Math Anxiety Simplified

We all know people who, it seems, just can’t get mathematics.  I was once in a store, buying a $10.00 item that was on sale at 15% off.  The salesperson got into a bit of a dither looking for a calculator to perform this abstruse computation, and was amazed when I said the price should be $8.50.   As she rang up the sale, she told me, “I was good at math, but I never could do fractions.”

According to some new research, described in a press release from the University of Waterloo, this story is more representative than it might at first appear.  Traditionally, the view of educators has been that “math anxiety” hindered students in intermediate to advanced courses (from algebra onwards), but didn’t affect their ability to do simple arithmetic tasks  But, according to this research (abstract), even very basic tasks, like counting, are affected.  The study compared matched groups of students that had high and low scores on a math anxiety test, and evaluated their performance in a simple test that involved counting the number of black squares displayed on a computer screen.

In two experiments, 28 undergraduate students – 14 with low math anxiety and 14 with high math anxiety – were shown a set of black squares on a computer screen. The squares ranged in number from one to nine and participants were simply asked to identify the number of squares.

An interesting pattern was observed in the results.  When the number of black squares was in the range 1 to 4, both groups of students did equally well on the test.  However, when the number of squares was 5 or more, the high math anxiety group performed significantly worse.

The differences in the number of squares correspond to two apparently different mental processes that people use to answer the question “How many?”.  When the number is small (1-4), the person typically just “gets” the number, without consciously counting, a process called “subitizing” (cf. Latin subitus, “sudden”).  When the number is greater, the person consciously counts, and the process takes noticeably longer.  (It’s interesting, and perhaps related, that all human languages, even in very primitive societies, have words for small integers, such as “one”, “two”, “three”.  Once the number gets bigger than 5-7, most primitive languages just have a word for “many”.)

It appears that, once conscious effort is involved, math anxiety kicks in with visible effects.  The researchers go on to suggest, in the paper’s conclusions, that math anxiety somehow compromises the capacity for working memory.  In any case, gaining a better understanding of the sources and effects of math anxiety is a worthwhile goal for educators.  You can’t always count on finding your calculator.

Cyber Shockwave

Last Tuesday, a group of former government officials participated in an exercise called “Cyber Shockwave”, a large-scale simulated cyber-attack on the United States.   The exercise, which was organized by the Bipartisan Policy Center, a Washington DC  think tank, was essentially a war game, set in July, 2011.  The scenario is described in an article in the Washington Post:

A massive cyber attack has turned the cellphones and computers of tens of millions of Americans into weapons to shut down the Internet. A cascading series of events then knocks out power for most of the East Coast amid hurricanes and a heat wave.

According to the “plot”, the malware used in the attack had been planted on cellphones and computers months earlier, bundled with a “March Madness” NCAA basketball program.  (I have to say that I think parts of this scenario are not very realistic, particularly when it is assumed that failures can cascade across different networks: for example, from the cellular network to the power grid and the conventional phone network.  That sort of thing is not impossible, but it’s highly unlikely, in my judgment.)

The participants, former government officials who acted as players in security positions, did not do terrible well, according to an article on the exercise at the Dark Reading security site:

In a press release issued today, the Bipartisan Policy Center (BPC) — which organized “Cyber Shockwave” using a group of former government officials and computer simulations — concluded the U.S is “unprepared for cyber threats.”

The simulated attack and response revealed several areas of weakness.  The participants had a great deal of difficulty identifying the source of the attack.  More significantly, the exercise highlighted a legal and institutional gray area that would likely be a handicap in a real crisis.  The nation’s technology infrastructure is owned and controlled by many different parties.  Even though the government might be called on to respond in a crisis, the initial sources of the simulated attack were cellphones and personal computers, where both the devices and the networks connecting them are largely in private hands.  No one is suggesting that we nationalize our technical infrastructure, but the exercise demonstrated the potential for confusion and conflict in the absence of any clear technical, operational, or legal standards for how response to an attack should be coordinated.  The issue is not just who should be in charge; without some coordination mechanism, it is quite possible that government attempts to counter the attack might conflict with, for example, attempts by cellular carriers to protect their networks, even though both groups ultimately are on the same side.

Carrying out this kind of exercise could have significant value if it leads to these issues being considered before there is a crisis.

CNN has a special program on “Cyber Shockwave” which will be shown tonight, Sunday 21 February, at 8:00 PM EST.  (It was also shown last night.)