Blind people use brain’s visual cortex to help do maths

From New Scientist, by Colin Barras, September 2016:

“It’s actually hard to think of a situation when you might process numbers through any modality other than vision,” says Shipra Kanjlia at Johns Hopkins University in Baltimore, Maryland.

But blind people can do maths too. To understand how they might compensate for their lack of visual experience, Kanjlia and her colleagues asked 36 volunteers – 17 of whom had been blind at birth – to do simple mental arithmetic inside an fMRI scanner. To level the playing field, the sighted participants wore blindfolds.

We know that a region of the brain called the intraparietal sulcus (IPS) is particularly active when sighted people process numbers, and brain scans revealed that the same area is similarly active in blind people too.

“It’s really surprising,” says Kanjlia. “It turns out brain activity is remarkably similar, at least in terms of classic number processing.”

Read the whole story. Journal reference: PNAS, DOI: 10.1073/pnas.1524982113

From the original research paper:

Significance

Human numerical reasoning relies on a cortical network that includes frontal and parietal regions. We asked how the neural basis of numerical reasoning is shaped by experience by comparing congenitally blind and sighted individuals. Participants performed auditory math and language tasks while undergoing fMRI. Both groups activated frontoparietal number regions during the math task, suggesting that some aspects of the neural basis of numerical cognition develop independently of visual experience. However, blind participants additionally recruited early visual cortices that, in sighted populations, perform visual processing. In blindness, these “visual” areas showed sensitivity to mathematical difficulty. These results suggest that experience can radically change the neural basis of numerical thinking. Hence, human cortex has a broad computational capacity early in development.

Abstract

In humans, the ability to reason about mathematical quantities depends on a frontoparietal network that includes the intraparietal sulcus (IPS). How do nature and nurture give rise to the neurobiology of numerical cognition? We asked how visual experience shapes the neural basis of numerical thinking by studying numerical cognition in congenitally blind individuals. Blind (n = 17) and blindfolded sighted (n = 19) participants solved math equations that varied in difficulty (e.g., 27 − 12 = x vs. 7 − 2 = x), and performed a control sentence comprehension task while undergoing fMRI. Whole-cortex analyses revealed that in both blind and sighted participants, the IPS and dorsolateral prefrontal cortices were more active during the math task than the language task, and activity in the IPS increased parametrically with equation difficulty. Thus, the classic frontoparietal number network is preserved in the total absence of visual experience. However, surprisingly, blind but not sighted individuals additionally recruited a subset of early visual areas during symbolic math calculation. The functional profile of these “visual” regions was identical to that of the IPS in blind but not sighted individuals. Furthermore, in blindness, number-responsive visual cortices exhibited increased functional connectivity with prefrontal and IPS regions that process numbers. We conclude that the frontoparietal number network develops independently of visual experience. In blindness, this number network colonizes parts of deafferented visual cortex. These results suggest that human cortex is highly functionally flexible early in life, and point to frontoparietal input as a mechanism of cross-modal plasticity in blindness.

Alexandra O Fradkin: Exploring Rectangles

Friday is a special day in our math classes at the Main Line Classical Academy.  We read and discuss mathematical stories and we do exploration projects.  Here is the project that we did with the 2nd-4th grades last Friday.

It began with one of my favorite questions to discuss with kids: What is a rectangle?  Some of the kids in each class had participated in previous discussions with me on this topic, but this was close to 2 years ago and so probably had very little effect on the outcome.

Here is what the boards looked like after the 2nd grade and the 3rd/4th grade discussions respectively:

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The kids used a lot of hand motions in their initial descriptions, but I told them to pretend that we were talking on the phone and I couldn’t see them.  They would also sometimes come up with very long and convoluted explanations, which I also refused to write on the board.  After each initial set of properties, I’d try to draw a shape on the board that fit them all but was not a rectangle or did not fit some of them and was a rectangle (some of the shapes unfortunately did not make it into the pictures).  The kids had a lot of laughs when I would draw a silly shape and ask them “is this a rectangle?”  In the end though, I believe that we settled on a set of properties that succinctly characterized rectangles.

The second part of the class consisted of making all possible rectangles out of a given number of squares.  The kids had to make them out of snap cubes and then draw them on graph paper.  The second graders all got 12 snap cubes while the 3rd/4th graders initially got 12 and then each their own different number between 18 and 32.

I was very surprised that no one tried to draw the same rectangle in different orientations.  Some kids did, however, try to make and draw rectangles with holes in them.  A few of the second graders initially had trouble because the squares on the graph paper were smaller than the snap cubes, so tracing the structure did not work.  However, after a brief discussion, they were all able to make the one-to-one correspondence between the cubes and the squares.

Here are some pictures of the process:

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In the end, we discussed with both groups how to make sure that we have made all the possible rectangles.  One of the older kids pointed out the connection with factors/divisors of a number.  None of the kids had formally studied area or multiplication (although most know what those are to various degrees), but those will both be big topics in the 3rd/4th grade class this year.  I think that this served as a good indirect introduction to them.

Olivier Gerard: Learning mathematics as a Russian interpreter

You might be interested in reading How I Rewired My Brain to Become Fluent in Math, by Barbara Oakley, in Nautilus, October 2, 2014.

A quote:

“Time after time, professors in mathematics and the sciences have told me that building well-ingrained chunks of expertise through practice and repetition was absolutely vital to their success. Understanding doesn’t build fluency; instead, fluency builds understanding. In fact, I believe that true understanding of a complex subject comes only from fluency.

In other words, in science and math education in particular, it’s easy to slip into teaching methods that emphasize understanding and that avoid the sometimes painful repetition and practice that underlie fluency. “

How to raise a genius: lessons from a 45-year study of super-smart children

How to raise a genius: lessons from a 45-year study of super-smart children, by Tom Clynes, 07 September 2016, in Nature | News Feature.

On a summer day in 1968, professor Julian Stanley met a brilliant but bored 12-year-old named Joseph Bates. The Baltimore student was so far ahead of his classmates in mathematics that his parents had arranged for him to take a computer-science course at Johns Hopkins University, where Stanley taught. Even that wasn’t enough. Having leapfrogged ahead of the adults in the class, the child kept himself busy by teaching the FORTRAN programming language to graduate students.

Unsure of what to do with Bates, his computer instructor introduced him to Stanley, a researcher well known for his work in psychometrics — the study of cognitive performance. To discover more about the young prodigy’s talent, Stanley gave Bates a battery of tests that included the SAT college-admissions exam, normally taken by university-bound 16- to 18-year-olds in the United States.

Read the rest of the story

Tim Gowers: In case you haven’t heard what’s going on in Leicester …

[Reposted from Tim Gowers’ Blog, 15 Sept 2016]

Strangely, this is my second post about Leicester in just a few months, but it’s about something a lot more depressing than the football team’s fairytale winning of the Premier League (but let me quickly offer my congratulations to them for winning their first Champions League match — I won’t offer advice about whether they are worth betting on to win that competition too). News has just filtered through to me that the mathematics department is facing compulsory redundancies.

The structure of the story is wearily familiar after what happened with USS pensions. The authorities declare that there is a financial crisis, and that painful changes are necessary. They offer a consultation. In the consultation their arguments appear to be thoroughly refuted. The refutation is then ignored and the changes go ahead.

Here is a brief summary of the painful changes that are proposed for the Leicester mathematics department. The department has 21 permanent research-active staff. Six of those are to be made redundant. There are also two members of staff who concentrate on teaching. Their number will be increased to three. How will the six be chosen? Basically, almost everyone will be sacked and then invited to reapply for their jobs in a competitive process, and the plan is to get rid of “the lowest performers” at each level of seniority. Those lowest performers will be considered for “redeployment” — which means that the university will make efforts to find them a job of a broadly comparable nature, but doesn’t guarantee to succeed. It’s not clear to me what would count as broadly comparable to doing pure mathematical research.

How is performance defined? It’s based on things like research grants, research outputs, teaching feedback, good citizenship, and “the ongoing and potential for continued career development and trajectory”, whatever that means. In other words, on the typical flawed metrics so beloved of university administrators, together with some subjective opinions that will presumably have to come from the department itself — good luck with offering those without creating enemies for life.

Oh, and another detail is that they want to reduce the number of straight maths courses and promote actuarial science and service teaching in other departments.

There is a consultation period that started in late August and ends on the 30th of September. So the lucky members of the Leicester mathematics faculty have had a whole month to marshall their to-be-ignored arguments against the changes.

It’s important to note that mathematics is not the only department that is facing cuts. But it’s equally important to note that it is being singled out: the university is aiming for cuts of 4.5% on average, and mathematics is being asked to make a cut of more like 20%. One reason for this seems to be that the department didn’t score all that highly in the last REF. It’s a sorry state of affairs for a university that used to boast Sir Michael Atiyah as its chancellor.

I don’t know what can be done to stop this, but at the very least there is a petition you can sign. It would be good to see a lot of signatures, so that Leicester can see how damaging a move like this will be to its reputation.

Alexandra O Fradkin: Who’s the Oldest: Conversation with Kindergartners

[Reposted from Alexandra O Fradkin’s blog Musings of a Mathematical Mom]

Yesterday, I overheard a wonderful conversation between our Kindergarten teacher and the Kindergartners.  The kids needed to line up to exit the classroom and the teacher told them to line up by age, oldest to youngest.  Immediately, one of the kids (K1 from now on) had a question.  “But how can we do it?  I’m five, K2 is five, and K3 is 5, so that means we’re all the same age!”

Teacher: Are you all the exact same age?
K1: Yes.
Teacher: So you were all born on the exact same day?
K1: Noooo. (giggling from the other kids)
Teacher: Ah, so some of you were born before others.  When are your birthdays?
K1: July.
K2: May.
K3: May.
Teacher: When in May?
K2: May 5.
K3: May 17.
Teacher: So who is older, who was born first?
K1: K2 is older.
Teacher: Why?
K1: Because she is taller!
Teacher: So taller people are always older than shorter ones?
All kids: Noooo.
Teacher: So in order to figure out who is older we need to determine what comes first, May 5 or May 17?
Silence.
Teacher: Well when you count, do you say 5 or 17 first?
K1: 17.
Teacher: So we count 1, 2, 3, 4, 17, and then five comes at some point later?
K1 (after much giggling): Noooo, it’s 1,2,3,4,5.
Teacher: So who’s older?
All kids: K2!

After that conversation it still took them a moment to get into the correct order, but they did it, and off they went! I love hearing kids of this age group reason because they are, for the most part, still not afraid of being wrong and they will say whatever comes to mind. This allows you to analyze how they think and is just plain lots of fun.

Alexandra O Fradkin: Story Math in Kindergarten: Two of Everything

[Reposted from Alexandra O Fradkin’s blog Musings of a Mathematical Mom]

Friday is story day in our Kindergarten math class.  For our first book we read Two of Everything, a Chinese folktale.  We then had a wonderful discussion and the kids asked some very insightful questions.

Here is a brief synopsis of the story: A poor elderly couple find an old brass pot in their garden and it turns out to be magic.  Whenever you put something into the pot, two of that thing come out!  The couple started doubling everything and soon became very rich.  One day, the husband accidentally pushed his wife into the pot and then fell into it himself.  After some initial arguing, the two couples realized that they could become the best of friends and use the pot to create two of everything, one for each couple.

At the end of the story, one of the kids asked, “But would there also be two pots?”  What a great question!  I said that I thought there would be only one pot, but some kids disagreed.  They spent several minutes debating whether it was possible to put the pot inside of itself to create a second one.

The discussion then moved on to how one would make lots of something.  The kids suggested that you could just keep putting the same object into the pot over and over again, creating one more each time.  I then asked them what would happen if we put two of the same object into the pot at the same time.  They all immediately yelled out that you would get three of that object.  My next question was whether only one of the objects would be doubled or both.  This led one of the kids (and then the rest) to realize that in fact, four of that object would come out.

I wanted to ask them about putting three or more objects into the pot, but it was time to move on.  Perhaps that was for the best because they already had a lot to take in.  I hope to come back to this topic and can’t wait to read more stories with them.  I feel that stories engage this age group like nothing else does.  And I absolutely love the questions and thoughts that the kids come up with!

Alexandra O Fradkin: Playing Math Detectives: First Week of Second Grade Math

[Reposted from Alexandra O Fradkin’s blog Musings of a Mathematical Mom]

The first week in our second grade class we did lots of time traveling. We played the role of math detectives and helped people from different time periods solve problems. We also learned about some ancient number systems.

On the first day we went back to several million years ago. (The idea for our scenario was taken from the wonderful book by Julia Brodsky, Bright, Brave, Open Minds: Engaging Young Children in Math Inquiry.) During this time period, there lived ferocious saber-toothed tigers with sharp teeth, crocodiles with awful jaws, and the first cave people, who had no strong jaws, long teeth, or sharp claws. How could we help those early people survive in their unfriendly world of dangers? The kids came up with making weapons out of sticks and stones, building fires, hiding in caves, running away, and climbing trees. I think they would have had a good chance of survival!

On the second day, we went back just 10-20-30 thousand years, to a time before numbers were invented but people had a need for keeping track. The kids’ task was to help a farmer determine whether his shepherd was bringing back all of his sheep at the end of day or whether he was stealing or losing some along the way.

The kids were split up into groups and each group got a bag of coins (which stood for sheep). They were told that when they were ready, I would take the “sheep” on a walk and bring them back. They would have to determine whether any were missing. The main rule was that they were not allowed to count in any way!

Here is their solution:img_2955

They made holes/homes for each of the coins/sheep, and when I brought back two fewer coins than they gave me, they were easily able to detect that because they had two empty holes. I was very impressed with their inventiveness. We then discussed and looked at pictures of how people actually did use dots, tally marks, stones, and knots to keep track of animals, money, and anything else they needed to.

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Finally, on the third day we went back only several thousand years, to several locations around the globe. We visited the Babylonians, the Mayans, and the Romans, and learned how they wrote the numerals 1 through 10 in their number systems. The detective work consisted of helping them decide how they should write 11.

The kids examined the patterns closely, made suggestions, and discussed the merits of each one. In the end, they came up with versions that I think the ancient people would have been happy with.

Here is a picture of what it looked like (the Babylonian version did later get modified to be a horizontal wedge followed by a vertical one).

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Next week we will begin our in-depth exploration of the Hindu-Arabic number system. More specifically, we’ll focus on the usefulness and meaning of place value and the importance of zero!

The Humanistic Mathematics Network Newsletter

The Humanistic Mathematics Network Newsletter (HMNN) was founded by Alvin White in the summer of 1987. The Newsletter was later renamed The Humanistic Mathematics Network Journal (HMNJ). The last issue of the HMNJ was published in 2004. The open access digital archive of the full run of the HMNN/HMNJ (1987-2004) is now available at http://scholarship.claremont.edu/hmnj/.

This journal does not accept new content. A related current journal is the Journal of Humanistic Mathematics.

The 7 biggest problems facing science

The 7 biggest problems facing science, according to 270 scientists, by Julia Belluz, Brad Plumer, and Brian Resnick on September 7, 2016 in Vox.

I love this example of deductive resoning:

  •  Academia has a huge money problem
  • Too many studies are poorly designed
  • Replicating results is crucial — and rare
  • Peer review is broken
  • Too much science is locked behind paywalls
  • Science is poorly communicated
  • Life as a young academic is incredibly stressful

Conclusion:

  • Science is not doomed

Read the whole text.