Thursday, November 16, 2017

Mathematics and the Supreme Court

American 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 “prosecutor’s fallacy."

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 September and October of 2006.

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.

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 November 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.

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.

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 Boston Gazette 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 Gerry and mander to create the new word gerrymander. 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 gerrymandering score, G, is defined by:
G = gP/A, where
g: the district’s boundary length, minus natural boundaries (like coastlines and rivers)
P: the district’s total perimeter
A: the district’s area
The higher the score, the wilder is the apportionment as a geographic region, and hence the more likely to have been gerrymandered.

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.

The Wisconsin Republicans’ algorithm searched through thousands of possible maps looking for one that would look reasonable according to existing criteria, but would favor Republicans no matter what the election day voting profile might look like. As such, it would be a statistical outlier. 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.

One promising way to identify gerrymandering is with a simple mathematical formula suggested in 2015, called the “efficiency gap." 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.

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.

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 efficiency gap. If that works out to be greater than 7%, the systems developers suggest, the districting is unfair.

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.”

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.

There were 2,598M votes cast in total. So the efficiency gap is 279K/2598K = 10.7% in favor of the Democrats.

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 the results 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.

Another approach, developed by a team at Duke University, 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 statistical outlier. 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.

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.

Wednesday, October 11, 2017

Monty Hall may now rest in peace, but his problem will continue to frustrate

Monty Hall with a contestant in Let's Make a Deal.
The news that American TV personality Monty Hall died recently (The New York Times, September 30) 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.

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.

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.)

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.

I have discussed this problem in Devlin’s Angle on at least two occasions, the most recent being December 2005, 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.

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.

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.

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.

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 think 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.

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.

The late Carl Sagan warned of this danger back in 1995, in his book The Demon-Haunted World:Science as a Candle in the Dark, writing:
“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. ...”
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.





Wednesday, September 20, 2017

The Legacy of Jonathan Borwein


Keith Devlin and Jonathan Borwein talk to host Robert Krulwick on stage at the World Science Festival in 2011.

At the end of this week I fly to Australia to speak and participate in the Jonathan Borwein Commemorative Conference 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 obituary written by his long-time research collaborator David Bailey in their joint blog Math Drudge. For more details, check out his Wikipedia entry.]

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 The Computer as Crucible, published in 2008.

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.”

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.

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 Goodbye Descartes) 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.]

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 InfoSense, 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.”)

It was our two very different perspectives on the deliberative blending of mathematics and computers that made our book The Computer as Crucible such a fascinating project for the two of us.

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 Notices, sent out to all members, called “Computers and Mathematics,” the purpose of which was both informational and advocacy.

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.

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.

The “Computers and Mathematics” section of the AMS Notices 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 Experimental Mathematics was published.

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 Notices website has a complete index.] 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.

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.

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.

Monday, August 7, 2017

What 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.

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.

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.)

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.

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.)

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.)

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.

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.

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.

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.

The educational importance of having an active mathematics research group in a university is particularly important in today’s world. As I noted in an article in The Huffington Post 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 Wolfram Alpha. 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.

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.

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.

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.

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.)


Thursday, July 13, 2017

The Power of Simple Representations

The 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,” Disquisitiones Arithmeticae (1801), Article 76.]

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 notation does not matter, a representation can make a huge difference. The distinction is that developing or selecting a representation for a particular mathematical concept (or notion) involves deciding which features of the concept to capture.

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.

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).

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.

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).

In contrast, the shapes 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
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 form of the numerals is very much in Gauss’s “unimportant notations” bucket.

On the other hand, the huge difference a representation system 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 The Man of Numbers.

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.

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 explain it to a graduate math class in an early scene from the 1980s movie It’s My Turn.]

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.

A particularly well known example of such use are the Feynmann Diagrams, simple visualizations to help physicists understand the complex behavior of subatomic particles, introduced by the American physicist Richard Feynmann in 1948.



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.


A discussion of one use of this diagram can be found in a survey article I wrote in the volume Handbook of the History of Logic, Volume 7, edited by Dov Gabbay and John Woods (Elsevier, 2008, pp.601-664), a manuscript version 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 links to publications are on my homepage.)

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.

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.

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.

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. But that is why it can be so effective in assisting a learner arrive at that level of understanding! All it took to convince me was to watch Tanton’s lecture video on Vimeo. Like Tanton, and I suspect almost all other mathematicians, it took me many years of struggle 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.

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!

Wednesday, June 14, 2017

Classroom Clickers Are Good; Except When They Are Not

Many 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. (Shapiro et al, Computers & Education 111 (2017), 44–59) 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 EdSurge put out a feature article on the paper on May 25, which is how I learned of the result.

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.

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.

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.

As the researchers expected, both kinds of questions resulted in improved performance in fact- based questions on a test administered at the end.

Neither kind of question led to improved performance in a problem-based test questions that required conceptual understanding.

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 worse on conceptual, problem solving questions.

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.

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 to the detriment of acquiring the deeper understanding required to solve problems.

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.

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.

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 December 4, 2015 Devlin’s Angle 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 drop in performance on a problem solving test.

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.

That’s not just the secret to learning math. It’s the secret to success in almost any walk of life.

Monday, May 22, 2017

The Math Gift Myth

My 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.

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.

First, my Stanford colleague Professor Jo Boaler sent out a group email pointing to a New York Times article that quoted her, and which, she noted, she helped the author to write. Titled "No Such Thing as a Math Person," 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.

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 The Huffington Post, "All The Mathematical Methods I Learned In My University Math Degree Became Obsolete In My Lifetime," and its follow up article (same date), "Number Sense: the most important mathematical concept in 21st Century K-12 education.")

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 The New York Times article:

“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.”

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.

In her most recent book, Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching, 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.

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 The Friendly Atheist. His post on May 22 was titled "Years Later, the Mother Who 'Audited' an Evolution Exhibit Reflects on the Viral Response." Knowing Mehta’s work (for the record, I have also been interviewed by him on his education-related podcast), that title hooked me at first glance. I could not resist diving in.

As with The New York Times 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 from within our (current!) belief system, 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.

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.

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.)

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 – if we want to. That, surely is the message we educators should be telling our students, be they in the K-8 classroom or the postgraduate seminar room.

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.