Developing and Using Meaningful Math Tasks

1. Developing and Using Meaningful Math Tasks The Key to Math Common Core Take a moment to record on a sticky: What is a meaningful Math Task?

2. Norms Courtesy • Be on time • Cell phones on silent, vibrate, or off • Be mindful of side-bar conversations • Focus on the task at hand • Collaborative • Promote a sense of inquiry • Frame meaningful questions • Pay attention of self and others • Assume positive intentions • Be reflective

3. Today’s Outcomes • Participants will have a better understanding of what they need to expect from their students in math. • Participants will have a better understanding how to select and set up a challenging math task. • Participants will have a better understanding of how to analyze a problem or task. • Participants will have a better understanding of how to increase the cognitive demands of a math task.

4. Believe! The power to believe in your students is the cornerstone for change. The greater danger for most of us lies not in setting our aim too high and falling short but in setting our aim too low and achieving our mark.Michelangelo

5. 3 Beliefs • Students are capable of brilliance. • Understanding takes time. • There is more than one way. Remember the learning community is a culture, not a structure!

6. What are we asking our students to: • Think about? • Talk about? • Understand? “Never memorize something you can look up” Einstein

7. Challenging Tasks • They are about understanding, not coverage and should be our goals as teachers. • Challenging tasks generate and demonstrate understanding. • We need to devote our class time to making meaning of those juicy tasks to cultivate comprehension.

8. Selecting a Math Task • What are your goals for this lesson? • What mathematical content and processes do you hope students will learn from their work on this task? • In what ways does this task build on students’ previous knowledge? • What definitions, concepts, or ideas do students need to know in order to begin to work on the task?

9. Setting Up a Math Task • What are all the ways the task can be solved? • How will you ensure that students remain engaged in the task? • What are your expectations for students as they work on and complete this task? • How will you introduce students to the activity so as not to reduce the demands of the task? • What will you hear that lets you know students understand the task?

10. Supporting Students’ Exploration • What questions will you ask to focus their thinking? • What will you see or hear that lets you know how students are thinking about mathematical ideas? • What questions will you ask to assess students’ understanding?

11. Comparing Two Math Tasks • Martha’s Carpeting Task • The Fencing Task Think privately about how you would go about solving each task (solve them). (See the TASK handout.)

12. Comparing Two Math Tasks • How are Martha’s Carpeting Task and the Fencing Task the same and how are they different?

13. Mathematical Tasks:A Critical Starting Point for Instruction Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking. Stein, Smith, Henningsen, & Silver, 2000

14. Mathematical Tasks:A Critical Starting Point for Instruction There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics. Lappan & Briars, 1995

15. Mathematical Tasks:A Critical Starting Point for Instruction 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

16. High Level vs. Low Level Tasks:Different Opportunities for Thinking • Read Vignettes 1 (Mr. Patrick) and 2 (Mrs. Fox) (see VIGNETTE handout) • Consider the following question: What opportunities did students have to think and reason in each of the two classes?

17. High Level vs. Low Level Tasks:Different Opportunities for Thinking • Low-Level Tasks – memorization (e.g., formulas for area and perimeter) – procedures without connections (e.g., Martha’s Carpeting Task) • High-Level Tasks– procedures with connections (e.g., use a drawing to explain why area = length x width)– doing mathematics (e.g., The Fencing Task)

18. Task Analysis • Use the Task Analysis Guide: think of how high- and low-level tasks differ and why the differences matter. • What are your own experiences with using high-level tasks ? • What extent does your textbook support using high- level tasks? • What are other possible sources of such tasks?

19. Do High Level Tasks Decline? • Read Vignette 3 (Mrs. Jones) (see VIGNETTE handout) • Consider the following question: What factors might account for the decline of the task in Mrs. Jones classroom?

20. Conclusion • Not all tasks are created equal—they provide 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.

21. Increase the Cognitive Demand of the Task • Increase complexity • Introduce ambiguity • Synthesize strand of mathematics • Invite conceptual connections • Require explanation and justification • Propose solutions that reveal misconceptions or common errors

22. Does the task have all of the following? • Multiple entry points • Various possible approaches • A need for higher order thinking • Opportunities to synthesize, justify, and explain

23. What is 6 + 4 • How can you make this into a challenging task? • Ten children went to the movie. How many were girls? How many were boys? Explain your answer. Could there be other answers?

24. Change this into a Challenging Task. • A rectangle has a length of 4 units and a width of 3 units. What is the area? • Create rectangles with an area of 24 square units. How are they alike and different? Are their perimeters the same? Explain your observations.

25. Change this into a Challenging Task. • Three children shared a pizza. They each had the same amount. What fraction did each child have? • Three children are sharing a pizza. How might they share it? What fraction of the pizza could each child get? Justify your answer.

26. Make a task more challenging by changing the questions you ask. • Low- Three children fairly shared a box of 15 chocolates. How many chocolates are in each child’s share? • Middle- Same problem. How many chocolates are in each child’s fair share? Explain how you answered the question. • High- Same problem. Same first question. How much of the box of chocolates is one child’s share? How many times larger than one child’s share is the box of chocolates? Explain how you answered the questions.

27. Investigations • Find a task in your Investigations manual. • How can you make it more cognitively challenging for students? • Is the rigor of the tasks you are doing with students high enough?

28. Invite students to: • Describe their process • Reflect on their decisions • Explain their vigilance • Confirm their thinking • Make connections • Promote discourse

29. Questions or Concerns? Fill out exit slip