The mathematics generation gap

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. 

Comments

[Determinant]

Here's my countertheory: Economics does not recruit the top students with math ability into its first year. Really math-able students choose the natural sciences (physics, chemistry) or Engineering.

In my university Calculus for Engineering was marginally ahead of Calculus for Science in terms of difficulty, both were considered considerably more difficult than Calculus for Social Science.

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.

[Neil]

I was in the last class in Alberta that wasn't required to own graphing calculators. And I never have. Haven't a clue how to use them. But I recall being asked to help a friend who was a year behind me with her homework, and being completely baffled by what she was doing. Something involving circles. Who knows, maybe there's an important life skill that I'm missing.

"Today an average Canadian can live a happy and fulfilled life without being able to compute $4892.16+$5860.03+$512.41+$8967.35."

I disagree. Despite appearances, being able to do some basic arithmetic in your head really is a useful life skill. When going around a grocery store, I keep a tally, usually accurate to within $5, of everything I'm buying. It lets me make an informed choice when deciding if I should buy that cake I want or not. Similarly, at stores that don't give a nice unit price on the tag, I can compare different brands and packaging sizes that might be undifferentiable to others.

Yes, you could just punch the numbers into your cell phone and find out, but I don't think I've ever seen someone comparing prices this way. I'm under the impression that people who can't do mental arithmetic just grab stuff that looks good and hope for the best. Which may help explain the number of people who find day to day life unaffordable, despite having decent incomes.

At more advanced levels, it seems like understanding the underlying concept is more important than being able to compute numbers. And the calculator's focus on a numerical result can get in the way of understanding the concept. A friend who teaches university-level math got the following answer when asking what the differential represents: "This issue is hotly contested amongst mathematicians, with one camp claiming that the differential as the slope of the tangent, and the other claiming it is the rate of change." While my friend is quick to criticise his own teaching as leading to this answer, it's quite possible that a focus on numberical answers might have obscured that these are the same thing.

[Nick Rowe]

As an old-skool (traditionally taught) immigrant prof of a certain age, this is depressing. But you might be right.

An alternative theory: there has been more grade inflation in other high school subjects than in math, so: students who want to get into university rationally avoid math (moral hazard); and students who don't avoid math don't get the high grades needed to get into university (adverse selection). (I make this hypothesis without knowing the data, as usual).

[Moebius_strip]

Excellent post. I teach math at the postsecondary level, and I can tell you that your observations about calculators impeding the development of number sense are dead on. Back when I taught financial math, I'd always ask some basic interest rate question: "Joe invests $100 at 2% interest per year, compounded monthly. How much does he have after five years?" Without fail, I'd have at least one student merrily report that Joe would have eleven billion dollars after five years. It didn't occur to them that this couldn't possibly be right. I never got the name of the bank that provided such attractive incentives for opening savings accounts, alas.

It's only been ten years since I was an undergraduate math student, but I can see that the ubiquity of powerful calculators and software are allowing for curricula which, for better or for worse, include more and more sophisticated applications - in less depth - in the same number of teaching hours. A generation ago it was unthinkable for students to be allowed to bring in (or be given) formula sheets for exams. You had a handful of formulas that you'd have burned into your brain by the end of the semester, and use them as appropriate. Now, with computations so quick, the focus has shifted; I handed out an 8-page package of formulas for an exam I just gave. During class time, I had no choice but to present those formulas with minimal or no justification, so I couldn't expect students to be able to derive (or even remember) them themselves. (This was a class for engineers.) I allowed another group - statistics students - to bring in their own formula sheets. I didn't give any restrictions on length, and many students brought in entire notebooks in which pairs of formulas such as "a+b=c" and "a=c-b" were listed separately. To some students, each formula is a separate fact; the concepts unifying those facts escapes them.

Alas, my department, by and large, has given up on banning graphing calculators. It's a losing battle. Often, my only restriction on calculator use is "you are not allowed to use any calculator that can talk to other calculators." This after my department head caught a student using the Integration App, for which I believe he'd paid $0.99.

[david]

What sort of calculators do you use that can't add 1/3 + 1/3 = 2/3? As in, display the result 2/3? A bog-standard scientific calculator - not a graphical one, just a standard Casio high-school calculator - should be able to do fractions, too. A somewhat fancier scientific calculator will even support displaying it vertically instead of as "2 / 3".

An even fancier one can do the symbolic algebra for you, too - 1/x + 1/x = 2/x - but that is generally banned, even if it is not programmable.

But I can tell you where calculators hurt: they stop requiring you to be able to intuit simple algebra; instead, at a high-school level, they reward recognizing standard forms and plugging it in until the student arrives at a result that the scientific calculator can process. This isn't new; previously one recognized standard forms and plugged them in until one got results that a slide rule and a trig table could process. But because scientific calculators can process a much wider array of forms, one gets students that are better at algebra than at visualizing magnitudes and graphical intuition.

Great for the engineers, who promptly wander off into the depths of applied calculus. Not so good for the economists. The student never expects to have to stop and visualize the algebraic forms at hand; no, the training is toward recognizing possible symbolic manipulations and recognizing when the calculator can take over, if at all (preferably as soon as possible, given time pressure under exam conditions - the engineers just change this to recognizing more and more integrals). That is what you have to do to be good at high-school math (and undergraduate engineering). But undergraduate economics is not about algebraic manipulation or the plugging of numbers into the results - imagine teaching supply and demand by handing students some explicit utility and production functions, with the goal being to calculate the simultaneous solution! Because that's pretty much what they are trained to do.

The whole idea of comparative statics is completely new to a first-year, even if the math is theoretically already all there. The fresh undergraduate has, almost certainly, never been asked to compare the graphs of two similar equations, never mind comparing two similar graphs while keeping their underlying equations unspecified: it may be just a linear transform, but they don't think in terms of graphical transforms! If you hand a student a graph, what the student wants to do is extract some series of symbols and apply rules to it until it spits out an answer. Graphical comparative statics is wholly alien, as is appealing to intuition about how the solution to some (unspecified) simultaneous equations might change under a (not-symbolically-specified change) in variables.

[david]

Cont: A Casio scientific calculator, from Wikipedia.

Moebius_strip's description seems about right. Students are not getting worse at manipulating fractions, if anything, rapid symbolic manipulation through applying some given set of rules is what they excel at. We encourage it so much that we ask them to do this under time constraints, so that thinking about anything else besides symbolic recognition is actively penalized. And then we're surprised if they only dimly grasp what the underlying concepts might be? Ha.

[Bill Bell]

"... students can now solve problems that were previously too time-consuming to attempt, and can focus on underlying concepts ..."

Let's pretend that it's not too late to reverse this nonsense, if nonsense it be. This statement seems to promise that students will now learn things that they didn't before. Would it be reasonable to ask educational authorities to show us their experimental evidence?

I'm neither a teacher nor a scholar of education. However, I would say that we would want to know which style of education equips people more effectively for (say) undergraduate economics, undergraduate economics, life as an electrician ... or buying groceries. I wonder if anyone has troubled to check?

[Frances Woolley]

Determinant: "Sorry, you don't recruit the math high-fliers." I teach a third year economics course that is required for public policy majors. In any given year, my best student might be majoring in math or journalism or poli sci or business or even, sometimes, econ. No, they're not the math high fliers, and they don't have to be. But they want to learn econ, and I want to teach them.

Neil: "Despite appearances, being able to do some basic arithmetic in your head really is a useful life skill. When going around a grocery store, I keep a tally, usually accurate to within $5, of everything I'm buying. "

I don't keep a tally in my head, but I can usually guess, to the nearest $20, what the bill is going to be, just by looking at the cart. No idea how I do it. I think - and perhaps Moebius_strip or other math teachers who read the blog can weigh in here - that we don't yet know the connection between practicing arithmetic, gaining basic number sense (not calculating 3,4798x2,043 precisely but rather knowing that the answer is about 7,000,000), and doing higher-order mathematics.

Nick, my sense is that there's more variance in math than in other subjects, because answers can be right or wrong. Do you remember talking a while ago about trying to teach students who simply *could not* understand how to graph things? I was going to write about that, but nothing's gelled yet.

Moebius: "I can see that the ubiquity of powerful calculators and software are allowing for curricula which, for better or for worse, include more and more sophisticated applications - in less depth - in the same number of teaching hours."

But you're teaching at the post-secondary level. What (or who) is driving these curricular changes?

[Phil Koop]

Well, I am one of those elderly people who was drilled in mental arithmetic, and I don't really see how people can get by without it. Perhaps I am naive, but I have to think that an inability to solve 3x5 must be due to learned helplessness; surely anyone who actually tries to work it out must stumble across the answer.

Nevertheless, being lazy I "cheat" all the time in may daily work. For instance, I often use a symbolic system like Mathematica to solve an integral for me, or check definitions in Wikipedia (what are the exact technical requirements that must be satisfied in order to apply Cameron-Martin/Girsanov again?) So far I have not been struck by lightning. It really is true that computers make practicable techniques that would be too time-consuming by hand. Would you want to solve a finite difference grid, or a monte carlo simulation, or a large, many-variable regression by hand?

But it is also true that a profound understanding of mathematical concepts subsists upon a shallow facility with boring, mechanical symbol manipulations. Nowhere is this more evident than with computer programming itself. The trend is inexorably toward higher levels of abstraction; but the people who are good programmers today would have made fine assembly programmers. A good short-term memory and the ability to concentrate very narrowly continue to be indispensable.

Finally, graphing a function can be a very useful way to understand how it behaves - but only if you understand what you are looking at! It is more or less impossible to have this understanding unless you can go in the other direction: knowing the properties of a function, you ought to be able to sketch its shape. That is why one of our interview questions is often to sketch the properties of some function such as ln(x)/x (e.g. asymptotes, maximum, zero crossing.) And you can't use your graphing calculator then!

[Frances Woolley]

david: "What sort of calculators do you use that can't add 1/3 + 1/3 = 2/3?" A 30+year old solar powered HP calculator. It works perfectly. For anything else, I use excel or stata.

You make some fascinating observations.

[Randy E]

I've been teaching first and second year calculus and linear algebra courses (and one course in cryptography) to a variety of students (mainly engineering, but also econ, bio, business, etc.) since I finished my math PhD in the fall of 2007. I can relate to a lot what's been said here, both in the post itself and the responses.

The first couple of years teaching were difficult for me because I simply did not comprehend how much students didn't know. For example, I didn't spend much time going over trig calculations, such as evaluating trig functions or using trig identities, when I was working through integration problems, because they were beside the point and I assumed that they already knew and understood the trig functions well enough that they could fill in the details. But when it came down to test time, many students bombed any questions involving trig.

The quote from the Ontario curriculum is interesting. On the face of it, it seems reasonable, and I largely agree with the spirit of it. The concepts are more important than mechanical computations that follow from those concepts, because, based on my own experience, it seems that knowing how to apply mathematical ideas to "real world" problems depends on a fairly deep understanding of concepts. If you spend less time figuring out how to do trig computations, then you can spend more time understanding what trigonometry is, and if you understand the concept better, you understand better when it should be used.

In my experience, though, it doesn't seem to have worked out the way the Ontario curriculum suggests. Rather than getting a better understanding of concepts, it seems like calculator use has resulted in conceptual understanding being bypassed altogether. The value of the sine function is not the ratio of sides of a triangle or the y-coordinate of a certain point on the circle, but rather, it's the number you get when you push the "sin" button on your calculator.

Something else just came to mind while I was typing this. Often the people teaching math in elementary school are people who weren't comfortable with math in the first place (I don't know this for sure. I'm inferring it from the fact that math is not a very popular major). I would imagine these people are less comfortable with concepts than they are with computations. If calculators are cheaply and readily available, they might not be likely to see calculators as a giving them time to explain the theory better, but rather as an opportunity to avoid explaining the mechanics as well as the theory.

[david]

Reverse Polish Notation input required in those older HP calculators may encourage more engagement than newer calculators, too... recent Casios permit inserting parenthesis, fractions, powers, etc. as one might do in Mathtype, which probably just encourages rigid adherence to symbolic similarity.

[Moebius_strip]

But you're teaching at the post-secondary level. What (or who) is driving these curricular changes?

I can't speak for other post-secondaries - I teach at a polytechnic, to students who have very clear ideas of what they plan to do after graduation - but where I am, to a large extent, it's industry. Technological advances have led industry to use, and hence require, a broader set of skills. A deeper (or equally deep) set of skills would be nice too, but if we're going to stick with 4-year degree programs (or 2-year diploma programs), something has to give. Time will tell whether, on balance, my school and others have made the right choice.

Interesting question about the connection between basic number sense and doing higher order math, and one on which I'm not sure I have much professional insight. But to extend Phil Koop's last comment - that graphing a function can be useful, but only if you have some notion of what to expect - I'd say that "knowing what to expect" is something that ought to be, and no longer is, developed from a young age. Better to train the younguns recognize that 3,4798x2,043 is about 7,000,000 before expecting them to figure out that the function y=ln(x)/x should look "approximately like so" before invoking their TI-84+'s.

[Brett]

My son's in grade 4 in Ontario, and he's had to memorize addition, subtraction, multiplication and division up to 12 such that he can do each set (e.g., the from 8 x 1 to 8 x 12) in under 30 seconds. They also have units on estimating that integrate well with their other units as a means of predicting and double-checking. The curriculum and the specific teaching of it haven't been perfect, but they're not leaving yawning gaps in their knowledge. I'll also note that the EQAO tests (grade 3) also allow calculators, but my son's daily classes haven't typically employed them.

[Vladimir]

If I may be permitted to voice some sharply critical opinions, speaking as a non-economist with a decently strong background in math and with a lot of amateur interest in economics, I have always been struck with the way economists (of all generations) often use math, and not in a positive way. When I observe math-based discussions between mathematicians or physicists, and when I compare them to the way economists commonly do it, I can't help but often see a huge difference in the quality, precision, and soundness of the discourse.

Here I don't mean that economists are generally bad at math in the sense that they would flunk an algebra, calculus, or statistics exam. Rather, the crucial thing that I often observe them lacking is the ability to clearly state what the assumptions of their models are, what is the justification for assuming that they correspond to reality, and under what exact assumptions the model breaks down and diverges from reality. This makes their math-based models (let alone their debates) full of confusion and misunderstanding, instead of the tools for clear and precise expression of complex ideas that they're supposed to be.

If two physicists come to different conclusions about what a physical theory predicts, they can normally work their disagreement back to the exact point where one of them made a mistake or where they made different assumptions, all the time being aware of the difference between what is a part of the model and what of the actual physical reality. As far as I see, economists are frequently unable to do so in their disagreements, leading to confused debates that never get down to the underlying logic, the limits of the models involved, and the difference between abstract concepts used in the model and things that actually exist in the real world. (Of course, one could find counterexamples for both physicists and economists, but the general trends seem clear.)

Considering all this, I think the issues discussed in the original post are interesting but of much lesser concern than the more fundamental problems of the entrenched ways in which math is (mis-)used in economics. If there is one thing where mathematical education for economists is lacking, it is the sort of epistemological discipline and precision that many of them could (and should) learn from physicists.

[Terry McGarty]

Frances

Great post as usual, I am reminded of the famous statement by McLuhan that the medium defines what is knowledge, truth, etc. Thus can we expect that what we see as "truth" changing as we use other methods of calculation? Or lack thereof.

I will write a bit on this but the point is that math is for many the ability to analyze what is presented, the typical example is looking at a spread sheet and immediately seeing a mistake. I once hired an IIT/U Chicago grad (Indian Institute of Tech) who on an interview found a mistake in a spread sheet I used to explain an acquisition.

There is also the Feynman effect that some people intuit the answer and then apply the math.

Both examples result from understanding the problem in depth. Having an almost visceral understanding of the interrelationships and dynamics.

Matlab has destroyed for many engineering and science students the understanding of looking at boundary conditions and making analytical approximations.

On the other hand for many problems there may be no model as of yet. This may very well be the case in economics, since we lack the ability to deal with the human response.

This is worth some thought as are all your posts. It I believe is also generational.

[Frances Woolley]

Brett: "My son's in grade 4 in Ontario, and he's had to memorize addition, subtraction, multiplication and division up to 12 such that he can do each set (e.g., the from 8 x 1 to 8 x 12) in under 30 seconds."

Some questions from my garage sale 1950s grade 5 arithmetic textbook: calculate 3695 divided by 54. Calculate 4 1/2 - 2 5/16 (i.e. four and a half minus 2 and five sixteenths). "A large school system uses 1875 T of coal a year. By buying all of it from one dealer a saving of $0.96 a ton can be made. Find the total saving that can be made."

Compare with the questions on the grade 6 province-wide Ontario EQAO exam here: Consider the fractions shown below: 3/4, 18/25, 14/20, 75/100. Which fractions represent equal values?

Now the 1950s grade 5 textbook has no discussion of geometry, patterns - indeed the bulk of the things in the Ontario grade 6 test. Perhaps they had other books for those things, I don't know.

So better - perhaps. Worse - perhaps. One thing is for sure, though: in education a huge amount just comes down to having a good teacher, and if your son is so blessed, that's wonderful.

Vladimir - I think economics-math is a little bit less sloppy than it used to be, but in the basic (micro) ECON 1000 course, demand curves are drawn with the dependent variable (quantity demanded) on the horizontal axis, and we act as if the coordinate plane has only one quadrant. It drove me crazy for a couple of years, but then I became accustomed to the strange rites of my tribe. Perhaps mathematical sloppiness is a barrier to students still.

Terry, thanks. The McLuhan parallel is interesting - that would make a good post, the one I imagine you're writing right now, if you haven't written it already!

[Marion]

Very interesting post. Although I'm not used to thinking of myself as the "older generation", I've always been pretty good at basic math. As someone else pointed out, I also have a scientific calculator (a solar-powered Casio), that I've used since high school and is capable of doing fractions in the proper way.

I found your comment on Google Maps interesting. I use Google Maps all the time, but I use it mostly to look at the map (or the directions on the map). The written directions are a reference to help me remember the map.

I used to be quite good at math in high school, but around my third year, I kind of lost track of what those formulae were supposed to *do*. Working with physics or chemistry formulae was more interesting, because I knew what I was trying to find out. I would like to take a refresher course, and maybe some advanced math, but I'm not sure where those would be available.

[[email protected] Sadowski]

Really good comments. I'm almost reluctant to add anything. However, I'm a doctoral candidate/adjunct instructor in economics and former high school mathematics teacher, so here are some observations based on my own experience.

1) Most people don't realize just how mathematical economics is. Often students are attracted to the field for very superficial reasons (something to do with money). Just to put this into proper perspective, consider the average GRE scores by field. Those with an intended major of economics average 708. This is higher than every other major with the exception of Mathematics (732), Physics (735), Banking and Finance (708) and four out of six fields of Engineering (608-726). When students moan about me using simple algebraic equations to explain economics concepts I solemnly tell them they're in the wrong field.

2) The arithmetic gap shocks me. I learned arithmetic just before calculators became affordable (and consequently made proper use of them). I find that the easiest thing to do to make students' jaws drop is to perform rapid fire mental arithmetic calculations at the board. They literally have no conception of how a person can do that without a calculator. It really blows their minds. And that's sad. As a result they have no quantitative intuition, which means they have no idea if their arithmetic results (achieved by simply hitting buttons on a keypad) even make sense.

3) As for graphing calculators, you should know that the National Council of Teachers of Mathematics (NCTM) is very influential. Their standards are universally recognized and emphasize high school mathematics instruction with graphing calculators. Thus it is very confusing for students making the transition to college to find that use of graphing calculators is often prohibited (as I do myself). This is not really a criticism of the NCTM's standards. I believe that graphing calculators are wonderful instruments provided you have the mathematical foundations to make good use of them. Unfortunately few of today's students do. Exposing students to graphing calculators at a later date, and requiring better pencil and paper skills at an earlier age, would be an improvement on the current NCTM standards.

[Determinant]

Frances wrote:

Determinant: "Sorry, you don't recruit the math high-fliers." I teach a third year economics course that is required for public policy majors. In any given year, my best student might be majoring in math or journalism or poli sci or business or even, sometimes, econ. No, they're not the math high fliers, and they don't have to be. But they want to learn econ, and I want to teach them.

That's what I expected. I can't see the basis for complaining about math skills when the sample you are using is not the cream of the crop. A typical Economics student doesn't like math as much as they should. This leads right in to Vladimir's post which is where I was coming from.

As a follow on to Randy F, my OAC calculus course and first-year calculus courses burned trig identities onto my brain forever.

As a sidenote, Nick made a post a few months ago where he agonized over evaluating a limit function of the form x/infinity or something like that. He never got to the answer, being scared away by the form. The short answer is you invoke L'Hopital's Rule and take the derivative until you get to final condition for the limit. Nick plain out forgot L'Hopital's Rule. This is first-year calculus.

Why are we going on about fractions again?

Anybody want to start solving differential equations with Laplace transforms? Anybody?

[Frances Woolley]

Sadowski: "you should know that the National Council of Teachers of Mathematics (NCTM) is very influential"

And have also been criticized by mathematics researchers, see, for example, this open letter. See also the math wars link at the end of the post.

I use that mental-arithmetic-to -convince-my-students-I'm-brilliant trick also.

[[email protected] Sadowski]

"I use that mental-arithmetic-to -convince-my-students-I'm-brilliant trick also."

Yes, I admit, it's fun watching their eyes bulge with star filled wonder. (And it guarantees that students give you very high marks for subject knowledge in evaluations, which in my case butters my bread.)

[Andrew F]

What kind of mental math magic are we talking about?

I had a prof in first year who could calculate roots in his head as fast as he could write them on the board. That was the only mental mathemagic that made my jaw drop (but my undergraduate was a math degree).

I had another prof who gave the same lecture three times in a row. Two friends and I were taking that course in a section that was otherwise dedicated to accounting students and I'm pretty sure no one else noticed. They all seemed terrified to participate in the lecture when the prof asked questions. The same prof would also try to do matrix algebra or reduce them to solve a system of equations without preparing them in advance. He would inevitably after 10 minutes and four boards realize he made an arithmetic error (which is really easy with matrices). The standard excuse/joke is that his PhD was in math, not arithmetic.

It's important to be able to do these calculations by hand, but thank god for matlab.

[Frances Woolley]

Andrew F - "What kind of mental math magic are we talking about?"

The type of mental math that would put you in the top 1/4 of a pre-1975 elementary school math class. Nothing fancy.

[[email protected] Sadowski]

Andrew F.
You wrote:
"He would inevitably after 10 minutes and four boards realize he made an arithmetic error (which is really easy with matrices)."

Actually if your arithmetic is up to par, it's difficult to make mistakes in most matrix computations. It's sad that they let people earn PhDs in math these days without first requiring them to master arithmetic.

[Kevinmilligan]

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.

[Andrew F]

Can't say I agree, Sadowski. They were silly little mistakes, but when you're trying to grind through a lot of arithmetic in a reasonable amount of time, it's likely that'll you'll make an error. Especially with a matrix that hasn't been contrived to have a series of pleasant operations required to get it to the desired form.

[btg]

"Here's my theory: Some students struggle with economics because they do not fully understand the mathematical tools economists use."

My theory is that too much of economics is about abstract mathematical models and theories, and not enough about actual evidence of what works or can be observed in the real world.

i graduated from high school in th elate 70s - yet my eyes glaze over whenever i try to read something on economics that is full of equations... and I have an MBa and did a few economics courses.

I also have a theory about dinosaurs...

[Linda]

Determinant wrote: "I can't see the basis for complaining about math skills when the sample you are using is not the cream of the crop."
I am in solid agreement with Frances, and other posters, on this one: my basic issue with students' math skills is the inability to do grade 10 math (or what was grade 10 math in BC when my daughter took it 5-6 years ago). So much of how we teach basic econ is reliant on some ability for abstract analysis - a graph or equations with parameters, not numbers - that communicating with students who find this difficult if very frustrating. (My first year teaching I had an intro class for engineers. I'd been assured they were good at math; turns out they were good at manipulating numbers.) Part of the difficulty, I think, is that we (my generation?) relied on constant repetition for these abilities - no math beyond grade 11? Who had that choice? - whereas current students might have gone 2-3 years without math. BUT: we want, and need, to be able to teach the fundamentals of economics to these students - not the high-flyers.
As a side-note: I recall a conversation years ago with a prof of Spanish literature, who was talking about the difficulty of trying to teach students who had no background in Catholicism.

[Randy E]

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."

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. Ex. Find the square root of 169. Sol. Well, we know 10^2 = 100 and 169>100, so the square root is greater than 10. Since 169 is odd, the square root must also be odd (assuming it's an integer). Therefore, try squaring odd numbers greater than 10. 11^2=121\not=169, so 11 is not the square root. Bummer. 13^2=169, so the square root of 169 is 13. Huzzah!

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.

[Frances Woolley]

Determinant wrote: "I can't see the basis for complaining about math skills when the sample you are using is not the cream of the crop."

Linda, thanks for your comments!

This post was about the kinds of knowledge profs assume students have but that, in fact, students lack. Like a deep, automatic, sense of which way is clockwise.

There are all sorts of things that students can do that profs can't.

For example, video-game generation people can perceive things that I simply *cannot* see. The movie Fight Club has spliced in frames/images that I could not see until the movie was slowed down and played frame-by-frame (caution: do not try this at home; the images are disturbing). The person I was watching with could see the images that flashed up for a fraction of a second perfectly, and couldn't figure out why I couldn't see them.

And who knows what Portal 2 and that Mario game where you navigate around all of those little planets do for people's spatial ability.

[Frances Woolley]

Randy E - we were taught to calculate square roots by: "This is the algorithm. Apply it." Totally pre-constructionist. And since I've completely forgotten the algorithm, I'm not sure what good the exercise did, except for instilling in me a sense that yes, I could calculate these things if I tried hard enough. (Of course for people who failed to apply the algorithm correctly, the message was "no, you can't do this, give up and go home.")

[rabbit]

One of the striking things of modern society is the growth of consumer and government debt in the western nations.

We can all agree that debt is a very useful thing. But we can also all agree that taken to extremes, debt can be horrendously destructive.

From my experience from debates on web sites, it appears that many people don't GET debt. It seem to have no meaning to them. They consider it some abstraction, dreamed up by conservatives to force people to their political will.

Could the lack of appreciation for the destructive side of debt be due to a growing mathematical illiteracy?

[Jacques René Giguère]

I learned l'Hopital's rule in Cegep back in 72.
A natural evolution of working with fractions.
What is interesting about fractions is ,in french "simplification", ( in english I think it is "striking out" or something...). Not only simplify computations but teaches you to avoid situations where you will be mired in singularities and boundary problems. Might even avoid the need for L'Hôpital's rule later on.
Working with fractions is great to learn algorithmics ( the way you structure a problem before beginning a computation). Today,only programmers routinely study that art.
Also great in learning dimensionnal analysis, a skill routine ( at least in my youth) for engineers and sadly lacking to economists ,then and now. When I switched from physics to economics, my new-found colleagues were always surprised to see me check the variables before starting the computing part.

Frances:
"And who knows what Portal 2 and that Mario game where you navigate around all of those little planets do for people's spatial ability."

Just see
http://en.wikipedia.org/wiki/Everything_Bad_Is_Good_for_You

It seems that the increase in IQ of 4 points/decade is mostly due to an increase in the spatial form recognition tests.

[bfuruta]

Frances, thanks for the link to JUMP. I think there is something missing in that program. It teaches children to compute the right answer, but there isn’t enough emphasis on conceptual understanding and critical thinking. Did you see the John Hopkins University evaluation of math programs? They divided programs into three categories: curriculum, CAI, and teaching approach. They concluded teaching approach made a difference--cooperative learning is best.

http://www.bestevidence.org/math/elem/elem_math.htm

Harvard University professor Eric Mazur has a long YouTube video that shows how he brought cooperative learning into a large university classroom.

Eric Mazur: "I thought I was a good teacher until I discovered my students were just memorizing information rather than learning to understand the material. Who was to blame? The students? The material? I will explain how I came to the agonizing conclusion that the culprit was neither of these. It was my teaching that caused students to fail! I will show how I have adjusted my approach to teaching and how it has improved my students' performance significantly."

http://www.youtube.com/watch?v=WwslBPj8GgI

I don’t believe it is, as Mazur implies, that students can explain the concepts to other students better than an expert (the professor). It think it is the debating, the back and forth discussion and deep thinking that goes into that group problem-solving that results in real understanding. Creating good questions, that don’t just involve computing an answer, is part of the challenge for the instructor. The more you understand how students think and what misconceptions they are likely to have, the better you are at writing good questions.

[K]

Bfuruta: I Think JUMP is exactly right in its approach. There is way to much emphasis on understanding in my kids elementary school curriculum. Understanding is not something transferable. Only knowledge is transferable. Understanding is something that happens automatically after endless quantities of repetition which eventually make the individual steps so innate that the problem can be overviewed in its entirety. This was first explained to me by a complex analysis prof: "Copy every theorem and proof over and over until you have it memorized. Then solve 20 related problems. Stop whining about 'not getting it' and get to work!" Understanding is a detail once you can *do* it.

[Frances Woolley]

bfuruta - John Mighton talks a lot about this, and argues that in fact his approach is conceptual and does build understanding.

Think about my student who couldn't work out the answer to 3x5. What Mighton does when he works with kids is say: o.k., can you count by twos on your fingers? By threes on your fingers? Then he says, o.k., 3x5 means count by three five times. So count on your fingers: 3 6 9 12 15.

It seems to me that this is pretty conceptual in that it's really getting at the idea of what multiplication means, and relating it to a fundamental underlying concept, addition. And I also really like the idea of linking mathematical understanding to physical memory and giving students tools - i.e. their fingers - that they always have with them. Indeed, the connection of counting with fingers gives students a really fundamental insight into why we have a base 10 number system.

Although I agree that understanding is important, measuring understanding is hard.

Take a look at, for example, that EQAO fractions question, and the way that understanding is assessed: "Consider the fractions shown below: 3/4, 18/25, 14/20, 75/100. Which fractions represent equal values?"

Students need to "explain". What's an explanation? 3/4=75/100 because 3/4= 3*25/4*25=75/100. What more can one say? (Does one also need to show every 3/4 is not equal to 18/25, 18/25 is not equal to 14/20, 14/20 is not equal to 3/4?)

Mighton argues that one problem with the idea that students should be able to explain things in English (or using little pictures or...) is that mathematics is, itself, a language - and children's understanding of the mathematical language might exceed their ability to translate that language into English.

[Patrick]

Many students today are exposed to programming languages at a young age. This teaches valuable skills that can't be learned through other mediums. The ability to think of an algorithm as an abstract entity. The ability to apply modular organization to data or programs. The ability to see how information can be both data and program at the same time.

These are skills that haven't mattered in a lot of professions in the past. These are skills that professors who were taught mathematics via arithmetic memorization only acquired later in life, if at all. But I suspect that these skills will be integral to economics in the future and will probably be the source from which many new ideas derive.

Over time, we lose some skills and we gain new ones. I don't think we should get too normative about the process without strong evidence and good reason.

[Michael D]

Many people above have noted the need for repetition in order to develop a deep understanding of mathematics. I can tell you that, even at a university level, more focus on repetition would be welcomed.

I'm currently taking (and procrastinating studying for) a second-year Econ-math course. In front of me is my textbook. As an example, I can look at the section on definite integrals - a topic for which the course assumes little to no background knowledge. Taking a glance at the chapter questions, I can tell you that this section has a total of... eight practice questions, three of which are application questions; four of which have final answers in the back of the book, and none of which correspond well to the examples and theorems which precede them. The next section deals with properties and economic applications (again, about six questions of practice) before moving on to interest rates and leaving integrals behind for a few chapters. This, I've found, is pretty common even among university-level textbooks.

There seems to be a temptation to gloss over the nuts-and-bolts of computations and go straight to demonstrating all the really neat things one can do with these newfound tools - problem is, as I've found, you could go through each and every question in the practice sections and barely use the same technique twice. And that's a problem if you've little past experience in the subjects, as you never feel comfortable doing what ought to be familiar operations. I know that's the biggest difficulty in these Econ-math courses for my haphazard study group and I.

[Burt]

Frances, you might be surprised to learn that the underlying concept for multiplication is not addition. Repeated addition is a good strategy for computation with whole number multiplication, just as counting up is a good strategy for whole number addition. But conceptually, multiplication is not a form of addition, just as addition is not a form of counting. Mathematician Keith Devlin started a controversy when he asked teachers to stop telling their students that multiplication is repeated addition. See his latest post on multiplication here:

http://www.maa.org/devlin/devlin_01_11.html

Part of the problem is that math is too often taught as only computation, so being able to compute the right answer is what is considered understanding, and it is understanding of the computational procedure, but not necessarily understanding of the concept. Watch the video of Eric Mazur and you’ll see how his students could solve complex problems by computation but didn’t understand the fundamental concepts.

Getting back to multiplication as scaling, you could show that 3/4 is 75/100 by changing the scale, the unit size, from 1 to 25. You multiplied by 25 as a computation. That is equivalent to, but conceptually different from scaling the unit. Conceptually, think of 3/4 as 3 units out of 4 units. If the unit size is 1, you have 3 out of 4. If the unit size is 25, you have 75 out of 100. Singapore Math does this with block diagrams. The blocks represent units, and the units can be of different sizes—i.e. scaled.

[Frances Woolley]

Michael D: "I can tell you that this section has a total of... eight practice questions, three of which are application questions; four of which have final answers in the back of the book, and none of which correspond well to the examples and theorems which precede them."

This is a really good point. I've noticed a huge improvement in my students' learning (and my teaching evaluations) since I started using powerpoint, which allows me to put up a question after each concept, and get the students to answer them in class, or watch me work through the answers on the blackboard. (If I was the other side of the math generation gap, I'd have reprogrammed the notes to use clickers already, but that's not happening for a while yet). It's good for the students, but it also gives me immediate feedback on what students know and what students don't know.

But, from the other side of the fence, I know that making up questions is really time consuming, and the question that seems brilliant on your own computer screen sometimes ends up being impossibly obscure for students. On-line programs like Aplia are, IMHO, more useful than a traditional textbook, but they're expensive for students. And there's always the temptation for students to attempt to memorize every possible permutation of every single functional form the prof is ever likely to use and then just re-gurgitation answers to the practice questions on the exam.

Burt: "Multiplication is not a form of addition"

What that post illustrates, I think, is the inherent difficulty of using the English language or any other language to explain mathematical concepts. Mathematics is, itself, a language. To me, multiplication just "is", and I would have real problems putting it into words.

What Devlin actually wrote is: "So what is my mental conception of multiplication? It's a holistic amalgam of all the above and several variants I have not listed. That's why I say multiplication is complex and multi-faceted."

In other words, multiplication is addition. It is scaling. It is finding an area. It is distributing things.

But how does a student begin the journey towards that deeper understanding of multiplication? Memorizing time tables? (probably not). Drawing those 10x10 grids that allow you to read the time tables off them? (never liked doing those).

Starting off by teaching students a method by which they can work out what 3x5 is without a calculator seems to me to be a useful first step on the path.

Interestingly, John Mighton's JUMP math approach spends a lot of time on fractions, first adding them, then multiplying them, which actually gets at a math-as-scalar concept relatively quickly.

Patrick: "Over time, we lose some skills and we gain new ones. I don't think we should get too normative about the process without strong evidence and good reason."

Agreed. Someone needs to write a post on 'things students know that profs don't.' Unfortunately I'm not in a position to do so!

[vimothy]

This is certainly a big problem here in the UK. Most technical subjects now have proactive strategies for ensuring that their students are up to scratch. E.g., I know of one institution whose physics undergrad course starts with nearly a full year of maths (what they should have been taught--as far as the programme is concerned, anyway--at A level); all accredited undergrad engineering programmes are required by the Engineering Council to diagnostically test new students for mathematical background and offer remedial classes where necessary, etc.

[Titus Barik]

Memorizing multiplication tables and such is indeed useless if we're actually talking about understanding. Knowing that 3x5 = 15 gives you speed in computation, but it is probably far more useful to understand that this can represent 3 bags of 5 items. Or 5 bags of 3 items. Drawing and counting these out gives us fifteen. We all have an algorithm for applying a rule to perform computations such as 43 x 15, but most (including myself) manage to make it through high school and sometimes college without realizing that the algorithm is just a shortcut application of the distributive property (43 = 40 + 3, 15 = 10 + 5) so that 43 x 15 = (40 + 3)(10 + 5).

Similarly, memorizing 6 / 3 = 2 is quick, but understanding that division can represent both partitioning and measurement (and when one model works better than the other) is probably more useful for understanding more advanced mathematics.

[Shangwen]

As one who has both suffered and excelled in various classes and work environments requiring math, I think part of the issue is that there are always many people in schools and industry--usually technology hyperbolists--promoting the appealing notion that technology is rendering basic math skills superfluous, and that this is good because it will liberate the fun, creative, intuitive side of people and allow those people to get rich in the future. Unfortunately the opposite has been true. As the automation of math has grown, the need to understand what it is we are automating, and to manipulate its findings, have grown increasingly valuable. If you had told someone 20 years ago that the world's largest marketing and advertising company would be named after a number (Google) and be one of the largest private employers of mathematicians, statisticians, and computer engineers, they would have thought you were nuts.

Nonetheless, the belief still persists that computerization will dispense with the need for math skills and make creative skills more valuable, and I think that is mostly because it offers a thin beacon of hope to the mathematically ill-equipped, who are mostly looking at an ever-bleaker private sector job market.

[John Golden]

I think one issue missing from most of the above discussion is that the existence of this technology and its descendants, like Wolfram|Alpha, not only change how we teach, but should change what is taught. Few real life problems can be solved with pencil and paper technology. Real data takes computing power. What understanding is necessary to use it, evaluate it, and innovate with it? It seems unlikely that banning it will lead to that understanding.

My experience has been that requiring computation and over-assessing it leads to decreased conceptual understanding because students focus on what is assessed. If we want problem solving and creativity, back it up with lessons on that, practice of that and assessment of that.

[Frances Woolley]

John: "the existence of this technology and its descendants, like Wolfram|Alpha, not only change how we teach, but should change what is taught"

Absolutely. I'm teaching a fourth year econ seminar course this fall - basically the course involves writing a paper. A huge amount of the course will be about finding data, formulating a research question, sketching a theoretical framework, etc. There is just *so much* data out there that was unavailable five, ten, 20 years ago. Really cool data on sports or multi-player role playing games or whatever. The idea that data+(easy to use packages)+(cheap computing power)=(real econometrics in undergrad courses) is pretty standard, I think.

But what about the way we teach undergrad intro/intermediate/advanced theory? How does that change? E.g. can the Portal 2 generation "see" three-dimensions in a way that I can't? Solve difficult optimization problems with hand-held computing devices?

[Randy E]

Frances Woolley: "E.g. can the Portal 2 generation "see" three-dimensions in a way that I can't? Solve difficult optimization problems with hand-held computing devices?"

One of the bigger topics in first year linear algebra is dimension (well, subspaces, spanning sets, and bases, which eventually leads to dimension). I always had an intuitive idea of what it was and never had difficulty with the mathematical definition. Most of my students are from the Portal 2 generation, and dimension is one of the most difficult concepts for them to grasp. Either it's not helping or Portal 2 and the like has made it the concept so obvious to them that they don't understand why to bother with a formal definition at all.

[Adam P]

"has made it the concept so obvious to them that they don't understand why to bother with a formal definition at all"

how can something be so obvious that it doesn't need a definition?

I once tried to tutor someone in math who wanted to use a calculator to multiply 1/3 times 3. To most of us it's very obvious that 1/3 times 3 equals 1.

I sometimes wonder if everyone understands that the reason for this is that the solution to the equation 3x=1 is the very definition of the symbol "1/3".

[Randy E]

"How can something be so obvious that it doesn't need a definition?"

If I think I already know the definition of a word, I won't go look it up in a dictionary and won't think I need to be told what it means. If the concept seems obvious to them, then they think they already know it, so they don't think they need it to be defined for them. I'm not saying it doesn't need a definition. I'm just saying from their perspective, they probably don't understand why it's necessary to go through such a long process to get to something they already have an intuitive grasp of.

[Whitfit]

Randy E:

I don't think that "not needing a definition" works for any real inquiry into anything. If you really are sure you understand a word, and you don't need to consider a definition, then you might be missing something important about that word, and certainly would get yourself into trouble when doing something important with words, like drafting a contract. Similarly, when considering problems in linear algebra or calculus, you need to be able to formalize your concept of space in order to really understand it and do useful work.

Although, I admit that when my instructor had finished dealing with the three dimensional space, and I was comfortable with saddle points and other such things, then was told to just imagine that concept going in to four dimensions, I got confused. But maybe that is where the formal understanding and definition helps - maybe they get three dimensions really well from playing three dimensional video games, but does that help them understand four dimensions? Do three dimensional video games help more than just being in physical space? What is it that they intuitively grasp from those interactions that I haven't been able to because I stopped playing video games when I got rid of my greyscale Game Boy in 1990?

[K]

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.

[Adam P]

"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.

[Adam P]

"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.

[rabbit]

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.

[janoma]

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.

[ColinTGraham]

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.

[K]

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.  

[rabbit]

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.

[Frances Woolley]

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.

[K]

rabbit: OK. But, by "axiom" I really meant "axiom," not "false proposition."

[hosertohoosier]

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?

[Thomas De Long]

"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!

[John]

good job on the freakeconomics link

[Mike B]

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.

[Min]

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?

[Nick Rowe]

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!

[Min]

Nick Rowe: "One big haystack, dummie!"

;)

[Mandos]

"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...

[andrew]

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!"

[Peter Lowen]

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.

[henrik]

"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

[Frances Woolley]

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.

[Randy E]

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.

[Richard S]

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.

[Randy E]

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.