In the early 1990s, anime brother Van Lendert lost his sight in his eyes. The tragic outcome of diabetes. She has played a role in the path that van Lendert has taken in her work ever since. It supports students with visual impairments and their teachers in science subjects. For the past six years, she has been looking to improve the reading and understanding of mathematical expressions in Braille. She got her PhD last month.

Van Lindert tells us about it over a computer screen and it quickly becomes clear how passionate she is. “I once heard someone say about a blind girl of about 12 years old: Leave that hard math, this kid is already going through such a hard time. As long as she’s happy.” It makes Van Lindert sad and angry. “Would you be happy if you dropped out of school? Of course not. You didn’t want to be behind geraniums when you were a kid.”

In the Netherlands there are about eighty braille readers in secondary education. Van Lindert: Half of them attend regular education. Some of them are taking a math B exam, which I think is great.”

The way mathematical symbols are displayed can be compared to a programming language

How does math work in braille?

Nowadays, almost every braille reader works with a laptop. Text written on the screen is converted to braille on a touch-screen braille. The way mathematical symbols are displayed can be compared to a programming language. Take a radical sign for example. In programming code, square root is short for “square root.” You can convert these four characters into four braille characters.

This is especially difficult with long expressions.

“Of course. If a formula naturally fits into one line, you quickly need multiple lines in Braille.” Van Leendert shows a formula that was once part of the VWO Exam in Mathematics A. It has ten fractures, four parentheses. “This formula takes three lines in braille. But on a braille screen, you can only read one line at a time. So you have to remember a lot. Going back to the beginning of the formula, what did that look like again?”

Height isn’t the only complication. “People with good vision just look at the formula and immediately see what it’s about. Something with a fraction, or square root, power, to name a few. Braille readers have to feel very accurately before they can determine something like that.” Van Lindert gives the fraction (x + 3) / (x – 2) as an example. “Braille readers feel that first arc, but only when they get to the fraction line do they know it’s a fraction. It’s hard to get an overview.

“There are countries where separate braille letters are used to mark the beginning and end of a fraction. This helps braille readers to read and understand mathematical expressions more efficiently.”

The braille reader has to build an overview by moving his or her fingers from left to right

In the Netherlands, the start and end symbols are not used?

“No, different braille symbols are used for mathematical symbols all over the world. When we came up with a linear notation similar to a programming language in the Netherlands, about ten years ago, we simply didn’t think about start and end symbols, for example. The notation seemed very nice, and finally Teachers and students were able to communicate well with each other, because math teachers understand this code language, which they can read on the screen. Now I realize: for a braille reader, this notation is not nice at all. He has to build an overview by moving his fingers From left to right “.

What were your experiences before you began your PhD research?

“I noticed that teachers often had low expectations. It made a huge impact on me, and made me work harder.

“Then I got my master’s degree from the University of Greenwich. I researched how to “learn to learn” for students with a visual impairment. I then received a grant from the NWO to explore how you can stimulate collaboration between braille readers and students with good vision and how you can help braille readers On reading concrete graphs. Teachers who have a braille reader in the classroom go the extra mile to accurately describe a graph or graph. The amazing thing is that visually impaired students have also benefited from this.”

During my research in recent years, I have worked with braille readers and sighted people. What came out of it?

„we have with *finger tracing* I looked at how braille readers read mathematical expressions. We compare this to the results *eye tracking* in the sighted. What turned out? The pattern of braille readers’ finger movements depends primarily on the person, not the expression itself. This was different for the blind: the trajectories of scanning eye movements were completely different depending on the type of expression.”

If you need to add two fractions, it is useful to compare the denominators by touch.

And what can you conclude from that?

“Anyway, reading strategies for sighted students usually have something to do with the structure of the expression and the desired way of solving. This turns out to be more difficult for Braille readers. I thought it would help them a lot.”

How can you help them understand such an expression?

Braille readers must learn different tactile strategies to read different types of expressions. In addition, the combination of speech synthesis and Braille can produce a lot. Using speech, they can quickly get an overall view of the expression. Then they can look at it from a deeper level in Braille. Of course they have to learn how to get an overall overview through the ear, because that’s not an obvious thing. In the pictures of Johan Cruyff: You will hear it only when you realize it.

How new is your research?

Existing research places a strong emphasis on technical aspects. How do you convert math to braille accurately and efficiently? My research focuses more on educational aids, which are relatively new. How can you help a student understand something better? If you have to add two fractions, it is useful to compare the denominators by touch. You read, at about the same time, the denominator on your index finger and the denominator on your other index finger. Thus the finger movements depend strongly on the structure of the expression.

“We tested haptic strategies with different types of commands. The effect was that Braille readers got 30% faster. But we don’t yet know if they really got it better. This requires more research, with more complex tasks. Best way to record intermediate steps? How do you explain the relationship between equation and graph intersections? Much remains unclear on this topic. Good work for years to come.”

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

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