Not so fast, says Sara Fulmer over at *The Learning Scientist;*

Although manipulatives can increase students’ attention, this attention may not benefit their learning. In fact, the very aspect of manipulatives that capture students’ attention—bright colors, visual appeal, realistic features—may be their downfall. Manipulatives that are more visually interesting or realistic can increase off-task behavior, such as building or sorting (1). This is especially true if students interact with that object in other contexts, such as during play time or outside of the classroom.

Students who learn with manipulatives can become too reliant on the object and context, and as a result, have difficulty transferring their knowledge to new contexts, different testing formats, or to abstract representations (e.g., algebraic expressions) of the problem

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Here is piece composed by Jeanell Carrigan in honor of Smullyan:

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“Psychologists from the University of Liverpool and the University of Milan conducted a study involving more than 2,000 students in Italy aged between six to 17 years and asked them to complete a number of mathematical tasks, including simple arithmetic and problem-solving.

In the study, the participants’ degree of handedness was ascertained by the Edinburgh Handedness Inventory, a questionnaire assessing how much an individual is right- or left-handed (or ambidextrous). The researchers then analysed the results in relation to the extent to which they were right or left-handed.

Liverpool psychologist, Giovanni Sala, who conducted the study, said: “This study found there is a moderate, yet significant, correlation between handedness and mathematical skill. Moreover, the amount of variance in the maths scores explained by handedness was about 5-10%, a surprisingly high percentage for a variable like handedness.”

This was a conference paper, I will reserve judgment until I read the actual research.

Note: This video claims that Gandhi was left handed. When I looked on line for images of him writing, the ones I could find show him writing with his right hand. Of course, it’s always possible that the images were flipped. Does anyone know a credible source about Gandhi’s handedness?

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“The team used functional magnetic resonance imaging (fMRI) to scan the brains of 15 professional mathematicians and 15 nonmathematicians of the same academic standing. While in the scanner the subjects listened to a series of 72 high-level mathematical statements, divided evenly among algebra, analysis, geometry and topology, as well as 18 high-level nonmathematical (mostly historical) statements. They had four seconds to reflect on each proposition and determine whether it was true, false or meaningless.

The researchers found that in the mathematicians only, listening to math-related statements activated a network involving bilateral intraparietal, dorsal prefrontal, and inferior temporal regions of the brain.”

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“Now, a new study proves that people who are good at reading are also quite naturally talented at math.”

Actually, not really. The study, in and of itself, is quite interesting, but this blogger, apparently relying on an *LA Times* piece rather than the original paper, gets a number of the details wrong.

You can read the original paper here. The abstract reads:

“Dissecting how genetic and environmental influences impact on learning is helpful for maximizing numeracy and literacy. Here we show, using twin and genome-wide analysis, that there is a substantial genetic component to children’s ability in reading and mathematics, and estimate that around one half of the observed correlation in these traits is due to shared genetic effects (so-called Generalist Genes). Thus, our results highlight the potential role of the learning environment in contributing to differences in a child’s cognitive abilities at age twelve.”

One of the big confusions that accompanies a study like this is the misunderstanding of the concept of heritability. Heritability is *not* a measure of the percentage of a trait is caused by genes. It is not even an individual level measure of a trait, it is a measure of how much of the variance in a population is genetic.

One of the consequences of this is that heritablity is not fixed, it will be affected by the level of environmental variance. We can see this by a simple thought experiment. Imagine a society with total absolute equality between individuals. In that circumstance, since there would be no difference in environment, heritablity would be one hundred percent. This conclusion, strikes many as counterintuitive but if you contemplate it for a while you will see that this must be true.

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Going through some of my notes I rediscovered this mnemonic for the quadratic formula:

From square of b, take 4ac;

Square root extract, and b subtract;

Divide by 2a; you’ve x, hooray!

I am not sure of the origins of this mnemonic. The earliest version I could find was from J. S. Mackay in 1894, but he says the mnemonic was given to him by a member of the Edinburgh Mathematical Society.

Here’s another mnemonic for the quadratic formula:

And let’s not forget the Quadratic Formula Rap:

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I own many books by Smullyan. Including these:

I have also been working through his book *Set Theory and the Continuum Problem*

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