Frances Woolley has written an interesting piece as usual and this time it is regarding the changing understanding and usage of mathematics in students. She states:

*Here's my theory: Some students struggle with economics because they do not fully understand the mathematical tools economists use. Profs do not know how their students were taught mathematics, what their students know, what their students don't know - and have no idea how to help their students bridge those gaps.*

*The arithmetic gap is the most obvious one: profs over a certain age (and some immigrant profs) were drilled in mental math; Canadian students under a certain age haven't been. Some implications of the arithmetic gap are familiar: profs who can't understand why students insist on using calculators; students who can't understand why their profs are so unreasonable.*

I believe it is even worse than that. Estimating was once the art which followed measuring. You went out and measured again and again until you had the understanding of what distance was and then you could look at say a tree, a lot, or almost anything and estimate. You could look at a jar of jelly beans and estimate to within a good margin of error what was in there. So if you are being attacked by a giant dinosaur you were able to determine rather quickly how far to hurl the bolder to scare the fellow away. Not so in today’s grammar schools.

Technology has entered. Frances uses the map example and she states:

*Other technologies also create generation gaps. Today's undergrads have been carrying a cell-phone since their early teens, if not earlier. They rarely wear watches. Some will struggle to read an analogue clock - even if there's a clock on the wall in the exam room, they might not know how much time they have left to write the exam. The disappearance of analogue clocks, however, means that profs risk confusing students when they use clock-based language: "Rotate counter-clockwise." "Turn clockwise." "At 2 o'clock" (as in, 60 degrees to your right).*

*Maps are another rapidly changing technology. Google maps was launched in 2005, in other words, when an undergraduate entering university this Fall was 11 or 12 years old. She has always been able to navigate by reading a list of instructions from google maps, she might never have had to locate two points on a map and plan a route from one to the other. Yet maps imbed spatial concepts very similar to those used in economics. An indifference curve or iso-profit line is, conceptually, similar to a contour line on a topographical map. What forms of understanding do students lose -and what do they gain - when they rely on google maps rather than map-reading?*

This goes to the classic McLuhan statement that as media changes what we perceive as knowledge, read truth, change to resonate with the new knowledge.

Take maps. I started companies in twenty countries and in each case I did three things first. I learned 100 words, such as please, thank you, where is the bath room, here and there. Second, I read a history of the country, so for example in Prague I knew all about the Thirty Years War. In Greece, well, you had a lot. In Korea, you better know about Japan and China. Third, I got maps, I would look at them again and again, it was the culture, it was the history, it was the present. I now use Google Maps but I always change the route, and have second exists. I also from time to time use my Garmin, but alas, if you were born and raised in New York, you never follow Garmin! It must have been programmed by someone in Duluth! They always send you on the Cross Bronx Expressway, the parking lot of the city. Culture counts.

Take watches, yes, most use the cell phone, but even there I have a pocket watch, yes a pocket watch, from an old Russian friend. It stops the conversation every time. But a watch is more than a time teller, it is as Frances says an artifact of saying lots of things, like clockwise and counter clock wise.

The technology is change what we know as knowledge, it is splitting a generation.

But I take her a step further. In mathematics some fifty years ago before computers we had to solve problems by approximation. We had to have a gestalt about the equation, its movement as we changed variables. We understood the equation as a living thing. We looked at boundary conditions and extrema to see if they made sense. The classic example here is Richard Feynman. Feynman had this art down pat, he could feel the answers. He could see a problem and intuit the answer, and then fill in the details. He lived in the equations as a particle, moving and rotating. His approach led to his work on quantum electrodynamics and his symbolic math which got him the Noble Prize. Very few students can even near that now, they all use MatLab and they have totally lost a feel, the no longer go down the alleys of mistakes that lead to enlightenment. Pity. The technology has removed the opportunity of discovery by mistake.

Since getting beck peripherally into academe, at MIT, I have been spending time trying to model cell genetic pathways and cancer dynamics. Not that it will solve world hunger but it is an interesting problem which may or may not be solved by such models. The more I work, the less I know, and the harder it gets. It is akin to economics, just when you think you know something you discover a new path that someone has discovered which changes your model.

For the folks in the cancer modeling area, they understand and await the next change, the stem cell, the micro RNAs, the epigenetic factors, and try again. They are always change models to match the flow of new facts.

Economics is NOT that way however. They add new theories to explain what the result should be and the facts be damned. The Phillips curve and CAPM is an example. So why are economists so different, is it their love of the math qua math, and has the technologies changed any of that. Is there an economist like Feynman who can intuit the right answer and everyone agrees. It is not Krugman, that is for sure. Does the technology make any difference for economics. It has for genetic pathway research, both in generating data and in modifying models.

Economics however is so much of an understanding of the human condition, and more importantly of humans with dramatically differing world views. The homogeneous assumptions of markets often fail to incorporate that factor. It also fails to incorporate the fact that humans change, in random and intermittent ways. Perhaps economics shall never adapt to that. Perhaps the goal of Isaac Asimov and his

*psychohistory world view*, as beloved by Krugman, will not be achieved.

Finally, even

Mark Thoma and the followers there seem to have resonated with the post by Frances. My comments still stand as one who uses mathematics to solve real problems, fundamentally as an engineer. It is a tool, a tool to understand reality, to project forward and to live with reality. The tool just for the sake of the tool, or the tool used according to the old dictum, "if elephants had wings..." world, has much lesser value. When employing mathematics it is done to seek truth, employ facts, and predict outcomes. It is not to see what the proverbial elephant looks like.