What is Evidence-based Sentencing?
For several decades, parole and probation departments have been using research-backed assessments to determine the best supervision and treatment strategies for offenders to try and reduce the risk of recidivism. In recent years, state and county justice systems have started to apply these risk and needs assessment tools (RNA’s) to other parts of the criminal process.
Of particular concern is the use of automated tools to determine imprisonment terms. This relatively new practice of applying RNA information into the sentencing process is known as evidence-based sentencing (EBS).
What the Models Do
The different parameters used to determine risk vary by state, and most EBS tools use information that has been central to sentencing schemes for many years such as an offender’s criminal history. However, an increasing amount of states have been utilizing static factors such as gender, age, marital status, education level, employment history, and other demographic information to determine risk and inform sentencing. Especially alarming is the fact that the majority of these risk assessment tools do not take an offender’s particular case into account.
This practice has drawn sharp criticism from Attorney General Eric Holder who says “using static factors from a criminal’s background could perpetuate racial bias in a system that already delivers 20% longer sentences for young black men than for other offenders.” In the annual letter to the US Sentencing Commission, the Attorney General’s Office states that “utilizing such tools for determining prison sentences to be served will have a disparate and adverse impact on offenders from poor communities already struggling with social ills.” Other concerns cite the probable unconstitutionality of using group-based characteristics in risk assessments.
Where the Models Are Used
It is difficult to precisely quantify how many states and counties currently implement these instruments, although at least 20 states have implemented some form of EBS. Some of the states or states with counties that have implemented some sort of EBS (any type of sentencing: parole, imprisonment, etc) are: Pennsylvania, Tennessee, Vermont, Kentucky, Virginia, Arizona, Colorado, California, Idaho, Indiana, Missouri, Nebraska, Ohio, Oregon, Texas, and Wisconsin.
The Role of Race, Education, and Friendship
Overwhelmingly states do not include race in the risk assessments since there seems to be a general consensus that doing so would be unconstitutional. However, even though these tools do not take race into consideration directly, many of the variables used such as economic status, education level, and employment correlate with race. African-Americans and Hispanics are already disproportionately incarcerated and determining sentences based on these variables might cause further racial disparities.
The very socioeconomic characteristics such as income and education level used in risk assessments are the characteristics that are already strong predictors of whether someone will go to prison. For example, high school dropouts are 47 times more likely to be incarcerated than people in their similar age group who received a four-year college degree. It is reasonable to suspect that courts that include education level as a risk predictor will further exacerbate these disparities.
Some states, such as Texas, take into account peer relations and considers associating with other offenders as a “salient problem”. Considering that Texas is in 4th place in the rate of people under some sort of correctional control (parole, probation, etc) and that the rate is 1 in 11 for black males in the United States it is likely that this metric would disproportionately affect African-Americans.
Sonja Starr’s paper
Even so, in some cases, socioeconomic and demographic variables receive significant weight. In her forthcoming paper in the Stanford Law Review, Sonja Starr provides a telling example of how these factors are used in presentence reports. From her paper:
For instance, in Missouri, pre-sentence reports include a score for each defendant on a scale from -8 to 7, where “4-7 is rated ‘good,’ 2-3 is ‘above average,’ 0-1 is ‘average’, -1 to -2 is ‘below average,’ and -3 to -8 is ‘poor.’ Unlike most instruments in use, Missouri’s does not include gender. However, an unemployed high school dropout will score three points worse than an employed high school graduate—potentially making the difference between “good” and “average,” or between “average” and “poor.” Likewise, a defendant under age 22 will score three points worse than a defendant over 45. By comparison, having previously served time in prison is worth one point; having four or more prior misdemeanor convictions that resulted in jail time adds one point (three or fewer adds none); having previously had parole or probation revoked is worth one point; and a prison escape is worth one point. Meanwhile, current crime type and severity receive no weight.
Starr argues that such simple point systems may “linearize” a variable’s effect. In the underlying regression models used to calculate risk, some of the variable’s effects do not translate linearly into changes in probability of recidivism, but they are treated as such by the model.
Another criticism Starr makes is that they often make predictions on an individual based on averages of a group. Starr says these predictions can predict with reasonable precision the average recidivism rate for all offenders who share the same characteristics as the defendant, but that does not make it necessarily useful for individual predictions.
The Future of EBS Tools
The Model Penal Code is currently in the process of being revised and is set to include these risk assessment tools in the sentencing process. According to Starr, this is a serious development because it reflects the increased support of these practices and because of the Model Penal Code’s great influence in guiding penal codes in other states. Attorney General Eric Holder has already spoken against the practice, but it will be interesting to see whether his successor will continue this campaign.
Even if EBS can accurately measure risk of recidivism (which is uncertain according to Starr), does that mean that a greater prison sentence will result in less future offenses after the offender is released? EBS does not seek to answer this question. Further, if knowing there is a harsh penalty for a particular crime is a deterrent to commit said crime, wouldn’t adding more uncertainty to sentencing (EBS tools are not always transparent and sometimes proprietary) effectively remove this deterrent?
Even though many questions remain unanswered and while several people have been critical of the practice, it seems like there is great support for the use of these instruments. They are especially easy to support when they are overwhelmingly regarded as progressive and scientific, something Starr refutes. While there is certainly a place for data analytics and actuarial methods in the criminal justice system, it is important that such research be applied with the appropriate caution. Or perhaps not at all. Even if the tools had full statistical support, the risk of further exacerbating an already disparate criminal justice system should be enough to halt this practice.
Both Starr and Holder believe there is a strong case to be made that the risk prediction instruments now in use are unconstitutional. But EBS has strong advocates, so it’s a difficult subject. Ultimately, evidence-based sentencing is used to determine a person’s sentencing not based on what the person has done, but who that person is.
This is a guest post by Marc Joffe, a former Senior Director at Moody’s Analytics, who founded Public Sector Credit Solutions in 2011 to educate the public about the risk – or lack of risk – in government securities. Marc published an open source government bond rating tool in 2012 and launched a transparent credit scoring platform for California cities in 2013. Currently, Marc blogs for Bitvore, a company which sifts the internet to provide market intelligence to municipal bond investors.
Obama administration officials frequently talk about the need to improve the nation’s infrastructure. Yet new regulations published by the Federal Reserve, FDIC and OCC run counter to this policy by limiting the market for municipal bonds.
On Wednesday, bank regulators published a new rule requiring large banks to hold a minimum level of high quality liquid assets (HQLAs). This requirement is intended to protect banks during a financial crisis, and thus reduce the risk of a bank failure or government bailout. Just about everyone would agree that that’s a good thing.
The problem is that regulators allow banks to use foreign government securities, corporate bonds and even stocks as HQLAs, but not US municipal bonds. Unless this changes, banks will have to unload their municipal holdings and won’t be able to purchase new state and local government bonds when they’re issued. The new regulation will thereby reduce the demand for bonds needed to finance roads, bridges, airports, schools and other infrastructure projects. Less demand for these bonds will mean higher interest rates.
Municipal bond issuance is already depressed. According to data from SIFMA, total municipal bonds outstanding are lower now than in 2009 – and this is in nominal dollar terms. Scary headlines about Detroit and Puerto Rico, rating agency downgrades and negative pronouncements from market analysts have scared off many investors. Now with banks exiting the market, the premium that local governments have to pay relative to Treasury bonds will likely increase.
If the new rule had limited HQLA’s to just Treasuries, I could have understood it. But since the regulators are letting banks hold assets that are as risky as or even riskier than municipal bonds, I am missing the logic. Consider the following:
- No state has defaulted on a general obligation bond since 1933. Defaults on bonds issued by cities are also extremely rare – affecting about one in one thousand bonds per year. Other classes of municipal bonds have higher default rates, but not radically different from those of corporate bonds.
- Bonds issued by foreign governments can and do default. For example, private investors took a 70% haircut when Greek debt was restructured in 2012.
- Regulators explained their decision to exclude municipal bonds because of thin trading volumes, but this is also the case with corporate bonds. On Tuesday, FINRA reported a total of only 6446 daily corporate bond trades across a universe of perhaps 300,000 issues. So, in other words, the average corporate bond trades less than once per day. Not very liquid.
- Stocks are more liquid, but can lose value very rapidly during a crisis as we saw in 1929, 1987 and again in 2008-2009. Trading in individual stocks can also be halted.
Perhaps the most ironic result of the regulation involves municipal bond insurance. Under the new rules, a bank can purchase bonds or stock issued by Assured Guaranty or MBIA – two major municipal bond insurers – but they can’t buy state and local government bonds insured by those companies. Since these insurance companies would have to pay interest and principal on defaulted municipal securities before they pay interest and dividends to their own investors, their securities are clearly more risky than the insured municipal bonds.
Regulators have expressed a willingness to tweak the new HQLA regulations now that they are in place. I hope this is one area they will reconsider. Mandating that banks hold safe securities is a good thing; now we need a more data-driven definition of just what safe means. By including municipal securities in HQLA, bank regulators can also get on the same page as the rest of the Obama administration.
I was was having a wonderful ramen lunch with the mathbabe and, as is all too common when two broad minded Ph.D.’s in math get together, we started talking about the horrible state math education is in for both advanced high school students and undergraduates.
One amusing thing we discovered pretty quickly is that we had independently come up with the same (radical) solution to at least part of the problem: throw out the traditional sequence which goes through first and second year calculus and replace it with a unified probability, statistics, calculus course where the calculus component was only for the smoothest of functions and moreover the applications of calculus are only to statistics and probability. Not only is everything much more practical and easier to motivate in such a course, students would hopefully learn a skill that is essential nowadays: how to separate out statistically good information from the large amount of statistical crap that is out there.
Of course, the downside is that the (interesting) subtleties that come from the proofs, the study of non-smooth functions and for that matter all the other stuff interesting to prospective physicists like DiffEQ’s would have to be reserved for different courses. (We also were in agreement that Gonick’s beyond wonderful“Cartoon Guide To Statistics” should be required reading for all the students in these courses, but I digress…)
The real point of this blog post is based on what happened next: but first you have to know I’m more or less one generation older than the mathbabe. This meant I was both able and willing to preface my next point with the words: “You know when I was young, in one way students were much better off because…” Now it is well known that using this phrase to preface a discussion often poisons the discussion but occasionally, as I hope in this case, some practices from days gone by ago can if brought back, help solve some of today’s educational problems.
By the way, and apropos of nothing, there is a cure for people prone to too frequent use of this phrase: go quickly to YouTube and repeatedly make them watch Monty Python’s Four Yorkshireman until cured:
Anyway, the point I made was that I am a member of the last generation of students who had to use slide rules. Another good reference is: here. Both these references are great and I recommend them. (The latter being more technical.) For those who have never heard of them, in a nutshell, a slide rule is an analog device that uses logarithms under the hood to do (sufficiently accurate in most cases) approximate multiplication, division, roots etc.
The key point is that using a slide rule requires the user to keep track of the “order of magnitude” of the answers— because slide rules only give you four or so significant digits. This meant students of my generation when taking science and math courses were continuously exposed to order of magnitude calculations and you just couldn’t escape from having to make order of magnitude calculations all the time—students nowadays, not so much. Calculators have made skill at doing order of magnitude calculations (or Fermi calculations as they are often lovingly called) an add-on rather than a base line skill and that is a really bad thing. (Actually my belief that bringing back slide rules would be a good thing goes back a ways: when that when I was a Program Director at the NSF in the 90’s, I actually tried to get someone to submit a proposal which would have been called “On the use of a hand held analog device to improve science and math education!” Didn’t have much luck.)
Anyway, if you want to try a slide rule out, alas, good vintage slide rules have become collectible and so expensive— because baby boomers like me are buying the ones we couldn’t afford when we were in high school – but the nice thing is there are lots of sites like this one which show you how to make your own.
Finally, while I don’t think they will ever be as much fun as using a slide rule, you could still allow calculators in classrooms.
Why? Because it would be trivial to have a mode in the TI calculator or the Casio calculator that all high school students seem to use, called “significant digits only.” With the right kind of problems this mode would require students to do order of magnitude calculations because they would never be able to enter trailing or leading zeroes and we could easily stick them with problems having a lot of them!
But calculators really bug me in classrooms and, so I can’t resist pointing out one last flaw in their omnipresence: it makes students believe in the possibility of ridiculously high precision results in the real world. After all, nothing they are likely to encounter in their work (and certainly not in their lives) will ever need (or even have) 14 digits of accuracy and, more to the point, when you see a high precision result in the real world, it is likely to be totally bogus when examined under the hood.
This is a guest post written by Stephanie Yang and reposted from her blog. Stephanie and I went to graduate school at Harvard together. She is now a quantitative analyst living in New York City, and will be joining the data science team at Foursquare next month.
Last week’s hysterical report by the Daily Show’s Samantha Bee on federally funded penis pumps contained a quote which piqued our quantitative interest. Listen carefully at the 4:00 mark, when Ilyse Hogue proclaims authoritatively:
“Statistics show that probably some our members of congress have a vested interested in having penis pumps covered by Medicare!”
Ilya’s wording is vague, and intentionally so. Statistically, a lot of things are “probably” true, and many details are contained in the word “probably”. In this post we present a simple statistical model to clarify what Ilya means.
First we state our assumptions. We assume that penis pumps are uniformly distributed among male Medicare recipients and that no man has received two pumps. These are relatively mild assumptions. We also assume that what Ilya refers to as “members of Congress [with] a vested interested in having penis pumps covered by Medicare,” specifically means male member of congress who received a penis pump covered by federal funds. Of course, one could argue that female members congress could also have a vested interested in penis pumps as well, but we do not want to go there.
Now the number crunching. According to the report, Medicare has spent a total of $172 million supplying penis pumps to recipients, at “360 bucks a pop.” This means a total of 478,000 penis pumps bought from 2006 to 2011.
45% of the current 49,435,610 Medicare recipients are male. In other words, Medicare bought one penis pump for every 46.5 eligible men. Inverting this, we can say that 2.15% of male Medicare recipients received a penis pump.
There are currently 128 members of congress (32 senators plus 96 representatives) who are males over the age of 65 and therefore Medicare-eligible. The probability that none of them received a federally funded penis pump is:
In other words, the chances of at least one member of congress having said penis pumps is 93.8%, which is just shy of the 95% confidence that most statisticians agree on as significant. In order to get to 95% confidence, we need a total of 138 male members of congress who are over the age of 65, and this has not happened yet as of 2014. Nevertheless, the estimate is close enough for us to agree with Ilya that there is probably someone member of congress who has one.
Is it possible that there two or more penis pump recipients in congress? We did notice that Ilya’s quote refers to plural members of congress. Under the assumptions laid out above, the probability of having at least two federally funded penis pumps in congress is:
Again, we would say this is probably true, though not nearly with the same amount of confidence as before. In order to reach 95% confidence that there are two or moreq congressional federally funded penis pump, we would need 200 or more Medicare-eligible males in congress, which is unlikely to happen anytime soon.
Note: As a corollary to these calculations, I became the first developer in the history of mankind to type the following command:
git merge --squash penispump.
Today’s guest post was written by Amie, who describes herself as a mom of a 9 and a 14-year-old, mathematician, and bigmouth.
Nota bene: this was originally posted on Facebook as a spontaneous rant. Please don’t miscontrue it as an academic argument.
Time for a rant. I’ll preface this by saying that while my kids are creative, beautiful souls, so are many (perhaps all) children I’ve met, and it would be the height of arrogance to take credit for that as a parent. But one thing my husband and I can take credit for are their good manners, because that took work to develop.
The first phrase I taught me daughter was “thank you,” and it’s been put to good use over the years. I’m also loathe to tell other parents what to do, but this is an exception: teach your fucking kids to say “please” and “thank you”. If you are fortunate to visit another country, teach them to say “please” and “thank you” in the native language.
After a week in paradise at a Club Med in Mexico, I’m at some kind of breaking point with rude rich people and their spoiled kids. And that includes the Europeans. Maybe especially the Europeans. What is it that when you’re in France everyone’s all “thank you and have a nice day” but when these petit bourgeois assholes come to Cancun they treat Mexicans like nonhumans? My son held the door for a face-lifted Russian lady today who didn’t even say thank you.
Anyway, back to kids: I’m not saying that you should suppress your kids’ nature joie de vivre and boisterous, rambunctious energy (though if that’s what they’re like, please keep them away from adults who are not in the mood for it). Just teach them to treat other people with basic respect and courtesy. That means prompting them to say “please,” “thank you,” and “nice to meet you” when they interact with other people.
Jordan Ellenberg just posted how a huge number of people accepted to the math Ph.D. program at the University of Wisconsin never wrote to tell him that they had accepted other offers. When other people are on a wait list!
Whose fault is this? THE PARENTS’ FAULT. Damn parents. Come on!!
P.S. Those of you who have put in the effort to raise polite kids: believe me, I’ve noticed. So has everyone else.
This is a guest post by Leopold Dilg.
There’s little chance we can underestimate our American virtues, since our overlords so seldom miss an opportunity to point them out. A case in point – in fact, le plus grand du genre, though my fingers tremble as I type that French expression, for reasons I’ll explain soon enough – is the Cadillac commercial that interrupted the broadcast of the Olympics every few minutes.
A masterpiece of casting and directing and location scouting, the ad follows a middle-aged man, muscular enough but not too proud to show a little paunch – manifestly a Master of the Universe – strutting around his chillingly modernist $10 million vacation house (or is it his first or fifth home? no matter), every pore oozing the manly, smirky bearing that sent Republican country-club women swooning over W.
It starts with Our Hero, viewed from the back, staring down his infinity pool. He pivots and stares down the viewer. He shows himself to be one of the more philosophical species of the MotU genus. “Why do we work so hard?” he puzzles. “For this? For stuff?….” We’re thrown off balance: Will this son of Goldman Sachs go all Walden Pond on us? Fat chance.
Now, still barefooted in his shorts and polo shirt, he’s prowling his sleak living room (his two daughters and stay-at-home wife passively reading their magazines and ignoring the camera, props in his world no less than his unused pool and The Car yet to be seen) spitting bile at those foreign pansies who “stop by the café” after work and “take August off!….OFF!” Those French will stop at nothing.
“Why aren’t YOU like that,” he says, again staring us down and we yield to the intimidation. (Well gee, sir, of course I’m not. Who wants a month off? Not me, absolutely, no way.) “Why aren’t WE like that” he continues – an irresistible demand for totalizing merger. He’s got us now, we’re goose-stepping around the TV, chanting “USA! USA! No Augusts off! No Augusts off!”
No, he sneers, we’re “crazy, hardworking believers.” But those Frogs – the weaklings who called for a double-check about the WMDs before we Americans blasted Iraqi children to smithereens (woops, someone forgot to tell McDonalds, the official restaurant of the U.S. Olympic team, about the Freedom Fries thing; the offensive French Fries are THERE, right in our faces in the very next commercial, when the athletes bite gold medals and the awe-struck audience bites chicken nuggets, the Lunch of Champions) – might well think we’re “nuts.”
“Whatever,” he shrugs, end of discussion, who cares what they think. “Were the Wright Brothers insane? Bill Gates? Les Paul?… ALI?” He’s got us off-balance again – gee, after all, we DO kinda like Les Paul’s guitar, and we REALLY like Ali.
Of course! Never in a million years would the hip jazz guitarist insist on taking an August holiday. And the imprisoned-for-draft-dodging boxer couldn’t possibly side with the café-loafers on the WMD thing. Gee, or maybe…. But our MotU leaves us no time for stray dissenting thoughts. Throwing lunar dust in our eyes, he discloses that WE were the ones who landed on the moon. “And you know what we got?” Oh my god, that X-ray stare again, I can’t look away. “BORED. So we left.” YEAH, we’re chanting and goose-stepping again, “USA! USA! We got bored! We got bored!”
Gosh, I think maybe I DID see Buzz Aldrin drumming his fingers on the lunar module and looking at his watch. “But…” – he’s now heading into his bedroom, but first another stare, and pointing to the ceiling – “…we got a car up there, and left the keys in it. You know why? Because WE’re the only ones goin’ back up there, THAT’s why.” YES! YES! Of COURSE! HE’S going back to the moon, I’M going back to the moon, YOU’RE going back to the moon, WE’RE ALL going back to the moon. EVERYONE WITH A U.S. PASSPORT is going back to the moon!!
Damn, if only the NASA budget wasn’t cut after all that looting by the Wall Street boys to pay for their $10 million vacation homes, WE’D all be going to get the keys and turn the ignition on the rover that’s been sitting 45 years in the lunar garage waiting for us. But again – he must be reading our mind – he’s leaving us no time for dissent, he pops immediately out of his bedroom in his $12,000 suit, gives us the evil eye again, yanks us from the edge of complaint with a sharp, “But I digress!” and besides he’s got us distracted with the best tailoring we’ve ever seen.
Finally, he’s out in the driveway, making his way to the shiny car that’ll carry him to lower Manhattan. (But where’s the chauffer? And don’t those MotUs drive Mazerattis and Bentleys? Is this guy trying to pull one over on the suburban rubes who buy Cadillacs stupidly thinking they’ve made it to the big time?)
Now the climax: “You work hard, you create your own luck, and you gotta believe anything is possible,” he declaims.
Yes, we believe that! The 17 million unemployed and underemployed, the 47 million who need food stamps to keep from starving, the 8 million families thrown out of their homes – WE ALL BELIEVE. From all the windows in the neighborhood, from all the apartments across Harlem, from Sandy-shattered homes in Brooklyn and Staten Island, from the barren blast furnaces of Bethlehem and Youngstown, from the foreclosed neighborhoods in Detroit and Phoenix, from the 70-year olds doing Wal-mart inventory because their retirement went bust, from all the kitchens of all the families carrying $1 trillion in college debt, I hear the national chant, “YOU MAKE YOUR OWN LUCK! YOU MAKE YOUR OWN LUCK!”
And finally – the denouement – from the front seat of his car, our Master of the Universe answers the question we’d all but forgotten. “As for all the stuff? That’s the upside of taking only two weeks off in August.” Then the final cold-blooded stare and – too true to be true – a manly wink, the kind of wink that makes us all collaborators and comrades-in-arms, and he inserts the final dagger: “N’est-ce pas?”
This is a guest post by Manya Raman-Sundström.
If you talk to a mathematician about what she or he does, pretty soon it will surface that one reason for working those long hours on those difficult problems has to do with beauty.
Whatever we mean by that term, whether it is the way things hang together, or the sheer simplicity of a result found in a jungle of complexity, beauty – or aesthetics more generally—is often cited as one of the main rewards for the work, and in some cases the main motivating factor for doing this work. Indeed, the fact that a proof of known theorem can be published just because it is more elegant is one evidence of this fact.
Mathematics is beautiful. Any mathematician will tell you that. Then why is it that when we teach mathematics we tend not to bring out the beauty? We would consider it odd to teach music via scales and theory without ever giving children a chance to listen to a symphony. So why do we teach mathematics in bits and pieces without exposing students to the real thing, the full aesthetic experience?
Of course there are marvelous teachers out there who do manage to bring out the beauty and excitement and maybe even the depth of mathematics, but aesthetics is not something we tend to value at a curricular level. The new Common Core Standards that most US states have adopted as their curricular blueprint do not mention beauty as a goal. Neither do the curriculum guidelines of most countries, western or eastern (one exception is Korea).
Mathematics teaching is about achievement, not about aesthetic appreciation, a fact that test-makers are probably grateful for – can you imagine the makeover needed for the SAT if we started to try to measure aesthetic appreciation?
Why Does Beauty Matter?
First, it should be a bit troubling that our mathematics classrooms do not mirror practice. How can young people make wise decisions about whether they should continue to study mathematics if they have never really seen mathematics?
Second, to overlook the aesthetic components of mathematical thought might be to preventing our children from developing their intellectual capacities.
In the 1970s Seymour Papert , a well-known mathematician and educator, claimed that scientific thought consisted of three components: cognitive, affective, and aesthetic (for some discussion on aesthetics, see here).
At the time, research in education was almost entirely cognitive. In the last couple decades, the role of affect in thinking has become better understood, and now appears visibly in national curriculum documents. Enjoying mathematics, it turns out, is important for learning it. However, aesthetics is still largely overlooked.
Recently Nathalie Sinclair, of Simon Frasier University, has shown that children can develop aesthetic appreciation, even at a young age, somewhat analogously to mathematicians. But this kind of research is very far, currently, from making an impact on teaching on a broad scale.
Once one starts to take seriously the aesthetic nature of mathematics, one quickly meets some very tough (but quite interesting!) questions. What do we mean by beauty? How do we characterise it? Is beauty subjective, or objective (or neither? or both?) Is beauty something that can be taught, or does it just come to be experienced over time?
These questions, despite their allure, have not been fully explored. Several mathematicians (Hardy, Poincare, Rota) have speculated, but there is no definite answer even on the question of what characterizes beauty.
To see why these questions might be of interest to anyone but hard-core philosophers, let’s look at an example. Consider the famous question, answered supposedly by Gauss, of the sum of the first n integers. Think about your favorite proof of this. Probably the proof that did NOT come to your mind first was a proof by induction:
Prove that S(n) = 1 + 2 + 3 … + n = n (n+1) /2
S(k + 1) = S(k) + (k + 1)
= k(k + 1)/2 + 2(k + 1)/2
= k(k + 1)/2 + 2(k + 1)/2
= (k + 1)(k + 2)/2.
Now compare this proof to another well known one. I will give the picture and leave the details to you:
Does one of these strike you as nicer, or more explanatory, or perhaps even more beautiful than the other? My guess is that you will find the second one more appealing once you see that it is two sequences put together, giving an area of n (n+1), so S(n) = n (n+1)/2.
Note: another nice proof of this theorem, of course, is the one where S(n) is written both forwards and backwards and added. That proof also involves a visual component, as well as an algebraic one. See here for this and a few other proofs.
Beauty vs. Explanation
How often do we, as teachers, stop and think about the aesthetic merits of a proof? What is it, exactly, that makes the explanatory proof more attractive? In what way does the presentation of the proof make the key ideas accessible, and does this accessibility affect our sense of understanding, and what underpins the feeling that one has found exactly the right proof or exactly the right picture or exactly the right argument?
Beauty and explanation, while not obvious related (see here), might at least be bed-fellows. It may be the case that what lies at the bottom of explanation — a feeling of understanding, or a sense that one can ”see” what is going on — is also related to the aesthetic rewards we get when we find a particularly good solution.
Perhaps our minds are drawn to what is easiest to grasp, which brings us back to central questions of teaching and learning: how do we best present mathematics in a way that makes it understandable, clear, and perhaps even beautiful? These questions might all be related.
Workshop on Math Beauty
This March 10-12, 2014 in Umeå, Sweden, a group will gather to discuss this topic. Specifically, we will look at the question of whether mathematical beauty has anything to do with mathematical explanation. And if so, whether the two might have anything to do with visualization.
If this discussion peaks your interest at all, you are welcome to check out my blog on math beauty. There you will find a link to the workshop, with a fantastic lineup of philosophers, mathematicians, and mathematics educators who will come together to try to make some progress on these hard questions.
Thanks to Cathy, the always fabulous mathbabe, for letting me take up her space to share the news of this workshop (and perhaps get someone out there excited about this research area). Perhaps she, or you if you have read this far, would be willing to share your own favorite examples of beautiful mathematics. Some examples have already been collected here, please add yours.