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Thinking about Mathematical Tasks
- 1. Thinking about the ways students encounter mathematics
- 2. Why Focus on Tasks? Classroom instruction is generally organized and orchestrated around mathematical tasks The tasks with which students engage determine what they learn about mathematics and how they learn it The inability to enact challenging tasks well is what distinguished teaching in the U. S. from teaching in other countries that had better student performance on TIMSS
- 3. The Importance of Mathematical Tasks “…a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity.
- 4. The Importance of Mathematical Tasks But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a task for, and some means of, independent thinking.” Polya, 1973/1945
- 5. The Importance of Mathematical Tasks “Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.” thinking Stein, Smith, Henningsen, & Silver, 2000
- 6. Task-Focused Activities Distinguishing between high and low cognitive demand mathematics tasks Maintaining the cognitive demands of high-level tasks during instruction
- 7. Task-Focused Activities Distinguishing between high and low cognitive demand mathematics tasks
- 8. Distinguishing Popsicle Sticks Empty Number Line
- 9. Popsicle Sticks Example Say, “We have nine ones. Let’s put the nine ones using the popsicle sticks in the ones place (cup). The problem is 9 + 3. Let’s count three more popsicle sticks and add them to the nine ones.”
- 10. Popsicle Sticks Example Say, “Oh no! What’s wrong with the ones place?” Students respond, “There can only be nine ones in the ones place. We have to regroup by making a group of ten.” Count the ten popsicle sticks with the class and make a group of ten.
- 11. Popsicle Sticks Example Say, “How many groups of ten do we have in the tens place? How many ones do we have in the ones place? We have 9 + 3 which equals one group of ten and two ones.”
- 12. Popsicle Sticks Solve Summarize: “We learned addition with regrouping one digit, how we can do regrouping with popsicle sticks, and how we use regrouping every day.”
- 13. Jerry made a drawing using an empty number line to solve this problem Zena also used an empty number line but her solution does not look like Jerry’s solution. Explain how Jerry solved the problem. Explain how Zena solved the problem. Are both solutions correct? How do you know?
- 14. Reflecting….. How are the two tasks similar? How are the two tasks different?
- 15. Importance of Distinguishing Low Cognitive Demand Tasks Popsicles Sticks High Cognitive Demand Tasks Empty Number Line
- 16. Sorting Tasks You will need the complete set of the Task Sort Card set (File name: Task Cards (PV)082909). Before you continue, make sure you have each task as a separate sheet so you can move them individually as you sort them. Also make sure you have the Task Sort Chart to record your decisions.
- 17. Sorting Tasks Make two piles: those tasks that your group considers to be high cognitive demand tasks and those tasks that your group considers to be low cognitive demand tasks. As you sort, take notes on your thoughts about what makes a task high or low in terms of cognitive demand.
- 19. Some Features of High Cognitive Demand Tasks A high cognitive demand task begins where the students are (zone of proximal development; scaffolding) The problematic or engaging aspect of a high cognitive demand task is due to mathematics that students are to learn. A high cognitive demand task requires justifications and explanations for answers and methods.
- 20. Task-Focused Activities High cognitive demand tasks or activities are the vehicle through which the curriculum can be developed. Maintaining the cognitive demands of high-level tasks during instruction affects the learning that occurs.
- 21. Maintaining The Mathematical Tasks Framework Student Learning TASKS as they are summarized implemented set up in appearby the by students teachers curricular/ teacher and students instructional materials or are designed by teachers Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4
- 22. Maintaining The Mathematical Tasks Framework Student Learning TASKS as they are summarized implemented set up in appearby the by students teachers curricular/ teacher and students instructional materials or are designed by teachers Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4
- 23. Maintaining The Mathematical Tasks Framework Student Learning TASKS as they are summarized implemented set up in appearby the by students teachers curricular/ teacher and students instructional materials or are designed by teachers Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4
- 24. Maintaining The Mathematical Tasks Framework Student Learning TASKS as they are summarized implemented set up in appearby the by students teachers curricular/ teacher and students instructional materials or are designed by teachers Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4
- 25. Maintaining The Mathematical Tasks Framework Student Learning TASKS as they are summarized implemented set up in appearby the by students teachers curricular/ teacher and students instructional materials or are designed by teachers Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4
- 26. Maintaining The Mathematical Tasks Framework Student Learning TASKS as they are summarized implemented set up in appearby the by students teachers curricular/ teacher and students instructional materials or are designed by teachers Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4
- 27. Maintaining The Mathematical Tasks Framework Student Learning TASKS as they are summarized implemented set up in appearby the by students teachers curricular/ teacher and students instructional materials or are designed by teachers Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4
Editor's Notes
- First, classroom instruction is generally organized and orchestrated around mathematical tasks. That is, students’ day-to-day work in mathematics classrooms consists of working on a tasks, activities, or problems. For example, an analysis of 100 eighth grade lessons revealed that the delivery of content “was accomplished primarily by working through problems” (NCES, 2003, p. 144) .Second, the tasks with which students engage determines what they learn about mathematics and how they learn it. According to Doyle (1983, p.161), “tasks influence learners by directing their attention to particular aspects of content and by specifying ways of processing information.” Third, in the TIMSS video study involving seven countries including US (all who out performed US on TIMSS) although the percentage of hl tasks used was within range of other countries (17%) NONE of the tasks were enacted at a high level.
- Refer participants to the blue sheet -- which contains these two problems. Give participants a few minutes to review the problems and to indicate how they are different or how the same. Get at the fact that content same - processes required to solve are different. We call these high level vs. low level tasks.
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