High levels of cognitive demand must be maintained in order to ensure high levels of student learning. A high-demand task (procedures with connections or doing mathematics) in the written curriculum is not necessarily implemented at that level. So, part of ensuring student learning is maintaining high levels of cognitive demand. Research describes four specific trajectories for high-demand tasks when they are implemented. These include:
Maintenance of high-level demands
Decline into procedures without connection to meaning
Decline into unsystematic and nonproductive exploration
Decline into no mathematical activity
READ the file Tasks in Implementation (in the module contact) for more detail about each of these trajectories. Given these possibilities, it is important to think about how this can happen. What does it look like for a high-demand task to devolve into something else? For this activity, read the file Classroom Vignettes. Answer the following questions:
For each vignette, identify which of the four categories best describes what happened to the high-demand task during implementation.
For each vignette, provide a justification for the description you chose in #1.
What specific details from the vignette led you to categorize the case as you did?
POST ONCE. RESPOND TWICE. Do you agree with the categories/justifications provided by your classmates? If so, why? If not, why not
Full Answer Section
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Specific Details: The teacher's prompt to "find the formula" and the students' subsequent focus on calculation rather than conceptual understanding indicate a shift away from high-level thinking. The emphasis on the answer, rather than the process, further reinforces this decline.
Vignette 2: The Fraction Task
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Category: Decline into no mathematical activity.
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Justification: The task, which had the potential for rich mathematical discussion and exploration of fraction concepts, was completely derailed by classroom management issues. The focus shifted entirely away from the mathematics and onto student behavior. No meaningful mathematical work occurred.
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Specific Details: The teacher's repeated redirection of student behavior and the complete absence of any mathematical engagement demonstrate the decline into no mathematical activity. The task was abandoned, and no learning took place.
Vignette 3: The Locker Problem
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Category: Maintenance of high-level demands.
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Justification: This vignette illustrates successful implementation of a high-demand task. Students were actively engaged in problem-solving, exploring different strategies, and justifying their reasoning. The teacher facilitated the discussion without providing direct answers, encouraging students to think critically and make connections. The task remained challenging and open-ended, promoting deep mathematical thinking.
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Specific Details: The students' collaborative work, their discussion of patterns, and their attempts to generalize a solution all suggest sustained engagement with the high-level demands of the task. The teacher's role as a facilitator, rather than a direct instructor, further supports this categorization.
Vignette 4: The Function Task
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Category: Decline into unsystematic and nonproductive exploration.
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Justification: While the task had the potential for exploring function concepts, the implementation lacked structure and direction. Students seemed to be randomly trying different approaches without any clear strategy or connection to the underlying mathematical ideas. The teacher's interventions were not effective in guiding students toward productive exploration.
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Specific Details: The students' seemingly random guessing and the teacher's inability to focus the exploration suggest a decline into unsystematic activity. The lack of clear direction and the absence of productive mathematical discourse indicate that the task, while potentially high-demand, was implemented in a way that did not lead to meaningful learning.
Sample Answer
Here's an analysis of the classroom vignettes, categorized and justified:
Vignette 1: The Candy Box Problem
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Category: Decline into procedures without connection to meaning.
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Justification: The task, while initially presented as a problem-solving activity involving maximizing candy box volume, quickly devolved into a rote application of a formula. The teacher's focus shifted from understanding the underlying mathematical principles (relationship between dimensions and volume) to simply finding the "right" equation and plugging in numbers. Students were not encouraged to explore different box dimensions or reason about why the formula worked; they were primarily focused on getting the answer.