1 From Theory to Practice: Teaching mathematics through problem solving Misfer Saud AlSalouli AlHasa Teachers’ College King Abdulaziz City for Science and Technology

2 Definition of teaching mathematics through problem solving As facilitating the classroom and striving to maximize opportunities for students to construct and understand concepts, and allowing discussions among students themselves and the teacher, creating an opportune learning environment, encouraging students to express different ideas, and assessing understanding (AlSalouli, 2005).

3 What would you do? Imagine that one of your students comes to class very excited. He tells you that he has discovered a mathematical rule that you have not told the class. He says that he had determined that as the perimeter of a figure increases, the area also increases. To “prove” that his discovery is true, he shows you these drawings:

4 What would you do? 4 cm 8 cm 4 cm Perimeter = 16 cm Perimeter = 24 cm Area = 16 cm² Area = 32 cm² What would you say to this student?

5 Characteristics of effective tasks:  What is problematic MUST be the mathematics  Tasks MUST be accessible to students  Tasks MUST require students to give justifications or explanations for their solutions and methods

6 Signposts for classroom teacher:  Allow mathematics to be problematic for students.  Focus on the methods used to solve problems.  Tell the right things at the right times.

7 Activity  Ask students to solve the same problem for a different amount of fencing. Ahemd’s class will raise rabbits. They have 24 meters of fencing to build a rectangular rabbit pen. If the students want their rabbits to have as much room as possible, how long would each side of the pen be?

8 Extending the activity  Ask students to explore a variation on the original problem.  What if the students have 24 meters of fencing, but the shape of the rabbit pen need not be rectangular? Does this change your answer? Why?

9 Extending the activity  Ask students to describe a general approach to determining the solution for any amount of fencing.  How would you determine the pen with the most room for any amount of fencing?

10 Extending the activity Change the focus of the activity from finding the shape with maximum area to exploring the nature of the relation between perimeter and area.

11 What Research & Theory Tell Us Traditional Teachers: Math is procedures & facts Understanding is NOT important Non-Traditional Teachers: Math is conceptual & creative Deep understanding is important

12 Benefits of teaching through problem solving  Motivating understanding  Promoting more understanding  Enhancing memory  Enhancing transfer  Influencing attitudes and beliefs  Developing autonomous learners

13 Challenges to using this approach  Need for support (other teachers, principals, parents, policy makers, even the students)  Traditional textbooks  Teacher math knowledge  Teacher must really believe!

14 Activity 44$ 30$ How much does a t-shirt cost? How much is a drink? Explain how you got your answer

15 Formalized algebra steps  2S + 2D = 44; so S + D = 22.  S + 3D = 30, which can be written as (S + D) + 2D = 30.  Through substitution, D = 30.  By solving for D, 2D = 8; so D = 4.  By solving for S, S + 4 = 22; so S = 18

16 Student 1:  What I did to figure this out is I took the top picture, and I took 1 t-shirt and 1 drink away. It was $22. so the bottom t-shirt and 1 drink cost $22, but you still had two drinks left that cost $8 total so 1 drink cost $4 and 1 t-shirt cost $18

17 Student 2:  To do this problem, I first crossed out the items common to both pictures. All that was left a shirt in the $44 box and a drink in the $30.00 box. This told me that the shirt cost $14.00 more than the drink (y). I used this information and the original $30 box to make this problem:

18 3y + (y +14) = 30 I modeled the problem like this: yyy y 14 = I crossed out the 14 from both sides and got: yyyy = 16 If 4y = 16, then y = 4. and = 18. Therefore, a drink costs $4 and a shirt costs $18. Follow student 2:

19 Teaching through problem solving occurs when problems are posed that are just within students’ reach, allowing them to struggle to find solutions. Conclusion

20 Thank you