Improving Student Understanding of Multivariable Calculus Concepts Using VisualizationM. VanDieren (Robert Morris University), D. Moore-Russo (University at Buffalo, SUNY), P. Seeburger (Monroe Community College)

Blog: https://calcplot3dblog.wordpress.com/

Website: http://web.monroecc.edu/calcNSF/

Abstract

CalcPlot3D is an award-winning, interactive, Java appletthat allows students to visually explore and gain a deeper un-derstanding of multivariable calculus concepts. This projectbuilds upon the success of CalcPlot3D which is used by morethan 85 instructors at over 68 institutions worldwide.

Goals of the Project

• Design and test a series of new discovery-based activities.• Expand the features of CalcPlot3D to addressapplications in physics and engineering.

• Reprogram CalcPlot3D to work on tablets and phones.• Investigate how student understanding of multivariablecalculus concepts changes through the use ofvisualization and dynamic concept explorations.

• Extend and diversify the user base through a blog and aSpanish language version of the applet.

Pedagogical Challenges

Multivariable Calculus (aka Calc 3) aims to extend the ideasfrom the first two semesters of calculus to three-dimensionalspace. Even those students with a strong understanding of Calc1 and 2 are challenged by the transition to three dimensionsbecause they are not able to visualize mathematical concepts in3D (Trigueros & Martinez-Planell, 2007). Furthermore severalCalc 3 concepts involve adding motion to what were stagnantconcepts in Calc 1 and 2.

3D Visual Reasoning Skills

For students to succeed in Calc 3, they need to develop three-dimensional reasoning skills (Trigueros & Martinez-Planell,2007). These skills will also enhance students’ ability to suc-ceed in engineering, physics, chemistry, biology, and medicine(Hungwe, Sorby, Drummer, & Molzon, 2007; Sorby, 2009).

Students need to be able to see the problems clearly inorder to be able to solve them creatively and with enoughinsight to innovate new approaches and solutions.

Left: Position, velocity and acceleration of a particle. Right: Cross sectionof f (x, y) = cos(x) sin(y).

Intervention

Sorby (2009) found that software and workbook exercises, cov-ering concepts such as 2D to 3D transformations; rotations;reflections; cutting planes; and solids of revolution, can improvestudent spatial visualization. All are features of CalcPlot3D.

Conjecture

The CalcPlot3D applet and explorations mayhelp students to transition from an action-viewof multivariable functions, with a limited under-standing of the "Cartesian connection" (Knuth2000) between the graph and equation of afunction, to a connected process-view with arich understanding of the relationships betweenthe symbolic, verbal, numeric, and graphicalrepresentations of multivariable concepts.

Counterclockwise From Upper Left: Riemann prisms; Vector field and surface in 3D; Partial derivative and tangent line; Contour plot of constraint and surface;3D view of constraint and surface; Taylor polynomials; Screenshot of an animation of a flow line in a time-dependent vector field; Parametrically-defined surface.

Theoretical Framework

Concept Image: total cognitive structure associated with a concept, including all mental pictures and associated propertiesand processes (Tall & Vinner, 1981). Zandieh (2000) additionally examines different aspects of theconcept, relationships between aspects, and representations of the aspects.

Representational Fluency: ability to call on a variety of representations and choose the most appropriate for the situation or task.Individuals who easily make connections between different representations understand a concept moredeeply than those who are missing connections or for whom the connections between representations areweak or not easily made (Moore-Russo & Viglietti, 2012).

Variation Theory: design of curriculum to help students discern a target aspect in a concept by developing a way to keepsome critical aspects invariant while varying others (Bowden & Marton, 2005).

Need for an Assessment Instrument

Currently there is no assessment instrument that combines multivariable calculus concepts (like the single-variable analogs inthe Calculus Concept Inventory) with 3D visual reasoning skills (like those in the Purdue Spatial Visualization Test) for vectors,vector-valued functions, single-valued functions of several variables, and vector-valued functions of several variables

First Stage of Research: Vectors

• Analyze existing data set of mostly quantitative data on pre- and post-tests of dot and cross product explorations to determinequestion reliability and validity and to gauge gains in student understanding of concepts.

• Consult the results of the data analysis, the Physics Vector Concept Inventory, and our Board of Advisors to create aMathematics Vector Concept Inventory.

• Collect qualitative data on a small group of students conducting the vector explorations.• Pilot test the Mathematics Vector Concept Inventory.• Adjust the Mathematics Vector Concept Inventory and report results of how students use visualization to understand concepts.

Applet Design

Clements (2000) proposes principles of design for software de-veloped to improve student understanding of mathematics:• Students are encouraged to test ideas and play withmathematical concepts and immediately see what happens.

• Students can directly manipulate mathematical objects tomake connections between the formal algebraicrepresentations of the concepts and their geometricrepresentations.

Features

• Surfaces in 3D with multiple views including contour plots• Animated parametric curves• Vector fields, implicit functions, and parametric surfaces• Symbolic, numeric, and graphical representations of calculusconcepts

• Cartesian, cylindrical, and spherical coordinate systems• Animations• 3D-printable files• Built-in discovery-based, self-contained labs

Comparison with Alternative Programs

• Cost = FREE.• Accessible from any desktop or laptop browser withouthaving to download any extra add-ons like WolframDemonstration Project.

• Easy to use: natural syntax for students who grew up usingthe TI 83/84 calculators. Menu-driven makes it easier to usethan Maple and Mathematica.

• Easy to adopt: Built-in examples and discovery-based labs.• Versatile: Scripting language to save work.• More colorful, faster, and more accurate than the TI-Inspirecalculator or online graphers like Apple Grapher.

Drawbacks of CalcPlot3D are the reliance on Java and the lim-itation of concepts covered compared to more comprehensiveprograms. Both of these are being addressed as part of thisproject.

References

• J. Bowden and F. Marton. (2005). The University of Learning: Beyond Quality and Competence. NewYork: Routledge Farmer.

• Clements, D.H. From exercises and tasks to problems and projects: Unique contributions of computers toinnovative mathematics education. Journal of Mathematical Behavior, Vol. 19, (2000) pp. 9-47.

• Hungwe, K., S. Sorby, T. Drummer, & R. Molzon. Preparing K-12 Students for Engineering Studies byImproving 3D Spatial Skills. The International Journal of Learning, Vol. 14, No. 2, (2007) pp. 127-135.

• E. J. Knuth. Student understanding of the Cartesian connection: An exploratory study. Journal forResearch in Mathematics Education, Vol. 31, No. 4, (July, 2000) pp. 500-507.

• D. Moore-Russo and J. M. Viglietti. Using the K5 Connected Cognition Diagram to analyze teachers’communication and understanding of regions in three-dimensional space. Journal of MathematicalBehavior, Vol. 31, No. 2, (2012) pp. 235-251.

• Sorby, S. Educational Research in Developing 3?D Spatial Skills for Engineering Students. InternationalJournal of Science Education, Vol. 31, No. 2, (2009) pp. 459-480.

• D. Tall and S. Vinner. Concept Image and Concept Definition in Mathematics with Particular Referenceto Limits and Continuity. Educational Studies in Mathematics, Vol. 12, No. 2, (May, 1981) pp. 151-169.

• Trigueros, M. and R. Martinez-?Planell. Visualization and abstraction: Geometric representation offunctions of two variables. In T. Lamberg L.R. Wiest (Eds.), Proceedings of the 29th Annual Conferenceof the North American Chapter of the International Group for the Psychology of Mathematics Education(2007) pp. 100-107.

• M. Zandieh. A Theoretical Framework for Analyzing Student Understanding of the Concept of Derivative.Research in Collegiate Mathematics Education, Vol. 8, No. 4, (2000) pp. 103-127.

This material is based upon work supported by the National ScienceFoundation under a collaborative research Grant No. 1523786. The firstauthor was also supported by an AWM-NSF Mentoring Travel Grant.