This is a guest post by Ernie Davis Professor of Computer Science at NYU. Ernie has a BS in Math from MIT (1977) and a PhD in Computer Science from Yale (1984). He does research in artificial intelligence, knowledge representation, and automated commonsense reasoning. He and his father, Philip Davis, are editors of Mathematics, Substance and Surmise: Views on the Ontology and Meaning of Mathematics, published by Springer.
Any government which genuinely cared about education would see to it that a globe map, at present an expensive rarity, was accessible to every school child.
— George Orwell, “As I Please” February 11, 1944
The decision by the Boston school system to replace maps of the world using the Mercator projection with maps using the Gall-Peters projection has garnered a lot of favorable press from outlets such as NPR, The Guardian, Newsweek, and many others.
The pros and cons of these two maps have been debated extensively for many years; there was even an episode of West Wing that dealt with the subject. I would give the current collection of news articles a B for clarity and accuracy. If you read the Guardian article carefully from beginning to end, you can get a clear idea of the issues. But if you only skim the beginning, then phrases like “more accurate map”, and “amend[ing] 500 of distortion” are likely to leave you with the impressions, first, that the Gall-Peters map is indisputably more accurate, and, second, that the Mercator map was devised as an expression of Eurocentrism, neither of which is correct.
I hope that that is not what the students in Boston are being taught about the two maps. Still more, I hope that they are not being taught that these two maps are two competing theories about the geography of the world and that choosing between them is all a matter of your point of view and your political preferences, and that there is no actual truth of the matter. I wish I felt more confident of that.
The well-known truth is this: The Gall-Peters map accurately displays relative area, whereas the Mercator map grossly distorts relative area. However, the Mercator map accurately displays shapes and direction, whereas the Gall-Peters maps substantially distorts those. Each has their strengths and weaknesses.
As an illustration, consider Suriname and Iceland. Both are roughly squarish countries. Suriname’s area is 161,470 square km; it has an east-west span of about 460 km and an almost equal north-south span. Iceland is smaller; it has an area of 102,775 square km; its east-west span is about 390 km and its north-south span is about 300 km (these spans were hand-measured from a map and are not precise).
If you take the Gall-Peters map of the world and cut out the maps of Suriname and Iceland, then the relative areas will be correct; Suriname will be about 1.6 times as large as Iceland. However, both will have bad distortion in the aspect ratio in opposite direction; the north-south span Suriname will appear substantially larger than its east-west span; and the north-south span of Iceland will appear very much smaller than its east-west span. Both of these are terrible maps of their individual countries. If you are used to looking at maps of Iceland and then look at Iceland on the Gall-Peters map, it will look seriously wrong, for good reason.
On the other hand, if you take a Mercator map of the world and cut out the maps of Suriname and Iceland, then each one by itself is exactly the right shape; each is a fine map of its individual country. However, Iceland will be 3 times the area of Suriname instead of 2/3 the area.
The explanation of both of these distortions is simple. At the equator, the 460 km east-west span of Suriname corresponds to 4 degrees of longitude. At 64 degrees latitude, the 390 km east-west span of Iceland corresponds to 9 degrees of longitude. However, on both the Gall-Peters map and the Mercator map draw parallels of longitude as vertical lines, so Iceland ends up measuring 2-1/2 times as wide as Suriname on both maps; and, generally, an east-west mile in Iceland is displayed about 2-1/2 times as long as one in Suriname. The two maps adjust the north-south scale in opposite ways, depending on their different purposes. The Gall-Peters map, to preserve the area relation, must make a north-south mile in Suriname 2-1/2 times as long as one in Iceland; it does this, partly by stretching Suriname north-south and partly by shrinking Iceland north-south. The Mercator map, to preserve shape, must keep the ratio of the north-south mile to east-west mile always equal to 1; therefore, a north-south mile in Iceland is also 2-1/2 times as long as a north-south mile in Suriname.
There are other kinds of area-preserving maps besides Gall-Peters, and there are other kinds of shape-preserving maps besides Mercator. And there are many other kinds of maps; this article by Max Galka surveys the Miller, Winkel-Triple, and the Authagraph, which gets his vote. (The Authagraph is quite accurate as regards the land masses; the distortion gets pushed off onto the oceans, so the relative positions of the continents is bizarre.) However, there is absolutely no planar map of the world that can succeed in being both shape preserving and area preserving. There is just no way to perfectly flatten out the surface of a sphere.
Most educational debates do not have any final, ideal answer. What should be taught in literature, in history, in science, even in math, are matters of eternal debate, with no possible final resolution that is uncontentious, or apolitical, or value-free. However, this question of the proper way to display the geography of the earth is an exception. The obvious, perfect solution is — drumroll — a globe. A globe is an (essentially) perfect scale model of the geography of the earth, with no distortion of any kind. A child or adult who gets used to consulting a globe on questions of large scale geography, can get an exactly right idea of relative sizes and shapes and relative positions. (They should also have a good atlas, for small-scale geography).
As quoted above, George Orwell said that in 1944 globes were an “expensive rarity”. Presumably in 1944, getting globes to schoolchildren was not the top priority of the British government. But now they are really not expensive. You can get an inexpensive 6-inch globe for $10. You can get a good 11-inch globe for $30. I do find it surprising that none of the articles I’ve seen about the choice of maps even mentions this, best, option.
Of course if you have a globe for reference then it becomes enormously easier to explain how the Mercator, Gall-Peters, and other flat maps work, and what they get right and wrong.
In addition to its huge value in teaching geography, there are all kinds of cool things you can do and teach with a globe, particularly if you take it out of its stand:
- Great circle. You can easily illustrate the great circle path from any point to any other point by stretching a string between them and pulling it tight. No possible planar map correctly represents large scale geodesics.
- Seasons. Apparently a surprisingly large fraction even of college educated people think that the earth is closer to the sun in summer and further in winter. A globe makes it easier to illustrate both that the days are longer and that the light is more direct in the summer than in the winter. You also easily explain the significance of the poles, the equators, and of the polar and tropical circles.
- Astronomy of the earth and the moon. With a second, smaller ball, you can illustrate the interaction of the earth and the moon, and explain things like eclipses. (In fact, you can make a scale model; if you have an 11 inch globe for the earth, then the moon is a 3 inch ball, about 23 feet away.)
- Other celestial astronomy. With some additional props, you can show how the appearance of the night sky changes with latitude and with the time of year. You can explain why latitude has always been easy to determine, whereas the determination of longitude on board ship was one of the major problems for eighteenth century science. You can explain the significance of the ecliptic and the zodiac and the precession of the equinoxes.
- Geometry. Purely from the standpoint of teaching geometry, a globe has the amazing property of being a sphere on which there are thousands of easily identifiable points with memorable names. So an enterprising high school math teacher with good students can use it as a source of examples for an introduction to spherical geometry and thus non-Euclidean geometry. You can also illustrate three-dimensional rotations and the fact that they don’t commute. The original meaning of “geometry”, after all, is “measuring the earth.”
At the minimum, hopefully, early and extensive exposure to globes will deter students from growing up to believe that the earth is flat.
Artie has kindly allowed me to post his thoughtful email to me regarding my NYU conversation with Julia Angwin last month.
This is a guest post by Arthur Doskow, who is currently retired but remains interested in the application and overapplication of mathematical and data oriented techniques in business and society. Artie has a BS in Math and an MS that is technically in Urban Engineering, but the coursework was mostly in Operations Research. He spent the largest part of my professional life working for a large telco (that need not be named) on protocols, interconnection testing and network security. He is a co-inventor on several patents. He also volunteers as a tutor.
Dear Dr. O’Neil and Ms. Angwin,
I had the pleasure of watching the livestream of your discussion at NYU on February 15. I wanted to offer a few thoughts. I’ll try to be brief.
- Algorithms are difficult, and the ones that were discussed were being asked to make difficult decisions. Although it was not discussed, it would be a mistake to assume a priori that there is an effective mechanized and quantitative process by which good decisions can be made with regard to any particular matter. If someone cannot describe in detail how they would evaluate a teacher, or make a credit decision or a hiring decision or a parole decision, then it’s hard to imagine how they would devise an algorithm that would reliably perform the function in their stead. While it seems intuitively obvious that there are better teachers and worse teachers, reformed convicts and likely recidivist criminals and other similar distinctions, it is not (or should not be) equally obvious that the location of an individual on these continua can be reliably determined by quantitative methods. Reliance on a quantitative decision methodology essentially replaces a (perhaps arbitrary) individual bias with what may be a reliable and consistent algorithmic bias. Whether or not that represents an improvement must be assessed on a situation by situation basis.
- Beyond this stark “solvability” issue, of course, are the issues of how to set objectives for how an algorithm should perform (this was discussed with respect to the possible performance objectives of a parole evaluation system) and the devising, validating and implementing of a prospective system. This is a significant and demanding set of activities for any organization, but the alternative of procuring an outsourced “black box” solution requires, at the least, an understanding and an assessment of how these issues were addressed.
- If an organization is considering outsourcing an algorithmic decision system, the RFP process offers them an invaluable opportunity to learn and assess how a proposed system is designed and how it will work – What inputs does it use? How does its decision engine operate? How has it been validated? How will it cover certain test cases? Where has it been used? To what effect? Etc. Organizations that do not take advantage of an RFP process to ask these detailed questions and demand thorough and responsive answers have only themselves to blame.
- While a developers’ code of ethics is certainly a good thing, the development, marketing and support of a proposed solution is a shared task for which all members of the team must share responsibility – coders, system designers and specifiers, testers, marketers, trainers, support staff, executives. There is no single point of responsibility that can guarantee either a correct or an ethical implementation. Perhaps, in the same way that a CEO must personally sign off on all financial filings, the CEO of a company offering an evaluative system should be required to sign off on the legality, effectiveness and accuracy of claims made regarding the system.
- Software contracts are notoriously developer-friendly, basically absolving the developer of all possible consequences arising out of the use of their product. This needs to change, particularly in the case of systems sold as “black box” solutions to a purchaser’s needs, and contracts should be negotiated in which the developer retains significant responsibility and liability.
- As I think was pointed out, there is a broad range of analysis and modeling techniques, ranging from expert systems that seek to encode human knowledge, to heuristic learning system such as neural nets. While heuristic systems have the potential to ferret out non-intuitive relationships, their results obviously require a much higher degree of scrutiny. Part of me wonders how IBM and Watson would do at developing decision systems.
- Extensive testing and analysis should be required before any system “goes live”. It is disappointing to hear that “algorithm auditing” does not seem to be a thriving business, and, depending on the definition of “algorithm auditing”, I may be suggesting even more. Perhaps “algorithm testing” would be a more attractive sounding service name. Beyond requiring an analytical assessment of underlying data requirements and assessment algorithms, systems should be tested using an extensive set of test cases. Test cases should be assessed in advance by other (e.g., human expert) means, and system results should be examined for plausibility and for sanity. Another set of test cases should assess performance with extreme (e.g., best case, worst case) scenarios to check for system sanity. Another possibility is “side by side” testing, in which the system will “shadow” the current implementation, either concurrently or in retrospect and the results will be compared.
- Psychological and other pre-employment tests, described in Weapons of Math Destruction, are problematic in two ways. First is whether it is appropriate to conduct them at all, and second is whether they are effective in their stated purpose (i.e., to select the best prospective employees, or those best matched to the position in question). Certainly, competency testing is an appropriate part of candidate selection, but whether psychological characteristics are a component of competency is arguable, at best. At the very least, however, such testing should be assessed as to whether it predicts what it claims to predict, and whether that characteristic is emblematic of work effectiveness. How to conduct such testing would require some creativity. Testing could be conducted on an “incoming class” of employees, whether prior to hiring, or after hiring with the test results being sequestered (neither reported to company management nor used in any evaluation process). After some period (1 – 2 years), the qualitative measures of employee performance and effectiveness could be compared to the sequestered test results and examined for correlation. Another possibility would be to identify a disinterested company with employees performing similar work. (By disinterested, I mean disinterested in using the evaluative test in question.) Employees of that company could be asked to undergo “risk free” testing, with results again being sequestered from their employer. The quantitative test results could then be compared to the qualitative measures of employee performance and effectiveness used by that employer. Whatever one thinks of such testing, as Weapons of Math Destruction correctly points out, to the extent to which it is used, efforts should be made to test and improve its efficacy. To the extent that such testing is promoted by an outside party, that party should be ready, willing and able to demonstrate observed effectiveness.
- An interesting alternative to a proprietary black box system would be what might be called a meta-system, a configurable engine which would allow its procurer to specify the inputs, weightings and the manner in which they are used to formulate a decision, perhaps offering a drag and drop software interface to specify the decision algorithm. Such a system would leave the fundamentals of the decision algorithm design to the purchasing company, but simply facilitate its implementation.
- One must always be cautious the possibility of inherent bias in data. As a simple example, recidivism is most easily estimated by the proportion of released convicts who are re-arrested. But if recidivism is actually defined by the percentage of released convicts who return to criminal life, then the estimate is likely skewed in several ways. Some recidivists will be caught; others will not. For example, some types of crime are more heavily investigated than others, leading to higher re-arrest rates. Further, even among perpetrators of the same crime, investigation and enforcement may well be targeted more to some areas than to others.
- As was pointed out during the discussion, being fair, being humane may cost money. And this is the real issue with many algorithms. In economists’ terms, the inhumanity associated with an algorithm could be referred to as an externality. Optimization has its origins with the solutions to problem in the inanimate world, how to inspect mass produced parts for flaws, how to cut a board to obtain the most salable pieces of lumber, how to minimize the lengths of circuit traces on a PC board. There were problems that touched on human behavior, scheduling issues, or traveling salesman type problems, but not to the extent that they ignored humane considerations. We are now to the point where we have human beings being compared to poisonous Skittles, and where life altering decisions of great import (hiring, firing parole, assessment, scheduling, etc.) are being subjected to optimization processes, often of questionable validity, which objectify people, view them as resources or threats, and give little or no consideration to the very human consequences of their deployment. Assuming that your good work can drive to this consensus, there is a fork in the road as to how it can be addressed. One way would be to attempt to implement humane costs, benefits and constraints into the models being deployed and optimize on that basis. The other is to stand back and monitor applications for their human costs and attempt to address them iteratively. Or, as Yogi said, you can come to the fork and take it.
This is just out on Bloomberg:
And this came out Monday:
One of the most ridiculous aspects of the “blame the individual” approach to obesity is the overall trend of fatness throughout the country and the world over the past few decades.
The rates of obesity have skyrocketed and we simply don’t know why. Here are a few reasons that are consistently trotted out:
- People have become lazy. (Actually this one’s easy. In fact we’re getting more exercise and it doesn’t seem to help. And for that matter, exercise doesn’t make you lose weight.)
- We’re eating too much fast food.
- We’re drinking too much soda and generally eating too much sugar.
- We’re watching too much TV/ playing too many video games.
- There are too many food options constantly surrounding us.
- Our portion sizes are too big.
- We’re just eating too much of everything. In some sense this is a tautology. The question is why.
- Glyphosates in our grains are making us fat.
- Our internal stomach biomes are messed up and make us fat.
- Bad genetics make us fat. This one’s easy too, since our gene pool hasn’t changed that much recently.
- Dieting itself makes us fat.
So, there are tons of reasons, and I’m sure I missed some. The tricky thing is, all of them sound plausible, and none of them are likely the single answer. Likely it’s a combination of a bunch of them.
But the truth is, we don’t know. And people hate not knowing stuff, so they pretend they know. That’s not helpful. We need to be scientists and try testing out hypotheses.
The biggest problem with testing the above hypotheses is that many of them are hard to avoid environmental factors of modern life. You can take the soda machine away from a high school but then the kids will just buy soda at the nearby corner store.
Until we’ve figured it out, I’d like us to admit I don’t know why we’re all so fat. And I’d like us to stop blaming individuals, especially children.
There are lots of ways to get statistical thinking wrong, not so many ways to get it right. Here’s a series of examples from wrong to right:
- I did this, and it’s not a “diet,” it’s a lifestyle change, and it works for me!
- I know people who live or interact with the world in a certain way, and it seems to work for them! After all, French women are thin. We should all do what they do.
- There was a study of volunteers, and for the people who stayed in the study to the end, they lost weight doing such and such lifestyle change!
- There was a study of volunteers, and they tracked people down who tried to leave the study, and the average weight gain was still real, among the people they found!
- There was a study of doctors giving advice or enrolling people in programs to help overweight people lose weight, and 97% of people lost no weight and plenty of people gained weight, maybe even more than half.
What I’d love is for people to understand how much difference there is between a personal experience (1) and advice we’d have on public health (5).
Here’s the golden standard: if you can come up with something to tell Medicare about how to have a population of morbidly obese people become a population of regular weight people, then you win. Otherwise, if you’re tempted to tell me about a lifestyle change that worked for you, please don’t, because that’s not statistical.
Also, I’d like a word about the theory that with enough discipline and willpower, anyone can lose weight. I think it’s fair to say I have discipline and willpower. In fact, I’m a fucking poster child for them. I wrote a Ph.D. as one of few women in a male-dominated field. I wrote a book or two. I’ve had three kids and I’ve never struck one of them in anger. In fact I’m pretty nice to people most of the time, even though I’m relatively often filled with rage at the unfairness of the world. That’s hard. It takes willpower.
I even ran a sprint triathlon at 275 pounds, really fast, which took months of ridiculous training. Also, I know all about healthy habits, I don’t eat “emotionally,” just when I’m hungry, and I love brussel sprouts and other healthy foods. I just get really fucking hungry, often.
Readers, I’m the fucking center of the disciple in willpower universe over here.
Given all of that, if anything I’d argue my willpower is one reason I’m so heavy. When I was 22 or so, I went on a fat-free diet, on the advice of my doctor, that fucked me up; I lost 30 pounds but then gained something like 75. I think I messed up my insulin resistance. In fact I believe that also happened to me on my first starvation diet when I was 14.
I’m guessing I’d be thinner if I’d had less willpower, in other words. I wouldn’t be better off, though, because I kind of like my books and my Ph.D. and my kids who don’t fear their parents.
Anyway, from now on let’s talk statistically, shall we?
There’s a lot of statistical evidence that dieting doesn’t work. I’ll postpone the documentation of the highlights of that evidence for a later post, but you can google it for yourself (avoid, if you can, the links that are trying to get you to buy something).
And when I say “diets don’t work,” here’s what I mean. I mean that, statistically speaking, people who go on diets don’t successfully lose and keep off weight for more than about six months. So, after two years or so, the average weight is about the same or higher in a group of dieters.
Can we take that as a given for now? Thanks. We can argue about it later if you want.
Here’s the thing. That statement confounds lots of people, I think because it’s statistical in nature. They will always imagine that, because they are themselves examples of someone who has lost weight and kept it off for more than two years through dieting, dieting does in fact work, and we should all try what they’ve tried.
It’s annoying to be told this over and over again, especially when you’re someone who’s tried a million things. And believe me, almost every fat person I know has tried a million things. For that reason I’d appreciate no more such advice, although in a later post I will be asking for zany pseudo-scientific theories about why fat people stay fat (there are so many!).
So yeah, people don’t understand statistical facts. But I think there’s something more going on here. Namely, the illusory nature of choice when it comes to dieting.
Because diets do seem to work short term, people think they’ve gotten control over their eating, at least temporarily. And then, at some point, people drop off their diets. They sometimes do it with a “what the fuck” attitude, but my guess is most of them don’t even remember doing it. It’s a kind of momentary amnesia, and before they know what’s happened they’re eating something they shouldn’t have. That is certainly my experience.
From the outsider’s perspective, that’s a person who has chosen to go off their diets, and in a certain sense it’s obviously true, since for example anyone who was locked in a cell with no food would not have the ability to go off a diet, nor would someone who cannot feed themselves. Indeed, it requires the access to food and the action of eating to go off a diet. So in that sense it takes a certain amount of freedom.
But, there’s another sense in which, I’d argue, there’s no choice in the matter at all. After all, dieting requires a positive declaration of a desire to lose weight. Sometimes it even requires forking over cash, maybe a lot of cash. People are trying hard to lose weight, in other words, and yet they can’t, and even statistically speaking they cannot.
Said another way: if 1000 people went to a lot of trouble to do something, and they all tried but 990 of them failed to do it, would we decide they had made the choice not to do it?
I’m ready to say there’s something else at work here, something more basic than free will. It’s like our choice to breathe. We can’t decide not to do it. Or we can, but only for a bit.
Commenters, please stick to the question of the nature of choice in dieting. I will delete other stuff, thanks!
Readers, I’ve got two announcements today.
First, I’ll be giving a TED talk in April in Vancouver. And yes, for those of you remember, I haven’t always been the biggest fan of such things. But I’ve changed my mind/ sold out/ decided that it might just be great.
As a friend of mine explained to me, sometimes things get so douchey they come out the other side and are super cool. Also, I’m giving a talk in the section called Our Robotic Overlords, so that’s a very good sign.
Second, I’ve decided to undergo bariatric surgery. I’m jumping through the many insurance-qualifying hoops for now but if all goes well it will happen later this year, possibly as soon as July.
And… I’m planning to chronicle my journey on mathbabe. If that kind of thing doesn’t interest you, feel free to never come back, but if that kind of thing does interest you, then buckle up!
I’m not planning to keep myself to the subject of the bariatric surgery; in fact that’s just an excuse to think about a lot more, specifically:
- the nature of scientific understanding and how it does or does not percolate throughout society as a whole,
- how money and shame corrupt our understanding of scientific evidence,
- how bad data and bad technologies and biased academic publishing prevent us from learning optimally,
- the nature of individual choice, willpower, and control,
- my historical self-image as a dieter, a fat person, a woman, a feminist, and a thinker,
- how I gathered evidence and made this decision, and of course
- the process itself.
So I’m thinking kind of big and I’m going to have fun with it. Please feel free to comment, I’d love your help!