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Student Understanding of the Concept of Limit. Rob Blaisdell Center for Research on STEM Education University of Maine, Orono. Research Questions. What do teachers know about student difficulties with the concept of limit in calculus? - PowerPoint PPT Presentation
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Student Understanding of the Concept of Limit
Rob BlaisdellCenter for Research on STEM Education
University of Maine, Orono
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 limitFresh dataPick tasks from various sources to use in
surveyUnexpected data
Knowledge of student thinking about limit can inform and improve practice
Calculus is foundational and difficultOehrtman, M. (2002), Williams, S. (1991), Davis, R., & Vinner, S. (1986)
Questions used may affect student responsesCarlson, M. (1998)
Data may be influenced by question representationCarlson, M. (1998), Hawkins, J., Thompson, J., Wittmann, M., Sayre, E., & Frank, B.
(2010)
Introduction
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
today
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)
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)
Student thinking about limit
111 students in first semester calculus course
Students completed survey mid-semester
Survey conducted at a public university in the
northeast
Research design
Tasks for the student surveys were:
Taken from literature
Williams (1991), Oehrtman, M. (2002)
Modified from literature
Bezuidenhout, J. (2001)
Created
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
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)
Research design
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?
Let’s do some math
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)?
Representation comparison3) 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) 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)
Graphical 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)
Mathematial Notation Tasks
Correct Responses for Particular Questions
Graphical vs. Mathematical Notation Responses
#3 – Mathematical Notation Question
#5 & #6 – Graphical Questions
Data
Q#3 - Notation
Q#4 - Notation
Q#5 - Graphic
al
Q#6 - Graphical
Q#5 & Q#6Graphical
Q#8 – Definition
19.8% 1.8% 67.5%
65.7% 55.7% 21.6%
Correct Responses #3 Correct #5 and 6 Correct
#5 or #6 Correct 22.1 % X
#5 and #6 Correct 28.1 % X
#3 Correct X 85.7 %
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
Mathematial Notation Tasks
Responses selected on Q#4Numbe
r Pct.Numbe
r Pct.
Bezuidenhout n=100 Blaisdell n=111
Selecting 4A but not 4B 18 18 19 17
Selecting 4A but not 4C 19 19 19 17
Selecting 4C but not 4B 20 20 29 26
Selecting 4E but not 4A 7 7 24 22
Selecting 4E but not 4B 16 16 20 18
Data comparison
Bezuidenhout, J. (2001)
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)
Definition tasks
Student Concept ModelTrue
False
Best
True
False
Best
All students (in pct.) Williams N=341 Blaisdell N=111
1 – Dynamic/Theoretical 80 19 30 88 12 35
2 - Boundary 33 67 3 18 82 2
3 - Formal 66 31 19 78 22 22
4 - Unreachable 70 30 36 53 47 24
5 - Approximation 49 50 4 66 34 2
6 – Dynamic/Practical 43 57 5 69 31 12
Data comparison
Williams, S. (1991)
Student Concept ModelTrue
False
Best
True
False
Best
Selecting #3 as true (in pct.)
Williams N=226 Blaisdell N=87
1 – Dynamic/Theoretical 82 18 29 90 10 34
2 - Boundary 28 72 2 12 88 0
3 - Formal 100 0 29 100 0 26
4 - Unreachable 65 35 31 52 48 21
5 - Approximation 53 46 3 71 39 2
6 – Dynamic/Practical 45 55 5 71 39 11
Data comparison
Williams, S. (1991)
Data
Multiple Model Analysis
Contradictory Responses
Questions One model choice
Question #7 3.6 %
Question #7 and #8 0.9 % (correct)
Mathematical Notation
Contradictory Responses
Questions #3 and #4 18 %
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
Conclusions & Remaining Questions
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.
Implications
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.
References
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?
Discussion Question
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?
Discussion Questions