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