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.

But the mental arithmetic gap has more subtle implications. Mental calculations often require intuition about, and comfort with, the use of fractions. Pre-calculator: 1/3+1/3=2/3. Calculator era: 0.3333....+0.3333....=0.6666.... Pre-calculator: "To multiply by twenty-five, divide by four and add two zeros (25*Y=1/4*100*Y)" Calculator: Multiply by twenty-five. Back in the day, fractions were easier than - or at least not much more difficult than - decimals. Calculators make fractions obsolete.

An economic concept that requires a deep understanding of how to use and manipulate fractions is elasticity: the percentage change in X/percentage change in Y. I wonder: how many students struggle with elasticity formulas because they struggle to manipulate and understand fractions?

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 consequences are neatly described in an "Alex" cartoon I have on my fridge about a proposal to ban the use of calculators in school. "Faced with home work which requires him to work out simple sums in his head today's lazy seven-year-old will instinctively turn to the quick and easy method of arriving at the answer... i.e. asking his dad, who, embarrassingly also wouldn't have a clue without a calculator."

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:

The development of sophisticated yet easy-to-use calculators and computers is changing the role of procedure and technique in mathematics. Operations that were an essential part of a procedures-focused curriculum for decades can now be accomplished quickly and effectively using technology, so that students can now solve problems that were previously too time-consuming to attempt, and can focus on underlying concepts. “In an effective mathematics program, students learn in the presence of technology. Technology should influence the mathematics content taught and how it is taught. Powerful assistive and enabling computer and handheld technologies should be used seamlessly in teaching, learning, and assessment.”

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. After all, according to the OECD rankings, Canada scores highly in international comparisons of mathematics performance. (Particularly notable, according to this detailed OECD study of Canada's exemplary international performance, is Canada's welfare state, teacher selectivity, and success with immigrant children.)

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 - 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?

I originally titled this post "bridging the mathematics generation gap." I've been reading about mathematics pedagogy, particularly the JUMP math approach founded by John Mighton, and about the history of math wars. But I need to work out where the mathematics generation gap lies, and what its consequences are, before writing about how to bridge it.