Newbury Park, CA: Sage publications. Stewart, J. (2012). Calculus: Early transcendentals (7th ed.). Stamford, CT: Brooks/Cole Cengage Learning. References.
Location-Thinking and Value-Thinking: An Empirical Study and Framework of Two Ways Students Reason about Graphs Erika David and Morgan Sellers Under the advisement of Dr. Kyeong Hah Roh
Motivation & Research Question
Methodology
• Visual representations of formal statements in undergraduate mathematics are commonly used in textbooks to support student understanding.
• Clinical Interviews (Clement, 2000) with nine undergraduate students, three from Calculus I, Introduction to Proof, and Advanced Calculus • Participants evaluated the following statements alone, then using various graphs below
• Ex: The Intermediate Value Theorem (IVT)
Suppose that f is a continuous function on [a, b] with f(a)≠f(b). Then, for all real numbers N between f(a) and f(b), there exists a real number c in (a, b) such that f(c)=N.
Characteristics of Location-Thinking In Figure 7 (right): • Zack, a location-thinker, considered N to be locations along the graph • Labeled points as outputs • Incorrectly claimed f(a) ¹ f(b) on this constant function
Statement 1 Suppose that f is a continuous function on [a, b] such that f(a)¹f(b). Then, for all real numbers c in (a, b), there exists a real number N between f(a) and f(b) such that f(c)=N. Statement 2 Suppose that f is a continuous function on [a, b] such that (IVT) f(a)¹f(b). Then, for all real numbers N between f(a) and f(b), there exists a real number c in (a, b) such that f(c)=N. Statement 3 Suppose that f is a continuous function on [a, b] such that f(a)¹f(b). Then, there exists a real number N between f(a) and f(b), such that for all real numbers c in (a, b), f(c)=N. Statement 4 Suppose that f is a continuous function on [a, b] such that f(a)¹f(b). Then, there exists a real number c in (a, b), such that for all real numbers N between f(a) and f(b), f(c)=N.
Figure 7. Zack’s labels on Graph 5
Table 1. Statements given to participants
In Figure 8 (left): • Nate, another location-thinker also labeled points as outputs • Claimed N’s he labeled on the graph were between f(a) and f(b) • Claimed point marked on far right as an output not between f(a) and f(b)
Figure 4. Graphs given to participants
Figure 1. Visual Representation of IVT (Stewart, 2012)
Research Question:
What are characteristics of students’ visual reasoning in the context of statements from Calculus? Specifically, • How do students interpret outputs of the function, points on the graph, and graphs as a whole? • How do students’ visual reasoning impact their understanding and evaluation of the Intermediate Value Theorem and similar statements?
Data Analysis & Results • Consistent with grounded theory, we used a process of open coding and axial coding in which our visual reasoning framework emerged (Corbin & Strauss, 2014).
Visual Reasoning
Value-Thinker
Our Original Theoretical Framework Value-Thinking
Aspects of a Graph
Visual Reasoning Output of Function
Location-Thinking
Evidence
▪ Labels output values on output The resulting value from inputting a value axis ▪ Speaks about in the function output values
The coordinated values of the input and ▪ Labels points as Point on Graph output represented ordered pairs together ▪ Speaks about points as the result of coordinating an A collection of Graph as a coordinated values of input and output Whole the input and output value
Visual Reasoning
Evidence
The resulting location in the Cartesian plane from inputting a value ▪ Labels output on the graph in the function A specified spatial location in the Cartesian plane
▪ Labels point as output
• We categorized each of the nine students according to our visual reasoning framework
Location-Thinker
Student Name
Math Level
Final Evaluation S1 S2 S3 S4
Jay
Advanced Calculus
F
T
F
F
James
Advanced Calculus
F
T
F
F
Nikki
Introduction to Proof
F
T
F
F
Ron
Introduction to Proof
T
T
F
F
Mike
Introduction to Proof
F
F
F
F
Zack
Calculus
ST
ST
ST
ST
Nate
Advanced Calculus
T
T
F
F
Hannah
Calculus
T
T
T
T
Marie
Calculus
T
T
T
T
Note: students in italics switched from value-thinking to location-thinking
Table 2. Participants’ visual reasoning and final statement evaluations
Characteristics of Value-Thinking
▪ Speaks about points as a result of an input A collection of spatial into the function (e.g., locations in the “an input maps to a Cartesian plane point on the graph”) associated with input values
In Figure 5 (left): • Jay, a value-thinker, considered N to be a value from the y-axis • Labeled points as ordered pairs
Figure 3. Sample labeling of a location-thinker
Significance of Findings • Five students were classified as value-thinkers and four students as location-thinkers • Three value-thinkers and no location-thinkers evaluated all four statements correctly • Value-Thinking and Location-Thinking highlight distinctions in visual reasoning in students’ meanings for: • Claimed N’s he labeled on the graph were between f(a) and f(b) • Outputs of functions • Points on graphs • Graphs as a whole • Students may not interpret graphs as intended • Different interpretations of graphs can have significant consequences on students’ understanding of the given concept
Implications for Instruction • When teaching the Intermediate Value Theorem, instructors might incorporate graphs like Graph 1 to differentiate in students’ visual reasoning • In general, when teaching how to interpret graphs in the Cartesian coordinate system, instructors might seek opportunities to distinguish between values represented at points and locations of points in space in instruction.
Figure 5. Jay’s hand-drawn graph explaining Statement 1
Figure 2. Sample labeling of value-thinker
Figure 8. Nate’s labels on Graph 1
•
In Figure 6 (right): • Jay labeled a portion of the y-axis on Graph 1 to indicate possible values of N between f(a) and f(b) • He explained that the output at the highlighted point would not be between f(a) and f(b)
For instance, when referring to and labeling outputs of a function on its graph in the Cartesian plane, refer to the value as represented on the output axis.
• Our research is situated in the context of IVT, in which value-thinking was the intended visual reasoning for students. The role of value-thinking and locationthinking, though, may be different in different contexts, such as geometry or graphs in different coordinate systems. Figure 6. Jay’s labels on Graph 1
References Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In Handbook of research methodologies for science and mathematics education (pp. 547-589). Hillsdale, New Jersey: Lawrence Erlbaum. Corbin, J., & Strauss, A. (2014). Basics of qualitative research: Techniques and procedures for developing grounded theory. Newbury Park, CA: Sage publications. Stewart, J. (2012). Calculus: Early transcendentals (7th ed.). Stamford, CT: Brooks/Cole Cengage Learning.
