Student Understanding of the Concept of Limit

1. Student Understanding of the Concept of Limit Rob Blaisdell Center for Research on STEM Education University of Maine, Orono

2. Research Questions What do teachers know about student difficulties with the concept of limit in calculus? What difficulties do students have with the concept of limit in calculus? • Teacher knowledge of student ideas about limit • Fresh data • Pick tasks from various sources to use in survey • Unexpected data

3. Introduction • Knowledge of student thinking about limit can inform and improve practice • Calculus is foundational and difficult Oehrtman, M. (2002), Williams, S. (1991), Davis, R., & Vinner, S. (1986) • Questions used may affect student responses Carlson, M. (1998) • Data may be influenced by question representation Carlson, M. (1998), Hawkins, J., Thompson, J., Wittmann, M., Sayre, E., & Frank, B. (2010)

4. today • What is known about student thinking about limit • Research design • We’ll do some math • Present survey tasks, data and findings • Conclusions and implications • Discussion and questions

5. Student thinking about limit • Student models of thinking as suggested in the literature: • Limit as a boundary (cannot pass) • Limit acting as an approximation Conflicting example: , evaluate • Limit as unreachable (cannot reach) Conflicting example: Ex. f(x) = 3 evaluate • Limit as dynamic (theoretical and practical) i.e. “idealization of evaluating the function at points successively closer to a point of interest” • Limit as formal Williams (1991), Tall, D. & Vinner, S. (1981), Orton, A. (1983), Davis, R., & Vinner, S. (1986), Oehrtman, M. (2002)

6. Student thinking about limit • Students may use multiple models to solve problems • Students resist changing their model of understanding Williams (1991), Tall, D. & Vinner, S. (1981), Orton, A. (1983), Davis, R., & Vinner, S. (1986)

7. Research design • 111 students in first semester calculus course • Students completed survey mid-semester • Survey conducted at a public university in the northeast

8. Research design • Tasks for the student surveys were: • Taken from literature Williams (1991), Oehrtman, M. (2002) • Modified from literature Bezuidenhout, J. (2001) • Created

9. Research design • Type of question and representation of chosen tasks • Describe what limit means, definition task • Consider two different multiple choice, mathematical notation tasks • Answer questions about and explain two graphical representations of limit, graphical tasks • Answer true/false, multi-part question involving various definitions of limit, definition task

10. Research design • Responses to tasks taken from other researchers’ studies were analyzed using the same approach Williams (1991), Bezuidenhout, J. (2001) • Ground Theory approach was used to analyze tasks for student model of thinking Strauss, A., & Corbin, J. (1990) • Responses were also examined for inconsistencies among and between different representations. Bezuidenhout, J. (2001)

11. Let’s do some math • Do problems number #3 and #5 DISCUSS WITH SOMEONE NEARBY: • If a student answers number #3 correctly how likely is it that the student would answer #5 correctly? • If a student answers number #5 correctly how likely is it that the student would answer #3 correctly?

12. Representation comparison 3) Given an arbitrary function f, if , what is f (3)? a. 3 b. 4 c. It must be close to 4. d. f (3) is not defined. e. Not enough information is given. ANS:______________ From the CCI – Calculus Concept Inventory 5) a) What is the value of the function at x = 2? b) How did you figure out your answer to (a)? c) Does the function have a limit as x approaches 2? d) How did you figure out your answer in (c)?

13. Graphical tasks 5) 6) a) What is the value of the function at x = 2? b) How did you figure out your answer to (a)? c) Does the function have a limit as x approaches 2? d) How did you figure out your answer in (c)? (Note: a through d were asked for each graph)

14. Mathematial Notation Tasks 3) Given an arbitrary function f, if , what is f (3)? a. 3 b. 4 c. It must be close to 4. d. f (3) is not defined. e. Not enough information is given. ANS:_________________ From the CCI – Calculus Concept Inventory (numbers modified)

15. Data Correct Responses for Particular Questions Graphical vs. Mathematical Notation Responses #3 – Mathematical Notation Question #5 & #6 – Graphical Questions

16. Mathematial Notation Tasks 4) In this question circle the number in front of your choice(s). Which statement(s) in A to E below must be true if f is a function for which ? Circle letter F if you think that none of them are true. A. f is continuous at the point x = 2 B. f (x) is defined at x = 2 C. f (2) = 3 D. E. f (2) exists F. None of the above-mentioned statements. Bezuidenhout, J. (2001) - Modified

17. Data comparison Bezuidenhout, J. (2001)

18. Definition tasks 7) Mark the following six statements about limits as being true or false. A. A limit describes how a function moves as x moves toward a certain point. B. A limit is a number or point past which a function cannot go. C. A limit is a number that the y-values of a function can be made arbitrarily close to by restricting x-values. D. A limit is a number or point the function gets close to but never reaches E. A limit is an approximation that can be made as accurate as you wish. F. A limit is determined by plugging in numbers closer and closer to a given number until the limit is reached. 8) Which of the above statements best describes a limit as you understand it? (Circle one) A B C D E F None Williams, S. (1991)

19. Data comparison Williams, S. (1991)

20. Data comparison Williams, S. (1991)

21. Data Multiple Model Analysis Contradictory Responses

22. Conclusions & Remaining Questions • Students did better with limit questions in graphical than notation or definition format • Low student correct responses to notation and definition questions • Inconsistent student responses to questions formated using the same and different representations • Students may have multiple models of the limit concept as suggested by other researchers

23. Implications • Question and representation selections may affect: • student model of thinking • what student difficulties arise • data Hawkins, J., Thompson, J., Wittmann, M., Sayre, E., & Frank, B. (2010), Carlson, M. (1998) • Fundamental conceptual difficulties with limit may affect: • student understanding of calculus • attitudes towards calculus in general.

24. References • Bezuidenhout, J. (2001). Limits and Continuity: Some Conceptions of First-year Students. International Journal of Mathematical Education in Science and Technology, 32:4, 487-500. • Carlson, M. (1998). A Cross-Sectional Investigation of the Development of the Function Concept. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Research in Collegiate Mathematics Education (Vol. 1, pp. 114-162). Washington, DC: American Mathematical Society. • Davis, R., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281-303. • Hawkins, J., Thompson, J., Wittmann, M., Sayre, E., & Frank, B. (2010). "Students’ Responses to Different Representations of a Vector Addition Question." Paper presented at the Physics Education Research Conference 2010, Portland, Oregon, July 21-22, 2010. • Monk, S. (1983). Representation in School Mathematics: Learning to Graph and Graphing to Learn. A Research Companion to Principals and Standards for School Mathematics, (Chapter 17). • Oehrtman, M. (2002). Collapsing Dimensions, Physical Limitations, and other Student Metaphors for Limit Concepts: An Instrumentalist Investigations into Calculus Students’ Spontaneous Reasoning. PhD Thesis, The University of Texas at Austin. • Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14, 235-250. • Smith, S.P. (2006). Representation in School Mathematics: Children’s Representationsof Problems A Research Companion to Principals and Standards for School Mathematics, (Chapter 18). • Tall, D. & Vinner, S. (1981). Concept Image and Concept Definition in Mathematics with Particular Reference to Limits and Continuity. Educational Studies in Mathematics, 12, 151-169. • Williams, S. (1991). Models of Limit Held by College Calculus Students. Journal for Research in Mathematics Education, 22, 219-236.

25. Discussion Question • The next phase of this project is to examine college mathematics instructors' knowledge of student thinking about limit. What questions might be asked of these instructors to tap into their knowledge of the student thinking, including their knowledge of the impact of these format differences on students' performance on tasks?

26. Discussion Questions • If interviews were to be conducted with students who took the survey, what questions might help uncover the sources of the discrepancies of how they respond to questions? • Are there additional questions or question formats that should be included in future surveys if the goal is to further examine these patterns in student responses?