1. 2010-2011 Workshop Series for YISD Teachers of 6 th Grade Math 2010-2011 Workshop Series for YISD Teachers of 6 th Grade Math Fostering Conceptual Understanding and Mathematical Thinking Kien Lim Kien Lim Dept. of Mathematical Sciences, UTEP

2. Goals: Strengthen our mathematical knowledge for teaching concepts related to fractions, decimals, percents, and ratios Get a sense of what it takes to teach in a manner to foster conceptual understanding and mathematical thinking Increase our commitment to help our students develop mathematical thinking

3. Objectives for Workshop #1: Strengthen our mathematical knowledge for teaching concepts related to fractions (comparing fractions, addition/subtraction, etc.) Get a sense of how we can use problems as means for students to develop (a) mathematical understanding (i.e., concepts) (b) mathematical thinking (i.e., process) Understand how problems are designed and sequenced to extend prior knowledge and introduce “new” concepts/ideas/procedures

4. Plan of Activities: Introduce Problem-based Approach Solve, analyze and improve a sequence of problems involving fractions Highlight the key points in the article: Developing Understanding through Problem Solving Making connections among fractions, decimals, and percents Discuss needs for next workshops

5. Why is Understanding Mathematics so Important? One’s knowledge are interconnected and one can reconstruct them when needed. One can apply those ideas flexibly to new situations. One can apply those ideas flexibly to new situations. One feels great! Really understanding something is the most satisfying experiences one can have One feels great! Really understanding something is the most satisfying experiences one can have

6. How Can Classrooms be Designed to Promote Understanding?

7. “Where To Sit” Problem

8. Suppose you are a chocolate lover and is very popular among your friends. During lunch break, two of your best friends, Katrina and Claudia, are each trying to get you to sit in her table.

9. Suppose you are a chocolate lover and is very popular among your friends. During lunch break, two of your best friends, Katrina and Claudia, are each trying to get you to sit in her table. On each table, there is a chocolate bar to be shared equally among students sitting on the table. Mimi, Nancy, and Lanna are in Katrina’s table. Anita is in Claudia’s table. Claudia Anita ? Katrina Mimi Nancy Lanna ?

10. Suppose you are a chocolate lover and is very popular among your friends. During lunch break, two of your best friends, Katrina and Claudia, are each trying to get you to sit in her table. On each table, there is a chocolate bar to be shared equally among students sitting on the table. Mimi, Nancy, and Lanna are in Katrina’s table. Anita is in Claudia’s table. Katrina says she will give you an additional 1/2 of a chocolate bar if you sit in her table. Claudia says she will give you 1/3 of a chocolate bar if you sit in her table. Claudia gives you 1/3 bar Anita ? Katrina gives you 1/2 bar Mimi Nancy Lanna ?

11. Suppose you are a chocolate lover and is very popular among your friends. During lunch break, two of your best friends, Katrina and Claudia, are each trying to get you to sit in her table. On each table, there is a chocolate bar to be shared equally among students sitting on the table. Mimi, Nancy, and Lanna are in Katrina’s table. Anita is in Claudia’s table. Katrina says she will give you an additional 1/2 of a chocolate bar if you sit in her table. Claudia says she will give you 1/3 of a chocolate bar if you sit in her table. You don’t really care which table you sit as long as you get to have more chocolate. Which table will you sit? (Explain your reasoning. You may draw diagrams.) Handout Claudia gives you 1/3 bar Anita ? Katrina gives you 1/2 bar Mimi Nancy Lanna ?

12. How do you think your students will solve this problem? Claudia gives you 1/3 bar Anita ? Katrina gives you 1/2 bar Mimi Nancy Lanna ?

13. Questions for Discussion Do you think students will find this problem interesting? Why, or why not? Are your students able to solve this problem on their own? If not, what can we do?

14. Classroom Version of the Problem What is the advantage of splitting the problem into two parts and asking 6 th graders to solve only part a?

15. Classroom Version of the Problem

16. What mathematical conceptions can this problem help students develop? What types of ways of thinking can this problem help to foster? What types of ways of thinking can this problem help to foster? Handout

17. Questions for Discussion What attributes of this problem are good? Why? What attributes of this problem are good? Why? Why is a breakable-part-type of chocolate bar used for this problem? Why is a breakable-part-type of chocolate bar used for this problem? Why were those numbers chosen? Why were those numbers chosen? Should we provide students with any manipulatives or materials to facilitate their learning and/or problem solving? If so, what? Should we provide students with any manipulatives or materials to facilitate their learning and/or problem solving? If so, what? Handout What will be a good follow-up problem? What will be a good follow-up problem? An online applet to demonstrate how fractions are added and subtracted: http://nlvm.usu.edu/en/nav/frames_asid_106_g_3_t_1.html http://nlvm.usu.edu/en/nav/frames_asid_106_g_3_t_1.html

18. 2.Three more friends just joined Claudia and Anita. Claudia knows you love chocolate and says she will give you 2/3 of a chocolate bar (instead of 1/3). When Katrina hears that, Katrina says she will give 3/5 of a chocolate bar (instead of 1/2). Now which table will you sit and what fraction of a chocolate bar will you get? (Explain your reasoning. You may draw diagrams.) Handout Katrina gives you 3/5 bar Mimi Nancy Lanna ? Claudia gives you 2/3 bar Anita ? Friend 1 Friend 2 Friend 3

19. Questions for Discussion What do you think is the intent of this follow-up problem? What mathematical concepts and ways of thinking can be fostered through this problem?

20. 3.Your teacher puts another chocolate bar in each table. Now which table will you sit and what fraction of a chocolate bar will you get? (Explain your reasoning. You may draw diagrams.) Handout Katrina gives you 3/5 bar Mimi Nancy Lanna ? Claudia gives you 2/3 bar Anita ? Friend 1 Friend 2 Friend 3

21. Questions for Discussion What do you think is the intent of this follow-up problem? What mathematical concepts and ways of thinking can be fostered through this problem?

22. 4. Suppose the situation is as shown in the diagram. Which table will you sit and what fraction of a chocolate bar will you get? (Explain your reasoning. You may draw diagrams.) Handout Alex will give you 1/5 bar Friend 1 Friend 4 ? Friend 5 Friend 2 Friend 3 John will give you 1/8 bar Friend 6 Friend 7 ?

23. Questions for Discussion What do you think is the intent of this follow-up problem? What mathematical concepts and ways of thinking can be fostered through this problem?

24. 5. Suppose the situation is as shown in the diagram. Which table will you sit and what fraction of a chocolate bar will you get? (Explain your reasoning. You may draw diagrams.) Handout Jose will give you 1/4 bar Friend 3 Friend 4 ? Friend 5 Mario will give you 2/5 bar Friend 1 Friend 2 ?

25. Questions for Discussion What do you think is the intent of this follow-up problem? What mathematical concepts and ways of thinking can be fostered through this problem?