When I first read Greg Mankiw's "Ten Principles of Economics" I thought they were embarrassing. But I really liked the rest of the book, and decided to become a Canadian co-author (along with Ron Kneebone and Ken McKenzie of The University of Calgary). I thought that when teaching I would just skim over the 10 principles quickly to get past them. But I changed my mind. I now spend more time (minutes per page) teaching that first chapter than anything else in the book.
This post is about why I changed my mind, and how I teach them.
Then I go totally off-topic and talk about high school math and political correctness.
It was the very idea of "10 Principles" that put me off initially. It sounds too much like "10 Commandments". Nothing in economics is written in stone; everything has exceptions, and is up for debate. And why precisely 10, for heaven's sake? Why not 9, or 11? I kept thinking of the holy hand-grenade of Antioch: "Three shall be the number thou shall count, and the number thy counting shall be three…". What is it with these magic numbers, like 10, or 3?
But when I came to teach them, right at the beginning of the course, I found myself spending much more time on them than I had planned. And the students seemed receptive, and to learn a lot from them.
I finally realised why they worked when I spent 3 hours in Jamaica (with Carleton colleague and Cuban economy blogger Arch Ritter). We went to a restaurant for lunch while waiting for our connecting flight to Havana. I looked down the very unfamiliar menu, and chose the "tourist platter". The tourist platter had a little bit of everything. Obviously designed for people like me, who were ignorant of Jamaican cuisine and wanted a quick introduction.
Greg Mankiw's 10 principles is the tourist platter of economics.
You can define economics as the study of how societies allocate scarce resources. Or you can define it as the study of human interaction based on methodological individualism, rational choice, and equilibrium. Or you can define economics as what economists do; and the best way to explain to new students what economists do is to give them some examples. The 10 principles are just examples of how we think, not holy commandments. And 10 is just a nice round number.
So I tell the students that the 10 principles are the tourist platter of economics. And I tell them to approach each one with a critical eye. Can you think of exceptions where this principle might not make sense? (Does jerk chicken taste good?) Quite apart from anything else, thinking of possible exceptions makes you understand the principles better. Think of an example where rational people would not think at the margin, because the relevant choice is an all-or-nothing choice.
Now I'm going to go off-topic.
There are three ways to understand economics: words, diagrams, and math. In first year we use all three, but with less emphasis on the math, and we keep the math simple. If we think of words, diagrams, and math as three languages, students need to understand the three languages, but they also need to translate back and forth between the three languages. What does this graph mean in words? What does this equation look like in a graph?
And the biggest problem students have is not with math; it's with translating between words and math. Here's a (to me) shocking example.
I do a very simple model of a Production Possibility Frontier, and explain it in words, math, and graphs, to give them practice (and to explain trade-offs and opportunity costs). 10 acres of land, which can grow either apples or bananas. One acre of land planted with apple trees grows one ton of apples per year, 2 acres grows 2 tons, etc. Translate into math, where A is tons of apples, and La is acres growing apples. A=La.
Then I say "One acre of land produces two tons of bananas per year (and 2 acres produces 4 tons, etc). Is that: Lb=2B; or B=2Lb?" I write it all down, let them think about it for half a minute, with no discussion, then hold a vote. Democratic math.
For the last three years I have done this experiment, about 75% of the students vote for the first (wrong) option. Maybe 10% vote for the second option. I glare at the minority, and ask them if they are really sure about their answer. Some hands drop, and the remaining students with their hands up look uncomfortable, but determined. Then I tell them they are right, and the majority is wrong, and explain why.
This teaches us two things: first about political correctness — how hard it is to look a fool by dissenting from majority opinion, even when you know you are right; and that something is seriously wrong with the majority of students' understanding of math. They might be able to "do" algebra, and rearrange Lb=2B as B=(1/2)Lb, but they don't have a clue what it means. They can't translate between words and math. Many have difficulty even after I plug in some numbers and show them it's wrong. And algebra doesn't get much simpler than y=2x.
Why? What's going on in high schools?
Worthwhile Canadian Initiative 
I think Michael Weiss has probably got to the root of it. Plus, he backs it up with some research.
agree
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?”
yes.
Adam,
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.
Michael,
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. 🙂
“Exactly…”
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).