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Tony PhillipsTony Phillips' Take on Math in the Media
A monthly survey of math news

This month's topics:

The pain of math anxiety

An article published in PLOS ONE (October, 2012) by the math anxiety specialists Ian Lyons and Sian Beilock (Psychology, Chicago) was picked up by the national media, including The Atlantic. The title was a grabber: "When Math Hurts: Math Anxiety Predicts Pain Network Activation in Anticipation of Doing Math." Lyons and Bellock ask, in a study of high-math-anxiety (HMA) subjects, "Are [their] feelings about math merely psychological epiphenomena, or is their anxiety grounded in simulation of a concrete, visceral sensation--such as pain--about which they have every right to feel anxious?" They tested 14 HMAs and 14 low-math-anxiety (LMA) individuals.

  • Participants completed a word task and math task (block-design) while neural activity was measured using fMRI. Thirty-two blocks of each task-type (16 hard blocks and 16 easy blocks; 4 trials/ block) were randomly interleaved and spread over 8 functional runs. In the math task, participants verified whether arithmetic problems of the form $(a\times b)-c = d$ were correct, where $a\neq b$, $c>0$, $d>0$. For hard math problems, $5\leq a\leq 9$, $5\leq b\leq9$ ($a\times b \geq 30)$, $15\leq c\leq 19$; subtracting $c$ from $a\times b$ always involved a borrow operation ... . For easy math problems, $1\leq a\leq 9$, $1\leq b\leq 9$ ($a\times b \leq 9$), $1\leq c\leq 8$; subtracting $c$ from $a\times b$ never involved a borrow operation. In the word task, participants verified whether a word, if reversed, spelled an actual word.
  • Crucially, before each task-block, a cue (yellow circle or blue square) indicated whether the math-task or word-task would follow.

The conclusion: "high levels of math anxiety predict increased pain-related activity during anticipation of doing math, but not during math performance itself." (The LMA individuals showed no such effect). The Atlantic's Lindsay Abrams, writing on November 2 ("Study: Worrying About Math Can Activate Pain Areas In the Brain"), calls attention to the last point: "But the results also show that math itself isn't painful, raising the possibility that people can be helped to get over their anxiety and go on to embrace math, signing up for math classes and even pursuing math-related careers.

Information theory in biological cells

A perspective piece in Science for October 19, 2012 tells how information theory can be adapted to the analysis of cell signaling capabilities. The article, by Matthew Brennan, Raymond Cheong and Andre Levchenko (Biomedical Engineering, Johns Hopkins) begins by reminding us that biochemical networks have been traditionally studied by measuring the aggregate response of a population of cells, whereas detailed studies have shown that individual cells can have quite varied and hence unpredictable "decision-making" behaviors. They conclude that we should seek to learn "the limits to how well cell signaling can enable decision-making, given a cell's uncertain response to changes in the environment." In particular, "a new 'language' may be needed to understand and quantify the impact of noise (variability) on a cell's functionality." "Mathematics turns out to have just the right theory." This is information theory, invented by Claude Shannon in 1948 to quantify the efficiency and reliability of human communications networks. "Conveniently, its general formulation permits analysis of many complex systems, including those found in biological signaling." For example, "under evolutionary pressure, it's expected that signaling systems are optimally matched to the sources of information they have evolved to process. Indeed, examples from neuroscience (such as sensory perception) and developmental biology (such as embryonic patterning) show that biological systems usually have a capacity that is minimally sufficient for the information they process. This optimality principle can answer long-standing questions that cannot currently be addressed through models or direct experiments."

On arithmetic and the unconscious

"Reading and doing arithmetic nonconsciously," by Ran Hassin and five colleagues at the Hebrew University, appeared in the Proceedings of the National Academy of Sciences for November 12, 2012. In the abstract, the authors "report a series of experiments in which we show that multiple-word verbal expressions can be processed outside conscious awareness and that multistep, effortful arithmetic equations can be solved unconsciously." The team's experiments use Continuous Flash Suppression (CFS), "a cutting edge masking technique that allows subliminal presentations that last seconds. CFS is a game changer in the study of the unconscious, because unlike all previous methods, it gives unconscious processes ample time to engage with and operate on subliminal stimuli." They explain: "CFS consists of a presentation of a target stimulus to one eye and a simultaneous presentation of rapidly changing masks to the other eye. The rapidly changing masks dominate awareness until the target breaks into consciousness. Importantly, this suppression may last seconds." For their investigation of arithmetic, the authors had initially determined in a (conscious) pilot experiment that 2-step subtraction problems (e.g. $9-3-4$) took much longer to solve, so they began with "these more difficult equations." In Experiment 6, participants were presented with a CFS-masked 2-step single-digit equation (e.g., $9-3-4 =$) for a certain period of time. Then they were expected to pronounce the name of a number (e.g., the number 2) which would appear normally on the screen. The times it took for the participant to recognize and say the number were compared for the two cases: "compatible" where the number shown was the solution to the subliminally presented equation and "incompatible," when it was not. In their most striking experiment, when the subliminal equation had been presented for 1700ms., the mean time for compatible reaction was 697ms. versus 718ms. for the incompatible. The mean difference, which the authors attribute to the presence in the compatible case of the unconsciously calculated answer, was 21ms, with a standard deviation of 17.75 (this is reported in the Supplementary Information for the article). According to the authors, this difference is significant at the $P=0.001$ level. The report also describes related experiments, some requiring explanation. For example with 2-step addition problems (e.g. $3+2+4=$) with the same conditions as above, the facilitation effect was reversed.

American Educator takes on the Khan Academy

American Educator is a quarterly published by the American Federation of Teachers, AFL-CIO. Their Fall 2012 issue contains "Khan Academy: The Hype and the Reality" by Karim Kai Ani, a former middle school teacher and math coach. The Khan Academy ("A free world-class education for anyone, anywhere") offers free online mini-lessons on many topics, including Arithmetic, Pre-Algebra and Algebra. The mathematics offerings are under review here. Ani sketches the history of the Khan Academy, and then zooms in on Khan's definition of slope. It may be useful to compare this critique with the actual lesson.

  • "Take Khan's explanation of slope, which he defines as 'rise over run.' An effective math teacher will point out that 'rise over run' isn't the definition of slope but merely a way to calculate it. In fact, slope is a rate that describes how two variables change in relation to one another: hor a car's distance changes over time (miles per additional hour); how the price of an iPod changes as you buy more memory (dollars per additional gigabyte). To the layperson, this may seem like a trivial distinction, but slope is one of the most fundamental concepts in secondary math. If students don't understand slope at the conceptual level, they won't understand functions. If they don't understand functions, they won't understand algebra. And if they don't understand algebra, they can't understand calculus."

There is more, but Ali's basic criticism seems to be that the Khan Academy is successful, and that its success has made us believe in simple solutions to complex pedagogical and educational problems, losing sight of "what really needs to be done." In the American Educator it is not surprising to read what that means: investing in professional development, providing teachers with more support and resources, giving teachers time to collaborate and create appropriate content, helping new teachers to figure out classroom management.

Tony Phillips
Stony Brook University
tony at math.sunysb.edu