Using visuals

Visuals can be a powerful tool for helping the audience to understand a concept, but can also unintentionally mislead. The following resources are about using visuals in mathematics. Most were found by M.I.T.’s undergraduate researcher Noor Doukmak.

Some of these websites require javascript or java. If you have trouble viewing images, check that your browser has javascript enabled and check whether it supports java.

Examples of effective mathematical visuals

• Visual Insight: Mathematics made visible
  This AMS blog is “…a place to share striking images that help explain advanced topics in mathematics.”
• MIT Mathlets
  “Here you will find a suite of dynamic [javascript] applets for use in learning about differential equations and other mathematical subjects, along with examples of how to use them in homework, group work, or lecture demonstration, and some of the underlying theory.”
• TeXample.net has a gallery of TikZ visuals from many areas of mathematics, science and other technical areas. Includes links to pdf and tex files.
• Visual Mathematics, a math visualization website with visuals for over fifty topics in algebra, geometry, trigonometry, and other subjects. Requires Macromedia Flash Player.
• Gizmos are “virtual manipulatives” (like applets) of math concepts that can aid in the teaching and learning of concepts from third grade to pre-calculus.
• Editor’s fun pick: Blooms: Strobe-Animated Sculptures by John Edmark and Charlie Nordstrom. These rotating, strobe-animated, 3D-printed sculptures were perhaps not intended as visualizations of mathematics, but they raise interesting and beautiful possibilities for visualizing mathematics.
• The Bridges Organization: art and mathematics holds conferences and has online galleries of mathematical art.
• Interactive java tutorials for teaching concepts in discrete math, from the Applied Math Department of the Polytechnic University of Madrid (Spanish language site, but there are webpage translators online.)
• JavaSketchpad Gallery [Requires Java] Java Sketchpad is a version of Geometers’ Sketchpad that can be used to post interactive geometric constructions on the internet. The JavaSketches on this page include Pythagoras’ Theorem, Desargues’ Theorem from projective geometry, Lissajous Curves, and a hypercube for visualizing n-dimensional space.
• See also examples in the following sections of this page.

Proofs without words

• Proofs Without Words: Exercises in Visual Thinking, and Proofs Without Words II: More Exercises in Visual Thinking, by Roger B. Nelson. These books present examples of proofs without words related to geometry, algebra, trigonometry, calculus, analytic geometry, inequalities, integer sums, and sequences and series.
• Mathoverflow’s Proofs Without Words, a discussion to which users have contributed many examples of proofs without words, from the forum mathoverflow. Includes a discussion of whether proofs without words can be more misleading than non-picture proofs.
• The MAA’s College Mathematics Journal and Mathematics Magazine contain proofs without words.
• Q.E.D.: Beauty in Mathematical Proof, by Burkard Polster, Wooden Books, 2006.
• Alex Bogomolny’s “Proofs Without Words,” [requires Java] provides an argument for the value of proofs without words including a brief summary of relevant history, and includes links to several examples. From Alex Bogomolny’s blog “Cut the Knot!” interactive essays using Java applets.
• The Art of Problem Solving has a gallery of proofs without words.

Tools for visualizing math

Various commercial tools are available for creating a variety of different kinds of math visualizations. Here are links to just a few: Maple, Mathematica, MATLAB, Geometer’s Sketchpad (geometry), Cabri (geometry).

Below are some free visualization tools and resources about using visualization tools:

• Geogebra has many of the capabilities of Geometer’s Sketchpad and Cabri: it supports geometric constructions as well as graphing algebraic relations.
• Open source software package takes aim at high-cost math programs, by Michael Cooney
  This 2007 article describes Sage, whose mission is to create a viable, free, open-source alternative to Magma, Maple, Mathematica, and MATLAB.
• Desmos has a graphing calculator, geometry construction tool, etc., as well as associated teaching resources.
• 3D-ExplorMath
  “The program presents itself as series of galleries of different categories of interesting mathematical objects, ranging from planar and space curves to polyhedra and surfaces to ordinary and partial differential equations, and fractals. Morever, the carefully chosen default parameters and viewing options may be changed by the user so that each gallery is turned into a experimental lab. Every exhibit has its own online documentation with suggestions for how to explore it further. We hope that in this way the program will be useful to the interested layperson, the teacher, and the research scientist.”
• “Reading and Writing Mathematics” MSP: MiddleSchoolPortal
  This annotated bibliography for middle school includes links to tools for visualizing mathematics (e.g., online tools for making various types of graphs).
• “MathForum >> Discussions >> Software” The Math Forum@Drexel
  These online discussion threads address ways of teaching using visualization software as well as the behavior of specific programs in specific situations. Includes discussion threads for Mathematica, MATLAB, Fathom, Tinkerplots, and such geometry software programs as The Geometer’s Sketchpad and Cabri Geometry II.
• Mathematical Visualization Toolkit [requires Java] of the Department of Applied Math at the University of Colorado, Boulder
  This toolkit includes a variety of plotting and solving applets for visualizing relations in cylindrical and spherical coordinates, contours, 3-variable ODE vector fields, etc. This site was selected as the 2005 MERLOT Classics Award winner for the Mathematics discipline.
• See also additional tools in the section below on including LaTeX labels on figures.

Including LaTeX labels on figures

A page about how to include LaTeX labels on figures is here.

Including visuals in a mathematics paper

How to combine text and visuals is illustrated by this annotated journal article:

• “Maximum Overhang” by Paterson et al (annotated), American Mathematical Monthly 116, December 2009. See pages 116 and 117 (pages 6 and 7 of the pdf) for illustrations of how to refer to figures in the text, how to cite the source of a figure, brief suggestions for when to include a figure, etc.

Teaching mathematical visualization: theory and practice

The following resources present pedagogical strategies, theory, research, &/or experience, both for teaching students to communicate math via visualizations, and for using visualizations to help students learn math.

• Herrera, T., “Reading and Writing Mathematics” MSP: MiddleSchoolPortal
  This clickable, annotated bibliography for middle school educators includes links to activities for visualizing mathematics.
• Math Made Visual: Creating Images for Understanding Mathematics, by Claudi Alsina and Roger B. Nelson, MAA 2006.
  From the back cover: “The objective of this book is to show how some visualization techniques may be employed to produce pictures that have both mathematical and pedagogical interest.” Includes examples of math visualizations from geometry through analysis (Lipschitz condition, uniform continuity), challenges for the reader, and suggestions for classroom use.
• Visualization in Teaching and Learning Mathematics, Steve Cunningham and Walter S. Zimmerman Eds., MAA Notes #19.
• Visualization in Mathematics: Claims and Questions towards a Research Program, by Walter Whiteley, 2004
  “a series of claims about visualization in mathematics and mathematics education, and some possible research questions for future work on visualization in mathematics and mathematics education.”
• Intuition and Visualization in Mathematical Problem Solving by Valeria Giardino, 2010
  This article discusses the relationship between visualization and intuition in math and alludes to the potential dangers of using visuals when attempting to solve math problems.
• Visualizations in Mathematics, by Brating and Pejlare, 2008
  “In this paper we discuss visualizations in mathematics from a historical and didactical perspective. We consider historical debates from the 17th and 19th centuries regarding the role of intuition and visualizations in mathematics. We also consider the problem of what a visualization in mathematical learning can achieve. In an empirical study we investigate what mathematical conclusions university students made on the basis of a visualization. We emphasize that a visualization in mathematics should always be considered in its proper context.”
• Interactive Mathematical Visualisations: Frameworks, Tools and Studies, by Kamran Sedig, 2009
  “This chapter discusses issues pertaining to the design of interactive mathematical visualizations in the context of performing epistemic activities. To this end, this chapter presents the following. It explains what mathematical visualizations are and their role in performing epistemic activities. It discusses the general benefits of making mathematical visualizations interactive. It emphasizes the need for having and using frameworks in order to describe, analyze, design, and evaluate the interactive features of mathematical visualization tools…Finally, it offers some suggestions for future lines of research in this area.”
• Seeing the Value of Visualization by Ho Siew Yin, 2010
  This article aims to answer several questions about visualization in math. “Why is visualization important in mathematics? What are the factors that influence students’ choice of problem-solving method? How does visualization help students in mathematical problem solving?” Addressing this last question, the article suggests five processes of visualization and seven roles of visualization in students’ problem solving.
• Visual Thinking in Mathematics: An Epistemological Study, by Marcus Giaquinto, Oxford University Press, 2007. (MAA Review)
• Visualization on Learning Mathematics Concepts for Engineering Education by Sanchez-Torrubia et al., 2007
  This paper presents the authors’ experiences using Java tutorials to teach first-year undergraduate math and suggests the requirements such tools should meet.
• Using visualisation in maths teaching, a teacher-training unit from the Open University
• Mathematical Visualization, a course taught by John Sullivan at the Berlin Mathematical School in 2011
• Visualization for Mathematics, Science, and Technology Education taught by Violeta Ivanova at M.I.T. in 2011

Using visuals to teach mathematics

We can model for students how to use visuals to communicate mathematics by using visuals ourselves when we teach specific concepts.

• Aatish Bhatia, “The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face,” Nautilus, Nov. 6, 2013.
  In this blog post, Aatish Bhatia uses visuals by Lucas V. Barbosa, Matthew Henderson, and others to explain the Fourier transform.
• M. A. Nyman, D. A. Lapp, D. St John and J. S. Berry, “Those do what? Connecting Eigenvectors and Eigenvalues to the Rest of Linear Algebra: Using Visual Enhancements to Help Students Connect Eigenvectors to the Rest of Linear Algebra” The International Journal of Technology in Mathematics Education, Vol 17, No 1, 2010.
  Abstract “This paper discusses student difficulties in grasping concepts from Linear Algebra – in particular, the connection of eigenvalues and eigenvectors to other important topics in linear algebra. Based on our prior observations from student interviews, we propose technology-enhanced instructional approaches that might positively impact student understanding of concepts within a problem-solving context. We discuss some barriers to student understanding and suggest ways of using geometrical interpretations and visualisations of eigenvectors and eigenvalues to address these barriers. These technological interventions can readily implement this geometric approach, permitting students to easily experiment with a visual interpretation of eigenvectors and eigenvalues.”

This topic is a new addition to this website so we have not yet conducted a literature review. Please feel free to suggest new additions to this list. Particularly helpful would be an annotated bibliography of resources about using visuals to teach specific mathematical concepts.

General resources (e.g., color blindness)

• The Coblis–Color Blindness Simulator lets you to upload an image and see how it looks to people with various types of color blindness.