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I am interested in how blind people learn mathematics at any level, but particularly before college. Math is often taught using a lot of visualization; how does this work with blind people?

My interest in this is a little round-about. I have nonverbal learning disabilities (NLD) and am writing a book about it. I am good at math (I work as a statistician) but very poor at visualization. Most people with NLD are quite poor at math (ranging from difficulties at the very earliest levels to problems with upper-elementary school "word problems" to problems with geometry (high school level).

My idea is that most people learn math visually, but I learn it largely non-visually. But NLD is relatively little known. So, I thought that the methods used to teach blind people might be useful for others who have visual difficulties, even if we can see.

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    $\begingroup$ Related (but not an answer): matheducators.stackexchange.com/questions/992/… $\endgroup$ – Benjamin Dickman Jun 7 '14 at 13:04
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    $\begingroup$ Also, I've reposted this on the blindmath mailing list to see if anyone there wants to contribute. $\endgroup$ – Loop Space Jun 7 '14 at 14:53
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    $\begingroup$ @PeterFlom With regard to your last paragraph ("My idea is that most people learn math visually, but I learn it largely non-visually") you might be interested in some of the quotations I pulled from Hadamard's (1945) classic work in this MESE response: matheducators.stackexchange.com/a/1801/262 $\endgroup$ – Benjamin Dickman Jun 7 '14 at 15:21
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    $\begingroup$ Here is an article to get you started: "The World of Blind Mathematicians" from Notices of the AMS ams.org/notices/200210/comm-morin.pdf $\endgroup$ – Mikko Korhonen Jun 9 '14 at 15:56
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    $\begingroup$ Based on very limited personal experience teaching a person with vision impairment: It is not that people learn math visually, it is that they learn it spatially. So a person with a vision impairment can learn math using tactile senses. Manipulatives are great for this purpose. $\endgroup$ – Anonymous Physicist Oct 15 '14 at 2:36
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This may not directly answer your question, but perhaps it will help.

The brain is not born preprogrammed. It evolves during development (ie conception to adulthood) and gradually adapts to its environment. As such, it is really a misconception that the brain has particular areas that we are born with that must automatically carry out certain functions. Of course, it would be ludicrous to say that there aren't certain distinct regularities. But one must first appreciate just how adaptive brains are. Let me illustrate this.

For example, in all mammalian brains, there is a particularly important area called the thalamus. Destruction of this region results in a loss of consciousness. So it essentially controls the flow of sensory information into the various higher order regions. Researchers who studied the thalamus have run experiments with different mammals and found that, if you connect their visual organs to the traditional auditory regions during development, that brain area will actually grow into a visual processing center instead of an auditory one. In other words, it's no longer an auditory processor, but now a visual one. That's extraordinary. It means an area that one would think would be devoted completely to one sense (ie vision) can actually develop into a completely different kind of circuit (ie auditory processing). Thus, it is really inaccurate to say that a particular brain region by necessity has to correspond to a particular function. It all depends on how the system evolves from birth.

Because most humans grow up with fairly similar sensory inputs and overall wiring patterns, the average human brain tends to share similar patterns of functional circuitry. So it isn't a free for all. Nature gives our brains strong biases and incentives so that there's a tendency to prefer certain circuit relationships. That's why for most people, the auditory circuits don't get mixed up with visual. It's not a free for all, but a guided self-organizing process. So it works out that most of the time, but on occasion you do see irregularities (eg synesthesia).

Now, as for mathematical areas in particular, in humans, the brain has select areas that tend to participate in mathematical calculation and reasoning. For example, the angular gyrus is one such area. Here's a general article about the different regions: http://newsoffice.mit.edu/1999/math-0512 You can read the original papers if you're interested. It's not hard to find general information about math and the brain. However, there are so many kinds of math and types of math thinking, they can't study all of them. So the best you can do is google around and hope someone has done experiments relating to your topic of interest.

Let me elaborate on this point further. The brain doesn't just have a "math center". Quite the contrary. Each type of math may have induce different kinds of firing patterns spread out in different regions. Geometry may induce different patterns than algebra. Algebra may be different from mere arithmetic. For instance, just because I can add 5 + 5 doesn't mean my brain understands the concepts of inverses, commutativity, associativity, etc. So don't assume that all math is processed the same way.

Finally, to make meaningful comments about how the brain regions that participate in a blind person, nothing substitutes for real data. I'm not familiar with this topic, so I don't know if there are any studies about math, blindness, and visualization in particular. But I do know that people who are blind from birth have very different wirings from those who go blind after having sight. People who go blind after having seen retain the ability for visual imagination. They can still dream visually and presumably do other things like spatial rotation. This is not true for those who are born blind. In all case though, most blind brains start to channel extra processing through other senses. For instance, they may start to show augmented spatial processing capacities in their auditory cortical areas. Same for haptic (ie touch). To get a sense for why this would happen, turn off your lights in your room at night and you'll immediately start to rely on these additional sense to give you cues. But ---interestingly --- unlike a blind brain, your brain is still getting visual data. Well, a blind brain realizes that its not getting data and starts to adapt. It says, "what a waste of neurons! I might as well use them for other stuff." So the circuits may shift some of the functions to other regions like auditory areas. This is similar to how some people develop savant capabilities after brain damage. You might google the term plasticity, although beware since it's an overused term and thus is only so useful.

Hope this helps!!

EDIT:

In case your main question was about the practical strategies people use, I'll add the following remarks:

Since different kinds of math use different brain areas, you might expect people to display deficits associated with the particular senses they have trouble with. So if visualization is a problem, then geometry might be a weakness or harder to learn whereas algebra and numbers might come more easily. I have a friend who is terrible with visualization. Can't even hold up four fingers properly! But she can understand the concept of a vector algebraically. So if someone stinks at algebraic thinking, teach them vectors with arrows. Conversely, if a person can't visualize a vector, introduce it algebraically through axiomatic vector space theory. Math is cool because you can teach a concept many different ways and some of the coolest math isn't visual at all.

I also know that business and communications people have argued for a while that its best to target all senses while teaching to an audience through what is called "neuro-linguistic programming" (NLP). You may find this literature provides you with ideas for how to convert a teaching method in one modality into another.

Finally, there are many books on learning disabilities that involve systematic studies on how to help people with them learn best. You might consider consulting that type of literature. It's hit or miss but there are definitely intelligent and systematic people researching practical strategies for helping LDs of all kinds.

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    $\begingroup$ Google "seeing with your tongue" for some relevant material on the plasticity of the brain. $\endgroup$ – Tom Copeland Mar 1 '15 at 3:43
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The post begins:

I am interested in how blind people learn mathematics at any level, but particularly before college. Math is often taught using a lot of visualization; how does this work with blind people?

The following information concerns not the main question, about the use (or nonuse) of visualization in teaching mathematics to blind people, but rather about new findings in how blind people understand mathematics; in particular, with respect to the part of the brain largely responsibly for visualization (i.e., the visual cortex). To this end, I point to an article recently published in PNAS:

Kanjlia, Shipra, et al. "Absence of visual experience modifies the neural basis of numerical thinking." Proceedings of the National Academy of Sciences (2016): 201524982. Link 1; Link 2.

Here are a couple of findings about mathematical understanding from the aforecited taken out of a mainstream media piece (Futurity):

People blind from birth appear to do math in a part of the brain typically devoted to vision, a study finds.

Researchers using functional MRI watched the visual cortex in the brains of congenitally blind people as they solved algebra problems in their heads.

The visual cortex didn’t merely respond, the researchers say. The more complicated the math, the greater the activity they saw in the vision center. The same did not happen in the brains of sighted people who—with masks covering their eyes—did the same math exercises as the blind subjects.

The aforelinked continues:

The study contradicted conventional wisdom in another way, says lead author Shipra Kanjlia, a graduate student in psychological and brain sciences. Humans as young as babies have a basic number sense: “My dish has more carrot sticks than apple slices,” or “This picture has more dogs than cats.” Many believe this number sense evolves from seeing the world and trying to quantify all the sights. But seeing has nothing to do with it; the study showed that the brain network behind this sort of numerical reasoning is identical in blind and sighted people.

“The number network develops totally independently of visual experience,” Kanjilia says. “These blind people have never seen anything in their lives, but they have the same number network as people who can see.”

I reiterate that I do not know how visualization is incorporated into mathematics education for the blind, but I suspect that current/future efforts to create materials in this direction would do well to take into account findings such as the one described above.

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  1. I hate to be the one, but...a Google search gives several articles on the topic. Didn't see one killer one. But also did not see them all a waste. Think skimming a few would help (and more content than in QA here).

  2. Quora has several responses also. Being even more the one.

  3. One interesting insight in some articles is that it appears there is "cannibalization" of the visual cortex by blind people to use it for math! Seems to be just with people blind from birth (presumably others don't since their bodies hope to get the sight back). Also I don't mean to say they have an advantage. Fer sure, still many hurdles for a blind man. But still...fascinating that this accommodation is made.

  4. Some other articles comment on the obvious issues with following lectures (helps if teacher speaks out the equations). Q&A helpful to blind person but there can be concern on slowing lecture down. Office hours needed more, used more to get dedicated instruction.

  5. Nemeth Braille is one system for showing math characters: https://en.wikipedia.org/wiki/Nemeth_Braille

  6. For some reason, blind R&D mathematicians tend to be more often geometers! Not sure why this is (or even the proof through statistics, but several people mention it). Maybe it is just perplexed combatitiveness? Or maybe it is that the relationships of advance geometry are a strain for anyone to visualize?

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