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Analyzing Mathematical Tasks

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Analyzing Mathematical Tasks

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    1. Math & Science Collaborative Analyzing Mathematical Tasks and the Mathematical Task Framework

    2. Math & Science Collaborative Analyzing Mathematical Tasks At the heart of teaching well is the core challenge of getting learners engaged in productive work. Mathematical tasks are the sites for engaging students in core mathematical activity. At the heart of teaching well is the core challenge of getting learners engaged in productive work. Mathematical tasks are the sites for engaging students in core mathematical activity.

    3. Math & Science Collaborative What are mathematical tasks? We define mathematical tasks as a set of problems or single complex problem the purpose of which is to focus students’ attention on a particular mathematical idea. Mathematical tasks can be examined from a variety of perspectives including the number and kinds of representations evoked, the variety of ways in which they can be solved, and their requirements for student communication.Mathematical tasks can be examined from a variety of perspectives including the number and kinds of representations evoked, the variety of ways in which they can be solved, and their requirements for student communication.

    4. Math & Science Collaborative Why focus on mathematical tasks? Tasks form the basis for students’ opportunities to learn what mathematics is and how one does it; Tasks influence learners by directing their attention to particular aspects of content and by specifying ways to process information; and The level and kind of thinking required by mathematical instructional tasks influences what students learn. *The day-in and day-out cumulative effect of classroom-based tasks leads to the development of students’ implicit ideas about the nature of mathematics - about whether mathematics is something about which they can personally make sense and about how long and how hard they should have to work to do so. * If we choose to give students well defined problems in which they will utilize previously learned procedures, this is what they will come to understand as doing mathematics. When they find themselves in a problem situation which is somewhat ambiguous, they will quickly give up and come to believe that they cannot deal with mathematically ambiguous problems. Will this approach to mathematics serve well our 21st century students of mathematics?*The day-in and day-out cumulative effect of classroom-based tasks leads to the development of students’ implicit ideas about the nature of mathematics - about whether mathematics is something about which they can personally make sense and about how long and how hard they should have to work to do so. * If we choose to give students well defined problems in which they will utilize previously learned procedures, this is what they will come to understand as doing mathematics. When they find themselves in a problem situation which is somewhat ambiguous, they will quickly give up and come to believe that they cannot deal with mathematically ambiguous problems. Will this approach to mathematics serve well our 21st century students of mathematics?

    5. Math & Science Collaborative Comparing Two Mathematical Tasks

    6. Math & Science Collaborative Comparing Two Mathematical Tasks Solve Two Tasks: Martha’s Carpeting Task The Fencing Task

    7. Math & Science Collaborative Comparing Two Mathematical Tasks Martha was recarpeting her bedroom which was 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase? Stein, Smith, Henningsen, & Silver, 2000, p. 1

    8. Math & Science Collaborative Comparing Two Mathematical Tasks If Ms. Brown's students want their rabbits to have as much room as possible, how long would each of the sides of the pen be? How long would each of the sides of the pen be if they had only 16 feet of fencing? How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it. Stein, Smith, Henningsen, & Silver, 2000, p. 2

    9. Math & Science Collaborative Comparing Two Mathematical Tasks How are Martha’s Carpeting Task and the Fencing Task the same and how are they different? (Consider your own experience in solving the tasks, the “mathematical possibilities” of the tasks, or the PA Content Standards that can be addressed through the use of each of the tasks.)

    10. Math & Science Collaborative Comparing Two Mathematical Tasks Do the differences between the Fencing Task and Martha’s Carpeting Task matter? Why or Why not?

    11. Math & Science Collaborative Comparing Two Mathematical Tasks “Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.” Stein, Smith, Henningsen, & Silver, 2000 This is the math classroom version of “You reap what you sow.” Do we want our students to be mathematical problem solvers ? What types of mathematical tasks will build problem solving skills?This is the math classroom version of “You reap what you sow.” Do we want our students to be mathematical problem solvers ? What types of mathematical tasks will build problem solving skills?

    12. Math & Science Collaborative Comparing Two Mathematical Tasks “The level and kind of thinking in which students engage determines what they will learn.” Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997

    13. Math & Science Collaborative Characterizing Tasks We would like to engage you in an activity which is about characterizing mathematical tasks. The goal of the activity is for you to participate in a thoughtful analysis of the tasks. We would like to engage you in an activity which is about characterizing mathematical tasks. The goal of the activity is for you to participate in a thoughtful analysis of the tasks.

    14. Math & Science Collaborative Sort Tasks A – P into two categories [high level and low level] Develop a list of criteria that describe the tasks in each category

    15. Math & Science Collaborative Categorizing Tasks “If we want students to develop the capacity to think, reason, and problem solve then we need to start with high-level, cognitively complex tasks.” Stein & Lane, 1996 Tasks that require students to perform a memorized procedure in a routine manner lead to one type of opportunity for student thinking; tasks that demand engagement with concepts and that stimulate students to make purposeful connections to meaning or relevant mathematical ideas lead to a different set of opportunities for student thinking.Tasks that require students to perform a memorized procedure in a routine manner lead to one type of opportunity for student thinking; tasks that demand engagement with concepts and that stimulate students to make purposeful connections to meaning or relevant mathematical ideas lead to a different set of opportunities for student thinking.

    16. Math & Science Collaborative Categorizing Tasks Are all high-level tasks the same? [Is there an important difference between Tasks B and J?] Are all low-level tasks the same? [Is there an important difference between Tasks I and O?]

    17. Math & Science Collaborative Levels of Cognitive Demand & The Mathematical Tasks Framework

    18. Math & Science Collaborative Linking to Literature/Research: The QUASAR Project Low-Level Tasks High-Level Tasks When determining the level of cognitive demand provided by a mathematical task, it is important not to become distracted by superficial features of the task and to keep in mind the students for whom the task is intended. Low-level tasks, for example, can appear to be high-level when they have characteristics of reform-oriented instructional tasks such as requiring the use of manipulatives; using real-world contexts; involving multiple steps, actions or judgments; an/or making use of diagrams. ( use 2 elementary tasks as an example) It is also possible for tasks to be designated low-level when in fact they should be considered high level. (use 2 elementary tasks as an example) Another consideration when deciding the level of challenge provided be a task is the students and the norms and expectations for work in their classroom. The age, grade-level, prior knowledge and experiences need to be taken into consideration when deciding whether the task is likely to provide an appropriate level of challenge for their students. When determining the level of cognitive demand provided by a mathematical task, it is important not to become distracted by superficial features of the task and to keep in mind the students for whom the task is intended. Low-level tasks, for example, can appear to be high-level when they have characteristics of reform-oriented instructional tasks such as requiring the use of manipulatives; using real-world contexts; involving multiple steps, actions or judgments; an/or making use of diagrams. ( use 2 elementary tasks as an example) It is also possible for tasks to be designated low-level when in fact they should be considered high level. (use 2 elementary tasks as an example) Another consideration when deciding the level of challenge provided be a task is the students and the norms and expectations for work in their classroom. The age, grade-level, prior knowledge and experiences need to be taken into consideration when deciding whether the task is likely to provide an appropriate level of challenge for their students.

    19. Math & Science Collaborative Linking to Literature/ Research: The QUASAR Project Low-Level Tasks memorization procedures without connections (e.g., Martha’s Carpeting Task) High-Level Tasks procedures with connections doing mathematics (e.g., The Fencing Task) Talk through the characteristics given in the task analysis guide. A copy of this guide should probably be given to participants. Discuss how the high level tasks constitute rich materials and resources which would support student learning. Also discuss how these types of tasks should not be the add-ons but rather the content of core instruction (RTI Tier I) Talk through the characteristics given in the task analysis guide. A copy of this guide should probably be given to participants. Discuss how the high level tasks constitute rich materials and resources which would support student learning. Also discuss how these types of tasks should not be the add-ons but rather the content of core instruction (RTI Tier I)

    20. Math & Science Collaborative Linking to Literature/ Research: The QUASAR Project The Mathematical Tasks Framework The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers. As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. We will now consider each phase of the math task framework.The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers. As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. We will now consider each phase of the math task framework.

    21. Math & Science Collaborative Linking to Literature/ Research: The QUASAR Project The Mathematical Tasks Framework The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers. As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. We will now consider each phase of the math task framework. The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers. As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. We will now consider each phase of the math task framework.

    22. Math & Science Collaborative Linking to Literature/ Research: The QUASAR Project The Mathematical Tasks Framework The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers. As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. We will now consider each phase of the math task framework. This section and the next are where we consider the type of instruction and assessment needed throughout the class period.The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers. As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. We will now consider each phase of the math task framework. This section and the next are where we consider the type of instruction and assessment needed throughout the class period.

    23. Math & Science Collaborative Linking to Literature/ Research: The QUASAR Project The Mathematical Tasks Framework The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers. As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. We will now consider each phase of the math task framework.The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers. As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. We will now consider each phase of the math task framework.

    24. Math & Science Collaborative Linking to Literature/ Research: The QUASAR Project The Mathematical Tasks Framework The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers. As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. We will now consider each phase of the math task framework.The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers. As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. We will now consider each phase of the math task framework.

    25. Math & Science Collaborative Linking to Literature/ Research: The QUASAR Project The Mathematical Tasks Framework The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers. As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. We will now consider each phase of the math task framework.The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers. As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. We will now consider each phase of the math task framework.

    26. Math & Science Collaborative To highlight how the math task framework plays out in the United States – this TIMSS research shows what types of task we use in the United States as well as how they are implemented. Although we are in line with other high achieving countries in terms of the number of high level tasks we use, we do not implement any of them at a high level. We tend to take the struggle out of the mathematics in our country. The results of the recent TIMSS video study provide additional evidence of the relationship between the cognitive demands of mathematical tasks and student achievement. In this study, a random sample of 100 8th grade mathematics classes from each of six countries (Australia, the Czech Republic, Hong Kong, Japan, the Netherlands, Switzerland) and the United States, were videotaped during the 1999 school year. The six countries were selected because each performed significantly higher than the U.S. on the TIMSS 1995 mathematics achievement test for eighth grade (Stigler & Hiebert, 2004). The study revealed that the higher-achieving countries implemented a greater percentage of making connections tasks in ways that maintained the demands of the task. With the exception of Japan, higher-achieving countries did not use a greater percentage of high-level tasks than in the U.S. All other countries were, however, more successful in not reducing these tasks into procedural exercises. Hence, the key distinguishing feature between instruction in the U.S. and instruction in high achieving countries is that students in U.S. classrooms “rarely spend time engaged in the serious study of mathematical concepts” (Stigler & Hiebert, 2004, p. 16). Approximately 17% of the problem statements in the U.S. suggested a focus on mathematical connections or relationships. This percentage is within the range of many higher-achieving countries (i.e., Hong Kong, Czech Republic, Australia). Virtually none of the making-connections problems in the U.S. were discussed in a way that made the mathematical connections or relationships visible for students. Mostly, they turned into opportunities to apply procedures. Or, they became problems in which even less mathematical content was visible (i.e., only the answer was given). Other findings from the TIMSS research are addressed on the next slides.To highlight how the math task framework plays out in the United States – this TIMSS research shows what types of task we use in the United States as well as how they are implemented. Although we are in line with other high achieving countries in terms of the number of high level tasks we use, we do not implement any of them at a high level. We tend to take the struggle out of the mathematics in our country. The results of the recent TIMSS video study provide additional evidence of the relationship between the cognitive demands of mathematical tasks and student achievement. In this study, a random sample of 100 8th grade mathematics classes from each of six countries (Australia, the Czech Republic, Hong Kong, Japan, the Netherlands, Switzerland) and the United States, were videotaped during the 1999 school year. The six countries were selected because each performed significantly higher than the U.S. on the TIMSS 1995 mathematics achievement test for eighth grade (Stigler & Hiebert, 2004). The study revealed that the higher-achieving countries implemented a greater percentage of making connections tasks in ways that maintained the demands of the task. With the exception of Japan, higher-achieving countries did not use a greater percentage of high-level tasks than in the U.S. All other countries were, however, more successful in not reducing these tasks into procedural exercises. Hence, the key distinguishing feature between instruction in the U.S. and instruction in high achieving countries is that students in U.S. classrooms “rarely spend time engaged in the serious study of mathematical concepts” (Stigler & Hiebert, 2004, p. 16). Approximately 17% of the problem statements in the U.S. suggested a focus on mathematical connections or relationships. This percentage is within the range of many higher-achieving countries (i.e., Hong Kong, Czech Republic, Australia). Virtually none of the making-connections problems in the U.S. were discussed in a way that made the mathematical connections or relationships visible for students. Mostly, they turned into opportunities to apply procedures. Or, they became problems in which even less mathematical content was visible (i.e., only the answer was given). Other findings from the TIMSS research are addressed on the next slides.

    27. Math & Science Collaborative Boaler & Staples (2008) The success of students in the high-achieving school was due in part to the high cognitive demand of the curriculum and the teachers’ ability to maintain the level of demand during enactment through questioning. The previously cited research focused on middle school classrooms. The Boaler & Staples study focused on high school classrooms and produced similar findings. In a recent study Boaler and Staples report the results of a five-year longitudinal study of 700 students in three high schools. Students at one high school, Railside, used a standards-based curriculum designed by teachers around key concepts (e.g., What is a linear function?) and featuring groupworthy tasks drawn from curricula such as College Preparatory Mathematics (CPM) and the Interactive Mathematics Program (IMP) and a textbook of activities that use algebra manipulatives. The students at the other two high schools used conventional curricula. The researchers report that students at Railside achieved at higher levels than those at other schools. In particular by the end of the second year Railside students significantly outperformed all other students in a test of algebra and geometry. Boaler and Staples indicate that one factor contributing to the success of students at Railside was the high cognitive demand of the curriculum and the teachers’ ability to maintain the level of demand during enactment through questioning. The previously cited research focused on middle school classrooms. The Boaler & Staples study focused on high school classrooms and produced similar findings. In a recent study Boaler and Staples report the results of a five-year longitudinal study of 700 students in three high schools. Students at one high school, Railside, used a standards-based curriculum designed by teachers around key concepts (e.g., What is a linear function?) and featuring groupworthy tasks drawn from curricula such as College Preparatory Mathematics (CPM) and the Interactive Mathematics Program (IMP) and a textbook of activities that use algebra manipulatives. The students at the other two high schools used conventional curricula. The researchers report that students at Railside achieved at higher levels than those at other schools. In particular by the end of the second year Railside students significantly outperformed all other students in a test of algebra and geometry. Boaler and Staples indicate that one factor contributing to the success of students at Railside was the high cognitive demand of the curriculum and the teachers’ ability to maintain the level of demand during enactment through questioning.

    28. Math & Science Collaborative Conclusion Not all tasks are created equal -- they provided different opportunities for students to learn mathematics. High level tasks are the most difficult to carry out in a consistent manner. Engagement in cognitively challenging mathematical tasks leads to the greatest learning gains for students. Professional development is needed to help teachers build the capacity to enact high level tasks in ways that maintain the rigor of the task.

    29. Math & Science Collaborative Link to SAS How do you see the Math Task Framework linking to the Standards Aligned System? Ask participants how they see the Math Task Framework linking to the Standards Aligned System. Possible responses include: Task Analysis Guide could be used to sort the types of tasks embedded in the lesson plans in the materials and resources segment of SAS. These types of tasks should not be the add-ons but rather the content of core instruction (RTI Tier I) The content of the selected task should be linked to the standards and curriculum framework segments of SAS Participants should consider the instruction and assessment segments of SAS during the setting up and implementation sections of the Math Task Framework.Ask participants how they see the Math Task Framework linking to the Standards Aligned System. Possible responses include: Task Analysis Guide could be used to sort the types of tasks embedded in the lesson plans in the materials and resources segment of SAS. These types of tasks should not be the add-ons but rather the content of core instruction (RTI Tier I) The content of the selected task should be linked to the standards and curriculum framework segments of SAS Participants should consider the instruction and assessment segments of SAS during the setting up and implementation sections of the Math Task Framework.

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