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Nick, isn't it going on about 'political correctness' just another form of political correctness?

On your more substantive point: high school math teaching is generally not so bad. I think the real problem is what happens in elementary schools.

Elementary school teachers rarely love math. Students pick up on those emotions, and the teachers' lack of deep understanding translates into mechanical instruction.

The curriculum sets low standards and doesn't spend much time on math.

There isn't nearly enough time spent on basic numeracy - students are questions about probability when they haven't got a really solid grasp of fractions. Somehow one has to get a sense of how numbers work.

Highly recommended: JUMP http://thevarsity.ca/articles/27709 and many-talented founderJohn Mighton. It might sound politically correct, some of it, but a lot of it comes down to starting with very simple problems and gradually building and building - much like you do with your land and bananas example.

Try changing your example from farms producing apples/bananas to factories producing cell phones/ipods. Farming is too abstract. I bet the numbers will flip. :)

"Explain in Words, Explain in Math, Explain in Diagrams"

Good first principle for teaching economics, Nick. (I'm trying to commit that one to memory because I don't think that I do it enough.)

Can anyone suggest 9 more?

I once tried to earn some extra cash by tutoring, in math, someone who was a student at Seneca college. He wanted to use a calculator to evaluate 3 times 1/3. I didn't try again.

You can't grow apples or bananas on the same land. Apples are a cold climate fruit, bananas a hot climate fruit.

That's like comparing apples and oranges ....

Adam P: But if you have any facility at all with mental math, you can stun students (friends, enemies, almost anyone under 30) with your brilliance by doing basic mental arithmetic without a calculator;-) Last term I had a student reach for a calculator to work out 3 times 5.

But seriously: people who grew up in the calculator era don't get fractions. That was the problem your student faced. You saw the problem as 3*1/3, cancel the threes, you obviously get 1. He saw the problem as 3*1/3 or 3*0.333333333333333 - better get a calculator to work that one out.

I think Frances has a key point about looking at elementary schools, but I'd extend what to look for.

First, the elementary curriculum has been compressed so much over the last few years that it is very hard for any teacher to fit in more than one learning style. You get this - explain in words, math, and diagrams. There just isn't time. Some facets - such as classic rote and drill - don't seem to happen at all, possibly because they take more time than is available. It's possible this is happening at the high school level as well -- I'll let you know in a couple of years.

Second - harder to assess, but potentially much more serious - is the question of whether or not some of the math elements have been pushed to too early an age. It's certainly the case there will always be some math that can be taught at for any given age, and that some kids will be ready earlier than others. But for some subjects, like algebra, there is a concern that pushing it down to lower grades, some of it to grade two, may be unproductive for many children.

I think most people would agree that one of the best ways to turn someone off a subject is to have them fail at it. If we attempt to stuff some of the math concepts into brains that aren't developmentally ready, we are almost guaranteeing that a significant fraction of the kids will come away with bad and negative experiences.

Finally - Nick, what are your school's entrance requirements in math for Economics? I recall discussing with one of my Physics profs that they finally had to add high school algegbra as a mandatory requirement, because there was an almost perfect correlation between not having it and failing first year algebra. If students are implicitly told they don't need algebra for economics, don't be surprised if they show up without it.

1: try "The Hurried Child", by David Elkind
2: for a view of the changes in Ontario math curriculum, see http://www.ocup.org/resources/documents/MATH_Grade2_SidebySide.pdf

I don't think this is a calculator problem. This is a fundamental problem with understanding how variables works.
Students look at "One acre of land produces two tons of bananas per year" and do this:
One = 1
acre of land = Lb
produces = =
two = 2
tons of bananas per year = B
TRANSLATING.... ...."1Lb = 2B"
That process doesn't involve a calculator. But it does involve mistaking variables (which students are not familiar with) with units (which is obvious to pretty much anyone that may encounter natural numbers, which is to say pretty much everyone outside of certain isolated tropical tribes).

Heck, to this day I still have to make sure I don't do mistakes like that.

I don't think the error is surprising at all. It took me three read-throughs to get it myself.

The sentence is: 1 acre of land produces 2 bananas. If you just follow that, you get 1acre = 2Bananas. That's straightforward.

It is very subtle to figure out that you have to reverse things to express it as a production functon, F(X) = X, which leads you then to Bananas = 2 * land. I think that this may be natural to those who have spent the last 20 years thinking about production functions; perhaps not to those who haven't.

Try this experiment: Next year, say "The number of bananas you get is 2 times the number of acres"

I suspect your proportions will reverse.

I think the language you are using to frame the question is deceptive and trickier than you think.

20 years ago economists used to ignore the pyschology of framing effects of questions; how language matters. Haven't we just spent a lot of time over the last 20 years showing how/why people's choices can (in some cases) be totally reversed by how they are framed?

We all know that many of these behavioural paradoxes "shouldn't" happen if people behave like little economists. But we ought not to grasp tenaciously to models of how people *should* make decisions rather than how they actually *do* make decisions. Thus, we should be conscious that framing matters and not be so quick to castigate.

On the other hand, you are in fact trying to train 'little economists' aren't you! So teaching them to see past the framing and into the substance is a good goal. I just wouldn't expect them to get that in the first week they are taking economics. Getting them to think like an economist should be your learning goal, not a pre-requisite for the course.

> I think the language you are using to frame the question is deceptive and trickier

I'm pretty sure that's the point. Being able to deal with deceptive language is a useful skill.

Hi Leo. Yes, and that is a good learning outcome. It is what we hope we are able to teach people to do when they study economics. But we shouldn't expect that of them the first day. It takes time to build a homo economicus.

This is precisely the error that so many economists make--they assume that because everyone **should** think like an economist they actually **do**. Hey, maybe that would be nice, but it's not reality.

Try offering the correct answer first in the list of possibilities and see what happens...

Speaking of subtle language:

This is precisely the error that so many economists make--they assume that because everyone **should** think like an economist they actually **do**. Hey, maybe that would be nice, but it's not reality [ how economists think].


Greg Mankiw was George W. Bush's Chairman of Economic Advisers, and is in part responsible for the collapse of the American economy. He should not be listened to.

Frances: "Nick, isn't it going on about 'political correctness' just another form of political correctness?"
It can be. I can think of one example where it was. When the Carleton students got dumped on for switching the Shinerama funds "away from just white males". And I was pleased to see the Carleton Pres telling people to lay off them. It's not the only case where people feel a lot of pressure to conform to the "official majority" view. But in a university setting, it's the most important and salient example for me. Because university is about ideas not shoes. When I look at those students, they look like they feel just like I did in the 1980's or 90's voting against the establishment of Women's Studies (20 years later they are now halfway towards my position, at least nominally, by calling it "Women's and Gender Studies"). If looks could kill, I would be long dead!

Frances and Chris S: I think that's part of it. I had to review the hi skool economics curriculum, years ago. It was far too advanced. Far more advanced than I teach in first year. It made me realise they weren't actually teaching the students to *understand* ideas; they just wanted them to be "exposed to" ideas.

JvfM: I'm just so stuck on apples and bananas! But maybe I'll try it. Trouble is, there's probably big economies of scale in cell phones etc.

Simon: Nick's second principle of teaching principles: stolen from Dierdre McCloskey, and it works even better as a principle of teaching than of economical writing):

Define "comparative advantage" this way: "If the opportunity cost of an apple is 3 bananas in Canada and 4 bananas in Brazil, then Canada has a CA over Peru in apples". Only *after* you have done that can you talk about two goods A and B and two countries C and P, etc.

This works for explaining everything, not just comparative advantage.

Go from the concrete specific to the abstract general, not vice versa. Which, if you look, is exactly what I have just done in explaining my second principle. I start with a specific example, comparative advantage, and then generalise.

Dierdre was right.

Adam and Frances: that reminds me of a graffiti on a Carleton washroom:

Yes, I discourage calculators. They don't learn, and probably make more mistakes in the long run. (But I keep the numbers very simple).

asp: When I teach National Income Accounting, I tell students that economists *can* add apples and oranges (produced) to get GDP. (And then when we do the substitution bias in the CPI, I say well, maybe we can't.)

Kelvin and Kevin: That seems to be exactly how most students do approach the translation, and the framing may be the issue. But most ways of framing the question could cause the same sort of problem. "One ton of Bananas requires half an acre of land" Umm, B=(1/2)Lb? And some of them are still asking me about it. And, most importantly, they have no difficulty working out what the English sentence implies for a table with B in one column and Lb in the other. And that table doesn't push a specific framing. Maybe next year I will start from the table?

But then they won't realise just how tricky it can be to translate from words to math, and as Leo says, they need to learn that.

If I tried Patrick's suggestion, and had them vote on the right answer first, that might affect the majoritarian pressure thing.

el chief: you are obviously a lefty; so i will stop listening to you ;-)

Of course framing matters. I did undergrad Engineering (not at Carleton) and first year Physics is replete with these problems. English isn't as precise as we like to think it is. You always need to go back and check the units on your formula to make sure they are correct.

Another example is the logical function "OR". "OR" includes both as choice, so A OR B will include {A, B, A+B...}. XOR, Exclusive OR is when you choose between A/B but can't have both. In conversation though the word "or" usually means "exclusive or".

I'll hazard that Nick, Frances and the other pros on this board made similar first-year mistakes in their day. You learned, so will your students.

Besides, the real math pro's are in hard Science or Engineering. "Calculus for Soc. Sci" was always seen as softer than Science or Engineering calculus at my university.

I'm a little thrown off by the statement 1 Lb = 2 B (which, if I understand the notation correctly, means something like 1 acre of land that produce bananas = 2 bananas). If I understand that correctly, that doesn't seem right because lands of acres aren't strictly bananas so I'm not sure how they could equal.

I would have written it this way

Y B = (Rate of production) * X Lb


Rate of banana production = 2 B / 1 Lb

Err, I guess what I'm saying is that the "2" in your formula should probably have units (B/Lb) as that would make it more consistent and intuitive.

That way if you wanted to know how many acres you'd need given how many bananas you want to produce (for example, 2 bananas),
2 B = (Rate of banana production) * X Lb
2 B = 2 B / 1 Lb * X Lb
X Lb = 1 Lb

I hate to dwell on petty things but.. I've always been taught that units matter! And they do help in understanding equations.

That aside, I've done tutorials and help desks for a 1st year general physics course and I've found that just a lot of students don't value math as a language and don't realize they have to practice it often. No doubt some of them haven't taken it since grade 10 and so they forget a lot. I had one student who didn't realize that m^3 = m*m*m, which I think says a little bit about how alien sometimes some of the math that they see on the board must seem. All year round the general physics course stressed problems like "if x = 2y and you double x, what is y?" and still all year round there were students who had problems with it.

That and units. We stress units.

Nick, not sure if this is what you meant or not but it is true that 1 = .99999...
(the nines continue forever).

And here's a proof from Wikipedia! (Several, actually)

I once walked up to a store checkout to pay for something that cost 97 cents. I handed the cashier-in-training $1.22 whereupon he looked stupefied. His mentoring supervisor took over and rung up the tendered amount on the cash register which calculated for him the amount to return to me. The trainee, bewildered, asked the supervisor, "How does he do that?" "Oh, some people," came the reply, "can do that in their head."

The calculator is a great invention, but sometimes I wonder if it hasn't diminished basic numeracy.

Gregory: "The calculator is a great invention, but sometimes I wonder if it hasn't diminished basic numeracy."

And the question is: does basic numeracy have any bearing on the ability to do higher order math problems, e.g. describe the relationship between the number of bananas, the banana yield per acre, and the number of ares.

Those proofs are neat! I expect it means that 0.9999999 recurring means something like "in the limit, as we keep on adding 9's". Right?

Just passing: yes, i think the "2" should strictly have the units "tons per year per acre" (tons/(yearxacre)), because B is in tons per year, and Lb is in acres.

Nick: I love the teaching style. But are you not being a little hard on the students? The ability of trained economists to use math to clarify models and theory building is unique among the social sciences and ranks right up there with other technically sophisticated disciplines. Look at the engineers and other math-sophisticated professionals who cannot follow a simple market model if their life depended on it. They keep looking for social accounting matrix type relationships or simple reduced-form, deterministic models. These people are math-sophisticates.

This is our discipline's comparative advantage. (It is not fine tuning discretionary fiscal policy......) Of course it is hard. Look at the naive empiricism and junk thinking that comes out of sociology, geography and social-psychology! On a distinctly related theme: How many hard scientists with graduate degrees do you believe have a good grasp of basic probability and statistical theory? Less than 2% maybe? 10% at the outside? At least undergraduate economics students are trying whereas most in the social sciences gave up a long time ago.

In my experience, if college and undergraduate university students are not actively studying or using math, even simple things like calculating and understanding percentages is not always straightforward. At that age, highly specialized human capital appears to decline quickly.

Determinant speaks to my own experience when he observes that young undergraduate physics students fail to grasp the importance of mathematics as a language (as well as make all those silly mistakes). I used to polish off math exercises in high school during class while both quickly reading the textbook and listening to the instructor. I took the soft business and social science math course in 1st year and easily coasted through it. If I knew then what I know know, I would have front-loaded 1st year with at least 2 serious, demanding math courses. I really had no idea just how useful math would be for both measurement and modelling. Even non-mathematical writing seems to benefit from formal math training and exercises.

Nick: You obviously think George Bush's economic adviser is worth listening to, so you are not.

I thought I'd find some good economics on here, but then I saw you taught at Carleton, and now I know.

I would like to leave my impression of my first couple of classes with Nick Rowe.
My class was the first being subject to the "apples, bananas and acres" problem.
I was part of the 75% (I think it was more than that even) of the class who had it wrong. And that was good! I left the class wondering about what exactly had happened there! Those are the classes I love most... wondering brings (me) motivation to study and be ready for the next trap, which we never will be if we're lucky enough to have good "trappers" through out our studies...

You should consider allowing 1 minute group of 2:3 discussion, before polls, students bring to light their reasons with each other. Have a better sense of why they think something, and smaller sample to be judged by. They are then primed for the answer. My chemistry and history classes both used it, not sure the proper term for it.

I think the main problem with math is simply a problem of self-selection. Most people when picking majors, do not know that economics requires a bit of mathematical analysis. Certainly when I picked it (finance actually, but close enough) I had no idea that graduate training in economics requires real analysis. I picked economics instead of the physical sciences is because I was not good at math in high school (although it turned out that a) economics requires more math than some sciences and b) I got better at math).

However, I do think that most people pick economics because they think it's an easy route to investment banking and other lucrative business careers (but, there's nothing wrong about it). Since most US colleges (esp. LACs) still lack undergraduate business programs, students who are not interested in theory (or even applied economics) still end up taking economics to prepare for their CPA or CFA. From my experience, students who are good at math (and have done AP Calculus) generally go into the sciences. Therefore, I don't think economics students (in general) are a good representation of the average math ability for college-bound HS students.

Sorry, I just realized that you are a Canadian professor. UG business programs are more prevalent in Canada than the US.

"Those proofs are neat! I expect it means that 0.9999999 recurring means something like "in the limit, as we keep on adding 9's". Right?"

yes, remember a decimal expansion is just shorthand for a series (infinite sum). .99 means (9/10) + (9/10^2) + (0/10^3) + ...

.999... means just keep going with 9 instead of zero so it as series of the form [sum (n=1 to infinity) of (9/10^n)]. The definition of what this means is: take the limit as N goes to infinity of S(N) = [sum (n=1 to N) of (9/10^n)].

In the case of .9999... we can see imediately that the partial sums S(N) are monotonically increasing. It should aslo be clear that for N, S(N) < 1. These two facts taken together inform us of the happy fact that the limit in question exists, that is the series has a value.

Once we know the series can be summed to a real number we are free to make statements like: let x = .9999... and we know that x is a well defined number that follows the usual rules. Hence we can proceed as follows:

10x = 9.9999999...
so 10x - x = 9x = 9.
so x = 1.

(You could also proceed directly by choosing any x < 1 and proving that for all large enough N, S(N) > x which combined with S(N)<1 for all N would imply S(N) = 1.)

Adam P .. there is also a wiki entry on this


yes, was just looking at that (Stephen Gordon linked above). They give the "proof" that I just gave but leave out the part where you show that .999... is actually a meaningful quantity.

This is important, you can't say let x = .999... and then multiply x by 10 following the usual rules until you've shown that the series converges to a real number, the rules of multiplication only apply to number after all.

This is part of what drove me so crazy about the guy who wanted to use a calculator to evaluate 3 times (1/3). I think everyone should no that this is the very *definition* of the number (1/3). The number (1/3) is defined as the solution to 3x = 1.

I think everyone should know...

I got the link to this very interesting article via Greg Mankiw's very own blog. The "simple" math problem left me wondering for a bit myself until I finally recalled one formula from my business maths course. I guess permitting the calculator in schools is both a blessing and a curse. Since I was allowed to use it in 8th grade Bavarian grammar school my "manual calculation skills" rapidly deteriorated. The lack of numeracy skills is also haunting Germany's education system.

While the problem itself is very easy, I think one cannot expect a majority of modern students to be excellent "stand-up" mathematicians, who can recall every formula in class at once. I'm sure they'll improve in the course of your class.

And some are just glad to pass the more mathematical-focused tests to focus on their actual talents.

el chief: "Nick: You obviously think George Bush's economic adviser is worth listening to, so you are not.
I thought I'd find some good economics on here, but then I saw you taught at Carleton, and now I know."

Apart from the obvious political and ignorant anti-Carleton bias, there's something more importantly wrong about this comment.

I am capable of judging the quality of Greg Mankiw's writing in economics from reading it. I don't need to know who he worked for. I don't even need to know his name. I know his stuff on height tax is very good, for example. It's good because it forces us to think deeply about some questions we don't normally think about. And to take one example going the other way, I know that one of his very early macro papers, though it's good overall, has a serious flaw, because he says it's a Keynesian model but it isn't really Keynesian at all (even though everyone else thinks it was). Because it contradicts Keynes' classical postulate number 1, which keynes agreed with, and conforms to keynes' classical postulate number 2, which keynes disagreed with.

One fact I should have made clear: it's an intro economics course, but few of the students are economics majors. It's a required course for lots of business and policy majors. And most students are taking it as an option. So they are from all over the university.

Maybe I'm being too tough, or expecting too much. But, just to repeat, it's not their math that worries me. It's translating between math and words. It's setting up the problem not solving it, that's the key issue. At a pinch, you can always find a mathematician who can solve a problem, if you can only: set it up in math for him (translate from words to math); understand the answer (translate back from math to words). It's the translation that's the problem. That's what I think we don't teach enough of.

When I was a kid in hi skool, we did a lot of applied math problems where we were given the question in words. "A train 100 metres long passes through a station 400 metres long at 100km/hour. How many seconds between the front of the train being at one end of the station until the back of the train leaves the other end?" Setting up the problem was hard. Solving the math was the easy bit.

Do they still do those sort of problems?

"Elementary school teachers rarely love math. Students pick up on those emotions, and the teachers' lack of deep understanding translates into mechanical instruction."

True, though I think it's important to understand why. A big reason - increased employment opportunities for women.

My Mom likes to joke that when she was growing up, there were only two job opportunities for her - nurse or teacher. An exaggeration, perhaps, but not much of one. She jokes that she chose 'teacher' because she does not like the sight of blood.

Anyhow, Mom is *brilliant* at math. Won a gold medal as the top graduating math student in her class at UWO. Ended up spending her career teaching at a rural elementary school.

I don't think this is an isolated case - I can think of at least two terrific female math teachers I had in elementary school.

This simply doesn't happen any more. If Mom was born in the 1970s instead of the 1940s, should probably would have ended up teaching at a University or working for Statscan or for industry. But those weren't reasonable options for a farm girl in the mid-1960s.

Our elementary schools are worse off because of it. But society as a whole certainly isn't.

Mike: I've blogged on this point: Has female empowerment caused a decline in teacher quality?.

Nick: have you tried drawing pictures of bananas and islands and asking people to derive the relationship based on those pictures given them a minimum of verbal clues?

Frances: no, I haven't. I'm a bit scared to. They might really think I've lost it! But my guess is that yes, if I drew a graph of B against Lb, with numbers, more of them would get the right equation. Though maybe more abstentions in the vote?

Not being an economist but thinking that professional students who failed to take economics as undergraduates need some, I've searched the web for similar sets of economic "thumb rules" that I could provide them to stimulate their interest in pursuing the topic outside their programs.

Such sets of principles are scarce on-line; I've only found three - Mankiw's Ten are here http://www.swlearning.com/economics/mankiw/principles2e/principles.html

"Common Sense Economics: What everyone should know about wealth and prosperity" has 41 principles in 4 groups in the PowerPoints at http://commonsenseeconomics.com/instructor-resources/

"Ten Key Ideas: Fundamental economic concepts that open the door to the economic way of thinking" by Russell Roberts are here - http://www.econlib.org/library/Topics/Guides/TenKeyIdeas.html


Alas, I am too late to add anything intelligent (or perhaps even intelligible) to the many comments you've already received. I will share my own experience, teaching another section of 1000, that I spend a great deal of time on the ten principles, for similar reasons. In fact, I spent the whole first class on them, when I had planned to skim them, because there was a lot of feedback from the students.

Vivek: same story as me. When you say the whole first class, do you mean a whole 3 hour class? That's still less than me. But then i do tend to wander off on tangents.

Anon: yes, it seemed to spark a lot of other lists. Maybe copyright stuff prevented more?


Yep, most of a three hour class, minus a general intro to the course and a break. I also go off on lots of tangents, including a longish one on when marginal thinking doesn't make sense for the fair number of all-or-nothing decisions we face and another longish one on how scarcity only make sense relative to human wants, with a further digression on the blue people....

I guess my first post was wrong since I assumed that the class was filled with mostly economics students. I actually don't think that there's anything wrong with high school math. When I was in high school (not that long along) we had to transform a word problem into simple differential equations for AP Calculus. For example, we did things like: you are given the rate of change for volume, you are given the relationship between volume and length of a side, what's the rate of change for the length? But I think AP Physics was the best course to learn how apply formulas to word problems.

However, I think the "regular" (ie. non-AP/IB/honours, no offense) courses involve a lot more memorization rather than analytical thinking. I used to think that everyone who'd gone to university at least took some AP/IB or honour classes. But, I guess there may be a lot of heterogeneity among Canadian high schools (despite being free).

The high school I was at (in BC) had around 14 AP classes: the 3 sciences, calculus, statistics, 2 economics, 2 history, human geography, psychology, 2-3 foreign languages, English, and fine art. However, some schools don't have any AP courses (esp. rural schools).

Maybe it's not obvious to many of your students what the equations are meant to be expressing. They naturally see a ratio, Lb:2B. But you're instead (hence the equal sign!) wanting them to express the tons of bananas per year *in terms of* (not *per*) the acres of land. This requires doing a conversion (xX:yY <--> yX = xY) to get to B=2Lb.

I wonder if maybe it's because the math (in Mankiw, most other introductory texts, and therefore more intro courses) is too simple. Economics concepts are complex, and when you try to explain them using watered down math, it becomes sort of like voodoo (not in the sense of voodoo economics, just incomprehensible black-box formulas).

Think of elasticity: in Mankiw and elsewhere, the concept is explained in words, and then implemented using algebra: ∆Q/∆P * P/Q. But, this only gives you a point elasticity if the curve is linear (which in real-world models it often isn't, but that's another problem), and can be very hard for the student to convert the math symbols, which they probably understand very well, back into the economics words. On the other hand, you can just introduce it as a derivative of Q with respect to P, multiplied by a factor to eliminate units, and you get dQ/dP * P/Q, which at least (for me) was much easier to understand than the algebra version. In fact, you can make calculation elasticity very easy for the student if you explain that Q=f(P), so e=df/dP * P/f(P), and the student can much more easily convert the math and the words at that stage (at least, when studying economics, I could).

Mathematics is the language of economics. Trying to teach economics concepts without economics language is like trying to explain Islam without Arabic.

I was never very good at algebra, but could show almost anything with graphs. When I worked on homework problems with other students in grad school, I'd often show a graphical proof. They'd then ask me to prove it mathematically. I'd have to patiently explain that geometry is mathematics.

This is in response to your "words, graphs, and math" comment.

PS, Thanks for reminding me how to get Mankiw to link to my blog.

Probably not the forum to admonish the success of high school math, however I would like to express my opinion on student's math skills in economics and basic business math... we have found at our institution that student's math skills have been declining significantly over the past few years. Like others, I don't know definitively why, but I do have my ideas. First, I think that teaching math (or any subject) in a "silo" where students can learn a concept, be tested on it and then be allowed to forget about it because it won't be tested or used again produces only short term retention. This is why we see students not being able to do simple Grade 9 math tasks -- they haven't used it in four years! Second, I don't think students are being taught how to think critically or to problem solve to the extent that they need to in a post-secondary education. Their learning mode more closely resembles a memory/regurgitation process, rather than learning how to apply their knowledge to problem solving.

One aspect of the problem is that students interpret the symbol B as meaning "bananas", or "tons of bananas", rather than "the number of tons of bananas". So then "2B" means "two tons of bananas", and equates that to "1 Lb" which means "one acre of banana-producing land".

This phenomenon is well-known in the math ed research community. John Clement documented it and provided a comprehensive analysis back in 1982 in the Journal for Research in Mathematics Education (Vol. 13, No. 1, pp. 16-30). You can find a direct link to the PDF at http://www-unix.oit.umass.edu/~clement/pdf/algebra.pdf. (It's the first hit in Google Scholar if you search for "education variables word problems".)

Clement studies students' performance on the word problem "Write an equation using the variables S and P to represent the following statement: 'There are six times as many students as professors at this university.' Use S for the number of students and P for the number of professors." About 1/3 of students consistently produced the equation 6S = P, rather than the correct S = 6P.

A key excerpt from the middle of the Clement article: "It is difficult for a mathematically literate person to imagine how an equation could be viewed otherwise. However, from a naive point of view, the correct equation is strange in the sense that a comparison of two unequal groups must be 'force-fit' somehow into the notation of an equation showing two equal groups. This analysis exposes some tacit assumptions and meanings underlying our conventions for algebraic notation: [the] correct equation... does not describe the situation at hand in a literal or direct manner; it describes an equivalence relation that would occur if one were to perform a particular hypothetical operation, namely, making the group of professors six times larger than it really is."

I think Michael Weiss has probably got to the root of it. Plus, he backs it up with some research.


Forgot to add: the subjects in Clement's study were 1st year Engineering students.

This is a nice post, but I have to agree with the other commenters here: Your students aren't confused because they don't understand algebra, they're confused because you're not explaining the problem very well. That's not meant to be an insult, just a (blunt, I admit) critique. I studied economics and statistics in college and now work at a health care consultancy, which means I spend all day trying to explain econometric methods to MDs. The first thing I learned in this job is that you must, must, must explicitly (and almost pedantically) define all the terms you're using, even those that to you may seem obvious. Your problem here, as another commenter pointed out, is that you don't explicitly define A, La, B, and Lb as place-holder variables. People who don't have daily contact with mathematics will naturally interpret those terms as units, not variables. That's why they think Lb = 2B is the right answer; they see it and read "One acre of land producing bananas (1 Lb) produces (=) 2 tons of bananas (2B). Bottom line: The kids are all right.

Jay: that's the conclusion I've been coming to too. What they (or most of them) mean by B, and what I (and economists) mean by B, are very different.

Policy implication, though, is less obvious. Should I explain it pedantically first; or explain it pedantically after I do my little experiment? The advantage of the second is that is does shock them into realising that it doesn't mean what they might think it means.

Do you explain Bayes theorem to the MDs?

Setting them up to get it wrong is always a pedagogically powerful, but risky, maneuver. You can generate a lot of surprise and interest but also a lot of anxiety and confusion.

Here's one way of handling it.

Post the problem the way you are doing it now -- have them vote, but don't tell them (yet) whether they are right or wrong.
Then sharpen the question by providing some numbers: "So, if we plant 10 acres of land with bananas, Lb would be 10." (Write on board: Lb = 10). "And we would get 20 tons of bananas." (Write: B = 20).

Then ask students: Does the equation they chose produce a true equality when they plug in those values for Lb and B?
If now, what went wrong?
And then this leads in to a discussion of the meaning of the variables -- they stand for the number of acres and the number of tons of bananas, so that "2 Lb" doesn't mean "2 acres of land" but "2 times the number of acres of land", etc.

A little late to the game, but...
In reference to 1 = .999..., one thing that I love about math is that it is never as simple as explained but never more complicated than is explainable.

1 is equal to .999... only because of the way we frame our number system. If we use an altered number system that includes infinitesimals we get a more intuitive result. Thinking in terms infinitesimals made calculus concepts much easier to grasp for me.

Thank you Michael Weiss for confirming the labeling trap that I fell into when first read Nick Rowe's post. I hope many teachers follow your prescription.

I've taken to using units in calculation exercises with MBA students. One useful tip. Do NOT refer to "tons per acre per year" when you mean "tons per acre-year". The English hyphen substitutes for the math multiplication symbol, as in man-hour. Much incorrect inverting and multiplying can be avoided with proper lingo.

The problem with "tons per acre per year" is not, IMO, the use of the second "per", but rather the ambiguity caused by the absence of explicit grouping symbols. That is to say: If you parse it as "(tons per acre) per year", you have it exactly right, and this is in fact identical to "tons per (acre-year)". On the other hand if you hear it as "tons per (acre per year)" then you are talking about something else entirely, which would be equivalent to "ton-years per acre" -- and I'm not sure if such a quantity has any real significance.

In some cases the ".... per ... per ..." formulation is more intuitive than the (more concise) "... per ....-...." formulation. In Physics, for example, students are often baffled by the customary MKS units for acceleration, "meters per second^2". But "(meters per second) per second" makes some intuitive sense: If your velocity increases by 10 meters per second, and it takes 2 seconds to do it, then it goes up an average of (5 meters per second) per second.

A few more thoughts, by the way, on the bananas/land problem:
(1) I am pretty sure that even if you do interpret B and Lb as units, you are still on shaky mathematical ground. No amount of bananas is equal to an amount of land. To translate the word "produces" with an equals sign is a pretty big leap.
(2) Students learn to do this from years of exposure to problems with a set-up like: "David is twice as old as Joanne." They learn to represent this with the equation "D = 2J". If you ask them what D and J stand for they are apt to say "D stands for David, J stands for Joanne". But this is nonsense: David is not two Joannes. D does not stand for David, it stands for David's age. We customarily let them get away with saying "D is for David" because, well, we all speak somewhat carelessly in the name of brevity, and after all it leads to the right equation, so why not? But of course then you get to the bananas and land problem, or the students and professors problem, and those habits turn to bite you.

"1 is equal to .999... only because of the way we frame our number system"

This is just false. As I explained above, the very definition of the symbol .999... is as representing a certan infinite series that can be shown to converge and explicitly summed. The sum is 1.

Michael Weiss: "Forgot to add: the subjects in Clement's study were 1st year Engineering students."

That makes sense. That explains why only one third got the wrong answer.

On the 1=0.99999... thing:

When I saw that graffiti on the washroom wall, I couldn't figure out if anything was wrong with the proof, but it just looked so .... weird. But thinking about it as the limit makes so much more intuitive sense to me, that it no longer looks weird. It now seems natural.

Maths, and intuitive understanding of maths. Are proofs really proofs if they don't really convince us fully? Another topic.

"Are proofs really proofs if they don't really convince us fully?"



You wrote that Eric was "just false", and I repectfully disagree. "The very definition of the symbol .999...." is a matter of convention. Of course the normal definition of .999.... is that it is the limit of a sequence of rational numbers (the 'partial decimals', if you will), and under that definition it is easy to prove that .999.... = 1. However, Eric's point appears to be that other definitions, other conventions, are possible. Indeed this point was recently made by mathematicians Karin and Mikhail Katz in a recent issue of the Montana Mathematical Enthusiast (Vol. 7, No. 1, pp. 3–30, 2010, available at http://www.math.umt.edu/tmme/vol7no1/.

Of course from one point of view Katz & Katz and just being silly: The definition of an infinite decimal expansion is about as universally agreed to as anything you can name, and it seems perverse to say "Oh, but you can define it otherwise!" But I don't think they are saying just that. Rather, I think they are saying:

(1) The formal definition of an infinite decimal is not normally presented in K-12 mathematics classes, so any "proof" presented in that context .999... = 1 is fundamentally flawed (How can you prove that two things are equal when one of them hasn't even been defined yet?)
(2) There is at least one mathematical system (nonstandard mathematics) in which the existence of infinitesimals can be rigorously justified, and one can prove the existence of numbers strictly less than 1 with decimal expansions that begin with more than any finite number of 9s.
(3) Students' intuition about the meaning of infinite decimal representations is to regard such a representation as meaning not the limit of a sequence but rather the sequence itself. This intuition can be made rigorous, and within tnat framework .999... is most definitely not the same as 1.

Nick: So you never encountered the .999.... thing before? That's so surprising to me -- I think of that as something that people see in elementary school.


We're sort of arguing semantics now but my point was exactly that, within the real number system, a definition is not a matter of convention. Now, the fact that the same symbol is used to represent a different object that lives in an entirely different number system doesn't change that.

By the same token, this is not a matter of the "framing" of the number system. It is a matter of which number system the number that symbol represents lives in.

I did not dispute that their are other number systems, I'm aware of many.

As for your numbered points:

(1) This is a point I was making above, you first have to define the symbol .99... as the series, then show it converges to a real number and only then do you find that the number equals 1.

(2) The fact that in another number system their exists numbers strictly less than 1 with decimal expansions that begin with more than any finite number of 9s does not change the fact that none of those number live in the Real number system.

(3) The intuitive understanding of students, though important pedagogically, does not change the definitions. A sequence is a well defined object that does not live in the Real numbers, you map the sequence to the real number in some well defined way. The way that the sequence of decimal expansions is mapped to a real number is by identifying the members of the sequence with the partial sums of a series. The *sequence* is not the same as 1 but Real number that the sequence is mapped to is 1.

Agreed that this is mostly a matter of semantics. But I would still maintain that, if no definition has been specified (which is always the case in the K-12 context) then there isn't one, and therefore any argument about what is "really" true is silly. Nearly every "proof" that you can show a high school student (or write on a wall) glosses over that entirely, and hence (IMO) proves nothing really.

with this I agree. What have I said that you though contradict the statement that " no definition has been specified (which is always the case in the K-12 context) then there isn't one"?

How does this bear on Eric's statement though?

Michael: No, I hadn't encountered 0.999..=1 before. What's worse, I have 'A'-level maths in the UK (OK, I only got a grade of D, but UK A-levels were serious stuff, because most people going on to university only took 3 A-levels in those days). Worse, I went to a very good secondary school (same as Stephen Hawking, and we probably had the same math and physics teachers too). Maybe I slept through that class. I wasn't the most serious student, in those days.

Eric said "1 is equal to .999... only because of the way we frame our number system. If we use an altered number system..."

I take him to mean: If you assume that we are talking about real numbers, then yes, 1 = .999....
But if you are talking about another number system, maybe not.

My point is that in school we never say explicitly what kind of numbers we are talking about (because defining what a "real number" is goes way beyond what is reasonable in K-12). We just do lots of hand-waving and appealing to intuition.

Therefore there is no way of rigorously justifying whether 1 = .999... or not, especially when the very intuition that we routinely appeal to is quite capable of supplying a different interpretation of the claim.

Ok Michael, I'd say we agree. I had implicitly assumed that unless otherwise stated the term "number" meant "real number".

Really, at the heart of it I was objecting to the use of the word "framing". Within the real number system one can prove from the defining properties of real numbers that .9999... = 1 (as well as supply a sensible definition of the symbol ".999..." in the first place).

A different number system is not just a different way of framing the real number system. It's a set of entirely different objects.

Exactly. An "altered number system", you might say. :)


No, not exactly. The term "framing", at least as used in economics, refers to a situation where *what* is being described is unchanging, only *how* it's described changes.

But that is not the case here, when switching to a different number system, for example one in which:

"the existence of infinitesimals can be rigorously justified, and one can prove the existence of numbers strictly less than 1 with decimal expansions that begin with more than any finite number of 9s."

the object being described has changed, not just the description.

This not just about how we "frame" numbers.

If it makes sense to say that "0.33333. . ." is the same thing as 1/3, then it makes equal sense to say that "0.99999 . . ." is the same thing as 1. How could they be different?

Not to beat a dead horse, but I really think you're fixating on one word and ignoring the rest of Eric's paragraph.

"1 is equal to .999... only because of the way we frame our number system. If we use an altered number system that includes infinitesimals we get a more intuitive result."

I think the second sentence makes it clear that Eric knew he was describing a different number system, that more than the description has changed. He, and I, and you are all in agreement: starting with different postulates leads to a different number system, one in which .999... is not 1. He uses (or seems to be) the word "framing" to refer to that different set of postulates, which seems reasonable to me. To you, the word "framing" has a pre-existing meaning that is different from the use to which he puts it. That's fine.

"...as used in economics..." I think this is the problem right here. In economics you are trying to model an external reality, and "framing" refers to the way the external reality is described mathematically. But we are not discussing economics here, but pure mathematics. There is no external reality under debate. The number system is not a "real" thing to be "framed"; the model is the thing. Change the model, you change the thing.

... and now we have left mathematics entirely and are into ontology. Probably best to leave it there.

I'll beat the dead horse yet again:

"Translate into math, where A is tons of apples, and La is acres growing apples. A=La."

But translating into math, of course that's not true. The units don't match.

Do agree on the math literacy problem. My math teacher spouse says it's because they are letting students use calculators way too early.

Fascinating thread. Thanks to everyone for all the references. Here is my small contribution:

1. About mathematical illiteracy and its (sometimes hilarious) consequences get yourself Innumeracy: Mathematical Illiteracy and Its Consequences by John A. Paulos.

2. Calculators are like cell phones: they can be very useful but used in excess can be impoverishing. A convincing dissertation against calculators in the classroom can be found in the fascinating booklet Mathematics for the Curious by Peter M. Higgins.

3. Last but not least, and just to agree with Rowe and Weiss above... Teaching statistics or econometrics I make my students solve exercises whose statement avoids algebra completely: exercises in which "solving the math [is] the easy bit".

Why are these exercises so difficult?

Because the difficult part here is the mathematical modelling of the real world, not the mathematics. That explains the (apparent) paradox of mathematicians going through so much trouble in graduate school in economics. It also explains why basic physics is so easy compared to basic economics or statistics; in physics you model conceptually very simple systems compared to stats or econ. It further gives you a sense of the grandeur of mathematicians like Euler who dedicated their lives to create the mathematics needed to solve real-world problems; or economists like Samuelson who translated so much bla-bla into mathematical expressions (and rigor).

As a teacher of economics and statistics I have always tried to keep this in mind: it is going back and forth between formulas and real world concepts that is so extremely difficult. I would even say that this skill is what explains the success of economics compared to other social sciences; in a sense this is Lazear's point in his paper "Economic Imperialism" (Quarterly Journal of Economics, 115(1), 99-146).

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