The number under the cursor

We spotlight a skill of the analog world.

Our tutor has several students working in a math chapter that involves picking numbers off a graph.  That is, a curve is drawn against two labeled axes, and they’re supposed to find the abscissa given the ordinate, or vice versa.  Often there’s much more to the problem than that, but he finds the students having more trouble just reading the graph than he expected.  It’s only an impression, and he’s been reading graphs of many types for so long that maybe it’s a false one.  But just reading a number off a scale seems to be hard for them.

Well, theirs is a digital world.  They “dial” a telephone by pressing separate number keys; none has actually seen a telephone dial.  When they call up the weather on their smartphones, all the numbers (temperature, wind speed) are digits, not scales.  There are many fake-analog meters, from sound intensity to car gas tank status, but they are made up of a number of discrete LED units or liquid-crystal bars.

He’d already noticed that some students had trouble with questions involving analog clocks.  This type of clock certainly hasn’t disappeared; in the tutoring center there are several; but students were making mistakes (confusing the hour with the minute hand, for example) that simply wouldn’t have happened before the advent of digital watches.

For a long time calculators have been digital.  This means that the accuracy of a calculation doesn’t depend, as it used to, on the ability of the operator to estimate correctly the number under the hairline cursor.  This is, as we’ve noted, in general a good thing.  But it appears that, without the incentive, student’s aren’t developing the skill of reading a quantity off a continuous scale.  And that has implications.

First, much information still comes to us in analog or pseudo-analog form, in charts and graphs. Failure to notice the scales or read them properly results in wrong answers.

Second, not having a picture of the number in question, say a segment of a circular dial, the student doesn’t develop a grasp of the relationship between numbers. In particular, it’s difficult to realize just how small the difference between 3.26 and 3.28 actually is. We saw this long ago in chemistry class, when students would copy ten full digits off their calculators when only two had any justification.

Third, living in an entirely digital world it’s very hard for students to grasp the basic difference between practical calculation and rigorous proof, the difference we’ve highlighted between the Greek and the Babylonian approaches to mathematics.

And we wonder what other differences may arise between the older citizens, who can measure, and the younger ones who can only count.