Thales and the Nine-point Conic

New paper in the De Morgan Gazette:

David Pierce, Thales and the Nine-point Conic, The De Morgan Gazette 8 no. 4 (2016)  27-78. bit.ly/2hlyHzZ. ISSN 2053-1451

Abstract: The nine-point circle is established by Euclidean means; the nine-point conic, Cartesian.Cartesian geometry is developed from Euclidean by means of Thales’ s Theorem. A theory of proportion is given, and Thales’s Theorem proved, on the basis of Book I of Euclid’s Elements, without the Archimedean assumption of Book V. Euclid’s theory of areas is used, although this is obviated by Hilbert’s theory of lengths. It is observed how Apollonius relies on Euclid’s theory of areas. The historical foundations of the name of Thales’s Theorem are considered. Thales is thought to have identified water as a universal substrate; his recognition of mathematical theorems as such represents a similar unification of things.

Call for Nominations for the 2017 ICMI Felix Klein and Hans Freudenthal Awards

From IMCI Newsletter November 2016:

[All nominations must be sent by e-mail to the Chair of the Committee (annasd >>at<< edu.haifa.ac.il, sfard >>at<< netvision.net.il) no later than 15 April 2017.]

Since 2003, the International Commission on Mathematical Instruction (ICMI) awards biannually two awards to recognise outstanding accomplishments in mathematics education research: the Felix Klein Medal and the Hans Freudenthal Medal.

The Felix Klein medal is awarded for life-time achievement in mathematics education research. This award is aimed at acknowledging excellent senior scholars who have made a field-defining contribution over their professional life. Past candidates have been influential and have had an impact both at the national level within their own countries and at the international level. We have valued in the past those candidates who not only have made substantial research contributions, but also have introduced new issues, ideas, perspectives, and critical reflections. Additional considerations have included leadership roles, mentoring, and peer recognition, as well as the actual or potential relationship between the research done and improvement of mathematics education at large, through connections between research and practice.

The Hans Freudenthal medal is aimed at acknowledging the outstanding contributions of an individual’s theoretically robust and highly coherent research programme. It honours a scholar who has initiated a new research programme and has brought it to maturation over the past 10 years. The research programme is one that has had an impact on our community. Freudenthal awardees should also be researchers whose work is ongoing and who can be expected to continue contributing to the field. In brief, the criteria for this award are depth, novelty, sustainability, and impact of the research programme.
For further information about the awards and for the names of past awardees (seven Freudenthal Medals and seven Klein Medals, to date), see http://www.mathunion.org/icmi/activities/awards/the-klein-and-freudenthal-medals/

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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.