Lower-Level
Demands
Memorization
Tasks
~involve either
reproducing previously learned facts, rules, formulae, or definitions OR
committing facts, rules, formulae or definitions to memory.
~cannot be solved
using procedures because a procedure does not exist or because the time frame
in which the task is being completed is too short to use a procedure.
~are not
ambiguous- such tasks involve exact reproduction of previously seen material,
and what is to be reproduced is clearly and directly stated.
~have no connection
to the concepts or meaning that underlie the facts, rules, formulae, or
definitions being learned or reproduced.
Procedures
Without Connections
~are
algorithmic. Use of the procedure is
either specifically called for its use is evident based on prior instruction,
experience, or placement of the task.
~require limited
cognitive demand for successful completion.
~have no
connection to the concepts or meaning that underlie the procedure being used
are focused on
producing correct answers rather than developing mathematical understanding
~require no
explanations, or explanations that focus solely on describing the procedure
that was used.
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Higher-Level
Demands
Procedures
With Connections Tasks
~focus students’
attention on the use of procedures for the purpose of developing deeper
levels of understanding of mathematical concepts and ideas.
~suggest pathways
to follow (explicitly or implicitly) that are broad general procedures that
have close connections to underlying conceptual ideas as opposed to narrow
algorithms that are opaque with respect to underlying concepts.
~usually are
represented in multiple ways. Making
connections among multiple representations helps develop meaning.
~require some
degree of cognitive effort. Although
general procedures may be followed, they cannot be followed mindlessly. Students need to engage with conceptual
ideas that underlie the procedures in order to successfully complete the
tasks and develop understanding
Doing
Mathematical Tasks
~require complex and nonalgorithmic thinking (i.e.,
there is not a predictable, well-rehearsed approach or pathway explicitly
suggested by the task, task instructions, or a worked-out example).
~require students to explore and understand the
nature of mathematical concepts, processes, or relationships
~demand self-monitoring or self-regulation of one’s
own cognitive processes
~require students to access relevant knowledge and
experiences and make appropriate use of them in working through the tasks
~require students to analyze the task and actively
examine task constraints that may limit possible solution strategies and
solutions.
~require considerable cognitive effort and may
involve some level of anxiety for the student because of the unpredictable
nature of the solution process required.
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