In December of 2013, Marion Stevens, assistive technology specialist at the Office of Disability Services, contacted me about a Tree Mabry, a blind student who was close to finishing his undergraduate degree and needed to satisfy the core mathematics requirement. His major, communications, did not require advanced mathematics, and he neither knew nor wished to learn the Braille versions of mathematical symbols. Marion knew I teach PHL 195: Introduction to Deductive Logic, which has substantial mathematical content, and he wondered if this might do the trick. His idea was good.
The software we use in that course replaces all special logic symbols with characters on the QWERTY keyboard. For example, instead of ∃ and ∀, both standard logic symbols, the software uses $ and @. Tree already knew the QWERTY keyboard, so at least he would not have to learn the Braille mathematical symbols for only one class.
PHL 195 seemed to fit in another way too: students work at their own pace, making use of excellent (free) teaching software and lots of one-on-one tutoring. The course is divided into seven modules, and students take the exams when they have mastered the material, module-by-module. This format would allow Tree to work at his own pace with plenty of individual help. However, as I explained to Marion, this was completely new territory for me. I had never taught mathematics or logic to a blind student. Up until that point, all my logic instruction was heavily vision-based. But I also said I was eager to try.
When I met Tree, one of the first things he told me was that he wanted to take exactly the same tests as the other students and be held to the same standards. This simplified things somewhat, because it meant I would not have to devise new assessments. But it also set the bar even higher: the tests involve long complex formulas and I give no partial credit (students can, however, re-take tests as many times as they like).
My initial plan was simple: I assumed that Tree would explain to me how he learned mathematics in high school and that I would just use that model. However, it turned out that Tree lost his vision after he completed his high school mathematics requirements. He had no experience doing symbolic, mathematical work as a blind student. Seeing no reason to re-invent the wheel, I tried to find resources for instructing the blind in formal logic. But I found little that would help me teach Tree the exact same material that the other students were learning.
Still, I thought, how hard could it be to adapt what I normally do? Normally, I teach the material in two stages. I begin by using relatively examples to illustrate the basic concepts. After that, the students use the software to work exercises, getting one-on-one help whenever they like; the class meets in a computer lab and we hire undergraduates to work as teaching assistants. In the end, I was able to adapt that same teaching method to teach Tree the material. But it was challenging.
The first major challenge was teaching him the syntax of sentential logic. Sentential logic is one of two artificial languages in which the class’s logic problems is primarily done. Students usually learn the basic concepts pretty quickly. But that is because I can illustrate the syntax rules easily on the board or on paper. How was I to convey that information to a blind person?
I do not think I could have done this alone, but fortunately I was only one member of a three-person team. The other two were Rebecca (Becky) Kerley, a teaching assistant and philosophy major, and Julianne (Julie) Wilson, the former philosophy department secretary who happens to have had a career as a school teacher.
Here is what we did.
Julie went to Hobby Lobby and bought wooden letters and materials (foam rubber, various fabrics and glues) to fabricate three-dimensional logic symbols. Each symbol she made had a special texture. I then cleared my office desk and used these 3D symbols in much the same way that I use 2D written symbols for sighted students. Tree felt the symbols with his hands and learned to recognize them by touch. After several sessions, he learned the basic syntactical rules.
At that point, Tree was ready to begin doing complex exercises on the computer. This too presented challenges. To read computer text, he uses Voiceover, an Apple read-aloud program, and the logic software was not designed to be used in conjunction with that program. Even so, Tree, Becky, and I brainstormed the problem and quickly figured out that we could copy the problems and paste them into files that work well with Voiceover (such as TextEdit, MS Word and Apple’s Numbers), and then email them to Tree. Tree then was able to come up with attempted solutions and get feedback from Becky by email, in class, and in private tutoring sessions. It worked.
We used a similar two-stage approach to teach Tree all the material he learned. The first stage took place on the desk of my office, with me showing him the basic concepts using 3D symbols Julie fabricated. At the second stage, we would take problems the software generates and copy them into a format that worked for him. There were obstacles at each stage, but together we were able to overcome them. Tree passed the course. He took exactly the same tests as the other students, and he was graded using the same standards. It was a rewarding experience.
Early in the semester, I contacted Colin Allen, who co-wrote the software we use, about adapting it for teaching the blind. He recommended that we take field notes during the semester and eventually write a paper describing the experience. Becky, Julie, and I are now about three-quarters done writing up that paper.
So, what is the pedagogical upshot of this?
Four things come to mind.
First, we proved definitively that blind students can fully master deductive logic (or enough of it to pass our Introduction to Deductive Logic course)—not just in theory but in practice.
Second, I learned how extremely vision-centric my own teaching is, at least for teaching formal subjects. Awareness of this has, I believe, made me a better logic teacher for all students; after all, not everyone is a visual learner.
Third, we learned something that we probably should have known — that courses such as logic that require detailed symbolic work are intimidating for blind students. In Tree’s opinion, more blind students would major or minor in philosophy if they realized they could get through the logic requirement. So, we learned we should publicize this in some way.
Fourth, we learned there were limitations to our approach. In particular, we exploited the fact that Tree lost his sight relatively late in life and therefore could recognize letters of the English alphabet, along with other characters such as @ and $, by their shapes. As Tree pointed out, to teach the same material to someone who is congenitally blind would be more of a challenge (I am unfamiliar with the Braille characters).
Torin Alter is a professor and undergraduate advisor in the Department of Philosophy.