Unique Perceptions of Limits in Calculus Dr. Margaret Adams - - PowerPoint PPT Presentation

Unique Perceptions of Limits in Calculus

• Dr. Margaret Adams

Visiting Assistant Professor of Mathematics Francis Marion University

PhD in Curriculum & Instruction—Math Education

The University of North Carolina at Charlotte

Background

• Qualitative Dissertation Study (2 parts)
• Constructivist Epistemology based on work of

Jean Piaget and Richard Skemp’s notions of relational vs instrumental understanding.

• Original design of traditional (textbook) and

non-traditional tasks (from graph to function).

• In-depth interviews, problem solving,

interpreting limits and limiting behavior.

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 2

1. Phase I Pilot Study: In what ways do students perceive and interpret limits? 2. Phase II In-Depth Investigation: What unique perceptions and interpretations do students have about limits?

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 3

Significant Findings

Expected versus “unique” perceptions.

• appropriate schemas (correct as expected)
• altered schemas (incorrect/unique)

Perceptions of limit are based on one’s reality (neurophysiological and genetic epistemology)

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 4

Model of Understanding Limits

Shared Knowledge Structures Appear in the Intersections

Altered Schemas Instrumental and “Semi-Relational” Understanding

(non-traditional

• utcomes)

Appropriate Schemas Relational Understanding

(expected outcomes)

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 5

Significant Findings Unique Perceptions of Limits

• Definition of Limit & Limit Notation
• Role of the Domain
• Limits at a Point
• Limits at Infinity
• Infinite Limits (Limits that Do Not Exist)
• Limits that Do Not Exist (oscillations)

We start by considering 2 main graphs and examples of students’ work.

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 6

Graph of Piecewise Function

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 7

Linear, Discontinuous & Piecewise Functions

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 8

Definition of Limit

• Appropriate Schema: a limit is a number and can

exist if there is a hole or function value. (Brendon & NS)

• Altered Schemas:

– Limit can be a number or it can be infinity. It cannot exist if there is a hole but exists if it equals infinity. (Amanda) – Limit can only approach but not equal a function value. Therefore, a limit can only exist if there is a hole. Jean’s interpretation f(x) cannot equal L. (Jean & CL) – Limit cannot exist where there is a hole because nothing is there. (Linsey) – A limit is a hole one can fall into. Also, barrier/restraint of (vertical asymptote) that one cannot go past. (CL)

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 9

Limit Notation

• Appropriate Schema: The arrow beneath the

“lim” notation does not imply direction from the left.

• Altered Schema: The arrow beneath “lim”

implies direction only from the left (Carrie).

• Altered Schema: What appears below “lim” is

the actual limit (Carrie) .

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 10

2

1 lim ( 2)

x

x





3

lim ( ) or lim ( )

x x

f x f x

 

Role of Domain for Limits at a Point

• Appropriate Schema: As xa, “a” need not be

in the domain for limit to exist. (Brendon)

• Altered Schema: As xa, “a” cannot be in the

domain, given def. of limit and “approaching but not equals”. (Jean)

• Altered Schema: As xa, “a” must be in the

domain for a limit to exist. This suggests why when they see the hole they say the limit does not exist. (Linsey and Amanda)

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 11

Finite Interval Domains of Arccosine: Limits at a Point

• Appropriate Schema: As x approaches 1 or -1, the limit exists

in both cases. (Brendon)

• Altered Schema: As x approaches 1 or -1, the limit only exists

with solid dots (points). (Amanda and Linsey)

• Altered Schema: As x approaches 1 or -1, the limit exists

where there are holes, but not at solid dots (points). (Jean)

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 12

Finite Interval Domain: Limits at Infinity

• Appropriate Schema: As x approaches infinity for graphs D &

E, the limits do not exist because there are no x-values in the domain for x<-1 or x>1 or no function values to consider. (Brendon, Jean, Amanda)

• Altered Schema: As x  the limit does not exist in Graph E

because of the holes. (Linsey)

• Other Altered Schema: Limits equal pi and 0, or -1 & 1 for

graph D, and equal infinity for graph E.

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 13

lim arccos

x

x





Is the point (2,6) on the graph of the function? Identify this function.

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 14

2

lim ( )

x

f x



Significant Findings: Piecewise Functions

• Piecewise-defined functions are misidentified.

– Misconception - Point (2,6) is not on the graph of the function and is not part of the function.

• Function often not identified as piecewise.
• Graph often considered to be .
• Difficulty extracting original function from graph.
• Skill deficits with vertical & horizontal

shifts/translations.

15

2

6 if x=2 ( )

• (x-2) +4 if x

2 f x     

2

( ) f x x  

Another Piecewise Function

• Appropriate Schemas:

– Function value exists at (3,3). – Limit exists (separately on each side via continuity definition) – Both sides are NOT equal; therefore limit d.n.e. for whole function. Inappropriate Schemas:

• The limit exists at (3,3) b/c of the solid dot.
• The limit exists at x=3 b/c of imaginary vertical line (brick wall).
• The limit does not exist because there is a hole.

16

Unique Perceptions of Limits at a Point Examples of Carrie’s Altered Schemas

Limits are holes you can fall into. Holes are limits exist, so they exist. Once inside hole, the limit does not exist b/c you keep falling down. Solid dot means no limit b/c you can walk across it and not fall in.

Limits are “barriers, restraints,

• r brick walls you can’t go past.

Limit is under “lim” notation, so no math needed.

17

Limits at a Point: Instrumental Understanding Evidence of Altered Schema (Carrie)

• Limits are barriers, restraints or brick walls that
• ne cannot go past (vertical asymptotes).
• Limits are holes that one could fall into.
• The limit exists where there is a hole because one

could fall into it.

• Once in the hole, the limit did not exist because
• ne could keep falling down and keep going.

18

Unique Perceptions Carrie’s Interpretation of Limit Notation

No math necessary. Just look beneath “lim” and whatever x is approaching, is the limit. If x approaches infinity, the limit would be “infinity”.

Notation also implies direction from the left since that’s how the arrow points. x -> 2 only done from the left.

19

1 lim

x

x



Unique Perceptions of Limits at Infinity Altered Schemas

• As limit exists since it equals infinity

Reason: it keeps going and never touches the x-axis.

• There are 2 limits: At -1 and 1.

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 20

1 lim

x

x



x 

limcos & lim cos

x x

x x

 

Carrie’s Work Altered Schemas for Limits at Infinity

Limit exists, equals infinity. 2 limits that exist: y=1 & y=-1.

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 21

Unique Perceptions of Limits at Infinity Finite Interval Domains

• Appropriate Schema: As x approaches infinity for graphs D &

E, the limits do not exist because there are no x-values in the domain for x<-1 or x>1 or no function values to consider. (Brendon, Jean, Amanda)

• Altered Schema: As x  the limit does not exist in Graph E

because of the holes. (Linsey)

• Other Altered Schema: Limits equal pi and 0, or -1 & 1 for

graph D, and equal infinity for graph E.

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 22

lim arccos

x

x





Damped Cosine End Behaviors

• Appropriate Schema: As x approaches minus infinity, limit

does not exist due to oscillatory behavior. As x approaches plus infinity, function converges to 0 so limit exists. (Brendon, Jean, Linsey)

• Altered Schema: As x approaches minus infinity, the limit

exists and is minus infinity. (Amanda).

• Other Altered Schema: As x approaches plus infinity, the

limit exists and equals . (CL & 0thers)

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 23

lim cos

x x

e x

 



Perceptions of Infinite Limits

• Altered Schema: A limit exists if it equals infinity.

– Comment: should be limit d.n.e. b/c function values increase without bound. – Comment: The symbol exists; not the limit.

• Altered Schema The left and right hand infinite

limits must be compared to decide if limit exists.

– Comments: Left & right hand limits are evaluated separately. – Incorrect to conclude

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 24

  



1 1 lim lim

x x

x x

 

 

   

Example of Nicole’s Conceptual Structures

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 25

Example of Nicole’s Altered Schemas

Combination of Appropriate and Altered Schemas. Mathematical

• perations with infinity. LHL exists (= ), RHL exist ( = )

but so lim dne.

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 26



LHL RHL 



1 x 

Example of Nicole’s Altered Schemas

• Factored a sum of squares

resulting in incorrect solutions

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 27

End Behaviors Infinite Limits Different Perceptions

• Appropriate Schema: Limits do not exist if

they equal infinity. (Brendon, Jean)

• Altered Schema: Limits exist if they equal
• infinity. (Linsey, Amanda, Carrie, Nicole)
• Appropriate Schema: do not compare the left

and right hand infinite limits. (Brendon, Jean)

• Altered Schema: compare left and right hand
• limits. (Linsey, Amanda, Nicole)

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 28

1 lim

x

x



1 lim

x

x



Limits that just Do Not Exist Behavior near 0

• Appropriate Schema : Limit does not exist due to
• scillatory behavior near 0. (Brendon, Jean)
• Altered Schemas: Limit d.n.e. since function undefined when

x=0. No function value, no limit. (Note: x at 0 is not relevant to the limit) (Amanda, Linsey) 2 limits, 1 & -1 (Carrie)

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 29

1 limcos

x

x



Proposed Explanations for Unique Perceptions

• Content taught deviates from info. perceived.
• Individual differences in perception & learning.
• Information perceived gets organized into

appropriate or altered schemas.

• Math content connected to students’ realities.
• Personal experiences and vocab describe limits.
• Definitions in math not emphasized enough.
• Think infinity is a large number. (not unboundedness)
• The symbol on graph “exists”. (so limit must exist)
• Limits and function values are indistinguishable.

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 30



Implications for Teaching & Learning

• Emphasize “infinity” is not a large number; it’s a

symbol for unboundedness of function values.

• Reinforce definitions in mathematics.
• A “limit” is a number but the symbol is not.
• A limit must be a number in order to exist; not .
• Acknowledge individual differences in learning and

perception as students construct knowledge.

• What’s taught can differ from what’s learned.
• Math content is assimilated into knowledge

structures and organized into appropriate or altered schemas.

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 31

 

Teaching Recommendations

• Identify misconceptions ahead of time and design

lesson plans accordingly.

• Graphic Organizers (visual & KLW’s)
• Implement Literacy Toolkit for Calculus and

Anticipation Guide for Limits (see next slide).

• Assess frequently for conceptual understanding –
• ral and written.
• Develop technology and implement instructional

strategies to enhance quantitative literacy.

• Utilize cooperative groups and inquiry-based

learning to engage diverse learners.

• Textbooks: review & explain inconsistent, vague

language.

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 32

33

4/30/2015

• Dr. Margaret Adams, Francis Marion

University 34

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