American Action Fund for Blind Children and Adults
Future Reflections Summer 2025 ACCESS
The Square of the Hypotenuse
by Deborah Kent Stein
From the Editor: When I reflect on the challenges faced by blind students pursuing higher education in the STEM fields, I recall my experience back in high school, studying plane geometry. We had no high-tech equipment back in the 1960s. We used protractors, rulers, compasses, and, of course, hardcopy textbooks. In my case, that meant textbooks in Braille. Here are some thoughts and memories I’d like to share.
At the close of each school year during high school, I used to box up the dozens of hand-copied Braille volumes that comprised the textbooks I had used in my classes since September. Mom and I would load the cartons into the station wagon and drop them off at the post office to go back to the New Jersey Commission for the Blind so someone else could use them next year.
A number of the books I needed were produced on sound recordings, but the Commission made sure I had the rest in Braille—English grammar and vocabulary, Spanish, and math. Most of my Braille textbooks were transcribed by volunteers from organizations such as B’nai B’rith and the American Red Cross. The transcriber’s name always appeared on the title page of each Braille volume.
My math class in tenth grade was plane geometry, and my fourteen-volume Braille textbook contained not dozens, but hundreds of raised-line diagrams and figures. Not one of those squares, circles, octagons, bisected angles, and trapezoids was printed on swell paper or churned out by a 3-D printer. Each one was created by a volunteer who used a tracing wheel, copying the print figure by hand onto a sheet of Braille paper. The figure had to be copied precisely in reverse of the figure in the print textbook. When the page was flipped over, the raised-line figure appeared right side up. Each diagram was drawn with meticulous attention to detail, the angles and axes labeled in Braille just as they were labeled in the print textbook used by my classmates.
As we packed up my books at the end of tenth grade, my mother glanced over my shoulder. “Let me see that,” she said, and I handed her a random geometry volume.
Mom flipped through the pages, studying one raised-line figure after another. “Who transcribed this book?” she asked. “It’s wonderful!”
“Not wonderful,” I grumbled. “Just geometry.” But I turned to the front page and read, “Transcribed in Standard English Braille by Pauline Packard, American Red Cross, Montclair, New Jersey.”
“Do you have any idea how long it must have taken her to Braille this whole book, with all these diagrams in it?” Mom asked.
I had no idea. I hadn’t given it a moment’s thought.
“You really ought to thank her,” Mom said. “Write and tell her how much you appreciate having all these drawings. Let her know it helped you understand what you were studying in class.”
“Oh Mom!” I groaned. “Nobody else has to be grateful for their geometry book! It’s just something you’re supposed to have in tenth grade.”
My mother was very persistent. She pointed out that Pauline Packard, whomever she was, had spent uncountable hours creating this book for me, a student she would never even meet. She made sure I had the same textbook my classmates were using, complete with all the diagrams I needed to gain some basic understanding of Euclidian geometry.
Dutifully and under duress, I wrote a letter thanking Pauline Packard. I sent it to her care of the American Red Cross in Montclair and moved on. Life swept me along through the decades, and I didn’t give Pauline Packard another thought.
I was shocked a few years ago when I learned that blind high school students in some school districts were being excused from studying plane geometry. Despite all the technological advances in Braille production today, providing textbooks with the necessary graphics was considered too costly and time-consuming. I heard the underlying message that blind students really don’t need to study geometry. When would they ever have to understand the Pythagorean Theorem? Most likely they would never enter a STEM field anyway.
It’s hard to tally up all the flaws behind this reasoning. If educators believe sighted high school students should learn the basics of plane geometry, whatever their career aspirations, then plane geometry is equally important for blind students. Working out a proof in a geometric problem calls for a student to use reasoning based on established corollaries and theorems. Such systematic reasoning is invaluable as we solve problems, mathematical and non-numerical, throughout our lives. Besides, there is an intrinsic beauty in the geometric order of the universe: parallel lines never meet; a straight line is the shortest distance between two points; the whole is equal to the sum of its parts.
There aren’t many Pauline Packards in today’s world, women and men with the time and commitment to transform hundreds of figures by hand from a print textbook into raised-line drawings. Perhaps new devices such as the Monarch, with its multi-line tactile display, will meet some of the need. So far such high-tech devices are dauntingly expensive, only available to a handful of the students who could benefit from their use.
It’s been a very long time since anyone called on me to recite the Pythagorean Theorem. Yet I still remember that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the hypotenuse. The theorem goes beyond mere words for me. In my mind I carry the tactile image of a right triangle with squares sprouting from each of its three sides. It’s an image I explored with my fingertips again and again when I studied plane geometry long ago in tenth grade.
I wish I could write to Pauline Packard today and express my real, heartfelt thanks for her months of dedicated effort all those years ago. She and thousands of other volunteer Braillists helped me and other blind students learn alongside our sighted peers. Surely they deserve a place of honor in the history of the blind people’s movement.