? Use small groups.

We need stronger Tier 1 instruction? Use small groups.

Small group instruction feels more personal, more responsive, and more manageable than whole-class teaching, so its appeal is understandable. With fewer students in front of them, teachers believe that they can see more, hear more, prompt more quickly, and correct errors before they spread. But what happens is that small group instruction is often mistakenly treated as if the format itself carries the instructional weight, as if decreasing the number of students in front of the teacher somehow automatically increases the quality of instruction and, by extension, the amount of student learning. It does not.

Small group is not an instructional practice, it is a structure. It tells us how many students are sitting with the teacher. It tells us nothing about the substance of the instruction. It does not tell us what students are being asked to think about, whether the task is cognitively demanding, or whether the teacher is pressing students to make meaning rather than funneling them toward an answer. Treating small group instruction as a mechanism for learning is a category error (ie. a mistake in thinking where you treat something as if it belongs to one kind of thing when it actually belongs to another). At best, a small group setting can make strong instruction more targeted. At worst, it takes time and planning away from effective instruction.

The visible features of small group instruction are easy to mistake for the deeper conditions of learning. A group can look focused, students can be compliant, the teacher can be busy, and the work can get finished. But none of that tells us whether student understanding is better. The question, then, is not whether students are in a small group, but rather what kind of opportunity to learn the small group creates.

The variability in quality of small group interventions is well documented in the literature. For example, Lou and colleagues’ meta-analysis (1996) of within-class grouping found a modest effect favoring small-group learning, but the findings between studies were inconsistent. The conclusion was that small groups were most effective when teachers adapted instructional methods and materials for small-group learning. In other words, the group itself was not the mechanism, the instructional design was. The Education Endowment Foundation reached a similar conclusion in its summary of small-group instruction. They found that small-group instruction can have positive effects, but the likely mechanisms are increased feedback, sustained engagement, and instruction more closely matched to students’ needs. Again, the explanation is not simply that there are fewer students. Rather, the smaller setting may matter because it makes certain instructional interactions more feasible.

This is not a blanket argument against small group instruction. Small group instruction can increase opportunity to learn when it provides:

targeted feedback,
more opportunities for students to explain their thinking, and
stronger connections to important content.

Conversely, it can reduce opportunity to learn when it leads to excessive prompting, greater adult dependence, more proceduralized tasks, lower expectations, or less access to core instruction.

In these contrasting examples, the size of the group is not the critical variable. Rather, opportunity to learn, which consists of time, task, and teaching (Nobili et al., 2026), is the key determinant of whether instruction produces learning.

As a result, we need to talk about small group structures with much more specificity. When a school or district says, “Our focus this year is small group instruction,” they need to be able to expand upon that claim by answering a few essential questions:

For what purpose?
With what content?
Using what resources?
In line with what model of equitable and ambitious teaching?
Connected to what larger learning goal?

The issue of confusing format and mechanism when it comes to small group instruction is especially clear when we consider cognitive demand. Stein and colleagues have shown that the cognitive demand of a task is not static. A task moves through different phases: the task as it appears in the curriculum, the task as set up by the teacher, the task as enacted by students, and the learning that results (Stein et al., 2000). A task with high potential cognitive demand can easily decline during instruction when the teacher reduces ambiguity, provides too much direction, turns the task into a procedure, or prioritizes completion over sensemaking (Henningsen & Stein, 1997).

The challenge is that the small group setting may make loss of cognitive demand more likely. When a teacher sits with just five students, the teacher is very aware of how they are responding to the content and the instruction, which makes it much harder for the teacher not to step in and help, thereby taking some of the intellectual demand of the work away from the students. Furthermore, this may look like effective teaching. Students complete more work, they make fewer visible errors, and they appear more successful. The problem is that this appearance of success in the short term may undermine learning over time.

The short-term versus long-term distinction is what Soderstrom and Bjork (2015) call the difference between performance and learning. Performance is what students can do during instruction, under the conditions present at the time. Learning refers to relatively durable changes in knowledge, understanding, or skill that support retention and transfer. Their review makes clear that current performance is often an unreliable indicator of long-term learning. (See CoM #17 for more on this distinction).

This is also why additional adult support has to be examined carefully. In many classrooms, small group instruction is made possible by adding adults. The teacher takes one group and a paraprofessional, coach, or interventionist takes another. On the face of it, this looks like more support. But more adults do not automatically produce more learning. Research on teaching assistants presses on the assumption that adult proximity is inherently beneficial. Blatchford et al., (2012) found that students receiving the most teaching assistant support often made less academic progress than similar students receiving less support. The issue was not that teaching assistants were inherently harmful. Rather, the issue was how they were deployed. Students often had less interaction with the classroom teacher, more task-completion support, and fewer opportunities for independent engagement with content.

A related concern is grouping by level. Small groups are often organized around perceived ability. This is usually done in the name of differentiation. Students are given work that is supposedly “right” for them. But the result can be a classroom-level version of tracking. The ‘high group’ gets access to challenging tasks and ambitious instruction, while the ‘low group’ gets low level work and procedural instruction. Research on ability grouping has long warned that grouping students by perceived level often leads to unequal learning opportunities. Oakes (1985) argued that tracking does not merely sort students; it changes the quality of the educational experiences students receive. Lleras and Rangel (2009), found that African American and Hispanic students placed in elementary reading ability groups learned less than demographically similar students who were not grouped by ability. Importantly, the concern was not simply the existence of groups, but the kind of instruction lower groups tended to receive: more rote, more routinized, and less intellectually ambitious.

Like any intervention, small group instruction has an opportunity cost. If the teacher is sitting with six students, what are the other twenty students doing? Are they engaged in meaningful work they can do with independence? Or are they doing low-value activities designed mainly to keep them busy while the teacher runs the group? Tim Shanahan makes this point in relation to reading instruction. In his blog he explains, a study might compare thirty minutes of small-group instruction to thirty minutes of whole-class instruction and find benefits for the small group. But in actual classrooms, a teacher often rotates through several groups. That means each child may receive only a fraction of the teacher-led instruction, while the rest of the time is spent in independent or center-based work of varying quality. Small group may increase the intensity of a slice of instruction while reducing students’ total access to content and teacher response. This matters because small group instruction can create a density illusion. The minutes with the teacher feel more focused, but students may actually receive less instruction overall.

To reiterate, the argument is not that teachers should never pull small groups. That would simply replace one oversimplification with another. Small group instruction is useful when there is a clear instructional reason for the group. For instance, it can be useful when students need targeted support with a foundational skill, such as decoding, fluency, computation, or vocabulary, when the teacher has evidence that a few students share a particular misconception or similar unfinished learning, or when a group needs feedback that would not make sense for the whole class.

But in every case, the small group is not the mechanism.

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Small group instruction should be planned with the same level of intentionality as any other part of the instructional arc. The structure should follow the instructional purpose, not the other way around.

Before pulling a small group, teachers should ask four questions.

First, what is the instructional purpose?

Small group instruction is strongest when the purpose is specific enough to guide the teacher’s decisions. For example, in math, a teacher might briefly pull a group because several students are unable to get much traction on a multi-step word problem, not because the problem is conceptually out of reach, but because computation is consuming so much attention that they cannot focus on the relationships in the task. The purpose of the group is not to abandon the problem and practice facts in isolation. It is to provide just enough support with the needed calculations so students can re-enter the task. If the problem requires several partial products, the teacher might use an area model or known facts to help students derive the combinations they need. The goal is to clear a path back into the original problem, not replace the reasoning with computation practice.

In reading, a teacher might briefly pull a group because several students are struggling to access a complex passage due to a small set of vocabulary words, confusing syntax, or a missing piece of background knowledge. The purpose of the group is not to remove students from the common text and give them something easier. It is to provide enough support so they can re-enter the shared reading, discussion, and meaning-making work of the class.

Second, what is the opportunity cost?

If the teacher is working with five or six students, what are the other students doing? This question is often under-planned. Too often, the rest of the class is given work that functions mainly as classroom management: centers, packets, worksheets, or low-value independent tasks designed to keep students busy while the teacher runs the group. That is not a neutral decision. It changes students’ opportunity to learn.

Students away from the teacher should be engaged in work they can do with enough independence to be productive, but that still matters. In math, this might mean applying a recently developed idea to a related problem, comparing two strategies, or revising an explanation, In reading, this might mean rereading a common text with a specific purpose, annotating for evidence, preparing a written response for discussion, or revising a claim with text evidence.

Third, what support will increase access without reducing the intellectual work?

This question gets at the heart of the issue. If small group instruction is being used to provide more targeted support, then the quality of that support matters tremendously. The goal is not simply to make the work easier. The goal is to help students access the important thinking without removing the need to think.

In practice, this means attending to both access and demand. What is the cognitively demanding work in this task, text, or problem? What student thinking do I need to make visible? What am I listening for? What representation, question, prompt, or tool might help students organize their thinking without oversimplifying the task?

Fourth, how will this connect back to the core learning experience?

A small group should not become a parallel curriculum. It should help students return to the core work of the class with greater access, agency, and understanding.

For example, If students are pulled during a math lesson, how will the group help them re-engage with the shared problem? If students are pulled during reading, how will the group help them return to the common text or writing task? If students are pulled during science or social studies, how will the group help them participate more fully in the investigation or source analysis?

There are also two practical cautions worth keeping in mind.

First, small groups should have a short half-life. Groups should form around evidence of student thinking and dissolve when the instructional need changes. A group might exist for ten minutes, one lesson, or a few days. But when groups become stable labels — the low group, the high group, the intervention group — support begins to look more like tracking.

Second, the teacher must not disappear into one group for 15-20 minutes as the default model. Sometimes a group may need sustained attention, but small group instruction should often be more fluid. The teacher launches the group, listens, presses, offers a targeted prompt, steps away, checks the rest of the room, returns to gather evidence, and then connects the learning back to the class. This builds independence in the group and preserves teacher access for everyone else.

Ultimately, small group instruction should be judged by what it makes possible for students. Does it help students access important content? Does it make their thinking more visible? Does it help them connect new information to what they already know? Does it preserve cognitive demand? Does it support durable understanding rather than short-term performance? Does it return students to the shared intellectual work of the classroom? If the answer is yes, then the small group is serving its purpose. If not, then we have not solved the problem; we have only made it smaller.

The Core of the Matter Issue #17

Thomas Nobili — Sat, 23 May 2026 20:10:53 GMT

Most educators can relate to the following scenario. You walk into a class feeling pretty good about the prior day. The lesson went as planned, all the students finished their work, and a review of the exit tickets showed they seemed to have met the objective. However, as the current lesson unfolds, it becomes painfully obvious that students do not have the understanding that you thought they did. During lunch, you lament to your colleagues about what transpired, punctuated by a perplexed plea sounding something like, “I don’t get it, when I taught it yesterday they all knew it, but today it was like they never learned anything.”

Unfortunately, this scenario is not uncommon. In fact, it is probably the most common category error that exists in education: to confuse learning and performance. Bjork and Soderstrom (2015) distinguish between the two as follows:

“The primary goal of instruction should be to facilitate long-term learning—that is, to create relatively permanent changes in comprehension, understanding, and skills of the types that will support long-term retention and transfer. During the instruction or training process, however, what we can observe and measure is performance, which is often an unreliable index of whether the relatively long-term changes that constitute learning have taken place.”

In other words, learning refers to relatively durable changes in knowledge, understanding, or skill. Performance, in contrast, is what students can do in the moment. It is more immediate, more visible, and often more unstable. Bjork and Soderstrom go on to explain why the distinction is so critical:

“The distinction between learning and performance is crucial because there now exists overwhelming empirical evidence showing that considerable learning can occur in the absence of any performance gains and, conversely, that substantial changes in performance often fail to translate into corresponding changes in learning.”

It’s worth pausing to take in the implication of this. It means that very typical presumed indicators of learning, such as the artifacts students produce as a result of engaging with an instructional task, are very unreliable predictors of what students can be shown to have learned when tested weeks or months later.

My work affords me the opportunity to be in and out of numerous classrooms and I see this phenomenon often. It is what I call the performance-learning trap, whereby instruction is designed to value performance over learning, and the short term performance of students often gets mistakenly identified as sustained learning. [As a side note, there is a separate related phenomenon known as the illusion of competence, whereby a learner overestimates what they know as a result of inferior study techniques (Dunlosky et al., 2013), however this post will only address the performance-learning trap.]

There are a number of reasons why we confuse performance with learning, but I want to focus on three that I see most often. The most common, and therefore the one I will discuss first, is that in too many classrooms, success is more about compliance than learning. Students quickly learn that success involves finishing the assignment, following the teacher’s example, avoiding mistakes, and getting the right answer. Over time, they internalize the message that task completion matters more than sense-making, that deferring to the teacher’s thinking is safer than developing their own, and that struggle is a sign that something has gone wrong. The classroom produces visible signs of success, but those signs may tell us more about students’ ability to comply with the routines of school than about the durability or depth of their understanding.

A second reason for the performance-learning trap is that performance is easier to measure than learning. Performance expectations often map neatly onto rubrics, checklists, and “I can” statements. For example, we can easily quantify if a student wrote five sentences in each paragraph, if they showed their answers in more than one way, whether or not they used three vocabulary words in their response, if they included evidence from the text, performed a procedure, or completed the graphic organizer. Those things are all visible; and therefore easy to observe, but they do not necessarily tell us whether students have actually learned anything. A student can write five sentences without developing a coherent argument. A student can show an answer in more than one way without understanding the relationship between the representations. Likewise, a student can use academic vocabulary without understanding the concept, or follow the steps of a procedure without understanding why the procedure works. To be clear, the problem is not measurement itself. The problem is that we often measure what is easiest to see and then treat it as evidence of what is hardest to know.

Finally, learning and performance are often conflated because learning itself is counterintuitive. It seems like short-term success ought to be a reliable sign of long-term learning. If students are successful during the lesson, it feels reasonable to conclude that they are learning. But the research suggests that almost the opposite is true. Robert Coe (2013) puts the point bluntly: “Learning happens when people have to think hard.” In other words, learning often does not feel like learning while it is happening. It may feel effortful, uncertain, or inefficient. But that effort is the point. Without some degree of cognitive strife in the short term, there is almost no chance of developing robust knowledge in the long term. Soderstrom (2019) captures this paradox well:

“It’s a fascinating paradox in education: students can be wildly successful on tasks in class but learn virtually nothing; conversely, students can do relatively poorly on those same tasks but learn quite a lot.”
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If the performance-learning trap is created by mistaking short-term performance for durable understanding, then the solution is not to ignore performance. Performance matters as teachers need to know what students can do in the moment. But we need to reorder the evidence we value as well as how we make sense of it. Instead of looking primarily for signs of performance and treating learning as something we infer from those signs afterward, we should design experiences that privilege learning from the start. Then, when students perform well, that performance is more likely to be the byproduct of durable understanding, rather than the result of short-term performance. In other words, the goal is not simply to produce students who can appear successful during today’s lesson, but rather to design instruction so that success is increasingly rooted in challenging thinking leading to sustained understanding. Designing against the performance-learning trap requires three major shifts.

1. Treat tasks as diagnostic. A task is not just something for students to complete. It is an opportunity to learn about what prior knowledge students hold, how they are building on that knowledge, and what their thinking reveals about not only what they currently understand, but what understandings are on the horizon for them.

Some tasks reveal very little about student thinking. If every problem looks like the example given, the task may tell us that students can follow a pattern. If the steps are already laid out, the task may tell us that students can comply with a procedure. If the question tells students exactly which strategy to use, the task may tell us that students can respond to a cue. Other tasks reveal much more. They ask students to choose a strategy, connect representations, compare examples, justify a claim, identify an error, or explain why an approach works. These tasks help teachers see behind performance. They make student thinking visible for interpretation. In other words, a learning task is only as useful as the thinking it makes visible.

2. Treat learning as progressive development. If learning means relatively durable changes in knowledge, understanding, and skill, then it rarely happens fully in one lesson. A lesson can introduce an idea, surface a relationship, create a need, or help students make an important connection, but durable learning develops over time. As Graham Nuthall (2007) pointed out, students need repeated opportunities to return to ideas after time has passed. In addition, they need to use ideas in varied situations, to compare cases, confront non-examples, explain their reasoning, and revise partial understandings.

This is why “they had it yesterday” can be such a misleading way of thinking. Yesterday may have produced the beginning of learning. It may have produced a fragile connection or a first approximation. This is why it is much more useful to think of understanding as existing on a continuum rather than as a binary construct. In classrooms, we often talk about the students who get it and those that don’t. However, we don’t often take the time to actually define what we mean by “it.”

In cognitive science, learning is often understood as the development of robust schemas: connected networks of knowledge that help learners organize information, recognize when ideas apply, and make sense of new situations. If we accept schema acquisition as a key product of learning, then we also have to recognize that robust schemas do not typically form in a single lesson. They develop as multiple understandings, both new and old, begin to coalesce over time.

Designing against the performance-learning trap, then, means planning for progressive development. It means not treating the individual lesson as the unit in which learning is completed, but instead understanding learning as something that develops across a sequence of lessons. A single lesson may introduce an idea, surface a relationship, or help students make an important connection, but durable understanding becomes more stable as students return to ideas, use them in varied situations, connect them to prior knowledge, and refine them over time.

3. Treat evidence as conditional. Classroom evidence does not speak for itself. It has to be interpreted in relation to the conditions under which it was produced. A correct answer is not simply a correct answer. Its meaning depends on what students had to do in order to produce it.

For instance, did the student solve the problem immediately after watching a nearly identical example? Did every problem on the page follow the same pattern? Did the teacher point them toward the relevant strategy? Did the graphic organizer already organize the thinking for them? These questions matter because the same performance can represent very different kinds of learning.

This does not mean we should discount classroom evidence. It means we should interpret it more carefully. We may see correct answers, completed work, or smooth participation and assume that understanding has developed, but if students did not have to think hard, make meaning, or carry any of the intellectual weight, then what we are seeing may be performance under favorable conditions—not learning that will last.

Taken together, these three shifts change how we interpret classroom success. Tasks become windows into thinking, not just assignments to complete. Understanding becomes something we examine across contexts, not something we declare after one successful performance. Evidence becomes something we interpret in relation to the supports, cues, and other enabling conditions present, and learning becomes a developmental process, not a single-day event.

The Core of the Matter Issue #16

Thomas Nobili — Tue, 07 Apr 2026 23:03:59 GMT

What a learner already knows is often referred to as prior knowledge, and as far as I know, all models of learning include prior knowledge as a central component. As such, If you walk into any school on any given day, you are likely to hear conversations about prior knowledge: what background knowledge students are missing, the prerequisites required for a course, or how teachers plan to “activate” prior knowledge at the start of a lesson.

It is this last idea, activating prior knowledge, that is the focus of this post. And the big idea that I want to emphasize is that the way we often talk about activating prior knowledge obscures an important point: prior knowledge does not influence learning simply because it exists, or because a teacher has attempted to surface it. It matters only when learners bring it into play themselves, through the act of making meaning of new ideas.

We tend to equate activating prior knowledge as a separate routine that happens before instruction, as a precursor to new learning. However, this is not in line with what we know about how the brain works. Schema Theory (Bartlett, 1932), posits that learning involves the modification of existing schemas or the creation of new ones to accommodate new information. In other words, learning occurs when learners actively engage in constructing meaning by connecting new information to their existing knowledge base. My colleagues and I use a visual we call, A Simple Model for Learning (pictured below), to illustrate the dynamic relationship between new and prior knowledge (Nobili et al., 2026). Therefore, it is more accurate to think of prior knowledge activation as part of the meaning making process as opposed to a separate event.

Source: Nobili et al., 2026

In a previous post, I wrote about the alarming amount of time spent on review in many classrooms. In my own work, one of the rationales I hear frequently for spending excessive time on previously taught material is that the teacher is activating prior knowledge before moving into new content. However, I would argue that the typical ways teachers attempt to activate prior knowledge are out of sync with what we know about how people learn, and are therefore taking away from students’ opportunity to learn. Background knowledge is not something a teacher can turn on for students—like flipping a switch—while students sit back and receive it. In order to support future learning, students must actively retrieve and use their prior knowledge while making meaning of new content.

Students carry vast amounts of prior knowledge and experience in long-term memory, but that knowledge is inert unless it is brought into working memory and applied to the task at hand (Sweller, 1988). From a cognitive science perspective, for prior knowledge to influence learning, three things must occur: (a) the learner must recognize that relevant knowledge is available; (b) they must retrieve it from long-term memory; and (c) they must connect it to what they are currently thinking about. This process occurs inside the head of the learner, in working memory—it is, in other words, cognitive work performed by the learner.

No explanation, no matter how clear, can place knowledge into a student’s working memory on their behalf. At best, instruction can provide cues that may support retrieval, but whether that retrieval occurs depends on the learner. This is why so many “activation” routines fall flat. Reviewing yesterday’s lesson, asking a few recall questions, or re-explaining key ideas may create the appearance of activation without ensuring that any student has actually retrieved and used prior knowledge in a meaningful way.

Wittrock’s (1974) Generative Learning Theory sharpens this point. His research showed that learning is most likely to occur when learners are required to generate relationships between new information and what they already know. This includes processes such as explaining, predicting, inferring, and connecting. Later work by Fiorella and Mayer (2015) reinforces that these generative activities are what drive understanding. They argue that simply presenting information—even clearly and efficiently—does not ensure that students will connect it to existing knowledge. The key takeaway is that background knowledge is not activated through exposure, but through use, which requires students to actively do something with it.

Hammer et al., (2005) extend this idea by showing that it is not just whether students have prior knowledge that matters, but whether they recognize a situation as one in which that knowledge is relevant. They argue that learners possess many small pieces of knowledge, or “resources,” that are activated differently depending on how they frame the task; that is, how they interpret what kind of thinking is expected. If students frame a task as answer-getting or procedure-following, they may not draw on relevant prior knowledge, even if they have it. If they frame it as sensemaking, they are far more likely to retrieve and use what they know. From this perspective, the challenge is not simply activating prior knowledge, but creating conditions in which students see that knowledge as necessary and bring it to bear.

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The research discussed above makes one thing clear: activating prior knowledge requires active cognitive work from the learner and cannot be done to someone. As a result, educators should move away from passive practices, such as excessive review and front-loading, in the name of activation. Instead, we should think about activation in terms of retrieval and application. Below are some concrete movies to employ.

Design tasks that require students to use prior knowledge. This is probably the biggest lever we can pull. If students can complete a task without having to integrate what they already know, they probably will. As such, tasks should be designed in a way where: prior ideas are necessary for success, students must interpret, not just execute, and multiple strategies or lines of thinking are possible.

For example, we often give workshop participants a one-page excerpt from the novel A Single Shard. We place them in random groups and ask them to discuss two questions: (1) What can you say about where and when the story is set? and (2) What can you say about the characters and their relationship? When we debrief the task, participants frequently comment on how challenging it was, largely because the excerpt is so short. This creates a useful opportunity to highlight how much they had to rely on their ability to make inferences and draw on prior knowledge to engage with the text. In this case, the task itself is designed to force learners to select and retrieve relevant background knowledge; we, as facilitators, do not review or front-load anything during the launch.

Use launches, to cue, not tell. The way a task is launched is pivotal in shaping how students frame the thinking they will do. It is also a moment where over-scaffolding can easily reduce cognitive demand. An effective launch orients students, provides an entry point, and maintains the challenge.

One way to provide an entry point is to prompt students to think about what they already know—not through telling (e.g., “Remember, slope is rise over run…”), but through cues that support retrieval (e.g., “What do you notice about how these lines are changing?”). The difference is subtle but important: one does the thinking for students, while the other positions them to bring their own knowledge into the forefront.

Ask questions that press for retrieval and connection. Not all questions activate prior knowledge; many simply prompt short answers or surface-level recall. Questions that press for reasoning, however, require students to bring what they know into the work. For example, instead of asking “What’s the slope?” a teacher might ask, “How can you tell which one is growing faster?” and “What are you using to make that decision?” Follow-up questions such as “Where have we seen that idea before?” or “Is that visible in both representations?” press students to retrieve and connect prior knowledge across contexts.
Delay telling. This is one of the most powerful—and most difficult—shifts in practice. When students are stuck, the instinct is to step in and explain. But in doing so, we often replace the very thinking we are trying to develop. Explaining can short-circuit retrieval and sensemaking by removing the need for students to draw on what they already know. Instead, when students are stuck, we can press more productively by asking a question, pointing to a representation, or prompting a connection.

For example, if a student is unsure how to compare two fractions, rather than demonstrating a procedure, a teacher might ask, “What do you already know about how these fractions relate to 1?” or “Can you represent them in a way that makes their size easier to see?” These moves keep the cognitive work with the learner, positioning them to retrieve and use prior knowledge rather than receive it.

Prior knowledge is not something we can deliver to students, nor is it something that becomes useful simply because we have reviewed it. It matters only when learners retrieve it, connect it, and use it in the act of meaning making. The real work, then, is not activating knowledge, but designing for its use. When we create classrooms where students must bring their knowledge into play, learning becomes not just remembering, but sensemaking grounded in what they already know, and extended through what they are figuring out.

The Core of the Matter Issue #15

Thomas Nobili — Sat, 07 Mar 2026 19:06:20 GMT

Student discourse is often highlighted in conversations about High Quality Instruction. As a result, classrooms with a great deal of student dialogue are frequently assumed to be places where a great deal of learning is happening. However, talk alone is not sufficient to induce learning. Rather, student discourse should be viewed as a means to an end. In other words, talk by itself is not the goal. The goal is shared cognition through dialogue.

And when it comes to collective sensemaking, not all talk is created equal. Research on classroom dialogue has identified different forms of talk that emerge when students interact with one another, and these forms vary significantly in their potential to support learning. In this post, we will examine the different types of classroom talk and the research behind them. In addition, we will explore the teacher practices and classroom conditions most likely to foster the kind of dialogue that leads to learning.

Mercer (1995) identified three distinct types of classroom dialogue:

Disputational Talk. This type of talk is best described as surface-level. Students take short turns characterized by offering answers with little or no justification. Responses tend to be curt—e.g., “I agree” or “No, that’s wrong”—and the tone can sometimes be competitive. Ideas are asserted and countered, but rarely examined.

Cumulative Talk. In cumulative talk, students take turns sharing their thinking with one another, though often in a disconnected way. For example, after Student 1 shares an idea, Student 2 acknowledges the contribution and then offers their own thinking, which may not be directly connected to what was previously said. Even when students do build on one another’s ideas, the interaction typically takes the form of agreement”—“yeah, and…”—rather than critique, probing, or evaluation of the reasoning being offered.

Exploratory Talk. This is the form of talk most strongly associated with learning. Specifically, exploratory talk is strongly correlated to higher attainment, improved reasoning, and transfer to individual problem solving (Mercer & Littleton, 2007). In exploratory talk, claims are justified with reasons. Students challenge ideas rather than people. Alternatives are considered and evaluated, and reasoning becomes visible through language. Knowledge is constructed collaboratively as students respond directly to and build on one another’s thinking.

Let’s take a closer look at exploratory talk. What about this type of discourse makes it conducive to learning? Research suggests three main mechanisms associated with exploratory talk that are hypothesized to lead to deeper thinking and sustained learning: cognitive elaboration, cognitive conflict, and co-construction of knowledge. Let’s examine each of these in more detail.

Cognitive elaboration occurs when students explain their thinking to others. In order for their explanation to be coherent they are forced to organize and clarify their ideas. In the process, they often identify gaps in their own understanding and refine their reasoning. Explaining ideas also requires students to connect new information to what they already know, strengthening the underlying conceptual structure of their knowledge (Chi, 2000). These findings are also underscored in a related research strand on the power of self explanation (Chi, et al., 1994).

Mechanism number two is cognitive conflict (Howe, 2013), which takes place when learners encounter explanations that differ from their own thinking and are confronted with a discrepancy that must be resolved. This tension can prompt students to reconsider their assumptions, seek additional information, and test alternative explanations.

Finally, exploratory talk enables co-construction of knowledge. Rather than simply exchanging answers, students build understanding together by acknowledging, clarifying, correcting, and extending one another’s ideas. In some cases, groups develop explanations or strategies that no individual member possessed at the outset (Barron, 2003).

Taken together, these processes help explain why exploratory talk is so powerful. Mercer describes the process these mechanisms stimulate as interthinking—the use of language to think together. Through exploratory talk, students collectively examine ideas, test explanations, and construct understanding in ways that would be difficult to achieve individually (Mercer & Littleton, 2007).

However, like most other classroom behaviors, exploratory talk is emergent in nature. Simply placing students in groups and asking them to discuss something is not sufficient to produce learning. Barron (2003) found that successful and unsuccessful groups did not differ in prior achievement, the number of conversational turns, or even how often correct ideas were proposed. What distinguished the more successful groups was their willingness to engage with those ideas. In productive groups, proposals were taken up, examined, and discussed. In less productive groups, ideas were frequently ignored, rejected without explanation, accepted uncritically, or met with silence. This finding underscores a key feature of exploratory talk: learning depends not simply on the generation of ideas, but on the collective examination of those ideas.

Additionally, teacher practice plays a significant role in shaping whether these kinds of interactions occur. Research examining teacher interventions during collaborative work suggests that certain types of support are more likely than others to promote productive dialogue. Chiu (2004), for example, analyzed the explicitness of teacher help provided: the study utilized a continuum that ranged from focusing students’ attention on aspects of the task to explaining solution procedures. The results showed that more explicit forms of teacher help were negatively associated with students’ engagement immediately after the intervention, as well as with the group’s problem-solving performance. Similarly, teacher commands directing students toward specific steps tended to reduce the amount of productive problem solving that occurred after the teacher left the group. These findings suggest that when teachers provide explanations or directives too readily, responsibility for the intellectual work of the task shifts away from the students and dialogue is stifled.

Other studies reinforce the importance of focusing teacher support on process rather than product. Dekker and Elshout-Mohr (2004) compared classrooms in which teachers provided mathematical explanations with classrooms in which teachers focused on encouraging students to explain and critique one another’s reasoning. In the latter condition, referred to as process help, the teacher’s role was to prompt students to show, justify, and reconstruct their thinking while refraining from supplying mathematical solutions. Students in these classrooms demonstrated significantly greater gains in mathematical understanding than students in classrooms where teachers provided more direct content help.

Research on teacher questioning during collaborative work reveals a similar pattern. Webb and colleagues (2008) examined how teachers’ questioning practices influenced the explanations students offered to one another during peer discussions. Classrooms in which teachers consistently pressed students to clarify and elaborate their thinking were characterized by more complete explanations, greater student participation in reasoning, and higher levels of achievement. In contrast, when teachers accepted partial explanations or quickly moved on after a correct answer was given, students were less likely to engage deeply with one another’s ideas in small group discussions.

Taken together, these studies suggest that exploratory talk is not simply a byproduct of placing students in groups. Rather, it emerges when teachers establish classroom norms and instructional practices that position students as responsible for examining, refining, and building on one another’s ideas. Teacher moves that press students to explain their reasoning, respond to peers, and evaluate competing explanations help create the conditions in which dialogue becomes a tool for collective thinking.

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This section will explore additional foundational moves teachers can implement to create the conditions for exploratory talk.

Set clear ground rules for student talk. Mercer’s work has shown the benefits of making student expectations for talk explicit. These norms should be posted, discussed, and revisited regularly. Example ground rules include:

Share all relevant information
Ask for reasons
Give reasons
Challenge respectfully
Try to reach agreement

Use of cognitively demanding tasks. In order for students to engage in productive dialogue, there needs to be something to talk about. Thus using group-worthy tasks is a critical component in engendering exploratory talk. Mercer’s criteria for such tasks includes: ambiguity, multiple possible strategies, no one student can finish instantly, and explanation is necessary for progress. For more on cognitively demanding tasks check out CoM #9.

Structure interdependence. Group interdependence is a key variable in collaborative success. It refers to the degree that each individual sees their success tied to that of the group. Structuring group work to elicit more interdependence is associated with higher quality talk. Ways to create interdependence include: one shared product, one representation space, and roles tied to reasoning not logistics. For more on interdependence see CoM #5.

Group/self assessment. Periodic reflection on group processes has been shown to positively impact group interaction. For example, students can evaluate how well they explained ideas, responded to peers, and justified their reasoning using a rubric or during a whole class debrief (Johnson & Johnson, 1999; Gillies, 2007; Liljedahl, 2021).

Teacher stance. As discussed above, the way a teacher interacts with groups has a major impact on the quality of dialogue. Premature evaluation, supplying solution steps, or asking leading questions have all been shown to close down discussion. Conversely, pressing for reasoning, asking students to respond to one another, revoicing student thinking, withholding correctness, and prompting for comparison have all been shown to sustain exploratory talk.

Language Modeling. Teacher discourse also shapes expectations for explanation. In classrooms where the teacher consistently pushed students to clarify and fully articulate their thinking—even when answers were correct—students produced more complete explanations during peer discussions and demonstrated higher levels of achievement.

Student discourse is often treated as a visible indicator of High Quality Instruction. However, as the research reviewed in this article suggests, the presence of talk alone is not what drives learning. What matters is the nature of that talk, and the extent to which it supports students in examining, refining, and building on one another’s ideas. Exploratory talk provides one of the most powerful mechanisms for this kind of collective sensemaking. Through explanation, challenge, and negotiation of meaning, students engage in what Mercer describes as interthinking—the process of using language to think together

At the same time, exploratory talk does not emerge automatically from group work. The quality of student dialogue is shaped by the norms, structures, and teacher practices that govern how ideas are treated in the classroom. Tasks must create genuine opportunities for reasoning, students must see their success as connected to that of their peers, and teachers must adopt a stance that keeps the intellectual work with the learners.

The Core of The Matter Issue #14

Thomas Nobili — Tue, 17 Feb 2026 16:53:38 GMT

In most classrooms, long-standing norms dictate how students and teachers interact with one another. As a result, the flow of classroom activity tends to follow those expectations for what a good teacher should do, and teachers’ expectations of what effective instruction entails. In visiting numerous classrooms it is clear that the dominant mental model of an effective teacher is one who is maximally helpful. Our work has led us to a different conclusion: We see the primary role of the teacher as maintaining cognitive demand (Nobili et al., 2026). As such, this issue will examine the ways in which teachers respond to student questions and the implications for those responses on student thinking, student perceived self efficacy, and classroom norms.

One of the hallmarks of the maximally helpful model is the question–response cycle. Students are encouraged, and often expected, to ask questions when they reach a sticking point, and teachers are trained to respond by providing immediate precise answers. Within this pattern, confusion is treated as something to be resolved immediately, and struggle as evidence that the teacher should intervene. The responsibility for meaning making no longer resides with the learner and the cognitive demand of the task erodes. Over time, students learn which questions are worth asking and what kinds of responses those questions are likely to produce. These repeated exchanges establish what students can expect from the teacher and what they are expected to do themselves.

If asking “Is this right?” leads to an immediate response from the teacher—validation when it is correct and correction when it is not—then that question becomes reinforced as something to be repeated, as it leads to progress with minimal effort. Similarly, if, “What do I do next?” reliably leads to the teacher providing the next step without the student having to do any additional thinking, students internalize it as the best path to task completion.

Conversely, if teacher responses to these same questions redirect students to knowledge that exists independent of the teacher, for example, “Have you asked your group yet?” or prompt them to plan their own path forward, “What do you think your next step should be?” then sustained thinking and meaning making become the default as students learn that these types of questions don’t lead to teacher uptake. This pattern is consistent with research on academic culture, which shows that students orient their efforts toward the behaviors that are most efficient for completing work (Doyle, 1983; Doyle, 1988). When the most efficient path to completion is to ask the teacher for validation or direction, students will do exactly that. When the most efficient path is to construct an argument, test a conjecture, or consult peer reasoning, those behaviors become the norm (Cobb & Yackel, 1996; Kazemi & Stipek, 2001).

Classroom conditions are powerful communicators of what students believe the teacher believes they are capable of. Each time a teacher answers a question by supplying a step, they signal to a student that they do not believe they could have figured it out on their own. However, each time a teacher redirects a question back to the student to figure out, they communicate the idea that sensemaking is the student’s responsibility and that they are capable of working through it. Over time, these patterns shape perceived self-efficacy. Students who become accustomed to receiving immediate assistance may come to believe they cannot proceed without it, whereas students who are consistently positioned to make meaning for themselves develop greater independence and agency (Zimmerman, 2002). In psychology, this is known as learned helplessness (Seligman, 1975).

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Peter Liljedahl’s (2021) work provides a useful frame for teacher decision-making when responding to student questions. In Building Thinking Classrooms, he identifies three types of student questions and offers guidance on how teachers should or should not respond. He explains that the two most common types are stop-thinking questions and proximity questions.

Stop-thinking questions are asked by students in order to avoid further cognitive effort. “Is this right?” is the most universal example, but others include, “Do we have to multiply or add for this problem?” and “Can we use a rational expression to solve this?” These questions function as attempts to transfer decision-making and meaning making from the student to the teacher. Proximity questions are asked simply because the teacher is nearby. Students pose them not out of genuine uncertainty, but because their mental model of a “good student” is one who asks questions. These are questions students would not ask if the teacher were not present, either because they already know the answer or can easily obtain it, for example, by looking at the board or consulting a partner. Typical proximity questions include, “Do we have to show our work?” and “Is this going to be on the test?”

Liljedahl contends that teachers should not answer stop-thinking or proximity questions. Doing so not only reduces the cognitive demand of the task, but it also reinforces students’ reliance on teacher validation and direction. In addition, answering these questions signals to students that these are questions they should be asking, therefore increasing their frequency.

The third type, which he calls keep-thinking questions, serves a different function. These are questions students ask in order to continue engaging with the task and sustain their reasoning. Examples include, “When you say the smallest value, does that include negative numbers?” or “Are we including the parent function in this list, or only transformations?” Unlike stop-thinking and proximity questions, these questions signal ongoing sensemaking and therefore may merit a response, as they support continued engagement without transferring the intellectual work to the teacher. It is important to note, however, that even when directly responding to a keep-thinking question is warranted, answering directly should not become the default. As a general rule of thumb these types of questions should only be answered if the information does not exist elsewhere in the room, for example, through a peer, prior notes, or a previous task.

So what should a teacher do when a question, if answered directly, would reduce the cognitive demand of the task? Teachers need a repertoire of alternative moves that sustain engagement, reduce unproductive frustration, and advance learning. What follows are examples of responses that keep students thinking while maintaining the challenge of the task.

Smile and walk away. This move is one that Liljedahl popularized as a response to stop thinking and proximity questions. A variation of this move is affirming and walking away (i.e. That’s a great question, I can’t wait to see what you come up with.) and then walking away.
Answering with a question. This is an effective way to put the responsibility for sensemaking back on the student, as well as to build their ability to be metacognitive. Examples include, “ “What do you think you should do?” and ”What do you think I’m going to tell you?”
Directing a student to use the knowledge in the room. This is particularly powerful during group worthy tasks as answering an individual question in this context can both reduce cognitive demand and undermine collaborative structures.. As a result, if a student asks a question when working on a group worthy task a high leverage move is to simply direct them back to their group. (i.e. “Is this an individual or a group question, because I only answer group questions.”) Similarly, if it is a group level question, direct them to another group.
Redirecting authority. These responses are designed to shift authority from the teacher to the mathematics. They are especially effective for “Is this right?” type questions as they push students to self-regulate. Examples include: “How could you check that?” and “What would convince a peer that this works?”

It is important to note that although only the smile and walk away move explicitly names leaving the interaction, the walking away strategy applies to all of the techniques above. Remaining with the student after posing these prompts can lead to two undesirable outcomes: students may rephrase the same question several times in hopes of eliciting an answer, and group discourse can quickly reorient as students stop talking to each other and start talking to the teacher.

The way teachers respond to student questions is a key factor in maintaining or reducing the cognitive demand of a task. When teacher responses consistently provide verification, correction, or next steps, students learn that progress depends on external validation. Conversely, when responses require students to reflect, think flexibly, and consult available resources, students learn that progress depends on their own reasoning. By altering the way we respond to questions, students internalize that sensemaking, rather than answer-seeking, becomes the expected pathway to success.

The Core of the Matter Issue #13

Thomas Nobili — Mon, 29 Dec 2025 19:58:10 GMT

Selecting and implementing cognitively demanding tasks is one of the longest levers we have to close opportunity gaps and raise achievement for all students. Yet, as I have argued previously (Com #9, and #11), it has proven to be one of the most elusive reforms to bring to scale. Despite broad agreement on the importance of challenging tasks, classrooms too often fail to sustain the level of thinking those tasks were designed to elicit.

This issue builds on the previous two by examining one of the most consistent features distinguishing lessons that maintain cognitive demand from those where demand declines: teacher press.

Teacher press refers to the intentional moves teachers make to push students to explain, justify, elaborate, or connect ideas—rather than settle for superficial or incomplete thinking.

High-quality press typically includes:

Asking students to explain their reasoning, not just give answers;
Pressing for connections between representations, ideas, or strategies;
Encouraging students to persist when the work becomes challenging;
Holding students accountable to make meaning, not completion.

Although the construct originated in the mathematics education research on maintaining cognitive demand (Henningsen & Stein, 1997; Silver & Stein, 1996), subsequent studies have documented its relevance across disciplines, including science (Akcil-Okan & Tekkumru-Kisa, 2020; Jiménez-Aleixandre, Rodríguez, & Duschl, 2000), English language arts (Applebee et al., 2003), and History (Bain, 2005).

One of the most influential and detailed analyses of teacher press comes from Kazemi and Stipek’s (2001) study of 4th- and 5th-grade classrooms learning about fractions. Notably, the researchers did not contrast extreme examples of “good” and “bad” teaching. All classrooms were working on the same cognitively demanding task, and all were led by teachers who appeared supportive and student-centered. As the authors explain:

A casual observer of the lessons we video taped would have seen students working together, led by positive, supportive, caring teachers. On the surface students appeared to be focused on understanding mathematics, A deeper analysis of these lessons, however, revealed important differences in the quality of mathematical discourse. (p.64).

Across multiple episodes, the researchers documented systematic differences between high-press and low-press interactions. For example, while teachers in both types of classrooms asked students to explain their thinking, the nature of those explanations differed substantially. In high-press classrooms, students were pushed beyond recounting procedural steps and were expected to link their strategies to underlying mathematical ideas. Over time, students internalized that “explaining” meant providing justification, not narration. They learned to support their reasoning by coordinating verbal explanations with graphical, pictorial, and numerical representations. In contrast, low-press exchanges typically involved students describing what they did to solve a problem, with little follow-up to connect those actions to mathematical concepts.

A similar pattern emerged during whole-class discussions. In both high- and low-press classrooms, students regularly shared their solutions publicly. However, in high-press classrooms, teachers pressed students to examine similarities and differences across strategies and to consider why particular approaches worked. In low-press classrooms, solutions were often presented sequentially, with discussion focused primarily on surface features such as correctness or neatness rather than mathematical relationships.

The handling of errors also distinguished the two environments. In low-press classrooms, incorrect or incomplete solutions were treated as a natural part of learning, but teachers frequently either bypassed them in favor of a correct response or supplied the explanation themselves. In high-press classrooms, student errors were leveraged as opportunities for collective sensemaking—through peer debate, justification, and the deliberate selection of imperfect solutions for whole-class analysis..

Finally, although students worked in groups for substantial portions of every lesson, the norms governing collaboration differed sharply. In high-press classrooms, teachers established clear expectations for interdependence and shared accountability. Groups were responsible for reaching consensus, and each member was expected to understand and explain the group’s strategy. As a result, the distribution of labor was consistently equitable, with all students engaged in the intellectual work of the task.

In low-press classrooms, by contrast, students were told to work together, but few structures were in place to promote interdependence. Without shared accountability, the distribution of work was often uneven, with some students carrying the cognitive load while others remained peripheral.

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The work of Kazemi and Stipek is noteworthy for many reasons, but particularly because it clarifies the grain size at which teacher press operates. Press is not a discrete strategy that can simply be added to instruction; it is a constellation of mutually reinforcing practices including feedback, group norms, task enactment, discourse routines, and teachers’ beliefs about who is capable of doing rigorous mathematical work.

Importantly, effective teacher press depends on several enabling conditions:

Sufficient wait time. Students must be given time to think and formulate responses.
Equitable participation. Press must apply to all students, not just volunteers or high performers. Equity requires that every student be asked to engage with challenging ideas, which in turn requires deliberate techniques for broad participation.
Replacement, not addition. Press moves are not something teachers layer on top of existing practice. Rather, they require letting go of common responses—such as validating an answer too quickly or stepping in to demonstrate—that often lower cognitive demand before students have fully engaged in the thinking.

That said, there is a practical justification for zooming in on the types of prompts teachers can use to press student thinking. Below are six categories of press that have been shown to be effective across studies. However, the specific category is less important than the effect of the response: extending, probing, or challenging student thinking.

Press for explanation
“Why does that make sense?” “How did you know?
Press for connections
“How does this representation match what you just said?”
Press for justification
“Convince us.” “What evidence supports that?”
Press for precision
“Can we say that more clearly?” “What do you mean by…?”
Press for comparison
“How is your method similar to or different from theirs?”
Press for generalization
“Would this always work?” “What happens if…?”

The teacher’s goal is NOT to use every type of press, but to choose a question or comment that connects the student’s thinking with the learning intention of the task.

Research has consistently shown that teacher press is a key driver in maintaining—or allowing the decline of—a task’s challenge over the course of a lesson. Findings such as those of Kazemi and Stipek highlight that improving instruction at scale requires moving beyond surface-level changes and focusing instead on the sustained, high-quality interactions teachers create with students day in and day out.

The Core of the Matter #12

Thomas Nobili — Thu, 06 Nov 2025 19:10:09 GMT

Every classroom is a dynamic, unpredictable learning environment. Even the most carefully crafted lessons encounter variation from plan to implementation. Students arrive with diverse prior knowledge, confidence levels, and interests. Teachers must navigate these differences in real time, making numerous instructional decisions within a single lesson. While some decisions are pre-planned, many arise spontaneously in response to what students say, do, and reveal during learning. This is the essence of adaptive teaching: responding to student thinking during the act of instruction itself and not at a later time.

Adaptive teaching can be understood as a continuous, real-time instructional cycle in which teachers actively elicit, interpret, and respond to student thinking—both across the class and for individual learners. As Nuthall (2007) argues in The Hidden Lives of Learners, teaching is not deterministic: no single teaching method reliably produces learning of an intended objective for every student because countless variables shape classroom experiences, including students’ prior knowledge, interests, needs, and social dynamics. Teaching, therefore, he argues, requires “sensitivity and adaptation.” Teachers must continuously respond to what happens in real time rather than rely on fixed responses or rigid lesson plans. As such, teaching is more accurately described as a probabilistic practice. In other words, while research shows certain practices are more likely to lead to learning than others, nothing works every time, for every student, in every context. If teaching was deterministic, if certain techniques guaranteed learning, there would be no need to gather and interpret feedback during lessons. However, because learning is variable and unpredictable, what matters most is the teacher’s capacity to notice and respond to students’ experiences during instruction.

When working with teachers and leaders my colleagues and I used to refer to this aspect of teaching as formative assessment. However, we made the shift in terminology due to the many misconceptions associated with the formative assessment. Similarly, Dylan Wiliam (2013) has since reflected that he and Paul Black, in their groundbreaking work, Inside the Black Box, might have better captured their intent with the phrase responsive teaching; as it avoids the baggage often associated with “assessment,” which many educators interpret narrowly as testing.

When done well, adaptive teaching involves three interwoven processes:

Eliciting student thinking during instruction;
Interpreting that thinking to understand what students know and how they are reasoning; and
Responding in ways that move learning forward—individually, in small groups, or with the whole class.

These processes form a recursive loop rather than discrete steps. The teacher continuously elicits, interprets, and responds, adjusting the lesson in real time. At its core, adaptive teaching is both contingent and divergent. It is contingent because teacher moves depend on what students actually say, do, or write; they cannot be fully scripted in advance. Yet, this does not diminish the importance of planning. Instead, it elevates the need for anticipatory planning—designing tasks, questions, and contingency pathways: If students interpret the prompt this way, I’ll ask this follow-up. If they struggle here, I’ll offer this scaffold. These decision points are prepared in advance but activated responsively, based on emerging classroom data. Adaptive teaching is also divergent because there are often multiple reasonable instructional responses available, and which one is “best” may vary based on student needs, the learning goals, or the timing within the lesson.

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For good reason, the field has placed considerable emphasis on the first phase of adaptive teaching: eliciting evidence of student thinking, or making thinking visible. Ron Ritchhart and his colleagues have written extensively on this practice, and resources such as Harvard’s Project Zero provide dozens of routines designed to externalize students’ thinking. Given this attention, the focus of the remainder of this post is on the two other, often less-developed aspects of adaptive teaching: interpreting and responding.

Despite its importance, interpreting student thinking during instruction has been shown to be extremely challenging (Gehrtz, Branter, & Andrews, 2022; Luna & Selmer, 2021). Classrooms are dynamic, information-rich environments where teachers must make constant split-second decisions while juggling multiple inputs simultaneously, including: student talk, body language, pacing, content goals, and behavioral cues, to name just a few. The ability to make meaning from these inputs and act on them accordingly is a central quality of expert teaching. This skill, often described as situational awareness or professional vision (Goodwin, 1994), enables teachers to discern which student cues are instructionally relevant and which are peripheral. It is a form of sensemaking on the fly, requiring cognitive efficiency, emotional regulation, and instructional expertise.

A critical enabler of this sensemaking is strong pedagogical content knowledge (PCK) (Shulman, 1986), which is defined as the intersection between a teacher’s knowledge of the content they are teaching and their understanding of how students actually learn said content; including deep knowledge of concept specific learning trajectories and typical misconceptions. For example, a teacher with strong PCK in fractions might quickly identify whether a student’s mistake stems from a misunderstanding of part-whole relationships or from procedural confusion, allowing them to then tailor their response accordingly. Strong PCK allows teachers to map student thinking onto developmental trajectories and efficiently identify productive next steps.

In addition to content knowledge, interpretive stance matters. Research shows that expert teachers interpret student thinking differently from novices. Experts attend to students’ reasoning processes, noticing conceptual understanding, productive thinking, and naïve conceptions (Gehrtz, Branter, & Andrews, 2022). Rather than evaluating correctness, they ask, “What does this tell me about how the student is making sense of the concept?” This shift from evaluation to interpretation is key to unlocking the formative potential of student thinking. In contrast, teachers with less expertise or limited PCK often default to binary judgments of right or wrong, missing opportunities to build on partial or flawed ideas. Effective teachers view student thinking not as something to fix, but as something to build from.

Critically, the process of noticing is never neutral. What teachers notice and thus how they interpret what they see is shaped by their implicit beliefs, prior experiences, and expectations for students. As Rubie-Davies and Hattie (2024) emphasize, these expectations are powerful. Teachers tend to look more closely at, engage more deeply with, and interpret more generously the thinking of students they perceive as capable. These implicit patterns may go unnoticed, but can significantly impact which students receive meaningful attention and feedback and which are ignored and passed over.

Teacher response is the pivotal point where student thinking is transformed into instructional action. It is not simply about evaluating answers or offering praise, rather it is a skillful and strategic move that connects what students have revealed about their thinking to what happens next in the lesson. In this sense, teacher response is where instruction becomes concretized, agile, and consequential. It is the moment where learning can accelerate, or stall, based on how adeptly and equitably the teacher navigates the nuances of student thinking.

Responding in real time is one of the most cognitively demanding and uncertain aspects of teaching (Gehrtz, Branter, & Andrews, 2022; Luna & Selmer, 2021). Even when student thinking has been successfully surfaced and interpreted, it’s common for teachers to default to evaluating correctness, or to insert their own opinions rather than working from students’ actual ideas. The reflex to “fix” errors or steer students to the right answer can crowd out more generative possibilities like turning a misconception into a learning opportunity, or using a student idea to pose a new question to the class.

Beyond content, teachers must also determine the appropriate scale of response—individual, small-group, or whole-class. This decision depends on patterns in student thinking revealed through earlier routines. Diagnostic practices allow teachers to survey the landscape of understanding: who is stuck, who is ready to move on, and whose thinking needs further exploration. Once that thinking has been elicited and analyzed, the teacher faces a crucial choice:

Whole-class response: appropriate when a common misconception must be addressed, when a public model can serve as a shared reference point, or when a single student idea can advance the collective conversation. This response creates coherence and maintains a shared trajectory of learning.
Small-group response: useful when a subset of students shares a specific challenge or need. This approach allows for targeted support while maintaining high cognitive demand and encouraging collaborative reasoning among peers.
Individual response: most effective when a student’s thinking is unique, when affective factors (e.g., frustration, confidence) must be addressed, or when a contribution is still tentative and not yet ready for public discussion. These moments require sensitivity and strategic support, ensuring that the student feels seen and supported without being isolated or singled out.

The instructional move itself, or the teacher response, is the way the teacher operationalizes their interpretation of student thinking. These moves vary in form, but share a common function: to connect instruction to students’ actual ideas in order to move learning forward. To accomplish this an effective response must:

Align with students’ current thinking—not what the teacher hoped to hear;
Build on what students understand—not merely correct what they don’t, and;
Advance learning in real time—at the right scale, with the right support.

Crucially, teacher response is not only about instructional progress; it is also a moment of interpersonal and intellectual signaling. The fashion in which a teacher responds within a discretionary space communicates to students what kinds of thinking are valued, whose contributions matter, and whether struggle is treated as an obstacle to learning or a catalyst for it. A well-timed, thoughtful response can reposition a student from the margins to the center of the learning community. It can affirm a student’s intellectual identity, demonstrate that their thinking is worthy of public engagement, and model the idea that learning is a process of refinement, not perfection.

Ultimately, how a teacher responds to student thinking, especially during moments of uncertainty, serves as both a mirror and a lever. It reflects the teacher’s core beliefs about learning and students, and it shapes the opportunities students will have to engage with meaningful, challenging, and equitable instruction. As such, adaptive teaching is both a method for supporting learning and a lever for advancing equity. It positions student thinking as the primary guide for instructional decision-making. And it makes clear that the work of teaching is not just to deliver content, but to actively build a culture in which every student’s ideas are visible, taken seriously, and used to drive collective understanding forward. This work demands deep pedagogical content knowledge, a strong sense of situational awareness, and flexible planning. But perhaps most critically, it demands that teachers hold ambitious expectations for all students, and that they are willing to adapt their instruction in ways that affirm what students can do—especially when that thinking appears incomplete, tentative, or unexpected.

The Core of the Matter Issue #11

Thomas Nobili — Mon, 25 Aug 2025 00:33:15 GMT

This issue explores an important challenge for teachers: how to provide support for students during instruction without lowering the cognitive demand of the task? This brings together two concepts that I’ve discussed before—cognitive load and cognitive demand. This is a crucial and difficult challenge as research consistently highlights how hard it is for teachers to strike this balance effectively and consistently (Sullivan & Mornane, 2014). In fact, Stein et al. (1996) found that in their sample of math classrooms, scaffolds that reduced cognitive demand were provided 64% of the time during task enactment. But first, a quick review of these concepts.

Cognitive demand refers to the kind of thinking a task requires for students to be successful (Doyle, 1988). It captures how much students must figure out; how hard they must think to complete the task. When a task is aligned to important learning goals, this demand reflects germane cognitive load, or the mental effort relevant to the intended learning. For more on cognitive demand, see Issue #9.

Cognitive load, on the other hand, is concerned with how to best manage the limited capacity of working memory during the learning process. Unlike our long-term memory, our working memory has an extremely limited capacity and can become overloaded quickly. During knowledge acquisition, our working memories deploy resources to think about all kinds of things, also known as cognitive load, some of which are connected to what we are trying to learn and some that are not. For more on cognitive load, see Issue #4. Therefore, another way of looking at maintaining cognitive demand is to think about it in terms of maximizing the germane load of a task while minimizing extraneous load that distracts or overwhelms.

Further, in relation to cognitive load, scaffolding refers to any instructional move that reduces extraneous cognitive load. That is, unnecessary mental effort that distracts from the core learning goal. Cognitive Load Theory is a helpful lens for evaluating whether a proposed support is a true scaffold: one that reduces extraneous load without interfering with germane load, or the mental effort directly tied to the learning objective. If a support reduces germane load, it likely lowers the cognitive demand of the task and would not be considered effective in promoting meaningful learning.

As discussed in Issue 4, scaffolds typically take two main forms: tools and strategic prompts, both of which can help reduce extraneous load. Tools might include resources like books on tape, graphic organizers, or math manipulatives. Strategic prompts, on the other hand, guide novice learners’ attention toward the most germane aspects of a task, helping them stay focused on the core reasoning at the heart of the work.

The figure below maps the relationship between cognitive demand, cognitive load, and scaffolding. To maintain a task’s cognitive demand, a teacher must provide just enough scaffolding to prevent students from becoming cognitively overloaded without reducing the effort students must exert to engage with the task. In other words, because scaffolding reduces cognitive load, too much can inadvertently lower the task’s cognitive demand. This inverse relationship makes maintaining high demand a delicate balancing act, with little margin for error.

© Thomas Nobili 2025

When it comes to maintaining cognitive demand, it’s not always the content of the scaffold that matters, it’s often the timing. Teachers frequently jump in too early, sometimes at the first sign of struggle because we see it as our job to be helpful. But this well-intentioned instinct can inadvertently rob students of the opportunity to engage in meaning-making, the process of connecting new ideas to prior knowledge, which is essential for building robust mental schemas.

In many classrooms, struggle is treated as an obstacle to learning rather than a catalyst for it. Yet a substantial body of research shows that students need to grapple meaningfully with challenging mathematical ideas in order to learn deeply (Bjork, 1994; Hiebert & Wearne, 1993; Hiebert & Grouws, 2007; Warshauer, 2015). That said, the term productive struggle is not without controversy. It is sometimes misrepresented as simply letting students flounder by engaging in unstructured discovery with little to no teacher guidance. To avoid such ambiguity, I adopt Hiebert and Grouws’ (2007) operational definition: "The intellectual effort students expend to make sense of mathematical concepts that are challenging but fall within their reasonable capabilities."

According to this view, productive struggle involves students working through challenges that are within reach, a conceptual space described by Vygotsky as the zone of proximal development (ZPD), while being supported thoughtfully and given sufficient time. In fact, research on human-assisted tutoring by VanLehn et al. (2003) found that learning was more likely when students hit an impasse—a moment of not knowing—than when they completed tasks with ease. The study concluded that allowing students to attempt a step, even if they might make a mistake, was more beneficial for learning than preemptively showing them how to do it. This suggests that, at times, the best response to a student who is stuck is to do less, not more. As Peter Liljedahl (2021) puts it, sometimes the most effective move is simply to smile and walk away.

However, when students genuinely need support, responding in the moment is one of the most challenging and uncertain aspects of teaching (Gehrtz, Branter, & Andrews, 2022; Luna & Selmer, 2021). Even when a teacher has successfully surfaced and made sense of student thinking, a difficult task in itself, the next step isn’t straightforward. In these moments, there’s often a strong instinct to correct errors or guide students toward the “right” answer. This reflex is understandable: it’s fast, familiar, and feels helpful. But it can also undermine richer learning opportunities. By jumping in too quickly to correct or direct, we risk missing the chance to build on a partial idea, transform a misconception into a productive turning point, or pose a question that deepens the whole class’s engagement. Learning to pause that instinct and instead respond to student thinking as it is, not as we wish it were, is a complex and essential part of effective practice.

One of the most common ways cognitive demand is unintentionally reduced is through what Henningsen and Stein (1997) refer to as “task devolution”: when the teacher, in an effort to clarify or help, reshapes a rich task into a procedural one. This might involve rephrasing a question to make it simpler, narrowing the number of acceptable strategies, stepping in with a model too soon, or providing hints that remove the need for sensemaking. These responses often emerge from a desire to protect students from failure or frustration, but they come at a cost as they strip away the very cognitive work the task was designed to elicit.

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Students often reach different kinds of impasses as they engage with challenging mathematical tasks. These points of difficulty are not only expected but can be leveraged as opportunities for meaningful learning. However, the challenge lies in knowing how to support students in ways that manage cognitive load without diminishing the task’s cognitive demand. In this section, I explore two common moments of struggle during task enactment: getting started and carrying out a solution process. For each, I offer specific scaffolds that help students move forward while preserving opportunities for reasoning and sense-making.

Students may experience difficulty early in the lesson, particularly when they struggle to find an entry point into the task. When this happens, it can serve as valuable feedback to the teacher about the effectiveness of the task launch. One of the primary goals of a strong launch is to frame the problem in a way that provides all students with a clear entry point into the work. If multiple students are unable to get started, it may indicate that the launch failed to illuminate a meaningful starting point.

That said, there are also times when a task has been set up for broad accessibility, but students don’t recognize that they can get started. In these cases, teachers have several options for supporting students in ways that preserve the task’s cognitive demand.

First, whenever possible, cognitively demanding tasks should be embedded in a meaningful context. This serves several functions, but most relevant here is that the context can act as a scaffold by making abstract mathematical ideas more concrete, familiar, or meaningful thus reducing cognitive load and helping students access the problem. Let’s use the following example to illustrate this point. Suppose we presented the following question to a third grade class:

You're designing a dog pen for your new puppy. You have 24 feet of fencing. What are some different rectangular shapes you could make for the pen? Which shape gives your puppy the most space to run around?

In this example the context can scaffold student thinking in a few different ways:

The goal (giving the puppy space to run) gives meaning to maximizing area, helping students make sense of why that matters.
The fixed amount of fencing (24 ft) sets a real-world perimeter, providing structure without needing to define "perimeter" explicitly.
Students might start by drawing random sized rectangles or draw on prior experience (i.e. we have a fenced in space for our dog so I’ll start with sketching that shape), before there is a need for formal mathematical reasoning.

Contrast this with,a decontextualized version of this task like "List all rectangles with a perimeter of 24 units. Which has the greatest area?" which may feel overly abstract or confusing to students who haven’t yet developed fluency with the relationship between perimeter and area. So, the context doesn't just decorate the task, it helps students enter it.

Second, teachers can reframe the conversation from what students don’t know to what they do know. For example, after a task launch, a student might raise their hand and say, “I don’t get any of this.” Rather than jumping in with a directive, the teacher might respond, “Tell me one thing you do know about the problem.” Starting from what the student can articulate helps lower extraneous cognitive load and guides them toward initial steps they can take.

Finally, teachers can leverage peer thinking as a scaffold. If a student or group is stuck, the teacher might say, “I see you’re having trouble figuring out how to begin. Go talk to Eddie’s group about how they got started and see if that gives you any ideas.” This kind of strategic redirection not only supports the struggling student, but also promotes knowledge mobility across the classroom.

Another common point during task enactment where students may need support is in carrying out a process that leads to a solution. When this happens, a teacher’s first move should often be to reduce extraneous cognitive load. Specifically, any unnecessary mental effort that interferes with students’ ability to focus on the mathematics at hand. As discussed in my earlier post on cognitive load, this might involve providing tools like a multiplication chart or a list of relevant formulas, so students can devote their cognitive resources to reasoning rather than retrieval.

Sometimes, reducing extraneous load also means supplying a key piece of information necessary for students to engage in the kind of mathematical thinking germane to the lesson. For instance, my colleague Isobel and I recently facilitated a workshop where the topic of background knowledge came up. The group of teachers we were working with expressed concern about how students could engage with a measurement task if they didn’t know how many inches are in a foot. This concern highlights an important instructional distinction: when students are unable to carry out a process because of a missing fact, and they clearly recognize the need for that fact, providing it directly is not a case of reducing cognitive demand, rather it’s clearing the path for deeper engagement. In this case, the struggle is not conceptual; it’s about having access to information required for meaningful sense-making.

This is where the distinction between background knowledge as schema versus background knowledge as isolated facts becomes crucial. When we talk about the importance of activating prior knowledge and its role in managing cognitive load, we are referring to schema, or well-organized networks of connected knowledge, not simply isolated facts.

For example, a student may not know the fact that there are 12 inches in a foot, but this does not mean they lack a schema for the concept of measurement. In fact, they may understand quite a bit: that different units are used to measure different attributes, that some units can be converted into others, and that there are distinctions between standard and non-standard units of measure. So if they are working on a measurement task and get stuck because they don’t know how many inches are in a foot, we should simply tell them because the very fact that they realize they need that information suggests that they are actively reasoning within a robust schema. In this case, they are engaging in measurement conversion, which is a form of proportional reasoning. Withholding that fact in the name of “productive struggle” would misinterpret the nature of maintaining cognitive demand, they don’t need to discover it; they need it as a tool to advance deeper thinking.

In situations where students are struggling to carry out a solution process, but are not overwhelmed by extraneous cognitive load, prompting them to consider a particular strategy or representation can help focus their cognitive effort on the germane aspects of the problem. The key is to avoid simply telling them what to do. For example, if students are stuck on a ratio problem involving the relationship between pounds of wheat and total cost, a teacher might be tempted to say, "You should use a ratio table so you can see how the price increases at the same rate per pound." While this may help the student get an answer, it likely robs them of the opportunity to make sense of the mathematical relationships themselves.

Instead, we want to use the thinking the student has already done as a bridge to deepen their reasoning. Suppose a student has already figured out that 2 pounds of wheat cost $4.46 and 4 pounds cost $8.92, but is struggling to determine the cost of 10 pounds. Rather than introducing a new strategy or representation, we might organize their existing work in a ratio table and ask, “If 2 pounds cost $4.46 and 4 pounds cost $8.92, what do you think 8 pounds would cost?” Since the student already used a doubling strategy from 2 to 4 pounds, they’re likely to double again to find the cost of 8 pounds. From there, we can ask, “So now you have the cost for 2, 4, and 8 pounds. Can you use that information to figure out 10 pounds?” This approach honors the student’s thinking, reinforces the structure in the mathematics, and supports strategic sense-making—all without lowering the cognitive demand of the task.

It’s important to emphasize that the scaffold provided, namely organizing the student’s work into a table and use of strategic prompting, offers access to germane features of the task without constraining the student to a single line of reasoning. The information in the table can be interpreted and extended in multiple ways, depending on how the student is making sense of the problem. For example, a student might:

Combine the costs of 8 pounds and 2 pounds to find the cost of 10 pounds;
Scale the cost for 2 pounds by a factor of 5;
Halve the cost for 2 pounds to find the cost for 1 pound, then multiply by 10;
Use the fact that 10 is 2.5 times 4 to scale up accordingly;
Notice that the cost in each row is consistently 2.23 times the number of pounds and apply that relationship to 10 pounds.

Each of these approaches reflects significant mathematical thinking: unit rate, equivalence, the Distributive Property, scaling, and Constant of Proportionality, and all contribute to building deeper conceptual understanding. The point is not to guide students toward a single “correct” method, but to support and expand the reasoning they’ve already begun.

Supporting students during moments of struggle is one of the most critical and nuanced aspects of teaching. It requires a deep understanding of the relationship between cognitive load and cognitive demand, and a commitment to keeping students intellectually engaged. Whether the challenge lies in getting started or progressing through a solution, the teacher’s role is not to rescue but to reframe, redirect, and refocus while drawing on students’ existing thinking as a bridge to new understanding. When done well, scaffolding doesn’t reduce the challenge of the task; it makes that challenge more accessible. And in doing so, it helps students build the kind of mathematical agency and resilience that meaningful learning requires.

The Core of the Matter Issue #10

Thomas Nobili — Wed, 21 May 2025 19:01:32 GMT

One of the staples of many mathematics classrooms is spending time reviewing previously taught material. This often takes the form of going over homework, completing Do-Now or warm-up problems, or having students work on computer programs after finishing the day’s assignment. In fact, a comprehensive analysis of the TIMSS Video Study (1999) found that teachers in U.S. classrooms spend 53% of lesson time reviewing previously taught content—the second highest proportion among the countries studied, just behind the Czech Republic (see figure below for stats on all countries sampled). Moreover, the study reported that 28% of lessons analyzed consisted entirely of review, a figure tied for first place in the sample. At first glance, this may be hard to believe: are we really spending over half of our instructional time on review? And that roughly one out of every four lessons—equivalent to about 50 days in a 180-day school year—is devoted solely to review rather than new content?

Source: Teaching Mathematics in Seven countries: Results from the TIMMS 1999 video study.

Some readers may be shaking their heads and thinking, “Okay, sure—but that study is over 25 years old. Maybe things are different now.” Unfortunately, more recent research continues to corroborate the same pattern. A growing body of evidence shows that a substantial portion of instructional time in U.S. math classrooms is still spent reviewing content from prior grade levels, particularly at the beginning of the school year. (TNTP, 2021; NCTM, 2014; WestEd, 2017). This aligns with findings from TNTP’s (2018) large-scale Opportunity Myth study, which analyzed over 30,000 assignments and found that students spent more than 500 hours per school year—nearly six months—on work that was not grade-level appropriate, much of it focused on remediation. These trends are especially pronounced in classrooms serving historically marginalized students, where extended time on review can delay access to grade-level mathematics and exacerbate opportunity gaps.

Additional large-scale research further supports the claim that excessive review remains a systemic issue. Polikoff (2012), analyzing data from over 7,000 teachers across the United States, found that instructional redundancy in math is prevalent, with up to 60% of instructional time in a given grade spent on content already taught in the previous year. By comparing the instruction of teachers in consecutive grades within the same schools, Polikoff showed that this redundancy often exceeds what is prescribed by state standards. When translated to a 180-day school year, this amounts to between 68 and 90 days of instruction—depending on the grade level—focused on previously taught material.

Further analysis of these redundant instructional units, referred to as Skill-Expectation-Content (SEC) cells, reveals that the majority are procedural in nature. Of the 24 redundant SEC cells highlighted, all but two were classified at level C, “Perform Procedures,” with 18 at level D and only one at level E. None were at level F, which indicates conceptual or problem-solving skills. This pattern demonstrates that the bulk of redundant instruction emphasizes rote procedural skills rather than deeper conceptual understanding. Certain topics, such as procedures for adding and subtracting whole numbers and integers, were repeatedly taught across multiple consecutive grade levels (K–1, 1–2, 2–3, and 4–5). Overall, the most repetitively taught content clustered around number sense and procedural fluency, underscoring a systemic focus on repetitive skill practice at the expense of conceptual growth.

Together, these findings illustrate that instructional overlap is not an isolated phenomenon but a persistent structural feature of U.S. math education, which significantly limits time for grade-level learning and access to thinking critically about content.

The question that naturally stems from such findings is: why are so many instructional minutes devoted to review? One plausible explanation is that teachers feel it is necessary—believing that without review, students will forget what they have learned. This belief is often reinforced by their own classroom experiences. For example, a teacher may assign review problems from previous lessons or units only to find that many students still struggle, prompting additional review in subsequent lessons. Many teachers have experienced informal conversations with colleagues lamenting, “I had to go over everything we did last week again. It was like I never taught it.”

But this leads to a deeper question: why are students having such difficulty retaining or applying prior knowledge? Findings from the TIMSS (Trends in International Mathematics and Science Study) provide insight. In U.S. eighth-grade math classrooms, approximately 70% of problems students worked on involved using procedures, with 67% of those classified as low complexity. Only 17% of the problems required students to make connections or engage in higher-level thinking. Furthermore, 75% of individual or group work time was spent repeating procedures, and only 34% of problems focused on application—the lowest proportion among the countries sampled. Additionally, the average time spent per independent problem in the U.S. was about 5 minutes, compared to 15 minutes in Japan, which had the highest average. These patterns suggest that instructional emphasis on low-complexity procedural work may contribute to students’ struggles, perpetuating the cycle of repeated review and limiting opportunities for deeper conceptual understanding.

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To be clear, I am not suggesting that no instructional time should be devoted to review. But I am arguing that if less than half of our instructional time is focused on teaching and learning new material, then we are spending too much time on review. Here are some suggestions for reducing that time.

One common practice in many classrooms is devoting the lesson before a test or quiz to reviewing the material that will be assessed. There are multiple reasons for this, including student and parent expectations, and the fact that the test is likely for a grade. But if the primary purpose of a test or quiz is to determine which students have mastered the content and which require additional instruction, then we would test without review, knowing that the test itself diagnoses which students need further reinforcement of a concept. Treating every student as if they need the same review is inefficient.

Another inefficient use of instructional time is the routine of going over homework in class. Like test review, this practice is due for retirement. How students approach homework varies widely: some work diligently to complete it, others skip it altogether, some copy answers from peers, and others receive substantial help from parents or older siblings. This variation makes in-class homework review inefficient and inequitable. The only students likely to benefit are those who completed the work independently and may need clarification on a few minor misconceptions. A more effective alternative is to provide worked solutions—either digitally or in print—that students can review on their own time if needed. This frees up class time for more meaningful engagement with new material.

A third, widespread practice, as mentioned earlier, is spending weeks at the beginning of the school year reviewing content from previous grades. This approach is often inefficient and ineffective because the review is done in isolation rather than connected to current grade-level material. A far more effective strategy is to review prior content on a just-in-time basis. For example, rather than spending time at the start of sixth grade reviewing how to multiply fractions—a fifth-grade standard—you could provide targeted re-teaching of this skill precisely when students are learning to divide fractions, a sixth-grade concept. This approach not only saves instructional time but also reinforces the relationship between multiplication and division, extending understanding from whole numbers to rational numbers in a meaningful, connected way.

Finally, giving students consistent opportunities to make sense of new content—by connecting it to what they already know, rather than practicing isolated procedures—strengthens their long-term understanding. This deeper learning reduces the likelihood that concepts will need to be retaught, creating a more efficient and meaningful learning experience over time.

In sum, while review plays an important role in supporting student learning, the evidence shows that U.S. math classrooms currently devote an excessive amount of instructional time to revisiting prior content—often procedural and low-level—at the expense of engaging with new, grade-appropriate material. This overemphasis on review reflects both systemic instructional redundancy and teachers’ well-intentioned responses to student difficulties. However, to truly support student growth and close opportunity gaps, it is essential to balance review with meaningful, connected learning experiences that promote deeper understanding. By shifting to strategic, just-in-time support and fostering opportunities for students to actively make sense of new concepts, educators can better ensure that instructional time advances students’ knowledge and reduces the need for repeated reteaching. This approach holds promise for creating more equitable and effective mathematics instruction that prepares all students for success.

The Core of the Matter Issue #9

Thomas Nobili — Thu, 03 Apr 2025 16:42:40 GMT

The quality of the tasks students are given, coupled with the quality of instruction they receive, has been shown to be one of the most powerful predictors of student achievement. Together these two constructs are called opportunity to learn, which has also been shown to be the single most important factor in explaining differences in achievement among different groups of students (Hiebert & Grouws, 2007). Relatedly, level of cognitive demand has proven to be a useful way to describe both the tasks teachers select and how they are implemented. Walter Doyle (1988) classified tasks as the context for students thinking both before and after instruction. In addition, he defined cognitive demand as the cognitive processes required of a student to be successful with a task.

In mathematics education there exists a strong correlation between giving students frequent opportunities to solve challenging mathematical problems, justify their reasoning, and chances to make connections between mathematical ideas, and increased student outcomes. It has also been shown that in order for students to engage in these practices, they need regular opportunities to work on cognitively demanding tasks (Stein & Lane, 1996). This includes evidence that cognitively demanding tasks increase learning opportunities for all students, not just those who were previously high-achieving (Zohar & Dori, 2003). Tasks with high cognitive demand tend to be open-ended, require students to make connections to underlying mathematical ideas, and engage students in the disciplinary practices mentioned above. In contrast, tasks with low cognitive demand require students to memorize or reproduce facts, or to perform relatively routine procedures without making connections to any underlying mathematical concepts.

Stein et al., (2000) expanded the notion of task by proposing that the cognitive demand inherent to a task is not static, but rather a dynamic construct that is affected by the interactions teachers and students have with each other, as well as with the content in a classroom environment. The Task Framework (pictured below), depicts tasks as existing in different phases.

Source: Tekkumru‐Kisa, M., Stein, M.K., & Doyle, W. (2020).

The first phase, shown in the far left rectangle, represents the potential intellectual work, or cognitive demand of a task, as it exists in the curriculum, unit plan, or some other source. The next phase, shown immediately to the right of the initial rectangle, represents the task as set up or launched by the teacher. In other words, how a teacher takes the task as it exists on paper and brings it to life in the classroom. The oval between these two phases represents the factors associated with either reducing or maintaining the cognitive demand of the task as it moves from curriculum to classroom. The third rectangle in the figure (moving from left to right), represents the actual intellectual work that students are engaged in while working on the task. Again, the oval between the setup phase and this final phase represents the factors associated with either reducing or maintaining the cognitive demand of the task through transition. The far right triangle in the framework represents the actual student learning that occurs as a result of engaging in the task, with the goal being that the closer the final cognitive demand level of the task is to the intended demand level, the more learning will have taken place.

Despite its importance, maintaining the cognitive demand of a task through all phases of instruction has proven to be extremely challenging (Boston & Smith, 2009; Jackson, 2013; Munter & Haynes, 2019). Multiple studies have documented both the relatively few opportunities American students have to engage in cognitively demanding tasks, as well as the frequent reduction of demand that often occurs during implementation. For example, an analysis of a random sample of classroom video that was part of the 1999 TIMMS study found that American teachers selected tasks with a high level of potential cognitive demand only 17% of the time, and that of the tasks selected, the level of cognitive demand was maintained during implementation less than 1% of the time. Similarly, Jackson’s (2013) analysis of a large sample of middle school mathematics lessons revealed that the cognitive demand of many tasks was reduced in the launch phase (64% of the time), and only 6.7% of tasks that were launched effectively also maintained their potential level of cognitive demand through all phases of the lesson. Other research has shown that marginalized students receive fewer opportunities to engage in high cognitive demand tasks than their more privileged peers (Rubie-Davies, et al., 2014), and that students of color are more likely than their white classmates to perceive the reduction of the cognitive demand of a task over the course of a lesson to be related to teacher expectations based on their race (Munter & Haynes, 2019).

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There are many factors related to maintaining or reducing the cognitive demand of a task during instruction. This post will not get into them all, but I will focus on a few that have proved to be both pernicious and prevalent. The factor most associated with reducing the demand of a task is the teacher removing some element of challenge for students in order to make the task easier and often much more procedural (Henningsen & Stein 1997). Catalysts for this reduction in challenge have been traced to things like teachers acquiescing to student pleas to make the task less ambiguous, which often removes opportunities for student sensemaking to occur. Other factors driving this lowering of demand also include teacher expectations, and teachers prioritizing task completion over productive struggle (Stein,Grover, & Henningsen, 1996).

Conversely, one of the factors most associated with the maintenance of cognitive demand is teacher press for explanation and justification (Menzies, Schunn, & Stein, 2024). This often involves teachers asking questions to probe student thinking, as well as prompting students to provide reasons or justifications for their answers or proposed next steps. These types of interactions have been shown to successfully surface student misconceptions, engage students in metacognitive processes, and support students in connecting mathematical actions and procedures to overarching concepts and ideas (Warshauer, 2015). Additionally, Wilhelm (2014) found that a teacher's ability to maintain the cognitive demand of a task throughout all phases of instructions is highly correlated to their understanding of effective instruction, level of mathematical knowledge for teaching, and views on the capabilities of struggling students.

While there is no doubt that tasks with high potential cognitive demand are necessary for sustained student success, they are in no way sufficient. As this post has illustrated, a student's opportunity to learn also hinges on whether the cognitive demand of a task is maintained throughout instruction. Unfortunately, maintenance of cognitive demand continues to be difficult to achieve. Without careful attention to how tasks are launched, facilitated, and consolidated, the cognitive demand can easily be diminished, limiting the depth of student learning. Supporting teacher efficacy in this area is crucial, as it serves as a powerful leverage point in creating more equitable outcomes and greater overall student success in mathematics.

The Core of the Matter Issue #8

Thomas Nobili — Tue, 11 Mar 2025 15:17:17 GMT

My work as a consultant takes me into classrooms in a number of districts. And a frequent observation when on these travels is that students are doing a lot more of their work standing at vertical whiteboards. This is of course in response to Peter Liljedahl’s revolutionary Thinking Classrooms framework, which has blazed through the K-12 landscape over the past 4 years. Yet, this influx of what Peter calls vertical non permanent surfaces (VNPSs), has not necessarily led to drastic shifts in the way teachers teach and students experience mathematics. Now don’t get me wrong, my work routinely affords me the pleasure of seeing plenty of fantastic educators brilliantly delivering equitable and adaptive instruction; however, I am also frequently in the position of viewing a lesson featuring students working at VNPSs where the instruction does not necessarily leverage the potential of the board structure to engage students in cognitively demanding work.

This tension between the ideal version of an instructional reform versus the way it is implemented in practice has historically been one of the largest barriers to sustainable instructional improvement. One of the most vivid examples of this phenomenon is illustrated in David Cohen’s 1990 account of Mrs. Oublier’s perceived instructional transformation in teaching mathematics. After informing us of Mrs O’s recent attendance at a workshop based on California’s recently released math reform framework, Cohen tells us:

“Though her revolution began while the framework was still being written, it was inspired by many of the same ideas. She reports that her math teaching has wound up where the framework intends it to be. Yet as I watched and listened in Mrs. O's classroom, things seemed more complicated. Her teaching does reflect the new framework in many ways. For instance, she had adopted innovative instructional materials and activities, all designed to help students make sense of mathematics. But Mrs. O seemed to treat new mathematical topics as though they were a part of traditional school mathematics. She used the new materials, but used them as though mathematics contained only right and wrong answers. She has revised the curriculum to help students understand math, but she conducts the class in ways that discourage exploration of students' understanding (p. 312).

He continues:

“That melange is part of the fascination of Mrs. O's story. Some observers would agree that she has made a revolution, but others would see only traditional instruction. It is easy to imagine long arguments about which is the real Mrs. O, but they would be the wrong arguments. Mrs. O. is both of these teachers. Her classroom deserves attention partly because such mixtures are quite common in instructional innovations-though they have been little noticed. As teachers and students try to find their way from familiar practices to new ones, they cobble new ideas onto familiar practices (p. 312).”

The last sentence in the above quote, “...they cobble new ideas onto familiar practices,” is precisely how we get a classroom where students are standing at vertical whiteboards, yet not necessarily collaborating with peers to negotiate meaning about mathematics, or engaging in novel and challenging tasks.

This is of course because BTC is not about the boards! Rather, the essence of Building Thinking Classrooms is about getting more students to think harder for longer amounts of time. This is accomplished through giving students challenging things to ponder over, the space to make meaning, and creating the conditions for collective and individual knowledge generation. The boards are certainly a key component of a larger catalyzing process, but they are not the star of the show.

In statistical terms the boards are considered a moderating variable. In other words, if the goal of BTC is to increase mathematics learning, working at the boards is not powerful enough to do this directly. However, they are important in that they make it more likely that the mediator, or the variable that does directly influence the goal, has a better chance of working. In the case of BTC, the mediating variables are mathematical thinking and meaning making.

In order to avoid the creation of future Mrs Os it is important that educators are supported in deconstructing a new instructional reform in order to separate what components are moderators, which can be categorized in terms of format and characteristics, from the mediator, or the actual mechanism that leads to increased student learning. Two things that a district can do to help educators to successfully identify the mechanism in a given instructional practice are to create a shared understanding of high quality instruction across the system, and to make sure that all staff have a deep understanding of how learning happens.

A shared vision of what good looks like is essential. Without a shared target you are creating a space where everyone will operate according to their own mental model of what good teaching is, with no guarantee of alignment across classrooms, grade levels, or schools. This is part of the Mrs. O conundrum. In reading the article it is quite evident that Mrs.O and the piece's author, David Cohen, have two different pictures in their heads of what high quality math instruction looks like. This is why I cringe every time I hear an administrator or coach say something to the effect of, “you are already doing most of these things” when addressing a group of teachers regarding some kind of instructional reform. I try to point out nicely: “No, they are not already doing it, because if they were we wouldn't be having this conversation right now.” What they are doing, by trying to provide reassurance that the change they are asking for is going to be easy, is actually attempting to minimize the conceptual distance between current practice and desired practice. I am also reminded of the wisdom of my colleagues in these situations, who often use the line, “you cannot talk someone into a new mental model.” Instead people need experiences that have the potential to push on their existing schema and force them to think differently. As such, part of cultivating a shared vision of effective instruction must include multiple experiences where people can see and experience the shared target.

Relatedly, having an understanding of how students learn is crucial in being able to decipher the mechanism from form and characteristics when it comes to effective instruction. Once again, the case of Mrs. O provides a particularly salient example. Cohen explains:

“The teacher used a new mathematics curriculum, but used it in a way that conveyed a sense of mathematics as a fixed body of right answers, rather than as a field of inquiry in which people figure out quantitative relations. It is easy to see the framework's ideas in Mrs. O's classroom, but it also is easy to see many points of opposition between the new policy and Mrs. O's approach (p.313).

It is evident from the above quote, that Mrs O views learning as transmission and thus teaches in accordance with that belief. However, learning involves integrating new knowledge with old. It occurs when deep schemata are developed and encoded into long term memory. Schema acquisition is the result of a meaning making process facilitated by the act of thinking. Therefore, if you want learning to occur you need to employ instructional practices whose mechanisms engender deep thinking.

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What is the mechanism? This is the question that needs to remain top of mind as we work to shift and improve existing practice. More specifically, we need to be able to connect the mechanism of a given practice to current models of learning. To accomplish this will require major shifts in how we currently approach instructional reform. For example, rather than individual teachers trying these things in isolation, we need structures that allow teams of teachers to work collaboratively on instructional improvement. Further, in order to avoid Mrs O’s fate of improvement at the margins, we need teachers to have ample opportunity to experience ambitious and equitable instruction for themselves, as well as with their own students, in order to challenge potential limiting beliefs about what certain students are capable of achieving. In addition, we need to stop using evaluation tools and look-for docs, which often contain detailed descriptions of the moderators of a given practice (i.e. students standing at whiteboards), with no mention of the mediator. This tends to lead to follow-up actions and conversation focused on things that are easy to measure, but have limited impact.

Mrs. O serves as a cautionary tale. She teaches us that the process of transforming instructional practices requires more than just the adoption of new tools and techniques, such as vertical whiteboards. True transformation occurs when the focus moves toward understanding and fostering the mechanisms that drive meaningful learning. The story of Mrs. O illustrates the complexities that arise when teachers simply layer new practices over traditional ones without fundamentally changing their underlying beliefs about learning. To avoid such outcomes, districts must support educators in distinguishing between the surface-level changes (moderators) and the core mechanisms (mediators) of effective instruction. This requires fostering a shared vision of high-quality instruction, providing teachers with ample opportunities for collaborative learning, and ensuring they have a deep understanding of how students learn.

The Core of the Matter Issue #7

Thomas Nobili — Fri, 07 Feb 2025 20:27:13 GMT

Thin slicing is perhaps the hottest new term in math education. Coined by Peter Liljedahl in his book, Building Thinking Classrooms (2021), it refers to, “A carefully sequenced series of problems utilizing small, incremental changes to support students’ development of new mathematical knowledge building from their current ways of understanding” (Frazee et al., 2023, p. 338). In my work as a consultant I can certainly attest to its ubiquity across the classrooms I visit. However, as with every promising new practice in education, the explosion in interest in teachers working to design thin sliced task sequences has left the practice vulnerable to misinformation, misconceptions, and misapplication. (The term “lethal mutation” is often used to describe a practice that has been altered beyond its original intent and/or limits, such that it is no longer effective or useful.) Therefore, this post is an attempt to set the record straight regarding thin slicing. We will examine what is, what it is not, its theoretical underpinnings, and its applicability in relation to task design and implementation.

To understand thin slicing, you first must understand the theory on which it is grounded. Variation Theory arose mainly from the work of Ference Marton and colleagues, who viewed learning as the result of discernment. Specifically, they theorized learning to be the outcome of a learner successfully discerning the critical and non critical aspects of a given object of study. As such, they suggested that discernment was more likely to occur when certain aspects of a given object were varied against a backdrop of other aspects that remained unvaried. For example, if you were on a quest to make the fluffiest possible pancakes, you might start with a basic pancake recipe and then experiment by only changing one ingredient at a time (i.e. flour type, presence of baking powder, milk-to-flour ratio, mixing time, etc.) in order to discern the critical and non-critical aspects of pancake fluffiness. Al-Murani et al. (2019) specify that the aspects of a concept presented, and subsequently varied or not varied, can be either defining, such as the size of the angle if the object of learning is a right angle, or non-defining such as the length of the two lines of the angle, because both defining and non-defining aspects can be critical for learning. Thus perhaps the most critical tenet of Variation Theory is that an aspect of a given object (whether defining or non-defining) is more likely to be discerned if its variation is foregrounded against the invariance of other features. Further, according to Watson and Mason (2006) an object could be a symbol, text, diagram, theorem, a line of a theorem, a graph, an equation, and so on.

An important question for those designing thin sliced sequences is, “What aspects of an object should vary, and which should stay the same?” Of course this is largely dependent on what you want learners to discern; however, the research has demonstrated that examples of different types facilitate student learning more than the use of multiple examples of the same type (Hatala et al.2003; Kornell and Bjork 2008; Rohrer and Pashler 2010; Schmidt and Bjork 1992; Taylor and Rohrer 2010). The working hypothesis for this finding is that when different types of examples are mixed, learners are forced to distinguish between them and thus get better at making sense of unfamiliar examples. Relatedly, Kullberg, Kempe, and Marton (2017) explain it as the structure of the exercise as a whole, not the individual items, that promotes mathematical sensemaking. A learner must have ample opportunities with an object in order to internalize which aspects are fixed, which are varied, and how they are varied.

Variation Theory separates the concept of variance into four patterns: contrast, separation, generalization, and fusion.

Contrast entails giving examples that differ in one critical aspect while all other factors remain constant. An example of a contrast can be seen when determining the definition of a shape called a triangle. The triangle must be compared with other shapes such as circles or squares in order to make sense of the meaning of the triangle shape itself (Baskoro, 2021).
Separation involves the learner becoming aware of the critical features/dimensions of variation. The critical features of the given object become separated out. Separation can be thought of as focusing on differentiating parts within a whole. For example, when identifying right triangles students have to separate a right triangle from a triangle in general in order to discern the critical and non- critical features of all right triangles.
Generalization enables learners to discern the various effects of each variable on the overall object, fostering their capacity to recognize underlying linkages and dependencies. An illustration of this type of variation can be seen when a math teacher demonstrates “geometric transformations” by using various shapes (such as triangles and rectangles) and the way they look after being translated, rotated, and reflected. Students can generalize that all translations preserve lengths, all rotations preserve angles, and all reflections produce mirror images by finding the common characteristics of each transformation (Hasana et al., 2023).
Fusion involves the strategic combination of numerous interconnected concepts or elements into a single instance. Fusion patterns allow learners to experience variation in several critical aspects simultaneously. For example, when learning how to compare fractions, a teacher might start by giving examples where the numerator varies, but the denominator does not (i.e 1/4 2/4 3/4), in order for students to internalize the idea that the bigger the numerator, the larger the fraction when the denominator is common. From here, the teacher might expose students to examples where the denominator varies but the numerator does not (i.e. 1/6, 1/7, 1/10), in order for students to understand the idea that the bigger the denominator, the smaller the fraction when the numerator is common. If the teacher stopped here, it would end up being quite problematic for students when they had to compare fractions such as, 5/7, 3/4, 6/8, 11/10, where both the numerator and denominators are varying. Thus,it is clear that in order to deeply understand the concept of comparing fractions, fusion is necessary, where both the numerator and denominator are eventually varied simultaneously (Baskoro, 2021).

Source: Hassan, 2024

Finally, Gu (2017) makes a useful distinction between what he calls procedural and conceptual variation. Procedural variation involves creating problem sequences designed to enable learners to develop connections among different concepts step by step or from multiple approaches. Procedural variation patterns lead students to determine varied procedures or situations so that students can develop multiple methods for solving a problem, or generalize a solution method to a particular class of problems. Conceptual variation involves varying the representation of a concept in order for learners to be able to discern its core aspects. Essentially, conceptual variation focuses on the "what" of a concept by varying its appearance, while procedural variation focuses on the "how" by changing the steps involved in solving (Ballantine, 2018).

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In his book, Building Thinking Classrooms, Peter Liljedahl explains that thin sliced sequences utilize two main principles of variation theory. First, “...That we can only see variation against a backdrop of invariance” (p.152), which is what Marton refers to as discernment. And second, “Only one thing can be varied at a time” (p.152) which is what Marton refers to as contrast. These are really important points for educators to understand because what they mean in practice is that although pretty much any concept can be taught using variation theory, not every concept should be thin sliced.

For example, suppose we were teaching fractions to our fourth grade class. We already saw how using variation theory would be an effective way to teach students about comparing fractions. Additionally, thin slicing a comparison task, where students received a bunch of sets of fractions to compare where the denominators were common and the numerators varied, before experiencing several examples where the denominator remained fixed and the numerators varied, then working through several examples where both the numerator and denominator vary simultaneously, before finally experiencing a mix of all types of variance, would likely be an effective way for students to discern which aspects are critical when comparing fractions. However, suppose that prior to teaching comparison we wanted students to internalize some big ideas about equivalence. Specifically, why multiplying or dividing a fraction by the same number preserves the ratio between the numerator and denominator. We may rely on the tenets of variation theory when designing learning experiences focused on this idea, however thin slicing this content is probably not the best way to go about it. In this example, our intent is not for students to be able to simply find equivalent fractions by multiplying or dividing the numerator or denominator by the same number, or to figure out if two fractions are equivalent by testing to see whether their numerators and denominators are common factors or multiples, but rather to understand why this process produces two fractions that are equivalent. This will likely involve students making meaning through a richer task that involves a mathematizable context, as well as multiple opportunities for students to connect concrete representations to more abstract ones.

Another key understanding about thin slicing that sometimes gets overlooked in practice is that all thin slicing sequences should be thinking tasks ( As the first practice in the BTC framework is to give thinking tasks). According to Liljedahl (2021), “Good problem solving tasks require students to get stuck and then to think, to experiment, to try, to fail, and to apply their knowledge in novel ways to get unstuck” (p.20). Liljedahl goes on to say that thinking is what we do when we don’t know what to do, so a thinking task must be one in which students don’t immediately know what to do. Therefore, a thin sliced task, whether designed to promote conceptual or procedural variation, should be one where students do not have a firm grasp on the concepts or procedures going into the task.

When I work with teachers who are attempting to implement thin sliced sequences they often will express frustration at the fact that students aren’t collaborating well when working at the boards. This may take the form of limited dialogue between students in a group, a situation where one student takes the marker and does the first problem in the sequence, then passes it to the next student who does the second problem and so on, or students seeming bored and disengaged as they watch the person with the marker work through the sequence. The first thing I ask in these instances is not about their micro-moves for moving the marker or promoting discourse, but rather, what was the task? This is always my initial question because, a majority of the time, the thin sliced sequence the teacher implemented was not novel; it was instead what I call practice problems given one at a time. The problem with this is that it is not a group-worthy task. When students already have familiarity with the concepts or procedures targeted in the sequence they can work through it independently, they have no reason to collaborate (see this post where I explain when it is best to group students and when it is optimal to have kids work on their own). To use BTC language, we should not give students check-your-understanding questions, which are designed for students to work on individually in order to build fluency and automaticity, as a thin-sliced task, where the purpose is to work collaboratively to develop shared meaning around a mathematical concept or procedure.

Thin slicing is a powerful practice grounded in Variation Theory, which can deeply enhance students' mathematical understanding by promoting discernment through carefully sequenced tasks. However, it is essential to remember that not all content or concepts are suited to thin slicing, and the focus must always be on ensuring that the tasks encourage thinking, exploration, and collaboration. When designed and implemented effectively, thin-sliced sequences can foster a rich learning environment where students are actively engaged in meaning-making. Teachers must be mindful of the task design to avoid falling into the trap of using it for simple practice or routine problems. Ultimately, the success of thin slicing lies in its ability to challenge students with tasks that are novel and thus require collaboration, helping them construct a deeper understanding of mathematics.

The Core of the Matter Issue #6

Thomas Nobili — Tue, 31 Dec 2024 19:56:00 GMT

John Hattie and Dylan Wiliam have both made the point that almost anything in education works to some degree. As a result, basing pedagogical decisions on the answer to the question, “does it work?” is overly simplistic and misguided. Instead, when evaluating the efficacy of a given instructional practice, the more nuanced question, attributed to the philosopher and sociologist Luc Boltanski, is “not what works, but what works for whom, under what conditions, and why,” should be our north star. In my last post I looked at which conditions matter most when trying to decide if students should work independently or collaboratively. In this related post, I look at which conditions matter most when trying to decide if students should be engaged in direct instruction or inquiry based learning. This is one of the most long-standing and hotly contested topics among current and past instructional idealogues.

Of course, as I have already pointed out, this is the wrong question. There is ample evidence in the literature to support the efficacy of direct instruction as well as certain forms of inquiry-based teaching. The question this post seeks to answer is under which conditions is inquiry-oriented instruction more effective than direct instruction and vice versa. The reframing of thinking about direct instruction and inquiry not as a dichotomy, but instead as a continuum, has significant empirical support. For example, Chen and colleagues (2017), in their analysis of 2015 PISA data, found the highest average score increases to be associated with inquiry learning in combination with forms of direct instruction. Further, in a review of the literature on inquiry-based and direct instruction in the teaching of science, Djong et al. (2023) concluded that many moderating factors influence the effectiveness of each approach, such as the type of content to be learned. As such they offer the following guidance on when each type of pedagogy is most effective:

So, our first design recommendation is to employ inquiry-based methods if the instructional goal requires students to develop deep and transferrable conceptual understanding of topics that are open-ended or susceptible to misconceptions. Direct instruction is generally more appropriate and probably more efficient for acquiring well-structured and foundational (surface) knowledge, and it will not necessarily lead to deep conceptual understanding in less-well-structured domains (p.7).

As a point of clarification, the use of the term “surface” to describe certain types of knowledge is not meant to be pejorative; it simply indicates that it is not necessarily part of elaborate schema or sophisticated understanding. At the same time, you cannot build elaborate schema without a fund of surface knowledge with which to make connections.

In addition to the either/or thinking that has dominated the inquiry v. direct instruction debate, proponents at either extreme have drawn a caricature of the pedagogy they protest. For example, hard-core inquiry advocates typically describe direct instruction as boring lecture that treats students as empty vessels waiting to be filled with knowledge by the teacher. However, effective direct instruction is quite dyadic, characterized by frequent checks for understanding and corrective feedback. The work of Barak Rosenshine and Siegfried Engelmann is foundational to this type of teaching.

Similarly, direct instruction die-hards typically characterize inquiry-based instruction as children wasting time and experiencing frustration while trying to discover everything on their own, without any type of teacher intervention. Although there was a time where some in the field advocated for this type of pure discovery, current incarnations of inquiry-based methods involve significant teacher intervention in the form of scaffolding and the facilitating of productive discussions. To clarify that this approach involves skillful guidance by teachers, the term “guided inquiry” is often used in present day literature. Mayer (2004) clearly summarizes the benefits of what he calls guided discovery compared to pure discovery and direct instruction.

So my reading of the research is that students should experience a mix of both inquiry-based and direct instruction, as most standards in our curriculum require students to obtain both surface level factual knowledge and skills, as well as a deeper understanding of concepts. The instructional design questions have to do with the balance between direct instruction and inquiry, and the instructional sequence when employing them both. (And there is another subset of this research that deals with the amount of scaffolding needed during inquiry, but that will have to wait for another time.)

A major design factor to be considered is the sequence. In other words, should students receive direct instruction before trying problems on their own, or is it more beneficial for students to engage in some exploratory problem solving before consolidating their learning via direct instruction?

The answer to this question, in keeping with the theme of this and my previous post, is that both sequences are effective depending on the task and learning goals for the lesson. The most robust research we have on the sequencing of problem solving and direct instruction comes from the “productive failure” literature. Productive failure is an instructional approach characterized by students first engaging in a problem solving phase, which affords them opportunities to activate prior knowledge and generate and refine solutions. Since no direct teaching has taken place this phase often ends in failure, which is defined as not generating the “canonical” solution to the task. In a subsequent consolidation phase, an expert teacher builds upon the student generated solutions to teach the targeted concepts through direct instruction.

There are four principles that separate a productive failure design from other forms of inquiry. Namely:

The task must be challenging, but not so challenging that the learner gives up;
The task must allow for multiple solutions, strategies, or representations;
The problem should activate the learner’s prior knowledge; and
The teacher should build upon student-generated solutions by comparing and contrasting them with the correct solution.

There is a substantial body of resources supporting this approach, particularly in the domains of math and science, including this recent meta-analysis by Sinha & Kapur (2021), which compares the effects of a problem solving before instruction sequence to the more traditional instruction to problem solving sequence. And this study by Kapur (2014), which examines the efficacy of a productive failure approach in learning math content. One of the key takeaways from the productive failure research is that it is most effective when the task is relatively complex and the learning goals are focused on conceptual understanding and transfer.

But why is this the case? What about switching the instructional sequence facilitates deeper learning? The research posits that several mechanisms are likely at work. These include:

The activation of student prior knowledge. Since students receive minimal instruction before engaging in the problem solving phase, they must rely on their prior knowledge to make meaning about the task;
Making students aware of their knowledge gaps; in order for students to update their mental models and initial conceptions or naive misconceptions they need to be made aware of their flaws;
Because students are aware of their knowledge gaps they will be more likely to be able to distinguish the deep features and structure of the solutions presented in the consolidation phase.
It has also been theorized that because students have invested effort in attempting to solve the problem they likely have developed what Harel (2013) calls an intellectual need to understand the solution. Thus, they will be more motivated to engage during the direct instruction phase of the lesson.

Another pivotal design feature educators need to consider is one of dosage. In other words, how long should the direct instruction or inquiry phase of a lesson be? Unfortunately, the research is not as clear on this point as it is on sequence. In general, there are no hard and fast rules as to how long an inquiry or direct instruction phase of a lesson should be. Advocates for direct instruction typically champion shorter chunks of instruction followed by guided practice for younger students, although the pace should be brisk, and there should be ample opportunities for student interaction no matter the age of the students.

Likewise, the productive failure research does not attach time frames to each phase of the lesson. However, as a core principle it is suggested that the exploration phase be long enough to allow students sufficient time to activate prior knowledge, struggle productively, and begin to make meaning. However, there are certain subsets of the guided inquiry/productive failure genre that offer more specific guidance. For example, Peter Liljedahl, creator of the Building Thinking Classrooms Framework, which is an instructional approach that first affords students the opportunity to collaboratively explore novel problems before the teacher leads a discussion focused on teaching key math strategies and ideas, advocates for teachers to devote at least a third of the lesson time to this consolidation phase. Additionally, in Japan teachers routinely dedicate half the class to consolidating learning through presenting and discussing student solution methods.

In conclusion, the effectiveness of instructional methods like inquiry-based learning and direct instruction depends not on choosing one over the other, but on understanding when and how each approach is most effective in relation to the specific content, learning goals, and task complexity. Direct instruction excels in helping students acquire foundational knowledge and procedural fluency, particularly when tasks are well-structured. On the other hand, inquiry-based methods are particularly effective for fostering deep conceptual understanding and encouraging critical thinking, especially when dealing with complex, open-ended problems. The key, as emphasized by productive failure research, is to carefully sequence instruction and problem-solving experiences, providing students with opportunities to activate prior knowledge, struggle, and confront knowledge gaps before formal instruction consolidates their learning. Rather than adhering to rigid pedagogical doctrines, educators should strive to create learning environments where both inquiry and direct instruction coexist in a complementary relationship, enhancing students' ability to acquire, transfer, and apply new knowledge.

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In practice, the decision a teacher needs to make is not whether to use inquiry or direct instruction, but rather when to use each, and for how long. This question about timing and dosage should be informed by the type of knowledge being taught, the learning goals for the lesson, and the task being used.

Let’s examine two opposing examples. Suppose a kindergarten teacher’s intention is for her students to develop letter-sound-correspondence (a type of factual knowledge). It would be absurd to send students off to try and discover the sound the letter “a” makes before providing direct instruction. However, it may be more effective to have students explore whether certain objects sink or float when placed in water, prior to having a consolidating conversion, where the teacher provides direct instruction on the concepts of buoyancy and density (in kindergarten-friendly terms, of course). In the latter example, the learning intention is for students to develop conceptual knowledge, in order to generalize this understanding to other objects that they don't directly test.

Although there is still some debate in the literature as to whether instruction followed by problem solving is superior to problem solving followed by instruction for the acquisition of all types of procedural knowledge, there is general consensus that instruction followed by problem solving is more effective for learning most types of procedural knowledge. For instance, direct instruction would be the most efficient and effective method to teach middle school students about the different forms of an equation of a line (i.e. slope intercept, point-slope, and standard form). Once students had this knowledge, we might send them off to figure out which equation to use given different starting information. In this scenario, it makes more pedagogical sense to give students the information they need before we ask them to apply it.

However, let's suppose we were teaching a fifth grade class about division of fractions. In this example, allowing students to explore via problem solving first is the better choice. Students need opportunities to make meaning about what it means to divide, for example ¾ by ⅔, before learning the canonical invert and multiply method. In this example, conceptual understanding should precede procedural fluency, thus a problem solving first approach is warranted.

The Core of the Matter Issue #5

Thomas Nobili — Sat, 30 Nov 2024 04:04:25 GMT

Teachers have to make a tremendous number of decisions when planning any given lesson, spanning from the more foundational, such as establishing the learning goal, to the more nuanced, such as figuring out when understanding might break down and what to do about it. Somewhere in this bevy of choices is the topic of this post: Should students work in groups or individually? The answer, like many things involving teaching and learning is… it depends. But on what? What variables should we be paying attention to so that our choices are intentional and backed by research, rather than based on preference or personal style? It turns out that there are three main factors we should be considering when deciding whether to have students work individually or collaboratively, namely: the complexity of the task, the prior knowledge students have on the topic of the lesson, and the interdependence that exists within groups. Let’s dig into each of these ideas in more detail.

Task complexity refers to the level of cognitive demand a task imposes on a learner. Features of a complex task can include things like: number of subtasks, the amount of information to manage, the need for complex decision-making, the level of uncertainty involved, and how different elements within the task interact with each other. In the literature, complex tasks are sometimes referred to as non-routine or problem solving tasks. Conversely, less complex tasks are known as routine or recall tasks.

Research shows that for recall tasks, collaboration impedes performance, because the cognitive effort that it takes to collaborate takes away from the effort available to perform the task. Counting, for example, is a fairly low-level skill; but think how much harder it is to count if someone is talking to you, or even if there is noise in the background.

However, when engaging in problem solving tasks, groups outperform individuals (Laughlin, Hatch, Silver,& Boh, 2006). The explanation for this finding is that while the additional cognitive load of working in a group becomes extraneous when the task is relatively straightforward (as in the counting example), when the task is more novel and complex, the members of a group can pool their individual working memories, which allows the group to be able to handle a higher level of cognitive load. This collective working memory enables cognitive capacity to be freed up at the individual level, which can be used to build higher quality schema than could be developed if working alone. For example, if the task is to read a challenging technical text, it is likely that having a small group of students working together on figuring out what it means will lead to higher performance.

Another relevant factor when deciding if students will learn more from working in groups or by themselves is the level of prior knowledge students have about the topic. Students with little relevant prior knowledge have been found to learn more when working in a collaborative group compared to on their own. (Zambrano et al., 2019). This finding supports the theory of the collective working memory effect, which suggests greater cognitive capacity allows collaborative groups to acquire better mental representations from complex information.

What about a situation in which working in a group would benefit students with weak prior knowledge, but they would be working with peers who have strong prior knowledge. Working in a collaborative group needlessly increases the extraneous cognitive load of higher prior knowledge peers due the fact that the extra mental effort needed to work with others is not actually helpful, since whatever they would potentially learn from the group is likely redundant. However, the research suggests that higher prior knowledge students perform at similar levels when working individually or collaboratively on complex tasks despite this increase in cognitive load. Therefore if some students would benefit from working in a group, that should be the teacher’s decision.

The final variable that needs to be considered when deciding whether to have students work alone or in groups is the level of interdependence that can be created within the group. Positive interdependence reflects the extent to which group members must depend on one another for effective group performance. In other words, the conditions for collaboration need to be such that each group member is responsible for the work of the group as a whole, and the group as a whole is responsible for the learning of each individual group member. Group interdependence is realized when the relevant knowledge held by each individual group member is communicated and coordinated within the group, leading to the construction of shared mental models (Kirschner, Sweller, Kirschner, Zambrano, 2018). Basically, what this is saying is that when making meaning about complex tasks, collaboration becomes a scaffold for an individuals’ knowledge acquisition. For example, during collaborative learning, some information comes from collaborators rather than other sources, and that information is likely to become available exactly when it is needed, resulting in increased learning.

Other explanations posited for the effects of collaborative learning when using complex tasks include the notion that social interaction stimulates elaboration of conceptual knowledge. Research has found that groups working on a task generate more elaborative talk (i.e. students verbalizing their thinking) than students who work individually, which in turn leads to higher learning outcomes (Boxtel, Linden, & Kanselaar, 2000). One theory proposed for this finding is that talking to someone else creates deeper understanding, because the person speaking is thinking hard about how to get the listener to understand their thinking, resulting in a more coherent explanation. Other explanations put forth for the relationship between conceptual discourse and achievement focus on the process of negotiating meaning. The premise being that the process of negotiating a shared meaning about a concept likely requires participants to engage in elaboration, justification, questioning, building upon ideas, reflection, and the integration of different viewpoints, all of which involve sustained deep thinking about a concept.

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Classroom Connections

Now that we have some background on each of the three variables that impact the potential for students to learn more when working individually or in a group, let’s examine the implications of each on classroom practice. Let’s suppose a grade 4 teacher is planning on having her students find the product of 25 x 12. How should she think about what kind of task to design?

Is this a complex or routine task? It depends on the learning intention the teacher has for the lesson.
1. If the teacher wants the students to understand how to perform the standard algorithm to solve multi-digit multiplication problems then the task and related learning intention are pretty routine, and the teacher should have students work individually.
2. However, if the teacher has a more conceptually based learning intention in mind, such as understanding how both the associative and distributive properties can be used to create equivalent and easier to compute expressions, then the task becomes much more complex, and requires a more collaborative approach.
What do students already know about the subject?
1. If her students are unfamiliar with the topic, collaborative learning is advantageous because it distributes cognitive demand, allowing students to build better mental models through shared knowledge and interaction.
2. However, if her students already know quite a bit about the topic (perhaps they have already learned a lot about the associative and distributive property) then having them work individually would be a better choice, since they may not gain as much from collaboration, as group work can introduce redundant or unnecessary effort.
Let’s say that the teacher decides on the more conceptual learning intention, and her students have little prior knowledge of the topic. She decides to have students work collaboratively, and now needs to think about how to foster positive interdependence within classroom groups.
1. There are two types of group interdependence that can be created. The first is called resource interdependence, which occurs when each group member has access to partial information which is needed to complete the task. This type of interdependence can be created by activities that require active communication and coordination, such as brainstorming sessions, debates, or jigsaw readings.
2. A second type of interdependence, called task or goal interdependence, can also be generated in order to maximize the benefits of collaboration. This type of interdependence is developed when each member of the group is dependent on every other member of the group to complete the task or reach a shared goal. Activities that require students to justify their reasoning, integrate multiple perspectives, and refine their ideas collaboratively are best suited to stimulate this type of interdependence.
3. Techniques for fostering task or goal interdependence include having groups share one writing utensil, requiring all group members to demonstrate understanding before assigning the next subtask or extension, teaching students how to effectively collaborate through modeling (i.e. fishbowls), and the use of collaboration rubrics.

This post aims to give you an overview of key theories and research on when it's most effective for students to work in groups versus individually. I then provided a practical example to demonstrate how these ideas can be applied in the classroom. Specifically, it offers insight into the decision-making process of a typical fourth-grade teacher planning the "25 x 12" task. See below for additional resources. If you'd like to continue the discussion or explore additional resources on this topic, don't hesitate to reach out at tnobili314@gmail.com.

The Core of the Matter Issue #4

Thomas Nobili — Tue, 29 Oct 2024 02:12:50 GMT

Dylan Wiliam once tweeted, “I've come to the conclusion that Sweller's Cognitive Load Theory is the single most important thing for teachers to know.” Yet, in my work I have found that many educators have never even heard of it. So in this issue, I dig into Cognitive Load Theory and breakdown why it is hard to argue with Dylan Wiliam.

Cognitive scientists define learning as a change in long term memory. However, in order for new knowledge to connect with your prior knowledge and be organized in long term memory (also known as schema building), meaning making processes must occur in your working memory. The challenge is that unlike our long-term memory, our working memory has an extremely limited capacity and can become overloaded quickly. During knowledge acquisition, our working memories deploy resources to thinking about all kinds of things, also known as cognitive load, some of which are connected to what we are trying to learn and some that are not.

Cognitive load theory is concerned with how to best manage the limited capacity of working memory during the learning process. As the theory has evolved, three types of cognitive load have been identified:

Intrinsic cognitive load is a product of the material being learned. The more complex the material, the greater the intrinsic cognitive load. Intrinsic load is dependent on the background knowledge of each learner. For example, if an accomplished baker was presented with a new souffle recipe it probably wouldn’t contribute much intrinsic load due to the fact that the expert baker has deep schema in their long term memory to draw from. However, if I presented the same recipe to a person who has never baked anything before, and thus has little background knowledge to draw from, it would likely lead to a high amount of intrinsic load on working memory. Therefore, intrinsic cognitive load, sometimes called the base load, cannot be reduced as a result of instruction. It can only be lessened either by making the task less complex, or by the learner acquiring related schema in long term memory.
Extraneous cognitive load is considered extraneous because it interferes with schema acquisition and thus actually hinders learning. Extraneous load arises from elements in the learning environment or instructional materials that may not be directly related to the content being learned, but still require mental resources to process. For example, let’s suppose a student is working on how to solve multi-digit multiplication and division problems in math with the instructional goal being for the student to understand the role place value plays in these types of computations. If the student is not fluent with their basic facts and thus must devote a lot of their cognitive resources to computation, little would be left to notice anything about place value. Unlike intrinsic load, extraneous load can be reduced during instruction, which mainly occurs through teacher scaffolding. In the above example, the teacher may provide the student with a multiplication chart in order to reduce extraneous load, which would allow them to use more of their limited working memory to make meaning about place value.
The third type of cognitive load is called germane cognitive load. Germane cognitive load is necessary for learning as it signifies the cognitive resources required for schema acquisition and skill automation. Optimizing germane cognitive load involves promoting activities and instructional practices that encourage learners to make meaning of new ideas and knowledge, make connections between new and prior learning, and engage in associated thinking processes.

Source: Nobili, 2024

Classroom Connection

Cognitive Load Theory clearly has significant implications for instructional design and implementation. For instance, when presenting material to students, we want to make sure their attention is on the most salient information, therefore we should avoid over-cluttering slides or other visuals with distracting fonts or graphics. Additionally, we should not play music during times when we ask our students to read or write or think, because the act of listening recruits some of their cognitive bandwidth that would otherwise be available for learning. There are numerous other examples for which Cognitive Load Theory is a useful frame; however, for this post, I want to focus on teacher scaffolding and how Cognitive Load Theory is critical to informing this practice.

Scaffolding is one of the most ubiquitous practices in K-12 classrooms. However, like many other education terms, it suffers from definitional ambiguity. In other words, if you were to ask ten teachers to define what scaffolding is you would get eleven different answers. This lack of a shared understanding of scaffolding has led many educators to justify any support given to a learner as a type of scaffold. As a result, many practices employed under the guise of scaffolding actually serve to reduce the level of thinking required of a student, or flat out do the thinking for students, both of which are antithetical to learning. When I work with educators, I talk to them about the different types of cognitive load and then I define scaffolding as anything the teacher does to reduce the extraneous cognitive load of their students.

There are several pedagogical implications of this definition of scaffolding. First and foremost, the teacher has to be extremely clear on what the learning intention is for a given lesson or lesson segment. They need to have a clear answer to the question: what meaning-making do I want my students to engage in? This is especially critical because it is very difficult to categorize what is germane to learning and what is extraneous without a clear understanding of the knowledge we want our students to acquire. In fact, the same support could be classified as both a scaffold and as over-teaching, depending on the goal of instruction. For example, allowing a child who struggles with decoding grade level text to listen to a text is an appropriate scaffold if the instructional intent is about character analysis. In this scenario, it is likely that the cognitive resources the learner would have to devote to decoding the text would leave little left for thinking about the characters, but the oral recitation of the story frees up cognitive resources the child would otherwise have had to use to decode, allowing the student to use them to make meaning about the characters. Conversely, allowing the learner to listen to a story would not be an appropriate scaffold if the instructional goal was for the student to improve at decoding. In this case, the cognitive resources the child employs to decode the text are actually germane to the intended learning and therefore should not be reduced.

Once a teacher has clarity on the schema they want students to build, they can plan for the scaffolds they might use if their students should need them. Cognitive Load Theory is very useful to consider when deciding whether the proposed support is a true scaffold (i.e., it reduces extraneous load), or if it actually reduces germane load. If it reduces germane load, it would not be classified as scaffold at all and instead be labeled over-teaching.

Scaffolds come in two main forms and both can be useful in reducing extraneous load and maximizing germane load. Tools are a common form of scaffold a teacher might use. In fact, all of the scaffolding examples I have used so far in this post fall under this category, which includes things like multiplication charts, books on tape, graphic organizers, math manipulatives, and Google Translate. Effective scaffolds can also come in the form of a prompt or strategy. Examples of this type of scaffold include:

The teacher prompting students to organize their findings to a math problem about ratios in a table so that they will be better able to focus on the numerical patterns that emerge;
A teacher asking a student to consider the different ways in which a character’s actions could be interpreted in an attempt to focus thinking on how their prior experiences and mental models shape their perceptions;
Not stacking questions (i.e., asking them one at a time) so that students can focus on each one deeply and remain in the present.

Understanding and applying Cognitive Load Theory is essential for educators aiming to enhance student learning. By recognizing the intricacies of intrinsic, extraneous, and germane cognitive loads, teachers can make informed instructional decisions that optimize the cognitive resources of their students. Effective scaffolding, when rooted in this theory, ensures that supports are designed to reduce extraneous load while promoting meaningful engagement with content.

The Core of the Matter Issue #3

Thomas Nobili — Wed, 18 Sep 2024 15:16:27 GMT

My colleague, Isobel, has this saying that has always stuck with me, “It’s all about the debrief.” At this point I would probably go as far as to categorize it as a mantra. Even though she is referring to the work we do with adults, it rings every bit as true when talking about students. In fact, I contend the debriefing of learning, which goes by many names in math, such as consolidation (Liljedahl, 2021), math congress (Fosnot & Dolk, 2001), or Neriage (in Japan), is the most important part of any lesson. It allows for students’ informal thinking to be formalized, and for their learning to be solidified and encoded into long term memory. Peter Liljedahl (2024) refers to this process as going from, “meaning making to meaning made.”

However, the consolidation process, as powerful as it is in supporting student learning, is an extremely difficult thing for a teacher to do well, as it requires extensive expertise. In a math classroom, for example, a teacher must possess a strong understanding of standards, learning progressions, and the development of student thinking. Furthermore, they have to be skilled in facilitating student discourse and keeping all students engaged and thinking. On top of that they need to have a high level of tacit knowledge in order to best navigate all the decision points that will come up during the course of the discussion, what Deborah Ball (2018) calls discretionary spaces.

At this point I think it would be prudent to clarify some things about what a consolidation or congress is and is not. The first thing to clear up is that it is not a show and tell, or just a time to share. If students are simply providing a narrative of their solution process, or just sharing their answer, nothing is being consolidated. Rather, it is a much more intentional activity, where the teacher is actively selecting what pieces of student work to highlight during the discussion and in what order. The goal of a consolidation is focused on weaving together the key mathematical ideas that emerged throughout the lesson in order to help students make sense of those ideas, in a way that connects new learning to prior knowledge.

A second and related clarification point is that the consolidation is not just about discussing different ways students got to the answer. Again, simply hearing about a bunch of different solution strategies on its own will not lead to anything being consolidated. The teacher needs to press students to make connections between the various strategies presented, and then relate these similarities to the underlying mathematical ideas, properties, or principles from which they were derived. This idea of students starting with informal strategies and ideas, and then gradually moving towards more abstract and formal ways of thinking is referred to as Progressive Mathematization. The term stems from the work of Hans Freudenthal, a Dutch mathematician, who focused much of his career on researching how to best teach the subject of mathematics. Later, Adrian Treffers, a mathematics curriculum researcher and disciple of Freudenthal, made a further distinction within the construct of progressive mathematization by delineating two different types of mathematical activity, which he called horizontal and vertical mathematizing.

According to Treffers, horizontal mathematizing occurs when a learner comes up with strategies to organize and solve math problems embedded within a real world context. Freudenthal describes horizontal mathematizing as, “going from the world of life to the world of symbols.” Treffers goes on to explain that horizontal mathematization occurs when any of the following activities can be identified: identifying or describing specific mathematics in a general context, formalizing and visualizing a problem in different ways, recognizing relations and regularities, recognizing related aspects in different problems, and transferring a contextual problem into a mathematical problem. Treffers describes vertical mathematizing as going beyond what was constructed during the horizontal mathematizing phase. It is defined as the process of organizing within the mathematical system itself. Freudenthal described vertical mathematization as, “moving within the world of symbols.” Treffers adds vertical mathematizing occurs when the following activities can be identified: reorganizing within a mathematical system, representing a relation in a formula, proving regularities, refining and adjusting models, using different models, combining and integrating models, or formulating or generalizing a mathematical model.

Source: Drijvers, 2022

It is important to note that moving from horizontal to vertical mathematizing should not be seen as a linear process, but rather as dynamic and cyclical in nature. Additionally, Freudenthal cautioned educators not to place more importance on one over the other. Rather, he argued that both were important for students to experience as a part of progressive mathematization.

Classroom Connection

The construct of vertical and horizontal mathematizing is important for teachers to understand, specifically as it relates to the instructional goals during the consolidation phase of a lesson. A teacher must have a plan to select and sequence student work in such a way that allows for both horizontal and vertical mathematizing to occur.

For example, let’s suppose we are teaching a third grade multiplication unit. We have chosen a task from the Context for Learning Mathematics (CFLM) unit, The Big Dinner, where students work to find the price for different amounts of apples and carrots for a Thanksgiving dinner. While the class is working on the problem we notice that many students have made use of a partial products strategy to find the various prices. For example, some students added the price for 5 pounds to the price for 1 pound to get the price for six pounds, and others doubled the price of 5 pounds to get 10 pounds, etc. Our goal then during the consolidation could be to select student work that highlights the use of partial products (horizontal mathematizing) as a means to examine the conjecture that decomposing a factor and combining the subsequent partial products will work for all multiplication problems. Internalizing this idea leads to a generalized understanding of the distributive property (vertical mathematizing).

Let’s take another example. This time let’s imagine we are teaching a grade 7 class and students are working on the Lewis Carroll Cats and Rats Problem. The task asks students to figure out how many cats are needed to kill 100 rats in 50 minutes if we know that 6 cats can kill 6 rats in 6 minutes. There are a lot of different ways to approach this problem, and I will not go through all of them, but what they all have in common is that they make use of the relationship between the different variables in the problem. The goal for consolidation would be to select work that highlights strategies which leverage these relationships such as repeated addition, scaling in tandem, or doubling and halving (horizontal mathematizing), in order to push students to identify and generalize different types of proportional relationships (vertical mathematizing). For instance, you might select a student work sample that used a table to show that 1 cat can kill 1 rat in 6 minutes leading to the conclusion that 1 cat can kill 8 rats in 48 min, 2 cats can 16 rats in 48 min, and so on until they arrive at 12 cats killing 96 rats in 48 minutes. The work sample shows that from here they decide they need an extra cat in order to ensure all 100 rats are killed in 50 minutes. After discussing the utility of this strategy, the focus could shift to analyzing the proportional relationship between cats and rats. Specifically, that 1 cat can kill 1 rat in 6 minutes. This also allows for a discussion on the idea of the minutes being an invariant that mediates the relationship between cats and rats. In other words, the ratio between cats and rats is 1:1 given the minutes remain constant at 6. Next, you could show a solution where the students concluded that 6 cats can kill 1 rat in 1 minute and scaled up to figure out that 6 cats can kill 100 rats in 100 minutes. From here they may have figured out that in order to kill 100 rats in half the time they need to double the number of cats to 12. This opens the door to focus on the relationship between cats and minutes, which are inversely proportional to one another. An increase in cats will lead to a decrease in the minutes needed to kill a constant or invariant number of rats, and an increase in minutes will require a decrease in cats needed.

The Core of the Matter Issue #2

Thomas Nobili — Mon, 26 Aug 2024 13:23:33 GMT

As the new school year creeps closer, there are some things we can bank on being true as we welcome our new classes of learners. For example, there’s sure to be a beginning of the year fire drill at an inopportune time, someone will need you to lend them a pencil, and there will likely be a broad range of academic diversity within your classroom. In fact, new research shows that the percentage of classes that contain five-plus grade levels in terms of proficiency range is between 44-59% for math and a whopping 75-82% for ELA. The study also highlights that more variance exists in the achievement level between students within the same classroom (48-60% in mathematics and 51% for ELA) than between classes or even schools.

As flabbergasting as some of those statistics are, if you have been around education for any amount of time you are probably not that surprised. And if you are familiar with this dilemma then you are also probably familiar with the strategy most touted for addressing this vast array of needs, the ever pragmatic advice (cough, cough), “just meet them where they are.” I am not sure what this phrase is supposed to mean in theory, but in practice it manifests as, giving students remedial assignments designed to ‘catch them up,’ lowering our expectations for what students are capable of doing, and allowing them very limited, if any, access to grade level work. TNTP’s Learning Acceleration Guide (2020) expounds upon these concerns:

The typical approach to remediation—providing work better suited for earlier grades—won’t come close to catching students up and will likely compound the problem. In our recent study, The Opportunity Myth, we found this approach of “meeting students where they are,” though well intentioned, practically guarantees they’ll lose more academic ground and reinforces misguided beliefs that some students can’t do grade-level work. The students stuck in this vicious cycle are disproportionately the most vulnerable: students of color, from low-income families, with special needs, or learning English. In other words, doubling down on current strategies for catching students up will only widen opportunity and achievement gaps. Schools need to be ready on the first day back with a fundamentally different strategy for diagnosing lost learning and putting every student on a fast track back to grade level—a strategy designed to accelerate their exposure to grade-appropriate work, not delay it.

At this point you may be asking yourself, if remediation (aka: meeting students where they are) is ineffective, inequitable, and inefficient, then how am I supposed to address the needs of students with unfinished learning in my class? The answer lies in a process called acceleration. Now if you are a secondary math teacher, you may be a bit confused right now because the term acceleration has an alternate and much different meeting in math education. To clarify, I am not talking about the kind of acceleration typically found in middle and high school math courses where some students are placed in courses that teach them more content in a given year than non-accelerated courses, nor am I referencing instances when students skip over learning content altogether. The acceleration I am promoting is best understood through the work of Partners for Educational Leadership who define it as a strategy involving significant adjustments to curriculum and instruction to accelerate exposure to grade-appropriate work and advance the learning progress of all students. They go on to warn us that It is NOT “just good teaching”—while it is made up of strong teaching practices, it is not a random collection thereof, but an inter-connected, intentional and curated set. They add, it is grounded in principles of equity, and is integral to advancing the goals of equity.

In my own work I have found that putting acceleration into practice involves six key levers.

Determine priority grade/course level curricular content and related pathways for student access;
Giving all students access to high quality tasks;
Utilize formative assessment strategies to elicit evidence of potential unfinished learning and overall student progress;
Strategically address unfinished learning via just in time opportunities situated within grade/course level content;
Prioritize instructional strategies that privilege student thinking;
Collaborate/consult with coaches and other colleagues regularly to plan for and reflect upon teaching and learning;

While I won’t be digging into each lever in this post, I do want to focus on number four as I have found it to be particularly difficult for some people to wrap their head around in practice. So let’s concretize this idea by looking at how a teacher might address a student’s unfinished learning through grade level content.

Classroom Connection

The first barrier students often need to overcome in order to get access to grade level content is for the teacher to believe that they have the prerequisite knowledge needed to be successful at a given task. In my experience teachers almost always overestimate the prerequisites a student needs to enter a task. In fact, when my colleagues and I run workshops we will often put teachers through a difficult task and then highlight all the different entry points they used during the debrief. Next, we ask the same participants to list all the prerequisites they think students would need to have to be successful on the same task and their lists always contain superfluous items. For example, they'll say kids need to be able to do multi-digit multiplication, until we point out that many groups of adults just used a phone to do the calculations, or they’ll argue that kids need to understand what plot and theme are, to which we respond that we never used those terms. All we did was ask them to discuss what the story is literally about and what it is actually about. Take the Open Middle task pictured below,

You could certainly use the task as a means for students to learn about coefficients, reciprocals, and variability, just to name a few objectives, but students don’t need much background knowledge to be able to access that learning. All they really need to enter this task is to know what an equation is, have some basic number sense and be able to add and subtract within 10.

The other barrier that can keep students from having access to grade level content is time. Teachers have to constantly make decisions about when to move on to a different lesson or unit. This can be difficult because it is often the case that there are still at least some students in the class who would benefit from continued work on the topic currently being taught and another group who is ready to go forward. Relatedly, there are often students in our classes who need more time to master concepts from previous grades that are crucial for establishing foundational understanding. So the dilemma becomes how do we address those crucially important areas of unfinished learning while still moving forward with grade level content? Fortunately, the same strategy can be used to evade both of the aforementioned time barriers.

Mathematics is a cumulative and interconnected discipline. Things don’t really exist in isolation and so we need to leverage this as a part of our instruction. For example, let's suppose I have a group of kindergarten students who are still working toward the major milestones in counting e (i.e. one-to one correspondence, cardinality, conservation of number, hierarchical inclusion, sorting, tagging, 1-9 sequence, etc) when most of the class is ready to move into a geometry unit. Rather than continue to work on counting with those students in isolation, I could do so while also helping them make sense of the big ideas about shapes. For instance, I could ask students to count the sides of the shapes we are analyzing, or I could ask them to count out a certain number of pattern blocks as part of our work on composite shapes. I might add a shape or take away a shape from their set and prompt the students to tell me how many they have now. In addition, I could prompt students to begin to sort the shapes into sets to count by asking how many 4 sided shapes are in their bins, or how many blue squares they have.

The above example illustrates how one can utilize the nested nature of mathematics in order to intentionally address unfinished learning through grade level content as opposed to letting unfinished learning be a barrier to access. Below are some additional examples of how this can be done spanning different grade levels and mathematical domains and topics.

Utilize multi-digit multiplication and division to address unfinished place value learning for a fourth grader.
Leveraging early multiplication content in grade 3 to work on unfinished learning around unitizing, part whole relationships and addition strategies.
Take the opportunity to continue to build fluency with fraction and decimal operations during a unit on solving equations in grade 8.

The Core of the Matter #1

Thomas Nobili — Mon, 05 Aug 2024 15:46:47 GMT

For the inaugural issue of this newsletter I figured I might as well go after one of the white whales in primary math education, fact fluency. So let's dive in. It's safe to say that anyone who knows anything about early numeracy would agree that it is imperative for students to learn their “basic facts.” However, there has historically been and continues to be some debate on what being fluent means and how to best help students reach that goal.

One side of the argument promulgates the belief that students need to memorize their facts (i.e. commit them to long term memory) through practices such as using flashcards and timed tests. The rationale behind this position is that working memory capacity is limited, and if students are not fluent with their facts they will struggle with higher level math concepts because they will have to use a large chunk of their available working memory on retrieving basic facts, leaving little left to tackle more challenging concepts.

On the other end of the continuum are those who argue that fluency is not so much about memorization as it is about using relationships and derived fact strategies to think flexibly and efficiently. Proponents of this approach argue for students to learn their facts through strategy instruction, games, and routines like number strings. The rationale behind this approach is that there are hundreds of discrete fact combinations for students to master and it is inefficient and overly arduous to expect students to commit each of those to memory separately. They argue it is much more efficient and practical for students to know, for example, that 7+3 and 3+7 are both equal to 10 (due to the commutative property of addition), than to view them as two separate facts.

The research certainly supports aspects of both ideologies. The literature on the science of learning strongly supports the position that the goal of fact fluency should be for the single digit facts to be committed to one’s long term memory. Further, the research on strategic competence and number sense supports the claim that it is crucial for students to understand the relationships between numbers in order to derive unknown facts from facts they know. So what does that mean for how to best support students in becoming computationally fluent? Enter Baroody’s three phase framework for how students typically progress in the mastery of basic facts.

The first phase of Baroody’s framework is counting. In this phase students rely on using objects or verbal counting strategies to solve basic facts. This phase also includes more advanced counting strategies like counting-on or skip-counting as well. The last phase of Baroody’s framework is mastery (no you didn’t miss anything. I intentionally skipped phase 2 for reasons that will be clear as you read on). Here students can retrieve facts automatically and with little effort. This final phase is synonymous with the goals advocated by the ‘memorize your facts’ crowd.

Baroody contends that this phase 1 to phase 3 journey is the typical progression most students experience when learning their facts. They are introduced to an operation like addition or multiplication and begin to model and make sense of it with objects and counting strategies. Then at some point the focus shifts to where they are trying to become automatic with facts. However, he warns, this is where the problem lies. If students are pushed to the mastery phase prematurely they have to memorize each discrete fact separately because they have not yet made connections between them, and when they don’t know the answer to a basic fact they have to rely on tedious cognitively draining counting strategies to figure it out. Baroody recognized this and thus has a stage in his framework in between counting and mastery which he calls reasoning strategies. In this stage (stage 2) students use facts and arithmetic properties they know to figure out facts they don’t know. For example, a student who knows 2 x 7 and 5 x 7 can use that knowledge as well as an understanding of the distributive property to derive the answer to 7 x 7. As students gain proficiency and flexibility in their strategy use they begin to organize facts based on these relationships (called schema) which are easier to commit to long-term memory.

Classroom Connection

The question of, “how to best teach basic facts?” can only be answered when we know what phase of Baroody’s framework best matches where a student is in the fluency progression. Let’s look at three fictitious examples to illustrate this point.

Student A: This student is still counting all to figure out their addition facts to 20. For example, if they were trying to solve 4 + 7, they would count out 4, either with objects or their fingers, then count out 7, then count the total. Student A is in the beginning stages of the counting phase so the next step for them is to move to more sophisticated counting strategies, such as counting-on. Thus instruction should focus on constraining student A’s ability to count all. This could be done by doing various quick image routines. For instance, you could flash 5 red beads on a math rack (for 3 seconds) and ask how many they see. They may count to start off, but with repetition will realize that if you show all the red beads on the top row of the math rack it is always five. Next, you could flash all 5 red beads on the top row as well as 3 of the white beads. The goal here is to not let student A see it long enough to count all the beads. Eventually they will trust that when all the red beads are pushed over it is five and will count-on from there to find the total. (I have referenced using the math rack in this example, but this work could also be done with ten frames or dot cards if that is what you have available.)

Student B: This student is working toward fluency with single-digit multiplication facts and has a pretty good handle on multiplying by 2, 5, and 10. Instruction with this student should therefore focus on stage 2–developing reasoning strategies. Implementing number strings would be the way to go here, as well as having this student play fluency games that focus on the development of strategies like doubling, adding/subtracting a group, and partial products.

Student C: This learner is solidly in the reasoning strategy phase, meaning that they have internalized many derived fact strategies and properties of operations. In this case instruction should shift toward attaining automaticity. This can be done through the use of hint flashcards where students get shown a fact like 4 x 8 as well as a hint for the fact based on a reasoning strategy. So for example underneath the 4 x 8 it might say something like: Hint think 2 x 8. The back of the card would have the answer. If the student knows the answer to 4 x 8 from memory then the hint is irrelevant, however, if they forget they can use the hint to figure it out, without having to rely on a tedious counting strategy. Jason Zimba wrote about a similar flashcard approach here.

Thus the key to supporting students in becoming computationally fluent is to figure out where a student is within Baroody’s progression and then match the goal of instruction to helping them move to the next phase. Moving from counting to automaticity too quickly will result in students trying to commit facts to memory with no understanding of how they relate to similar facts, and lingering in the reasoning phase without ever striving for automaticity will result in students having to use valuable cognitive load to execute a strategy.

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