# Notes on Teaching Precalculus to a Blind Student in a College Precalculus Course

By Steven Schluchter

Steven Schluchter is an Affiliate Faculty member in the Department of Mathematical Sciences at George Mason University.

## Abstract

This article is a case study of the author’s experiences teaching a precalculus mathematics course for a class that included a blind student. This document details the challenges that were confronted and the solutions that were implemented. The challenges that were encountered included screen reader accessibility of online homework systems and computer algebra systems, the need to formulate appropriate mathematical diction in lecture and in office hours settings, and the need to develop better techniques to teach graphing. These challenges also included a decision on what course content and reinforcement was best left delegated to a regular one-on-one meeting with the student while there was still an imperative for the student to learn in an integrated setting with his sighted peers. We close with a Practitioner’s Notes section, which contains shortened takeaways from the rest of this article and a table of selected mathematical diction appropriate for a teacher to use when communicating with a blind learner.

## Keywords

Mathematics pedagogies, scribes, classroom techniques, one-on-one strategies, adaptive technologies

The existing literature on the subject of mathematics pedagogies for the blind seems to be a collection of case studies and advice geared toward learners and educators in primary or secondary level settings. Spindler (2006) details the case of a student in college, and implies that a small sample size of blind mathematics students makes it difficult to refine pedagogical methods over time. In this particular case, the student in question (“the student”) was in the author’s large lecture course (87 students) at George Mason University (GMU), a large public research university in Fairfax, Virginia. The student was also proficient in the Nemeth Braille Code for Mathematics and Science Notation (“Nemeth Code”), which is a mathematical braille add-on to conventional braille. A subset of blind mathematics learners are proficient in braille, and a subset of those are proficient in Nemeth Code. The student also came to the course with a reasonable programming background, which was leveraged to communicate adapted mathematical diction, and helped with the assignments in Mathematica (a computer algebra system, which we describe later).

A reader who is interested in more general advice and other case studies on teaching a blind student in a tertiary education setting is encouraged to consult Spindler (2006), Pilgrim and Kennedy (2014), and Godfrey and Loots (2015). Godfrey and Loots (2015) offer a wealth of advice and a large collection of references on teaching statistics to blind students, including preparatory advice for students and educators. Their references are quite extensive but devoted to statistics pedagogies. Specifically, this article aims to treat the pedagogy of more fundamental material, yet the challenges confronted by educators at all levels have common threads including the formulation of appropriate diction, the development of graphing pedagogies, and appropriately integrating a blind student with sighted learners, which challenge educators at all levels.

## Notation In Lecture and Classroom Environment

The single greatest challenge during lecture is communicating mathematics verbally in such a way that blind and sighted students can understand the material simultaneously. Sighted students will hear an instructor say “a plus b over c”, see $\frac{a+b}{c}$ on a whiteboard, and they will not be perturbed. However, a blind student will not know whether the expression “a plus b over c” should be taken to mean $\frac{a+b}{c}$ or $a+\frac{b}{c}$. The expression $\frac{a+b}{c}$ is better said as “c divided into the sum of a and b,” or as “a fraction whose numerator is a plus b and whose denominator is c”. The expression $a+\frac{b}{c}$ is better said as “a plus the fraction b over c.” Similar difficulties arose when treating expressions like $\sqrt{a+b}$ and ${\left(a+b\right)}^{2}$. The former can be described as “the square root of the sum of a and b”, and the latter can be described as “the square of the sum of a and b”. Saying “a plus b, quantity squared,” was not helpful and somewhat confusing when an expression like ${\left(a+b\right)}^{2}$ was part of a larger expression, even when a varied speaking cadence was used to imply the proper placement of parenthesis. When describing functions it was seemingly more necessary to include the parenthesis as part of the description of inputs; if an instructor says “f of a plus b,” then it is not obvious whether the intended notation is $f\left(a+b\right)$ or $f\left(a\right)+b$. One may say “f of the sum of a and b” or “f of open parenthesis a plus b close parenthesis.” Since the “open parenthesis” part of the previous sentences is implied, saying “f of a plus b close parenthesis” did not cause any problems.

The expression ${a}^{bc}d$ is also hard to distinguish from ${a}^{b}cd$ if one is trying to say “a to the power of b times c times d” or “a to the power of bc times d”. In this case, the author borrowed from Nemeth Code, and used the “back to base” character: we would read ${a}^{bc}d$ as “a to the power of b times c, back to base, times d,” which distinguishes it from ${a}^{b}cd$.

Of course, the more complicated the expression, the harder the description will be to articulate in such a way that the instructor is reaching everyone in the room. The author found that the sighted students appreciated some more lengthy and artificial verbiage because they knew its purpose; a blind student sitting in the front row of a lecture hall is highly visible, especially when he has a white cane and a braille notetaking device. In fact, there seemed to be a minor spirit of celebration in the room when the blind student would get called on and answer a question correctly; the entire class seemed interested in the student’s success.

It was not damaging to say words like “see”, “red”, “dark”, or any other characteristic qualifier that might leave the instructor feeling that the blind student is being excluded. Indeed, the student would use similar vocabulary when saying “Could you shed some light on that, please?” or “Thank you, I can see that now.”

However, it was inappropriate to rely on visual media as a necessary supplement to one’s writing. Phrases like “over there” cannot be totally understood without a frame of reference, even by a sighted individual. Such a phrase cannot be reliably understood without some other direction like a pointed finger or some other gesture, which is not appropriate for a class including a blind student.

The author is the kind of instructor who likes to maintain a lively dialog in class. It is not always required for students to raise their hands and get called on before speaking. It was necessary to maintain an awareness of the student’s desire to speak since he seemed to feel that his disability kept him from knowing when it was appropriate to speak up. He would only do so when the author would pose a question to the entire class. In this environment, the student was sometimes the only student in the room who would raise his hand.

Lastly, it is important to point out that having a thorough lesson plan is necessary but not sufficient when teaching a blind student. Verbal backtracking hurts everyone, but it especially hurts a blind student. A blind student will presumably take notes using a braille device. The sighted reader is encouraged to contemplate the consequences to the student’s train of thought if an instructor has to fix an error made in the early part of a calculation or derivation if the reader was forced to use a typewriter to take notes. The author felt that his usual lesson plans, which were already close to being “scripted”, were not sufficient, and that it was necessary to think through how to say the more complicated expressions before each class.

## Graphing In and Out of Class

### Graphs Drawn In Class

Graphing in class was done in a slightly more descriptive manner, with no use of the directional terms: left, right, above, below, toward, away, etc., without points of reference. Without eyesight it is a challenge to understand the macroscopic and microscopic features of a graph simultaneously since zooming in and zooming out cannot be effectively done verbally. A two-dimensional graph was started by plotting a point on the graph away from the x and y intercepts, and then describing the behavior of the graph as the independent variable increased. Since the course was a precalculus mathematics course, it was inappropriate to use calculus-based concepts (derivatives, points of inflection, etc.) to describe the shape of a graph. Instead a more intuitive description featuring sentences like: “The graph increases and gets steeper as x increases until reaching (a,b), and then the graph decreases and gets steeper until reaching the point (c,d).”. The reader will note the conflation of the term “the graph” and the actual y values on the graph. These were informally understood to mean the same thing for quicker communication. The x and y values of intercepts of the graph were always noted. It should be noted here that this increasingly descriptive method of describing a graph was helpful, but it was not a comprehensive solution. Graphs and their shapes had to be revisited again outside of class.

### Graphing Out of Class for Tactile Discovery

The specific kinds of tactile media we used to communicate graphs are addressed later in this section. The student’s answers to graphing questions on quizzes and exams had to be dictated, and this is also addressed later in this article.

Just as in most settings, graphing windows (the x and y ranges) should be selected so as to accentuate the distinguishing characteristics of the graph, and small features should be exaggerated (larger graphs were better than expected for our purposes): open dots versus closed dots, dashed lines versus solid lines, curves versus cusps or corners. Despite the blind student having a firm grasp of Nemeth Code, this was not good enough to explore representations of graphs, in part because the representations themselves had to be developed.

When presenting a graph to the student, the author would manually guide the student’s hands. Each graph would be discovered in the following manner (that evolved from a less sophisticated form as the semester went on), which was intended to reinforce a more localized sense of direction that better utilized points of reference. The reader should note the implied repetition of the mathematical content of the graphs both verbally and by touch in the steps that follow.

• The graph(s) would be weighted down so that they would not move when touched.
• The student would be told what kind of graph it was (a sketch drawn by hand, a 3Doodler 2.0 plot, or a 3D-printed graph).
• The student was told what functions or shapes were plotted, and a brief verbal description was given. The description included a list of what features were accentuated.
• If the student was feeling a 3D-graph, printed or doodled, the student would be instructed to extend only his pointer fingers so as to minimize unwanted and confusing contact with the graphs
• The student’s fingers would first be guided over the coordinate axes, one after the other, with one hand going left to right and the other going bottom to top. His index fingers would be paused while touching each other at the origin so that the intersection of the x and y axes would be stressed to the student. This provided a good frame of reference for the student.
• One of the student’s index fingers would be placed on a point of a plotted function, and the (x,y) coordinate pair would be read to the student. The other index finger would be kept at the origin as a point of reference.
• The student’s finger would be moved along the graph (for x increasing) pausing at points that accentuate the prominent features in the graph (intercepts, extrema, etc.).
• If there were multiple graphs on the same set of axes, then the behavior of one graph relative to another would be accentuated by following both curves simultaneously (after following both of them individually), maintaining consistent x or y coordinates as appropriate (sometimes for the purpose of accentuating a horizontal or vertical asymptote). The student would feel one graph with his right hand, and another graph with his left hand since using two fingers on the same hand to feel different graphs was not as effective.
• Varying portions of this procedure would be repeated at the student’s request so that he could get all the information he needed with the benefit of a guide.
• If the graph being felt was a 3D-graph, printed or doodled, the student would be left to rediscover the features of the graph on his own. At this point the student would be able to use more than just his index fingers to touch the graph without causing confusion.
• The student would be instructed to take notes in braille so that nothing important was forgotten.

### The 3Doodler 2.0

The 3Doodler 2.0 (3D-doodler) was used to produce tactile plots of functions when the 3D printer could not be utilized. The 3D-doodler is a device that is handled much like one would handle a pen, except one must fill it with filaments that the device would heat and convey through the tip onto the user’s chosen surface as they write. The 3D-doodler can be used if the goal is to have a dialogue with the student and spend less time making tactile graphics. However, even with this device, a good tactile plot does take a fair amount of time to make. The reader is advised to make the plots before consulting with the blind student. Since each feature of a tactile plot has to be drawn by hand, there is not much advantage to using the 3D-doodler for plots with gridlines unless one has a significant amount of time to draw each one by hand, and each stroke of the 3D-doodler does take time and careful attention.

### 3D Printing

We had access to a 3D printer, and it was used to produce tactile plots of functions. These printouts were no more than a few inches by a few inches, but it does have the advantage of being able to produce gridlines and to plot a function that was taller than the gridlines. The student preferred the graphs that were printed just barely higher than the gridlines so that he could feel the graph and the gridlines simultaneously with one finger (the use of gridlines provided the student with more points of reference than only highlighting extrema, intercepts, etc.).

Prof. Evelyn Sander (2015) lent her expertise in developing 3D-printed graphs. Her website contains a general set of instructions on how to make 3D-printed graphs. A problem with this process is that the printer sometimes fails, which can result in multiple attempts being required to produce a graph. However, this technology did produce the most useful graphs where the gridlines provided helpful points of reference that assisted the student in feeling the graph on his own after being guided through it. The texture of the graph was also kept uniform (which could not always be done effectively using other media), the thickness and height of the graph and the gridlines were easy to control and could be kept uniform, and the graphs themselves broke less easily than the 3D-doodler produced graphs. The student preferred the 3D-printed graphs to all other forms of graphing that were used in this course and those that he had previously experienced (e.g., thermoform, etc.).

### Graphs Drawn with Pencil on Paper

This is a medium that should only be used as a method of last resort since the other media led to much better results on quizzes and exams. Moreover, this medium does not lend itself to the more undirected learning that can take place with the previous two graphing media discussed in this article. Graphs drawn with pencil on paper were effective when getting the student to associate specific hand motions with specific graphs, but the more subtle details were not as transparent, and had to be covered multiple times. It is recommended that the plots of graphs be traced repeatedly and even more slowly.

## Testing and the Course Text

### Braille and Nemeth Code

The student was trained to read Nemeth Code before arriving at GMU, and his textbook was transcribed into Nemeth Code by an outside vendor with whom GMU contracted. The author was not involved in the contracting. The brailled course text was produced in electronic form, one chapter at a time, and this was made available to the student in the form of files stored on a flash drive, which the student could read on his braille-enabled computer.

One unanticipated challenge was that while the student was trained to read Nemeth Code, the student did not know all of the Nemeth Code that he would need for the course: some notation had to be learned as the semester progressed. The student had to learn a sufficient quantity of Greek letters, which, in Nemeth Code, are communicated using specific alphanumeric sequences. The author and the student both felt that the lack of certainty concerning the student’s knowledge of all of the Nemeth Code that he would read warranted his scribe being present for the full lengths of quizzes and exams, meaning the student could have any problem read to him if necessary.

### Scribes

Scribes were used for MyMathLab (an online homework system), Mathematica code writing, and quizzes/exams. A mathematically competent scribe is absolutely essential to the student’s success, no matter the capacity in which the scribe serves. There is a definite disadvantage to having a mathematically incapable scribe when blind. A sighted student (say, one with a mobility disability) who needs a scribe has the benefit of being able to see what the scribe has drawn, and then resolve any discrepancies between what was intended and what was drawn. Only a mathematically competent scribe can detect ambiguities and resolve any confusing statements by asking non-leading questions. Leading questions are to be avoided in quiz and test situations since they may give away answers or lead a student to an incorrect answer. Problems with scribes arose from the differences in the special mathematical verbiage used to communicate concepts with the student by each scribe (i.e., each person has his or her own way of attempting to formulate a nonstandard set of diction).

### Graphing

A scribe was needed to take verbal input from the student, and this scribe was trained to ask non-leading questions of the student if they spoke ambiguously. The author had to coach the student to give elegant and concise descriptions of the shapes of graphs since he began the course giving descriptions of graphs in terms of very coarse interpolations through sampled points. This coaching was a good opportunity to positively reinforce the effective verbal articulation of the techniques that were taught in the course.

The process of grading the work of a blind student is different from grading the work of a sighted student. All of the student’s written work was produced in braille and then sent to an outside contractor for transcription into written mathematical text. The author was not involved with this work. The following are the challenges that arose.

• The occasional braille typo did arise. One misplaced dot can have serious consequences as far as transcription is concerned. It was sometimes a challenge to decide when a typo was not a serious algebra error. This was usually judged by seeing if the typo was corrected on the subsequent line.
• The author would have to give alternative forms of feedback that were readable by a screen reader or otherwise comprehensible to the student. Emailed typewritten feedback worked for most quizzes and Mathematica assignments. Exam-related feedback was given privately and in an office-hours setting, which worked better since the problems were themselves more involved and more prone to student errors. Verbal feedback was more easily understood anyway given the author’s inability to read Nemeth Code, and the occasional inability of a screen reader to read mathematical notation.
• Turnaround time was a problem. Transcription of each quiz would usually take a few days, and each exam would usually take a week or more. This made for difficulties in providing timely feedback, which could be mitigated by having our customary meeting very shortly after getting the transcribed exam back.
• The student would sometimes produce multiple submissions of answers for the same problem. When using a typewriter and putting multiple solutions on the same page, it is impossible to scratch out work that the student wants the instructor to ignore or to discard the portion that contains the erroneous work. Since all typed and submitted braille was transcribed, the author had to skim the whole exam, decide which work was relevant, and then grade that work. This became less of an issue as the semester went on after the author instructed the student to begin each new solution on a new sheet of paper.

## Office Hours

Office hours for this student served several purposes aside from the usual question-and-answer sessions, including: providing feedback on exams (as stated earlier, emailed typewritten feedback worked well for Mathematica assignments and quizzes); providing additional guidance on things that the student may not have asked about during class; and spending additional time teaching graphing. It was necessary to have almost weekly meetings with the student. For the most part, the meetings had to be scheduled outside of regularly scheduled office hours to allow for one-on-one meetings between the blind student and the instructor.

## Reinforcement of Class Content

Long derivations done in class sometimes required an additional discussion during office hours, as the student would sometimes get lost and not ask to return to that point in the explanation. This was sometimes a function of the typewriter-like braille device being the medium of notetaking. Whereas a sighted student might be able to suspend his or her disbelief, continue writing, and ask a question later to fill in a gap in their thought process, the blind student would be hampered by not being able to see both the individual lines and the whole derivation at the same time (since his braille-enabled computer could only render one line of text at a time), and would therefore be less inclined to ask a question. The student had a difficult time identifying exactly where in the derivation his point of concern was. The author found the following techniques to be useful when talking the student through a long derivation in an office-hours setting.

• Prepare the derivation in advance, and contemplate how to say all steps appropriately. As a way to make it easier to think about how to say something, the author would sometimes pause to write a whole derivation or long solution, think about how to articulate the details more appropriately, and then talk the student through it. The student was aware of the author’s efforts to avoid verbal backtracking, and was very patient when the author would take a minute to think about how best to articulate the next step if the in-class explanation was not effective.
• Do not worry about making too much eye contact, and do not try to read too much into the student’s facial expressions. Like most teachers, the author was used to taking questions, giving answers, and reading facial expressions (since some students will not say when they are having trouble) as a way of sensing how the meeting was going. If a sighted student is one who is easily flustered, then being very selective about when to make eye contact is sometimes a good idea since it takes some unnecessary psychological pressure off of the student. In this case, the student did not make a wide variety of facial expressions and his tone of voice was a better indicator of how he was processing the author’s input. The student would sometimes smile, but the author was not able to make a clear distinction between a facial expression conveying pensiveness and a facial expression that conveyed frustration.
• Be prepared with even more technical verbal descriptions. The absence of other students made it less worrisome that the author was speaking in a manner that the author felt was too artificially mechanical for consumption by both the blind student and the class. Something like $\frac{\left[{\left(a+b\right)}^{2b}+1\right]c}{{\left({\left(4a\right)}^{2}+2b\right)}^{3}}$ can be described as: “fraction, numerator, open bracket, open parenthesis, a plus b, close parenthesis, to the 2b power back to base, plus 1, close bracket, times c, denominator, open parenthesis, open parenthesis, 4a, close parenthesis, squared, plus 2b, close parenthesis, cubed, end fraction”. The reader might liken this to speaking in adapted LaTeX (a markup language used for writing technical documents), which is how the author approached it. Speaking in terms of adapted code seemed to work well for the student because they were able to read and think character by character if necessary. The student came to the course with a programming background, which may have helped to make the more artificial language more intelligible. The student would make an effort to speak the mathematics in the manner in which the author would articulate them, and this was helpful for our discussions.
• Sometimes (as the author would do with students who needed only macroscopic advice), it was effective to summarize the next step, and have the student say what the result would be after the step was executed. For example, the author would say something like: “Complete the square on the left-hand side. What do you get?” This was done in keeping with the usual goals of getting a visiting student to find out that the problems assigned are not beyond their grasp and assessing where the student’s difficulties really are, which are often not exactly along the lines of their questions. Doing this with the student was not difficult since he is the kind of student who already wanted to do things on his own.

## Other Technology

### Mathematica

Mathematica is a computer algebra system that uses the resources of a computer to perform complicated tasks. Mathematica is very powerful and can accomplish computations suitable for almost any undergraduate mathematics course and many industry settings. Unfortunately, Mathematica was not accessible in its IDE (Integrated Development Environment, a graphic user interface) form. It was suggested that we try using the Wolfram Kernel (a command line interface), but we were unable to get that implemented in time. Mathematica assignments were transcribed into braille so that the student could read them, and then a scribe had to type the code (which was stated character by character by the student) and execute it. This braille transcription was not done by the author. While the Wolfram code documentation is otherwise very user friendly, the student was not able to access the documentation from within the Mathematica IDE because of the existing accessibility issues. However, he was able to use an internet search engine to search for the specific commands and access the documentation that way.

### MyMathLab

The course utilized MyMathLab, an online homework system. While some questions were accessible via the student’s screen reader, a complete assortment of questions were not accessible. The questions that the student did not find accessible were transcribed into braille, and the student’s answers were entered by a scribe.

### Course Documents

LaTeX was an effective tool for producing screen reader-accessible documents, and for producing documents that could be easily translated into braille (including Nemeth Code) using applicable technologies.

## Practitioner’s Notes

### A Table of Mathematical Diction

 Expression An Effective Reading $\frac{a+b}{c}$ the sum of a and b divided by c $a+\frac{b}{c}$ a plus the fraction b over c $\sqrt{a+b}$ the square root of the sum of a and b ${a}^{bc}d$ a to the power of b times c, back to base, times d $f\left(a\right)+b$ f of a, close parenthesis, plus b $f\left(a+b\right)$ f of the sum of a and b

### A Collection of Short Takeaways

• The author wishes to stress the need for continued collection and refinement of mathematics pedagogies for blind mathematics learners in tertiary settings. If so much of what is addressed in this article appears to be a reinvention of what is widely known, that is because it probably is the case. Any experimentation should be documented and published, and the collective body of work should be bound and studied thoroughly.
• While it is not harmful to use descriptive qualifiers that do not require a point of reference for location purposes, a phrase like “over there” is not effective without some other contextual point of reference.
• Correcting one’s mistakes and all other forms of backtracking hurts everyone, but especially a blind student. Instructors should be as close as they can be to being scripted for each class period.
• It is not harmful to use somewhat inelegant mathematical diction in one-on-one settings if it is still accurate and precise (the author felt that this was too awkward for classroom settings). One may speak in ways that do not seem natural and still effectively convey mathematical content. A reader who knows just the basics of computer programming may benefit from likening this kind of spoken mathematical language to how one would read a computer program; drawing inspiration from a mathematical markup language, like LaTeX, seemed particularly effective.
• It is recommended that students hear descriptions of graphs that incorporate how the shape of the graph changes as the independent variable increases. This is an opportunity to reinforce the language of the course.
• It was particularly effective to have the student associate hand motions with the shapes of a graph. The use of the 3D-printed and 3D-doodled graphs was particularly helpful towards this objective. Moreover, repetition was helpful, as was letting the student feel the graphs independently after having his hands guided a few times.

## Acknowledgments

The author thanks Evelyn Sander for her help with producing 3D-printed graphs.

The author thanks the editors and reviewers for their carefully considered feedback, and for steering this article to publication.

The author wishes to thank the student, whose dedication and bravery were quite inspirational.

## References

Godfrey, A. J. R., & Loots, M. T. (2015). Advice from blind teachers on how to teach statistics to blind students. Journal of Statistics Education, 23(3), 1-28. doi: 10.1080/10691898.2015.11889746

Pilgrim, M. E., & Kennedy, P. (2014). Accommodating blind students taking mathematics. MAA FOCUS, 34(1), 41-43. Retrieved from http://digitaleditions.walsworthprintgroup.com/publication/?i=198193#{%22issue_id%22:198193,%22page%22:0}

Sander, E. (2015, November 4). Tactile graphs for a blind math student. Retrieved from http://gmumathmaker.blogspot.com/2015/11/tactile-graphs-for-blind-math-student.html

Spindler, R. (2006). Teaching mathematics to a student who is blind. Teaching Mathematics and its Applications: An International Journal of the IMA, 25(3), 120-126. doi: 10.1093/teamat/hri028

The Journal of Blindness Innovation and Research is copyright (c) 2018 to the National Federation of the Blind.