In the early 1990s, a brother of anemic van Leendert lost his sight. The tragic consequences of diabetes. It has played a part in the trajectory of Van Leendert in her career since then. She supports visually impaired students and their teachers in science subjects. For the past six years she has been conducting research to improve reading and comprehension of Braille terminology. She earned her doctorate in it last month.

Van Leendert talks about it through the computer screen, and it soon becomes clear how emotional she is. “I once heard someone say about a blind girl of twelve: Give up that difficult math, that kid is already too hard. As long as she’s happy.” This makes Van Leendert sad and angry. Do not want to.

There are about eighty Braille readers in secondary education in the Netherlands. Van Leendert: “Half of them attend general education, and some of them write exams in Mathematics B, and I think it’s great.

The way mathematical symbols are displayed is comparable to a programming language

How does mathematics work in Braille?

“Right now, almost all Braille readers work with laptops. The text written on the screen is converted to Braille on a touching Braille display. The way mathematical symbols are displayed can be compared to a programming language. Take a radical symbol for example. In the programming code it is sqrt, short for ‘square root’. You can convert those four letters to four Braille letters.

This is especially difficult in longer expressions.

“Of course. If a formula usually fits into one line, you need several lines in Braille immediately. Van Leandert shows a formula that was once part of a VWO exam in mathematics. It has ten fractions, four brackets.” This formula takes three lines in Braille. But in the Braille display you can only read one line at a time. So you have to remember a lot. Back to the beginning of the formula, how did it look again? “

Not only length but also complexity. “People with good eyesight look at a formula and immediately see what it is. A fraction, or a square root, something with a power, just a few. Braille readers need to experience it very precisely before they can determine anything like that. Van Leendert gives the example fraction (x + 3) / (x – 2). “Braille readers feel the first parenthesis, but they only know that it is a fraction when they reach the fraction. It’s hard to get a review.

“There are countries that use special Braille to declare the beginning and end of a fraction. It helps Braille readers to read and understand mathematical terminology more efficiently.

A Braille reader needs to make an overview by moving his fingers from left to right

In the Netherlands, are not those beginning and end symbols used?

“No, different braille notations are used for mathematical symbols around the world. When linear and programming language-like notations were introduced in the Netherlands ten years ago, for example, we did not think about starting and ending symbols. The notation felt so beautiful that eventually the teachers and students were able to communicate well with each other because the math teachers understood the code language that could be read on the screen. Now I understand: that notation is not pleasant at all for a Braille reader. He needs to make a review by moving his fingers from left to right.

What were your experiences before you started your PhD research?

“I have noticed that teachers often have low expectations. It touched me so much, it made me work even harder.

“After that, I graduated from the University of Greenwich. I explored how visually impaired students can ‘learn to learn’. I later received a grant from the NWO to explore how we can stimulate collaboration between Braille readers and students and help Braille readers read touching charts. Teachers who have a braille reader in the classroom make an extra effort to accurately describe a drawing or graph. It is noteworthy that students with good outlook also benefited from this.

During your research in recent years, you have worked with Braille readers and visually impaired people. What came out of it?

“It simply came to our notice then *Finger tracking* Braille readers looked at how they actually read mathematical expressions. We compared this with the results of *Eye tracking* In visually impaired people. what happen? The pattern of Braille readers’ finger movements depends primarily on the individual, not on the expression itself. It was different for the visually impaired: the scanning paths of the eye movements were very different for each type of expression.

If you want to add two fractions, it helps to compare the two denominators by touch.

Also, what can you conclude from that?

“However, reading strategies of visually impaired students are usually related to the structure of the vocabulary and the required solution. This makes it more difficult for Braille readers. I doubted it would help them much.

How can you help them understand such an application?

“Braille readers need to learn different tactics to read different types of expressions. In addition, the combination of speech synthesis and Braille can produce a lot of results. With speech, they can quickly gain a global view of the word. Then they can look deeper into Braille. Of course they need to learn how to get a global overview by ear, because it is not an obvious one. In Johan Cruyff’s Imagery: You only hear it when you recognize it.

How new is your research?

“Existing research focuses heavily on technical aspects. How do you translate mathematics into Braille accurately and efficiently? My research focuses more on advice, which is relatively new. How can you help a student to understand something better? Almost simultaneously, you read one denominator with one index finger and the other denominator with the other index finger.

“We tested touch tactics with different types of commands. Braille readership grew by more than 30 percent. But we do not yet know if they really understand that. This requires more research with more complex assignments. What is the best way to record intermediate stages? How would you describe the relationship between an equation and graph intersections? Many people are still unclear about this. Great work in the years to come. ”

A version of this article also appeared in the NRC on the morning of December 6, 2021

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