Maybe this is why my econometrics class got so upset when I wouldn't allow calculators to be used during exams:

The mathematics generation gap, by Frances Woolley: 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;... 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. ... Another aspect of the mental arithmetic gap that is easily overlooked is its widening over time. Calculators became affordable in the mid- to late-1970s. Students in the 1980s were taught by teachers who had learned mathematics without calculators, and could do basic mental arithmetic. Students today might be taught by a teacher who is himself unable to work out 37+16 without help. ... The average professor might be unaware of just how ubiquitous calculators are in elementary and secondary schools. The Ontario province-wide grade 6 math test allows students to use calculators at all times. The use of calculators is mandated by the high school curriculum... School curricula reflect society at large. Back in the 1950s, grade 5 students were taught to answer questions such as "Joe picked 3/4 bu. [bushels] of apple while Jack picked 1/4 bu. How many more did Joe pick than Jack?" No amount of back-to-basics rhetoric will change the fact that the ability to subtract fractional apple bushels is a useless life skill. Today an average Canadian can live a happy and fulfilled life without being able to compute $4892.16+$5860.03+$512.41+$8967.35. So why teach those skills? Recent research is suggesting that deep understanding of mathematical concepts is related to basic number sense. A person who can look at two sets of dots and quickly determine which set is larger will also generally be better at abstract, conceptual, mathematical reasoning. I have had a student in my office who could not work out 3x5=? without a calculator. I wonder: what else was she missing out on? But perhaps the struggling students make a deeper impression on me than the competent ones. ... So maybe I'm just out of touch. Take graphics calculators, for example. I don't know precisely how they work or what they do, but I regard them with suspicion. Graphing the production function F(x)=ln(x) by entering the function into a graphics calculator and copying down the result just seems like cheating. And because I've heard these calculators are programmable, I ban their use in exams. It's another mathematics generation gap: between students who were taught from a curriculum that encourages - or even requires - graphics calculators, and their old-school profs. But I don't know what you do know and what you don't know, and I don't know how to teach you the basic mathematical concepts you require to understand economics. 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... The disappearance of analogue clocks ... 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? I originally titled this post "bridging the mathematics generation gap." ... But I need to work out where the mathematics generation gap lies, and what its consequences are, before writing about how to bridge it.

I think there is a lot of intuition that can be gleaned from a bottom up approach that emphasizes basic skills. During these rote exercises, the mind searches for shortcuts, and much intuition and insight can be gained from the aha moments when you find a way to shorten the exercise with a trick of some sort. E.g., to multiply any single digit number by nine, just add a zero to the end and subtract the number. Thus, 8*9 must be 80-8=72. Or, 4*9 = 40-4. Then, it's easy to generalize, 9 times any two digit number is the number with a zero attached minus the number. Thus, 28*9=280-28=252. Or, 9*32 = 320-32 = 288. Then extend further -- it works for a single digit times numbers of any size, e.g. four digit would be 9*1234=12340-1234 = 11,106, something you can do relatively easy in your head. This can be generalized further (it's best to do it for 8s, then 7s, until you see the pattern emerging). To find x*y, where x is one digit and y is any whole number, use x*y = 10*y - (10-x)*y. So, to find 187*7, it's 1870-3*187. But you say, the second number is too hard to multiply in my head! No problem, just apply the rule again, or better use some other trick, like finding 3*200-3*13 = 600-39=561. Thus, the answer is 1870-561=1,309, and it can all be done in your head. When I was a kid using tricks like this allowed me to speed up finishing worksheets for homework so I could do what I really wanted to do, go outside and play. This also leads directly to the proof -- just expand the right hand side, then cancel terms -- but how do you discover this rule, and learn how to take it to a proof, without rote exercises that force you to search for shortcuts? I understand that the response to all of the above is to use a calculator instead, these tricks aren't needed if you have a calculator at hand, but that isn't the point. The point is that these exercises lead to additional insights, proofs, etc. and those insights are critical for more advanced insights and more complex proofs.

I plan to remain hard-headed about this until I am convinced that abandoning the rote sorts of exercises done in, say, a linear algebra class (which can also be done on a calculator) does not hinder our ability to form intuition about how to do proofs, etc. And these skills are valuable in other settings as well. I don't know how many times I've written computer programs by brute force initially, then realized the programs can be shortened and made much more elegant by exploiting the patterns that emerge in the brute force program, and that usually leads to a very compact, linear algebra representation of the program (which can then lead to further insights about the underlying statistical model). The inductive type reasoning that emerges from these exercises is valuable in many settings -- I'd guess learning to find patterns is a skill that is useful beyond pure mathematics -- and I worry that an over reliance on calculators will erode the development of these skills. I am absolutely convinced, for example, that forcing people to do econometric and statistical exercises by hand develops intuition that you cannot get any other way, and this is a key to moving on to doing proofs.

But what is your view on all of this?