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 3×5=? 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.
Worthwhile Canadian Initiative 
Adam P: initially, as children, we conceive of math as discovery, akin to the revealed laws of physical science. Then, around age 16, we are taught to think of if as a construction. Write your definitions, pick your axioms, the rest just mechanically follows. The latter is the way that mathematics is formally expressed, but it’s not how it’s done. We perceive mathematical truths (propositions) before we prove them; the direction of advancement in maths is nothing like the mechanical application of the rules of propositional logic to the body of currently known axioms and theorems. That’s because there are powerful truths in our intuitions about the meanings of mathematical objects, which go deeper than the formal collection of rules and axioms. Math, like physics, is in this perspective revealed truth and there are deep truths in the childish perspective that some mathematicians prefer to reject. Like so many other things the meaning of division may be broader than what we were taught in our introductory real analysis course.
“We perceive mathematical truths (propositions) before we prove them”
Yes, we do. But for some reason or other proper mathemeticians insist on proving them.
We perceive mathematical objects before we define them yet for some reason or other mathematicians expend an enourmous amount of effort trying to actually formulate definitions.
Sometimes this takes many generations, I recall my first year Calculus prof giving a “definition” of a limit that Cauchy had offered and pointing out that usiing it would get you no credit in that course.
Perhaps this is because in Math, and on economics blogs, many of us perceive things to be true, obviously true, when in fact they aren’t.
“That’s because there are powerful truths in our intuitions about the meanings of mathematical objects, which go deeper than the formal collection of rules and axioms”
Perhaps this is true, or perhaps this statement reflects a failure to grasp the full depth of the formal collection of definitions and axioms.
Adam P, K:
It actually runs both ways. Mathematicians usually intuit a theorem before proving it, but sometimes a result falls out of the equations and mathematicians are left trying to figure out why it intuitively “means”.
Intuition and rigor are like yin and yang. They complement and feed off each other.
Sadowski wrote: “It’s sad that they let people earn PhDs in math these days without first requiring them to master arithmetic.”
I don’t see how that’s sad. That sounds to me like you’re missing “the good old days”, which is noble, but arithmetic is not an absolute requirement for today’s mathematics. As other people pointed out, A PhD in math is one thing, and doing arithmetic is another. Yes, some mathematicians use arithmetic in their research, but many of them do not. And when you do need to use it, you have the options of being extra careful, double checking, and asking colleagues to verify your results, not to mention using computers/calculators.
It’s not a question which is confined to economists. Maths teachers are also asking similar questions about the ‘gap’ that the increasing rate at which technology is being introduced is causing in the ability to do mathematics. I posted in March (http://colintgraham.com/2011/03/21/why-cant-you-help-me-with-my-maths-dad/) about some of the issues surrounding the problems that many parents have helping their children with homework. The search for machines to do calculations more quickly is not new (eg abacus). It is maybe the flexibility of the mind, to approach calculations based on arithmetic from a number of different directions, which is the advantage gained from mental arithmetic. Anyone who accepts an answer from any source, numerical or otherwise, without checking it elsewhere, deserves the errors which may or may not accompany it.
Arithmetic is only one part, and perhaps a small if fundamental one, of mathematics and what mathematicians do. Many problems come from people who teach mathematics in the early years not understanding the underlying concepts of what they teach and so revert to teaching by algorithm. Use of calculating machines is incidental if you don’t understand what it is you are calculating or whether what you have calculated is reasonable.
Adam P: “Perhaps this is true, or perhaps this statement reflects a failure to grasp the full depth of the formal collection of definitions and axioms.”
It seems you strongly believe, but can’t prove the latter. Which I find ironic affirmation of my point about the value of perceived truth. 🙂
As you say, unproven propositions, even strongly believed ones, often turn out to be wrong. I’m not suggesting intuition has a closer relationship to truth than mathematical rigour. But as you point out, sometimes we reject an axiom or definition. But not because mathematics tells us that it’s wrong. It can’t be “wrong”. We reject it because we find that we cannot derive from it things that are useful or which we find intuitively true. Sometimes, although the definitions that we have adopted have proven to be useful, they still may not capture the full intended meaning. Then it’s a good idea not to forget our intuitive understanding, in case the formalism ends up getting in the way of where we needed to go. Then, like the definition of the limit, we may have to change the formalism. If we didn’t have an independent understanding of truth, mathematics would be a random collection of axioms and rules of logic.
K:
An axiom can be “wrong” in at least two ways. First, it can — as you say — be derivable from existing axioms, making it unnecessary. But second, its negation can be derivable, making the entire mathematical system inconsistent and thus useless.
Sorry, my feet are too firmly planted on the ground to join this higher level math conversation. Not that I don’t long to fly….
Getting back to some of the earlier posts/threads, take a look at this, allegedly a 1895 grade 8 exam from Kansas: http://www.physicsforums.com/showthread.php?t=100461
What’s remarkable is the complete lack of any questions on geometry. I find this surprising, but others have commented on the secular increase over time in people’s ability to do abstract spatial-manipulation type tasks.
rabbit: OK. But, by “axiom” I really meant “axiom,” not “false proposition.”
I wonder if enhanced computing power may have a similarly stupefying effect on more high-level tasks. Easy-to-use statistical software opened up regression analysis to a vast number of people, while reducing the need for people to understand what goes into a regression. People run inordinately fancy models, without remotely understanding all of the assumptions that go into them (I’m in political science, so I imagine this phenomena is worse here than in economics). Increasingly the tendency is to learn specific skills (eg. a course on a hot technique, like multilevel modeling), and recipes of Stata functions, rather than general skills that can unify different techniques.
This seems problematic on a few levels. It means a greater quantity but lower average quality of quantitative articles. Normally the review process could catch bad articles, but the phenomena extends there as well. If fewer people understand the fundamentals, if more quantitative articles are being produced in disciplines with weak statistical knowledge and if there is a greater demand for quantitatively adept reviewers, you are going to get lower quality reviews. This, in turn, may feed into public policy.
To some extent, this is a general problem with a broad class of technology: devices that accomplish what would otherwise be many small tasks. We lose the understanding contained in those small tasks. Normally that isn’t such a big problem, because we are rewarded with high productivity. My vacuum cleaner allows me to clean much faster than if I relied on a broom. However, I’m not sure that you can apply that logic to endeavors in which quality is more important than quality. Academia seems like an obvious example – computing power allows us to do more analysis (and every journal editor I talk to notes an up-tick in submissions), but may not always enable better analysis. If we are crafts-people, should we really apply the logic of the assembly line to our field?
“the ability to subtract fractional apple bushels is a useless life skill”
As a farmer with a higher-than-average ability for mental math, I for one find this a little insulting!
good job on the freakeconomics link
Tell me about it. Not long ago I was in a class where the professor was using single precision floating point numbers and I was using double precision. He just wasn’t used to having the amount of memory available to modern computers.
Kevin Milligan: “I was the last class in my elementary school to be taught how to do square roots by pencil and paper. I was always a math lover, but I remember that technique as terribly painful.”
It was painful, wasn’t it? Painful in part because it was a mechanical manipulation of numerals, imparted without any explanation except this is how you do it.
I remember trying to come up with a similar method for cube roots. I did, to my satisfaction, but whether it was correct, who knows? I have long since forgotten both methods.
Randy E: “I dislike teaching square roots for the simple reason that I don’t know of a simple way of computing them (and suspect one doesn’t exist, because most of the time they are irrational, even when the inputs are not). To the best of my knowledge, there are two types of techniques to find square roots.
“1) ad hoc, educated guess work. . . .
“2) Approximation techniques such as a recursive formula like x_{n+1}=(x_n + a/x_n)/2 to find the square root of a, or a binomial/Taylor series expansion.
“Option 1 is okay for small numbers, but is unsatisfactory because it doesn’t generalize.
“Option 2 is perfect for a computer, but annoying for a human. At the grade school or high school level I don’t imagine it’s very instructive either, aside from the fact that it could make us appreciate computers more. The potential appreciation of computers, however, comes at the probable cost of resentment towards mathematics. In my opinion, techniques like this are better suited to a computer science course, where students can be asked to write programs that implement the various numerical methods.”
I feel that I must rise to the honor of Isaac Newton, who devised that recursive formula. When I read about it as a kid, far from being annoyed, I was in awe at the beauty of its conception. (Later on I appreciated is speed of convergence.) And once you get the idea, finding cube roots and other higher roots is a snap. 🙂
Why not teach kids Newton’s formula? It is certainly less painful than the method I had to learn. Why teach math as mindless drudgery? Why not lift the veil a little to reveal the hidden beauty?
I can’t resist re-telling that grade-school joke:
A farmer has 1/2 a haystack in one field, 1/4 a haystack in another field, and 1/3 a haystack in a third field. He puts them all together into one big haystack. How many haystacks has he got?
One big haystack, dummie!
Nick Rowe: “One big haystack, dummie!”
😉
“Sorry, you don’t recruit the math high-fliers. I put this down partly to economics being seen more as political science with money rather than a field of applied mathematics.”
That is because it kind of is more a political science…
i would also add that math illiteracy is a more general societal problem, and that it’s not just math but also statistical/probability-type thinking. for example, when there was the ruckus about the H1N1 vaccine. i would listen to fellow students at university talking about how they read a news article of some person who had previously been perfectly healthy being nearly killed and subsequently disabled by getting a vaccine.
my first question in reaction would be: “gee, thats horrible. is that case representative of the general population, though?” there would be a pause and then “well, no, it’s not representative. but you just HAVE to read the article, it’s so crazy! it’s scary!”
so the “scary” unrepresentative outlier which is something like 1-2% of all results is more attention-worthy then the 80-90% who do just great. you can also see this in how people who think vaccines cause autism react to the statistical studies that find no “statistically relevant” connections between the two with cries of “you’re stupid!” or “you’re lying, it’s all a conspiracy!”
To me, this debate is clear. There is a generation gap yes, but in a couple years this will not be a problem. Our societal institutions, especially education are often faulted for being unwieldy and slow to adapt to our ever-changing society. The mathematics department’s commitment to integrating technology into the classroom should be applauded. Only recently have other segments of education began to fully utilize newly developed technology. The perceived generation gap will disappear in only a matter of a few years as students who have grown up with calculators take on the role of the professor. It is up to the teachers to understand that students need to know the concepts underlying economic calculus.
“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.”
awful, just awful. as someone of this generation, i feel insulted on behalf of everyone
Henrik – ” i feel insulted on behalf of everyone”
No, it’s not the norm, but it happens. For a long time we had a clock on top of our TV with no numbers, just an hour and a minute hand, something like this. Visiting teenagers routinely struggled to read it, adults generally read it without difficulty. But there’s a whole stack of stuff I can’t do that people of my parent’s generation can do, e.g. use a slide rule, sew.
It’s important for profs to know that in a class of 100 students, the odds are reasonably good that one or two people might not be able to read the clock, and they need to accommodate the students in an exam situation, e.g. by writing the time on the blackboard.
Re. Clocks, I just got myself a new BlackBerry Torch. It shows the time numerically on the home page, but if you run the clock app, you get a clock with hands and a face.
I never even thought about whether or not students would know how to read the clock on the wall (besides, it’s usually wrong, often not even working, and sometimes completely absent). I usually give 10, or 5 minute warning and a 2 or 1 minute warning anyway.
Interesting post and debate. One thing that I’ve felt more and more strongly as I make my way through an electrical engineering undergraduate degree and work as a peer tutor in math and engineering is that in many cases students would benefit from a curriculum that included more utilisation of the technology we all use regularly. I think one of the best ways to do that would be to have students write their own computer programs to carry out the operations they are learning. This might not work as well for some areas (the “pure” mathematics of calculus and beyond), but for anything numeric in nature I feel like that would be an excellent way to help ensure that most students gain a decent understanding of what is going on and how the complex systems like Matlab, Mathematica, or Maple actually come up with the answers. It would also encourage problem solving and computer skills that would be of immense help to any engineer or scientist.
Last, I would say that being able to do quick mental calculations and having the ability to predict the outcome of a problem in general terms is important and useful, but ultimately any important problem solving should be run through a computer and peer-reviewed anyway so really those skills do have less importance than they once did.
Richard,
I think there could be some interesting and “pure mathy” programming exercises. Implementing Newton’s method, referenced above, for example, could be worthwhile, or writing programs for various approximation techniques for definite integrals (Midpoint Rule, Simpson’s Rule, etc.). The problem I’ve encountered in my classes is that, even in a second year class, I can’t assume that everyone had even elementary programming abilities, despite the fact that all students were supposed to take a computing class in first year.