tag:blogger.com,1999:blog-2516188730140164076Mon, 18 Dec 2017 07:33:02 +0000mathematical thinkingCommon CoreMOOCmathematics educationFibonacciLeonardo of PisaJo Boaleralgebraonline educationCourseraDragonBoxKhan AcademyMOOCsPisaSteve Jobseducational video gamesgame-based learningproblem solvingAlpine Road TrailApple MacintoshHindu-arabic arithmeticJordan EllenbergKeith DevlinMultiplicationPi DayStanford Universityapplications of mathematicsarithmeticconceptual understandingeducational startupsgolden ratioinfinitymathematical modelingmountain bikingnumber sensepiproofsquantitative literacyscience of patternstransition courseAdolf ZeisingAlex GibneyAndrew HackerAndrew NgArea of a circle formulaAronszajn treeAsa ButterfieldAspen Ideas FestivalBedtime MathBrexitCarl SaganCreateSpaceDan MeyerDaphne KollerDavid LeonhardtDerek MullerDowns paradoxETSEdge questionEinstein’s energy equationEric MazurErlwangerEuler’s IdentityEuler’s equationEuropean CommunityExploding dotsFibonacci sequenceFinnish educationGOFMGeorg CantorGeorge BooleGrant SandersonGregory’s SeriesH-STARHerman MehtaHilbert's HotelHypercubesIMOITSInquiry-Based LearningIs algebra necessaryJames Paul GeeJames TantonJohn BrockmanJon BarwiseJonathan BorweinK-12 educationKarim AniKen RobinsonKool AidKönig’s LemmaLaura OverdeckLearnLabLet’s Make a DealLiber abbaciLiping MaM.C. EscherMAAMTT2KMarilyn BurnsMath and musicMath moviesMonty HallMonty Hall ProblemMonument Valley gameMoore MethodNSANumberphileOECDPIAACPLCPaul CohenPeer InstructionPhil DaroPolyaPythagoras’ equationR L MooreRiccardiana LibraryRiemann’s HypothesisSERPSanfordSebastian ThrunSergio CorreaSimon's ParadoxStandard algorithmStatisticsSteven StrogatzStuxnetSugata MitraSupreme CourtTIMSSTeachingTest FlightUdacityUnited KingdomUniversity of ChicagoUri TreismanVeritasiumWikipedia contributionWisconsin state electionsWorld Science FestivalWorld Wide WebWuzzit TroubleYouCubedZeta functionabstractional-Khwarizmiamygdalaan approach that sucksanalytic continuationanalytic functionsanswer gettingappreciation of mathematicsbeauty in mathematicsboolean algebraboolean logiccalculuscollaborative workingcomplex numberscomputer science educationcomputer viruscomputer virusescomputers and mathematicscountable setcyberwarfarecyclingdistance learningdot diagramseducation researcheducational appseducational strandseducational tsunamieducational videoselection mathelectionselementary arithmeticequityexceptional performanceexperimental mathematicsfailureforcinggeometrygerrymanderinggifted and talentedgooglehigher educationinfinite seriesinfinite sumsinstructioninstructional videointelligence analysislarge cardinalslearning outcomeslearning scienceliteracymachine gradingmath and dancemath appsmath competitionsmath learningmath learning appmath learning video gamemath on TVmath video gamesmathematical cognitionmathematical mindsetmathematical modelmathematical proofsmathematical writingmathematicsmathematics and societymathematics outreachmathematics transition coursemathematics visualizationmemesmilkmind setmultiple-choice questionsmusicnature of mathematicsnumeracyonline learningonline magazinesparadox of votingpeer evaluationpercentagespie chartsplace-value arithmeticpop quizzespovertyprobability theoryprocedural masteryproof techniquespublish-on-demandquantitative reasoning requirementredistrictingrepeated additionrepeated addition (not)representations of mathematicsscience videoself-publishingset theorysoft mathematicstalent searchtextbookstreesuncountable setuniversitiesuniversity administrationuniversity educationuniversity researchvisual representationsvotingDevlin's AngleDevlin's Angle is a monthly column sponsored by the Mathematical Association of America. Mathematician Keith Devlin is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guyon NPR's Weekend Edition.http://devlinsangle.blogspot.com/noreply@blogger.com (Mathematical Association of America)Blogger77125tag:blogger.com,1999:blog-2516188730140164076.post-4646632644831926229Thu, 14 Dec 2017 20:37:00 +00002017-12-14T15:37:14.152-05:00mathematical cognitionpercentagespie chartsClash of representations<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-ZhAuLxqHGCk/Wi7xCCFyiiI/AAAAAAAAK_c/oWSNle3TL5IZbcWyEsU8cmZ_n25JO2N4gCLcBGAs/s1600/Original_tweet.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1326" data-original-width="960" height="400" src="https://2.bp.blogspot.com/-ZhAuLxqHGCk/Wi7xCCFyiiI/AAAAAAAAK_c/oWSNle3TL5IZbcWyEsU8cmZ_n25JO2N4gCLcBGAs/s400/Original_tweet.jpg" width="289" /></a></div><br />The pie chart in the above tweet jumped out of the page when it appeared in my twitter feed on September 14. My initial shock at seeing the figure 1% attached to a region of the pie chart that was evidently almost 25% of the total area of the disk did not last long, of course, since the accompanying text made it clear what the diagram was intended to convey. The 1% label referred to the section of the population being discussed, whereas the pie-chart indicated the share of taxes paid by that group. Indeed, the image was an animated GIF; when I clicked on it, the region labeled “1%” shrank, culminating with the chart on the right in the image shown below:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-n1mBg8RGILw/Wi7yQdbx0AI/AAAAAAAAK_0/WzRnrTYoETYWdxVcKRW2mOmXsCRNAcGswCLcBGAs/s1600/Pie_GIF.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="474" data-original-width="851" height="221" src="https://3.bp.blogspot.com/-n1mBg8RGILw/Wi7yQdbx0AI/AAAAAAAAK_0/WzRnrTYoETYWdxVcKRW2mOmXsCRNAcGswCLcBGAs/s400/Pie_GIF.jpg" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"></div>But here’s the thing. Even after I had figured out what the chart was intended to convey, I still found it confusing. I wondered if a lay-reader, someone who is not a professional mathematician, would manage to parse out the intended meaning. It was not long before I found out. The image below shows one of the tweets that appeared in response less than an hour later:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-g7-5kLuiETo/Wi7xl0FPZ2I/AAAAAAAAK_o/haf-Gqyva8MP-61rhIEnwiUHsDMPjEoWwCLcBGAs/s1600/Reply.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="169" data-original-width="925" height="71" src="https://2.bp.blogspot.com/-g7-5kLuiETo/Wi7xl0FPZ2I/AAAAAAAAK_o/haf-Gqyva8MP-61rhIEnwiUHsDMPjEoWwCLcBGAs/s400/Reply.jpg" width="400" /></a></div>As I had suspected, a common reaction was to dismiss the chart as yet another example of a bad data visualization created by an innumerate graphics designer. Indeed, that had been my initial reaction. But this particular example is more interesting. Yes, it is a bad graphic, for the simple reason that it does not convey the intended message. But not because of the illustrator’s innumeracy. In fact, numerically, it appears to be as accurate as you can get with a pie chart. The before and after charts do seem to have regions whose areas correspond to the actual data on the tax-payer population. <br /><br />This example was too good to pass up as an educational tool: asking a class to discuss what the chart is intended to show, could lead to a number of good insights into how mathematics can help us understand the world, while at the same time having the potential to mislead. I was tempted to write about it in my October post, but wondered if I should delay a couple of months to avoid an example that was at the heart of a current, somewhat acrimonious party-political debate. As it turned out, the September 30 death of the game-show host Monty Hall resolved the issue for me—I had to write about that—and then November presented another “must do” story (the use of mathematics in election jerrymandering). So this month, with the background political, tax votes now a matter of historical record, I have my first real opportunity to run this story. <br /><br />The two-month delay brought home to me just how problematic this particular graphic is. Even knowing in advance what the issue is, I still found I had to concentrate to “see” the chart as conveying the message intended. That “1%” label continued to clash with the relative area of the labeled region. <br /><br />It’s a bit like those college psychology-class graphics that show two digits in different font sizes, and ask you to point to the digit that represents the bigger integer. If the font sizes clash with the sizes of the integers, you take measurably longer to identify the correct one, as shown below:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-yx8NR9Vkn7A/Wi7x4Gw6ioI/AAAAAAAAK_s/YsP1uhGBpuslAZ2iCfYGR5xqwVwxoNn2ACLcBGAs/s1600/Stroop.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="393" data-original-width="759" height="206" src="https://1.bp.blogspot.com/-yx8NR9Vkn7A/Wi7x4Gw6ioI/AAAAAAAAK_s/YsP1uhGBpuslAZ2iCfYGR5xqwVwxoNn2ACLcBGAs/s400/Stroop.jpg" width="400" /></a></div>For me, the really big take-home lesson from the tax-proposal graphic is the power of two different mathematical representations of proportions: pie charts and numerical percentages. Each, on its own, is instant. In the case of the pie chart, the representation draws on the innate human cognitive ability to judge relative areas in simple, highly symmetrical figures like circular disks or rectangles. With percentages, there is some initial learning required—you have to understand percentages—but once you have done that, you know instantly what is meant by figures such as “5%” or “75%." <br /><br />But how do you get that understanding of the meaning of numerical percentages? For most of us (I suspect all of us), it comes from being presented (as children) with area examples like pie charts and subdivided rectangles. This sets us up to be confused, bigly, by examples where those two representations are used in the same graphic but with the percentage representing something other than the area of the segment (or what that area is intended to represent). <br /><br />The message then, from this particular example—or at least the message I got from it—is that powerful graphics are like any powerful tool, their power for good depends on using them wisely; if used incorrectly, they can confuse and mislead. And make no mistake about it, numbers are incredibly powerful tools. Their invention <b><i>alone</i></b> is by far the greatest mathematical invention in human history. That’s why in every nation in the world, math is the only mandated school subject apart from the native language. <br /><br />http://devlinsangle.blogspot.com/2017/12/clash-of-representations.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-1529472912946980232Thu, 16 Nov 2017 21:40:00 +00002017-12-11T15:52:16.423-05:00electionsgerrymanderingredistrictingSupreme CourtvotingWisconsin state electionsMathematics and the Supreme CourtAmerican courts have never appeared to be very receptive to mathematical arguments, in large part, some (including me) have assumed, because many judges do not feel confident evaluating mathematical reasoning and, in the case of jury trials, no doubt because they worry that unscrupulous, math-savvy lawyers could use formulas and statistics to fool jury members. There certainly have been some egregious examples of this, particularly when bogus probability arguments have been presented. Indeed, one classic misuse of conditional probability is now known as the “<a href="https://en.wikipedia.org/wiki/Prosecutor%27s_fallacy" target="_blank">prosecutor’s fallacy</a>."<br /><br />Another example where the courts have trouble with probability is in cases involving DNA profiling, particularly Cold Hit cases, where a DNA profile match is the only hard evidence against a suspect. I myself have been asked to provide expert testimony in some such cases, and I wrote about the issue in this column in <a href="https://www.maa.org/external_archive/devlin/devlin_09_06.html" target="_blank">September</a> and <a href="https://www.maa.org/external_archive/devlin/devlin_10_06.html" target="_blank">October</a>of 2006. <br /><br />In both kinds of case, the courts have good reason to proceed with caution. The prosecutor’s fallacy is an easy one to fall into, and with Cold Hit DNA identification there is a real conflict between frequentist and Bayesian probability calculations. In neither case, however, should the courts try to avoid the issue. When evidence is presented, the court needs to have as accurate an assessment as possible as to its reliability or veracity. That frequently has to be in the form of a probability estimate. <br /><br />Now the courts are facing another mathematical conundrum. And this time, the case has landed before the US Supreme Court. It is a case that reaches down to the very foundation of our democratic system: How we conduct our elections. Not how we use vote counts to determine winners, although that is also mathematically contentious, as I wrote about in this column in <a href="https://www.maa.org/external_archive/devlin/devlin_11_00.html" target="_blank">November</a> of 2000, just before the Bush v Gore Presidential Election outcome ended up before the Supreme Court. Rather, the issue before the Court this time is how states are divided up into electoral districts for state elections. <br /><br />How a state carves up voters into state electoral districts can have a huge impact on the outcome. In six states, Alaska, Arizona, California, Idaho, Montana, and Washington, the apportioning is done by independent redistricting commissions. This is generally regarded—at least by those who have studied the issue—as the least problematic approach. In seven other states, Arkansas, Colorado, Hawaii, Missouri, New Jersey, Ohio, and Pennsylvania, politician commissions draw state legislative district maps. In the remaining 37 states, the state legislatures themselves are responsible for state legislative redistricting. And that is where the current problem arises. <br /><br />There is, of course, a powerful temptation for the party in power to redraw the electoral district maps to favor their candidates in the next election. And indeed, in the states where the legislatures draw the maps, both major political parties have engaged in that practice. One of the first times this occurred was in 1812, when Massachusetts governor Elbridge Gerry redrew district boundaries to help his party in an upcoming senate election. A journalist at the <i>Boston Gazette</i> observed that one of the contrived districts in Gerry’s new map looked like a giant salamander, and gave such partisan redistricting a name, combining <i>Gerry</i> and <i>mander</i> to create the new word <i>gerrymander</i>. Though Gerry lost his job over his sleight-of- hand, his redistricting did enable his party to take over the state senate. And the name stuck. <br /><br />Illegality of partisan gerrymandering is generally taken to stem from the 14th Amendment, since it deprives the smaller party of the equal protection of the laws, but it has also been argued to be, in addition, a 1st Amendment issue—namely an apportionment that has the purpose and effect of burdening a group of voters’ representational rights. <br /><br />In 1986, the Supreme Court issued a ruling that partisan gerrymandering, if extreme enough, is unconstitutional, but it has yet to throw out a single redistricting map. In large part, the Supreme Court’s inclination to stay out of the redistricting issue is based on a recognition that both parties do it, and over time, any injustices cancel out, as least numerically. Historically, this was, generally speaking, true. Attempts to gerrymander have tended to favor both parties to roughly the same extent. But in 2012, things took a dramatic turn with a re-districting process carried out in Wisconsin. <br /><br />That year, the recently elected Republican state legislature released a re-districting map generated using a sophisticated mathematical algorithm running on a powerful computer. And that map was in an altogether new category. It effectively guaranteed Republican majorities for the foreseeable future. The Democrat opposition cried foul, a Federal District Court agreed with them, and a few months ago the case found its way to the Supreme Court. <br /><br />That the Republicans come across as the bad actors in this particular case is likely just an accident of timing; they happened to come to power at the very time when political parties were becoming aware of what could be done with sophisticated algorithms. If history is any guide, either one of the two main parties would have tried to exploit the latest technology sooner or later. In any event, with mathematics at the heart of the new gerrymandering technique, the only way to counter it may be with the aid of equally sophisticated math. <br /><br />The most common technique used to gerrymander a district is called “packing and cracking." In packing, you cram as many of the opposing party’s voters as possible into a small number of “their” districts where they will win with many more votes than necessary. In cracking, you spread opposing party’s voters across as many of “your” districts as possible so there are not enough votes in any one of those districts to ever win there. <br /><br />A form of packing and cracking arises naturally when better-educated liberal-leaning voters move into in cities and form a majority, leaving those in rural areas outnumbered by less-educated, more conservative-leaning voters. (This is thought to be one of the factors that has led to the increasing polarization in American politics.) Solving that problem is, of course, a political one for society as a whole, though mathematics can be of assistance by helping to provide good statistical data. Not so with partisan gerrymandering, where mathematics has now created a problem that had not arisen before, for which mathematics may of necessity be part of the solution. <br /><br />When Republicans won control of Wisconsin in 2010, they used a sophisticated computer algorithm to draw a redistricting map that on the surface appeared fair—no salamander-shaped districts—but in fact was guaranteed to yield a Republican majority even if voter preferences shifted significantly. Under the new map, in the 2012 election, Republican candidates won 48 percent of the vote, but 60 of the state’s 99 legislative seats. The Democrats’ 51 percent that year translated into only 39 seats. Two years later, when the Republicans won the same share of the vote, they ended up with 63 seats—a 24-seat differential. <br /><br />Recognizing what they saw as a misuse of mathematics to undermine the basic principles of American democracy, a number of mathematicians around the country were motivated to look for ways to rectify the situation. There are really two issues to be addressed. One is to draw fair maps—a kind of “positive gerrymandering.” The other is to provide reliable evidence to show that a particular map has been intentionally drawn to favor one party over another, if such occurs, and moreover to do so in a way that the courts can understand and accept. Neither issue is easy to solve, and without mathematics, both are almost certainly impossible. <br /><br />For the first issue, a 2016 Supreme Court ruling gave a hint about what kind of fairness measure it might look kindly on: one that captures the notion of “partisan symmetry,” where each party has an equal opportunity to convert its votes into seats. The Wisconsin case now presents the Supreme Court with the second issue. <br /><br />When, last year, a Federal District Court in Wisconsin threw out the new districting map, they cited both the 1st and 14th Amendments. It was beyond doubt, the court held, that the new maps were “designed to make it more difficult for Democrats, compared to Republicans, to translate their votes into seats.” The court rejected the Republican lawmakers’ claim that the discrepancy between vote share and legislative seats was due simply to political geography. The Republicans had argued that Democratic voters are concentrated in urban areas, so their votes have an impact on fewer races, while Republicans are spread out across the state. But, while that is true, geography alone does not explain why the Wisconsin maps are so skewed. <br /><br />So, how do you tell if a district is gerrymandered? One way, that has been around for some time, is to look at the geographical profile. The <i>gerrymandering score</i>, G, is defined by:<br /><i>G</i> = <i>gP/A</i>, where<br />g: the district’s boundary length, minus natural boundaries (like coastlines and rivers)<br />P: the district’s total perimeter<br />A: the district’s area<br />The higher the score, the wilder is the apportionment as a geographic region, and hence the more likely to have been gerrymandered. <br /><br />That approach is sufficiently simple and sensible to be acceptable to both society and the courts, but unfortunately does not achieve the desired aim of fairness. And, more to the point in the Wisconsin case, use of sophisticated computer algorithms can draw maps that have a low gerrymandering score and yet are wildly partisan. <br /><br />The Wisconsin Republicans’ algorithm searched through thousands of possible maps looking for one that would <b><i>look reasonable</i></b> according to existing criteria, but would favor Republicans <i><b>no matter what the election day voting profile might look like</b></i>. As such, it would be a statistical <i><b>outlier</b></i>. To find evidence to counter that kind of approach, you have to look at the results the districting produces when different voting profiles are fed into it. <br /><br />One promising way to identify gerrymandering is with a simple mathematical formula suggested in 2015, called the “<a href="https://www.brennancenter.org/sites/default/files/legal-work/How_the_Efficiency_Gap_Standard_Works.pdf" target="_blank">efficiency gap</a>." It was the use of this measure that caused, at least in part, the Wisconsin map to be struck down by the court. It is a simple idea—and as I noted, simplicity is an important criterion, if it is to stand a chance of being accepted by society and the courts. <br /><br />You can think of a single elector’s vote as being “wasted” if it is cast in a district where their candidate loses or it is cast in a district where their candidate would have won there anyway. The efficiency gap measures those “wasted” votes. For each district, you total up the number of votes the winning candidate receives in excess of what it would have taken to elect them in that district, and you total up the number of votes the losing candidate receives. Those are the two parties’ “wasted votes” for that district. <br /><br />You then calculate the difference between those “wasted-vote” totals for each of the two parties, and divide the answer by the total number of votes in the state. This yields a single percentage figure: the <b><i>efficiency gap</i></b>. If that works out to be greater than 7%, the systems developers suggest, the districting is unfair. <br /><br />By way of an example, let’s see what the efficiency gap will tell us about the last Congressional election. In particular, consider Maryland’s 6 th Congressional district, which was won by the Democrats. It requires 159K votes to win. In the last election, there were 186K Democrat votes, so 186K – 159K = 26K Democrat votes were “wasted,” and 133K Republican votes, all of which were “wasted.” <br /><br />In Maryland as a whole, there were 510K Democrat votes “wasted” and 789K Republican votes “wasted.” So, statewide, there was a net “waste” of 789K – 510K = 279K Republican votes. <br /><br />There were 2,598M votes cast in total. So the efficiency gap is 279K/2598K = 10.7% in favor of the Democrats. <br /><br />I should note, however, that the gerrymandering problem is currently viewed as far more of a concern in state elections than in congressional races. Last year, two social scientists published <a href="http://www-personal.umich.edu/~jowei/gerrymandering.pdf" target="_blank">the results</a> they obtained using computer simulations to measure the extent of intentional gerrymandering in congressional district maps across most of the 50 states. They found that on the national level, it mostly canceled out between the parties. So banning only intentional gerrymandering would likely have little effect on the partisan balance of the U.S. House of Representatives. The efficiency gap did, however, play a significant role in the Wisconsin court’s decision. <br /><br />Another approach, developed by a team at <a href="https://services.math.duke.edu/projects/gerrymandering/" target="_blank">Duke University</a>, takes aim at the main idea behind the Wisconsin redistricting algorithm—searching through many thousands of possible maps looking for ones that met various goals set by the creators, any one of which would, of necessity, be a <b><i>statistical outlier</i></b>. To identify a map that has been obtained in this way, you subject it to many thousands of random tweaks. If the map is indeed an outlier, the vast majority of tweaks will yield a fairly unremarkable map. So, you compare the actual map with all those thousands of seemingly almost identical, and apparently reasonable, variations you have generated from it. If the actual map produces significantly different election results from all the others, when presented with a range of different statewide voting profiles, you can conclude that it is indeed an “outlier” — a map that could only have been chosen to deliberately subvert the democratic process. <br /><br />And this is where we—and the Supreme Court—are now. We have a problem for our democracy created using mathematics. Mathematicians looking for mathematical ways to solve it, and there are already two candidate “partisan gerrymandering test” in the arena. Historically, the Supreme Court has proven resistant to allowing math into the courtroom. But this time, it looks like they may have no choice. At least as long as state legislatures continue to draw the districting maps. Maybe the very threat of having to deal with mathematical formulas and algorithms will persuade the Supreme Court to recommend that Congress legislates to enforce all states to use independent commissions to draw the districting maps. Legislation under pain of math. We will know soon enough. <br /><br />http://devlinsangle.blogspot.com/2017/11/mathematics-and-supreme-court.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-3498933541202426241Wed, 11 Oct 2017 21:45:00 +00002017-10-12T09:37:29.227-04:00Carl SaganLet’s Make a DealMonty HallMonty Hall Problemprobability theoryMonty Hall may now rest in peace, but his problem will continue to frustrate<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-uMMlSwZYqjw/Wd6Q2LQyGXI/AAAAAAAAK7w/bNey7uxYHNQh1ducy7jasljp7VEP915PwCLcBGAs/s1600/MontyHallDoors.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="312" data-original-width="516" height="241" src="https://1.bp.blogspot.com/-uMMlSwZYqjw/Wd6Q2LQyGXI/AAAAAAAAK7w/bNey7uxYHNQh1ducy7jasljp7VEP915PwCLcBGAs/s400/MontyHallDoors.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Monty Hall with a contestant in <i>Let's Make a Deal</i>.</td></tr></tbody></table>The news that American TV personality Monty Hall died recently (<a href="https://www.nytimes.com/2017/09/30/obituaries/monty-hall-dead-lets-make-a-deal.html?hp&amp;action=click&amp;pgtype=Homepage&amp;clickSource=story-heading&amp;module=second-column-region&amp;region=top-news&amp;WT.nav=top-news&amp;_r=0" target="_blank">The New York Times, September 30</a>) caused two groups of people to sit up and take note. One group, by far the larger, was American fans of television game shows in the 1960s and 70s, who tuned in each week to his show “Let’s Make a Deal.” The other group include lovers of mathematics the world over, most of whom, I assume, have never seen the show. <br /><br />I, and by definition all readers of this column, are in that second category. As it happens, I have seen a key snippet of one episode of the show, which a television documentary film producer procured to use in a mathematics program we were making about probability theory. Our interest, of course, was not the game show itself, but the famous — indeed infamous — “Monty Hall Problem” it let loose on an unsuspecting world. <br /><br />To recap, at a certain point in the show, Monty would offer one of the audience participants the opportunity to select one of three doors on the stage. Behind one, he told them, was a valuable prize, such as a car, behind each of the other two was a booby prize, say a goat. The contestant chose one door. Sometimes, that was the end of the exchange, and Monty would open the door to reveal what the contestant had won. But on other occasions, after the contestant had chosen a door, Monty would open one of the two unselected doors to reveal a booby prize, and then give them the opportunity to switch their selection. (Monty could always do this since he knew exactly which door the prize was hidden behind.) <br /><br />So, for example, if the contestant first selects Door 2, Monty might open Door 1 to reveal a goat, and then ask if the contestant wanted to switch their choice from Door 2 to Door 3. The mathematical question here is, does it make any difference if the contestant switches their selection from Door 2 to Door 3? The answer, which on first meeting this puzzler surprises many people, is that the contestant doubles their chance of winning by switching. The probability goes up from an original 1/3 of Door 2 being the right guess, to 2/3 that the prize is behind Door 3. <br /><br />I have discussed this problem in <i>Devlin’s Angle</i> on at least two occasions, the most recent being <a href="https://www.maa.org/external_archive/devlin/devlin_12_05.html" target="_blank">December 2005</a>, and have presented it in a number of articles elsewhere, including national newspapers. That on each occasion I have been deluged with mail saying my solution is obviously false was never a surprise; since the problem is famous precisely because it presents the unwary with a trap. That, after all, is why I, and other mathematics expositors, use it! What continues to amaze me is how unreasonably resistant many people are to stepping back and trying to figure out where they went wrong in asserting that switching doors cannot possibly make any difference. For such reflection is the very essence of learning. <br /><br />Wrapping your mind around the initially startling information that switching the doors doubles the probability of winning is akin to our ancestors coming to terms with the facts that the Earth is not flat or that the Sun does not move around the Earth. In all cases, we have to examine how it can be that what our eyes or experience seem to tell us is misleading. Only then can we accept the rock-solid evidence that science or mathematics provides. <br /><br />Some initial resistance is good, to be sure. We should always be skeptical. But for us and society to continue to advance, we have to be prepared to let go of our original belief when the evidence to the contrary becomes overwhelming. <br /><br />The Monty Hall problem is unusual (though by no means unique) in being simple to state and initially surprising, yet once you have understood where your initial error lies, the simple correct answer is blindingly obvious, and you will never again fall into the same trap you did on the first encounter. Many issues in life are much less clear-cut. <br /><br />BTW, if you have never encountered the problem before, I will tell you it is not a trick question. It is entirely a mathematical puzzle, and the correct mathematics is simple and straightforward. You just have to pay careful attention to the information you are actually given, and not remain locked in the mindset of what you initially <i><b>think</b></i> it says. Along the way, you may realize you have misunderstood the notion of probability. (Some people maintain that probabilities cannot change, a false understanding that most likely results from first encountering the notion in terms of the empirical study of rolling dice and selecting colored beans from jars.) So reflection on the Monty Hall Problem can provide a valuable lesson in coming to understand the hugely important concept of mathematical probability. <br /><br />As it happens, Hall’s death comes at a time when, for those of us in the United States, the system of evidence-based, rational inquiry which made the nation a scientific, technological, and financial superpower is coming under dangerous assault, with significant resources being put into a sustained attempt to deny that there are such things as scientific facts. For scientific facts provide a great leveler, favoring no one person or one particular group, and are thus to some, a threat. <br /><br />The late Carl Sagan warned of this danger back in 1995, in his book <i><a href="https://www.amazon.com/Demon-Haunted-World-Science-Candle-Paperback/dp/B00EQBY4TW/ref=sr_1_2?ie=UTF8&qid=1506948921&sr=8-2&keywords=the+demon+haunted+world" target="_blank">The Demon-Haunted World:Science as a Candle in the Dark</a></i>, writing:<br /><blockquote>“I have a foreboding of an America in my children’s or my grandchildren’s time — when the United States is a service and information economy; when nearly all the key manufacturing industries have slipped away to other countries; when awesome technological powers are in the hands of a very few, and no one representing the public interest can even grasp the issues; when the people have lost the ability to set their own agendas or knowledgeably question those in authority; when, clutching our crystals and nervously consulting our horoscopes, our critical faculties in decline, unable to distinguish between what feels good and what’s true, we slide, almost without noticing, back into superstition and darkness. ...”</blockquote>Good scientists, such as Sagan, are not just skilled at understanding what is, they can sometimes extrapolate rationally to make uncannily accurate predictions of what the future might bring. It is chilling, but now a possibility that cannot be ignored, that a decade from now, I could be imprisoned for writing the above words. Today, the probability that will happen is surely extremely low, albeit nonzero. But that probability could change. As mathematicians, we have a clear responsibility to do all we can to ensure that Sagan’s words do not describe the world in which our children and grandchildren live. <br /><br /><br /><br /><br /><br />http://devlinsangle.blogspot.com/2017/10/monty-hall-may-now-rest-in-peace-but.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-2543952525288897833Wed, 20 Sep 2017 18:02:00 +00002017-09-20T14:22:37.383-04:00computers and mathematicsexperimental mathematicsJon BarwiseJonathan BorweinThe Legacy of Jonathan Borwein<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-eabNsYY8UW4/WcKs4cjposI/AAAAAAAAK6A/wXD7oNry-8MJx2A2RprinA6kiFXemNzIwCEwYBhgL/s1600/keith%2Bdevlin.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="891" data-original-width="1600" height="222" src="https://4.bp.blogspot.com/-eabNsYY8UW4/WcKs4cjposI/AAAAAAAAK6A/wXD7oNry-8MJx2A2RprinA6kiFXemNzIwCEwYBhgL/s400/keith%2Bdevlin.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Keith Devlin and Jonathan Borwein talk to host Robert Krulwick on stage at the World Science Festival in 2011.</td></tr></tbody></table><br />At the end of this week I fly to Australia to speak and participate in the <a href="https://carma.newcastle.edu.au/meetings/jbcc/">Jonathan Borwein Commemorative Conference</a> in Newcastle, NSW, Borwein’s home from 2009 onwards, when he moved to the Southern hemisphere after spending most of his career at various Canadian universities. Born in Scotland in 1951, Jonathan passed away in August last year, leaving behind an extensive collection of mathematical results and writings, as well as a long list of service activities to the mathematical community. [For a quick overview, read the brief <a href="http://experimentalmath.info/blog/2016/08/jonathan-borwein-dies-at-65/">obituary</a>written by his long-time research collaborator David Bailey in their joint blog Math Drudge. For more details, check out his <a href="https://en.wikipedia.org/wiki/Jonathan_Borwein">Wikipedia entry</a>.] <br /><br />Jonathan’s (I cannot call him by anything but the name I always used for him) career path and mine crossed on a number of occasions, with both of us being highly active in mathematical outreach activities and both of us taking an early interest in the use of computers in mathematics. Over the years we became good friends, though we worked together on a project only once, co-authoring an expository book on experimental mathematics, titled <i><a href="https://www.amazon.com/Computer-Crucible-Introduction-Experimental-Mathematics/dp/1568813430/ref=asap_bc?ie=UTF8" target="_blank">The Computer as Crucible</a></i>, published in 2008. <br /><br />Most mathematicians, myself included, would credit Jonathan as the father of experimental mathematics as a recognized discipline. In the first chapter of our joint book, we defined experimental mathematics as “the use of a computer to run computations—sometimes no more than trial-and- error tests—to look for patterns, to identify particular numbers and sequences, to gather evidence in support of specific mathematical assertions that may themselves arise by computational means, including search.” <br /><br />The goal of such work was to gather information and gain insight that would eventually give rise to the formulation and rigorous proof of a theorem. Or rather, I should say, that was Jonathan’s goal. He saw the computer, and computer-based technologies, as providing new tools to formulate and prove mathematical results. And since he gets to define what “experimental mathematics” is, that is definitive. But that is where are two interests diverged significantly. <br /><br />In my case, the rapidly growing ubiquity of ever more powerful and faster computers led to an interest in what I initially called “soft mathematics” (see my 1998 book<i> <a href="https://www.amazon.com/Goodbye-Descartes-Logic-Search-Cosmology/dp/0471251860/ref=asap_bc?ie=UTF8" target="_blank">Goodbye Descartes</a></i>) and subsequently referred to as “mathematical thinking,” which I explored in a number of articles and books. The idea of mathematical thinking is to use a mathematical approach, and often mathematical notations, to gather information and gain insight about a task in a domain that enables improved performance. [A seminal, and to my mind validating, example of that way of working was thrust my way shortly after September 11, 2001, when I was asked to join a team tasked with improving defense intelligence analysis.] <br /><br />Note that the same phrase “gather information and gain insight” occurs in both the definition of experimental mathematics and that of mathematical thinking. In both cases, the process is designed to lead to a specific outcome. What differs is the nature of that outcome. (See my 2001 book <i><a href="https://www.amazon.com/Infosense-Turning-Information-Into-Knowledge/dp/0716741644/ref=asap_bc?ie=UTF8" target="_blank">InfoSense</a></i>, to get the general idea of how mathematical thinking works, though I wrote that book before my Department of Defense work, and before I adopted the term “mathematical thinking.”) <br /><br />It was our two very different perspectives on the deliberative blending of mathematics and computers that made our book <i>The Computer as Crucible</i> such a fascinating project for the two of us. <br /><br />But that book was not the first time our research interests brought us together. In 1998, the American Mathematical Society introduced a new section of its ten-issues- a-year <i>Notices</i>, sent out to all members, called “Computers and Mathematics,” the purpose of which was both informational and advocacy. <br /><br />Though computers were originally invented by mathematicians to perform various numerical calculations, professional mathematicians were, by and large, much slower at making use of computers in their work and their teaching than scientists and engineers. The one exception was the development of a number of software systems for the preparation of mathematical manuscripts, which mathematicians took to like ducks to water. <br /><br />In the case of research, mathematicians’ lack of interest in computers was perfectly understandable—computers offered little, if any, benefit. (Jonathan was one of a very small number of exceptions, and his approach was initially highly controversial, and occasionally derided.) But the writing was on the wall—or rather on the computer screen—when it came to university teaching. Computers were clearly going to have a major impact in mathematics education. <br /><br />The “Computers and Mathematics” section of the AMS <i>Notices</i> was intended to be a change agent. It was originally edited by the Stanford mathematician Jon Barwise, who took care of it from the first issue in the May/June 1988 Notices, to February 1991, and then by me until we retired the section in December 1994. It is significant that 1988 was the year Stephen Wolfram released his mathematical software package Mathematica. And in 1992, the first issue of the new research journal <i>Experimental Mathematics</i> was published. <br /><br />Over its six-and- a-half years run, the column published 59 feature articles, 19 editorial essays, and 115 reviews of mathematical software packages — 31 features 11 editorials, and 41 reviews under Barwise, 28 features, 8 editorials, and 74 reviews under me. [The <i>Notices</i>website has a <a href="http://www.ams.org/notices/199502/devlinsixyear.pdf" target="_blank">complete index</a>.] One of the feature articles published under my watch was “Some Observations of Computer Aided Analysis,” by Jonathan Borwein and his brother Peter, which appeared in October 1992. Editing that article was my first real introduction to something called “experimental mathematics.” For the majority of mathematicians, reading it was their introduction. <br /><br />From then on, it was clear to both of us that our view of “doing mathematics” had one feature in common: we both believed that for some problems it could be productive to engage in mathematical work that involved significant interaction with a computer. Neither of us was by any means the first to recognize that. We may, however, have been among the first to conceive of such activity as constituting a discipline in its own right, and each to erect a shingle to advertise what we were doing. In Jonathan’s case, he was advancing mathematical knowledge; for me it was about utilizing mathematical thinking to improve how we handle messy, real-world problems. In both cases, we were engaging in mental work that could not have been done before powerful, networked computers became available. <br /><br />It’s hard to adjust to Jonathan no longer being among us. But his legacy will long outlast us all. I am looking forward to re-living much of that legacy in Australia in a few days time. <br /><br />http://devlinsangle.blogspot.com/2017/09/the-legacy-of-jonathan-borwein.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-1953274983910512263Mon, 07 Aug 2017 18:44:00 +00002017-09-07T16:41:07.060-04:00universitiesuniversity administrationuniversity educationuniversity researchWhat are universities for and how do they work?Exactly 30 years ago, I and my family arrived in the U.S. from the U.K. to take up a one-year visiting position in the mathematics department at Stanford University. (We landed on July 28, 1987.) That one year was subsequently extended to two, and in the end we never returned to the U.K. A very attractive offer of a newly endowed chair in mathematics at Colby College in Maine provided the pull. But equally significant was a push from the U.K. <br /><br />The late 1980s were a bad time for universities in Britain, as Prime Minister Margaret Thatcher launched a full-scale assault on higher education, motivated in part by a false understanding of what universities do, and in part by personal vindictiveness stemming from her being criticized by academics for her poor performance as Minister for Education some years earlier. My own university, Lancaster, where I had been a regular faculty member since 1977, had been a source of some of the most vocal criticisms of the then Minister Thatcher, and accordingly was dealt a particularly heavy funding hit when Prime Minister Thatcher started to wield her axe. A newly appointed vice chancellor (president), with a reputation for tough leadership as a dean, was hired from the United States to steer the university through the troubled waters ahead. <br /><br />One of the first decisions the new vice chancellor made was to cut the mathematics department faculty by roughly 50%, from around 28 to 14. (I forget the actual numbers.) The problem he faced in achieving that goal was that in the British system at the time, once a new Lecturer (= Assistant Professor) had passed a three-year probationary period, they had tenure for life. The only way to achieve a 50% cut in faculty was to force out anyone who could be “persuaded” to go. That boiled down to putting pressure on those whose reputation was sufficiently good for them to secure a position elsewhere. (So, a strategy of “prune from the top,” arguably more productive in the garden than a university.) <br /><br />In my case, the new vice chancellor made it clear to me soon after his arrival that my prospects of career advancement at Lancaster were low, and I could expect ever increasing teaching loads that would hamper my research, and lack of financial support to attend conferences. As a research mathematician early in my career, with my work going well and my reputation starting to grow, that prospect was ominous. Though I was not sure whether he would ever actually follow through with his threat, it seemed prudent to start thinking in terms of a move, possibly one that involved leaving the U.K. <br /><br />Then, just as all of this was going on, out of the blue I got the invitation from Stanford. (I had started working on a project that aligned well with a group at Stanford who had just set up a new research center to work on the same issues. As a result, I had gotten to know some of them, mostly by way of an experimental new way to communicate called “e-mail,” which universities were just starting to use.) <br /><br />In my meeting with the vice chancellor to request permission to accept the offer and discuss the arrangements, I was told in no uncertain terms that I would be wise not to return after my year in California came to an end. The writing was on the wall. Lancaster wanted me gone. In addition, other departmental colleagues were also looking at opportunities elsewhere, so even if I were to return to Lancaster after my year at Stanford, it might well be to a department that had lost several of its more productive mathematicians. (It would have been. The vice chancellor achieved his 50% departmental reduction in little more than two years.) <br /><br />Yes, these events were all so long ago, in a different country. So why am I bringing the story up now? The answer, is that, as is frequently observed, history can provide cautionary lessons for what may happen in the future. <br /><br />Those of us in mathematics are deeply aware of the hugely significant role the subject plays in the modern world, and have seen with every generation of students how learning mathematics can open so many career doors. We also know sufficient mathematics to appreciate the enormous impact on society that new mathematical discoveries can have—albeit in many cases years or decades later. To us, it is inconceivable that a university—an institution having the sole purpose of advancing and passing on new knowledge for the good of society—would ever make a conscious decision to cut down (especially from the top), or eliminate, a mathematics department. <br /><br />But to people outside the universities, things can look different. Indeed, as I discovered during my time as an academic dean (in the U.S.), the need for mathematics departments engaged in research is often not recognized by faculty in other departments. Everyone recognizes the need for each new generation of students to be given some basic mathematics instruction, of course. But mathematics research? That’s a much harder sell. In fact, it is an extremely hard sell. Eliminating the research mathematicians in a department and viewing it as having a solely instructional role can seem like an attractive way to achieve financial savings. But it can come at a considerable cost to the overall academic/educational environment. Not least because of the message conveyed to the students. <br /><br />As things are, students typically graduate from high school thinking of mathematics as a toolbox of formulas and procedures for solving certain kinds of problems. But at university level, they should come to understand it as a particular way of thinking. To that end, they should be exposed to an environment where tasks can be approached on their own terms, with mathematicians being one of any number of groups of experts who can bring a particular way of thinking that may, or may not, be effective. <br /><br />The educational importance of having an active mathematics research group in a university is particularly important in today’s world. As I noted in <a href="http://www.huffingtonpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_us_58693ef9e4b014e7c72ee248" target="_blank">an article</a> in <i>The Huffington Post</i> in January, pretty well all the formulas and procedures that for many centuries have constituted the heart of a university mathematics degree have now been automated and are freely available on sites such as <a href="https://www.wolframalpha.com/examples/Math.html" target="_blank">Wolfram Alpha</a>. Applying an implemented, standard mathematical procedure to solve, say, a differential equation, is now in the same category as using a calculator to add up a column of numbers. Just enter the data correctly and the machine will do the rest. <br /><br />In particular, a physicist or an engineer (say) at a university can, for the most part, carry out their work without the need for specialist mathematical input. (That was always largely the case. It is even more so today.) But one of the functions of a university is to provide a community of experts who are able to make progress when the available canned procedures do not quite fit the task at hand. The advance of technology does not eliminate the need for creative, human expertise. It simply shifts the locus of where such expertise is required. Part of a university education is being part of a community where that reliance on human expertise is part of the daily activities; a community where all the domain specialists are experts in their domains, and able to go beyond the routine. <br /><br />It is easy to think of education as taking place in a classroom. But that’s just not what goes on. What you find in classrooms is instruction, maybe involving some limited discussion. Education and learning occur primarily by way of interpersonal interaction in a community. That’s why we have universities, and why students, and often their parents, pay to attend them. It’s why “online universities” and MOOCs have not replaced universities, and to my mind never will. The richer and more varied the community, the better the education. <br /><br />Lest I have given the impression that my focus is on topline research universities, stocked with award winning academic superstars, let me end by observing that nothing I have said refers to level of achievement. Rather it is all about the attitude of mind and working practices of the faculty. As long as the mathematics faculty love mathematics, and enjoy doing it, and are able to bring their knowledge to bear on a new task or problem, they contribute something of real value to the environment in which the students learn. It’s a human thing. <br /><br />A university that decides to downgrade a particular discipline to do little more than provide basic instruction is diminishing its students educational experience, and is no longer a bona fide university. (It may well, of course, continue to provide a valuable service. The university, my focus in this essay, is just one form of educational institution among many.) <br /><br /><br />http://devlinsangle.blogspot.com/2017/08/what-are-universities-for-and-how-do.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-8036995707127677168Thu, 13 Jul 2017 16:18:00 +00002017-07-13T12:51:06.793-04:00Exploding dotsHindu-arabic arithmeticJames Tantonmathematics educationplace-value arithmeticvisual representationsThe Power of Simple RepresentationsThe great mathematician Karl Freidrich Gauss is frequently quoted as saying “What we need are notions, not notations.” [In “About the proof of Wilson's theorem,” <i>Disquisitiones Arithmeticae</i> (1801), Article 76.] <br /><br />While most mathematicians would agree that Gauss was correct in pointing out that concepts, not symbol manipulation, are at the heart of mathematics, his words do have to be properly interpreted. While a <i>notation</i> does not matter, a <i>representation</i> can make a huge difference. The distinction is that developing or selecting a representation for a particular mathematical concept (or <i>notion</i>) involves deciding which features of the concept to capture. <br /><br />For example, the form of the ten digits 0, 1, … , 9 does not matter (as long as they are readily distinguishable), but the usefulness of the Hindu-Arabic number system is that it embodies base- 10 place-value representation of whole numbers. Moreover, it does so in a way that makes both learning and using Hindu-Arabic arithmetic efficient. <br /><br />Likewise, the choice of 10 as the base is optimal for a species that has highly manipulable hands with ten digits. Although the base-10 arithmetic eventually became the standard, other systems were used in different societies, but they too evolved from the use of the hands and sometimes the feet for counting: base-12 (where finger-counting used the three segments of each of the four fingers) and base-20 where both fingers and toes were used. Base-12 arithmetic and base-20 arithmetic both remained in regular use in the monetary system in the UK when I was a child growing up there, with 12 pennies giving one shilling and 20 shillings one pound. And several languages continue to carry reminders of earlier use of both bases — English uses phrases such as “three score and ten” to mean 70 (= 3x20 + 10) and French articulates 85 as “quatre-vingt cinq (4x20 + 5). <br /><br />Another number system we continue to use today is base-60, used in measuring time (seconds and minutes) and in circular measurement (degrees in a circle). Presumably the use of 60 as a base came from combining the finger and toes bases 10, 12, and 20, allowing for all three to be used as most convenient. <br /><br />These different base-number representation systems all capture features that make them useful to humans. Analogously, digital computers are designed to use binary arithmetic (base 2), because that aligns naturally with the two states of an electronic gate (open or closed, on or off). <br /><br />In contrast, the <i>shapes</i> of the Hindu-Arabic numerals is an example of a superfluous feature of the representation. The fact that it is possible to draw the numerals in a fashion whereby each digit has the corresponding number of angles, like this<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-YVDUy7CSxgo/WWeRZ91JdiI/AAAAAAAAK2c/FLqZuX8IVZogvX3K5mJN1OT1d89oYXWUwCLcBGAs/s1600/H-A_numerals.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="210" data-original-width="1558" height="43" src="https://3.bp.blogspot.com/-YVDUy7CSxgo/WWeRZ91JdiI/AAAAAAAAK2c/FLqZuX8IVZogvX3K5mJN1OT1d89oYXWUwCLcBGAs/s320/H-A_numerals.png" width="320" /></a></div>may be a historical echo of the evolution of the symbols, but whether or not that is the case (and frankly I find it fanciful), it is of no significance in terms of their use—the <i>form</i> of the numerals is very much in Gauss’s “unimportant notations” bucket. <br /><br />On the other hand, the huge difference a <i>representation system</i> can make in mathematics is indicated by the revolutionary change in human life that was brought about by the switch from Roman numerals and abacus-board calculation to Hindu-Arabic arithmetic in Thirteenth Century Europe, as I described in my 2011 book <i><a href="https://www.amazon.com/Man-Numbers-Fibonaccis-Arithmetic-Revolution/dp/0802778127/ref=sr_1_1?ie=UTF8&qid=1306990867&sr=8-1" target="_blank">The Man of Numbers</a></i>. <br /><br />Of course, there is a sense in which representations do not matter to mathematics. There is a legitimate way to understand Gauss’s remark as a complete dismissal of how we represent mathematics on a page. The notations we use provide mental gateways to the abstract notions of mathematics that live in our minds. The notions themselves transcend any notations we use to denote them. That may, in fact, have been how Gauss intended his reply to be taken, given the circumstances. <br /><br />But when we shift our attention from mathematics as a body of eternal, abstract structure occupying a Platonic realm, to an activity carried out by people, then it is clear that notations (i.e., a representation system) are important. In the early days of Category Theory, some mathematicians dismissed it as “abstract nonsense” or “mere diagram chasing”, but as most of us discovered when we made a serious attempt to get into the subject, “tracing the arrows” in a commutative diagram can be a powerful way to approach and understand a complex structure. [Google “the snake lemma”. Even better, watch actress Jill Clayburgh <a href="https://www.youtube.com/watch?v=etbcKWEKnvg" target="_blank">explain it</a> to a graduate math class in an early scene from the 1980s movie <i>It’s My Turn</i>.] <br /><br />A well-developed mathematical diagram can also be particularly powerful in trying to understand complex real-world phenomena. In fact, I would argue that the use of mathematical representations as a tool for highlighting hidden abstract structure to help us understand and operate in our world is one of mathematics most significant roles in society, a use that tends to get overlooked, given our present day focus on mathematics as a tool for “getting answers.” Getting an answer is frequently the end of a process of thought; gaining new insight and understanding is the start of a new mental journey. <br /><br />A particularly well known example of such use are the <a href="https://en.wikipedia.org/wiki/Feynman_diagram" target="_blank">Feynmann Diagrams</a>, simple visualizations to help physicists understand the complex behavior of subatomic particles, introduced by the American physicist Richard Feynmann in 1948. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-mOGUeuoP5MA/WWeR0IIEQbI/AAAAAAAAK2g/pzXCt-qIDssO22H9ykXN7mqbf5v3vN_aQCLcBGAs/s1600/Feynmann_Diagram.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="366" data-original-width="574" height="204" src="https://4.bp.blogspot.com/-mOGUeuoP5MA/WWeR0IIEQbI/AAAAAAAAK2g/pzXCt-qIDssO22H9ykXN7mqbf5v3vN_aQCLcBGAs/s320/Feynmann_Diagram.png" width="320" /></a></div><br /><br />A more recent example that has proved useful in linguistics, philosophy, and the social sciences is the “completion diagram” developed by the American mathematician Jon Barwise in collaboration with his philosopher collaborator John Perry in the early 1980s, initially to understand information flow. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-fEXr7O3UZjc/WWeR6upRpsI/AAAAAAAAK2k/oxW8sjjYlYw-C2FCa4s8GHSnpfNjJfJNgCLcBGAs/s1600/CompletionDiagram.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="558" data-original-width="892" height="250" src="https://1.bp.blogspot.com/-fEXr7O3UZjc/WWeR6upRpsI/AAAAAAAAK2k/oxW8sjjYlYw-C2FCa4s8GHSnpfNjJfJNgCLcBGAs/s400/CompletionDiagram.png" width="400" /></a></div><br />A discussion of one use of this diagram can be found in a survey article I wrote in the volume <i>Handbook of the History of Logic</i>, Volume 7, edited by Dov Gabbay and John Woods (Elsevier, 2008, pp.601-664), a <a href="https://web.stanford.edu/~kdevlin/Papers/HHL_SituationTheory.pdf" target="_blank">manuscript version</a> of which can be found on my Stanford homepage. That particular application is essentially the original one for which the diagram was introduced, but the diagram itself turned out be to be applicable in many domains, including improving workplace productivity, intelligence analysis, battlefield command, and mathematics education. (I worked on some of those applications myself; some <a href="https://web.stanford.edu/~kdevlin/papers.html" target="_blank">links to publications</a> are on my homepage.) <br /><br />To be particularly effective, a representation needs to be simple and easy to master. In the case of a representational diagram, like the Commutative Diagrams of Category Theory, the Feynmann Diagram in physics, and the Completion Diagram in social science and information systems development, the representation itself is frequently so simple that it is easy for domain experts to dismiss them as little more than decoration. (For instance, the main critics of Category Theory in its early days were world famous algebraists.) But the mental clarity such diagrams can bring to a complex domain can be highly significant, both for the expert and the learner. <br /><br />In the case of the Completion Diagram, I was a member of the team at Stanford that led the efforts to develop an understanding of information that could be fruitful in the development of information technologies. We had many long discussions about the most effective way to view the domain. That simple looking diagram emerged from a number of attempts (over a great many months) as being the most effective. <br /><br />Given that personal involvement, you would have thought I would be careful not to dismiss a novel representation I thought was too simple and obvious to be important. But no. When you understand something deeply, and have done so for many years, you easily forget how hard it can be for a beginning learner. That’s why, when the MAA’s own James Tanton told me about his “Exploding Dots” idea some months ago, my initial reaction was “That sounds cute," but I did not stop and reflect on what it might mean for early (and not so early) mathematics education. <br /><br />To me, and I assume to any professional mathematician, it sounds like the method simply adds a visual element on paper (or a board) to the mental image of abstract number concepts we already have in our minds. In fact, that is exactly what it does. But that’s the point! “Exploding Dots” does nothing for the expert. But for the learner, it can be huge. It does nothing for the expert because it represents on a page what the expert has in their mind. <i>But that is why it can be so effective in assisting a learner arrive at that level of understanding!</i> All it took to convince me was to watch Tanton’s <a href="https://vimeo.com/204368634" target="_blank">lecture video</a> on Vimeo. Like Tanton, and I suspect almost all other mathematicians, it took me <i>many years of struggle</i> to go beyond the formal symbol manipulation of the classical algorithms of arithmetic (developed to enable people to carry our calculations efficiently and accurately in the days before we had machines to do it for us) until I had created the mental representation that the exploding dots process capture so brilliantly. Many learners subjected to the classical teaching approach never reach that level of understanding; for them, basic arithmetic remains forever a collection of incomprehensible symbolic incantations. <br /><br />Yes, I was right in my original assumption that there is nothing new in exploding dots. But I was also wrong in concluding that there was nothing new. There is no contradiction here. Mathematically, there is nothing new; it’s stuff that goes back to the first centuries of the First Millennium—the underlying idea for place-value arithmetic. Educationally, however, it’s a big deal. A very big deal. Educationally explosive, in fact. Check it out! <br /><br />http://devlinsangle.blogspot.com/2017/07/the-power-of-simple-representations.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-1313249873137848776Wed, 14 Jun 2017 20:53:00 +00002017-06-14T16:53:23.620-04:00Classroom Clickers Are Good; Except When They Are NotMany math instructors use clickers in their larger lecture classes, and can cite numerous studies to show that they lead to more student attention and better learning. A recent research paper on clicker use devotes a page-long introductory section to a review of some of that literature. (<a href="http://www.sciencedirect.com/science/article/pii/S0360131517300726" target="_blank">Shapiro et al, Computers & Education 111 (2017), 44–59</a>) But the paper—by clicker aficionadas, I should stress—is not all good news. In fact, its main new finding is that when clickers are used in what may be the most common way, they actually have a negative effect on student learning. This finding was sufficiently startling that <i>EdSurge</i> put out a <a href="https://www.edsurge.com/news/2017-05-25-study-finds-classroom-response-clickers-can-impede-conceptual-understanding" target="_blank">feature article</a> on the paper on May 25, which is how I learned of the result. <br /><br />The most common (I believe) use of clickers is to provide students with frequent quiz questions to check that they are retaining important facts. (The early MOOCs, including my own, used simple, machine-graded quizzes embedded in the video lectures to achieve the same result.) And a lot of that research I just alluded to showed that the clickers achieve that goal. <br /><br />So too does the latest study. All of which is fine and dandy if the main goal of the course is retention of facts. Where things get messy is when it comes to conceptual understanding of the material—a goal that almost all mathematicians agree is crucial. <br /><br />In the new study, the researchers looked at two versions of a course (physics, not mathematics), one fact-focused, the other more conceptual and problem solving. In each course, they gave one group fact-based clicker questions and a second group clicker questions that concentrated on conceptual understanding in addition to retention of basic facts. <br /><br />As the researchers expected, both kinds of questions resulted in improved performance in fact- based questions on a test administered at the end. <br /><br />Neither kind of question led to improved performance in a problem-based test questions that required conceptual understanding. <br /><br />The researchers expressed surprise that the students who were given the conceptual clicker questions did not show improvement in conceptual questions performance. But that was not the big surprise. That was, wait for it: students who were given only fact-based clicker questions actually performed <b><i>worse</i></b> on conceptual, problem solving questions. <br /><br />To those of us who are by nature heavy on the conceptual understanding, not showing improvement as a result of enforced fact-retention comes as no big surprise. But a negative effect! That’s news. <br /><br />By way of explanation, the researchers suggest that the fact-based clicker questions focus the student’s attention on retention of what are, of course, surface features, and do so <b><i>to the detriment of acquiring the deeper understanding required to solve problems.</i></b><br /><br />If this conclusion is correct—and is certainly seems eminently reasonable—the message is clear. Use clickers, but do so with questions that focus on conceptual understanding, not retention of basic facts. <br /><br />The authors also recommend class discussions of the concepts being tested by the clicker questions, again something that comes natural to we concepts matter folks. <br /><br />I would expect the new finding to have implications for game-based math learning, which regular readers will know is something I have been working on for some years now. The games I have been developing are entirely problem-solving challenges that require deep understanding, and university studies have shown they achieve the goal of better problem-solving skills. (See the <a href="http://devlinsangle.blogspot.com/2015/12/life-inside-impossible-escher-figure.html">December 4, 2015</a> <i>Devlin’s Angle</i> post.) The majority of math learning games, in contrast, focus on retention of basic facts. Based on the new clickers study, I would hypothesize that, even if a game were built on math concepts (many are not), unless the gameplay involves active, problem-solving engagement with those concepts, the result could be, not just no conceptual learning, but a <i><b>drop</b></i> in performance on a problem solving test. <br /><br />Both clickers and video games set up a feedback cycle that can quickly become addictive. With both technologies, regular positive feedback leads to improvement in what the clicker- questions or game-challenges ask for. Potentially more pernicious, however, that positive feedback will result in the students thinking they are doing just fine overall—and hence have no need to wrestle more deeply with the material. And that sets them up for failure once they have to go beneath the surface fact they have retained. Thinking you are winning all the time seduces you to ease off, and as a result is the path to eventual failure. If you want success, the best diet is a series of challenges— that is to say, challenges in coming to grips with the essence of the material to be learned—where you experience some successes, some failures from which you can recover, and the occasional crash-and- burn to prevent over-confidence. <br /><br />That’s not just the secret to learning math. It’s the secret to success in almost any walk of life. <br /><br />http://devlinsangle.blogspot.com/2017/06/classroom-clickers-are-good-except-when.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-6323129377743866240Mon, 22 May 2017 19:19:00 +00002017-06-01T11:34:58.430-04:00gifted and talentedHerman MehtaJo Boalermathematics educationmind setThe Math Gift MythMy May post is more than a little late. The initial delay was caused by a mountain of other deadlines. When I did finally start to come up for air, there just did not seem to be any suitable math stories floating around to riff off, but I did not have enough time to dig around for one. That this has happened so rarely in the twenty years I have been writing Devlin’s Angle (and various other outlets going back to the early 1980s in the UK), that it speaks volumes against the claim you sometimes hear that nothing much happens in the world of mathematics. There is always stuff going on.<br /><br />Be that as it may, when I woke up this morning and went online, two fascinating stories were waiting for me. What’s more, they are connected – at least, that’s how I saw them. <br /><br />First, my Stanford colleague Professor Jo Boaler sent out a group email pointing to a <i>New York Times</i> article that quoted her, and which, she noted, she helped the author to write. Titled "<a href="https://www.nytimes.com/2017/05/15/well/family/trying-to-add-up-girls-and-math.html?_r=1" target="blank’">No Such Thing as a Math Person</a>," it summarizes the consensus among informed math educators that mathematical ability is a spectrum. Just like any other human ability. What is more, the basic math of the K-8 system is well within the capacity of the vast majority of people. Not easy to master, to be sure; but definitely within most people’s ability. It may be defensible to apply terms such as “gifted and talented” to higher mathematics (though I will come back to that momentarily), but basic math is almost entirely a matter of wanting to master it and being willing to put in the effort. People who say otherwise are either (1) education suppliers trying to sell products, (2) children who for whatever reason simply do not want to learn and find it reassuring to convince themselves they just don’t have the gift, or (3) mums and dads who want to use the term as a parental boast or an excuse. <br /><br />Unfortunately, the belief that mathematical ability is a “gift” (that you either have or you don’t) is so well established it is hard to get rid of. Part of the problem is the way it is often taught, as a collection of rules and procedures, rather than a way of thinking (and a very simplistic one at that). Today, this is compounded by the rapid changes in society over the past few decades, that have revolutionized the way mathematics needs to be taught to prepare the new generation for life in today’s – and tomorrow’s – world. (See my January 1 article in <i>The Huffington Post</i>, "<a href="http://www.huffingtonpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_us_58693ef9e4b014e7c72ee248" target="blank’">All The Mathematical Methods I Learned In My University Math Degree Became Obsolete In My Lifetime</a>," and its follow up article (same date), "<a href="http://www.huffingtonpost.com/entry/number-sense-the-most-important-mathematical-concept_us_58695887e4b068764965c2e0" target="blank’">Number Sense: the most important mathematical concept in 21st Century K-12 education</a>.") <br /><br />With many parents, and not a few teachers, having convinced themselves of the “Math Gift Myth,” attempts over the past several decades to change that mindset have met with considerable resistance. If you have such a mindset, it is easy to see what happens in the educational world around you as confirming it. For instance, one teacher commented on <i>The New York Times</i> article: <br /><br />“Excuse me? I'm a teacher and I refute your assertion. I have seen countless individuals who have problems with math – and some never get it. The same goes for English. But, unless you've spent years in the classroom, it takes years to fully accept that observation. The article's writer is a doctor, not a teacher; accomplishment in one field does not necessarily translate readily to another.” <br /><br />Others were quick to push back against that comment, with one pointing out that her final remark surely argues in favor of everyone in the education world keeping up with the latest scientific research in learning. We are all liable to seek confirmation of our initial biases. And both teachers and parents are in powerful positions to pass on those biases to a new generation of math learners. <br /><br />In her most recent book, <i><a href="https://www.amazon.com/Mathematical-Mindsets-Unleashing-Potential-Innovative/dp/0470894520/ref=sr_1_1?ie=UTF8&qid=1495502636&sr=8-1&keywords=mathematical+mindsets" target="blank’">Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching</a></i>, Prof Boaler lays out some of the considerable evidence against the Math Gift Myth, and provides pointers to how to overcome it in the classroom. The sellout audiences Boaler draws for her talks at teachers conferences around the world indicates the hunger there is to provide math learning that does not produce the math-averse, and even math-phobic, citizens we have grown accustomed to. <br /><br />And so to that second story I came across. Hemant Mehta is a former National Board Certified high school math teacher in the suburbs of Chicago, where he taught for seven years, who is arguably best known for his blog <i>The Friendly Atheist</i>. His post on May 22 was titled "<a href="http://www.patheos.com/blogs/friendlyatheist/2017/05/22/years-later-the-mother-who-audited-an-evolution-exhibit-reflects-on-the-viral-response/" target="blank’">Years Later, the Mother Who 'Audited' an Evolution Exhibit Reflects on the Viral Response</a>." Knowing Mehta’s work (for the record, I have also been <a href="http://www.patheos.com/blogs/friendlyatheist/2015/09/20/friendly-atheist-podcast-episode-73-dr-keith-devlin-mathematics-communicator-and-author/" target="blank’">interviewed by him</a> on his education-related podcast), that title hooked me at first glance. I could not resist diving in. <br /><br />As with <i>The New York Times</i> article I led off with, Mehta’s post is brief and to the point, so I won’t attempt to summarize it here. Like Mehta, as an experienced educator I know that it requires real effort, and courage, to take apart ones beliefs and assumptions, when faced with contrary evidence, and then to reason oneself to a new understanding. So I side with him in not in any way trying to diminish the individual who made the two videos he comments on. What we can do, is use her videos to observe how difficult it can be to make that leap from interpreting seemingly nonsensical and mutually contradictory evidence <b><i>from within our (current!) belief system</i></b>, to seeing it from a new viewpoint from which it all makes perfect sense – to rise above the trees to view the forest, if you will. The video lady cannot do that, and assumes no one else can either. <br /><br />Finally, what about my claim that post K-12 mathematics may be beyond the reach of many individuals’ innate capacity for progression along that spectrum I referred to? Of course, it depends on what you mean by “many”. Leaving that aside, however, if someone, for whatever reason, develops a passionate interest in mathematics, how far can they go? I don’t know. Based on a sample size of one, me, we can go further than we think. I look at the achievement of mathematicians such as Andrew Wiles or Terrence Tao and experience the same degree of their being from a different species as the keen-amateur- cyclist-me feels when I see the likes of Tour de France winner Chris Froome or World Champion Peter Sagan climb mountains at twice the speed I can sustain. <br /><br />Yet, on a number of occasions where I failed to solve a mathematics problem I had been working on for months and sometimes years, when someone else did solve it, my first reaction was, “Oh no, I was so close. If only I had tried just a tiny bit harder!” Not always, to be sure. Not infrequently, I was convinced I would never have found the solution. But I got within a hairsbreadth on enough occasions to realize that with more effort I could have done better than I did. (I have the same experience with cycling, but there I do not have a particular desire to aim for the top.) <br /><br />In other words, all my experience in mathematics tells me I do not have an absolute ability limit. Nor, I am sure, do you. Mathematical proficiency is indeed a spectrum. We can all do better – <b><i>if we want to</i></b>. That, surely is the message we educators should be telling our students, be they in the K-8 classroom or the postgraduate seminar room. <br /><br />Gifted and talented? Time to recognize that as an educational equivalent of the Flat Earth Belief. Sure, we are surrounded by seemingly overwhelming daily experience that the world is flat. But it isn’t. And once you accept that, guess what? From a new perspective, you start to see supporting evidence for the Earth being spherical. http://devlinsangle.blogspot.com/2017/05/the-math-gift-myth.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-114325093655230072Wed, 05 Apr 2017 17:24:00 +00002017-04-05T13:25:09.912-04:00Adolf ZeisingFibonacciFibonacci sequencegolden ratioFibonacci and Golden Ratio MadnessThe first reviews of my new book <i><a href="https://www.amazon.com/Finding-Fibonacci-Rediscover-Forgotten-Mathematical/dp/0691174865/" target="_blank">Finding Fibonacci</a></i> have just come out, and I have started doing promotional activities to try to raise awareness. As I expected, one of the first reviews I saw featured a picture of the Nautilus shell (no connection to Fibonacci or the Golden Ratio), and media interviewers have inevitably tried to direct the conversation towards the many fanciful—but for the most part totally bogus—claims about how the Golden Ratio (and hence the Fibonacci sequence) are related to human aesthetics, and can be found in a wide variety of real-world objects besides the Nautilus shell. [Note: the Fibonacci sequence absolutely <i><b>is</b></i> mathematically related to the Golden Ratio. That’s one of the few golden ratio claims that is valid! There is no evidence Fibonacci knew of the connection.] <br /><br />For some reason, once a number has been given names like “Golden Ratio” and “Divine Ratio”, millions of otherwise sane, rational human beings seem willing to accept claims based on no evidence whatsoever, and cling to those beliefs in the face of a steady barrage of contrary evidence going back to 1992, when the University of Maine mathematician George Markovsky published a seventeen- page paper titled "<a href="https://www.goldennumber.net/wp-content/uploads/George-Markowsky-Golden-Ratio-Misconceptions-MAA.pdf" target="_blank">Misconceptions about the Golden Ratio</a>" in the MAA’s <i>College Mathematics Journal</i>, Vol. 23, No. 1 (Jan. 1992), pp. 2-19. <br /><br />In 2003, mathematician, astronomer, and bestselling author Mario Livio weighed in with still more evidence in his excellent book <i><a href="https://www.amazon.com/Golden-Ratio-Worlds-Astonishing-Number/dp/0767908163/ref=sr_1_1?ie=UTF8&amp;qid=1490933233&amp;sr=8-1&amp;keywords=phi+astonishing" target="_blank">The Golden Ratio: The Story ofPHI, the World's Most Astonishing Number</a></i>. <br /><br />I first entered the fray with a Devlin’s Angle post in June 2004 titled "Good Stories Pity They’re Not True" [the MAA archive is not currently accessible], and then again in May 2007 with "The Myth That Will Not Go Away" [ditto]. <br /><br />Those two posts gave rise to a number of articles in which I was quoted, one of the most recent being "<a href="https://www.fastcodesign.com/3044877/the-golden-ratio-designs-biggest-myth" target="_blank">The Golden Ration: Design’s Biggest Myth</a>," by John Brownlee, which appeared in <i>Fast Company</i> <i>Design</i> on April 13, 2015. <br /><br />In 2011, the Museum of Mathematics in New York City invited me to give a public lecture titled "<a href="https://www.youtube.com/watch?v=JuGT1aZkPQ0" target="_blank">Fibonacci and the Golden Ratio Exposed: Common Myths andFascinating Truths</a>," the recording of which was at the time (and I think still is) the most commented-on MoMath lecture video on YouTube, largely due to the many Internet trolls the post attracted—an observation that I find very telling as to the kinds of people who hitch their belief system to one particular ratio that does not quite work out to be 1.6 (or any other rational number for that matter), and for which the majority of instances of those beliefs are supported by not one shred of evidence. (File along with UFOs, Flat Earth, Moon Landing Hoax, Climate Change Denial, and all the rest.)<br /><br />Needless to say, having been at the golden ratio debunking game for many years now, I have learned to expect I’ll have to field questions about it. Even in a media interview about a book that, not only flatly refutes all the fanciful stuff, but lays out the history showing that the medieval mathematician known today as Fibonacci left no evidence he had the slightest interest in the sequence now named after him, nor had any idea it had several cute properties. Rather, he simply included among the hundreds of arithmetic problems in his seminal book <i>Liber abbaci</i>, published in 1202, an ancient one about a fictitious rabbit population, the solution of which is that sequence.<br /><br />What I have always found intriguing is the question, how did this urban legend begin? It turns out to be a relatively recent phenomenon. The culprit is a German psychologist and author called <a href="https://en.wikipedia.org/wiki/Adolf_Zeising" target="_blank">Adolf Zeising</a>. In 1855, he published a book titled: <i>A New Theory of the proportions of the human body, developed from a basic morphological law which stayed hitherto unknown, and which permeates the whole nature and art, accompanied by a complete summary of the prevailing systems.</i><br /><br />This book, which today would likely be classified as “New Age,” is where the claim first appears that the proportions of the human body are based on the Golden Ratio. For example, taking the height from a person's naval to their toes and dividing it by the person's total height yields the Golden Ratio. So, he claims, does dividing height of the face by its width. <br /><br />From here Zeising leaped to make a connection between these human-centered proportions and ancient and Renaissance architecture. Not such an unreasonable jump, perhaps, but it was, and is pure speculation. After Zeising, the Golden Ratio Thing just took off. <br /><br />Enough! I can’t bring myself to continue. I need a stiff drink. <br /><br />For more on Zeising and the whole wretched story he initiated, see the article by writer Julia Calderone in business Insider, October 5, 2015, "<a href="http://www.businessinsider.com/the-golden-ratio-fibonacci-numbers-mathematics-zeising-beauty-2015-9" target="_blank">The one formula that's supposed to 'prove beauty' is fundamentally wrong</a>."<br /><br />See also the <a href="https://misfitsarchitecture.com/tag/adolf-zeising/" target="_blank">blogpost on Zeising</a> on the blog <a href="https://misfitsarchitecture.com/" target="_blank"><i>misfits’ architecture</i></a>, which presents an array of some of the battiest claims about the Golden Ratio. <br /><br />That’s it. I’m done. <br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />http://devlinsangle.blogspot.com/2017/04/fibonacci-and-golden-ratio-madness.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-4731062327114722175Wed, 08 Mar 2017 16:48:00 +00002017-03-08T11:51:20.847-05:00FibonacciLeonardo of Pisamathematics outreachFinding Fibonacci<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-9rmd71_ydyU/WL8io9bBagI/AAAAAAAAKyg/mGPU2s-sj_oocaNcZU-ZES4uz9vPKgrkgCLcB/s1600/Keith_Leonardo_statue.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://3.bp.blogspot.com/-9rmd71_ydyU/WL8io9bBagI/AAAAAAAAKyg/mGPU2s-sj_oocaNcZU-ZES4uz9vPKgrkgCLcB/s320/Keith_Leonardo_statue.jpeg" width="240" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Devlin makes a pilgrimage to Pisa to see the<br />statue of Leonardo Fibonacci in 2002.</td></tr></tbody></table>In 1983, I did something that would turn out to have a significant influence on the direction my career would take. Frustrated by the lack of coverage of mathematics in the weekly science section of my newspaper of choice, <i>The Guardian</i>, I wrote a short article about mathematics and sent it to the science editor. A few days later, the editor phoned me to explain why he could not to publish it. “But,” he said, “I like your style. You seem to have a real knack for explaining difficult ideas in a way ordinary people can understand.” He encouraged me to try again, and my second attempt was published in the newspaper on May 12, 1983. Several more pieces also made it into print over the next few months, eliciting some appreciative letters to the editor. As a result, when <i>The Guardian</i> launched a weekly, personal computing page later that year, it included my new, twice-monthly column MicroMaths. The column ran without interruption until 1989, when my two-year visit to Stanford University in California turned into a permanent move to the US.<br /><br />Before long, a major publisher contracted me to publish a collection of my MicroMaths articles, which I did, and following that Penguin asked me to write a more substantial book on mathematics for a general audience. That book, <a href="https://www.amazon.com/Mathematics-New-Golden-Keith-Devlin/dp/023111639X/ref=asap_bc?ie=UTF8" target="_blank"><i>Mathematics: The NewGolden Age</i></a>, was first published in 1987, the year I moved to America. <br /><br />In addition to writing for a general audience, I began to give lectures to lay audiences, and started to make occasional appearances on radio and television. From 1991 to 1997, I edited <i>MAA FOCUS</i>, the monthly magazine of the Mathematical Association of America, and since January 1996 I have written this monthly <i>Devlin’s Angle </i>column. In 1994, I also became the <i>NPR </i>Math Guy, as I describe in my latest article in the <i><a href="http://www.huffingtonpost.com/entry/how-i-became-the-npr-math-guy_us_58bb4169e4b0fa65b844b419" target="_blank">Huffington Post</a></i>. <br /><br />Each new step I took into the world of “science outreach” brought me further pleasure, as more and more people came up to me after a talk or wrote or emailed me after reading an article I had written or hearing me on the radio. They would tell me they found my words inspiring, challenging, thought-provoking, or enjoyable. Parents, teachers, housewives, business people, and retired people would thank me for awakening in them an interest and a new appreciation of a subject they had long ago given up as being either dull and boring or else beyond their understanding. I came to realize that I was touching people’s lives, opening their eyes to the marvelous world of mathematics. <br /><br />None of this was planned. I had become a “mathematics expositor” by accident. Only after I realized I had been born with a talent that others appreciated—and which by all appearances is fairly rare—did I start to work on developing and improving my “gift.”<br /><br />In taking mathematical ideas developed by others and explaining them in a way that the layperson can understand, I was following in the footsteps of others who had also made efforts to organize and communicate mathematical ideas to people outside the discipline. Among that very tiny subgroup of mathematics communicators, the two who I regarded as the greatest and most influential mathematical expositors of all time are Euclid and Leonardo Fibonacci. Each wrote a mammoth book that influenced the way mathematics developed, and with it society as a whole. <br /><br />Euclid’s classic work <i>Elements</i> presented ancient Greek geometry and number theory in such a well-organized and understandable way that even today some instructors use it as a textbook. It is not known if any of the results or proofs Euclid describes in the book are his, although it is reasonable to assume that some are, maybe even many. What makes <i>Elements</i> such a great and hugely influential work, however, is the way Euclid organized and presented the material. He made such a good job of it that his text has formed the basis of school geometry teaching ever since. Present day high school geometry texts still follow <i>Elements </i>fairly closely, and translations of the original remain in print. <br /><br />With geometry being an obligatory part of the school mathematics curriculum until a few years ago, most people have been exposed to Euclid’s teaching during their childhood, and many recognize his name and that of his great book. In contrast, Leonardo of Pisa (aka Fibonnaci) and his book <i>Liber abbaci</i> are much less well known. Yet their impact on present-day life is far greater. <i>Liber abbaci</i> was the first comprehensive book on modern practical arithmetic in the western world. While few of us ever use geometry, people all over the world make daily use of the methods of arithmetic that Leonardo described in<i> Liber abbaci</i>. <br /><br />In contrast to the widespread availability of the original Euclid’s <i>Elements</i>, the only version of Leonardo’s <i>Liber abbaci</i> we can read today is a second edition he completed in 1228, not his original 1202 text. Moreover, there is just one translation from the original Latin, in English, published as recently as 2002. <br /><br />But for all its rarity, <i>Liber abbaci </i>is an impressive work. Although its great fame rests on its treatment of Hindu-Arabic arithmetic, it is a mathematically solid book that covers not just arithmetic, but the beginnings of algebra and some applied mathematics, all firmly based on the theoretical foundations of Euclid’s mathematics. <br /><br />After completing the first edition of <i>Liber abbaci</i>, Leonardo wrote several other mathematics books, his writing making him something of a celebrity throughout Italy—on one occasion he was summonsed to an audience with the Emperor Frederick II. Yet very little was written about his life. <br /><br />In 2001, I decided to embark on a quest to try to collect together what little was known about him and bring his story to a wider audience. My motivation? I saw in Leonardo someone who, like me, devoted a lot of time and effort trying to make the mathematics of the day accessible to the world at large. (Known today as “mathematical outreach,” very few mathematicians engage in that activity.) He was the giant whose footsteps I had been following. <br /><br />I was not at all sure I could succeed. Over the years, I had built up a good reputation as an expositor of mathematics, but a book on Leonardo would be something new. I would have to become something of an archival scholar, trying to make sense of Thirteenth Century Latin manuscripts. I was definitely stepping outside my comfort zone. <br /><br />The dearth of hard information about Leonardo in the historical record meant that a traditional biography was impossible—which is probably why no medieval historian had written one. To tell my story, I would have to rely heavily on the <i>mathematical </i>thread that connects today’s world to that of Leonardo—an approach unique to mathematics, made possible by the timeless nature of the discipline. Even so, it would be a stretch. <br /><br />In the end, I got lucky. Very lucky. And not just once, but several times. As a result of all that good fortune, when my historical account <i><a href="https://www.amazon.com/Man-Numbers-Fibonaccis-Arithmetic-Revolution/dp/0802778127/ref=sr_1_1?ie=UTF8&amp;qid=1306990867&amp;sr=8-1" target="_blank">The Man of Numbers: Fibonacci’s Arithmetic Revolution</a></i> was published in 2011, I was able to compensate for the unavoidable paucity of information about Leonardo’s life with the first-ever account of the seminal discovery showing that my medieval role-model expositor had indeed played the pivotal role in creating the modern world that most historians had hypothesized. <br /><br />With my Leonardo project such a new and unfamiliar genre, I decided from the start to keep a diary of my progress. Not just my findings, but also my experiences, the project's highs and lows, the false starts and disappointments, the tragedies and unexpected turns, the immense thrill of holding in my hands seminal manuscripts written in the thirteenth and fourteenth centuries, and one or two truly hilarious episodes. I also encountered, and made diary entries capturing my interactions with, a number of remarkable individuals who, each for their own reasons, had become fascinated by Fibonacci—the Yale professor who traced modern finance back to Fibonacci, the Italian historian who made the crucial archival discovery that brought together all the threads of Fibonacci's astonishing story, and the remarkable widow of the man who died shortly after completing the world’s first, and only, modern language translation of<i> Liber abbaci</i>, who went to heroic lengths to rescue his manuscript and see it safely into print. <br /><br />After I had finished the <i>Man of Numbers</i>, I decided that one day I would take my diary and turn it into a book, telling the story of that small group of people (myself included) who had turned an interest in Leonardo into a passion, and worked long and hard to ensure that Leonardo Fibonacci of Pisa will forever be regarded as among the very greatest people to have ever lived. Just as <i>The Man of Numbers</i> was an account of the writing of <i>Liber abbaci</i>, so too <i>Finding Fibonacci</i> is an account of the writing of <i>The Man of Numbers</i>. [So it is a book about a book about a book. As Andrew Wiles once famously said, “I’ll stop there.”] <br /><br /><i>This post is adapted from the introduction of Keith Devlin’s new book</i> <a href="https://www.amazon.com/Finding-Fibonacci-Rediscover-Forgotten-Mathematical/dp/0691174865/"><i>Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World</i></a><i>, published this month by Princeton University Press.</i>http://devlinsangle.blogspot.com/2017/03/finding-fibonacci.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-8233984482849369276Wed, 08 Feb 2017 14:01:00 +00002017-02-08T09:01:01.742-05:00Hans Rosling, July 27, 1948 – February 7, 2017<div style="text-align: center;"><span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><a href="https://www.youtube.com/watch?v=RUwS1uAdUcI" target="_blank">The power of numbers to help us understand our world.</a></span></div><div style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/RUwS1uAdUcI/0.jpg" src="https://www.youtube.com/embed/RUwS1uAdUcI?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><div style="text-align: center;"><br /></div>http://devlinsangle.blogspot.com/2017/02/hans-rosling-july-27-1948-february-7.htmlnoreply@blogger.com (Mathematical Association of America)1tag:blogger.com,1999:blog-2516188730140164076.post-4240772258053024859Fri, 06 Jan 2017 05:02:00 +00002017-01-09T15:56:49.437-05:00analytic continuationanalytic functionscomplex numbersGrant Sandersoninfinite seriesinfinite sumsmathematics visualizationNumberphileRiemann’s HypothesisZeta functionSo THAT’s what it means? Visualizing the Riemann HypothesisTwo years ago, there was a sudden, viral spike in online discussion of the Ramanujan identity <br /><br />1 + 2 + 3 + 4 + 5 + . . . = –1/12 <br /><br />This identity had been lying around in the mathematical literature since the famous Indian mathematician Srinivasa Ramanujan included it in one of his books in the early Twentieth Century, a curiosity to be tossed out to undergraduate mathematics students in their first course on complex analysis (which was my first exposure to it), and apparently a result that physicists made actual (and reliable) use of. <br /><br />The sudden explosion of interest was the result of a <a href="https://www.youtube.com/watch?v=w-I6XTVZXww" target="_blank">video</a> posted online by Australian video journalist Brady Haran on his excellent <a href="https://www.youtube.com/user/numberphile" target="_blank">Numberphile</a> YouTube channel. In it, British mathematician and mathematical outreach activist James Grime moderates as his physicist countrymen Tony Padilla and Ed Copeland of the University of Nottingham explain their “physicists’ proof” of the identity. <br /><br />In the video, Padilla and Copeland manipulate infinite series with the gay abandon physicists are wont to do (their intuitions about physics tends to keep them out of trouble), eventually coming up with the sum of the natural numbers on the left of the equality sign and –1/12 on the right. <br /><br />Euler was good at doing that kind of thing too, so mathematicians are hesitant to trash it, rather noting that it “lacks rigor” and warning that it would be dangerous in the hands of a lesser mortal than Euler. <br /><br />In any event, when it went live on January 9, 2014, the video and the result (which to most people was new) exploded into the mathematically-curious public consciousness, rapidly garnering hundreds of thousands of hits. (It is currently approaching 5 million in total.) By February 3, interest was high enough for <i>The New York Times</i> to run a <a href="https://www.nytimes.com/2014/02/04/science/in-the-end-it-all-adds-up-to.html?_r=0" target="_blank">substantial story</a> about the “result”, taking advantage of the presence in town of Berkeley mathematician Ed Frenkel, who was there to promote his new book <i>Love and Math</i>, to fill in the details. <br /><br />Before long, mathematicians whose careers depended on the powerful mathematical technique known as <i>analytic continuation</i> were weighing in, castigating the two Nottingham academics for misleading the public with their symbolic sleight-of- hand, and trying to set the record straight. One of the best of those corrective attempts was another <a href="https://www.youtube.com/watch?v=0Oazb7IWzbA" target="_blank">Numberphile video</a>, published on March 18, 2014, in which Frenkel give a superb summary of what is really going on. <br /><br />A year after the initial flair-up, on January 11, 2015, Haran published a <a href="http://www.bradyharanblog.com/blog/2015/1/11/this-blog-probably-wont-help" target="_blank">blogpost</a> summarizing the entire episode, with hyperlinks to the main posts. It was quite a story.<br /><br />[[ASIDE: The next few paragraphs may become a bit too much for casual readers, but my discussion culminates with a link to a really cool video, so keep going. Of course, you could just jump straight to the video, now you know it’s coming, but without some preparation, you will soon get lost in that as well! The video is my reason for writing this essay.]] <br /><br />For readers unfamiliar with the mathematical background to what does, on the face of it, seem like a completely nonsensical result, which is the MAA audience I am aiming this essay at (principally, undergraduate readers and those not steeped in university-level math), it should be said that, as expressed, Ramanujan’s identity is nonsense. But not because of the -1/12 on the right of the equals sign. Rather, the issue lies in those three dots on the left. Not even a mathematician can add up infinitely many numbers. <br /><br />What you can do is, under certain circumstances, assign a meaning to an expression such as <br /><br />X<sub>1</sub> + X<sub>2</sub> + X<sub>3</sub> + X<sub>4</sub> + … <br /><br />where the X<sub>N</sub> are numbers and the dots indicate that the pattern continues for ever. Such expressions are called <i>infinite series</i>. <br /><br />For instance, undergraduate mathematics students (and many high school students) learn that, provided X is a real number whose absolute value is less than 1, the infinite series <br /><br />1 + X + X<sup>2 </sup>+ X<sup>3</sup> + X<sup>4 </sup>+ … <br /><br />can be assigned the value 1/(1 – X). Yes, I meant to write “can be assigned”. Since the rules of real arithmetic do not extend to the vague notion of an “infinite sum”, this has to be defined. Since we are into the realm of definition here, in a sense you can define it to be whatever you want. But if you want the result to be meaningful and useful (useful in, say, engineering or physics, to say nothing of the rest of mathematics), you had better define it in a way that is consistent with that “rest of mathematics.” In this case, you have only one option for your definition. A simple mathematical argument (but not the one you can find all over the web that involves multiplying the terms in the series by X, shifting along, and subtracting—the rigorous argument is a bit more complicated than that, and a whole lot deeper conceptually) shows that the value has to be 1/(1 – X). <br /><br />So now we have the identity <br /><br />(*) 1 + X +X<sup>2 </sup>+ X<sup>3</sup> + X<sup>4 </sup>+ … = 1/(1 – X) <br /><br />which is valid (by definition) whenever X has absolute value less than 1. (That absolute value requirement comes in because of that “bit more complicated” aspect of the rigorous argument to derive the identity that I just mentioned.) <br /><br />“What happens if you put in a value of X that does not have absolute value less than 1?” you might ask. Clearly, you cannot put X = 1, since then the right-hand side becomes 1/0, which is totally and absolutely forbidden (except when it isn’t, which happens a lot in physics). But apart from that one case, it is a fair question. For instance, if you put X = 2, the identity (*) becomes <br /><br />1 + 2 + 4 + 8 + 16 + … = 1/(1 – 2) = 1/(–1) = –1 <br /><br />So you could, if you wanted, make the identity (*) the definition for what the infinite sum <br /><br />1 + X + X<sup>2 </sup>+ X<sup>3</sup> + X<sup>4</sup> + … <br /><br />means for any X other than X = 1. Your definition would be consistent with the value you get whenever you use the rigorous argument to compute the value of the infinite series for any X with absolute value less than 1, but would have the “benefit” of being defined for all values of X apart from one, let us call it a “pole”, at X = 1. <br /><br />This is the idea of analytic continuation, the concept that lies behind Ramanujan’s identity. But to get that concept, you need to go from the real numbers to the complex numbers. <br /><br />In particular, there is a fundamental theorem about differentiable functions (the accurate term in this context is <i>analytic functions</i>) of a single complex variable that says that if any such function has value zero everywhere on a nonempty disk in the complex plane, no matter how small the diameter of that disk, then the function is zero everywhere. In other words, there can be no smooth “hills” sitting in the middle of flat plains, or even one small flat clearing in the middle of a “hilly” landscape—the quotes are because we are beyond simple visualization here. <br /><br />An immediate consequence of this theorem is that if you pull the same continuation stunt as I just did for the series of integer powers, where I extended the valid formula (*) for the sum when X is in the open unit interval to the entire real line apart from one pole at 1, but this time do it for analytic functions of a complex variable, then if you get an answer at all (i.e., a formula), <i><b>it will be unique</b></i>. (Well, no, the formula you get need not be unique, rather the function it describes will be.) <br /><br />In other words, if you can find a formula that describes how to compute the values of a certain expression for a disk of complex numbers (the equivalent of an interval of the real line), and if you can find another formula that works for all complex numbers and agrees with your original formula on that disk, then your new formula tells you <i><b>the</b></i> right way to calculate your function for any complex number. All this subject to the requirement that the functions have to be analytic. Hence the term “<b><i>analytic</i></b> continuation.' <br /><br />For a bit more detail on this, check out the <a href="https://en.wikipedia.org/wiki/Analytic_continuation" target="_blank">Wikipedia explanation</a> or the one on <a href="http://mathworld.wolfram.com/AnalyticContinuation.html" target="_blank">Wolfram Mathworld</a>. If you find those explanations are beyond you right now, just remember that this is not magic and it is not a mystery. It is mathematics. The thing you need to bear in mind is that the complex numbers are very, very regular. Their two-dimensional structure ties everything down as far as analytic functions are concerned. This is why results about the integers such as Fermat’s Last Theorem are frequently solved using methods of Analytic Number Theory, which views the integers as just special kinds of complex numbers, and makes use of the techniques of complex analysis. <br /><br />Now we are coming to that video. When I was a student, way, way back in the 1960s, my knowledge of analytic continuation followed the general path I just outlined. I was able to follow all the technical steps, and I convinced myself the results were true. But I never was able to visualize, in any remotely useful sense, what was going on. <br /><br />In particular, when our class came to study the (famous) <a href="https://en.wikipedia.org/wiki/Riemann_zeta_function" target="_blank">Riemann zeta function</a>, which begins with the following definition for real numbers S bigger than 1: <br /><br />(**) Zeta(S) = 1 + 1/2<sup>S</sup> + 1/3<sup>S</sup> + 1/4<sup>S</sup> + 1/5<sup>S</sup> + … <br /><br />I had no reliable mental image to help me understand what was going on. For integers S greater than 1, I knew what the series meant, I knew that it summed (converged) to a finite answer, and I could follow the computation of some answers, such as Euler’s <br /><br />Zeta(2) = π<sup>2</sup>/6 <br /><br />(You get another expression involving π for S = 4, namely π<sup>4</sup>/90.) <br /><br />It turns out that the above definition (**) will give you an analytic function if you plug in any complex number for S for which the real part is bigger than 1. That means you have an analytic function that is rigorously defined everywhere on the complex plane to the right of the line x = 1. <br /><br />By some deft manipulation of formulas, it’s possible to come up with an analytic continuation of the function defined above to one defined for all complex numbers except for a pole at S = 1. By that basic fact I mentioned above, that continuation is unique. Any value it gives you can be taken as <i><b>the right answer</b></i>. <br /><br />In particular, if you plug in S = –1, you get <br /><br />Zeta(–1) = –1/12 <br /><br />That equation is totally rigorous, meaningful, and accurate. <br /><br />Now comes the tempting, but wrong, part that is not rigorous. If you plug in S = –1 in the original infinite series, you get <br /><br />1 + 1/2<sup>-1</sup> + 1/3<sup>-1</sup> + 1/4<sup>-1</sup> + 1/5<sup>-1</sup> + … <br /><br />which is just <br /><br />1 + 2 + 3 + 4 + 5 + … <br /><br />and it seems you have shown that <br /><br />1 + 2 + 3 + 4 + 5 + . . . = –1/12 <br /><br />The point is, though, you can’t plug S = –1 into that infinite series formula (**). That formula is not valid (i.e., it has no meaning) unless S > 1. <br /><br />So the only way to interpret Ramanujan’s identity is to say that there is a unique analytic function, Zeta(S), defined on the complex plane (apart from at the real number 1), which for all real numbers S greater than 1 has the same values as the infinite series (**), which for S = –1 gives the value Zeta(–1) = –1/12. <br /><br />Or, to put it another way, more fanciful but less accurate, if the sum of all the natural numbers were to suddenly find it had a finite answer, <i><b>that answer could only be</b></i> –1/12. <br /><br />As I said, when I learned all this stuff, I had no good mental images. But now, thanks to modern technology, and the creative talent of a young (recent) Stanford mathematics graduate called <a href="http://www.3blue1brown.com/" target="_blank">Grant Sanderson</a>, I can finally see what for most of my career has been opaque. On December 9, he uploaded <a href="https://www.youtube.com/watch?v=sD0NjbwqlYw" target="_blank">this video</a> onto YouTube.<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/sD0NjbwqlYw/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/sD0NjbwqlYw?feature=player_embedded" width="320"></iframe></div><br /><br />It is one of the most remarkable mathematics videos I have ever seen. Had it been available in the 1960s, my undergraduate experience in my complex analysis class would have been so much richer for it. Not easier, of that I am certain. But things that seemed so mysterious to me would have been far clearer. Not least, I would not have been so frustrated at being unable to understand how Riemann, based on hardly any numerical data, was able to formulate his famous hypothesis, finding a proof of which is agreed by most professional mathematicians to be <i><b>the</b></i> most important unsolved problem in the field. <br /><br />When you see (in the video) what looks awfully like a gravitational field, pulling the zeros of the Zeta function towards the line x = 1/2, and you know that it is the only such gravitational field there is, and recognize its symmetry, you have to conclude that the universe could not tolerate anything other than all the zeros being on that line. <br /><br />Having said that, it would, however, be <i><b>really</b></i> interesting if that turned out not to be the case. Nothing is certain in mathematics until we have a rigorous proof. <br /><br />Meanwhile, do check out some of Grant’s other videos. There are some real gems! http://devlinsangle.blogspot.com/2017/01/so-thats-what-it-means-visualizing.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-2042228397204354363Tue, 13 Dec 2016 21:14:00 +00002016-12-14T11:35:37.149-05:00You can find the secret to doing mathematics in a tubeless bicycle tire<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-A9Nsf6YrkN8/WFBkEN1G73I/AAAAAAAAKuQ/_o6sZqb8xq4jK3E0galcpn1N3Y3YglUKACLcB/s1600/CountryView.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="https://2.bp.blogspot.com/-A9Nsf6YrkN8/WFBkEN1G73I/AAAAAAAAKuQ/_o6sZqb8xq4jK3E0galcpn1N3Y3YglUKACLcB/s400/CountryView.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">The author climbing the locally-notorious Country View Road just south of San Jose, CA</td></tr></tbody></table><br />As regular readers may know, one of my consuming passions in life besides mathematics is cycling. Living in California, where serious winters were wisely banned many years ago, on any weekend throughout the year you are likely to find me out on a road- or a mountain bike. <br /><br />Being also a lover of well-designed technology, I long ago switched to using tubeless tires on my road bike. Actually, it’s bikes, in the plural—my road bikes number four, all with different riding conditions in mind, but all having in common the same kind of ultra- narrow saddle that non-cyclists think must be excruciatingly painful, but is in fact engineered to be the only thing comfortable enough to sit on for many hours at a stretch. [Keep going; I am working my way to making a mathematical point. In fact, I am heading towards THE most significant mathematical point of all: What is the secret to doing math?] <br /><br />Road tubeless tires have several advantages over the more common type of tire, which requires an airtight innertube. One advantage is that you need inflate them only to 80 pounds per square inch, as opposed to the 110 psi or more for a tubed tire, which provides even more comfort over those many hours in the saddle.<br /><br />You need tire pressures 3 or more times that of a car tire because of the extremely low volume in a road-bike tire, which sits on a 700 cm diameter wheel with a rim whose width is between 21 mm and 25 mm. It is that high pressure that made the manufacture of tubeless wheels and tires for bicycles such a significant challenge. How can you ensure an almost totally airtight fit when the tire is inflated, and it still be possible for an average person to remove and mount a deflated tire with their bare hands. (Tire levers can easily damage tubeless wheels and tires.) We are almost to the secret to doing math. Hang in there. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-UEpD5QbsJ5o/WFBkSqYwhiI/AAAAAAAAKuU/W8n8OrKbUgo8KU-zr_Cg1W5PZhqNCJjigCLcB/s1600/Tubeless_Rims.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="152" src="https://3.bp.blogspot.com/-UEpD5QbsJ5o/WFBkSqYwhiI/AAAAAAAAKuU/W8n8OrKbUgo8KU-zr_Cg1W5PZhqNCJjigCLcB/s400/Tubeless_Rims.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Clever design: Tubeless rims and tires on a road bike wheel.</td></tr></tbody></table><br />The airtight fit is possible precisely because of that relatively high pressure inside the tire—80 psi is over five times the air pressure outside the tire. (An automobile tire is inflated to roughly twice atmospheric pressure, much lower.) The cross-sectional photo on the left shows how a tubeless tire has a squared-off ridge that fits into a matching notch in the rim. The more air pressure there is in the tire, the tighter that ridge binds to the rim, increasing the air seal. <br /><br />The problem is, as I mentioned, getting the tire on and off the rim. The tire ridge that fits into the rim-notch has a steel wire running through it, and its squared-off shape is designed to make it difficult for the tire to separate from the rim—that, after all, is the point. To solve the mounting/removal problem, the wheel has a channel in the middle, as shown more clearly in the photo on the right. <br /><br />To mount the tire, you push the two tire-rims into that channel, one after the other. By the formula for the circumference of a circle, when a tire rim is in that center channel, you have just over 3 times the depth of the channel of superfluous tire length to play with, roughly 12mm of tire looseness. The idea is to use that “looseness” to work your way around the wheel, pushing (actually rolling) first one tire edge over the wheel rim and into the channel, then the other. Once the tire is seated on the rim, inflating it with a hand pump forces the tire rims out of the channel into the notches. To remove the tire after it is deflated, you push the two tire rims into the channel and reverse the process. <br /><br />That, at least, is the theory. Putting theory into practice turns out to be quite a challenge. When I first started to use road tubeless tires, several years ago, I read several online manuals and watched a number of YouTube videos demonstrating how to do it,<i><b> and could never do it</b></i>. I usually ended up taking the wheel and tire to my local bike shop, where the mechanic would do it for me with seeming ease before my eyes. “Fifteen dollars, please.” <br /><br />But what would happen if I had a flat on one of the remote rides I regularly do in the mountains that surround Silicon Valley, where I live? One major advantage of tubeless tires is that, even if they puncture, usually the air leaks out only very slowly, and can generally be stopped by inflating the tire from a small pressure-can of air and liquid latex you carry in your back pocket, which seals the hole. Which is how I was always able to get to a bike shop where someone else could solve the problem for me. But a major puncture in the remote, with no cell phone access, could leave me dangerously stranded. Clearly, I had to learn how to do it myself. <br /><br />From now on, when I say “change a tubeless tire”, you can interpret it as “do mathematics”. The secret is coming up. <i>Moreover, it is coming with a moral that those of us in mathematics education ignore at our students’ peril</i>. <br /><br />What I find cool is that, for me I somehow stumbled on the secret to doing math fairly early in life, before math had become such a problem that I felt I could never do it. But taking up cycling later in life, when I had a fully developed set of metacognitive skills, I approached the problem of changing a tubeless tire in much the same way as many people—including, I suspect, the mechanics in my local bicycle shop—see math. Namely, people like me (and that smart kid sitting in the front row in the school math class) make doing math look effortless, but many people feel they could never master it in a million years. <br /><br />Nothing, surely, can look less requiring of skill or expertise than putting a tire on a bicycle wheel. (This is why I think this is such a great example.) Surely, you just need to read an instruction manual, or perhaps have someone demonstrate to you. But no matter how many times I read the instructions, no matter how many times I viewed—and re-viewed—those how-to YouTube videos, and no matter how many times I stood alongside the bike shop mechanic and watched as he quickly and effortlessly put the tire onto the wheel, I could never do it. <br /><br />Just think about that for a moment. For some tasks, <b><i>instruction (on its own) just does not work</i></b>. Not even for the seemingly simple task of changing a bicycle tire. And yet we think that forcing kids to sit in the math class while we force-feed <b><i>instruction</i></b> will result in their being able to do math! Dream on. <br /><br />What does work, in fact what is absolutely necessary, both for changing tubeless tires and doing math, is that the learner has to learn to <b><i>see things the way the expert does</i></b>. And, since instruction does not work, that key step has to be made by the learner. All that a good teacher can do, then, is find a way to help the learner make that key leap. [That short initial word “all” belies the human expertise required to do this.] <br /><br />Clearly, when I was, yet again, standing in the bicycle repair shop, watching the mechanic change my tire, what he was doing—more precisely, what he was <i><b>experiencing</b></i>—was very different from what I was doing and experiencing when I tried and failed. What was I not getting? <br /><br />My big breakthrough finally came the one time when the mechanic, holding the wheel horizontally pressed to his stomach, while manipulating the tire with both hands, told me what he was <b><i>really</i></b> doing. “You have to think of the tire as alive,” he said. “It wants to be sitting firmly on the rim” [that, after all, is what it was—expensively—designed for], “but it is not very disciplined. It’s like a small child. It moves around and resists your attempts to force it. You have to understand it, and be aware, through your hands, of what it is doing. Work with it—be constantly aware of what it is trying to do—so you both get what you want: the tire gets onto the wheel, where it belongs, and you can inflate it and get back on your bike (where you want to be).” <br /><br />Fanciful? Maybe. But it worked. And it continues to work. As a result, not only can I now change my tubeless tires, it has for me become “mindless and automatic,” as effortless (to me) as Picasso drawing a simple doodle on a restaurant napkin to pay the bill for his meal was to him. (I thought that if you got this far, you deserved a second example with greater cultural overtones.) <br /><br />It took many years for Picasso to learn to draw the way he did (and for the marketplace to assign high value to his work), but that does not mean his work was not skillful; rather, he simply routinized part of it. When I watch a film of him at work, I see superficially how he created, and it looks routine and effortless, but I do not see his canvas as he did, and I could not draw as he did. <br /><br />Likewise, my skill in fitting a tubeless tire, now effortless and automatic, is a result of my now seeing and understanding what earlier had been opaque. <br /><br />I admit that it is far easier to learn to mount a tubeless tire on a road bike wheel than to draw like Picasso. But I am less sure the difference is so great between changing a tubeless tire and being able to solve any one particular kind of math problem. Still, no matter how great the difference in the degree of skill required, it is possible to learn from the analogy. <br /><br />Given what I have said here, will reading this essay mean you can go out and immediately be able to change a tubeless tire? Have I just made a case for instruction working after all? It’s possible—for changing bicycle tires, but surely not for painting like Picasso. Instruction can and does work, and it is an important part of learning. But my guess is you would find my words are not enough. I think that the reason that one piece of bike-shop instruction was so instantly transformative for me was that I had spent an aggregate of many hours struggling to change my tire and failing. I had reached a stage where the effective key was <b><i>to get inside the mind of an expert</i></b>. But the ground had to be prepared for that simple revelation to work. <br /><br />In education, as in so many parts of life, there are no silver bullets. But given enough of the right preparation—enough experience acquired through repeated trying and failing—an ordinary lead bullet will do the job. <br /><br />---- <br /><br />This month’s column is loosely adapted from a passage of my forthcoming book <a href="https://www.amazon.com/Finding-Fibonacci-Rediscover-Forgotten-Mathematical/dp/0691174865/"><i>Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World</i></a>, due out in March. <br /><br />http://devlinsangle.blogspot.com/2016/12/you-can-find-secret-to-doing.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-8966259372490164051Fri, 04 Nov 2016 14:01:00 +00002016-11-04T13:51:18.978-04:00Downs paradoxelection mathgolden ratioKool Aidmathematical modelingmilkparadox of votingMathematical Milk and the U.S. Presidential Election<div style="text-align: right;"></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-9BWK4dkWBOA/WBzGZEjPp-I/AAAAAAAAKsU/AvrdcRR9-1YXKMNWmQ3r2AUFLYOIrMkmACLcB/s1600/Devlin_voting.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="https://1.bp.blogspot.com/-9BWK4dkWBOA/WBzGZEjPp-I/AAAAAAAAKsU/AvrdcRR9-1YXKMNWmQ3r2AUFLYOIrMkmACLcB/s400/Devlin_voting.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Keith Devlin mails his completed election ballot. What does math have to say about his act?</td></tr></tbody></table>With the United States is the final throes of a presidential election, my mind naturally turned to the decidedly tricky matter of election math. Voting provides a great illustration of how mathematics – which rules supreme, yielding accurate and reliable answers to precise questions, in the natural sciences and engineering – can lead us astray when we try to apply it to human and social activities.<br /><br />A classic example is how we count votes in an election, the topic of an earlier <i><a href="https://www.maa.org/external_archive/devlin/devlin_11_00.html" target="_blank">Devlin’sAngle</a> </i>post, in November, 2000. In that essay, I looked at how different ways to tally votes could affect the imminent Bush v. Gore election, at the time blissfully unaware of how chaotic would be the process of counting votes and declaring a winner on that particular occasion. The message there was, particularly in the kinds of tight race we typically see today, the different ways that votes can be tallied can lead to very different results.<br /><br />Everything I said back then remains just as valid and pertinent today (mathematics is like that), so this time I’m going to look at another perplexing aspect of election math: why do we make the effort to vote? After all, while elections are sometimes decided by a small number of votes, it is unlikely in the extreme that an election on the scale of a presidential election will hang on the decision of a single voter. Even if it did, that would be well within the range of procedural error, so it makes no difference if any one individual votes or not.<br /><br />To be sure, if a large number of people decide to opt out, that can affect the outcome. But there is no logical argument that takes you from that observation to it being important for a single individual to vote. This state of affairs is known as the Paradox of Voting, or sometimes Downs Paradox. It is so named after Anthony Downs, a political economist whose 1957 book <i><a href="https://www.amazon.com/Economic-Theory-Democracy-Anthony-Downs/dp/0060417501/ref=sr_1_1?ie=UTF8&amp;qid=1477921620&amp;sr=8-1&amp;keywords=an+economic+theory+of+democracy" target="_blank">An Economic Theory of Democracy</a></i> examined the conditions under which (mathematical) economic theory could be applied to political decision-making.<br /><br />On the face of it, Downs’ analysis does lead to a paradox. Economic theory tells us that rational beings make decisions based on <a href="https://en.wikipedia.org/wiki/Cost%E2%80%93benefit_analysis" target="_blank">expected benefit</a> (a notion that can be made numerically precise). That approach works well for analyzing, say, why people buy insurance year after year, even though they may never submit a claim. The theory tells you that the expected benefit is greater than the cost; so it is rational to buy insurance. But when you adopt the same approach to an election, you find that, because the chance of exercising the pivotal vote in an election is minute compared to any realistic estimate of the private individual benefits of the different possible outcomes, the expected benefits of voting are less than the cost. So you should opt out. [The same observation had in fact been made much earlier, in 1793, by Nicolas de Condorcet, but without the theoretical backing that Downs brought to the issue.] <br /><br />Yet, many otherwise sane, rational citizens do not opt out. Indeed, society as a whole tends to look down on those who do not vote, saying they are not "doing their part." (In fact, many countries make participation in a national election obligatory, but that is a separate, albeit related, issue.) <br /><br />So why do we (or at least many of us) bother to vote? I can make the question even more stark, and personal. Suppose you have intended to "do your part" and vote. You wake up on election morning with a sore throat, and notice that it is raining heavily. Being numerically able (as all <i>Devlin’s Angle</i> readers must be), you say to yourself, "It cannot possibly affect the result if I just stay at home and nurse my throat. I was <i><b>intending</b></i> to vote, after all. Changing my mind about voting <i><b>at the last minute</b></i> cannot possibly influence anyone else. Especially if I don’t tell anyone." The math and the logic, surely, are rock solid. Yet, professional mathematician as I am, I would struggle out and cast my vote. And I am sure many <i>Devlin’s Angle</i> readers would too – most of them, I would suspect. <br /><br />So what is going on? We can do the math. We are good logical thinkers. Why don’t we act according to that reasoning? Are we conceding that mathematics actually isn’t that useful? [SPOILER: Math is useful; but only when applied with a specific purpose in mind, and chosen/designed in a way that makes it appropriate for that purpose.] <br /><br />Which brings me to my main point. To make it, let me switch for a moment from elections to the Golden Ratio. In April 2015, the magazine <i>Fast Company Design</i> published an article titled "<a href="https://www.fastcodesign.com/3044877/the-golden-ratio-designs-biggest-myth" target="_blank">The Golden Ratio: Design’s Biggest Myth</a>," in which I was quoted at length. (The author also drew heavily on a <a href="https://www.maa.org/external_archive/devlin/devlin_05_07.html" target="_blank"><i>Devlin’s Angle</i> post</a> of mine from May 2007.) <br /><br />With a readership much wider than <i>Devlin’s Angle</i>, over the years the <i>Fast Company Design</i> piece has generated a fair amount of correspondence from people beyond mathematics academia, often designers who have not been able to overcome drinking Golden Ratio Kool-Aid during their design education. One recent email came, not from a designer but a high school math teacher, who objected to a statement the article quoted me (accurately) as saying, “Strictly speaking, it's impossible for anything in the real-world to fall into the golden ratio, because it’s an irrational number.” The teacher had, it was at once clear to me, drunk not just Golden Ratio Kool-Aid, but Math Kool-Aid as well. <br /><br />In the interest of full disclosure, let me admit that, in the early part of my career as a mathematics expositor, I was as guilty as anyone of distributing both Golden Ratio Kool-Aid and Math Kool-Aid, to whoever would drink it. But, as a committed scientist, when presented with evidence to the contrary, I re-examined my thinking, admitted I had been wrong, and started to push better, more honest products, which I will call Golden Ratio Milk and Mathematical Milk. I described Golden Ratio Milk in my 2007 MAA post and peddled it more in that <i>Fast Company Design</i> interview. Here I want to talk about Mathematical Milk. <br /><br />The reason why the Golden Ratio’s irrationality prevents its use in, say architecture, is that the issue at hand involves measurement. Measurement requires fixing a unit of measure – a scale. It doesn’t matter whether it is meters or feet or whatever, but once you have fixed it, that is what you use. When you measure things, you do so to an agreed degree of accuracy. Perhaps one or two decimal places. Almost never to more than maybe twenty decimal places, and that only in a few instances in subatomic physics. So it terms of actual, physical measurement, or manufacturing, or building, you never encounter objects to which a numerical measurement has more than a few decimal places. You simply do not need a number system that has fractions with denominator much greater than, say, 1,000,000, and generally much less than that. <br /><br />Even if you go beyond physical measurement, to the theoretical realm where you imagine having an unlimited number decimal places available, you will still be in the domain of the rational numbers. Which means the Golden Ratio does not arise. Irrational numbers arise to meet mathematical needs, not the requirements of measurement. They live not in the physical world but in the human imagination. (Hence my <i>Fast Company Design</i> quote.) It is important to keep that distinction clear in our minds. <br /><br />The point is, when we abstract from our experiences of the world around us, to create mathematical models, two important things happen. A huge amount of information is lost; and there is a significant gain in precision. The two are not independent. If we want to increase the precision, we lose more information, which means that our model has less in common with the real world it is intended to represent. Moreover, when we construct a mathematical model, we do so with a particular question, or set of questions in mind. <br /><br />In astronomy and physics, and related domains such as engineering, all of this turns out to be not too problematic. For example, the simplistic model of the Solar System as a collection of point-masses orbiting around another, much heavier, point-mass, is extremely useful. We can formulate and solve equations in that model, and they turn out to be very useful. At least they turn out to be useful in terms of the goal questions, initially in this case predicting where the planets will be at different times of the year. The model is not very helpful in telling us what the color of each planet’s surface is, or even if it has a surface, both of which are certainly precise, scientific questions. <br /><br />When we adopt a similar approach to model money supply or other economic phenomena, we can obtain results that are, mathematically, just as precise and accurate, but their connection to the real world is far more tenuous and unreliable – as has been demonstrated several times in recent years when those mathematical results have resulted in financial crises, and occasionally disasters. <br /><br />So what of the paradox of voting? The paradox arises when you start by assuming that people vote to choose, say, a president. Yes, we all say that is what we do. But that’s just because we have drunk Election Kool-Aid. We don’t actually behave in accordance with that statement. If we did, then as rational beings we would indeed stay at home on election day. <br /><br />Time to throw out the Kool-Aid and buy a gallon jug of far more beneficial Election Milk: (Presidential) elections are about <b><i>a society</i></b> choosing a president. Where that purpose impacts the individual voter is not who we vote for, but in providing social pressure <i><b>to be an active member of that society</b></i>. <br /><br />That this is what is actually going on is illustrated by the fact that U.S. society created, and millions of people wear, "I have voted" badges on election day. The focus, and the personal reward, is not "Who I voted for" but "I participated in the process." [For an interesting perspective on this, see the recent article in the <i>Smithsonian</i> <i>Magazine</i>, "<a href="http://www.smithsonianmag.com/smart-news/why-women-bring-their-i-voted-stickers-susan-b-anthonys-grave-180958847/?no-ist" target="_blank">WhyWomen Bring Their “I Voted” Stickers to Susan B. Anthony’s Grave</a>."] <br /><br />To be sure, you can develop mathematical models of group activities, like elections, and they will tend to lead to fewer problems (and "paradoxes") than a single-individual model will, but they too will have limitations. All mathematical models do. Mathematics is not reality; it is just a model of reality (or rather, it is a whole, and constantly growing, collection of models). <br /><br />When we develop and/or apply a mathematical model, we need to be clear what questions it is designed to help us answer. If we try to apply it to a different question, we may get lucky and get something useful, but we may also end up with nonsense, perhaps in the form of a "paradox."<br /><br />With both measurement and the election, as is so often the case, one benefit we get from trying to apply mathematics to our world and to our lives is we gain insight into what is really going on. <br /><br />Attempting to use the real numbers to model the acts of measuring physical objects leads us to recognize the dependency on the <b><i>physical activity of measurement</i></b>. <br /><br />Likewise, grappling with Downs Paradox leads us to acknowledge what elections are really about – and to recognize that choosing a leader is a <i><b>societal</b></i> activity. In a democracy, <b><i>who</i></b> each one of us votes for is inconsequential; <b><i>that</i></b> we vote is crucial. That’s why I did not just spend a couple of hours yesterday making my choices and filling in my ballot and leaving it at that. I also went out earlier today – in light rain as it happens (and without a sore throat) – and put my ballot in the mailbox. Yesterday I acted as an individual, motivated by my felt societal obligation to participate in the election process. Today I acted as a member of society. <br /><br />As a professional set theorist, I am familiar with the relationship between, and the distinction between, a set and its members. When we view a set in terms of its individual members, we say we are treating it extensionally. When we consider a set in terms of its properties as a single entity, we say we are treating in intensionally. In an election, we are acting intensionally (and intentionally) – at the set level, not as an element of a set. <br /><br />* A <a href="http://www.huffingtonpost.com/dr-keith-devlin/mathematical-milk-and-the_b_12740266.html" target="_blank">shorter version</a> of this article was published simultaneously in <i>The Huffington Post</i>.<br /><br /><br /><br />http://devlinsangle.blogspot.com/2016/11/mathematical-milk-and-us-presidential.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-5680996641846894034Wed, 12 Oct 2016 17:26:00 +00002016-10-12T16:42:07.758-04:00mathematical thinkingmathematical writingIt was Twenty Years Ago TodayThe title of the famous Beatles song does not exactly apply to <i>Devlin’s Angle</i>. The online column (now run on a blog platform, but unlike most blogs, still subject to an editor’s guiding hand) is in its twentieth year, but it actually launched on January 1, 1996. <br /><br />In <a href="http://devlinsangle.blogspot.com/2016/09/then-and-now-devlins-angle-turned.html" target="_blank">last month’s column</a>, I looked back at the very first post. It was a fascinating exercise to try to put myself back in the mindset of how the world looked back then, which was about the time when the World Wide Web was just starting to find its way onto university campuses, but had not yet penetrated the everyday lives of most of the world’s population. <br /><br />That period of intense technological and societal change – looking back, it is clear it was just beginning, in the first half of the 1990s being more evolutionary rather than the revolutionary that was soon to follow – and the strong sensation of change both underway and pending, is reflected in some of the topics I chose to write about each month in that first year. Here is a list of those first twelve posts, with hyperlinks.<br /><br /><ul><li>"<a href="http://www.maa.org/external_archive/devlin/devlin.html" target="_blank">Good Times</a>," January, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlinfeb.html" target="_blank">Base Considerations</a>," February, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/deepblue.html" target="_blank">Deep Blue</a>," March, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlinangle_april.html" target="_blank">Are Mathematicians Turning Soft?</a>," April, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_may.html" target="_blank">The Five Percent Solution</a>," May, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_june.html" target="_blank">Laws of Thought</a>," June, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_july.html" target="_blank">Tversky's Legacy Revisited</a>," July, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_aug.html" target="_blank">Of Men, Mathematics, and Myths</a>," August, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_sept96.html" target="_blank">Dear New Student</a>," September, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/october.html" target="_blank">Wanted: A New Mix</a>," October, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_nov96.html" target="_blank">Spreading the Word</a>," November, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_12_96.html" target="_blank">Moment of Truth</a>," December, 1996</li></ul><br />Along with essays you might find in a mathematics magazine for students (February, June, July, August, November, December), there are reflections on where mathematics and its role in the world might be heading in the next few years.<br /><br />January’s post, about the growth of computer viruses in the digital domain, was clearly in that Brave New World vein, as I noted last month, and in February I focused on another aspect of the rapid growth of the digital world, with a look at the ongoing debate about the future of Artificial Intelligence. Though that field has undoubtedly made many advances in the ensuing two decades, the core argument I summarized there seems as valid today as it did then. Digital devices still do not “think” in anything like a human fashion (though these days it can sometimes be harder to tell the difference).<br /><br />The posts for April, May, and October looked at different aspects of the “Where is mathematics heading?” question. Of course, I was not claiming then, nor am I suggesting now, that the core of pure mathematics is going to change. (Though the growth of <a href="https://www.amazon.com/Computer-Crucible-Introduction-Experimental-Mathematics/dp/B00EQCA1VG/ref=sr_1_sc_2?ie=UTF8&amp;qid=1475680338&amp;sr=8-2-spell&amp;keywords=computer+as+cricuble" target="_blank">Experimental Mathematics</a> in the New Millennium was a new direction, one I addressed in a <a href="https://www.maa.org/external_archive/devlin/devlin_03_09.html" target="_blank"><i>Devlin’s Angle</i> post</a> in March 2009.) Rather, I was taking a much broader view of mathematics, stepping outside the mathematics department of colleges and universities and looking at the way mathematics is used in the world. <br /><br />The October post, in particular, turned out to be highly prophetic for my own career. Shortly after the terrorist attack on the World Trade Center on September 11, 2001, I was contacted by a large defense contractor, asking if I would join a large team they were putting together to bid for a Defense Department contract to find ways to improve intelligence analysis. I accepted the offer, and worked on that project for the next several years. (From my perspective, that project and the work that followed did not end uniformly well, as I lamented in an <i>AMS Notices</i> <a href="http://www.ams.org/notices/201406/rnoti-p623.pdf" target="_blank">opinion piece</a> in 2014.) When that project ended, I did similar work for a large contractor to the US Navy and another project for the US Army. In all three projects, I was living in the kind of world I portrayed in that October, 1996 column.<br /><br />In fact, my professional life as a mathematician for the entire life of <i>Devlin’s Angle</i> has been in that world – a way of using mathematics I started to refer to as “mathematical thinking.” In a <i>Devlin’s Angle</i> <a href="http://devlinsangle.blogspot.com/2012/08/what-is-mathematical-thinking.html" target="_blank">post in 2012</a>, I tried to articulate what I mean by that term. (The term is used by others, sometimes with different meanings, though I see strong overlaps and general agreements among them all.) That same year, I launched the world’s first mathematical MOOC on the newly established online course platform <a href="https://www.coursera.org/" target="_blank">Coursera</a>, with the title “Introduction to Mathematical Thinking”, and published <a href="https://www.amazon.com/Introduction-Mathematical-Thinking-Keith-Devlin/dp/0615653634/ref=sr_1_5?ie=UTF8&amp;qid=1342652878&amp;sr=8-5&amp;keywords=devlin+mathematical+thinking" target="_blank">a book</a> with the same title.<br /><br />With the world as it is today, in particular the pervasive (though largely hidden) role played by mathematics and mathematical ideas in almost every aspect of our lives, I would hazard a guess that there are far more people using “mathematical thinking” than there are people doing mathematics in the traditional sense.<br /><br />If so, that would make the professions of mathematician and mathematics educator two of the most secure careers in the world. For there is one thing in particular you need in order to engage in (effective) mathematical thinking about a real world problem: an adequate knowledge of, and conceptual understanding of, mathematics. In fact, that need was ever so, but it often tended to be overlooked in the pre-digital eras, when doing mathematics meant engaging in a lot of paper-and- pencil, symbolic computations, which meant that the bulk of mathematics instruction focused on computation, with wide ranging knowledge and conceptual understanding often getting short shrift.<br /><br />But those days are gone. Today, we carry around in our pockets devices that give us instant access to pretty well all of the world’s mathematical information and computational procedures we might need to use. (Check out <a href="https://www.wolframalpha.com/examples/Math.html" target="_blank">Wolfram Alpha</a>.) But the thinking still has to be done where it always has: in our heads. <br /><br /><br />http://devlinsangle.blogspot.com/2016/10/it-was-twenty-years-ago-today.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-5188734630296998525Tue, 13 Sep 2016 19:16:00 +00002016-09-19T09:24:15.108-04:00computer virusesMAAonline magazinesWorld Wide WebThen and Now: Devlin’s Angle Turned Twenty This Year<i>Devlin’s Angle</i> turned 20 this year. The first post appeared on January 1, 1996, as part of the MAA’s move from print to online. I was the editor of the MAA’s regular print magazine <i>MAA FOCUS</i> at the time, continuing to act in that capacity until December 1997. (See the last edition of MAA FOCUS that I edited <a href="http://www.maa.org/sites/default/files/pdf/pubs/focus/past_issues/FOCUS_17_6.pdf" target="_blank">here.</a>)<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-Q5eh9C13eeE/V9hPlcIMF7I/AAAAAAAAKnU/Cx6pHvg1bEgiNd31AeDzlNS5Qnb7CxDkACLcB/s1600/oregon_math_summit%2B1997.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="242" src="https://2.bp.blogspot.com/-Q5eh9C13eeE/V9hPlcIMF7I/AAAAAAAAKnU/Cx6pHvg1bEgiNd31AeDzlNS5Qnb7CxDkACLcB/s400/oregon_math_summit%2B1997.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Keith Devlin at a mathematical exposition summit in Oregon in 1997. L to R: Ralph Abraham (Univ of California at Santa Cruz), Devlin, Roger Penrose (Univ of Oxford, UK), and Ivars Peterson (past MAA Director of Publications for Journals and Communications).</td></tr></tbody></table><br />One of the innovations I made when I took over as <i>MAA FOCUS</i> editor in September 1991 was the inclusion of an editorial (written by me) in each issue. Though my ten-times-a-year essays were very much my own personal opinion, they were subject to editorial control by the organization's Executive Director, supported by an MAA oversight committee, both of which had approved my suggestion to do this. Over the years, the editorials generated no small amount of controversy, sometimes based on a particular editorial content, and other times on the more general principle of whether an editor’s personal opinion had a proper place in a professional organization's newsletter. <br /><br />As to the latter issue, I am not sure anyone’s views changed over the years of my editorial reign, but the consensus at MAA Headquarters was that it did result in many more MAA members actually picking up <i>MAA FOCUS</i> when it arrived in the mail and reading it. That was why I was asked to write a regular essay for the new <i>MAA Online</i>. Though blogs and more generally social media were still in the future, the MAA leadership clearly had it right in thinking that an online newsletter was very much an organ in which informed opinion had a place. <br /><br />And so <i>Devlin’s Angle</i> was born. When I realized recently that the column turned twenty this year — in its early days we thought of it very much an online “column”, with all that entailed in the world of print journalism — I was curious to remind myself what topic I chose to write about in my <a href="http://www.maa.org/external_archive/devlin/devlin.html" target="_blank">very first post</a>. <br /><br />Back then, I would have needed to explain to my readers that they could click on the highlighted text in that last sentence to bring up that original post. For the <a href="https://en.wikipedia.org/wiki/World_Wide_Web" target="_blank">World Wide Web</a> was a new resource that people were still discovering, with 1995-96 seeing its growth in academia. Today, of course, I can assume you have already looked at that first post. The words I wrote then (when I might have used the term “penned”, even though I typed them at a computer keyboard) provide an instant snapshot of how the present-day digital world we take for granted looked back then.<br /><br />A mere twenty years ago. http://devlinsangle.blogspot.com/2016/09/then-and-now-devlins-angle-turned.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-8046285082487204651Mon, 01 Aug 2016 16:44:00 +00002016-08-01T12:44:01.903-04:00Alex Gibneycomputer viruscyberwarfarePLCStuxnetMathematics and the End of Days<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/--YZr0rH4uC8/V5ZQhG7ygQI/AAAAAAAAKkM/ess6v-TpFcEqrZW1STrL_Cbqc7cYyTCCgCLcB/s1600/zerodays.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="225" src="https://4.bp.blogspot.com/--YZr0rH4uC8/V5ZQhG7ygQI/AAAAAAAAKkM/ess6v-TpFcEqrZW1STrL_Cbqc7cYyTCCgCLcB/s400/zerodays.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">A scene from <i>Zero Days</i>, a Magnolia Pictures resease. Photo courtesy of Magnolia Pictures</td></tr></tbody></table><br />The new documentary movie<a href="http://www.zerodaysfilm.com/" target="_blank"> <i>Zero Days</i></a>, written and directed by Alex Gibney, is arguably the most important movie of the present century. It is also one of particular relevance to mathematicians for its focus is on the degree to which mathematics has enabled us to build our world into one where a few algorithms could wipe out all human life within a few weeks.<br /><br />In theory, we have all known this since the mid 1990s. As the film makes clear however, this is no longer just a hypothetical issue. We are there.<br /><br />Ostensibly, the film is about the creation and distribution of the computer virus <a href="https://en.wikipedia.org/wiki/Stuxnet" target="_blank">Stuxnet</a>, that in 2011 caused a number of centrifuges in Iran’s nuclear program to self-destruct. And indeed, for the first three-quarters of the film, that is the main topic.<br /><br />Most of what is portrayed will be familiar to anyone who followed that fascinating story as it was revealed by a number of investigative journalists working with commercial cybersecurity organizations. What I found a little odd about the treatment, however, was the degree to which the U.S. government intelligence community appeared to have collaborated with the film-makers, to all intents and purposes confirming on camera that, as was widely suspected at the time but never admitted, Stuxnet was the joint work of the United States and Israel.<br /><br />The reason for the unexpected degree of openness becomes clear as the final twenty minutes of the movie unfold. Having found themselves facing the very real possibility that small pieces of computer code could constitute a human Doomsday weapon, some of the central players in contemporary cyberwarfare decided it was imperative that there be an international awareness of the situation, hopefully leading to global agreement on how to proceed. As one high ranking contributor notes, awareness that global nuclear warfare would (as a result of the ensuing nuclear winter) likely leave no human survivors, led to the establishment of an uneasy, but stable, equilibrium, which has lasted from the 1950s to the present day. We need to do the same for cyberwarfare, he suggests.<br /><br />Mathematics has played a major role in warfare for thousands of years, going back at least to around 250 BCE, when Archimedes of Syracuse designed a number of weapons used to fight the Romans.<br /><br />In the 1940s, the mathematically-driven development of weapons reached a terrifying new level when mathematicians worked with physicists to develop nuclear weapons. For the first time in human history, we had a weapon that could bring an end to all human life.<br /><br />Now, three-quarters of a century later, computer engineers can use mathematics to build cyberwarfare weapons that have at least the same destructive power for human life.<br /><br />What makes computer code so very dangerous is the degree to which our lives today are heavily dependent on an infrastructure that is itself built on mathematics. Inside most of the technological systems and devices we use today are thousands of small solid-state computers called Programmable Logic Controllers (PLCs), that make decisions autonomously, based on input from sensors.<br /><br />What Stuxnet did was embed itself into the PLCs that controlled the Iranian centrifuges and cause them to speed up well beyond their safe range to the point where they simply broke apart, all the while sending messages to the engineers in the control room that the system was operating normally.<br /><br />Imagine now a collection of similar pieces of code that likewise cause critical systems to fail: electrical grids, traffic lights, water supplies, gas pipeline grids, hospitals, the airline networks, and so on. Even your automobile – and any other engine-driven vehicle – could, in principle, be completely shut off. There are PLCs in all of these devices and networks.<br /><br />In fact, imagine that the damage could be inflicted in such a catastrophic and interconnected way that it would take weeks to bring the systems back up again. With no electricity, water, transportation, or communications, it would be just a few days before millions of people start to die, starting with thousands of airplanes, automobiles, and trains crashing, and soon thereafter doubtless accompanied by major rioting around the world.<br /><br />To be sure, we are not at that point, and the challenge of a malicious nation being able to overcome the difficulty of bringing down many different systems would be considerable – though the degree to which they are interdependent could mitigate that “safety” factor to some extent. Moreover, when autonomous code gets released, it tends to spread in many directions, as every computer user discovers sooner or later. So the perpetrating nation might end up being destroyed as well.<br /><br />But Stuxnet showed that such a scenario is a realistic, if at present remote, possibility. (Not just Stuxnet, but the Iranian response. See the movie to learn about that.) If you can do it once (twice?), then you can do it. The weapon is, after all, just a mathematical structure; a piece of code. Designing it is a mathematical problem. Unlike a nuclear bomb, the mathematician does not have to hand over her results to a large, well-funded organization to build the weapon. She can create it herself at a keyboard.<br /><br />That raw power has been the nature of mathematics since our ancestors first began to develop the subject several thousand years ago. Those of us in the mathematics profession have always known that. It seems we have now arrived at a point where that power has reached a new level, certainly no less awesome than nuclear weapons. Making a wider audience more aware of that power is what Gibney’s film is all about. It’s not that we face imminent death by algorithm. Rather that we are now in a different mathematical era.http://devlinsangle.blogspot.com/2016/08/mathematics-and-end-of-days.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-8968782239150488048Fri, 15 Jul 2016 13:06:00 +00002016-07-20T16:20:44.467-04:00BrexitEuropean Communitynumber senseSimon's ParadoxStatisticsUnited KingdomWhat Does the UK Brexit Math Tell Us?The recent (and in many respects ongoing) Brexit vote in the United Kingdom provides a superb example of poor use of mathematics. Regardless of your views on the desirability or otherwise of the UK remaining a member of the European Community (an issue on which this column is appropriately agnostic), for a democracy to base its decision on a complex issue based on a single number displays a woeful misunderstanding of numbers and mathematics.<br /><br />Whenever there is an issue that divides a population more or less equally, making a decision based on a popular vote certainly provides an easy decision, but in terms of accurately reflecting “the will of the people”, you might just as well save the effort and money it costs to run a national referendum and decide the issue by tossing a coin—or by means of a penalty shootout if you want to provide an illusion of human causality.<br /><br />Politicians typically turn to popular votes to avoid the challenge of wrestling with complex issues and having to accept responsibility for the decision they make, or because they believe the vote will turn out in a way that supports a decision they have already made. Unfortunately, with a modicum of number sense, and a little more effort, it’s possible to take advantage of the power that numbers and mathematics offer, and arrive at a decision that actually <i><b>can</b></i> be said to “reflect the will of the people”.<br /><br />The problem with reducing any vaguely complex situation to a single number is that you end up with some version of what my Stanford colleague Sam Savage has referred to as the <a href="https://www.amazon.com/Flaw-Averages-Underestimate-Risk-Uncertainty/dp/1118073754">Flaw of Averages</a>. At the risk of over-simplifying a complex issue (and in this of all articles I am aware of the irony here), the problem is perhaps best illustrated by the old joke about the statistician whose head is in a hot oven and whose feet are in a bucket of ice who, when asked how she felt, replies, “On average I am fine.”<br /><br />Savage takes this ludicrous, but in actuality all-too- common, absurdity as the stepping-off point for using the power of modern computers to run large numbers of simulations to better understand a situation and see what the best options may be. (This kind of approach is used by the US Armed Forces, who run computer simulations of conflicts and possible future battles all the time.)<br /><br />A simpler way to avoid the Flaw of Averages that is very common in the business world is the well-known SWOT analysis, where instead of relying on a single number, a team faced with making a decision lists issues in four categories: strengths, weaknesses, opportunities, and threats. To make sense of the resulting table, it is not uncommon to assign numbers to each category, which opens the door to the Flaw of Averages again, but with four numbers rather than just one, you get some insight into what the issues are.<br /><br />Notice I said “insight” there; not “answer”. For insight is what numbers can give you. Applications of mathematics in the natural sciences and engineering can give outsiders a false sense of the power of numbers to decide issues. In science (particularly physics and astronomy) and engineering, (correctly computed) numbers essentially have to be obeyed. <b><i>But that is almost never the case</i></b> in the human or social realm.<br /><br />When it comes to making human decisions, including political decisions, the power of numbers is even less reliable than the expensively computed numbers that go into producing the daily weather forecast. And surely, no politician would regard the weather forecast as being anything more than a guide—information to help make a decision.<br /><br />One of mathematicians’ favorite examples of how single numbers can mislead is known as Simpson’s Paradox, in which an average can indicate <i><b>the exact opposite</b></i> of what the data actually says.<br /><br />The paradox gets its name from the British statistician and civil servant Edward Simpson, who described it in a technical paper in 1951, though the issue had been observed earlier by the pioneering British statistician Karl Pearson in 1899. (Another irony in this story is that the British actually led the way in understanding how to make good use of statistics, obtaining insights the current UK government seems to have no knowledge of.)<br /><br />A famous illustration of Simpson’s Paradox arose in the 1970s, when there was an allegation of gender bias in graduate school admissions at the University of California at Berkeley. The fall 1973 figures showed that of the 9,442 men and 4,321 women who applied, 44% of men were admitted but only 35% of women. That difference is certainly too great to be due to chance. But was there gender bias? On the face of it, the answer is a clear “Yes”.<br /><br />In reality, however, when you drill down <b><i>just one level </i></b>into the data, from the school as a whole to the individual departments, you discover that, not only was there no gender bias in favor of men, there was in actuality a statistically significant bias in favor of women. The School was going out of its way to correct for an overall male bias in the student population. Here are the figures.<br /><br /><div class="separator" style="clear: both; text-align: center;"> <a href="https://1.bp.blogspot.com/-EiYe8rIgGmo/V4PXUzbHgWI/AAAAAAAAKiI/IBTslKG3OPYWKVuRwUvj-kqVPX5wuh-lgCLcB/s1600/Devlins%2BAngle%2BBrexit.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="138" src="https://1.bp.blogspot.com/-EiYe8rIgGmo/V4PXUzbHgWI/AAAAAAAAKiI/IBTslKG3OPYWKVuRwUvj-kqVPX5wuh-lgCLcB/s400/Devlins%2BAngle%2BBrexit.PNG" width="400" /></a></div><br />In Departments A, B, D, and F, a higher proportion of women applicants was admitted, in Department A significantly so.<br /><br />There was certainly a gender bias at play, but not on the part of University Admissions. Rather, as a result of problems in society as a whole, women tended to apply to very competitive departments with low rates of admission (such as English), whereas men tended to apply to less-competitive departments with high rates of admission (such as Engineering). We see a similar phenomenon in the recent UK Brexit vote, though there the situation is much more complicated. British Citizens, politicians, and journalists who say that the recent referendum shows the “will of the people” are, either though numerically informed malice or basic innumeracy, plain wrong. Just as the UC Berekeley figures did not show an admissions bias against women (indeed, there was a bias in favor of women), so too the Brexit referendum does not show a national will for the UK to leave the EU.<br /><br />Britain leaving the EU may or may not be their best option, but in making that decision the government would do well to drill down at least one level, as did the authorities at UC Berkeley. When you do that, you immediately find yourself with some much more meaningful numbers. Numbers that tell more of the real story. Numbers on which elected representatives of the people can base an informed discussion as how best to proceed—which is, after all, what democracies elect governments to do.<br /><br />Much of that <a href="http://www.bbc.com/news/uk-politics-36616028" target="_blank">“one level down” data</a> was collected by the BBC and published on its website. It makes for interesting reading.<br /><br />For instance, it turned out that among 18-24 years old voters, a massive 73% voted to remain in the UK, as did over half of 25-49 years of age voters. (See Table.) So, given that the decision was about the <i>future</i> of the UK, the result seems to provide a strong argument to remain in the EU. Indeed, it is only among voters 65 or older that you see significant numbers (61%) in favor of leaving. (Their voice matters, of course, but few of them will be alive by the time any benefits from an exit may appear.)<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-6d_dLAJcDRc/V4PXfTffNPI/AAAAAAAAKiM/fu6OM4xjfCITfnTqGrdmPFSGfDig_XYRQCLcB/s1600/Table.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="262" src="https://4.bp.blogspot.com/-6d_dLAJcDRc/V4PXfTffNPI/AAAAAAAAKiM/fu6OM4xjfCITfnTqGrdmPFSGfDig_XYRQCLcB/s400/Table.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Source: <a href="http://www.bbc.com/news/uk-politics-%2036616028" target="_blank">http://www.bbc.com/news/uk-politics- 36616028</a></td></tr></tbody></table><br />You see a similar Simpson’s Paradox effect when you break up the vote by geographic regions, with London, Scotland, and Northern Ireland strongly in favor of remaining in the UK (Scotland particularly so).<br /><br />It’s particularly interesting to scroll down through the long chart in the section headed “Full list of every voting area by Leave”, which is ordered in order of decreasing Leave vote, with the highest Leave vote at the top. I would think that range of numbers is extremely valuable to anyone in government.<br /><br />There is no doubt that the British people have a complex decision to make, one that will have a major impact on the nation’s future for generations to come. Technically, I am one of the “British people,” but having made the US my home thirty years ago, I long ago lost my UK voting rights. My interest today is primarily that of a mathematician who has made something of a career arguing for improved public understanding of the sensible use of my subject, and railing against the misuse of numbers.<br /><br />My emotional involvement today is in the upcoming US presidential election, where there is also an enormous amount of misuse of mathematics, and many lost opportunities where the citizenry could take advantage of the power numbers provide in order to make better decisions.<br /><br />But for what it’s worth, I would urge the citizens of my birth nation to drill down one level in your referendum data. For what you have is a future-textbook example of Simpson’s Paradox (albeit with many more dimensions of variation). To my mathematician’s eye (trained as such in the UK, I might add), the referendum provides very clear numerical information that enables you to form a well-informed, reasoned decision as to how best to proceed.<br /><br />Deciding between the “will of the older population” and the “will of the younger population” is a political decision. So too is deciding between “the will of London, Scotland, and Northern Ireland” and “the will of the remainder of the UK”. What would be mathematically irresponsible, and to my mind politically and morally irresponsible as well, would be to make a decision based on a single number. Single numbers rarely make decisions for us. Indeed, single numbers are usually as likely to mislead as to help. A range of numbers, in contrast, can provide valuable data that can help us to better understand the complexities of modern life, and make better decisions.<br /><br />We humans invented numbers and mathematics to understand our world (initially physical and later social), and to improve our lives. But to make good use of that powerful, valuable gift from our forbearers, we need to remember that numbers are there to serve us, not the other way round. Numbers are just tools. We are the ones with the capacity to make decisions.<br /><br /><i>* A version of this blog post was also published on The Huffington Post.</i><br /><br />http://devlinsangle.blogspot.com/2016/07/what-does-uk-brexit-math-tell-us.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-5470449280026104921Tue, 07 Jun 2016 20:45:00 +00002016-06-08T09:12:33.013-04:00Aronszajn treecountable setHilbert's HotelinfinityKönig’s Lemmatreesuncountable setInfinity and IntuitionOn May 30, Gary Antonick’s always interesting Numberplay section in the New York Times featured a<a href="http://wordplay.blogs.nytimes.com/2016/05/30/frenkel-cantor/?_r=0"> contribution</a> by Berkeley mathematician Ed Frenkel on the difficulties the human mind can encounter when trying to come to grips with infinity. If you have not yet read it, you should.<br /><br />Infinity offers many results that are at first counter-intuitive. A classic example is Hilbert's Hotel, which has infinitely many rooms, each one labeled by a natural number printed on the door: Room 1, Room 2, Room 3, etc., all the way through the natural numbers. One night, a traveler arrives at the front desk only to be told be the clerk that the hotel is full. "But don't worry, sir," says the clerk, "I just took a mathematics course at my local college, and so I know how to find you a room. Just give me a minute to make some phone calls." And a short while later, the traveler has his room for the night. What the clerk did was ask every guest to move to the room with the room number the next integer. Thus, the occupant of Room 1 moved into Room 2, the occupant of Room 2 into Room 3, etc. Everyone moved room, no one was ejected from the hotel, and Room 1 became vacant for the newly arrived guest.<br /><br />This example is well known, and I expect all regular readers of <i>MAA Online</i> will be familiar with it. But I expect many of you will not know what happens when you step up one level of infinity. No sooner have you started to get the hang of the countable infinity (cardinality aleph-0), and you encounter the first uncountable infinity (cardinality aleph-1) and you find there are more surprises in store.<br /><br />One result that surprised me when I first came across it concerns trees. Not the kind the grow in the forest, but the mathematical kind, although there are obvious similarities, reflected in the terminology mathematicians use when studying mathematical trees.<br /><br />A tree is a partially ordered set (T,<) such that for every member x of T, the set {y in T : y < x} of elements below x in the tree is well ordered. This means that the tree has a preferred direction of growth (often represented as upwards in diagrams), and branching occurs only in the upward direction. It is generally assumed that a tree has a unique minimum element, called the root. (If you encounter a tree without such a root, you can simply add one, without altering the structure of the remainder of the tree.)<br /><br />Since each element of a tree lies at the top of a unique well ordered set of predecessors, it has a well defined height in the tree - the ordinal number of the set of predecessors. For each ordinal number k, we can denote by T_k the set of all elements of the tree of height k. T_k is called the k'th level of T. T_0 consists of the root of the tree, T_1 is the set of all immediate successors of the root, etc.<br /><br />Thus, the lower part of a tree might look something like this:<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-V6yTO9KD2dA/V1czATl5u3I/AAAAAAAAKhc/2rbB4C0uGcADNvAltnT6YoEyFDrqqN3OQCLcB/s1600/tree.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-V6yTO9KD2dA/V1czATl5u3I/AAAAAAAAKhc/2rbB4C0uGcADNvAltnT6YoEyFDrqqN3OQCLcB/s1600/tree.jpg" /></a></div><br /><br />(It could be different. There is no restriction on how many elements there are on each level, or how many successors each member has.)<br /><br />A classical result of set theory, sometimes called König's Lemma, says that if T is an infinite tree, and if each level T_n, for n a natural number, is finite, then T has an infinite branch, i.e., an infinite linearly ordered subset.<br /><br />It's easy to prove this result. You define a branch {x_n : n a natural number} by recursion. To start, you take x_0 to be the root of the tree. Since the tree is infinite, but T_1 is finite, there is at least one member of T_1 that has infinitely many elements above it. Let x_1 be one such element of T_1. Since x_1 has infinitely many elements above it and yet only finitely many successors on T_2, there is at least one successor of x_1 on T_2 that has infinitely many elements above it. Let x_2 be such an element of T_2. Now define x_3 in T_3 analogously so it has infinitely many elements of the tree above it, and so on. This simple process clearly defines an infinite branch {x_n : n a natural number}.<br /><br />Having seen why König's Lemma holds, it's tempting to argue by analogy that if you have an uncountable tree T (i.e., a tree whose cardinality is at least aleph-1) and if every level T_k, for k a countable ordinal, is countable, then T has an uncountable branch, i.e., a linearly ordered subset that meets level T_k for every countable ordinal k.<br /><br />But it turns out that this cannot be proved. It is possible to construct an uncountable tree, all of whose levels T_k, for k a countable ordinal, are countable, for which there is no uncountable branch. Such trees are called Aronszajn trees, after the Russian mathematician who first constructed one.<br /><br />Here is how to construct an Aronszajn tree. The members of the tree are strictly increasing (finite and countably transfinite), bounded sequences of rational numbers. The tree ordering is sequence extension. It is immediate that such a tree could not have an uncountable branch, since its limit (more precisely, its set-theoretic union) would be an uncountable strictly increasing sequence of rationals, contrary to the fact that the rationals form a countable set.<br /><br />You build the tree by recursion on the levels. T_0 consists of the empty sequence. After T_k has been constructed, you get T_(k+1) by taking each sequence s in T_k and adding in every possible extension of s to a strictly increasing (k+1)-sequence of rationals. That is, for each s in T_k and for each rational number q greater than or equal to the supremum of s, you put into T_(k+1) the result of appending q to s. Being the countable union of countably many sets, T_(k+1) will itself be countable, as required.<br /><br />In the case of regular recursion on the natural numbers, that would be all there is to the definition, but with a recursion that goes all the way up through the countable ordinals, you also have to handle limit ordinals - ordinals that are not an immediate successor of any smaller ordinal.<br /><br />To facilitate the definition of the limit levels of the tree, you construct the tree so as to satisfy the following property, which I'll call the Aronszajn property: for every pair of levels T_k and T_m, where k < m, and for every s in T_k and every rational number q that exceeds the supremum of s, there is a sequence t in T_m which extends s and whose supremum is less than q.<br /><br />The definition of T_(k+1) from T_k that I just gave clearly preserves this property, since I threw in EVERY possible sequence extension of every member of T_k.<br /><br />Suppose now that m is a limit ordinal and we have defined T_k for every k < m. Given any member s of some level T_k for k < m, and any rational number q greater than the supremum of s, we define, by integer recursion, a path (s_i : i a natural number) through the portion of the tree already constructed, such that its limit (as a rational sequence) has supremum q.<br /><br />You first pick some strictly increasing sequence of rationals (q_i : i a natural number) such that q_0 exceeds the supremum of s and whose limit is q.<br /><br />You also pick some strictly increasing sequence (m_i : i a natural number) of ordinals less than m that has limit m and such that s lies below level m_0 in the tree.<br /><br />You can then use the Aronszajn property to construct the sequence (s_i : i a natural number) so that s_i is on level m_i and the supremum of s_i is less than q_i.<br /><br />Construct one such path (s_i : i a natural number) for every such pair s, q, and let T_k consist of the limit (as a sequence of rationals) of every sequence so constructed. Notice that T_k so defined is countable.<br /><br />It is clear that this definition preserves the Aronszajn property, and hence the construction may be continued.<br /><br />And that's it.<br /><br />NOTE: <i>The above article first appeared in Devlin’s Angle in January 2006. Seeing Frenkel’s Numberplay article prompted me to revive it and give it another airing.</i>http://devlinsangle.blogspot.com/2016/06/infinity-and-intuition.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-4125738874907900223Wed, 04 May 2016 19:32:00 +00002016-05-04T15:33:15.004-04:00algebraCommon CoreDragonBoxITSLearnLabmathematical mindsetmathematics educationrepresentations of mathematicsAlgebraic Roots – Part 2What does it mean to “do algebra”? In Part 1, published here last month, I described how algebra (from the Arabic <i>al-Jabr</i>) began in 9th Century Baghdad as a way to approach arithmetical problems in a systematic way that scales. It was a way of thinking, using logical reasoning rather than (strictly speaking, in addition to) arithmetical calculation, and the first textbook on the subject explained how to solve problems that way using ordinary language, not symbolic expressions. Symbolic algebra was introduced later, in 16th Century France.<br /><br />Just as the formal algorithms of Hindu-Arabic arithmetic make it possible to do arithmetic in a purely procedural, rule-following way (without the need for any thought), so too symbolic algebra made it possible to solve algebraic problems by manipulating symbolic expressions using formal rules, again without the need for any thought.<br /><br />Over the ensuing centuries, schools focused more and more exclusively on the formal, procedural rules of arithmetic and symbolic algebra, driven in part by the needs of industry and commerce to have large numbers of people who could carry out computations for them, and in part for the convenience of the school system.<br /><br />Today, however, we have digital devices that carry out arithmetical and algebraic procedural calculations for us, faster and with greater accuracy, shifting society’s needs back to <i>arithmetical</i> and <i>algebraic thinking</i>. This is why you see the frequent use of those terms in educational circles these days, along with <i>number sense</i>. (All three terms are so common that definitions of each are easily found on the Web by searching on the name, as is also the case for the more general term <i>mathematical thinking</i>.)<br /><br />As more (and hopefully better) technological aids are developed, the nature of the activity involved in solving an arithmetical or algebraic problem changes, both for learning and for application. The fluent and effective use of arithmetical calculators, graphing calculators (such as <i>Desmos</i>), spreadsheets, computer algebra systems (such as <i>Mathematica</i> or <i>Maple</i>), and <i>Wolfram Alpha</i>, are now marketable skills and important educational goals. Each of these tools, and others, provides a different representation of numbers, numerical problems, and algebraic problems.<br /><br />One consequence of this shift that seemed to take an entire generation of parents off guard, is that mastery of the “traditional algorithms” for solving arithmetic and algebraic problems, which were developed to optimize human computations and at the same time create an audit trail, and which used to be the staple of school mathematics instruction, became a much less important educational goal. Instead, it is evidently far more valuable for today’s students to spend their time working with algorithms optimized to develop good arithmetical and algebraic thinking skills, that will (among other things) support fluent and effective use of the new technologies.<br /><br />I said “evidently” above, since to those of us in the education business, it was just that. With hindsight, however, it seems clear that in rolling out the Common Core State Standards, those in charge should have put much more effort into providing that important background context <i>that was evident to them</i> but, clearly, <i>not</i> evident to many people <i>not</i> working in mathematics education.<br /><br />I was not involved in the CCSS initiative, by the way, but I doubt I would have done any better. I still find it hard to wrap my mind round the fact that the “evident” (to me) need to modify mathematics education to today’s world is actually not at all evident to many of my fellow citizens—even though we all live and work in the same digital world. I guess it is a matter of the <i>educational perspective</i> those of us in the math ed business bring to the issues.<br /><br />But even those of us in the education business can sometimes overlook just how much, and how fast, things have changed. The most recent example comes from a highly respected learning research center, LearnLab in Pittsburgh (formerly called the Pittsburgh Science of Learning Center), funded by the National Science Foundation.<br /><br />The tweet shown below caught my eye a few weeks ago.<br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-EDt6JE0oojk/Vyoqv7wKDgI/AAAAAAAAKgs/YqCv_omwGB8uqA_STta-h6h8Ty0cCjbTwCLcB/s1600/Dragonbox.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="333" src="https://4.bp.blogspot.com/-EDt6JE0oojk/Vyoqv7wKDgI/AAAAAAAAKgs/YqCv_omwGB8uqA_STta-h6h8Ty0cCjbTwCLcB/s400/Dragonbox.PNG" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"></div><br />The tweet got my attention because I am familiar with DragonBox, and include it in the (very small) category of math learning apps I usually recommend. (I also know the creator, and have given occasional voluntary feedback on their development work, but I have no other connection to the company.)<br /><br />“Ineffective”? “#dragonboxfail”? Those are the words used in the tweet. <i>But neither can possibly be true</i>. DragonBox provides <i>an alternative representation</i> for linear equations in one unknown. Anyone who completes the game (for want of a better term) has demonstrated mastery of algebraic thinking for single variable linear problems. Period. (There is a separate issue of the representation that I will come to later.)<br /><br />Indeed, since the mechanics in DragonBox are essentially isomorphic to the rules of classical symbolic algebra (as taught in schools for the last four hundred years), completing the game demonstrates mastery of those mechanics too. From a logical perspective then, the tweet made no sense. All very odd for an official tweet from a respected, federally-funded research institute. Suspecting what must be going on, I looked further.<br /><br />The tweet was in response to a <a href="https://www.edsurge.com/news/2016-03-13-enter-the-dragonbox-can-a-game-really-teach-third-graders-algebra">review</a> of DragonBox, published by EdSurge. I recognized the name of the reviewer, Brady Fukumoto, a former game developer I had meet a few times. It was a well analyzed review. Overall, I agreed with everything Brady said. In particular, he spent some time comparing “doing algebra in the DragonBox representation” to “doing algebra using the traditional symbolic equations representation”, pointing out how much richer is the latter—but noting too that the former can result in higher levels of student engagement. Hardly the “promote” of a product that LearnLab accused him of. Indeed, Brady correctly summarized, and referenced (with a link) the Carnegie Mellon University <a href="http://www.cs.cmu.edu/~ylong/papers/Long&Aleven_ITS2014.pdf">study</a> the LearnLab tweet implicitly referred to.<br /><br />I recommend you read Brady’s review. It gets at many aspects of the “what does it mean to do algebra?” issue. As does playing DragonBox itself, which toward the end gradually replaces its initial “game representation” with the standard symbolic equation representation on a touch screen (a process often referred to as deconcretization).<br /><br />Unlike the tweet, the CMU paper was careful in stating its conclusion. The authors say, and Brady quotes, that they found DragonBox to be “ineffective in helping students acquire skills in solving algebra equations, <i>as measured by a typical test of equation solving</i>.” (The emphasis is mine.)<br /><br />Now we are at the root of that odd tweet. (One should not make too much of a tweet, of course. Twitter is an instant medium. But, rightly or wrongly, tweets in the name of an organization or a public figure are generally viewed as PR, presenting an authoritative, public stance.) The folks at LearnLab, their knowledge of educational technology notwithstanding, are assuming a perspective in which one particular representation of algebra is privileged; namely, the traditional symbolic one. (Which is the representation they adopt in developing their own algebra instruction app, an Intelligent Tutoring System called <i>Lynnette</i>.) But as I pointed out last month, that representation became the dominant one entirely by virtue of what was at that time the best available distribution technology: the printing press.<br /><br />With newer technologies, in particular the tablet computer (“printed paper on steroids”), other representations are possible, some better suited to learning, others to applications. To be sure, there are learning benefits to be gained from mastering symbolic algebra, perhaps even from doing so using paper-and-pencil, as Brady points out in his review. But at this stage in the representational technology development, we should adopt a perspective of all bets being off when it comes to how to best represent algebra in different contexts. I think it highly unlikely that we will ever again view algebra as something you learn or do exclusively by using a pen to pour symbols onto a page.<br /><br />Indeed, with his background in video game design, Brady ends his review by rating DragonBox according to three metrics:<br /><br /><b>Fun Factor – A:</b> I collected all 1,366 stars available in DragonBox 1 and 2 and had a great time.<br /><br /><b>Academic Value – B:</b> I worry that many will underestimate the effort needed to transfer DragonBox skills to practical algebra proficiency.<br /><br /><b>Educational Value – A+:</b> Anytime a kid leaves a game with thoughts like, “algebra is fun!” or “hey, I’m really good at math!” that is a huge win.<br /><br />The LearnLab researchers are locked into the second perspective: what he calls Academic Value. (So too is Brady, to some extent, with his use of the phrase “practical algebra proficiency” to mean “symbolic algebra proficiency.”)<br /><br />Make no mistake about it, transfer from mastery in an interactive engagement on a tablet to paper-and-pencil math is not automatic, as both Brady and the CMU researchers observe. To modify the old horse aphorism, DragonBox takes its players right to the water’s edge and dips their feet in, but still the players have difficulty drinking. (My best guess is that, for most learners it takes a good teacher to facilitate transfer.)<br /><br />I note in passing that initially I had difficulty playing DragonBox. My problem was, classical, symbolic algebra is a second language to me that I have been fluent in since childhood and use every day. I found it difficult mastering the corresponding actions in DragonBox. Transfer is difficult in both directions.<br /><br />At the present moment in time, those of us in education (or learning research) should absolutely not assume any one representation is privileged. Particularly so when it comes to learning. In that respect, Brady is right to note that DragonBox’s success in terms of his third metric (essentially, attitude and engagement) is indeed “a huge win.”<br /><br />In the world in which our students will live their lives, arithmetic, algebra, and many other parts of mathematics, should be learned, and will surely be applied, in multimedia environments. All the evidence available today suggests that mastery of the traditional symbolic representation will be a crucial ingredient in becoming proficient at arithmetic and algebra. But the more effective practitioners are likely to operate with the aid of various technological tools. Indeed, for some future practitioners, mastery of the traditional symbolic representation (which is, remember, just a user interface to a certain kind of thinking) may turn out to be primarily just a key step in the cognitive process of achieving conceptual understanding, not used directly in applications, which may all be by way of mathematical reasoning tools.<br /><br />Exactly when, in the initial learning process, it is best to introduce the classical symbolic representation is as yet unclear. What the evidence of countless generations of students-turned-parents makes abundantly clear, however, is that teaching only the classical symbolic approach is a miserable failure. That much is affirmed every time a parent posts on social media that they are unable to understand a Common Core math question that requires nothing more than understanding the place-value representation of integers. (Which is true of most of the ones I have seen posted.)<br /><br />There is some evidence (see for example Jo Boaler’s <a href="http://www.amazon.com/Mathematical-Mindsets-Unleashing-Potential-Innovative/dp/0470894520/ref=sr_1_1?ie=UTF8&qid=1462196234&sr=8-1&keywords=mathematical+mindsets">new book</a>) that a more productive approach is to use learning technologies to develop and sustain student engagement and develop a growth mindset, and provide learning environments for safe, productive failure, with the goal of developing number sense and general forms of creative problem solving (mathematical thinking), bringing in symbolic representations and specific techniques as and when required.<br /><br />**Full declaration: I should note that my own work in this area, some of it through my startup company <a href="http://www.brainquake.com/">BrainQuake</a>, adopts this philosophy. The significant <a href="http://documents.brainquake.com/backed-by-science/Stanford-Pope_Mangram_SUMMARY.pdf">learning gains </a> obtained with our <a href="http://wuzzittrouble.com/">first app</a> were in number sense and creative problem solving for a novel, complex performance task. Acquisition of traditional “basic skills” with our app comes about (intentionally, by design) as a valuable by-product. The improvement we see in the basic skills category is much more modest, and may well be better achieved by a tool such as LearnLab’s ITS. In a world where we have multiple representations, it is wise to make effective use of them all, according to context. It is not a case of an interface “fail”; to say that (with or without a hashtag) is to remain locked in past thinking. Easy to do, even for experts. Rather, in an era when algebra is being forced to return to its roots of being a way of thinking to help us solve practical problems, using all available representations in unison can provide us with a major win.http://devlinsangle.blogspot.com/2016/05/what-does-it-mean-to-do-algebra-in-part.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-7982488435981996827Mon, 04 Apr 2016 19:33:00 +00002016-04-04T15:54:32.484-04:00al-KhwarizmialgebraCommon Coremathematics educationAlgebraic roots – Part 1<div class="separator" style="clear: both; text-align: center;"></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/--sdywWmDWlA/VwKtXmK9fKI/AAAAAAAAKfo/XdGmL94Fr2Y9_nPd-OzAqsnUkrz2BEt2Q/s1600/Fig1.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="228" src="https://3.bp.blogspot.com/--sdywWmDWlA/VwKtXmK9fKI/AAAAAAAAKfo/XdGmL94Fr2Y9_nPd-OzAqsnUkrz2BEt2Q/s400/Fig1.jpeg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Fig. 1: A problem from the first ever algebra textbook.</td></tr></tbody></table>The first ever algebra text book was written in Baghdad around 830CE, by the Persian mathematician Muhammad ibn Musa al-Khwarizmi, our modern word “algebra” coming from the Arabic term al-Jabr, a technique for balancing an equation, described in the book. If you were a student – or a teacher – back then, the problem shown above (Figure 1) is the kind of thing you would be faced with in your math class. It is a direct translation from the original Arabic of a problem in al-Khwarizmi’s book.<br /><br />Most modern readers, on seeing this for the first time and being told it is an algebra problem, are surprised that there are no symbols. Yet it is clearly not an “algebra word problem” in the usual sense. It’s just about numbers. It is, in fact, a quadratic equation problem. Figure 2 below is the same problem as we would present it in an algebra textbook today.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-p5thdoqU-9I/VwKtnx4kZ5I/AAAAAAAAKfs/qp0hTRwi6GQn6qM852TwXSa60iPd0srtg/s1600/Fig2.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="196" src="https://4.bp.blogspot.com/-p5thdoqU-9I/VwKtnx4kZ5I/AAAAAAAAKfs/qp0hTRwi6GQn6qM852TwXSa60iPd0srtg/s320/Fig2.jpeg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Fig. 2: Al-Khwarizmi's quadratic equation in modern notation.</td></tr></tbody></table><br /><i>Symbolic </i>algebra, as we understand it today, was not introduced until the Sixteenth Century, when the French mathematician François Viète took what until then had been a discipline presented in prose, and turned it into the symbolic process we are familiar with today.<br /><br />This is not to say that mathematicians back in Ninth Century Persia did not use symbolic expressions in their work. They surely did. The issue is how they presented it in textbook form. In the days when books were handwritten and duplicated by hand-copying, the author of a mathematics book was faced with a problem that other writers did not have to worry about: faithful copying. Copying of manuscripts was largely done by monks in monasteries. While masters of the written word – they did, after all, “live by a book” — few monks mastered mathematics, and hence could not be relied upon to create an accurate copy of anything other than prose. Aware of this issue, authors of mathematics books wrote everything in prose.<br /><br />With the introduction of the printing press in the Fifteenth Century, however, everything changed. Indeed, one of the first printed books published after Gutenberg printed his famous edition of the Bible was an Italian book on practical arithmetic. True, to handle a symbolic textbook, you have first to master the linguistic rules for reading, writing, and manipulating symbolic expressions, but once you do, algebra becomes a whole lot easier to do, as a line-by-line comparison of Figures 1 and 2 makes abundantly clear. (Actually, it’s lines-by-line!)<br /><br />Notice, however, that the two presentations of the quadratic problem specify the same problem, and both solutions are, from a logical deduction point of view, the same. To some extent, the al-Khwarizmi’s prose version describes what goes on in your head when you solve the problem. At least — <i>and this is where I am going with this</i> — it does if you solve the problem by thinking about it. <br /><br />With the symbolic presentation, it is possible to reduce the solution of an algebra problem to the <i>mindless </i>(literally), algorithmically-specified manipulations of symbols. Ever since the invention of the printing press, generations of students quickly discovered that you can pass an algebra test by mastering a collection of symbolic-manipulation rules.<i> No understanding necessary</i>. Moreover, when taught this way, the teacher’s job became immeasurably easier. It is easier to teach rules to be followed than to develop thinking skills, and it is easy to evaluate a student’s work if the goal is simply to check that it accords with the rules and arrives at the correct answer. (Indeed, teachers soon realized that the quickest way to grade a student’s work was to first see if the answer is correct, and only if it is not look at the symbolic working.)<br /><br />While the student in Ninth Century Baghdad solved (linear and quadratic) equations by performing essentially the same steps as a student would today, with the problem presented in words, and the solution written out (presumably) in words, it can’t be carried out in a mindless fashion. The human mind can learn to follow rules for manipulating symbols, without knowing what they mean, but words are so much an integral part of human thinking that we cannot use them without their having meaning (albeit possibly a meaning other than the one intended by the author of an algebra book).<br /><br />There is, then, a potential loss in taking algebra from a prose presentation to a symbolic one: namely, the student can lose the appreciation that algebra is a <i>powerful way of thinking</i> with countless uses in the everyday world. Instead of algebra being a codification of human logical thinking that emerges from within, it becomes a set of externally imposed, and often arbitrary-seeming rules to be mastered by repetitive practice. The natural, relevant, and empowering becomes the artificial, pointless, and tedious. (Those of us who like symbolic algebra see beyond the rules.)<br /><br />“When will I ever use algebra?” today’s student justifiably asks. In terms of rule-based, symbol manipulation, the answer is, for most people (not all – and this is educationally significant), “Never.” But in terms of algebra, that codified way of thinking that has evolved and developed considerably since al-Khwarizmi’s day, the answer is, “All the time.” (Whenever you use a spreadsheet, for example.)<br /><br />In the introduction to his algebra book, al-Khwarizmi declared that he was presenting<br /><br /><i> “... what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.”</i><br /><br />This was cutting edge stuff back then. It doesn’t get much more practical than that! <br /><br />As al-Khwarizmi explains, he was asked to write his book by the Caliph, who recognized the importance — for trade and engineering in particular (both of which were crucial to the regional society at the time) — of making those new methods of calculation widely available. The Caliph’s reasoning was as sound and significant then as it would be today. When a society reaches a state of development where trade and commercial and financial activity go beyond two people engaging in one-off transactions, it needs a more efficient tool than basic arithmetic. What is required is <i>arithmetic-at-scale</i>. When you boil it down to its essence, that is what algebra is. Al-Khwarizmi’s book codifies and formalizes the numerical reasoning that people use in their daily personal and professional lives in a fashion that enables them to operate at scale.<br /><br />In the years since the printing press made it possible to produce algebra textbooks that used symbolic representations, the focus in the algebra class has gradually shifted from being about sophisticated reasoning about numbers to an often mindless game of symbol manipulation. For several centuries that could be justified on the grounds that the only effective way for society to be able to handle the arithmetic-at-scale required to advance was to train lots of people to carry out the necessary calculations. And for that, the most efficient way is to use rule-based, symbolic manipulation. The people carrying out those calculations no more had to understand what they were doing than the electronic calculator on your iPhone has to understand what it is doing. All that matters it that it – the human symbolic-algebraist or the calculator app — gets the right answer.<br /><br />But now that those of us in more advanced societies (and in a great many less advanced societies, for that matter) do have ready access to those powerful calculating devices, devices that in addition to performing numerical calculations can also solve algebraic problems (arithmetic-at-scale, aka the electronic spreadsheet), the once-important societal need for many human symbolic calculators has gone away. What is required today is that people can make effective use of those new tools. That has shifted the emphasis back from symbolic-rule-mastery to the kind of formalized, rigorous thinking about quantitative matters that, thanks to al-Khwarizmi, we call algebra. Only now, we are back to the realm, not of symbol manipulation, but codified, logical, rigorous thinking about issues in our lives and in the world we inhabit.<br /><br />To be sure, symbolic algebra is not going away. It is way too powerful to ignore. But whereas it used to be possible to provide a rationale for teaching algebra as pure, rule-based symbolic manipulation (albeit a societal rationale that views people as fodder for industry), it makes no sense to teach it that way today.<br /><br />Which is why the Common Core now directs the focus not on the symbolic rules that dominated math instruction for centuries past, but on sophisticated mathematical thinking skills that develop and require a deeper understanding of numbers. This is why there is now so much talk of “number sense” and why Mary and Johnny are coming home from school with homework questions that their parents often find strange and occasionally incomprehensible. <br /><br />In other words, algebra has returned to its roots. (Pun intended.)<br /><br /><b>END OF PART 1</b><br />In Part 2 of this commentary, to be published here next month, I will look at how those same digital technologies that have rendered obsolete much of what used to constitute K-12 algebra education, have provided new ways to teach the subject that are ideally suited to the way we use — and will increasingly use even more — algebra. After all, if the printing press turned algebra from prose to symbolic expressions, what will algebra look like now that the digital computer, and in particular the tablet device, has largely replaced the printing press?<br /><br /><b>NOTE</b>: I realize that there is little in this month’s post that is new to MAA members. But as I know from emails and comments I receive, <i>Devlin’s Angle</i> posts find their way to a wide variety of readers, occasionally onto the desks of governors, education administrators, and others who play a role in the nation’s education system. With so much media attention currently being given to a mathematics education proposal being made by an individual having little knowledge of mathematics or current mathematics education (see last month’s <a href="http://devlinsangle.blogspot.com/2016/03/the-math-myth-that-permeates-math-myth.html">column</a>), I thought it timely to bring us back to an appreciation of algebra (i.e., algebraic thinking) that was apparent to a Ninth Century Caliph in Baghdad, and which is even more relevant to our lives today than it was back then. <br /><br />I cannot avoid ending by observing that 2016 will surely go down as the year when the US media devoted more media space and time to individuals pontificating on topics they knew almost nothing about, than they did to experts, of which the United States has large numbers with global reputations. I think many editors would benefit from a (good) course in algebraic thinking.http://devlinsangle.blogspot.com/2016/04/algebraic-roots-part-1.htmlnoreply@blogger.com (Mathematical Association of America)3tag:blogger.com,1999:blog-2516188730140164076.post-5688277157532213592Tue, 01 Mar 2016 15:11:00 +00002016-03-02T10:53:47.183-05:00algebraAndrew HackerCommon CoreIs algebra necessarymathematics educationquantitative literacyThe Math Myth that permeates “The Math Myth”March 1 saw the publication of the book <i>The Math Myth: And Other STEM Delusions</i>, by Andrew Hacker. MAA members are likely to recognize the author’s name from an opinion piece he published in the New York Times in 2012, with the arresting headline "<a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html">Is Algebra Necessary?</a>"<br /><br />Yes, I thought you’d remember it! It’s almost up there with John Lennon’s murder in terms of knowing where you were at the time you first heard of it. But just to be sure we are all on the same page, let me recap that, in that essay, Hacker, a retired college professor of political science who over the years had taught some non-majors math courses, laid out a case for dropping algebra as a required course in K-12 and college.<br /><br />Before I dive into Hacker’s new book, you would be advised to refresh your memory of the case he presented in that article, since his book is essentially an extension of what he said then, expanded to cover the entire Common Core Mathematical Standards. Prior to writing this review, I wrote an <a href="http://www.huffingtonpost.com/dr-keith-devlin/andrew-hacker-and-the-cas_b_9339554.html">article for the Huffington Post</a> in which I summarized, with my commentary, his 2012 article, together with a <a href="http://chronicle.com/article/The-Case-Against-Mandating/235500">recent interview</a> he gave to the Chronicle of Higher Education.<br /><br />In my article, I noted that Hacker has no idea what algebra really is. His focus is entirely on school algebra as it is very often taught, as a collection of rules for manipulating symbolic expressions. What his argument actually establishes, with sound arguments and good examples, are two conclusions I would agree with:<br /><ol><li>Algebra as typically taught in the school system is presented as a meaningless game with arbitrary rules that does more harm than good.</li><li>There are strong arguments for teaching algebra as it was originally developed and how professional mathematicians today view it.</li></ol>I’ll leave you to read my HuffPost piece for more of the gory details. For the benefit of lay readers who may come to this site, I should though repeat here the brief summary I gave in that article of the difference between algebra (as mathematicians understand and practice it) and the rule-based-manipulation-of-symbolic-expressions that so often passes for algebra in our schools.<br /><br />First codified by the Persian mathematician al-Khwarizmi in his book <i>The Compendious Book on Calculation by Completion and Balancing</i> (balancing = <i>al-Jabr</i>), written in Bhagdad around CE 820, algebra is a powerful method for solving numerical problems more efficiently than by arithmetic. It does so by introducing two new ways of handling numerical problems.<br /><br />First, algebra provides methods for handling entire classes of numbers, rather than specific ones. (That’s where those x’s, y’s, and z’s come in, but that’s just an implementation detail introduced in France several centuries later.)<br /><br />Second, it provides a way to find numerical answers not by computing, which is often very difficult, but by reasoning logically to hone-in on the answer, using whatever information is available. Thus, whereas in arithmetic you work forwards, starting with numbers and computing with them to arrive at an answer, in algebra you work backwards, starting by postulating an answer and reasoning logically to figure out what it is. True, this powerful application of human logical reasoning capacity frequently gets boiled down to mastering various symbolic procedures to “Solve for x,” but again that’s just a particular implementation. Numerical forensics would be a sexier, and more descriptive, term for the real thing.<br /><br />The familiar symbolic expressions calculus usually taught in schools as “algebra” was a particular implementation of al-Khwarizmi’s ways of thinking introduced by the French mathematician François Viète in the 16th Century (700 years after algebra first began) to streamline paper-and-pencil problem solving. A more recent implementation of algebra is the computer spreadsheet.<br /><br />Since his new book follows the same line of attack as his 2012 opinion piece, but with his sights widened from school algebra to the Common Core, instead of crafting another analytic essay, I will do what Hacker himself does, and list a number of examples to make my case. More precisely, I’ll select some of the 20 instances (in a book of just over 200 pages) where I found a claim that is either plain wrong, wildly misleading, or otherwise problematic, and ask where he went wrong. In marking 20 pages, it’s likely I missed some. There were so many wild and inaccurate claims, I frequently found myself skimming through.<br /><br />First though, I should repeat what I said in my HuffPost article about his algebra piece. Just as his essay actually amounted to a strong argument in favor of teaching algebra to all students (albeit not the rule-based manipulations of formulas so often presented in place of algebra), so too his book includes a strong argument in favor of Common Core Math. In the same way that Hacker mischaracterized algebra in 2012, so too his portrayal of the CCSSM (Common Core State Standards for Mathematics) is totally at odds with the real thing—though not quite so far off if you turn your attention from the Standards themselves to some <i>implementations</i> of the CC.<br /><br />One of the book’s flaws is that Mr Hacker seems to get carried away with the flow of his rhetoric, since for the most part his argument consists of the erection of a series of straw men which he then, in time-honored tradition, proceeds to attack.<br /><br />“It’s a waste of time forcing kids to master azimuths and asymptotes,” he cries [not an exact quote] as early as page 2.<br /><br />I had to look up the word azimuth, since in my entire career as a mathematician and mathematics educator, I had never come across it. According to Wikipedia, azimuth is a “concept used in navigation, astronomy, engineering, mapping, mining and artillery.” I ran a search for the word on the entire, 93-page CCSSM document and, as I expected, it did not turn up. Straw man.<br /><br />Asymptotes are a different matter, of course, since a general sense of asymptotic behavior of functions is useful in many walks of life. The word is mentioned, but just once, in the CCSSM, in the section on Interpreting Functions (F-IF), where it says:<br /><br /><i>Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.</i><br /><br />That’s it. One mention, buried towards the end of the document, in the section that says the student should:<br /><ul><li>Understand the concept of a function and use function notation</li><li>Interpret functions that arise in applications in terms of the context</li><li>Analyze functions using different representations</li></ul>From the overall thrust of Hacker’s argument, I think it’s clear he believes this kind of knowledge is indeed important for everyone to have. But it’s also clear it is not a central pillar of the CC, to be used on page 2 to set the scene for what his book is about.<br /><br />Unfortunately, this example is indeed a good characterization of his overall argument: to knock down straw men.<br /><br />“<i>We’re told that if our nation is to stay competitive, on a given morning all four million of our fifteen-year-olds will be studying azimuths and asymptotes</i>,” he writes. (I am still on page 2, with over 200 more pages to go.) He provides no citation regarding who, exactly, is making this proclamation for the nation’s future. It’s not just disingenuously misleading, it’s about as far from reality as you could imagine, and not because of those azimuths. (See momentarily for the real story.)<br /><br />He continues, “<i>Then, to graduate from high school, they will face tests on radical notations and elliptical equations</i>.”<br /><br />To be sure, you will find mention of the word radical in the CCSSM, in the context of “Work with radicals and integer exponents” in the Section on Expressions and Equations (8.EE), which provides the helpful illustration,<br /><br />“<i>For example, estimate the population of the United States as 3 × 10<sup>8</sup> and the population of the world as 7 × 10<sup>9</sup>, and determine that the world population is more than 20 times larger</i>.”<br /><br />Again, this is exactly the kind of thing Hacker says (towards the end of his book) students should be able to do! And it is entirely reasonable that they be asked to demonstrate that ability on a test.<br /><br />“Elliptical equations” is another straw man.<br /><br />The point is, what Hacker keeps attacking are straw men. The CCSS is just what its name implies, a set of standards. It is not a curriculum, nor does it specify anything remotely like a daily, or even weekly timetable. How and when teachers across the land cover the various standards is for them, or perhaps their school district, to decide. As far as the CCSS are concerned, teachers can operate fluidly, depending on how their class progresses. (And no one will even suggest that they mention azimuths, let alone force the class to master them.)<br /><br />I would hazard a guess that Hacker has never looked at the CCSS document. Nor sat in on many math classes, as I have, and observed what actually goes on in today’s schools.<br /><br />Caveat: I get to see classes to which, for one reason or another, I have been invited to visit. Likely they are some of the best, since their teachers invite me along so their students can talk for a while with someone who has devoted a career to mathematical research. I hear enough stories to be prepared to believe things are often a lot worse. Perhaps even as bad as Hacker says. But his book is purported to be about educational policy, not what you can actually find in good or bad classrooms.<br /><br />Not only does Hacker give no indication he is familiar with the Common Core—the real one, not the azimuth-strewn, straw-man version he creates—he gives every indication he does not understand mathematics as it is practiced today. (He also does not know that pi is irrational, but I’ll come to that later.)<br /><br />Certainly, the examples he selects to illustrate the irrelevancy (in today’s world) of some of the test problems students are asked to solve simply demonstrate that he is lacking the basic, every-day, number sense he is arguing for. Let me give just three examples.<br /><br />On page 48, Hacker presents a question he took from an MCAT paper. It provides some technical data and asks what happens to the ratio of two inverse-square law forces between charges of given masses when the distance between them is halved. The context Hacker provides for this question is that medical professionals needs to be able to read and understand the mathematics used in technical papers. His claim is that this requirement does not extend to the physics of electrical and gravitational forces. In that, he is surely correct. But anyone with a grain of number sense will recognize at once that the setting is totally irrelevant. It’s a simple question about what happens to a ratio when the underlying scale is changed. The answer, of course, is nothing happens. It’s a ratio. The changes to the numerator and denominator cancel out. The ratio remains the same.<br /><br />What this question is asking for is, <i>Do you understand what a ratio is</i>? Surely that is something that any medical professional who will have to read and understand journal articles would need to know. Hacker completely misses this simple observation, and presents the question as an example of baroque mathematical testing run amok.<br /><br />On page 70, he presents a question from an admissions test for selective high schools. A player throws two dice and the same number comes up on both. The question asks the student to choose the probability that the two dice sum to 9 from the list 0, 1/6, 2/9, 1/2, 1/3. Hacker’s problem is that the student is supposed to answer this in 90 seconds. Now, I share Hacker’s disdain for time-limited questions, but in this case the answer can only be 0. It’s not a probability question at all, and no computation is required. It just requires you to recognize that you can never get a sum of 9 when two dice show the same number. As with the MCAT question, the question is simply asking, <i>Do you understand numbers</i>? In this case, do you recognize that the sum of two equal numbers can never be odd.<br /><br />Finally, on page 101, Hacker presents a list of mathematics requirements high school students must meet in order to study at Harvard and similar universities. The list includes the names of various kinds of analytic functions. As usual, Hacker seems overwhelmed by the technical terms, or worries that the students will be, but all the list is asking for is that students can read graphs and charts and know what they represent in terms of growth and change. An essential skill, surely, for anyone in today’s information-rich world, not just students at elite universities.<br /><br />You get the pattern surely? Hacker’s problem is he is unable to see through the surface gloss of a problem and recognize that in many cases it is just asking the student if she or he has a very basic grasp of number, quantity, and relationships. Yet these are precisely the kinds of abilities he argues elsewhere in the book are crucial in today’s world. He is, I suspect, a victim of the very kind of math teaching he rightly decries—one that concentrates on learning rules and mastering formal manipulations, with little attention to understanding.<br /><br />This, surely, explains why he would write (page 96), “<i>Reasoning mathematically may be a nice skill, but it is not relevant to most of life. We reason about many things: parenting, marriage, careers. Do we learn how to reason about these things by learning algebra?</i>”<br /><br />If he had asked instead if we learn such reasoning in a typical school algebra class, I would agree with his implied answer of “No.” But algebra arose by codifying the everyday reasoning people carried out—and still carry out today—about the numerical or quantity aspects of any human activity that involves them. (Trade, commerce, and civil engineering were the original applications.)<br /><br />From that historical perspective, it is absolutely clear that learning algebra can help us master such reasoning. It helps by providing an opportunity to carry out that kind of reasoning free of the complexities a problem generally brings with it when it arises in a real world context.<br /><br />The tragedy of <i>The Math Myth</i> is that Hacker is actually arguing for exactly the kind of life-relevant mathematics education that I and many of my colleagues have been arguing for all our careers. (Our late colleague Lynn Steen comes to mind.) Unfortunately, and I suspect because Hacker himself did not have the benefit of a good math education, his understanding of mathematics is so far off base, he does not recognize that the examples he holds up as illustrations of bad education only seem so to him, because he misunderstands them.<br /><br />The real myth in <i>The Math Myth</i> is the portrayal of mathematics that forms the basis of his analysis. It’s the same myth you see propagated in Facebook posts from frustrated parents about Common Core math homework their children bring home from school.<br /><br />In the interests of their overall cardiovascular health, I have to recommend that math educators do not read <i>The Math Myth</i>. But if you do, perhaps you should start with the final chapter, titled “Numeracy 101.” Here, at least, you will find things you are likely to agree with, as he lays out what he believes would be a good quantitative literacy course for college students.<br /><br />But even there, where all seems warm and friendly and positive, you will be jolted by Hacker’s fundamental lack of knowledge of mathematics. He writes,<br /><br />“<i>Along with phenomena like earthquakes and cyclones, nature also has some numbers that control or explain how the world works. One of them is pi, whose</i> 3.14159<i> goes on indefinitely, at least as far as we know</i>.”<br /><br />Yes, you read that last part correctly.<br /><br />“Few people writing today … can make more sense of numbers” proclaims the <i>Wall Street Journal</i> on the cover of Hacker’s book. Well, if that’s the view of the newspaper that purports to have the expertise to cover the nation’s financial markets, it is only a matter of time before we have another financial meltdown.http://devlinsangle.blogspot.com/2016/03/the-math-myth-that-permeates-math-myth.htmlnoreply@blogger.com (Mathematical Association of America)9tag:blogger.com,1999:blog-2516188730140164076.post-7574208066658684828Thu, 11 Feb 2016 15:49:00 +00002016-02-11T16:40:24.816-05:00exceptional performanceHypercubesmathematical modelingTheorem: You are exceptional“Everyone excels at something.” We hear it all the time, usually said to console someone who is miserable after underperforming at something. Parents, in particular, often fall back on it with their children. What few people realize, though, is that the statement can be mathematically verified. You need only consider a collection of 200 essentially independent human performance characteristics for 98% of people to measure as exceptional in at least one of them, where exceptional is defined as being in the top or bottom 1%. (The mathematics gives extremal values; if you want to effectively guarantee being in the top 1%, you need more characteristics. The phenomenon is asymptotic.) <br /><br />This result is a consequence of a rather surprising, but little known, observation about high-order hypercubes: as the dimension increases, the proportion of points in the interior (i.e., not on the bounding shell) decreases without limit.<br /><br />Here is how you can prove to your child, spouse, student, best friend, or whoever, that they—or you, as the circumstances may require—can or will excel at something.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-5aVnO2xvjz0/VrokChM7TsI/AAAAAAAAKc0/8EX50jzeub8/s1600/Fig1.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="118" src="https://2.bp.blogspot.com/-5aVnO2xvjz0/VrokChM7TsI/AAAAAAAAKc0/8EX50jzeub8/s200/Fig1.jpg" width="200" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small; text-align: start;">Fig 1. The bell curve (normal distribution)</span></td></tr></tbody></table><br />Everyone is familiar with the bell curve (normal distribution) showing the typical distribution of performance measures of a single characteristic across a sufficiently large population. This graph captures the fact that the scores for the majority of the population cluster around an “average”, middling value, with only a few individuals at either end (exceptionally poor or exceptionally good).<br /><br />For the purposes of the multi-dimensional computation, we can start with a geometrically simpler model, namely the closed interval [0,100], as in Figure 2. We define the <i>exceptional</i> points to be those in the unit intervals at each end. In this model, for a single characteristic, only 2% of the population are exceptional. The remaining 98% are “normal."<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-tCymcfRBOkU/VryqRi4oAHI/AAAAAAAAKds/DJQjtHJYhWw/s1600/Devlin%2Bnew%2Bfig.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://2.bp.blogspot.com/-tCymcfRBOkU/VryqRi4oAHI/AAAAAAAAKds/DJQjtHJYhWw/s1600/Devlin%2Bnew%2Bfig.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small; text-align: start;">Fig 2. A simple model of exceptionality in one characteristic</span><br /><div><br /></div></td></tr></tbody></table>Now consider two characteristics, X and Y (assumed to be independent). The distribution then is represented by a 100x100 square with an inner 98x98 cube, as in Figure 2.<br /><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-lYwIYcaoWWY/VryqmDLkSZI/AAAAAAAAKdw/94ISWc-PVoA/s1600/Devlin%2Bnew%2Bfigure.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://3.bp.blogspot.com/-lYwIYcaoWWY/VryqmDLkSZI/AAAAAAAAKdw/94ISWc-PVoA/s1600/Devlin%2Bnew%2Bfigure.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small; text-align: start;">Fig 3. A simple model of exceptionality in two characteristics</span></td></tr></tbody></table><br />An individual’s X measure is shown by the x-coordinate, their Y measure by the y-coordinate. The ordinary individuals are represented by points in the inner square; the exceptional individuals by points in the outer perimeter region.<br /><br />The total number of points is 100x100. The number of normal points is 98x98. So the number of exceptional points is 10,000 – 9,604 = 396.<br /><br />The proportion of exceptional points is thus 396/10,000 = 0.0396, i.e., 3.96%. Thus, more individuals are classified as exceptional when you consider two characteristics (3.96% as opposed to 2%).<br /><br />Going to three characteristics, X, Y, and Z, the model will be a 100x100x100 cube with an inner 98x98x98 cube, as in Figure 4.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-sNY53gvcS9M/Vrok_bk8b6I/AAAAAAAAKdc/Xu-qb4bYA_U/s1600/Devlin%2Bfig4.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://2.bp.blogspot.com/-sNY53gvcS9M/Vrok_bk8b6I/AAAAAAAAKdc/Xu-qb4bYA_U/s1600/Devlin%2Bfig4.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small; text-align: start;">Fig 4. A simple model of exceptionality in three characteristics</span></td></tr></tbody></table><br />The volume of the outer cube (representing the total population) is 1,000,000. The volume of the inner cube (representing the normal individuals) is 941,192. Thus the volume of the perimeter-region (representing the exceptional individuals) = 1,000,000 – 941,192 = 58,808. Hence, the proportion of exceptional individuals = 58,808/1,000,000 = 0.0588, i.e. 5.88%.<br /><br />So far, everything seems fairly straightforward and reasonable. Going beyond three characteristics, the model is a hypercube of four or more dimensions, and we can no longer provide meaningful illustrations. But by now we have grown familiar with the idea that the model represents exceptional individuals by points in the outer 1% shell. To see what this entails, let’s jump to 10 characteristics, X<sub>1</sub>,…,X<sub>10</sub>. In that case, our model will represent the situation as a 100<sup>10</sup> hypercube with an inscribed 98<sup>10</sup> hypercube.<br /><br />The volume of the outer hypercube (~ total population) = 100<sup>10</sup>. The volume of the inner hypercube (~ normal individuals) = 98<sup>10</sup>. Thus, the volume of the perimeter-region (~ exceptional individuals) = 100<sup>10</sup> – 98<sup>10</sup>, and the proportion of exceptional individuals = (100<sup>10</sup> – 98<sup>10</sup>)/100<sup>10</sup>. At this point, it’s time to bring in Wolfram Alpha to do the calculation. This gives the result that, with 10 characteristics, 18.29% of the population is exceptional.<br /><br />With 100 characteristics, X<sub>1</sub>,…,X<sub>100</sub>, our model gives: Volume of hypercube (~ total population) = 100<sup>100</sup>. Volume of inner hypercube (~ normal individuals) = 98<sup>100</sup>. Volume of perimeter- region (~ exceptional individuals) = 100<sup>100</sup> – 98<sup>100</sup>. Proportion of exceptional individuals = (100<sup>100</sup> – 98<sup>100</sup>)/100<sup>100</sup>. Calling on Wolfram Alpha again, we compute that with 100 characteristics, 86.74% of the population is exceptional.<br /><br />With 200 characteristics, X<sub>1</sub>,…,X<sub>200</sub>, our model gives: Volume of hypercube (~ total population) = 100<sup>200</sup>. Volume of inner hypercube (~ normal individuals) = 98<sup>100</sup>. Volume of perimeter-region (~ exceptional individuals) = 100<sup>200</sup> – 98<sup>200</sup>. Proportion of exceptional individuals = (100<sup>200</sup> – 98<sup>200</sup>)/100<sup>200</sup>. So with 200 characteristics, 98.24% of the population is exceptional. (Once again calling on the services of the normally unflappable Wolfram Alpha.)<br /><br />And there’s our result.<br /><br />Of course, we have been working with a model. As always, that entails making various assumptions and simplifications. If the final result surprises you, you have two choices. Either go back and modify your initial assumptions and generate another model. Or accept the result and modify the prejudices that led to your surprise. <br /><br />In this case, we have to accept that in higher dimensions, almost all the material in an equal-sided, rectangular, <i>solid</i> (!) box is on the outer shell. The (solid) inside is almost empty.<br /><br />When we consider more dimensions to a situation, the math can sometimes lead us to a counter-intuitive—but correct—conclusion we did not expect. Not everyone can accept that.<br /><br />Yes, in this US Election Season, this is a story with a moral. <a href="https://1.bp.blogspot.com/-vUhjWsgB9yc/Vryrb7op4BI/AAAAAAAAKeE/Kmjb2tYa-f4/s1600/simely.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://1.bp.blogspot.com/-vUhjWsgB9yc/Vryrb7op4BI/AAAAAAAAKeE/Kmjb2tYa-f4/s1600/simely.jpg" /></a><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br />http://devlinsangle.blogspot.com/2016/02/theorem-you-are-exceptional.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-3603082345084297799Fri, 01 Jan 2016 13:30:00 +00002016-01-28T11:01:05.113-05:00an approach that sucksBedtime MathLaura Overdeckmath appsmath learningStanford UniversityUniversity of ChicagoWuzzit TroubleDo your kids find learning math hard? There may be an app for that!<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-CV53K7QV8Es/VnHsJZHXpwI/AAAAAAAAKcE/U9gPiFFjWH4/s1600/BedtimeMath.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-CV53K7QV8Es/VnHsJZHXpwI/AAAAAAAAKcE/U9gPiFFjWH4/s320/BedtimeMath.png" width="213" /></a></div>If you are like me, you probably sigh and switch off when you read an article with a title claiming kids’ math scores show significant improvement after using some great new app for a few minutes each day. <br /><br />In which case, you may have paid little attention when a<a href="http://news.sciencemag.org/brain-behavior/2015/10/bedtime-problems-boost-kids-math-performance"> news article</a> came out in <i>Science</i> magazine recently, reporting a new study showing that after just one year of parents using a bedtime-story-telling app called<i> <a href="https://itunes.apple.com/us/app/bedtime-math/id637910701?mt=8">Bedtime Math</a></i> with their young children, those kids were three months ahead of fellow students whose parents were using an app to provide non-mathematical stories. In fact, children of math-anxious parents showed even greater improvement, ending up six months ahead. <br /><br />If you happened to see the article, you likely assumed it was essentially a piece of marketing, where a bogus “study” was carried out to produce the “results” the marketing folks wanted. After all, there are no magic bullets in the math ed world, right? <br /><br />My reaction was very different. I immediately wanted to know more. The reason being, as I recounted in <a href="http://devlinsangle.blogspot.com/2015/12/life-inside-impossible-escher-figure.html">last month’s column</a>, the same thing had happened a few months earlier with a math learning app I had created, a mobile game called <i><a href="https://itunes.apple.com/us/app/wuzzit-trouble/id600190128?mt=8">Wuzzit Trouble</a></i>. (Actually, I should have written “we” there. Paul McCartney may have sung that he had “got by with a little help from my friends,” but the reality was it took a lot of work by all four Beatles to make them a global phenomenon, and in the apps business it usually takes a whole team of highly talented people to create a great product. My team are listed <a href="http://www.brainquake.com/core/">here</a>.) A Stanford classroom study led by Prof Jo Boaler had found significant math learning after just two hours play of <i>Wuzzit Trouble</i> spread over four weeks.<br /><br />The <i>Bedtime Math</i> study was unlikely to have been a bogus marketing “study”, I felt, since it had been carried out at the University of Chicago, which is a great university. True, as the <i>Science</i> article noted, the study was funded by an entity called the Overdeck Family Foundation, whose chair, Laura Overdeck, a former astrophysics major at Princeton, established the nonprofit Bedtime Math Foundation, which created and supports the app. Some might read that and smell a rat – as some did when they first read of the Stanford <i>Wuzzit Trouble</i> study.<br /><br />But to my mind such a reaction says more about the reader than the researchers. We are talking about an educational app made and distributed for free by a nonprofit organization. Why would anyone want to fake data? Really!<br /><br />In fact, even if the app were for sale – say for a mind boggling $4.99 – the idea that seven researchers at a major university would fake a study about a small children’s app is simply not credible. As a number of news articles have made clear, the price for a university researcher faking a scientific study is dismissal from the university and the end of their career. When it does happen, the motivation is invariably massive career prestige and fame, or a huge follow-up research grant (or both). Neither of which are likely to result from an at best encouraging, small scale classroom study of <i>Bedtime Math</i>, <i>Wuzzit Trouble</i>, or any other kids’ app.<br /><br />If a foundation or a company wanted to run a fake study for marketing purposes, they could simply do it themselves, or else farm it out to an unscrupulous, individual researcher. Such people are to be found, sometimes associated with universities. (Google “intelligent design” or “climate change denial” for examples.) <br /><br />Certainly, James Stigler, a well known educational psychologist at UCLA, is not skeptical. <i>Science </i>quotes him as saying, "I think it's a fantastic study. But it is just the beginning." <br /><br />Another respected scholar, Andee Rubin, a mathematician and computer scientist at the nonprofit education R&D company TERC in Cambridge, Mass, has a similar reaction. <i>Science</i> quotes him as observing, "I'm interested in teasing it apart and seeing what makes this effective." <br /><br />Those are pretty much the same as the reaction I had to Stanford’s <i>Wuzzit Trouble</i> results, which prompted me to draw up the list of possible explanatory factors I published in my last column.<br /><br />With both the full paper and a cover article available in <i>Science</i>, all I will do here is provide a brief summary of the cover article.<br /><br />The Chicago team recruited 587 first-graders from 22 schools in the Chicago metropolitan area. The parents of each child were given a tablet computer with which to read to the child at bedtime. 420 families were told to use it to work through word problems related to counting, shapes, arithmetic, fractions, and probability using <i>Bedtime Math</i>. Another 167 families were instructed to use a reading app. With a standardized test, the researchers assessed all the subjects' mathematics performance at the beginning and end of the school year.<br /><br />As you would expect, use of the reading app made little difference to the children's math performance. In contrast, children who used the math app two or more times per week outpaced peers whose family rarely used it, ending up three months ahead.<br /><br />Perhaps most important, use of the app brought students whose parents said they were anxious about math up to par with those whose parents were at ease with the subject. Among children whose family rarely used the math app, those with math-phobic parents made only half as much progress as the children of parents comfortable with math. <br /><br />The researchers make some suggestions as to what may be going on. My own best guess, based on several months reflections on the Stanford and (subsequent) Finnish studies of <i>Wuzzit Trouble</i>, are consistent with what they think. Namely:<br /><br />We have created a system where learning is walled off from everyday life. Particularly in math. The “math classroom” operates according to its own rules. Even with a truly great teacher – and I have met many – there are many restrictions on what can be done. Not least because of an incessant rhythm of performance testing. <br /><br />Go into most math classrooms and what you see will most likely bring to mind a room full of clerks in the pre-computer age when companies employed large numbers of numerically-able people to crunch their numbers. (Young people will have to rely on old photographs or depictions in movies.) Which was, of course, what the system was set up to provide.<br /><br />The classroom certainly does not look remotely like a room full of professional mathematicians at work. The first words that might come to mind if you were to walk into such a room would be “fun”, “engagement”, “argument”, “passion”, “social interaction”. <br /><br />Nor does it look like the human activity that hundreds of thousands of years of natural selection have inbred into us to maximize learning in the young: play. (Some wise person once said that “play is the work of the child.” I agree.) <br /><br />And there is something else that evolution hard-wired into is: our love of stories. Effective political speeches are usually laden with stories of individual people, and for very good reason. Because they are powerful.<br /><br />At which point, it’s probably a good idea to do what we math instructors tell our student to do: look for patterns. Well, what do we see when we look at professional mathematicians at work and kids in their ideal learning activity?<br /><br />Fun, engagement, argument, passion, social interaction, play, stories. <br /><br />Those are all, I would argue, <b><i>essential</i></b> ingredients for good learning. Yet you would have difficulty finding <b><i>any</i></b> in many math classrooms. <br /><br />Indeed, society seems to have gotten into a mindset that these items are distractions that you have to eliminate to achieve good math learning. Even when good teachers do their best to inject some of those valuable features into their classes, they have to operate within a system that disapproves. And everyone knows that, most of all the kids.<br /><br />No wonder then, that when you have a well designed, engaging app – a game, a puzzle, a family-supporting bedtime story provider, or whatever – you will likely get good results. Because apps, if properly designed, create their own environment.<br /><br />Though I labored long and hard to create <i>Wuzzit Trouble</i>, and I am sure Laura Overdeck worked equally hard on <i>Bedtime Math</i>, all to good effect for sure, my strong suspicion is our apps work as well as they do <b><i>in such a short time</i></b> primarily because of <b><i>what they are not</i></b>. Namely they are not the typical school classroom approach to mathematics education.<br /><br />How else can you explain dramatic results after a few hours engaging with an app, other than it unlocking what had hitherto been shackled by the chains of an Industrial Revolution conception of mathematics learning? Learning something that is genuinely new takes time and a lot of effort. Freeing something that is already there – if only in embryonic form – is much quicker. If so, then this means that our received wisdom that it takes a lot of hard work, repetitive practice, frustration, tears, and pathological levels of anxiety to achieve competency in mathematics may simply be very strong evidence that our approach sucks.http://devlinsangle.blogspot.com/2016/01/do-your-kids-find-learning-math-hard.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-883903209776583603Fri, 04 Dec 2015 19:54:00 +00002015-12-16T17:40:34.303-05:00game-based learningJames Paul GeeJo BoalerM.C. EscherMonument Valley gameLife inside an impossible Escher figure<div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-JomIwhHa-6o/VmHfWQuhX5I/AAAAAAAAKb0/8aQUlnZ9rkc/s1600/MonumentValley2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="175" src="http://3.bp.blogspot.com/-JomIwhHa-6o/VmHfWQuhX5I/AAAAAAAAKb0/8aQUlnZ9rkc/s400/MonumentValley2.jpg" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"></div><span id="goog_2140169023"></span><span id="goog_2140169024"></span><br />When the M.C. Escher inspired puzzle video-game<i> <a href="http://www.monumentvalleygame.com/" target="_blank">Monument Valley</a> </i>came out last year, I knew I had to check it out. The more so when it started getting rave reviews and winning awards. But with so many other things to hold my attention, I never managed to get round to it. The recent decision of the creators to make a version available for free prompted me to finally take a quick look. Not that it had been expensive. Rather, a tweet about the new free version happened to come when I had an hour or so of free time on my hands. <br /><br />That free hour got immediately swallowed up, as did many more hours after that. I was hooked from the getgo. <br /><br />Anyone who is intrigued by those impossible figures that Escher made famous, even those of you with little or no interest in puzzle (video-) games, will surely be captivated by <i>Monument Valley</i>, where the solution to many of the puzzles involves orienting the figure to create an illusion of a continuous object. For when the player views the object as continuous, characters in the game that traverse the figures can move along it. Impossible chasms that prevented a character’s progress suddenly disappear as you rotate the entire figure just the right way.<br /><br />It’s not a learning game. I don’t see a player learning any new mathematics. But what it does is provide a rich, immersive experience of complex geometrical spaces <i>from the inside</i>. As a player, your task is to assist the princess on her quest, which involves finding her way through a fantasy world of Escher-like structures, the geometry of which you can sometimes change as you progress. By projecting yourself into the princess, you get a sense of what it would be like to live in such a world. <br /><br />And a beautiful world it is. The creators, based in the UK and operating under the name UsTwo, have crafted a series of truly gorgeous fantasy worlds, which you encounter one after another. It is not so much a game as a collection of interactive pieces of art where you play with, and experience, geometric shapes.<br /><br />In fact, it is the artistic creation that the developers bring to the work. The idea of taking Escher worlds and turning them into a puzzle game goes back to a 2007 video game called <a href="https://www.youtube.com/watch?v=QfICeBtVv8U" target="_blank">Echodrome</a>, designed for the Sony Play Station 3 by the Japanese designer Jun Fujiki. By all accounts it was fiendishly difficult, and never broke out beyond a small group of hard-core puzzle aficionados. <br /><br /><i>Monument Valley</i> shows the huge difference presentation can make. If you want to hold people’s attention, you often need to think carefully about the medium. The message on its own may not be enough. That holds in the math class or the math lecture hall as much as in a video game.<br /><br />Regular readers of this MAA blog or my other blog profkeithdevlin.org will know that I have a long-standing interest in video games, particularly so as an educational medium, where I am professionally active as a player, a learning researcher, and an entrepreneur. <br /><br />In fact, much of my career has involved looking for ways to use different media to make mathematics accessible to as many people as possible. I have authored many “<a href="http://www.amazon.com/Keith-Devlin/e/B000APRPC6/ref=ntt_athr_dp_pel_1" target="_blank">popular mathematics books</a>”, written for newspapers (MAA compilation of some of my articles<a href="http://www.amazon.com/All-Math-thats-Fit-Print/dp/0883855151" target="_blank"> here</a>), worked on television programs (including <a href="https://vimeo.com/127338218" target="_blank"><i>A Mathematical Mystery Tour</i></a>, BBC-tv 1984; <a href="http://www.amazon.com/Life-Numbers-Danny-Glover/dp/B000FVQM10" target="_blank"><i>Life by the Numbers</i></a>, PBS 1998; and <i><a href="http://www.amazon.com/Numb3rs-The-Complete-First-Season/dp/B000ERVJKE" target="_blank">NUMB3RS</a></i>, CBS, 2005-2008), and of course there is my regular <a href="http://profkeithdevlin.com/MathGuy.html" target="_blank"><i>Math Guy</i></a> radio gig for NPR, which started in 1994. More recently, in 2012, I launched the first ever <a href="https://www.coursera.org/course/maththink" target="_blank">math MOOC</a> on Coursera (the seventh session just ended). I even made a foray into using music, song, and dance, with the 2007 show <a href="http://profkeithdevlin.com/HE.html" target="_blank"><i>Harmonius Equations</i></a>. To me, video games are one more medium to carry mathematical content.<br /><br />In fact, when it comes to K-12 mathematics, video games are in many ways the most effective medium we currently have to provide good math learning, as I tried to articulate in a <a href="http://www.amazon.com/Mathematics-Education-New-Era-Learning/dp/1568814313/ref=sr_1_1?ie=UTF8&s=books&qid=1299701150&sr=8-1" target="_blank">book</a> I wrote in 2011, and a presentation I gave at the big Teaching and Learning 2014 conference in Washington D.C. last year, a 20-min video summary of which is available <a href="https://vimeo.com/89583763" target="_blank">here</a>.<br /><br />Until recently, there was relatively little research available to put any flesh onto educators’ beliefs/hopes/suspicions that video games could yield good math learning outcomes. That is starting to change. (Reports from two classroom studies, one of I was involved in, are due to be published in the <i><a href="http://journal.seriousgamessociety.org/" target="_blank">International Journal of Serious Games</a></i> this month. Preprints are available<a href="http://documents.brainquake.com/backed-by-science/Stanford-Pope-Mangram.pdf" target="_blank"> here</a> and <a href="http://profkeithdevlin.com/Papers/Kiili-Devlin_2015.pdf" target="_blank">here</a>.)<br /><br />Certainly, the results obtained in those two papers raise more questions than they answer. (Moreover, pending further, and substantially larger, studies, the results themselves have to be viewed as tentative.) What we are seeing is that, for mathematics in the K-8 range, significant learning outcomes can be observed after a video-game intervention of as little as two hours play spread over a month or so. (Some measures show an increase of 20% over a comparison group.)<br /><br />I’d seen reports earlier that made similar claims, and dismissed them as product- marketing masquerading as research. It was only when the first of the two particular studies I cited above came out in late 2014, carried out by Prof <a href="https://www.youcubed.org/" target="_blank">Jo Boaler</a>’s research group at Stanford University’s Graduate School of Education, using my own math learning video game <i><a href="http://www.brainquake.com/our-platform/" target="_blank">Wuzzit Trouble</a></i> as the intervention, that I sat up and really took notice. <br /><br />In fact, I did more than that. Together with a research colleague from Tampere University in Finland, Prof Kristian Kiili, who, like me, has founded a math-learning video game company, and who was spending the year at Stanford, we carried out our own study. (The second of the two papers I cited.) Kiili was developing a fractions learning game, <a href="http://kristiankiili.com/semideus/" target="_blank"><i>Semideus</i></a>, and wanted to see how well it could serve as an evaluation tool. So we repeated essentially the same study Boaler’s team had done, with <i>Wuzzit Trouble</i> as the intervention, but instead of a written pre- and post-test (which the Boaler team used), we used <i>Semideus</i>. The results were very similar to those obtained in the previous study. (With the added twist that this time we found transfer — in a game context — from the whole number arithmetic of <i>Wuzzit Trouble</i> to the fractional reasoning of <i>Semideus</i>.)<br /><br />Something is going on, that’s for sure. But what? It did not take long to come up with a fairly long list of possible factors. Among the many things that a (well designed) math learning game can offer, all which are known to have a positive impact on learning, are:<br /><br /><ul><li><a href="http://www.teachinglearning2014.org/media/blog/breaking-the-symbol-barrier" target="_blank">Breaking the Symbol Barrier</a> – human-friendly representation (not the traditional abstract symbols of math textbooks).</li><li>Focus on developing number sense and problem solving ability.</li><li>High level of engagement.</li><li>Instant feedback (both positive and negative).</li><li>Steady flow of dopamine – known to have positive impact on memory formation and consolidation.</li><li>Learning through failure – in a playful, safe environment.</li><li>“Failure” treated – and regarded – as “not yet succeeded”.</li><li>Constant sense of “I can do this on the next try.”</li><li>Lots of repetition – but at the demand of the student/player.</li><li>Student/player is in control.</li><li>Student/player has ownership.</li><li>Growth Mindset – good games encourage and develop this. (This is the important notion <a href="http://mindsetonline.com/whatisit/about/" target="_blank">Carol Dweck</a> is famous for.)</li><li>Fluid intelligence (Gf) – games require and develop this. (Loosely speaking, this is the ability to hold several pieces of information in the mind at the same time and reason fluidly with them.)</li><br />I have written about many (not all) of these factors in my series of video game learning articles in my blog <a href="http://profkeithdevlin.org/" target="_blank">profkeithdevlin</a>. (See also the many writings and videos on games and learning by <a href="http://www.jamespaulgee.com/" target="_blank">Prof James Paul Gee</a>.)<br /><br />My current guess is that all of these factors, and likely others, are at play in those dramatic learning outcomes. The only way to find out for sure, of course, is to do more research. A lot more. Prof Kiili, now back in Finland, is already hard at work on that, as am I and some of my colleagues at Stanford. And we are by no means alone. The field is wide open. Stay tuned. (Even better, get involved.) Truly, it’s an exciting time to be involved in mathematics education.<br /><br />Meanwhile, I have to sign off. <i>Monument Valley</i> is calling.<br /></ul>http://devlinsangle.blogspot.com/2015/12/life-inside-impossible-escher-figure.htmlnoreply@blogger.com (Mathematical Association of America)0