1. VES Mathematics PLC Session 2: We’ve Got Problems 11/13/12

2. Speaking of Problems... How do we know a good problem?

3. A Good Problem...  Involves a question that cannot be immediately answered  Is challenging to the learner  Holds the learner’s interest  Might have more than one answer  Might have only one answer, but many different approaches to the solution  Is often connected to real life

4. If it can be solved using a procedure or algorithm it is not a “problem” it is an exercise.

5. What is a problem at one time to one person may be an exercise to another.

6. Student Behaviors The Problem The Problem Teacher Behaviors Problem Solving

7. Classroom Environment  Risk taking  Provide enough time  Respect for each other’s thinking  Opportunity to discuss and challenge solutions  Values perseverance

9. The Role of the Teacher is crucial...  in choosing the problem  in determining how to present the problem  in determining how students will work  in order to assess what the students have learned  for facilitating student communication

10. Where do you find problems?

11. Intermediate Problem The John Hancock Building in Chicago is 1,127 feet tall. Another skyscraper in Chicago, the Amoco Building, is 1,136 feet tall. Which building is taller? Primary Problem There are 18 students in Mrs. Smith’s class. She wants to put them into pairs. How many pairs are there?

12. What challenges did you face as you rewrote the problem?

13. There is no other decision that teachers make that has a greater impact on students’ opportunity to learn and on their perceptions about what mathematics is than the selection of the tasks (or problems) with which the teacher engages the students in studying mathematics. Lappan and Briars, 1995

14. There are 25 sheep and 5 dogs in a flock. How old is the Shepherd?

16. How many rectangles are there on a standard 8 X 8 checkerboard? Count only those rectangles (including squares) whose sides lie on grid lines. For example, there are nine rectangles on a 2 X 2 board. 12 34 5 6 978 Problem Solving: Sample

17. The Value of this problem Deciding what determines a rectangle Examining a simpler problem Adding sequence of consecutive integers Problem lends itself to discovering mathematical applications based on number concepts Multiple correct solution strategies can be applied to arrive at a solution Variety of solutions

18. There are 9 cows in the barn. How many eyes are there?

20. Sergi’s basketball team is traveling for a tournament. There are 13 kids on the team. If each car holds 3 kids, how many cars are needed for the team to get to the tournament?