Tasks
“Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies." (Stein and Smith, 21)
This quote not only sets up the importance of tasks within a math classroom, but its reference to 'discussing tasks' also connects to another aspect of my framework: student voice.
In his work Making Sense: Teaching and Learning Math with Understanding, Heibert described characteristics of high-quality math tasks stating that they should:
With these characteristics in mind, Stein and Smith created a task analysis guide (TAG) which helps educators categorize the tasks they have their students work on. If students are to build strong conceptual understanding and make deep connections to the mathematics, they should be working on tasks that would fall under the “Procedures with Connections” or “Doing Mathematics” categories.
In my You-do We-do I-do model, when working on tasks, students have opportunities to use their 'tools' (Heibert's 2nd criteria) during the You-do stage. Since the teacher does not provide a demonstration for the students to emulate, the students are forced to use their tools to 'engage with conceptual ideas' (TAG: Procedures with Connections) and 'explore the nature of mathematical concepts, processes, or relationships' (TAG: Doing Mathematics). In the We-do stage, students are encouraged to communicate and reflect on their ideas (Hiebert's 1st criteria) with their classmates through pair or small-group work. Since solutions to these tasks, according to the TAG, can be 'represented in multiple ways' (Procedures with Connections) or 'require complex non-algorithmic thinking' (Doing Mathematics), collaboration becomes a vital part of students' work as they analyze and critique the various methods of their group members. During the class wide discussion (We-do) and the activity summary (I-do), students, with the guide of the teacher, move toward the learning goal of the lesson as they make connections among mathematical concepts and ideas. This would be an example of tasks meeting Hiebert's 3rd criteria - 'leaving behind important residue'.
Task features such as ‘reproducing previously learned facts, procedures, or formulae’ (TAG: Memorization) or ‘Use of the procedure is specifically called for or its use is evident based on prior instruction’ (TAG: Procedures without Connections), would describe the category of most tasks students engage in in the GRR model, since this model gives students opportunities to work on tasks only after the teacher demonstrates a similar example. As a result, the cognitive demand placed on students is limited to recall or memorization.
Use the following link to learn about how I implemented high-level demand tasks in my classroom and my observations of how my students responded: Task Artifacts
In his work Making Sense: Teaching and Learning Math with Understanding, Heibert described characteristics of high-quality math tasks stating that they should:
- encourage reflection and communication
- allow students to use tools (the problem-solving skills that the students possess)
- leave behind important residue - “the learning that students take with them from solving problems”
With these characteristics in mind, Stein and Smith created a task analysis guide (TAG) which helps educators categorize the tasks they have their students work on. If students are to build strong conceptual understanding and make deep connections to the mathematics, they should be working on tasks that would fall under the “Procedures with Connections” or “Doing Mathematics” categories.
In my You-do We-do I-do model, when working on tasks, students have opportunities to use their 'tools' (Heibert's 2nd criteria) during the You-do stage. Since the teacher does not provide a demonstration for the students to emulate, the students are forced to use their tools to 'engage with conceptual ideas' (TAG: Procedures with Connections) and 'explore the nature of mathematical concepts, processes, or relationships' (TAG: Doing Mathematics). In the We-do stage, students are encouraged to communicate and reflect on their ideas (Hiebert's 1st criteria) with their classmates through pair or small-group work. Since solutions to these tasks, according to the TAG, can be 'represented in multiple ways' (Procedures with Connections) or 'require complex non-algorithmic thinking' (Doing Mathematics), collaboration becomes a vital part of students' work as they analyze and critique the various methods of their group members. During the class wide discussion (We-do) and the activity summary (I-do), students, with the guide of the teacher, move toward the learning goal of the lesson as they make connections among mathematical concepts and ideas. This would be an example of tasks meeting Hiebert's 3rd criteria - 'leaving behind important residue'.
Task features such as ‘reproducing previously learned facts, procedures, or formulae’ (TAG: Memorization) or ‘Use of the procedure is specifically called for or its use is evident based on prior instruction’ (TAG: Procedures without Connections), would describe the category of most tasks students engage in in the GRR model, since this model gives students opportunities to work on tasks only after the teacher demonstrates a similar example. As a result, the cognitive demand placed on students is limited to recall or memorization.
Use the following link to learn about how I implemented high-level demand tasks in my classroom and my observations of how my students responded: Task Artifacts