Mathematical Minds

"Most mathematicians did not just take up math as a "job"...(most) get more pleasure out of mathematics than almost any other activity. And they often discovered this pleasure when they were young..."

While most people would agree that "math people" are not like "non-math people", it's not always easy for non-mathematical minds to recognize (and appropriately nurture) mathematical ones. The reasons for this are several - mathematical kids are often independent and internally-driven problem solvers who may or may not excel in the standard math tasks of the elementary school classroom (if he's such a math kid, how come he's getting C's on his timed drills?...) Many students with extreme talents in math may also be relatively verbal-poor, so are less obviously the "smart" children in class. Also they may be reluctant to show what they know or what they are interested in to relative strangers, and may have difficulty explaining how they arrived at answers. Many mathematical minds are dyslexic or twice exceptional in another areas, too, complicating their identification with standardized tests or screening tools.

Numbers Kids The numbers kids are perhaps the easiest to recognize - and they often come from families where one or both parents have a special affinity to mathematics (engineers, computer science, academics). It may start out with children interested in patterns and facts within mathematics (divisibility rules, cube roots, etc.), card and other games, recreational math topics (Fibonacci sequence, fractals, probability, solving problems for 'fun') or mathematics in the world of adults (e.g. Philip Davis' cousin who let him be bookkeeper at the age of 7, keeping track of a race horse's handicap and winnings).

Tinkering Kids Tinkering kids tend to enjoy conceptual science books, building and unbuilding (gears, taking apart ball point pens and toys, clocks, cameras, origami etc.), computer-related activities, projects (completed and incomplete), and beautiful and unbeautiful design.

By temperament, strong math minds will tend to be introverted and have high focus and task persistence for activities of intrinsic interest. This may mean they are difficult to direct in the traditional or even non-traditional classroom (prefer studying lines of own interest), and they may be benefited particularly by mentors (often relatives or math teachers at higher levels of education) willing to discuss topics, ideas, and problems far in advance of their years.

Silverman and Feldman have distinguished engineering / math-gifted individuals into sensor (likes facts, data, experimentation) and intuitor (prefers principles and theories) groups. Both were capable of becoming "fine engineers", but sensors with less direct success in traditional academics.

Recently, some investigators have begun to look at brain-related differences in mathematically-gifted students (to our knowledge this has not been done in professional mathematicians, engineers, physicists); in his study of mathematically-gifted adolescents, Michael O'Boyle has found that superior mathematics performance was correlated with increased bihemispheric activation (vs. unilateral activation) for mathematics tasks, enhanced involvement of the right hemisphere for information (including linguistic) processing, and strong prefrontal cortex activation. As seen in the figure above, math-gifted adolescents performing mental rotation tasks activate much more brain bilaterally than average math-performing peers.

The optimal educational pathways for young math thinkers may also vary widely. Some thrive with subject acceleration, while others plenty of free time to explore topics of personal interest - whether conceptual or technical.

Perhaps the most common feature seen in young mathematical minds is their interest is solving problems. If you have a young mathematical mind in your house and he or she hasn't seen the PBS special on Fermat's Last Theorem, check it out.It's great - sort of what Race for the Double Helix is to budding scientists. The PBS video on Fermat's Last Theorem (Youtube.com)

From Andrew Wiles:

" I loved doing problems in school. I'd take them home and make up new ones of my own. But the best problem I ever found, I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem—Fermat's Last Theorem. This problem had been unsolved by mathematicians for 300 years. It looked so simple, and yet all the great mathematicians in history couldn't solve it. Here was a problem, that I, a ten year old, could understand and I knew from that moment that I would never let it go. I had to solve it."

Learning Styles in Engineering Students
Discovering Mathematical Talent
Cognitive Profiles of Mathematical Precocity
Interhemispheric Interaction in Mathematically Gifted Adolescents pdf
Developing Mathematical talent
Parental roles of mathematically gifted students pdf
Aha Moments in Math
Riemann Hypothesis
Fermat's Last Theorem
Autism occurs more often in the families of physicists, engineers, and mathematicians pdf
Education of a Mathematician

Spreading Happiness Through Social Networks

From Harvard Medical School and Charles Dickens comes the news flash: happiness begets happiness, and unhappiness begets unhappiness through social networks.

Although it looked as if Fowler and Christakis were only iterated earlier work by Dickens, their data came from analyzing the massive Framingham heart study cohort.

"People who are surrounded by many happy people and those who are central in the network are more likely to become happy in the future. Longitudinal statistical models suggest that clusters of happiness result from the spread of happiness and not just a tendency for people to associate with similar individuals. A friend who lives within a mile (about 1.6 km) and who becomes happy increases the probability that a person is happy by 25% (95% confidence interval 1% to 57%). Similar effects are seen in coresident spouses (8%, 0.2% to 16%), siblings who live within a mile (14%, 1% to 28%), and next door neighbours (34%, 7% to 70%). Effects are not seen between coworkers. The effect decays with time and with geographical separation."

Happiness does seem to beget happiness in fMRI, but the effect is more pronounced for extroverts.

So if you find the holiday rush and weather hassles getting you down, gather your loved ones around you a catch some Christmas cheer. Merry Christmas and Happy Holidays to everyone!

Harvard magazine: Having happy friends can make you happy
BMJ: Dynamic spread of happiness
A Christmas Carol
Happy faces and fMRI pdf
Eide Neurolearning Blog: Christmas Cheer and a Positive Outlook are Good for Your Brain

The Myth of Multi-Tasking: The Problem of Listening while Driving

As we head into the holiday season, this study is a good reminder that cell phones and driving (and even listening to conversation while driving) has its risks. Using a driving simulation setup, Marcel Just and his colleagues saw that listening to sentences robbed from visual and spatial areas necessary for driving. This is also why hands-free phones are really not any better than conventional cell phones. The problem is not just having to hold the phone, it's diverting the brain's resources for visual and spatial awareness.

This principle applies to much more than driving while listening. It's especially hard not to multitask as we head into the holiday season, butits it's wise to realize it often comes with a price. For more on this general topic, check out Christine Rosen's The Myth of Multi-tasking

Driving while listening to someone speak fMRI pdf

Flash from the Past: Right-Brained Learner...Gifted Dyslexic?

His mother was a doctor who was concern her son would get sick in the epidemics, so he chose to school him at home. His unemployed uncle taught him, and he had many strong ideas about not liking rote learning: " He...avoided teaching me the alphabet and multiplication tables (even today they give me trouble). Most of the time we played chess and read maps." Today he still says he doesn't know his multiplication tables past fives.

But despite a disdain for conventional rote learning, this flash-from-the-past's uncle encouraged him to read and "pay attention to miscellaneous facts." His approach to learning was transdisciplinary, but this also had its toll on goal-directed learning and project completion.

"Every so often I was seized by the sudden urge to drop a field right in the middle of writing a paper, and to grab a new research interest in a field about which I knew nothing. I followed my instincts, but could not account for them until much, much later. "

Who was this?

None other than Benoit Mandlebrot, father of fractals and chaos theory in science mathematics, and economics.

Could any budding Mandlebrot's today able to be so omnivorous and transdisciplinary in their education?

Not surprisingly, not everyone could see with his broad brush: "Still, I remained an outsider in every field I worked in, and just couldn't get my interdisciplinary and philosophical views accepted. For instance, while working on economics, I was dying to mention that my methods were also pertinent to physics, but the referees of my papers told me to remove this broader philosophy. Later, when I studied turbulence (which, because of its unpredictability, resembled the stock market), my broader comments were again removed, and many papers were totally rejected."



Mandlebrot is a great example of an accomplished right hemispheric thinker - driven by intuitive search for connections, finding simple principles in complex material, trandisciplinary, and analogical. We do not know whether he may have been a stealth dyslexic, but his talent set would fit neatly (as would his weakness with rote mathematics and the alphabet). Today he says he isn't able to use a telephone book because he can't remember the alphabet.

Mandlebrot: Fruits of a wandering spirit
Fractal Wisdom
Fractals in Nature
Wikipedia: picture of Mandlebrot set
Eide Neurolearning Blog: Cross-disciplinary thinking
Eide Neurolearning Blog: Complex Thinkers
Eide Neurolearning Blog: Problem solving by insight
Eide Neurolearning Blog: Easy and hard problem solving

Christmas Cheer and a Positive Outlook are Good For Your Brain

We were called last week about what effects Christmas cheeris likely to have on the brain. This latest research from Northwestern provides additional proof that a positive outlook result fosters creativity and problem solving by insight.

"..positive mood enhances insight, at least in part, by modulating attention and cognitive control mechanisms via the ACC (anterior cingulate cortex), perhaps enhancing sensitivity to detect non-prepotent solution candidates...Analytic processing involves deliberate application of strategies and operations togradually approach solution. Insight, which is considered a type of creative cognition, is the process through which people suddenly and unexpectedly achieve solution through processes that are not consciously reportable. Insight solutions tend to involve conceptual reorganization, often occurring after solvers overcome an impasse in their solving effort, and are suddenly able to recognize distant or atypical relations between problem elements that had previously eluded them."

In the figure above, see how a strong performance on insight-related questions correlated quite closely with positive mood (and negatively with anxious mood). Researchers also found that the brain activations of successful insight-related problems solves closely matched patterns of activation seen with a positive mood.

If your style of problem solving leans more to insight (vs.logical-deductive, for instance), this is important to know. Happiness and optimism are not just optional add-ons or a frill; they may be absolutely essential for any high level insight-related creative work. We mustn't forget to find time for joy.

Eide Neurolearning Blog: Chance favors the prepared mind
Eide Neurolearning Blog: Easy and Hard Problem Solving
Teaching Optimism