1. Worthwhile Tasks

2. Four Fours and Operations Problem Use four 4s and some symbols +, x, -, ÷,and ( ) to give expressions for the whole numbers from 0 through 9: for example 5 = (4 x 4 + 4) ÷ 4. Solve the problem. Sharing solutions

3. Analyzing the Problem What mathematics did you use to solve the problem? When would you use this type of problem? What related problems are usually found in textbooks? What makes this task a worthwhile task?

4. Turning Traditional Textbook Problems into Open-Ended Problems Kabiri and Smith Article

5. Turning Traditional Textbook Problems into Open-Ended Problems (Jigsaw) Everyone: Introduction and conclusion Group 1: Number and Operations Group 2: Algebra Group 3: Geometry Group 4: Measurement Group 5: Data Analysis and Probability

6. On the transparency, each group should address the following: Discuss the major points from the section. Give an example of how a traditional problem was changed into a more open- ended problem. Tell how the section may impact the instruction of each member of the group

7. Summary of the “Worthwhile Tasks” Standard Professional Standards for Teaching Mathematics, NCTM, 1991.

8. The teacher of mathematics should pose tasks that are based on --- Sound and significant mathematics; Knowledge of students’ understandings, interests, and experiences; Knowledge of the range of ways that diverse students learn mathematics; And that engage students’ intellect; Develop students’ mathematical understandings and skills; Stimulate students to make connections and develop a coherent framework for mathematical ideas; National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM

9. Cont’d Call for problem formulation, problem solving, and mathematical reasoning; Promote communication about mathematics; Represent mathematics as an ongoing human activity; Display sensitivity to, and draw on, students’ diverse background experiences and dispositions; Promote the development of all students’ dispositions to do mathematics. Reference: National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM

10. How to Determine Worthwhile Tasks In selecting, adapting, or generating mathematical tasks, teachers must base their decision on three areas of concern: mathematical content, the students, and the ways in which students learn mathematics.

11. Mathematical Content Teachers should consider how appropriately the task represents the concepts and procedures entailed. Teachers must also use a curricular perspective, considering the potential of the task to help students progress in their cumulative understanding of a particular domain and to make connections among ideas they have studied in the past and those they will encounter in the future. Teachers must also assess what the task conveys about what is entailed in doing mathematics. Teachers must also consider how well a task helps in the development of appropriate skill and automaticity.

12. Students Teachers must consider what they know about students in deciding on the appropriateness of a given task. Teachers must consider what they know about students from psychological, cultural, sociological, and political perspectives. When selecting tasks, teachers must think about what their students already know and can do, what they need to work on, and how much they seem ready to stretch intellectually. Teachers must know their students interests, dispositions, and experiences.

13. Knowledge About Ways In Which Students Learn Mathematics The mode of activity, the kind of thinking required, and the way in which students are led to explore the particular content all contribute to the kind of learning opportunity afforded by a task. Teachers must be aware of common misconceptions about mathematical concepts. Teachers should deliberately select tasks that provide them with windows into students’ thinking. Reference: National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM

14. Which task is better? 1. Find means, medians, and modes for given sets of data. 2. Decide whether to calculate means, medians, or modes as the best measures of central tendency, given particular sets of data and particular claims that you would like to make about the data. Then calculate those statistics, and finally explain and defend your decision.

15. Which task is better? 1. Find the area and perimeter of each rectangle: 2. Suppose you had 64 meters of fence with which you were going to build a pen for your dog, Bones. What are some different pens you can make if you use all the fencing? What is the pen with the least play space? What is the biggest pen you can make--the one that allows Bones the most play space? Which would be best for running?

16. Tasks that foster skill development even as students engage in problem solving and reasoning “The Fraction Game” Turn over a card to reveal your “target fraction”. Move the markers so that the sum of your moves is less than or equal to the “target”. The object of the game is to get all of the markers to the right side of the game board, using as few cards as possible. From illuminations.nctm.org

17. Where can we find worthwhile tasks?

18. Good ideas can be found in articles in journals, such as Mathematics Teaching in the Middle School, Mathematics Teacher, Teaching Children Mathematics Problems can also be found in Principles and Standards for School Mathematics, and the Navigations series from NCTM. Resources can also be found on-line: the NCTM web site (nctm.org) the Illuminations web site (illuminations.nctm.org) Change the emphasis of tasks from products to explanations Collect and use a variety of contexts where mathematics can be done