Math with all the feels

The following is a guest post from David Austin. David is Professor of Mathematics at Grand Valley State University, author of the free OER text Understanding Linear Algebra, chair of the American Institute of Mathematics’ Open Textbook Initiative Editorial Board, and much more. He is also one of my closest friends and colleagues, and I’m delighted to share this post about the exciting work he and others are engaging to make braille versions of mathematics textbooks for unsighted users, especially regarding David’s work that creates tactile graphics. You can contact David at austind at gvsu dot edu.


If you’re reading Matt’s blog, chances are you already embrace a fundamental principle that motivates the open textbook community:  students should have access to everything they need to be successful on the first day of class.  This also means that mathematics textbooks should be accessible to our students regardless of their abilities.

In fact, accessibility is a key design feature of PreTeXt, the authoring system that Matt uses to create his great books.  For example, a visually-impaired reader can easily navigate the web versions of the Active books using a screen reader, and Rob Beezer, the founder of PreTeXt, recently made a braille version of Active Prelude directly from Matt’s PreTeXt source.  What’s missing, however, are the many figures that enrich a reader’s intuitive understanding of key concepts.

Your first thought may be, how can one communicate visual information to a visually-impaired reader?  In fact, the relatively new field of tactile graphic design intends to do just that. As described in a recent MIT Technology Review, tactile graphics communicate visual information by touch rather than by sight, similar to how braille’s raised dots express written language.  While tactile diagrams may be created using a variety of technologies, you could think of a collection of dots, raised by an embosser, that represents graphical information.  

Over the past year, I’ve been developing software to help authors create accessible mathematical diagrams following the PreTeXt paradigm.  In particular, an author produces a diagram by creating a textual description of the diagram’s components and how they relate to one another.  For instance, you could say that you want a diagram with a set of axes, the graph of a function, and the tangent line at a point.  Software will then convert this description into a requested format, such as an SVG file for inclusion in a web document or a tactile version suitable for embossing.

Here are two versions of a calculus diagram that were created from the same description.  The one on the left could be placed in a print version of a document or in an online version.  Besides the graphical description, an author can provide annotations that help a reader navigate the diagram using a screen reader and the graphs can be sonified to provide an aural description of the visual information, thanks to contributions from Volker Sorge.  These capabilities are demonstrated in our submission to the Accessibility Challenge at the Web4All conference last May.

A tactile diagram can be created from the image on the right when it is sent to an embosser.  The image has been resized to fit on a standard 11” x 11.5” embossed page, and the labels have been converted to Nemeth braille, a version of braille that communicates mathematical expressions.  Here’s a photograph of a similar embossed image:

Photo credit: Alexei Kolesnikov

The key point is that the author only concerns themself with the description of the diagram while the software handles all the details of producing the graphical output, whether tactile or not.

Last year, the disability resources office at Grand Valley State inquired to our department about accommodations for a blind student taking intermediate algebra because the course’s textbook publisher wanted $40,000 to produce a braille edition.  While a better option quickly appeared, we knew that this student would be using Active Prelude this fall and Active Calculus in the near future.  As a result, Rob got to work making a braille edition of Active Prelude, and I created tactile versions of almost all the diagrams in the book.  Our student has now begun the semester with a braille textbook, complete with tactile diagrams, produced at no cost.  

It’s sometimes said that attention to accessibility benefits everyone.  For example, SMS text messaging was originally developed as a means for the deaf to communicate, and most of us have probably been glad to use a wheelchair ramp at one time or another.  How does that principle apply here?

First, since they are meant to be read by touch, tactile graphics have a much lower resolution and braille labels typically require more characters.  As a result, space feels like it’s at a premium, even though the diagrams are embossed on 11” x 11.5” paper.  As I’ve started to create tactile diagrams, I’ve been reconsidering design principles I’ve used in diagrams for sighted readers, and I feel the urge to reduce visual clutter and communicate information more efficiently.  I’ve been thinking more carefully about diagrams and how a wider range of readers might interact with them, and my hope is that the new diagrams in Active Prelude, once they are included in the next edition of the text, are an improvement for all readers.  

Another issue arose when a blind mathematician was reading one of my tactile diagrams that illustrated a graph and its tangent line at a point.  He told me that his sense of touch had a hard time distinguishing between the graph and the tangent line.  In some sense, that’s the point the diagram is trying to convey, that a tangent line gives an excellent approximation to the graph on a small scale.  In this case, the very idea that I wanted to communicate made it difficult for a tactile diagram to express.  A solution is for the author to carefully ensure that the text and the diagrams are tightly integrated with one another.  In other words, we shouldn’t assume the point we’re trying to make is obvious to every reader, but instead we should strive to be explicit and to reinforce crucial ideas.  While this sounds like a simple principle in expository writing, expanding our understanding of how we can be misinterpreted is extremely powerful and a real gift.

In the introductory article to the recent accessibility edition of the MIT Technology Review, The future is disabled, Ashley Shew considers some of the problems we face as a society, including climate change and environmental racism, and concludes with:

We need more ways to be. Part of that involves looking to alternative ways of sensing, processing, moving, understanding, and communicating, and seeing those ways as good and worthwhile. Opening ourselves up to all-access thinking and disabled expertise will mean a more livable world—one that we all can inhabit. 

This work is part of a larger effort organized by the Raised Mathematics group.  Feel free to reach out to me with thoughts and questions: austind at gvsu dot edu.

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