## Saturday, August 11, 2007

### SAT challenge

Note: Figure not drawn to scale.

The circle shown above has center O and a radius of length 5. If the area of the shaded region is 20 pi, what is the value of x?

18
36
45
54
72

I like this challenging SAT problem because it demonstrates once again that background knowledge is needed to deploy critical and creative thinking. This should give pause to "critical thinking" advocates who disparage knowledge (known as "content" nowadays).

Here is the reasoning, based on background knowledge, offered by SAT (don't read past this point if you want to solve this problem yourself):

In order to find the value of x, you should first determine the measure of the angle that is located at point O in the right triangle. To determine this angle, you must calculate what fraction of the circle’s area is unshaded. The radius r of the circle is 5 and its area is pi r^2, or 25 pi. The area of the shaded region is 20 pi, so the area of the unshaded region must be 5 pi. Therefore, the fraction of the circle’s area that is unshaded is 5 pi/25 pi, or 1/5. A circle contains a total of 360 degrees of arc, which means that 1/5 of 360 degrees, or 72 degrees, is the measure of the angle at point O in the unshaded region. Since you now know that two of the three angles in the triangle measure 72 degrees and 90 degrees and that the sum of the measures of the three angles is always 180 degrees, the third angle must measure 18 degrees. Therefore, x = 18.

### More cold water on "learning styles"

The learning-styles craze has received another well-deserved drubbing from a leading scientist in England. Let's hope "senses working in unison" will gain the upper hand on non-sensical pigeonholing:

Pupils are instead given questionnaires to discover if they prefer to learn through "visual, auditory or kinaesthetic" (Vak) teaching. Once identified, the teacher will allow a visual child to learn through looking at cartoons, pictures and fast-moving computer programmes. A "kinaesthetic" learner will be allowed to spread their work on the floor, wander round while they are thinking or learn through dance and drama. In some schools, pupils' desks are even labelled to indicate their learning styles.

According to Susan Greenfield, however, the practice is "nonsense" from a neuroscientific point of view: "Humans have evolved to build a picture of the world through our senses working in unison, exploiting the immense interconnectivity that exists in the brain. It is when the senses are activated together - the sound of a voice is synchronisation with the movement of a person's lips - that brain cells fire more strongly than when stimuli are received apart.

Also see the article by cognitive scientist Willingham on teaching in the
subject's best modality.