# Brain finds true beauty in maths

From BBC: Brain finds true beauty in maths. A quote:

Brain scans show a complex string of numbers and letters in mathematical formulae can evoke the same sense of beauty as artistic masterpieces and music from the greatest composers.

Mathematicians were shown “ugly” and “beautiful” equations while in a brain scanner at University College London.

The same emotional brain centres used to appreciate art were being activated by “beautiful” maths.

The researchers suggest there may be a neurobiological basis to beauty.

The study in the journal Frontiers in Human Neuroscience says,in partucular, that

The formula most consistently rated as beautiful (average rating of 0.8667), both before and during the scans, was Leonhard Euler’s identity

$$1+e^{i\pi}=0$$

which links 5 fundamental mathematical constants with three basic arithmetic operations, each occurring once; the one most consistently rated as ugly (average rating of −0.7333) was Srinivasa Ramanujan’s infinite series for 1/π,

$$\frac{1}{π}=\frac{2\sqrt{2}}{9801}\sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4\cdot 396^{4k}}$$

which expresses the reciprocal of π as an infinite sum.

Other highly rated equations included the Pythagorean identity, the identity between exponential and trigonometric functions derivable from Euler’s formula for complex analysis, and the Cauchy-Riemann equations. Formulae commonly rated as neutral included Euler’s formula for polyhedral triangulation, the Gauss Bonnet theorem and a formulation of the Spectral theorem. Low rated equations included Riemann’s functional equation, the smallest number expressible as the sum of two cubes in two different ways, and an example of an exact sequence where the image of one morphism equals the kernel of the next .

– See more at: http://journal.frontiersin.org/Journal/10.3389/fnhum.2014.00068/full#sthash.7b7Pdf5a.dpuf

# EMPG 2014: Call for papers

You are cordially invited to attend the 2014 European Mathematical Psychology Group Meeting (EMPG 2014), held at the University of Tübingen, Germany, from Wednesday, July 30, 2014 until Friday, August 01, 2014.

Presentations

Proposals for paper and poster presentations as well as proposals for symposia related to all aspects of mathematical psychology are welcome. Relevant topics include:

• perception and psychophysics
• models of cognition and learning
• knowledge structures
• measurement and scaling
• psychometrics
• computational methods
• statistical methods
• mathematical models

Important dates

• Abstract submission opens:  soon
• Abstract submission closes:  April 30, 2014
• Notification of acceptance:  May 15, 2014
• Early registration deadline:  June 06, 2014
• Start of conference: July 30, 2014
• End of conference: August 01, 2014

Invited Symposia

Symposium in honor of Jean-Claude Falmagne celebrating his 80th birthday (organized by Michel Regenwetter and Jean-Paul Doignon).

Invited speakers

Andrew Heathcote, University of Newcastle, Australia
Ehtibar Dzhafarov, Purdue University, USA

For further information please consult the website of the EMPG 2014 (www.uni-tuebingen.de/psychologie/empg2014).

# Howdy!

Alexandre Borovik invited me to join the writers here at The DeMorgan Forum. (Thank you, Alexandre. I am honored.) He got me started by reposting my Top Ten Issues in Math Education post, which I have now edited. (There were links to dead blogs and disappeared posts, mentions of ‘last week’ four years ago, and opinions I’ve changed.)

I blog most often at Math Mama Writes, and will bring some of my favorite posts from there over here. If you want to hear more from me, please follow me there.

# My Top Ten Issues in Math Education

[Originally posted at Math Mama Writes. Revised for The DeMorgan Forum.]

10. Textbooks are trouble. Corollary: The one doing the work is the one doing the learning. (Is it the text and the teacher, or is it the student?)
Hmm, this shouldn’t be last, but as I look over the list they all seem important. I guess this isn’t a well-ordered domain. A few years back I read Textbook Free: Kicking the Habit, an article by Chris Shore on getting away from using a textbook (unfortunately no longer available online). I was inspired to take charge of my teaching in a way I really hadn’t before. Now I decide how to organize the course. I still use the textbook for its homework repositories, but I decide on my units and use the text as a resource. See dy/dan on being less helpful (so the students will learn more), and Bob Kaplan on becoming invisible.

9. Earlier is not better.
The schools are pushing academics earlier and earlier. That’s not a good idea. If young people learn to read when they’re ready for it, they enjoy reading. They read more and more; they get better and better at it; reading serves them well. (See Peter Gray‘s post on this.) The same can happen with math. Daniel Greenberg, working at a Sudbury school (democratic schools, where kids do not have enforced lessons) taught  a group of 9 to 12 year olds all of arithmetic in 20 hours. They were ready and eager, and that’s all it took.

In 1929, L.P. Benezet, superintendent of schools in Manchester, New Hampshire, believed that waiting until later would help children learn math more effectively. The experiment he conducted, waiting until 5th or 6th grade to offer formal arithmetic lessons, was very successful. (His report was published in the Journal of the NEA.)

8. Real mathematicians ask why and what if…
If you’re trying to memorize it, you’re probably being pushed to learn something that hasn’t built up meaning for you. See Julie Brennan’s article on Memorizing Math Facts. Yes, eventually you want to have the times tables memorized, just like you want to know words by sight. But the path there can be full of delicious entertainment. Learn your multiplications as a meditation, as part of the games you play, …

Just like little kids, who ask why a thousand times a day, mathematicians ask why. Why are there only 5 Platonic (regular) solids? Why does a quadratic (y=x2), which gives a U-shaped parabola as its graph, have the same sort of U-shaped graph after you add a straight line equation (y=2x+1) to it? (A question asked and answered by James Tanton in this video.) Why does the anti-derivative give you area? Why does dividing by a fraction make something bigger? Why is the parallel postulate so much more complicated than the 4 postulates before it? Then came “What if we change that postulate?” And from that, many non-Euclidean geometries were born.

7. Math itself is the authority – not the curriculum, not the teacher, not the standards committee.
You can’t want students to do it the way you do. You have to be fearless, and you need to see the connections. (Read this from Math Mojo.)

6. Math is not arithmetic, although arithmetic is a part of it. (And even arithmetic has its deep side.)
Little kids can learn about infinity, geometry, probability, patterns, symmetry, tiling, map colorings, tangrams, … And they can do arithmetic in another base to play games with the meaning of place value. (I wrote about base eight here, and base three here.)

5. Math is not facts (times tables) and procedures (long division), although those are a part of it; more deeply, math is about concepts, connections, patterns. It can be a game, a language, an art form. Everything is connected, often in surprising and beautiful ways.
My favorite math ed quote of all time comes from Marilyn Burns: “The secret key to mathematics is pattern.

U.S. classrooms are way too focused on procedure in math. It’s hard for any one teacher to break away from that, because the students come to expect it, and are likely to rebel if asked to really think. (See The Teaching Gap, by James Stigler.)

See George Hart for the artform. The language of math is the language of logic. Check out any Raymond Smullyan book for loads of silly logic puzzles, and go to islands full of vegetarian truthtellers and cannibal liars. Or check out some of Tanya Khovanova’s posts.

4. Students are willing to do the deep work necessary to learn math if and only if they’re enjoying it.
Which means that grades and coercion are really destructive. Maybe more so than in any other subject. People need to feel safe to take the risks that really learning math requires. Read Joe at For the Love of Learning. I’m not sure if this is true in other cultures. Students in Japan seem to be very stressed from many accounts I read; they also do some great problem-solving lessons. (Perhaps they feel stressed but safe. Are they enjoying it?)

3. Games are to math as picture books are to reading – a delightful starting point.
Let the kids play games (or make up their own games) instead of “doing math”, and they might learn more math. Denise’s game that’s worth 1000 worksheets (addition war and its variations) is one place to start. And Pam Sorooshian has this to say about dice.  Learn to play games: Set, Blink, Quarto, Blokus, Chess, Nim, Connect Four… Change the rules. Decide which rules make the most interesting play.

Besides games, consider puzzles, cooking, building, science, programming, art, math stories, and math history for ways to bring meaningful math into your lives. (Here’s a list of good games, puzzles, and toys.)  If you play around with all those, you can have a pretty math-rich life without ever having a formal math lesson.

2. If you’re going to teach math, you need to know it deeply, and you need to keep learning.
Read Liping Ma. Arithmetic is deeper than you knew (see #6). Every mathematical subject you might teach is connected to many, many others. Heck, I’m still learning about multiplication myself. In a blog conversation (at a wonderful blog that is, sadly, gone now), I once said, “You don’t want the product to be ‘the same kind of thing’.  …   5 students per row times 8 rows is 40 students. So I have students/row * rows = students.” Owen disagreed with me, and Burt’s comment on my multiplication post got me re-reading that discussion. I think Owen and I may both be right, but I have no idea how to do what he suggests and use a compass and straightedge to multiply. I’m looking forward to playing with that some day. I think it will give me new insight.

1. If you’re going to teach math, you need to enjoy it.
The best way to help kids learn to read is to read to them, lots of wonderful stories, so you can hook them on it. The best way to help kids learn math is to make it a game (see #3), or to make dozens of games out of it. Accessible mysteries. Number stories. Hook them on thinking. Get them so intrigued, they’ll be willing to really sweat.

That’s my list. What’s yours?
What do you see as the biggest issues or problems in math education?

[You may also enjoy reading the discussion my original post prompted back in 2010.]

# Play this book: “Moebius Noodles”

Moebius Noodles. Adventurous Math for the Playground Crowd

Text: Yelena McManaman and Maria Droujkova
Illustrations and design: Ever Salazar
Copyedits: Carol Cross

This brilliant book is published under Creative Commons Attribution-NonCommercial-ShareAlike license, and this allows me to reproduce the entire Introduction:

Why Play This Book

Children dream big. They crave exciting and beautiful adventures to pretend-play. Just ask them who they want to be when they grow up. The answers will run a gamut from astronauts to zoologists and from ballerinas to Jedi masters. So how come children don’t dream of becoming mathematicians?
Kids don’t dream of becoming mathematicians because they already are mathematicians. Children have more imagination than it takes to do differential calculus. They are frequently all too literate like logicians and precise like set theorists. They are persistent, fascinated with strange outcomes, and are out to explore the “what-if” scenarios. These are the qualities of good mathematicians!

As for mathematics itself, it’s one of the most adventurous endeavors a young child can experience. Mathematics is exotic, even bizarre. It is surprising and unpredictable. And it can be more exciting, scary, and dangerous than sailing on high seas!

But most of the time math is not presented this way. Instead, children are required to develop their mathematical skills rather than being encouraged to work on something more nebulous, like the mathematical state of mind. Along the way the struggle and danger are de-emphasized, not celebrated – with good intentions, such as safety and security. In order to achieve this, children are introduced to the tame, accessible scraps of math, starting with counting, shapes, and simple patterns. In the process, everything else mathematical gets left behind “for when the kids are ready.” For the vast majority of kids, that readiness never comes. Their math stays simplified, impoverished, and limited. That’s because you can’t get there from here. If you don’t start walking the path of those exotic and dangerous math adventures, you never arrive.

It is as tragic as if parents were to read nothing but the alphabet to children, until they are “ready” for something more complex. Or if kids had to learn “The Itsy-Bitsy Spider” by heart before being allowed to listen to any more involved music. Or if they were not allowed on any slide until, well, learning to slide down in completely safe manner. This would be sad and frustrating, wouldn’t it? Yet that’s exactly what happens with early math. Instead of math adventures – observations, meaningful play, and discovery of complex systems – children get primitive, simplistic math. This is boring not only to children, but to adults as well. And boredom leads to frustration. The excitement of an adventure is replaced by the gnawing anxiety of busy work.

We want to create rich, multi-sensory, deeply mathematical experiences for young children. The activities in this book will help you see that with a bit of know-how every parent and teacher can stage exciting, meaningful and beautiful early math experiences. It takes no fancy equipment or software beyond everyday household or outdoor items, and a bit of imagination – which can be borrowed from other parents in our online community. You will learn how to make rich mathematical properties of everyday objects accessible to young children. Everything around you becomes a learning tool, a prompt full of possibilities for math improvisation, a conversation starter. The everyday world of children turns into a mathematical playground.
Children marvel as snowflakes magically become fractals, inviting explorations of infinity, symmetry, and recursion. Cookies offer gameplay in combinatorics and calculus. Paint chips come in beautiful gradients, and floor tiles form tessellations. Bedtime routines turn into children’s first algorithms. Cooking, then mashing potatoes (and not the other way around!) humorously introduces commutative property. Noticing and exploring math becomes a lot more interesting, even addictive. Unlike simplistic math that quickly becomes boring, these deep experiences remain fresh, because they grow together with children’s and parents’ understanding of mathematics.

Can math be interesting? A lot of it already is! Can your children be strong at advanced math? They are natural geniuses at some aspects of it! Your mission, should you accept it: to join thrilling young math adventures! Ready? Then let’s play!

# Correlation for schoolchildren

A few comments on MEI‘s draft “Critical Maths” Curriculum. They list

Glossary of terms which students are expected to know and be able to use […]

Association: A tendency for two events to occur together.

Correlation: An association between two variables which is approximately linear.

This definition of correlation seems rather odd.  If $latex y = x^2$  aren’t $latex x$ and $latex y$  correlated?   What does “an association” mean here?  The suggested definition of association given above is for events, not “variables”.   Presumably the authors have in mind random variables.
There is a serious problem here in the use of language.  It needs to be made clear whether the notion being described is an intuitive one or a mathematical definition. I am not a statistician, but it seems to me that there are (at least) three common distinct types of usage of the word “correlation”,  none of which is captured by the “definition” proposed:
(1)  The vernacular usage. The  Merriam-Webster dictionary gives
“a relation existing between phenomena or things or between mathematical or statistical variables which tend to vary, be associated, or occur together in a way not expected on the basis of chance alone”
which seems to me a reasonable description of the vernacular or intuitive non-mathematical meaning of the term.    This is clearly much broader than the meaning suggested above.
(2)  The intended meaning proposed seems to correspond closest to the use of the  (Pearson) correlation coefficient  in statistics, although even then it is not  accurate, since  the correlation coefficient is not always a  reliable indicator of the existence of a linear relationship.   This meaning is that which tends to be used by a large class of people who have had some minimal exposure to statistics.
(3)  More generally correlation can be used to indicate a variety of mathematical measures of probabilistic interdependence  (e.g. mutual information).
On a separate point the very heavy concentration on statistical reasoning to the exclusion of other mathematics (including perhaps more elementary logical reasoning such as manipulation of quantifiers and logical connectives) rather worries me, since it may encourage the idea that  almost the only practical applications of mathematics are statistical.
Another  serious danger in my opinion is that statistics at this level tends to be more  like cookery than mathematics and it would have to be extremely well taught by a gifted and highly educated teacher if  conceptual precision is not going to be completely lost.  The danger is partially raised by Gowers in Objection 5 listed in his blog (though he doesn’t mention cookery), but I think his own answer is rather optimistic.
Somewhat in this connection there is an interesting passage in Noam Chomsky on Where Artificial Intelligence Went Wrong where Noam Chomsky is interviewed on various topics concerning science, in particular AI and  cognitive science, and what he clearly regards as a modern deviation from the classical scientific method, which has been indirectly caused by the power of modern computers .  The article is quite long, but I found his example of “how to justify the abolition of physics departments” very nice;  it could  equally well used to justify closing down everything in mathematics departments except statistics.

# Nesin Mathematics Village

The Nesin Mathematics Village is a small village of  about 13,5 acres, approximately 7,5 of which consist of olive groves. It is owned by the Nesin Foundation and is located 1 km away from the village of Şirince (tied to the Selçuk district of Izmir). Perched on a hillside and overflowing with greenery, it is a place where young and old learn, teach, and think about mathematics in peaceful remoteness. Unpretentious and unostentatious, the houses made out of rock, straw and clay give off a simple welcoming air.

Apart from the crickets, any factors which could prevent concentration and deep thought are kept away, there are no televisions, no music is publicly broadcasted. But traces of civilization such as electricity, warm water and wireless internet are nonetheless present. There is no shortage of insect life!

Most activities take place in the summer months; however in spring and autumn it is also an ideal environment for various types work groups, meetings and rest. It could for example be used as a place for an alumni reunion, a honeymoon in the “wild” or a mathematics workshop.

From teaching at the primary school level to the most advanced research, mathematical activities of any level can take place simultaneously at the village.

We now have the capacity to lodge 150 people, but there is the possibility of pitching tents if more capacity is required.