1. Using Mathematical Tasks To Provoke Intellectual Needs Kien Lim University of Texas at El Paso kienlim@utep.edu 57th Annual Conference for the Advancement of Mathematics Teaching July 16, 2010

2. What Constitutes a Good Mathematical Task? What is Intellectual Need? Why is it Important? How does an Intellectual-need- provoking Task Facilitate Mathematical Learning? ObjectivesObjectives

3. Sequence Of Activities 1.Find out what constitute a good mathematical task 2.Solve a few problems 3.Discuss 4.Conclude One More Problem?

4. What is “Mathematics”? Old-view Mathematics Collection of definitions, formulas, rules, and procedures The focus is on the “how” and the “answer” Students are taught the procedure and then practice it New-view Mathematics A subject that involves thinking and sense-making The focus includes the “why” and meaning Students engage in problem-solving, making connection, justifying, representing, and communicating

5. Purpose of Classroom Tasks To engage learners in thinking, sense- making, reflecting, and abstracting To engender discussions among learners to learn “new” mathematical knowledge To provide learners an opportunity to practice what they have learned Thompson, Carlson, and Silverman (2007)

6. Characteristics of a Good Task Mathematically driven Requires justifications and explanation Intrinsic to students Van de Walle (2003)

7. Tasks that Motivate Students Extrinsically Motivating  Something new (e.g., manipulatives)  Fun and exciting (e.g., game) Intrinsically Motivating  Intriguing

8. What Leads to Mathematical Learning? “Students are most likely to learn when they see a need for what we intend to teach them, where by ‘need’ is meant intellectual need, not social or economic need.” (Harel, 1998) “For students to learn what we intend to teach them, they must have a need for it, where by ‘need’ is meant intellectual need, not social or economic need.” (Harel, 2007)

9. Gremlins are rather smart. Although they have four fingers in each hand, they are able to represent 25 different numbers, from 0 to 24, with two hands. Can you figure out how they do it? Problem #1 for You

10. Do you find the task interesting? Did you experience some sort of puzzlement? What do you think is the mathematical concept for which this task is trying to provoke? Follow-up Questions

11. The line segment represents 1 2 / 3 km. Extend the line to represent 3 3 / 4 km. Be as accurate as you can without measuring the actual length of original line segment. Problem #2 for You 1 2 / 3 km

12. What do you think is the math concept for which this task is trying to provoke? What key ideas are necessary for solving this problem? Follow-up Questions #2 1 2 / 3 km 1 km How? Cut into 5 pieces. 1 km 3 / 4 km Why 5? 1 2 / 3 = 5 / 3 Referent unit (1km) Mixed num. - improper fraction conversion

13. Consider these 4 products of numbers: Problem #3 for You Without computing the actual value of each product, identify those products that have the same value. (i)2  5 2  6  11  12 (ii) 3  11  25  44 (iii) 5  12 2  55 (iv) 5  10  22  33

14. What do you think is the math concept for which this task is trying to provoke? Follow-up Question #3 = 2 4  3 2  5 2  11 = 2 2  3  5 2  11 2 = 2 4  3 2  5 2  11 = 2 2  3  5 2  11 2 (i)2  5 2  6  11  12 (ii) 3  11  25  44 (iii) 5  12 2  55 (iv) 5  10  22  33

15. How Does Intellectual-Need- Provoking Task Facilitate Learning? Students encounter a problematic situation due to the limitation of their existing knowledge They experience a desire to resolve the situation Their resolution of the situation may lead to construction of new knowledge or modification of existing knowledge

16. In Selecting a Need-Provoking Task, What Questions Can We Ask Ourselves? What mathematical learning can the task potentially provoke? Is the task intrinsic to students? Do students experience a limitation of their existing knowledge and a need for the new math idea in order to solve the problem?

17. Recapitulation What Problems Have We Worked On Today? 1. The Gremlin Problem 2. Extending 1 2 / 3 km Line Problem 3. Finding Equal Product Problem What Concept do these Problems Seek to Provoke? Place-Value Ruferent Unit & Procedure for Converting Mixed # to Improper Fraction Prime Factorization

18. Factors to Consider When Designing Need-Provoking Tasks The challenges associated with the math idea that we want our students to learn A typical student’s current knowledge Connection to past activities and future topics Classroom norms and practices

19. As a Math Teacher, You Can … Avoid problem-free tasks Provide opportunities for your students to experience the intellectual need for the math concepts that you want them to learn Select tasks that requires thinking, sense- making, exploring, justifying, and explaining

20. An Article Lim, K. H. (2009). Provoking intellectual need. Mathematics Teaching in the Middle School, 15(2), 92-99.

22. Presenter: Kien Lim Date: July 17, 2010 (Sat) Date: July 17, 2010 (Sat) Time: 11:45am – 12:45pm Time: 11:45am – 12:45pm Venue:101 A/B Venue:101 A/B The Hammer-and-Nail Phenomenon: Addressing Students’ Impulsive Disposition in Mathematics Classrooms

23. Thank You