Are Statistics Courses Accessible? Part II: Revisited with 2020 hindsight

Introduction

This work unashamedly builds on a similarly titled paper presented in late 2009 at the first Workshop on E-Inclusion in Mathematical Sciences (WEIMS) (see Godfrey 2009). Huge gains have been made in the overall access to statistics courses, but as I now suggest, the majority of impact is a consequence of how the statistical world has changed, as compared to the rate of development of relevant adaptive tools used by vision-impaired people. Much of the text from Godfrey (2009) is therefore reproduced with out of date material removed or updated accordingly.

The idea of having to take a statistics course often induces expressions of fear and trepidation. Many of the first year students we encounter hate the prospect of taking a statistics course and phrases like “I suck at math”, “I was useless in stats classes at school” and “I’m no good with numbers” are all too common. Anecdotal evidence suggests the blind students have fears too, but these are different, and in my opinion, should be different. It is all too clear that the dominating problem for the blind student is “how am I going to do what they do?”

In Godfrey (2009), I contended that “The author argues that the very real concerns blind students face are not adequately countered by sufficient solutions and encouragement; resources are lacking; role models are few and far between; and the current way Statistics is taught is making things worse not better.” Thankfully, times have changed in so many ways, but while the times are different, they are not uniformly better.

Introductory statistics classes are taught to a very wide range of students. Subject areas at many universities that expect or encourage their majoring students to take an introductory statistics course include all business majors, all technology and engineering majors, psychology, geography, economics, biological sciences such as ecology and veterinary science; Starting in 2020, all science students at Massey University are expected to take a course in statistics.

The timing of the introductory statistics course is far from consistent across subject areas and degree programmes. It’s often hidden away and guised as something like “Research Methodology for …” or “…Analysis”. Sometimes these courses are left to postgraduate level but most are compulsory elements of the majoring requirements for the related discipline. The point here is that while death and taxes are inevitable, having to take a statistics course isn’t far off. In fact, to be sure of not having to take a statistics course means the intending university student probably needs to enrol in a law degree, the fine arts, or one of the Humanities subjects.

Aside from the challenges in actually dealing with numbers, statistics lecturers force students to create and interpret graphs, use a (usually new) software application, and process the mathematical formulae with all that Greek. In addition, many statistics courses also force their students to look up statistical tables. Most mathematics courses do not deal with all of these key elements of the standard introductory statistics course. The vision impaired student, be they blind or having some residual vision, must be ready to overcome the challenges that each of these four elements presents before the semester begins. Let us now address these four elements in turn.

Element 1: Graphs

Godfrey (2009) said “There is no doubt that assessing, interpreting, and creating graphics has its challenges for the blind statistician. Working with graphs is also a challenge when assessment of knowledge is undertaken. How does the assessor know whether it is the student who knows that an outlier is highlighted in the histogram, or whether they were told it was there by their sighted reader/writer? Provision of tactile graphics is probably the only solution that will offer direct access to the graph to be examined.” Work on producing tactile graphics is well-advanced, but the availability and therefore uptake of tactile media frequently remains out of reach for many blind students today. The willingness to request tactile material seems to be lacking (anecdotal evidence), even though the tools to create tactile representations of graphs on the fly have existed for five years (Godfrey and Murrell 2016). Anecdotal evidence suggests that in addition to this lack of willingness , there is a shortage of personally owned braille embossers in the hands of students who might benefit from tactile images.

Efforts to provide a solution based on the exploration of graphs via a combination of synthetic speech and enlargement of portions of graphs (Godfrey, Murrell, and Sorge 2018) show promise to help anyone who understands a graph’s structure, to gain understanding of the particular data instance. This work builds on attempts to automatically generate text descriptions of graphs as they are created (Godfrey 2012).

Despite the existence of these modern tools, blind students continue to rely on sighted individuals to be able to give them the knowledge they would be able to gain were they sighted or had direct access to the graphs through such media as tactile graphs. This remains a concern for any blind person when producing graphs to meet publishing standards, but the vast majority of graphs in statistical analyses form part of the preparatory work, not just part of the final presentation. Ultimately, there is no problem with a blind student asking a sighted person if the graph is showing what was intended. Sighted students ask each other for their opinions, so why shouldn’t a blind person do likewise?

Modern teaching practices in statistics courses are increasingly focused on use of graphs as an intermediate step for developing ideas about a given data set. Without adequate substitutes for the graphs being used by sighted people, blind people are at an obvious disadvantage.

Godfrey (2009) stated “There are, however, alternatives to all graphics. Many graphs were designed to be used as highlights for quickly demonstrating a point discovered via analysis by other means. Calculation of the mean and the median as well as midpoints between quartiles would be used to gauge the skewness of a sample before graphs like histograms were so simply created by software. Now that software can create the graph so easily, we rely on this as the method for determining skewness. Exploratory data analysis techniques are prominent in introductory courses sometimes taking up as much as one quarter of the overall work. The old-fashioned ways are still relevant today but are less efficient for the sighted user. They are, of course, a better option for the blind user to consider, but seldom are they taught in the 21st century. Many of these numerical substitutes for graphs are now given practically no exposure courtesy of the ever-increasing use of graphics in every facet of statistics education. Hypothesis tests exist for analyzing the existence of skewness, but hypothesis tests are usually taught well after exploratory data analysis through graphs.” Little has changed.

The Godfrey (2009) claim that “Blind students need to learn how to do things differently to their sighted peers if they are to use statistical analyses in their ongoing study or careers. The standard introductory course may not actually provide the necessary content required by the blind student.” This remains relevant in 2021.

Element 2: Statistical Software

Statistical software has been an important part of most statistics courses in the 21st century. The choices made by teaching staff have a dramatic impact on the approach to teaching in 2021. The most modern courses make use of the software that interacts well with the other digital tools used in the education process.

Godfrey (2009) unashamedly promoted use of R (R Core Team 2020) for offering the best access to software for blind people. Promotion of this access was given by Godfrey (2013) to the R development community. This superiority remains today, but the latest developments available by and for R users are solving many old problems, while also creating new ones.

Access to standard classroom presentation material should not be an issue in 2021 unless the teaching staff are using what is now old-fashioned methods. However, if the staff use the latest interactive tools in R, there is a chance that blind students will be left in the position of being told what happened while sighted students see it for themselves. Teaching staff therefore need to think carefully about how to explain material in words that these demonstrations display. Demonstration using physical objects may well prove an invaluable approach even if it is seen as a retrograde or archaic solution.

The second (and probably most concerning) source of software issues for blind people is the use of software to generate statistical analyses for themselves. The material taught in introductory courses is changing as we are able to rely more and more on statistical software. Even introductory courses are now using code to analyse data in R, in preference to other software that relies on menus and dialogue boxes. Manipulation of data and the creation of graphics are much easier now too, and this might explain why we now have a greater focus on exploratory data analysis techniques in statistics courses. In addition, the days of generating a few histograms, boxplots, and maybe a scatter plot or two are behind us. As an example, the following code generates a graph (intentionally not included) that was well within the expectation of a first year statistics student at Massey University in 2020.

>  library(ggplot2) 
>  Graph1 = ggplot(mpg, aes(x=displ, y=hwy)) + geom_point() +
>      facet_grid(.~year) + geom_smooth(method="lm")    + 
>      xlab("Engine displacement (litres)") + ylab("Highway miles per gallon")

N.B. The data used in this example is an original component of the ggplot2 package (Wickham et al. 2020) used to generate the graph. Removing the irritation of the use of metric and imperial measurement scales is used as an example for improving the graph.

The R code used above is not all that human-readable especially to an R novice, but the BrailleR package (Godfrey et al. 2020) will go a long way to making sense of it, and therefore the associated graph. A sighted person who has access to the graph might well work out what each component of the code generates, but a reader of this paper is now in the same position as a blind student would find themselves. The BrailleR package generates the following description:

>  library(BrailleR) 
>  VI(Graph1)

This is an untitled chart with no subtitle or caption.
The chart is comprised of 2 panels containing sub-charts, arranged horizontally.
The panels represent different values of year.
Each sub-chart has x-axis 'Engine displacement (litres)' with labels 2, 3, 4, 5, 6 and 7.
Each sub-chart has y-axis 'Highway miles per gallon' with labels 10, 20, 30 and 40.
Each sub-chart has 2 layers.
Panel 1 represents data for year = 1999.
Layer 1 of panel 1 is a set of 117 big solid circle points of which about 57% can be seen.
Layer 2 of panel 1 is a 'lm' smoothed curve with 95% confidence intervals covering 6.4% of the graph.Panel 2 represents data for year = 2008.
Layer 1 of panel 2 is a set of 117 big solid circle points of which about 61% can be seen.
Layer 2 of panel 2 is a 'lm' smoothed curve with 95% confidence intervals covering 6.4% of the graph.

What we learn from BrailleR is that the code has actually generated two graphs, not just one, and that each of them has two “layers” of information. It is clear from the descriptions that a blind student could now know what the graph contains, but still does not get the benefits of the interpretation of its content. The BrailleR package is therefore likely to remain “under development” as efforts are made to catch up with the latest options being given to sighted R users.

The increasing use of markdown in conjunction with R is a huge leap forward in access to statistical analysis of data. Markdown files are plain text files and are processed into (very accessible) HTML, epub3 (also very accessible), or many other (less accessible) file formats (including the pdf version of this paper in the workshop proceedings). The use of R and markdown (Allaire et al. 2020) avoids the pain of copying and pasting text output and graphs, which is comparatively difficult for many blind people. This paper was written using R markdown in preference to LaTeX or MS Word. Reading the HTML version found at https://R-Resources.massey.ac.nz/papers/DEIMS21Access/ will show that the standard heading styles are included; easily read mathematics and alt tags for graphics are also features of R markdown documents. These are done as the default action so little to no user knowledge of accessibility is necessary. Given so many authors are now using R markdown and other associated tools as their normal way of working, the range of accessible statistics texts that demonstrate use of R is broadening. Many of these texts include a web-based search engine which makes the subject index somewhat redundant.

Element 3: Tables

Godfrey (2009) described scientific calculators used by sighted students as a common alternative to the old-fashioned statistical tables still in use. Times have changed and even use of those calculators in statistics courses is decreasing in popularity nowadays.

Godfrey (2009) stated “The student who does not have ready access to the simple tools will become comparatively inefficient.” Even though what is considered “simple” may have changed, this message remains sound. To be efficient and effective, a blind student may need to put extra time into the tools that are going to prove the simplest for them to manage; this decision could well be different to what a sighted student would choose if no advice was sought. My advice to sighted and blind students is practically the same now.

Godfrey (2009) also stated “Be ready to use tables before the course starts. It is unlikely that a software solution will be appropriate for use in assessment exercises such as tests or exams without prior negotiation.” Prior negotiation might be required, but the era of “reasonable accommodation” is upon us in so many situations in life, including education. I would now argue that making use of R or other suitable software (perhaps under supervision or with constraints) should be seen as a reasonable accommodation for a blind student even in circumstances when other students must do manual calculations.

Element 4: Mathematical Formulae

The situation promoted in Godfrey (2009) “Unfortunately there is no escaping horrible formulae, although the reliance on doing all the substitutions and integral calculus that was part and parcel of the introductory course in the 20th century is reducing. There is no way of avoiding the various symbols that arise in a statistics course. The intending statistics student needs to know how they will present work in a format the lecturer can handle.” The evolution of statistics courses mentioned previously has reduced, but not eliminated, the mathematical load on students.

Given the relative leaps forward in access to mathematical content in HTML via tools including Mathjax in particular, and the incredible increase in available accessible mathematical content, blind people working with mathematical content are substantially better off today than we have ever been before.

Accessibility of mathematical software is not a direct focus of this article, but options do exist should the statistical software not provide a solution for the more mathematical coursework at higher levels. This would include the need to perform analytic integration and differentiation for example, neither of which is a common tool for statistical software.

While authoring mathematical content is now relatively easy for a blind person, it does require effort. Use of standard LaTeX for mathematical expressions makes creation of documents in Microsoft Word (MathType is required) or markdown within reach of most students (blind or sighted), so the need to recommend that a blind student entering university should learn everything about LaTeX can now be set aside. One major criticism of LaTeX was always that it generated an inaccessible document so the blind author could not independently verify the outcome; this has been worked on, but the access is still below that offered by Microsoft Word or HTML generated from markdown. As an illustration, this paper was prepared in markdown so that it could be read by the author in HTML and submitted for publication in the organisers’ preferred format, because that is the way I do almost all of my work . Processing of a single file into multiple formats has vastly improved my ability to create documents independently.

It’s far from bad news

Aside from the fact that it is possible to avoid or meet the challenges of the four key elements as described above, there are reasons why blind students can succeed where their sighted peers may not. Statistics is all about the ability to predict the future with knowledge of the past; to apply probability; to deal with randomness and understand variability to name just a few points.

Blind people live life encountering things unexpectedly, even if they have some vague idea of where something should be (estimation) and there is always some amount of inaccuracy (error).

Part of any successful teacher-student relationship is the ability to connect the theory in a course to real life examples that resonate with the students. The experiences of the blind student may not match those of their sighted peers. In the author’s experiences, this mismatch exists in many different ways. For example, international students do not understand any examples based on events in New Zealand’s history without some serious explanation, but all students relate to world news events. Finding examples that match a blind student’s experiences will be difficult for the lecturer who has little knowledge of the blind person’s world. This can be overcome through developing rapport, but this requires some proactive effort by the student.

I firmly believe that some statistical ideas are best learnt from one’s blind peers. Some notions are so easy to frame up for blind students if only the right scenario can be found to help its explanation. Generally speaking, blind people have been able to form networks where they can pick up information from one another. This is especially true for assistance with our specific software just to name one example. In my opinion, those of us at the forefront of new endeavours by blind people must expect to support those that come along behind us. Thankfully, many blind people do this all the time.

Statistical software: The litany of failure

The inaccessibility of software has been recorded in many ways, including Godfrey and Loots (2014) and Godfrey (2013), but perhaps the personal commitment to attend conferences for the R development community and to make direct contact with developers has increased the general knowledge of the potential for R to be a viable solution for blind people. Godfrey and Loots (2014) compared four commonly used statistical software options; R came out on top. Although R as the base product was considered superior, Godfrey (2013) lamented the inaccessibility of the RStudio integrated development environment (IDE) which was becoming increasingly popular in the R user community.

Several statistical software companies needed to be contacted to make the best attempts to ensure fairness and accuracy in Godfrey and Loots (2014). Remarkably little has happened to change the situation between 2015 and 2020. In early 2020 though, RStudio, the developer of the previously inaccessible IDE of the same name, announced it was going to put effort into improving the IDE’s accessibility. This has proven to be welcome news to blind R users who want to work in the same fashion as their classmates and work colleagues. Initial work has shown great promise, in part due to the hiring of a blind person to take a key role in the project.

Role models do exist!

In 2009, I had no idea that I was the only totally blind person to gain full time employment as a Lecturer in statistics. I was an undergraduate in a pre-internet era; the arrival of the internet did not lead to my searching for others like me, until I stumbled upon the email lists promoted by the National Federation of the Blind for various interest groups, one of which was the Blindmath list. I found that I was soon contributing as much as I was receiving from joining that community. I believe my engagement in that community led to my attendance at WEIMS in 2009 and sharing my experiences in Godfrey (2009).

I was approached in 2010 by another blind person who wanted to get a job as a Lecturer in Statistics. He had gone looking for anyone to talk to and discovered that I was the only person to reach out to; this was quite a surprise to me. I always assumed that someone else had already been there and done that before me; I assumed that I just couldn’t find out who the others were, not that there were no others to find. The sharing of ideas with someone who could verify my ideas, challenge some of them, and complement others, was hugely beneficial in a professional sense and culminated in two major publications based on our experiences with statistical software Godfrey and Loots (2014) and as students and lecturers Godfrey and Loots (2015).

Over the years, I’ve kept in touch with a handful of blind people as they complete a PhD in Statistics, some blind people wanting to start an academic career in another STEM discipline or something closely related, and some others who forged their STEM careers before or at about the same time as I did. All of these people now form a community of role models and I no longer feel any pressure to be the only person proving that blind people can become statisticians. I’m proud that blind people working with statistical data are no longer a novelty.

As has now been demonstrated over and over again, there is potential for blind people to gain employment that makes use of their ability to analyse data. While some blind people have forged careers in academic institutions across a wide range of disciplines, others have found success in private industry.

Concluding remarks

Ten years ago, I discovered that I had a role to play in helping improve access to statistics education for blind people. Over those years, I’ve learned a lot from experimentation, trying to show blind people what they can learn, and seeing a lot of good ideas fall flat because they made work for the blind person or the staff that supported them.

In the 20th century, access to printed information was the primary limitation on the blind student. This is a considerably smaller problem for blind students undertaking statistics courses in 2021 than it was even ten years ago. In the author’s opinion, it is the modern ways of teaching statistics that are driving the inaccessibility in the 21st century. The ability of the blind student to take courses in qualitative disciplines and succeed is increasing with the improvements in technology. In many sciences, the opposite is true as we lose ground due to advancements in software and other technologies that are not created with the blind user in mind.

It is possible for blind students to succeed in statistics courses, as long as the right choices in software and the way work will be undertaken are made. We live in information-rich times, and the most current work being done in statistics that is relevant to introductory and advanced students is being presented in a very accessible format. Reliance on obtaining information from the inaccessible pdf format, even with the best enhancements available today, is an inferior way of working and leaves blind students at risk of suffering in a similar fashion to many others who preceded them. It will still take some serious preparation before the course starts, some additional work during the course, and some reasonable accommodations being made by the course staff. As much as the blind student is afraid of failing statistics, most academic staff are afraid of failing capable students.

It was necessary to prove that some things could be done, even if those actions were not eventually taken up by the thousands of blind students around the world. Numerous glass ceilings have been broken and some hugely innovative tools have been developed. The most enduring successes are those tools that provide improved access to blind people and have become embedded into the everyday software and practices being used by everyone. In statistics, this includes improvements in R and SAS, and in mathematics this includes MathJax. I put the availability of R markdown at the forefront of accessible tools that has made such a difference for R users, with both the consumption of printed information and the production of statistical analyses in HTML being key to this success.

With respect to other printed information, we need to be sure that research efforts get put into the tools that are the most practical and useful for blind people, instead of those tools that are the most commonly used by publishers. In my opinion, the efforts to improve access to content in pdf are important. The problem is that unless they become built-in so that they are not optional extras that require authors and publishers to learn how to build in access, they will not gain sufficient traction and therefore not be valued by blind consumers. Whether we like it or not, younger blind people do not want to read material in a fixed page format that is second rate when there is slightly better HTML based solutions ready to hand. They want access on a plethora of devices, running all operating systems, and in my experience, they want them to work right out of the box. It is difficult to see what (if any) benefits a blind person will enjoy from the most accessible pdf that do not already exist when working with documents available in accessible HTML.

The next ten years is going to be very exciting for blind people in STEM. We need to make sure the tools that currently work well continue to do so, and we also need to continue to develop more tools that improve the ability of blind people to succeed in more STEM fields. Access to information is a necessary problem to overcome, but access to modern tools will also continue to pose challenges.