This is a guest post from Jordan Ellenberg, a professor of mathematics at the University of Wisconsin. Jordan’s book, How Not To Be Wrong, comes out in May 2014. It is crossposted from his blog, Quomodocumque, and tweeted about at @JSEllenberg.
Cathy posted some cool data yesterday coming from the new visualization features of the magnificent Stacks Project. Summary: you can make a directed graph whose vertices are the 10,445 tagged assertions in the Stacks Project, and whose edges are logical dependency. So this graph (hopefully!) doesn’t have any directed cycles. (Actually, Cathy tells me that the Stacks Project autovomits out any contribution that would create a logical cycle! I wish LaTeX could do that.)
Given any assertion v, you can construct the subgraph G_v of vertices which are the terminus of a directed path starting at v. And Cathy finds that if you plot the number of vertices and number of edges of each of these graphs, you get something that looks really, really close to a line.
Why is this so? Does it suggest some underlying structure? I tend to say no, or at least not much — my guess is that in some sense it is “expected” for graphs like this to have this sort of property.
Because I am trying to get strong at sage I coded some of this up this morning. One way to make a random directed graph with no cycles is as follows: start with N edges, and a function f on natural numbers k that decays with k, and then connect vertex N to vertex N-k (if there is such a vertex) with probability f(k). The decaying function f is supposed to mimic the fact that an assertion is presumably more likely to refer to something just before it than something “far away” (though of course the stack project is not a strictly linear thing like a book.)
Here’s how Cathy’s plot looks for a graph generated by N= 1000 and f(k) = (2/3)^k, which makes the mean out-degree 2 as suggested in Cathy’s post.
Pretty linear — though if you look closely you can see that there are really (at least) a couple of close-to-linear “strands” superimposed! At first I thought this was because I forgot to clear the plot before running the program, but no, this is the kind of thing that happens.
Is this because the distribution decays so fast, so that there are very few long-range edges? Here’s how the plot looks with f(k) = 1/k^2, a nice fat tail yielding many more long edges:
My guess: a random graph aficionado could prove that the plot stays very close to a line with high probability under a broad range of random graph models. But I don’t really know!
Update: Although you know what must be happening here? It’s not hard to check that in the models I’ve presented here, there’s a huge amount of overlap between the descendant graphs; in fact, a vertex is very likely to be connected all but c of the vertices below it for a suitable constant c.
I would guess the Stacks Project graph doesn’t have this property (though it would be interesting to hear from Cathy to what extent this is the case) and that in her scatterplot we are not measuring the same graph again and again.
It might be fun to consider a model where vertices are pairs of natural numbers and (m,n) is connected to (m-k,n-l) with probability f(k,l) for some suitable decay. Under those circumstances, you’d have substantially less overlap between the descendant trees; do you still get the approximately linear relationship between edges and nodes?
This is a guest post by my friend Laura Strausfeld.
As an unlicensed psychotherapist, here’s my take on why Huma Abedin is supporting her husband Anthony Weiner’s campaign for mayor:
It’s all about the kid.
Jordan Weiner is 19 months old. When he’s 8 or 9—or 5, and wearing google glasses—maybe he’ll google his name and read about his father’s penis. Either that, or one of his buddies at school may ask him about his father’s penis. Jordan might then ask his mommy and daddy about his father’s penis and they’ll tell him either 1) your daddy was a great politician, but had to resign from Congress because he admitted to showing people his penis, which we recommend you don’t do, especially when you’re a grownup and on twitter; or 2) your daddy was a great politician and ran a very close race for mayor—that’s right, your daddy was almost mayor of New York City!—but he lost because people said he showed people his penis and that’s none of anybody’s business.
Let’s look at this from Huma’s perspective. She’s got a child for a husband, with a weird sexual addiction that on the positive side, doesn’t appear to carry the threat of STDs. But her dilemma is not about her marriage. The marriage is over. What she cares about is Jordan. And this is where she’s really fucked. Whatever happens, Anthony will always be her child’s father.
That bears repeating. You’ve got a child you love more than anything in the world, will sacrifice anything for, and will always now be stigmatized as the son of a celebrity-sized asshole. What are your choices?
The best scenario for Huma is if Anthony becomes mayor. Then she can divorce his ass, get primary custody and protect her child from growing up listening to penis jokes about his loser father. There will be jokes, but at least they’ll be about the mayor’s penis. And with a whole lot of luck, they might even be about how his father’s penis was a lot smaller in the mind of the public than his policies.
Weiner won’t get my vote, however. And for that, I apologize to you, Jordan. You have my sympathy, Huma.
This is a guest post by Peter Darche, an engineer at DataKind and recent graduate of NYU’s ITP program. At ITP he focused primarily on using personal data to improve personal social and environmental impact. Prior to graduate school he taught in NYC public schools with Teach for America and Uncommon Schools.
We all ‘know’ that money influences the way congressmen and women legislate; at least we certainly believe it does. According to poll conducted by law professor Larry Lessig for his book Republic Lost, 75% of respondents (Republican and Democrat) said that ‘money buys results in Congress.’
But what does that explanation really tell us? Yes, a congresswoman’s receiving millions dollars from an industry then voting with that industry’s interests reeks of corruption. But, when that industry is responsible for 80% of her constituents’ jobs the causation becomes much less clear and the explanation much less informative.
The real devil is in the details. It is in the ways that money has shaped her legislative worldview over time and in the small, particular actions that tilt her policy one way rather than another.
In the past finding these many and subtle ways would have taken a herculean effort: untold hours collecting campaign contributions, voting records, speeches, and so on. Today however, due to the efforts of organizations like the Sunlight Foundation and Center for Responsive Politics, this information is online and programmatically accessible; you can write a few lines of code and have a computer gather it all for you.
The last few months Cathy O’Neil, Lee Drutman (a Senior Fellow at the Sunlight Foundation), myself and others have been working on a project that leverages these data sources to attempt to unearth some of these particular facts. By connecting all the avenues by which influence is exerted on the legislative process to the actions taken by legislators, we’re hoping to find some of the detailed ways money changes behavior over time.
The ideas is this: first, find and aggregate what data exists related to the ways influence can be exerted on the legislative process (data on campaign contributions, lobbying contributions, etc), then find data that might track influence manifesting itself in the legislative process (bill sponsorships, co-sponsorships, speeches, votes, committee memberships, etc). Finally, connect the interest group or industry behind the influence to the policies and see how they change over time.
One immediate and attainable goal for this project, for example, is to create an affinity score between legislators and industries, or in other words a metric that would indicate the extent to which a given legislator is influenced by and acts in the interest of a given industry.
So far most of our efforts have focused on finding, collecting, and connecting the records of influence and legislative behavior. We’ve pulled in lobbying and campaign contribution data, as well as sponsored legislation, co-sponsored legislation, speeches and votes. We’ve connected the instances of influence to legislative actions for a given legislator and visualized it on a timeline showing the entirety of a legislator’s career.
Here’s an example of how one might use the timeline. The example below is of Nancy Pelosi’s career. Each green circle represents a campaign contribution she received, and is grouped within a larger circle by the month it was recorded by the FEC. Above are colored rectangles representing legislative actions she took during the time-period in focus (indigo are votes, orange speeches, red co-sponsored bills, blue sponsored bills). Some of the green circles are highlighted because the events have been filtered for connection to health professionals.
Changing the filter to Health Services/HMOs, we see different contributions coming from that industry as well as a co-sponsored bill related to that industry.
Mousing over the bill indicates its a proposal to amend the Social Security act to provide Medicaid coverage to low-income individuals with HIV. Further, looking around at speeches, one can see a relevant speech about the children’s health insurance. Clicking on the speech reveals the text.
By combining data about various events, and allowing users to filter and dive into them, we’re hoping to leverage our natural pattern-seeking capabilities to find specific hypotheses to test. Once an interesting pattern has been found, the tool would allow one to download the data and conduct analyses.
Again, It’s just start, and the timeline and other project related code are internal prototypes created to start seeing some of the connections. We wanted to open it up to you all though to see what you all think and get some feedback. So, with it’s pre-alphaness in mind, what do you think about the project generally and the timeline specifically? What works well – helps you gain insights or generate hypotheses about the connection between money and politics – and what other functionality would you like to see?
The demo version be found here with data for the following legislators:
- Nancy Pelosi
- John Boehner
- Cathy McMorris Rodgers
- John Boehner
- Eric Cantor
- James Lankford
- John Cornyn
- Nancy Pelosi
- James Clyburn
- Kevin McCarthy
- Steny Hoyer
Note: when the timeline is revealed, click and drag over content at the bottom of the timeline to reveal the focus events.
This is a guest post by Eugene Stern.
Now that I have kids in school, I’ve become a lot more familiar with high-stakes testing, which is the practice of administering standardized tests with major consequences for students who take them (you have to pass to graduate), their teachers (who are often evaluated based on standarized test results), and their school districts (state funding depends on test results). To my great chagrin, New Jersey, where I live, is in the process of putting such a teacher evaluation system in place (for a lot more detail and criticism, see here).
The excellent John Ewing pointed me to a pretty comprehensive survey of standardized testing called “Measuring Up,” by Harvard Ed School prof Daniel Koretz, who teaches a course there about this stuff. If you have any interest in the subject, the book is very much worth your time. But in case you don’t get to it, or just to whet your appetite, here are my top 10 takeaways:
Believe it or not, most of the people who write standardized tests aren’t idiots. Building effective tests is a difficult measurement problem! Koretz makes an analogy to political polling, which is a good reminder that a test result is really a sample from a distribution (if you take multiple versions of a test designed to measure the same thing, you won’t do exactly the same each time), and not an absolute measure of what someone knows. It’s also a good reminder that the way questions are phrased can matter a great deal.
The reliability of a test is inversely related to the standard deviation of this distribution: a test is reliable if your score on it wouldn’t vary very much from one instance to the next. That’s a function of both the test itself and the circumstances under which people take it. More reliability is better, but the big trade-off is that increasing the sophistication of the test tends to decrease reliability. For example, tests with free form answers can test for a broader range of skills than multiple choice, but they introduce variability across graders, and even the same person may grade the same test differently before and after lunch. More sophisticated tasks also take longer to do (imagine a lab experiment as part of a test), which means fewer questions on the test and a smaller cross-section of topics being sampled, again meaning more noise and less reliability.
A complementary issue is bias, which is roughly about people doing better or worse on a test for systematic reasons outside the domain being tested. Again, there are trade-offs: the more sophisticated the test, the more extraneous skills beyond those being tested it may be bringing in. One common way to weed out such questions is to look at how people who score the same on the overall test do on each particular question: if you get variability you didn’t expect, that may be a sign of bias. It’s harder to do this for more sophisticated tests, where each question is a bigger chunk of the overall test. It’s also harder if the bias is systematic across the test.
Beyond the (theoretical) distribution from which a single student’s score is a sample, there’s also the (likely more familiar) distribution of scores across students. This depends both on the test and on the population taking it. For example, for many years, students on the eastern side of the US were more likely to take the SAT than those in the west, where only students applying to very selective eastern colleges took the test. Consequently, the score distributions were very different in the east and the west (and average scores tended to be higher in the west), but this didn’t mean that there was bias or that schools in the west were better.
The shape of the score distribution across students carries important information about the test. If a test is relatively easy for the students taking it, scores will be clustered to the right of the distribution, while if it’s hard, scores will be clustered to the left. This matters when you’re interpreting results: the first test is worse at discriminating among stronger students and better at discriminating among weaker ones, while the second is the reverse.
The score distribution across students is an important tool in communicating results (you may not know right away what a score of 600 on a particular test means, but if you hear it’s one standard deviation above a mean of 500, that’s a decent start). It’s also important for calibrating tests so that the results are comparable from year to year. In general, you want a test to have similar means and variances from one year to the next, but this raises the question of how to handle year-to-year improvement. This is particularly significant when educational goals are expressed in terms of raising standardized test scores.
If you think in terms of the statistics of test score distributions, you realize that many of those goals of raising scores quickly are deluded. Koretz has a good phrase for this: the myth of the vanishing variance. The key observation is that test score distributions are very wide, on all tests, everywhere, including countries that we think have much better education systems than we do. The goals we set for student score improvement (typically, a high fraction of all students taking a test several years from now are supposed to score above some threshold) imply a great deal of compression at the lower end of this distribution – compression that has never been seen in any country, anywhere. It sounds good to say that every kid who takes a certain test in four years will score as proficient, but that corresponds to a score distribution with much less variance than you’ll ever see. Maybe we should stop lying to ourselves?
Koretz is highly critical of the recent trend to report test results in terms of standards (e.g., how many students score as “proficient”) instead of comparisons (e.g., your score is in the top 20% of all students who took the test). Standards and standard-based reporting are popular because it’s believed that American students’ performance as a group is inadequate. The idea is that being near the top doesn’t mean much if the comparison group is weak, so instead we should focus on making sure every student meets an absolute standard needed for success in life. There are three (at least) problems with this. First, how do you set a standard – i.e., what does proficient mean, anyway? Koretz gives enough detail here to make it clear how arbitrary the standards are. Second, you lose information: in the US, standards are typically expressed in terms of just four bins (advanced, proficient, partially proficient, basic), and variation inside the bins is ignored. Third, even standards-based reporting tends to slide back into comparisons: since we don’t know exactly what proficient means, we’re happiest when our school, or district, or state places ahead of others in the fraction of students classified as proficient.
Koretz’s other big theme is score inflation for high-stakes tests: if everyone is evaluated based on test scores, everyone has an incentive to get those scores up, whether or not that actually has much correlation with learning. If you remember anything from the book or from this post, remember this phrase: sawtooth pattern. The idea is that when a new high-stakes standardized test appears, average scores start at some base level, go up quickly as people figure out how to game the test, then plateau. If the test is replaced with another, the same thing happens: base, rapid growth, plateau. Repeat ad infinitum. Koretz and his collaborators did a nice experiment in which they went back to a school district in which one high-stakes test had been replaced with another and administered the first test several years later. Now that teachers weren’t teaching to the first test, scores on it reverted back to the original base level. Moral: score inflation is real, pervasive, and unavoidable, unless we bite the bullet and do away with high-stakes tests.
While Koretz is sympathetic toward test designers, who live the complexity of standardized testing every day, he is harsh on those who (a) interpret and report on test results and (b) set testing and education policy, without taking that complexity into account. Which, as he makes clear, is pretty much everyone who reports on results and sets policy.
If you think it’s a good idea to make high-stakes decisions about schools and teachers based on standardized test results, Koretz’s book offers several clear warnings.
First, we should expect any high-stakes test to be gamed. Worse yet, the more reliable tests, being more predictable, are probably easier to game (look at the SAT prep industry).
Second, the more (statistically) reliable tests, by their controlled nature, cover only a limited sample of the domain we want students to learn. Tests trying to cover more ground in more depth (“tests worth teaching to,” in the parlance of the last decade) will necessarily have noisier results. This noise is a huge deal when you realize that high-stakes decisions about teachers are made based on just two or three years of test scores.
Third, a test that aims to distinguish “proficiency” will do a worse job of distinguishing students elsewhere in the skills range, and may be largely irrelevant for teachers whose students are far away from the proficiency cut-off. (For a truly distressing example of this, see here.)
With so many obstacles to rating schools and teachers reliably based on standardized test scores, is it any surprise that we see results like this?
This is a guest post by Eugene Stern.
Sometimes you learn just as much from a bad analogy as from a good one. At least you learn what people are thinking.
The other day I read this response to this NYT article. The original article asked whether the Common Core-based school reforms now being put in place in most states are really a good idea. The blog post criticized the article for failing to break out four separate elements of the reforms: standards (the Core), curriculum (what’s actually taught), assessment (testing), and accountability (evaluating how kids and educators did). If you have an issue with the reforms, you’re supposed to say exactly which aspect you have an issue with.
But then, at the end of the blog post, we get this:
A track and field metaphor might help: The standard is the bar that students must jump over to be competitive. The curriculum is the training program coaches use to help students get over the bar. The assessment is the track meet where we find out how high everyone can jump. And the accountability system is what follows after its all over and we want to figure out what went right, what went wrong, and what it will take to help kids jump higher.
In track, jumping over the bar is the entire point. You’re successful if you clear the bar, you’ve failed if you don’t. There are no other goals in play. So the standard, the curriculum, and the assessment might be nominally different, but they’re completely interdependent. The standard is defined in terms of the assessment, and the only curriculum that makes sense is training for the assessment.
Education has a lot more to it. The Common Core is a standard covering two academic dimensions: math and English/language arts/literacy. But we also want our kids learning science, and history, and music, and foreign languages, and technology, as well as developing along non-academic dimensions: physically, socially, morally, etc. (If a school graduated a bunch of high academic achievers that couldn’t function in society, or all ended up in jail for insider trading, we probably wouldn’t call that school successful.)
In Cathy’s terminology from this blog post, the Common Core is a proxy for the sum total of what we care about, or even just for the academic component of what we care about.
Then there’s a second level of proxying when we go from the standard to the assessment. The Common Core requirements are written to require general understanding (for example: kindergarteners should understand the relationship between numbers and quantities and connect counting to cardinality). A test that tries to measure that understanding can only proxy it imperfectly, in terms of a few specific questions.
Think that’s obvious? Great! But hang on just a minute.
The real trouble with the sports analogy comes when we get to the accountability step and forget all the proxying we did. “After it’s all over and we want to figure out what went right (and) what went wrong,” we measure right and wrong in terms of the assessment (the test). In sports, where the whole point is to do well on the assessment, it may make sense to change coaches if the team isn’t winning. But when we deny tenure to or fire teachers whose students didn’t do well enough on standardized tests (already in place in New York, now proposed for New Jersey as well), we’re treating the test as the whole point, rather than a proxy of a proxy. That incentivizes schools to narrow the curriculum to what’s included in the standard, and to teach to the test.
We may think it’s obvious that sports and education are different, but the decisions we’re making as a society don’t actually distinguish them.
This is a guest post by Rachel Law, a conceptual artist, designer and programmer living in Brooklyn, New York. She recently graduated from Parsons MFA Design&Technology. Her practice is centered around social myths and how technology facilitates the creation of new communities. Currently she is writing a book with McKenzie Wark called W.A.N.T, about new ways of analyzing networks and debunking ‘mapping’.
Let’s start with a timely question. How would you like to be able to change how you are identified by online networks? We’ll talk more about how you’re currently identified below, but for now just imagine having control over that process for once – how would that feel? Vortex is something I’ve invented that will try to make that happen.
Namely, Vortex is a data management game that allows players to swap cookies, change IPs and disguise their locations. Through play, individuals experience how their browser changes in real time when different cookies are equipped. Vortex is a proof of concept that illustrates how network collisions in gameplay expose contours of a network determined by consumer behavior.
What happens when users are allowed to swap cookies?
These cookies, placed by marketers to track behavioral patterns, are stored on our personal devices from mobile phones to laptops to tablets, as a symbolic and data-driven signifier of who we are. In other words, to the eyes of the database, the cookies are us. They are our identities, controlling the way we use, browse and experience the web. Depending on cookie type, they might follow us across multiple websites, save entire histories about how we navigate and look at things and pass this information to companies while still living inside our devices.
If we have the ability to swap cookies, the debate on privacy shifts from relying on corporations to follow regulations to empowering users by giving them the opportunity to manage how they want to be perceived by the network.
What are cookies?
The corporate technological ability to track customers and piece together entire personal histories is a recent development. While there are several ways of doing so, the most common and prevalent method is with HTTP cookies. Invented in 1994 by a computer programmer, Lou Montulli, HTTP cookies were originally created with the shopping cart system as a way for the computer to store the current state of the session, i.e. how many items existed in the cart without overloading the company’s server. These session histories were saved inside each user’s computer or individual device, where companies accessed and updated consumer history constantly as a form of ‘internet history’. Information such as where you clicked, how to you clicked, what you clicked first, your general purchasing history and preferences were all saved in your browsing history and accessed by companies through cookies.
Cookies were originally implemented to the general public without their knowledge until the Financial Times published an article about how they were made and utilized on websites without user knowledge on February 12th, 1996 . This revelation led to a public outcry over privacy issues, especially since data was being gathered without the knowledge or consent of users. In addition, corporations had access to information stored on personal computers as the cookie sessions were stored on your computer and not their servers.
At the center of the debate was the issue on third-party cookies, also known as “persistent” or “tracking” cookies. When you are browsing a webpage, there may be components on the page that are hosted on the same server, but different domain. These external objects then pass cookies to you if you click an image, link or article. They are then used by advertising and media mining corporations to track users across multiple sites to garner more knowledge about the users browsing patterns to create more specific and targeted advertising.
In August 2013, Wall Street Journal ran an article on how Mac users were being unfairly targeted by travel site Orbitz with advertisements that were 13% more expensive than PC users. New York Times followed it up with a similar article in November 2012 about how the data collected and re-sold to advertisers. These advertisers would analyze users buying habits to create micro-categories where the personal experiences were tailored to maximize potential profits.
What does that mean for us?
The current state of today’s internet is no longer the same as the carefree 90s of ‘internet democracy’ and utopian ‘cyberspace’. Mediamining exploits invasive technologies such as IP tracking, geolocating and cookies to create specific advertisements targeted to individuals. Browsing is now determined by your consumer profile what you see, hear and the feeds you receive are tailored from your friends’ lists, emails, online purchases etc. The ‘Internet’ does not exist. Instead, it is many overlapping filter bubbles which selectively curate us into data objects to be consumed and purchased by advertisers.
This information, though anonymous, is built up over time and used to track and trace an individual’s history – sometimes spanning an entire lifetime. Who you are, and your real name is irrelevant in the overall scale of collected data, depersonalizing and dehumanizing you into nothing but a list of numbers on a spreadsheet.
The superstore Target, provides a useful case study for data profiling in its use of statisticians on their marketing teams. In 2002, Target realized that when a couple is expecting a child, the way they shop and purchase products changes. But they needed a tool to be able to see and take advantage of the pattern. As such, they asked mathematicians to come up with algorithms to identify behavioral patterns that would indicate a newly expectant mother and push direct marketing materials their way. In a public relations fiasco, Target had sent maternity and infant care advertisements to a household, inadvertedly revealing that their teenage daughter was pregnant before she told her parents .
This build-up of information creates a ‘database of ruin’, enough information that marketers and advertisers know more about your life and predictive patterns than any single entity. Databases that can predict whether you’re expecting, or when you’ve moved, or what stage of your life or income level you’re at… information that you have no control over where it goes to, who is reading it or how it is being used. More importantly, these databases have collected enough information that they know secrets such as family history of illness, criminal or drug records or other private information that could potentially cause harm upon the individual data point if released – without ever needing to know his or her name.
What happens now is two terrifying possibilities:
- Corporate databases with information about you, your family and friends that you have zero control over, including sensitive information such as health, criminal/drug records etc. that are bought and re-sold to other companies for profit maximization.
- New forms of discrimination where your buying/consumer habits determine which level of internet you can access, or what kind of internet you can experience. This discrimination is so insidious because it happens on a user account level which you cannot see unless you have access to other people’s accounts.
Here’s a visual describing this process:
What can Vortex do, and where can I download a copy?
As Vortex lives on the browser, it can manage both pseudo-identities (invented) as well as ‘real’ identities shared with you by other users. These identity profiles are created through mining websites for cookies, swapping them with friends as well as arranging and re-arranging them to create new experiences. By swapping identities, you are essentially ‘disguised’ as someone else – the network or website will not be able to recognize you. The idea is that being completely anonymous is difficult, but being someone else and hiding with misinformation is easy.
This does not mean a death knell for online shopping or e-commerce industries. For instance, if a user decides to go shoe-shopping for summer, he/she could equip their browser with the cookies most associated and aligned with shopping, shoes and summer. Targeted advertising becomes a targeted choice for both advertisers and users. Advertisers will not have to worry about misinterpreting or mis-targeting inappropriate advertisements i.e. showing tampon advertisements to a boyfriend who happened to borrow his girlfriend’s laptop; and at the same time users can choose what kind of advertisements they want to see. (i.e. Summer is coming, maybe it’s time to load up all those cookies linked to shoes and summer and beaches and see what websites have to offer; or disable cookies it completely if you hate summer apparel.)
Currently the game is a working prototype/demo. The code is licensed under creative commons and will be available on GitHub by the end of summer. I am trying to get funding to make it free, safe & easy to use; but right now I’m broke from grad school and a proper back-end to be built for creating accounts that is safe and cannot be intercepted. If you have any questions on technical specs or interest in collaborating to make it happen – particularly looking for people versed in python/mongodb, please email me: Rachel@milkred.net.
This guest post is by Sue VanHattum, who blogs at Math Mama Writes. She teaches math at Contra Costa College, a community college in the Bay Area, and is working on a book titled Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, which will be published soon.
Here’s the Pythagorean Theorem:
In a right triangle, where the lengths of the legs are given by and , and the length of the hypotenuse is given by , we have
Do you remember when you first learned about it? Do you remember when you first proved it?
I have no idea when or where I first saw it. It feels like something I’ve always ‘known’. I put known in quotes because in math we prove things, and I used the Pythagoeran Theorem for way too many years, as a student and as a math teacher, before I ever thought about proving it. (It’s certainly possible I worked through a proof in my high school geometry class, but my memory kind of sucks and I have no memory of it.)
It’s used in beginning algebra classes as part of terrible ‘pseudo-problems’ like this:
Two cars start from the same intersection with one traveling southbound while the other travels eastbound going 10 mph faster. If after two hours they are 10 times the square root of 24 [miles] apart, how fast was each car traveling?
After years of working through these problems with students, I finally realized I’d never shown them a proof (this seems terribly wrong to me now). I tried to prove it, and didn’t really have any idea how to get started.
This was 10 to 15 years ago, before Google became a verb, so I searched for it in a book. I eventually found it in a high school geometry textbook. Luckily it showed a visually simple proof that stuck with me. There are hundreds of proofs, many of them hard to follow.
There is something wrong with an education system that teaches us ‘facts’ like this one and knocks the desire for deep understanding out of us. Pam Sorooshian, an unschooling advocate, said in a talk to other unschooling parents:
Relax and let them develop conceptual understanding slowly, over time. Don’t encourage them to memorize anything – the problem is that once people memorize a technique or a ‘fact’, they have the feeling that they ‘know it’ and they stop questioning it or wondering about it. Learning is stunted.
She sure got my number! I thought I knew it for all those years, and it took me decades to realize that I didn’t really know it. This is especially ironic – the reason it bears Pythagoras’ name is because the Pythagoreans were the first to prove it (that we know of).
It had been used long before Pythagoras and the Greeks – most famously by the Egyptians. Egyptian ‘rope-pullers’ surveyed the land and helped build the pyramids, using a taut circle of rope with 12 equally-spaced knots to create a 3-4-5 triangle: since this is a right triangle, giving them the right angle that’s so important for building and surveying.
Ever since the Greeks, proof has been the basis of all mathematics. To do math without understanding why something is true really makes no sense.
Nowadays I feel that one of my main jobs as a math teacher is to get students to wonder and to question. But my own math education left me with lots of ‘knowledge’ that has nothing to do with true understanding. (I wonder what else I have yet to question…) And beginning algebra students are still using textbooks that ‘give’ the Pythagorean Theorem with no justification. No wonder my Calc II students last year didn’t know the difference between an example and a proof.
Just this morning I came across an even simpler proof of the Pythagorean Theorem than the one I have liked best over the past 10 to 15 years. I was amazed that I hadn’t seen it before. Well, perhaps I had seen it but never took it in before, not being ready to appreciate it. I’ll talk about it below.
My old favorite goes like this:
- Draw a square.
- Put a dot on one side (not at the middle).
- Put dots at the same place on each of the other 3 sides.
- Connect them.
- You now have a tilted square inside the bigger square, along with 4 triangles. At this point, you can proceed algebraically or visually.
- big square = small tilted square + 4 triangles
- Move the triangles around.
- What was is now
- Also check out Vi Hart’s video showing a paper-folding proof (with a bit of ripping). It’s pretty similar to this one.
This is an even more visual proof, although it might take a few geometric remarks to make it clear. In any right triangle, the two acute (less than 90 degrees) angles add up to 90 degrees. Is that enough to see that the original triangle, triangle A, and triangle B are all similar? (Similar means they have exactly the same shape, though they may be different sizes.) Which makes the ‘houses with asymmetrical roofs’ also all similar. Since the big ‘house’ has an ‘attic’ equal in size to the two other ‘attics’, its ‘room’ must also be equal in area to the two other ‘rooms’. Wow! (I got this language from Alexander Bogomolny’s blog post about it, which also tells a story about young Einstein discovering this proof.
Since all three houses are similar (exact same shape, different sizes), the size of the room is some given multiple of the size of the attic. More properly, area(square) = area(triangle), where is the same for all three figures. The square attached to triangle (whose area we will say is also ) has area , similarly for the square attached to triangle . But note that which is the area of the square attached to the triangle labeled . But , and , so and it also equals giving us what we sought:
I stumbled on the article in which this appeared (The Step to Rationality, by R. N. Shepard) while trying to find an answer to a question I have about centroids. I haven’t answered my centroid question yet, but I sure was sending out some google love when I found this.
What I love about this proof is that the triangle stay central in our thoughts throughout, and the focus stays on area, which is what this is really about. It’s all about self-similarity, and that’s what makes it so beautiful.
I think that, even though this proof is simpler in terms of steps than my old favorite, it’s a bit harder to see conceptually. So I may stick with the first one when explaining to students. What do you think?