*"Tasks that ask students to perform a memorized procedure in a routine manner lead to one type of opportunity for student thinking; tasks that require students to think conceptually and that stimulate students to make connections lead to a different set of opportunities for student thinking."*

(Stein & Smith, 1998, p.269)

## Low Cognitive Demand Tasks

“Low cognitive demand tasks involve stating facts, following known procedures, and solving routine problems.” (Van De Walle, Karp, & Bay-Williams, 2012, p.36) These tasks require minimum thinking or cognitive analysis, and rather focus on single, concrete answers that are solved using prior knowledge. Low demand tasks “lead to one type of opportunity for student thinking” (Stein & Smith, 1998, p.269), neglect to challenge students' ability to make mathematical connections within deeper concepts. These low-level demand tasks can be broken down into two different types: memorization and procedures without connections.

Memorization tasks involve pulling facts and formulas from prior memory in order solve the equation. These tasks are quick, and sometimes timed, resulting in the inability to use procedures to find an answer. Memorization tasks are not ambiguous, because they involve the exact replication of prior material. Lastly, this type of low cognitive demand task, requires no connections to the meaning of the information that is being learned. (Smith & Stein, 1998, p.348)

The second type of low cognitive demand tasks is procedures without connections. Tasks that fall under this category are algorithmic, meaning they follow a specific procedure from prior learning. They require little thinking of how to complete the task. A task that is labeled procedures without connections has no connections to concepts or to why a procedure is done. These tasks focus on only finding the correct answer and require no explanation or mathematical understanding. (Smith & Stein, 1998, p.348)

Memorization tasks involve pulling facts and formulas from prior memory in order solve the equation. These tasks are quick, and sometimes timed, resulting in the inability to use procedures to find an answer. Memorization tasks are not ambiguous, because they involve the exact replication of prior material. Lastly, this type of low cognitive demand task, requires no connections to the meaning of the information that is being learned. (Smith & Stein, 1998, p.348)

The second type of low cognitive demand tasks is procedures without connections. Tasks that fall under this category are algorithmic, meaning they follow a specific procedure from prior learning. They require little thinking of how to complete the task. A task that is labeled procedures without connections has no connections to concepts or to why a procedure is done. These tasks focus on only finding the correct answer and require no explanation or mathematical understanding. (Smith & Stein, 1998, p.348)

## High Cognitive Demand Tasks

“High cognitive demand tasks involve making connections,
analyzing information, and drawing conclusions.” (Smith & Stein, 1998) High-level
tasks require students to think abstractly and make connections to mathematical
concepts. These tasks can use procedures, but in a way that builds connections
to mathematical meanings and understandings. (Stein & Smith, 1998, p.270)
“When completing higher demanding tasks students are engaged in a productive
struggle, that challenges them to make connections to concepts and to other
relevant knowledge.” (Van De Walle, Karp, & Bay-Williams, 2012, p.37) Like,
low cognitive demand tasks, high-level tasks can be separated into two types:
procedures with connections and doing math.

Procedures with connections place emphasize the use of procedures, in order to develop a students’ deeper level of understanding of math concepts and ideas. Opposite of a standard algorithm, these tasks, suggest pathways for students to follow. These pathways are broader than typical procedures and have close connections to the fundamental conceptual ideas. These task help develop the meaning of mathematical ideas by the use of multiple representations (visual diagrams, manipulates, symbols, etc.). These higher-level tasks require some degree of thinking; students cannot solve them mindlessly. They must engage students with conceptual ideas, meaning the task triggers the procedure that is needed to complete the task, and develop understanding. (Smith & Stein, 1998, p.348)

Doing math is the second type of higher-level tasks. These tasks require multifaceted thinking. They are not algorithmic, meaning they are not predictable, and the exact plans to solve the task are not clearly proposed in the instructions. Doing math requires students’ to comprehend and understand mathematical connections. These tasks require students to monitor their own process of thinking, while using applicable knowledge to work through the task. In order to complete the task students must analyze the task, which requires considerable cognitive effort. These tasks may ensue apprehension for students, because there is no certain process to solve for the solution. (Smith & Stein, 1998, p.348)

Procedures with connections place emphasize the use of procedures, in order to develop a students’ deeper level of understanding of math concepts and ideas. Opposite of a standard algorithm, these tasks, suggest pathways for students to follow. These pathways are broader than typical procedures and have close connections to the fundamental conceptual ideas. These task help develop the meaning of mathematical ideas by the use of multiple representations (visual diagrams, manipulates, symbols, etc.). These higher-level tasks require some degree of thinking; students cannot solve them mindlessly. They must engage students with conceptual ideas, meaning the task triggers the procedure that is needed to complete the task, and develop understanding. (Smith & Stein, 1998, p.348)

Doing math is the second type of higher-level tasks. These tasks require multifaceted thinking. They are not algorithmic, meaning they are not predictable, and the exact plans to solve the task are not clearly proposed in the instructions. Doing math requires students’ to comprehend and understand mathematical connections. These tasks require students to monitor their own process of thinking, while using applicable knowledge to work through the task. In order to complete the task students must analyze the task, which requires considerable cognitive effort. These tasks may ensue apprehension for students, because there is no certain process to solve for the solution. (Smith & Stein, 1998, p.348)