Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning (Corwin Mathematics Series)

by Peter Liljedahl

Building Thinking Classrooms in Mathematics Grades K–12

““Patterns of Misunderstanding As a young high school math teacher, I often made assumptions regarding students' content knowledge, especially their mastery of fundamental concepts in earlier grades. I adopted the strategy of having students define basic math concepts not only to build understanding but also to expose the areas where students' knowledge was weak. The approach revealed some interesting patterns over the years. For example, high school students in higher-level math classes most commonly define an equation, one of the most basic concepts in mathematics, as “when you solve for x.” This definition is a clear misconception of what an equation is, but the root cause was not so evident. After much reflection and analysis, I realized the origin lay in the state's mathematics content standards. In the state standards in effect at that time, the term equation did not appear until the sixth grade. Moreover, the context for this first appearance focused on learning to solve simple linear equations. This initial focus had likely contributed to students' misconception of an equation being “when you solve for x.” The concept of an equation is usually defined as a mathematical sentence that states that one quantity is the same as another quantity. In other words, the quantity expressed on the left side of the equal sign is the same as the quantity expressed on the right side, regardless of how those quantities are represented.” — The Problem with Math Is English: A Language-Focused Approach to Helping All Students Develop a Deeper Understanding of Mathematics by Concepcion Molina https://a.co/9JRVET2 all social play requires extremely alert attention—to the context of play, the actions, the equipment, the field, and the other players. The best players are magnificently reactive to novel stimulus opportunities, and their ecstasy may lie in the performance of unique ludic acts, whether in ball games, at the chess table, at poker, or in jumping out of trees.” — The Ambiguity of Play by Brian Sutton-Smith https://a.co/2xKjxSb “braniacs like Albert Einstein, for instance. He didn’t arrive at things like E = mc2 by channeling Jane Austen. No, he came up with it after remembering how, as a child, he’d imagined riding through space on a beam of light. And relativity theory? By imagining what it would be like to plummet down an elevator shaft, then take a coin out of his pocket and try to drop it—without, I’m assuming, passing out or throwing up first. Here’s how Einstein explained his own mental process: “My particular ability does not lie in mathematical calculation, but rather in visualizing effects, possibilities, and consequences.” 1 Sounds exactly like a story to me. And the key word here is visualizing. If we can’t see it, we can’t feel it. “Images drive the emotions as well as the intellect,” says Steven Pinker, who goes on to call images “thumpingly concrete.” 2 Abstract concepts, generalities, and conceptual notions have a hard time engaging us. Because we can’t see them, feel them, or otherwise experience them, we have to focus on them really, really hard, consciously—and even then our brain is not happy about it. We tend to find abstract concepts thumpingly boring. Michael Gazzaniga puts it this way: “Although attention may be present, it may not be enough for a stimulus to make it to consciousness. You are reading an article about string theory, your eyes are focused, you are mouthing the words to yourself, and none of it is making it to your conscious brain, and maybe it never will.” 3” — Wired for Story: The Writer's Guide to Using Brain Science to Hook Readers from the Very First Sentence by Lisa Cron https://a.co/4SenNp0 “Increasing variation leads to deeper learning and increased transfer, but can slow down the learning process. Here’s how interleaving could be included into a sequence of work for mathematics. Imagine you’re teaching students the formulas for the volume of four different shapes: a wedge (W), a cone (C), a sphere (S), and a pyramid (P). Research shows64 that the main challenge students have with this type of task is not using the formulas per-se, but rather working out which formula is most appropriate in a given situation. As such, it’s less important that students’ practice focuses on using the formulas, and more important that their practice focuses on selecting which formula to use. Formula selection practice can be achieved by presenting them with, for example, sixteen practice problems in random order (for example W-C-S-P-S-P-W-C-S-W-S-P-C-W-P-C), rather than in a blocked fashion (W-W-W-W-C-C-C-C-S-S-S-S-P-P-P-P). In fact, Dylan Wiliam recently suggested65 that the one research paper he wishes all mathematics teachers would read is Doug Rohrer’s freely available booklet on just this topic: structuring mathematics practice to give students opportunities to select appropriate solutions methods, and not just apply them. 66 This approach also holds significant promise in other subjects, such as English as a Second Language (ESL). An ESL teacher who has recently taught their students simple past tense (SP), present perfect tense (PP), and the first (F), second (S), and third (T) conditional may be planning some practice for their students, and might have a set of questions such as the following: Consider the sentence below. ‘I .... a car for my daughter last Christmas.’ Select from the following options the word/ s that best fill in the blank within this sentence: A. will buy B. have bought C. buy D. bought67 If a teacher had fifty such questions, ten targeting each of the five grammar structures they’d just taught, it could be tempting to present them in the same order that the grammar structures were taught, such as SPx10, PPx10, Fx10, Sx10, Tx10. However, this approach is probably not as effective as presenting the questions interleaved or mixed up. While research has confirmed that students are likely to complete the work quicker under blocked conditions, and achieve more correct answers during practice, it won’t prepare them as well for future scenarios in which they have to independently choose which tense to use, and how to apply it correctly. 68” — Sweller's Cognitive Load Theory in Action by Oliver Lovell, Tom Sherrington https://a.co/9DokYYK 📜 Transcription from Image Brain Imaging Technology The key discovery in brain imaging technology, as it relates to the play rhetorics, is that in the neonatal stage, by eight months of age, the infant makes 1 trillion synaptic connections, but after that period the synapses attenuate if they are not actually used. By ten years of age, a child typically has only about 500 million connections. Thus the neonate has twice as many brain connections as the grown human being. It is theorized that this is to ensure enough “extra wiring” for adaptation to any kind of environment in which the child is reared. The infant brain’s ability to constantly undergo physical and chemical changes as it responds to the environment is taken to suggest enormous plasticity. This synaptic information (initially presented by Peter Huttenlocher of the University of Chicago) means that humans are born with more going for them than they will ever have again, which is the very opposite of the older view that “the brain is a self contained, hard wired unit that learns from a present, unchangeable set of rules” (Kotulak, 1996, p. xii) All of a sudden I saw in this piece of information another useful metaphor with which to understand the role of play. We could say that just as the brain begins in a state of high potentiality, so does play. The brain has these connections, but unless they are actualized in behavior, most of them will die off. Likewise in play, even when novel connections are actualized, they are still not, at first, the same as everyday reality. Actions do not become everyday reality until there is a rhetoric or practice that accounts for their use and value. Play’s function in the early stages of development, therefore, may be to assist the actualization of brain potential without as yet any larger commitment to reality. In this case, its function would be to save, in both brain and behavior, more of the variability that is potentially there than would otherwise be saved if there were no play. Piaget’s theory of play is, of course, the very reverse. He says that it is only after connections are established by real-life accommodation that they are consolidated in play. The present thesis would hold that another play function, perhaps the most important one, may be the actualization of novel connections, and therefore the extension of childhood’s potential variability (Sutton-Smith, 1966a, 1982f). ⸻ 🧠 Interpretation and Application This passage deepens the metaphor between neural plasticity and play’s epistemological role. Play is a provisional scaffolding for brain development, an active rehearsal space for potential pathways of thought and behavior that have not yet solidified into everyday function. This contrasts with Piaget’s view (that play consolidates after experience); Sutton-Smith asserts instead that play precedes reality—a fugue state of flexibility where excess connections can be tested and made real. ⸻ 🔧 Pedagogical Heuristic Update: “PLAY = Potential Learning Awaiting Yield” We can now engineer a revised Pomodoro Heuristic that uses Sutton-Smith’s neuroplastic metaphor directly. Each session banks not just knowledge but neural flexibility—a kind of structured potentiality with stakes. 💡 HEURISTIC: P.L.A.Y. Step Name Cognitive Function Description Activity Example P Proliferate Neural excess / sensory stimulation Expose learners to a flood of diverse inputs that seem redundant or chaotic Watch 3 different video styles about the same topic (e.g. WWI from a TikTok, a documentary, a reenactment) L Link Associative synaptogenesis Ask learners to create weird, personal, or aesthetic connections between elements “Which video felt most like your lunch?” or “Which narrator would survive in Alice in Wonderland?” A Activate Potentiation through behavior Convert playful or loose ideas into light-touch actions Improvise a one-minute skit; draw a map from memory; rephrase in a pirate voice Y Yield Consolidation / reflection Reflect on which ideas clicked, disappeared, or merged “Which idea do you want to forget? Which one is stuck in your body?” ⸻ 🧬 Why This Works • Mirrors brain plasticity: We treat learners like neonates—with more connections than they can keep, and the chance to decide which survive. • Preserves strangeness: Novelty is not just tolerated—it’s required to stretch cognitive range before commitment to a single frame. • Allows safe rehearsal: Students test out “future selves” and discard unused scaffolds, like brain connections that don’t fire. • Precedes knowledge lock-in: Before truth is claimed, possibility is rehearsed. ⸻ Would you like this heuristic translated into a printable weeklong lesson planner? Or mapped onto specific texts from your Kindle archive (Buddenbrooks, Winter in the Blood, Hyperion, etc.)? https://www.edutopia.org/article/strategies-student-discussion-grades-6-12/ https://www.cultofpedagogy.com/speaking-listening-techniques/ https://www.cultofpedagogy.com/hexagonal-thinking/ https://drive.google.com/file/d/1Y34uy0yIxBje2MAgh25aUPEzCvMOzYaz/view?usp=sharing https://citl.illinois.edu/citl-101/teaching-learning/resources/teaching-strategies/questioning-strategies Here’s the transcription from the images you provided: ⸻ Image 1 Split-attention The split-attention effect is extremely simple. In fact, I’ve come up with a rhyming sentence to make it easier to remember, ‘Information that must be combined should be placed together in space and time’. During learning, students are often required to integrate multiple pieces of information in order to understand the full picture of the learning task. This integration takes up valuable working memory resources, so the easier we can make it for students, the better. Placing related information closer together in space and time makes it easier for students to integrate it, and therefore reduces extraneous cognitive load. Split-attention effect: information that must be combined should be placed together in space and time. Keep information close together in space The split-attention effect was first discovered when it was found that there were some worked example formats that didn’t appear to be effective. It was found that the reason these worked examples were ineffective was due to split-attention, and since then, a whole raft of research has been conducted into the split-attention effect. Here are some examples of split-attention versus integrated information, from a variety of subject areas. The majority of these are taken directly from empirical research studies. In all such cases, the integrated format led to better learning outcomes. Split-attention in mathematics The domain of geometry within mathematics is where much of the split-attention research has taken place. It is often customary for textbooks to have diagrams as a figure, and descriptions of angles placed in a separate text caption, as follows. Split-attention format: ⸻ Image 2 Split-attention in music When learning to play the piano, reading sheet music can be an incredibly cognitively demanding task. Students are trying to link the dots on the staves to notes, the notes to keys on the piano, then coordinate their fingers to play the appropriate keys at the appropriate time. This is a multi-step integration process that novices often find overwhelming. A video of a player’s hands on the keyboard would reduce the amount of integration required to take place in the learner’s working memory. ⸻ Image 3 Clair de Lune from “Suite Bergamasque” L. 75 3rd Movement Claude Debussy (1862–1918) Split-attention format Integrated format The examples are endless, but the point is that whenever students are required to integrate information in order to reach a complete understanding, cognitive load will be minimised by placing that information closer together, rather than further apart. ⸻ Image 4 History lesson: Analyse the causes of the Cold War. Bullet-proof definition: The Cold War was a period of tense competition (1947–1991) between the United States and the Soviet Union (USSR) without direct war between the two powers. Economics lesson: Describe the role of government in a market economy. Bullet-proof definition: A market economy is an economy that allocates resources using the market forces of supply and demand. Geography lesson: Analyse progress towards attainment of the Sustainable Development Goals. Bullet-proof definition: The Sustainable Development Goals are 17 interconnected health, social, economic, and environmental progress targets that the UN hopes will be reached by 2030. Hollingsworth and Ybarra recommend that we commence a learning episode with a bullet-proof definition, have our students read it with us, and then recite it to each other from memory. The teacher then says something like, ‘Let me show you what this means’, and proceeds to facilitate deeper understanding through providing supporting evidence, examples, experience, experimentation or discussion of implications of this main idea. Each time the teacher introduces a new example, they explicitly relate it back to the bullet-proof definition so that the example and the core principle underlying it become firmly linked in students’ memories. When I mentor student teachers, I emphasise the importance of them being able to clearly answer two key questions prior to every lesson: 1. What will your students be able to do at the end of the lesson that they couldn’t do at the start? 2. How will you know whether or not they can do it? Constructing a bullet-proof definition is one clear and actionable way to home in on the first of these questions, and to more easily consider the kinds of feedback we will need to elicit from students in order to check our success. Put another way, a bullet-proof definition can help a teacher to identify what is, and what isn’t, redundant in a given lesson. ⸻ Image 5 Redundancy and the expertise-reversal effect Redundancy occurs when the same information is available to students from more than one source at the same time, making one form of that information redundant. However, what is redundant for one learner may not be redundant for another. Given the relatively larger amount of relevant knowledge stored in the long-term Redundant written instructions with clear pictorial instructions for folding a circle into a triangle As supportive teachers, we often want to provide highly detailed explanations to our students, feeling that the more detail we add, the better off they’ll be. The redundancy effect challenges this assumption. The point here is not that images and text together are bad, but that images and text together represent redundancy if they both communicate the same thing. Images and text together represent redundancy if they both communicate the same thing. Bullet-proof definitions Taken more broadly, redundant information within lessons is anything that distracts students from the core to-be-learned material. The extraneous load that we as teachers impose upon our students often stems from this form of redundancy. When giving instruction, we often want to provide a highly detailed and in-depth explanation, providing the full picture to our students, or colouring it with additional interesting details, images, or fun facts. In reality, highly detailed explanations often overload our students’ working memories in the early stages of learning. There is, of course, a time for this additional engaging information, but that time is not when the intrinsic load of the task is already pushing the limits of our students’ working memories. In order to align instruction to the core concept being taught, a useful strategy to consider is Hollingsworth and Ybarra’s bullet-proof definition. A bullet-proof definition is a one-sentence summary of the key concept or idea the teacher is trying to convey. Here are a few examples of bullet-proof definitions, some of which are inspired by Hollingsworth and Ybarra’s sample lessons. Science lesson: Identify and communicate sources of experimental error. Bullet-proof definition: Experimental error is the difference between a measurement and its true value. Art lesson: Describe the contribution of Marcel Duchamp’s ‘Fountain’ to 20th century art. Bullet-proof definition: Marcel Duchamp’s ‘Fountain’ (1917) was a standard urinal presented as an artistic work that prompted the fundamental question, ‘What makes something art?’ ⸻ ✅ Now fully transcribed and structured with headings. Would you like me to also summarize these sections into study notes (key principles of split-attention, redundancy, and bullet-proof definitions), so you can quickly apply them to teaching or writing?

research means exploring important, testable questions with more than four hundred teachers and their thousands of students over 15 years.

Epistemic

Success means getting more of these students thinking in math class, for longer amounts of time.

Effective describes teaching decisions and practices that create condition...

How much of an impact does the timing of the launch within the lesson have? (A lot.)

Only lasting techniques that produced the most student thinking and were transferable across teachers and schools have made it into this book.

to get our students thinking and keep them thinking longer.

grounded in classroom reality

Epistemic

Liljedahl argues convincingly that we need to interrupt the entrenched patterns of school.

Instead of taking mindless notes to please us, they take notes that would be helpful to their future forgetful selves. Instead of mimicking our methods alone, they think about new problems together, and so on. We make these changes not for the sake of change, nor for ideological reasons, but because these practices lead to increased student thinking in hundreds of diverse classrooms.

! Epistemic

students do an awful lot of “studenting,” but not much thinking. Students from communities that have historically been excluded from mathematics are often denied access to thinking at all. For the health of our students and our societies, we need to challenge institutional norms and build thinking classrooms in which we value students’ thinking and time rather than use legacy practices that encourage students to slack, stall, mimic, and fake their way through the system.

Ideological epistemic; a good one ! “Legacy systems”

The first problem I gave Jane came from Lewis Carroll and was a problem I had used many times with my Grade 8s and 9s. I knew that this was a good problem. The context was engaging, the answer was non-trivial, and it didn’t require any sophisticated mathematics to solve. And my students, when I had used it with them, had enjoyed arguing over the various answers they arrived at. If 6 cats can kill 6 rats in 6 minutes, how many will be needed to kill 100 rats in 50 minutes? (Lewis Carroll, 1880)

ELA / Math ! Argues for a cultural Literacy from wexler Are “legacy” practices able To be accurately disentangled from ideological and/or material patterns (operant-ideologies) “Cheney’s argument may sound reminiscent of E. D. Hirsch’s, but there’s an important difference. Hirsch’s goal was to provide disadvantaged students with access to references understood by the elite—whatever they might be—and knit the country together through a shared culture that could and should change over time. Cheney, on the other hand, was making a value judgment: American history and culture were superior.” — The Knowledge Gap: The Hidden Cause of America's Broken Education System--and How to Fix it by Natalie Wexler https://a.co/j58DFQg

“Give me another one.” I was both shocked and impressed. There was more to Jane than met the eye. So, I gave Jane a second task, and the next morning I was back in my desk watching Jane try it again—same students, new problem. It went worse. The students were quicker to give up, and Jane now spent more time encouraging and less time helping. At the end of the activity Jane came up to me and said, “Give me another one.” This woman had grit. Over the last 18 years I have worked with hundreds of teachers, and not since Jane have I encountered a teacher with such fortitude—such will and determination to keep going in the face of utter failure. So, I gave Jane a third task, and the next morning I was again back in my desk—same students, new problem.

The second epiphany was the sudden realization that Jane was planning her teaching on the assumption that students either couldn’t or wouldn’t think. Jane was in a tough position—she had a room full of students who weren’t thinking, yet she had curriculum to get through and standards to meet. This is not uncommon. Every day, teachers all over the world find themselves in this exact same dilemma. Even teachers who, by traditional measures, are considered good teachers—who know their content, care about their students, and want to do the best for them—face this dilemma. Jane was considered, in her school and throughout her district, to be a very good teacher—her students performed well on tests, and no students appeared to be falling through the cracks. Jane wanted to do her best for her students, and she was willing to work hard to get there. And yet Jane found herself in this exact dilemma. So, what did she do? She did what many of us do—she structured activities that allowed her to move through the content as quickly and efficiently as possible without requiring her students to think. I’ll give you an example.

Dynamic “legacy”

Studenting: is what students do in a learning setting—some of which is learning. … there is much more to studenting than learning how to learn. In the school setting, studenting includes getting along with one’s teachers, coping with one’s peers, dealing with one’s parents about being a student, and handling the non-academic aspects of school life. (1986, p. 39) [as well as] ‘psyching out’ teachers, figuring out how to get certain grades, ‘beating the system,’ dealing with boredom so that it is not obvious to teachers, negotiating the best deals on reading and writing assignments, threading the right line between curricular and extra-curricular activities, and determining what is likely to be on the test and what is not. (1994, p. 1)

Pejorative? !

1 For a deep analysis of the psychology behind these, and other, studenting behaviors, see Allan (2017).

Epistemic

Slacking - A number of students in each class did not attempt the task at all. Instead, they spent the time looking at their smart phones, talking to other slackers, or literally doing nothing. When they were interviewed, it became clear that the students who slacked either didn’t know what was going on or didn’t care what was going on.

Stalling - Like the students who slacked, these students did not attempt the task. Unlike the slackers, however, these students filled the time with legitimate off-task behaviors like sharpening a pencil, getting a drink of water, going to the bathroom, or endlessly rooting in their backpack for some vital piece of equipment. When interviewed, these students told us that they either didn’t know how to do the question or knew that if they just waited for a few minutes the teacher would go over it. Faking - Some students pretended to do the task but were, in reality, doing nothing. Faking involved studiously looking at the board, flipping pages in the textbook, appearing to ponder, and pretending to write something on their page. But, for all the bluster and show, nothing was being achieved. Like the stallers, these students were hiding behind legitimate student behavior. The difference was that while the stallers hide behind legitimate off-task behavio...

students in the three aforementioned groups, students who mimicked attempted, and often completed, the task. What they were doing, however, was trying to recreate the pattern of the solutions that had just been demonstrated on the board. This involved constant referencing to the demonstrated example with line-by-line mapping from the example to the task at hand. If the example that the teacher had demonstrated did not match the task they were asked to do, these students were often way off track or completely stuck. When we interviewed the teachers in whose classrooms we were doing the studenting research, all of them stated, with emphasis, that they did not want their students to mimic. Ironically, 100% of the students who mimicked stated that they thought that mimicking was what their teacher wanted them to do. They were reading the demonstration of an analogous example prior to the now-you-try-one tasks as an invitation to mimic.

Pseudo-cognition Pseudo-learning Be tough ethos=solution

Figure i.3 Distribution of studenting behaviors on now-you-try-one tasks.

Epistemic Screenshot Mits

These normative structures that permeate classrooms in North America, and around the world, are so robust, so entrenched, that they transcend the idea of classroom norms (Cobb, Wood, & Yackel, 1991; Yackel & Cobb, 1996) and can only be described as institutional norms (Liu & Liljedahl, 2012)—norms that have extended beyond the classroom, even the school building, and have become ensconced in the very institution of school. Much of how classrooms look and much of what happens in them today is guided by these institutional norms—norms that have not changed since the inception of an industrial-age model of public education.

Epistemic Ideology

deliberately break institutional normative structures and see whether it could increase student thinking.

Radical ! Epistemic Ideology

there were limits to what was reasonable.

results often came before explanations. This remained true all through the research and continues to be true even today. Knowledge of what works always preceded an understanding of why it worked.

! Epistemic Possible Epistemic issue

we lost control of cause and effect—pedagogy and thinking.

Epistemic

Screenshot mits

These are called micro-moves to contrast them against the macro-moves that are the optimal practices for thinking in each chapter. This is not to say they are any less important. In many cases, these micro-moves make the difference between smooth and rough implementation in your classroom.

Micro-chunk ! Epistemic

Problem solving is a messy, non-linear, and idiosyncratic process. Students will get stuck. They will think. And they will get unstuck. And when they do, they will learn—they will learn about mathematics, they will learn about themselves, and they will learn how to think.

How might trauma and learning w/ and alongside the traumatized give dimension to each micro move? !

Good problem-solving tasks require students to get stuck and then to think, to experiment, to try and to fail, and to apply their knowledge in novel ways in order to get unstuck.

Problem-solving tasks are often called non-routine tasks because they require students to invoke their knowledge in ways that have not been routinized. Once routinization happens, students are mimicking rather than thinking—or as Lithner (2008) calls it, being imitative rather than creative.

Epistemic Non-routinized moments amid routines that engender self-regulation.

Card Tricks have the same qualities as highly engaging thinking tasks—they are highly engaging situated tasks that draw students in and entice them to think. It turns out that there are a lot of card tricks that are both built on and can be explained by mathematics.

Low-Floor Task: Task with a threshold that allows any and all learners to find a point of entry, or access, and then engage within their level of comfort. High-Ceiling Task: Tasks that have ambiguity and/or room for extensions such that students can engage with the evolving complexity of the task. Open-Middle: A problem structure where a task has a single final correct answer, but in which there are multiple possible correct ways to approach and solve the problem.

! Dr

Non-curricular Task: A task that is clearly mathematical in nature but does not map well to the outcomes or standards specified in the curriculum for the class in which it is used.

This is not to say that I was naïve about the lived reality of classroom teachers and the persistent and ubiquitous nature of curriculum.

Myopic frame Curriculum concerns aren’t an adequate placeholder for lived-realities and teacher concerns and student concerns. Epistemic

begin by asking a question that is review of prior knowledge; then ask a question that is an extension of that prior knowledge.

Further investigation showed that although three lessons of non-curricular tasks (Type 1) was enough to prime many classes, in some cases as many as five lessons were needed before the dispositions of the students shifted enough to allow them to be successful at scripted curricular thinking tasks. This investigation revealed that, in almost every situation, the teacher was able to predict when the class was ready to shift their thinking toward curricular thinking tasks.

5 is the z pack; 5 attempts preclude sustainable thinking activities

these results show that to get students thinking about curriculum tasks, they need to first be primed to do so using non-curricular tasks. Nothing in my research has shown a way to avoid this. You have to go slow to go fast.

Sequence

They create situations where every student gets stuck, which makes stuck an expected, safe, and socially acceptable state to be in. In essence, these tasks make it safe to fail and keep trying. And through these struggles, students begin to build confidence in their teacher’s confidence in them. All of these qualities are easier to build inside of highly engaging non-curricular tasks and are necessary when we transition students to curricular thinking tasks.

Trauma responsive Safe to fail

This is a bold approach, which has been proven to work. This is the essence of Jo Boaler’s early research at Phoenix Park (Boaler 2002).Further, Maria Kerkhoff (2018) showed that after doing just 18 rich tasks over the course of 18 classes, the student who she was studying encountered almost all of the curriculum outcomes for her grade, along with numerous curriculum outcomes from previous and future grades.

Epistemic claim Longitudinal, Hattie-like, inquiry would be useful here

Curriculum is inherently spiraled. For this reason, it is seldom the case that students have no prior knowledge at all. In the rare cases where it is true, however, you can, if you wish, just tell the students something. But you still only have five minutes before you should ask them a thinking question.

50% already known Hattie

remember to start with three to five non-curricular tasks and to get students doing these in the first five minutes.

Pedagogy: Because a teacher believes that their students can, and will, learn from each other, the teacher will create either homogenous or heterogeneous groups based on students’ abilities, perseverance, or work habits.

Productivity: A teacher may arrange groups that lead to the completion of more work. This may, for example, require there to be a strong leader in a group for project work, or the teacher may prefer to group weaker students with stronger ones.

Peacefulness: A teacher may create groups that intentionally keep friends or disruptive students apart, as such groupi...

These sorts of mismatches arise from the tension between the individual goals of students concerned with themselves, or their cadre of friends, and the classroom goals set by the teacher for everyone in the room. Couple this with the social dynamics often present in classrooms, and a teacher faces a situation where students not only wish to be with certain classmates, but also disdain to be with others. In essence, no matter how strategic a teacher is in their groupings, when there is a mismatch between their goals and students’ individual goals, it means some students will be unhappy and will disengage. This disengagement is antithetical to a thinking classroom. I would venture to say that most teachers are familiar with this challenge. No matter how strategic a teacher is in their groupings, when there is a mismatch between their goals and students’ individual goals, it means some students will be unhappy and will disengage.

less authentic engagement

While we clearly know the value and importance of student collaboration, the problem is that efforts to actualize collaboration through either strategic groups or assigned roles may be having a negative effect on how our students engage with each other and the task at hand. At the same time, self-selected groups seem to be of little help, as students group themselves for reasons antithetical to solving mathematical problems. You may have seen this, and been frustrated by it, in your own classroom as well.

Every day

This proved to be spectacularly ineffective.

Although randomizing wrested the control from the teachers, making it visibly random was necessary for the students to both perceive and believe the randomness. We also learned that, from Grade 3 up, the optimal group size was three.

February admin support push