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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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