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. Dr. Margaret Adams, Francis Marion 4/30/2015 2 University

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? Dr. Margaret Adams, Francis Marion 4/30/2015 3 University

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) Dr. Margaret Adams, Francis Marion 4/30/2015 4 University

Model of Understanding Limits Shared Knowledge Structures Appear in the Intersections Altered Schemas Appropriate Instrumental and Schemas “Semi - Relational” Relational Understanding Understanding (non-traditional (expected outcomes) outcomes) Dr. Margaret Adams, Francis Marion 4/30/2015 5 University

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. Dr. Margaret Adams, Francis Marion 4/30/2015 6 University

Graph of Piecewise Function Dr. Margaret Adams, Francis Marion 4/30/2015 7 University

Linear, Discontinuous & Piecewise Functions Dr. Margaret Adams, Francis Marion 4/30/2015 8 University

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) Dr. Margaret Adams, Francis Marion 4/30/2015 9 University

Limit Notation • Appropriate Schema: The arrow beneath the “lim” notation does not imply direction from the left. 1 lim (   x 2) x 2 • Altered Schema : The arrow beneath “lim” implies direction only from the left (Carrie). • Altered Schema : What appears below “lim” is the actual limit (Carrie) . lim ( ) or lim f x f x ( )   x 3 x Dr. Margaret Adams, Francis Marion 4/30/2015 10 University

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) Dr. Margaret Adams, Francis Marion 4/30/2015 11 University

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) Dr. Margaret Adams, Francis Marion 4/30/2015 12 University

Finite Interval Domain: Limits at Infinity lim arccos x  x • 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. Dr. Margaret Adams, Francis Marion 4/30/2015 13 University

Is the point (2,6) on the graph of the function? Identify this function. lim f x ( )  x 2 Dr. Margaret Adams, Francis Marion 4/30/2015 14 University

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.  6 if x=2   f x ( )  2  -(x-2) +4 if x 2 • Graph often considered to be .   2 f x ( ) x • Difficulty extracting original function from graph. • Skill deficits with vertical & horizontal shifts/translations. 15

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. Limits are “ barriers, restraints, Holes are limits exist, so they exist. Once inside or brick walls you can’t go past. hole, the limit does not exist b/c you keep falling down. Limit is under “ lim ” notation, so Solid dot means no limit b/c you can walk across no math needed. it and not fall in. 17

Limits at a Point: Instrumental Understanding Evidence of Altered Schema (Carrie) • Limits are barriers, restraints or brick walls that one 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 one could keep falling down and keep going. 18

Unique Perceptions Carrie’s Interpretation of Limit Notation No math necessary. Just look beneath Notation also implies direction “ lim ” and whatever x is approaching, from the left since that’s how is the limit. the arrow points. If x approaches infinity, the limit x -> 2 only done from the left. would be “infinity”. 1 lim  x x 19

Unique Perceptions of Limits at Infinity Altered Schemas • As limit exists since it equals infinity x  Reason: it keeps going and never touches the x-axis. 1 lim  x x • There are 2 limits: At -1 and 1. limcos & lim cos x x   x x Dr. Margaret Adams, Francis Marion 4/30/2015 20 University

Carrie’s Work Altered Schemas for Limits at Infinity Limit exists, equals infinity. 2 limits that exist: y=1 & y=-1. Dr. Margaret Adams, Francis Marion 4/30/2015 21 University

Unique Perceptions of Limits at Infinity Finite Interval Domains lim arccos x  x • 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. Dr. Margaret Adams, Francis Marion 4/30/2015 22 University

 x lim e cos x Damped Cosine End Behaviors  x • 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) Dr. Margaret Adams, Francis Marion 4/30/2015 23 University

Perceptions of Infinite Limits 1 1     lim lim     x x x 0 x 0 • 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 Dr. Margaret Adams, Francis Marion 4/30/2015 24 University

Example of Nicole’s Conceptual Structures Dr. Margaret Adams, Francis Marion 4/30/2015 25 University