Sunday, February 1, 2015

The Greatest Math Teacher Ever?

Last month I wrote about the kind of mathematic learning experiences we need to design to prepare young people for life in the Twenty-First Century. I cited the hugely successful, pioneering educational work of the late Professor R L Moore of the University of Texas. This follow up article about Moore and his teaching method is a combination of two earlier Devlin’s Angle posts, from May 1999 and June 1999. Other than adding a short paragraph at the end leading to further information about Moore, the only changes to my original text are minor updates to adjust for the passage of time.

The set-up

Each year, the MAA recognizes great university teachers of mathematics. Anybody who has been involved in selecting a colleague for a teaching award will know that it is an extremely difficult task. There is no universally agreed upon, ability-based linear ordering of mathematics teachers, even in a single state in a given year. How much more difficult it might be then to choose a “greatest ever university teacher of mathematics.” But if you base your decision on which university teacher of mathematics has had (through his or her teaching) the greatest impact on the field of mathematics, then there does seem to be an obvious winner. Who is it?

Most of us who have been in mathematics for over thirty years probably know the answer. Let me give you some clues as to the individual who, I am suggesting, could justifiably be described as the American university mathematician who, through his (it’s a man) teaching, has had the greatest impact on the field of mathematics.

He died in 1974 at the age of 91.

He was cut in the classic mold of the larger-than-life American hero: strong, athletic, fit, strikingly good looking, and married to a beautiful wife (in the familiar Hollywood sense).

He was well able to handle himself in a fist fight, an expert shot with a pistol, a lover of fast cars, self-confident and self-reliant, and fiercely independent.

Opinionated and fiercely strong-willed, he was forever embroiled in controversy.

He was extremely polite; for example, he would always stand up when a lady entered the room.

He was a pioneer in one of the most important branches of mathematics in the twentieth century.

He was a elected to membership of the National Academy of Science, as were three of his students.

The method of teaching he developed is now named after him.

If you measure teaching quality in terms of the product - the successful students - our man has no competition for the title of the greatest ever math teacher. During his 64 year career as a professor of mathematics, he supervised fifty successful doctoral students. Of those fifty Ph.D.’s, three went on to become presidents of the AMS - a position our man himself held at one point - and three others vice-presidents, and five became presidents of the MAA. Many more pursued highly successful careers in mathematics, achieving influential positions in the AMS and the MAA, producing successful Ph.D. students of their own and helping shape the development of American mathematics as it rose to its present-day position of world dominance.

In the first half of the Twentieth Century, fully 25% of the time the president of the MAA was either a student or a grandstudent of this man.

Other students and grandstudents of our mystery mathematician served as secretary, treasurer, or executive director of one of the two mathematical organizations and were editors of leading mathematical journals.

After reaching the age of 70, the official retirement age for professors at his university, in 1952, he not only continued to teach at the university, he continued to teach a maximum lecture load of five courses a year - more than most full-time junior faculty took on. He did so despite receiving only a half-time salary, the maximum that state regulations allowed for someone past retirement age who for special reasons was kept on in a “part-time” capacity.

He maintained that punishing schedule for a further eighteen years, producing 24 of his 50 successful Ph.D. students during that period.

He only gave up in 1969, when, after a long and bitter battle, he was quite literally forced to retire.

In 1967, the American Mathematical Monthly published the results of a national survey giving the average number of publications of doctorates in mathematics who graduated between 1950 and 1959. The three highest figures were 6.3 publications per doctorate from Tulane University, 5.44 from Harvard, and 4.96 from the University of Chicago. During that same period, our mathematician’s students averaged 7.1. What makes this figure the more remarkable is that our man had reached the official retirement age just after to the start of the period in question.

Who was he?

The answer

His name is Robert Lee Moore. Born in Dallas in 1882, R. L. Moore (as he was generally known) stamped his imprint on American mathematics to an extent that only in his later years could be fully appreciated.

As I noted earlier, during 64 year career, the last 49 of them at the University of Texas, Moore supervised fifty successful doctoral students. Three of them went on to become presidents of the AMS (R. L. Wilder, G. T. Whyburn, R H Bing) -- a position Moore also held -- and three others vice-presidents (E. E. Moise, R. D. Anderson, M. E. Rudin), and five became presidents of the MAA (R. D. Anderson, E. Moise, G. S. Young, R H Bing, R. L. Wilder). Many more pursued highly successful careers in mathematics, achieving influential positions in the AMS and the MAA, producing successful Ph.D. students of their own -- mathematical grandchildren of Moore -- and helping shape the development of American mathematics as it rose to its present-day dominant position. That’s quite a record!

In 1931 Moore was elected to membership of the National Academy of Science. Three of his students were also so honored: G. T. Whyburn in 1951, R. L. Wilder in 1963, and R H Bing in 1965.

In 1973, his school, the University of Texas at Austin, named its new, two-winged, seventeen-story mathematics, physics, and astronomy building after him: Robert Lee Moore Hall. This honor came just over a year before his death. The Center for American History at the University of Texas has established an entire collection devoted to the writings of Moore and his students.

Discovery learning

Moore was one of the founders of modern point set topology (or general topology), and his research work alone puts him among the very best of American mathematicians. But his real fame lies in his achievements as a teacher, particularly of graduate mathematics. He developed a method of teaching that became widely known as “the Moore Method”. Its present-day derivative is often referred to as “Discovery Learning” or “Inquiry-Based Learning” (IBL).

One of the first things that would have struck you if you had walked into one of Moore’s graduate classes was that there were no textbooks. On each student’s desk you would see the student’s own notebook and nothing else! To be accepted into Moore’s class, you had to commit not only not to buy a textbook, but also not so much as glance at any book, article, or note that might be relevant to the course. The only material you could consult were the notes you yourself made, either in class or when working on your own. And Moore meant alone! The students in Moore’s classes were forbidden from talking about anything in the course to one another -- or to anybody else -- outside of class. Moore’s idea was that the students should discover most of the material in the course themselves. The teacher’s job was to guide the student through the discovery process in a modern-day, mathematical version of the Socratic dialogue. (Of course, the students did learn from one another during the class sessions. But then Moore guided the entire process. Moreover, each student was expected to have attempted to prove all of the results that others might present.)

Moore’s method uses the axiomatic method as an instructional device. Moore would give the students the axioms a few at a time and let them deduce consequences. A typical Moore class might begin like this. Moore would ask one student to step up to the board to prove a result stated in the previous class or to give a counterexample to some earlier conjecture -- and very occasionally to formulate a new axiom to meet a previously identified need. Moore would generally begin by asking the weakest student to make the first attempt -- or at least the student who had hitherto contributed least to the class. The other students would be charged with pointing out any errors in the first student’s presentation.

Very often, the first student would be unable to provide a satisfactory answer -- or even any answer at all, and so Moore might ask for volunteers or else call upon the next weakest, then the next, and so on. Sometimes, no one would be able to provide a satisfactory answer. If that were the case, Moore might provide a hint or a suggestion, but nothing that would form a constitutive part of the eventual answer. Then again, he might simply dismiss the group and tell them to go away and think some more about the problem.

Moore’s discovery method was not designed for -- and probably will not work in -- a mathematics course which should survey a broad area or cover a large body of facts. And it would obviously need modification in an area of mathematics where the student needs a substantial background knowledge in order to begin. But there are areas of mathematics where, in the hands of the right teacher -- and possibly the right students -- Moore’s procedure can work just fine. Moore’s own area of general topology is just such an area. You can find elements of the Moore method being used in mathematics classes at many institutions today, particularly in graduate courses and in classes for upper-level undergraduate mathematics majors, but few instructors ever take the process to the lengths that Moore did, and when they try, they rarely meet with the same degree of success.

Part of the secret to Moore’s success with his method lay in the close attention he paid to his students. Former Moore student William Mahavier (now deceased) addressed this point:

“Moore treated different students differently and his classes varied depending on the caliber of his students. . . . Moore helped his students a lot but did it in such a way that they did not feel that the help detracted from the satisfaction they received from having solved a problem. He was a master at saying the right thing to the right student at the right time. Most of us would not consider devoting the time that Moore did to his classes. This is probably why so many people claim to have tried the Moore method without success.”

Another famous (now deceased) mathematician who advocated -- and has successfully used -- (a modern version of) the Moore method was Paul Halmos. He wrote:

“Can one learn mathematics by reading it? I am inclined to say no. Reading has an edge over listening because reading is more active -- but not much. Reading with pencil and paper on the side is very much better -- it is a big step in the right direction. The very best way to read a book, however, with, to be sure, pencil and paper on the side, is to keep the pencil busy on the paper and throw the book away.”

Of course, as Halmos went on to admit, such an extreme approach would be a recipe for disaster in today’s over-populated lecture halls. The educational environment in which we find ourselves these days would not allow another R. L. Moore to operate, even if such a person were to exist. Besides the much greater student numbers, the tenure and promotion system adopted in many (most?) present-day colleges and universities encourages a gentle and entertaining presentation of mush, and often punishes harshly (by expulsion from the profession) the individual who seeks to challenge and provoke -- an observation that is made problematic by its potential for invocation as a defense for genuinely poor teaching. But as numerous mathematics instructors have demonstrated, when adapted to today’s classroom, the Moore method -- discovery learning -- has a lot to offer.

If you want to learn more about R. L. Moore and his teaching method, check out the web site:

But before you give the method a try, take heed of the advice from those who have used it: Plan well in advance and be prepared to really get to know your students. Halmos put it this way:

“If you are a teacher and a possible convert to the Moore method ... don’t think that you’ll do less work that way. It takes me a couple of months of hard work to prepare for a Moore course. ... I have to chop the material into bite-sized pieces, I have to arrange it so that it becomes accessible, and I must visualize the course as a whole -- what can I hope that they will have learned when it’s over? As the course goes along, I must keep preparing for each meeting: to stay on top of what goes on in class. I myself must be able to prove everything. In class I must stay on my toes every second. ... I am convinced that the Moore method is the best way to teach there is -- but if you try it, don’t be surprised if it takes a lot out of you.”

There is a caveat

When I wrote those two Devlin’s Angle posts back in 1999, I debated with myself whether to address a side to the Moore story that, particularly from a late Twentieth Century perspective, does not stand to his credit. The issue is race.

Moore’s racial attitude was nothing unusual for a white person who was born and lived most of his life in Texas in the late Nineteenth Century and the first three quarters of the Twentieth. When the Civil Rights Act was passed in 1964 (yes, that recently!), making racial discrimination illegal, Moore was already long past retiring age, and just five years short of actually vacating his university office. Moreover, no one who regularly reads a newspaper would believe that racial discrimination in America is a thing of the past. Moore’s racial views are still not unusual in Texas and elsewhere.

Were Moore not such a towering figure, his position on race (at least as demonstrated by his actions) would not merit attention. But like all great people, all aspects of his life become matters of scrutiny. Moore could have acted differently when it came to race, even back then, in Texas, but he did not. And from today’s perspective, that inevitably leaves an uncomfortable stain on his legacy.

In writing my two 1999 columns, I chose to focus on Moore the university teacher, in particular to raise awareness of discovery learning in mathematics. The focus of Devlin’s Angle is, after all, mathematics and mathematics teaching. I did not want to distract from that goal with what is clearly a side issue, particularly such an explosive one. Moore’s larger-than-life character was clearly a significant part of his success. His racism (or at least racist behavior) was not a part of that success story – if for no other reason than because he never accepted any Black students. So I did not raise the issue.

For the same reason, I have left this side of the Moore story to the end here. We can learn from Moore when it comes to designing good mathematics learning experiences, and even admire him as a highly gifted teacher, without condoning other aspects of his life, just as we can enjoy Wagner’s music without endorsing Nazism. I can however leave you with a pointer to an article posted online by Mathematics Professor Scott Williams on 5/28/99, about the same time my articles appeared (and possibly in response to the first of them). Like it or not, Williams’ post shines light on another side to the Moore story. We can learn things from great people in ways other than taking a class from them, and we can perhaps learn things they were not trying to teach us.


Paul R. Halmos (1975), The Problem of Learning to Teach: The Teaching of Problem Solving, American Mathematical Monthly, Volume 82, pp.466-470.
Paul R. Halmos (1985), I Want to Be a Mathematician, Chapter 12 (How to Teach), New York: Springer-Verlag, pp.253-265.
William S. Mahavier (1998), What is the Moore Method?, The Legacy of R. L. Moore Project, Austin, Texas: The University of Texas archives.

Thursday, January 1, 2015

Your Father’s Mathematics Teaching No Longer Works

Gender-challenged title courtesy of this famous 1988 Oldsmobile TV commercial:

The start of a New Year is traditionally a time when we resolve to make changes. Change is particularly imperative in US mathematics education, which is built on a (Nineteenth Century) pedagogic model that long since passed its expiry date.

In a nutshell, the school system we all grew up with was essentially developed in Nineteenth Century Britain to provided a global infrastructure to run the British Empire. In modern terms, the British Imperialists created an “Internet” and an “Internet of Things” using the best computational and manufacturing resources available at the time: people.

Controlled by an “Internet of human computers”: The British Empire in 1922.
Map from
While few of us in K-16 education today see it as merely a process to prepare young people for work, we inherited a system built to do just that.

Now we have an electronic, digital Internet, does it make sense to continue to use the old system?

What do Twenty-First Century citizens need from their education?

While not the only thing—not even close—equipping young people for work is still an important educational goal, both for the individual and for society as a whole. Accordingly, it makes sense for those of us in systemic education to be constantly aware of the skills that are actually required in the workplace. If those skills change, so should the education we provide.

A good place to start is by asking the leaders of the leading companies what they look for when hiring new employees. The table below shows us what skills the Fortune 500 companies were asking for in 1970, then again thirty years later in 1999.

Fortune 500 most valued skills, cited in
Linda Darling-Hammond et al, Criteria for High-Quality Assessment (2013)

It makes for dramatic reading. While the required skills ranked from 4 through 9 remained unaltered, the top three changed completely. The most important skill in 1970, writing, dropped to number 10, while skills two and three, computation and reading, respectively, dropped off the top ten list entirely.

The most important skill in the workplace at the start of the Twenty-First Century, according to those leading companies, is teamwork, which in a single generation had leapt up from number 10. The other two skills at the top, Problem Solving and Interpersonal Skills, were not even listed back in 1970.

Clearly, the world (of work) has changed, at least for those of us living in advanced societies. Unfortunately, for those of us in the United States, and many other parts of the world, our education system has failed to keep up.

In large part, this is because of the hard-to-avoid inertia that so often comes with national (or statewide) education systems. By and large, many politicians and bureaucrats are far less aware of rapidly changing workforce requirements than those in business, and politicians frequently pander to the often woefully uninformed beliefs of voters, who tend to resist change–especially change that will affect their children.

In the US, we see this dramatically illustrated by the widespread resistance to the Common Core State Standards. In the case of mathematics, just look at how closely the eight basic Mathematics Principles of the CCSS align to that Fortune 500 list of required Twenty-First Century skills:
  1. Make sense of problems and persevere in solving them.
  2. Reason abstractly and quantitatively.
  3. Construct viable arguments and critique the reasoning of others.
  4. Model with mathematics.
  5. Use appropriate tools strategically.
  6. Attend to precision.
  7. Look for and make use of structure.
  8. Look for and express regularity in repeated reasoning.
The fact is, any parent who opposes adoption of the CCSS is, in effect, saying, “I do not want my child prepared for life in the Twenty-First Century.” They really are. Not out of lack of concern for their children, to be sure. Quite the contrary. Rather, what leads them astray is that they are not truly aware of how the huge shifts that have taken place in society over the last thirty years have impacted educational needs.

Having lived through those changes, parents have (for the most part) been able to build on their own education and cope with new demands. “What worked for me will work for my children,” they say. (They say that even when it patently did not work for them!)

But the situation is very different for their children. They are being thrust straight into that new world. To prepare them for that, you need a very different kind of education: one based on understanding rather then procedural mastery, and on exploration rather than instruction.

One of the best summaries of this societal change, and the resulting need for educational shift, that I know is the 22 minutes TED Talk Build a School in the Cloud, given by the educational researcher Sugata Mitra, winner of the 2013 TED Prize.

In his talk, not only does Mitra explain why we need to make radical changes to education, he provides examples, backed by solid evidence, of how a “Fortune 500 oriented,” team-based, exploratory approach works. In the late 1990s and throughout the 2000s, Mitra conducted experiments in which he gave children in India access to computers. Without any instruction, they were able to teach themselves a variety of things, from English to DNA replication.

[See also Mitra’s earlier talk from 2010.]

Another good account of this need for educational change is provided by Sir Ken Robinson, also in a TED Talk (11 minutes).

These ideas are not new. Indeed, they are mainstream in educational research circles. They just have not permeated society at large.

For instance, Harvard physicist Eric Mazur has been teaching by Inquiry-Based Learning (IBL), to use one of several names for this general approach, for over twenty-five years, since he first noticed that instructional lectures simply do not work. He describes his approach, and the reasons for adopting it, in his 2009 talk Confessions of a Converted Lecturer (18 minutes, abridged version).

In mathematics, the IBL approach goes way back to the 1920s. I wrote about the best known proponent in two Devlin’s Angle posts back in 1999: May, June.

Moore’s ideas have been adapted and used successfully in present-day mathematics classrooms, as shown in the promotional video Creativity in Mathematics: Inquiry-Based Learning and the Moore Method (20 minutes).

Lest my account of R L Moore and that last video portrayal leaves you with the impression that IBL math is for bright college students, see also this WIRED magazine account of the success Mexican teacher Sergio Juárez Correa had when he took a Mitra inspired approach into a poor school in Matamoros, a city of half-a-million known more for its drug trade than for being at the forefront of Twenty-First Century mathematics education.

Remember, Bob Dylan sang this in 1964:


It’s long past time for the education system to catch up with the world outside the classroom. That should be our resolution for 2015.

Wednesday, December 10, 2014

How do you find good math learning apps?

There are approximately 20,000 math learning apps available on the App Store (classified as such by their creators). Google Play does not provide the corresponding figure for Android apps, but presumably there are a lot there as well.

Most of those apps do little more than provide repetitive practice of very basic skills, primarily about numbers. They are essentially just animated flash cards.

How can a parent, or a teacher, decide which apps are likely to benefit their child, or their students? I’ll come back to that later.

First, let me say that there is not necessarily anything wrong with an app that is essentially just an animated flash card – unless parents buy them (or just download them, as the majority are free) thinking that putting them on their children’s iPad or whatever is all they need to do to improve their performance in math.

In the days when the gateway to mathematics, and indeed much of everyday life, lay in mastering the multiplication tables and memorizing a few formulas for calculating areas and volumes, mastery of the basic number facts was indeed enough to start with. So it’s a pity those fun learning apps were not available back then. They would have made the acquisition of those fundamental facts and skills so much easier and far more enjoyable.

Unfortunately, the very digital technologies that have put those learning apps into eager young hands have also provided tools that have rendered procedural mastery of those basic skills all but irrelevant.

In today’s world, we use cheap, ubiquitous devices to do our calculations. It’s no longer important that all members of society have procedural mastery of basic arithmetic. What is required is the ability to make effective use of those digital devices, and what that depends upon is a good understanding of number – what is often referred to as number sense.

Roughly speaking, having number sense means being proficient with quantities and operations with numbers. A person with number sense is able to represent number concepts with models, words and diagrams, to communicate numerical ideas, and solve problems involving numbers. She or he can flexibly compose and decompose numbers for computation and solving problems. They can evaluate the reasonableness of solutions to numerical problems, and make connections between multiple solution methods. They can communicate their number sense verbally and in writing. They notice and explore number patterns, make connections and conjectures, and communicate their thinking to others. Number sense goes beyond solving word problems and memorizing basic facts and procedures. It involves engaging in numbers and operations in ways that develop a deep understanding of the content, which provides a firm foundation for mathematical success. In particular, a strong background in number sense sets the stage for later success in algebra and other parts of mathematics.

If that last paragraph sounds like something that emerged from a committee of mathematics education experts, it is because in essence it did. You find language like that in the National Research Council’s 2000 report Adding it Up, (which you can download for free from the National Academies Press) and in the preamble to the Common Core State Standards for Mathematics, which emphasize the development of number sense in young children.

For sure, you cannot have number sense without being able to solve an arithmetic problem and get the right answer. What has changed is that it is no longer important to solve that problem by the fastest method, or by a standard method that leaves a paper audit trail that others can check. Our calculating devices do those for us.

Much more important in today’s world is to be able to reason about the numbers in a problem from first principles, in a way that embodies the internal structure of the numbers. For as humans, we need to be able to operate when and where that calculator cannot: namely, when we are faced with a novel problem the real world has thrown up at us.

It was a lack of recognition that the world has changed fundamentally that led the consequently-Internet-famous “Jack’s Dad” to pen his satirical “letter to his son” that went viral on social media earlier this year. (See the next link below.)

Actually, Jack’s dad is an electronics engineer, so he was certainly aware of how much today’s world was different from the days of his own childhood. Unfortunately, as someone outside the world of education, he had just not connected the dots to understand what changes in education were required in order to properly prepare today’s kids to live, not just in our present world, but in the world they will help shape from it.

One of the best summaries of the issues behind that social media firestorm that I came across was the April 6 response to Jack’s Dad written by the math education blogger Christopher Danielson.

Danielson’s observations about different kinds of expertise rang very true to me. Having devoted the first part of my mathematics career to mathematical research, it was my appointment to serve on the Mathematical Sciences Education Board in 2000, and the close contact with leading experts in mathematics teaching that resulted, that brought home to me just how little I knew about how people learn mathematics, and how (consequently) we should teach it.

Put plainly, having a PhD in mathematics and a string of published research is absolutely nothing like enough background to speak with authority about K-12 mathematics learning. People like me can provide good advice on mathematical content; but not on mathematics teaching. That requires different knowledge and expertise.

My own university, Stanford, famous for its very high standards in research, apparently recognizes this when it comes to hiring new faculty in Education. While I cannot speak with authority for the School of Education’s policies, I have observed that no one gets appointed to the faculty who has not spent several years in K-12 teaching. (In addition to having done and published first class research!) Whether or not K-12 experience is official hiring policy, it certainly plays out that way, and it seems to me to be a sensible criterion to demand.

Going back to the standard algorithms and Jack’s Dad, a few months after his first post, on October 8, Danielson posted another excellent blog on the degree to which the position occupied by the standard arithmetic algorithms (in actual fact, there are many variations, so there is no such thing as “the standard algorithms) has changed in the educational landscape – from being the main focus as a method for daily use, to an interesting and historically important example of a set of highly efficient paper-and-pencil algorithms that quite literally changed the world. Their significance was a consequence of the dominant information storage and communication technology of the time: flat, static writing surfaces such as parchment, blackboards, and paper. (I describe that story in my book The Man of Numbers.)

I will note, in passing, that Danielson’s October post indicates that some math learning apps may in fact do harm to a child’s mathematics learning, an observation that should be coupled with my earlier remarks about choosing basic skills educational apps.

What put these thoughts onto my front burner recently were some discussions I was having with members of the Scientific Advisory Board for my educational technology startup company BrainQuake.

If you check out our company’s Team page, you will find we have recruited a number of world renown experts in mathematics education. Now you may think they are just there for marketing purposes – website name dropping. But you would be wrong. Each one is there because they bring very valuable, very specific expertise to the table.

To someone not an expert in mathematics learning, the arithmetic puzzles in our launch app, Wuzzit Trouble, may look as though they are just a series of problems we generated in an essentially random fashion, following the simple rule that the numbers should get "harder" the further a player goes in the game. But that is not the case. In a mathematics learning game, the mathematics ramp is just as critical as the level design of the game, and both require a lot of expertise to get it right.

(Interestingly, another name on our website, John Romero, is a world expert in level design – the ramping in game-play – but he joined forces with us only after we had brought out Wuzzit Trouble, so you will only see the results of his genius in future products we bring out.)

Which brings me back to my promise to provide advice on how to select good learning apps. It’s probably not a foolproof method, but a quick and easy way is to check out the website of the creators, and see who they have advising them on the learning side.

There is always the danger that some of the names are there for little more than window dressing, but the majority of education experts (indeed, experts in any domain) are not likely to lend their name to an enterprise they do not believe in. So the presence of names of distinguished mathematics educators should give you a lot of confidence in the product.

More to the point, the absence of such names should be taken as a serious warning. Quite frankly, it is not possible to design and build an educationally sound and effective learning app without a lot of expert input.

And I mean a lot of expert input. I bring years of my own expertise to BrainQuake, but Wuzzit Trouble would not have been anything like as educationally successful as it has, if it had just been me on the mathematics side.

There is your quick-and-easy quality check. If you use it, you will find that list of 20,000 apps suddenly shrinks down to a significantly smaller number. Fortunately, that number is not zero. There are some great math learning apps out there. You just have to choose wisely.

Wednesday, November 5, 2014

Against Answer Getting

"Correct answers are essential... but they're part of the process, they're not the product. The product is the math the kids walk away with in their heads." —Phil Daro
If you have not already watched Phil Daro's 17-minute video Against Answer Getting, you should do so right away. (I'll keep this post short to give you enough time to watch it in its entirety.)

Daro, a longtime mathematics educator and leading figure in the national mathematics education community, is currently director of the San Francisco field site of SERP, the Strategic Education Research Partnership. He was one of the mathematics educators who played a leading role in the formulation of the mathematics Common Core State Standards. (You know, one of those knowledgeable experts the StopCommonCore brigade keep claiming were not involved in CCSS development.)

The video is full of powerful insights that the mathematics education community has accumulated over many years of research. My opening quote sums up the focus of the video. Here is another one I like:
"Mathematics does not break down into lesson-sized pieces." Phil Daro
This particular quote resonates with me. I adopted the same principle in the design of my MOOC Introduction to Mathematical Thinking, currently about halfway through its fifth run.

Daro's focus, both in the video and in his work in general, is K-12 mathematics education. But it is very relevant to those of us in college-level mathematics education. When students come to college with a perception that mathematics is about "answer getting," we face the very uphill task of ridding them of that misleading mindset.

True, for hundreds of years, getting answers was a key component of learning and doing mathematics. But these days, if we want answers in mathematics, we generally use one of a number of digital technologies. The job of today's mathematician (or typical user of mathematics) is problem solving. The part that requires a human mind is when the problem has a novel aspect. It was precisely to put the focus on the thinking part that I named my MOOC the way I did.

The principle requirement for being able to solve a novel problem is conceptual understanding. That is why the issues Daro raises in that video are so central to the mathematics education of the citizens of tomorrow.

The outdated mindset about the purpose of mathematics that many students bring with them when they transition from school to college is not the only problem many have to overcome. A parallel issue manifests itself when they start to learn about mathematical proofs (if they follow the mathematics path).

My MOOC students are currently right in the middle of that part of the course (proofs), and many are having a very hard time coming to understand what role proofs play and what (therefore) constitutes a good proof.

The dominant perception is that proofs are what mathematicians produce in order to determine mathematical truth. That, of course, is true (at least in an idealistic sense that guides mathematical progress), but as with arithmetic answer getting, it is only part of the story. And in terms of actual mathematical practice, a very small part of the story.

As with answer getting in K-12 math, achieving a logically correct proof is a binary target (right or wrong), which make both very easy to evaluate for correctness and assign a numerical grade. (Ka-ching!)

But let's pause and ask ourselves how proofs work in practice. If you want to know if Fermat's Last Theorem is true, you consult a reliable source. Today, any moderately knowledgeable mathematician will tell you the answer: "Yes." Now you know.

But what if you want to know why it is true. That's when you need to look at a proof.

In terms of mathematical practice, proofs are about understanding. They are communicative devices we construct to convince ourselves and to convince others.

In my MOOC, because I cannot assume the students have access to individualized, expert feedback on their work, I do not ask them to construct proofs. But I do present them with a range of purported proofs, some correct, others not, and ask them to evaluate them. The evaluation is in terms both of logical correctness and communicative effectiveness.

To do this, I ask them to look at each purported proof in terms of five different factors: one logical correctness, the others focusing on communicative issues. Though the five factors are not independent variables, I ask them to treat them as such when evaluating a proof.

This is the part of the course where those students who have had some exposure to proofs in the K-12 system tend to do worse than those who are new to proofs. They are simply not able to approach a proof other than in the "answer getting" mode of "Is it logically correct?"

This shows up dramatically with extremal cases. When I present them with a carefully constructed argument that is logically correct but provides no explanation, they will give it high marks across the board. But faced with an argument that is superbly articulated but has a logical flaw, they are psychologically unable to evaluate the structure of the argument. "It's wrong," they keep saying. End of story (for them).

Of course, extremal examples are atypical, and often difficult to wrap our minds around. That's what makes them so valuable as learning devices. It's when the classroom rubber hits the road and we find ourselves using mathematical thinking in our lives or careers that it becomes important to have good communication skills.

Pick up a more advanced level mathematics book or research article and the chances are high that the arguments presented will contain errors. (Actually, the book does not have to be advanced. Euclid's Elements is littered with "proofs" that are not logically sound.) But if the arguments are well laid out, with adequate explanations, a suitably skilled reader can fix them as they go alongpossibly with help from someone else. (That's definitely the case with Elements, though it took two thousand years before David Hilbert noticed that Euclid's own arguments left a lot of work to be done to make them genuine "proofs.")

It's the same in software engineering. Any useful program will have bugs. But if the code is well structured, and adequately annotated, someone else can dive in and fix it whenever a flaw manifests. A good computer programmer is not someone who writes error-free, working code; it is someone who writes working code that can easily be fixed or modified.

I'll leave it as an exercise for the reader to identify the analogous issue in the natural sciences.

If those of us in the education business want to do the best we can to prepare our students for life in the 21st century, we need to recognize that in an era when technologies provide instant answers (facts), the one ability they will need above anything else is (creative, reflective) thinking. 

Tuesday, October 7, 2014

The Straw Teacher

When people argue for a position they hold because of political bias or some deep-rooted sense of conviction (as opposed to one arrived at by a process of reflection, weighing all sides of the issue), they often resort to straw-man tactics. This is particularly common in the U.S Math Wars, which these days are largely focused on the Common Core State Standards for mathematics.

A particularly popular straw man – more precisely, a "straw teacher" (a term that nicely gets us out of gender issues) – is a math teacher who spends class time exclusively discussing mathematics concepts (whatever that means) and pays no attention to helping the students master any procedures.

I guess there may be such a teacher, somewhere, but I have to confess I have yet to meet one. Ditto for the straw teacher who says getting the right answer (if there is one) is not important. Teachers just don't do either of those.

My colleagues who work in classroom teacher preparation do tell me that many math teachers do little else than drill on procedures (in some cases because they never set out to teach math, and don't really understand the concepts themselves), but in my walk of life I never meet them. I see the ones who became math teachers because they love mathematics and want to teach, and attend mathematics teacher conferences to exchange ideas and to learn more – which is where I meet them.

Anyone who has a working knowledge of (1) what mathematics (really) is and (2) how the brain works knows that learning math in a useful way requires both mastery of a set of basic procedures and conceptual understanding of the mathematical notions those procedures are built on.

In practical terms, you need to master basic procedures in order to develop conceptual understanding, and you need conceptual understanding in order to avoid any procedural mastery being brittle and short-lived.

So good math teaching involves both. And, for the record (yet again), both are called for in the CCSS.

Absent the CC connection, I've written about this issue on a number of occasions before in this column. For instance:

March 2006: How do we learn math?
September 2007: What is conceptual understanding?

Both articles were written long before the Common Core was developed. They were also written when I was just starting to become more actively involved in K-12 education issues. (And before I inadvertently ignited the "repeated addition" firestorm in the summer of 2008.) But having just re-read them for the first time in many years, I still stand by what I wrote. So I won't repeat myself here.

Tuesday, September 2, 2014

Will the Real Geometry of Nature Please Stand Up?

Is fractal geometry “the geometry of nature”? I was asked this question recently in an email from someone who had watched the PBS video Hunting the Hidden Dimension that I worked on, and appeared in, a few years ago.

It would have been easy to simply reply “Yes,” and for many audiences I would (and have) done just that—for this was by no means the first time I had been asked that question, or others very much like it. But the context in which this recent questioner raised the issue merited a less superficial response. So I wrote back to say that there is no such thing as the geometry of nature, or more generally, the mathematics of W, where W is some real world domain.

The strongest claim that can be made is something along the lines of “Mathematical theory T is the best mathematical description (or model) we currently have of the real world domain (or phenomenon) W.” But even then, this statement is less definitive than it might first appear: In particular, what do we mean by “best”?

Best in terms of understanding? (If so, then understanding by whom?)

Best in terms of building something in W? (If so, then building out of what, using what tools, and for what use?)

Best in terms of teaching someone about W? (If so, then teaching what kind of person in terms of age, background, education, motivation, etc.?)

Slightly edited and extended, the next few paragraphs are what I wrote back to my correspondent:

Nature is just what it is. Mathematics provides various ways to model our perception and experience of reality. Different parts of mathematics provide different models, some better than others. Fractal geometry provides one model that seems to accord with our observations, measurements, and experiences. But so too do the cellular automata models on which Steve Wolfram bases his “New Kind of Science.”

Many of us think fractal geometry does a better job than cellular automata in helping us understand the natural world by virtue of its nature, but that reflects an assumed patterns/relationship conception of what constitutes science.

I would prefer to call Wolfram’s framework a computational theory (of the world), rather than science. But the distinction is, I think, purely one of the meaning we attach to the relevant words (particularly “science”).

Both approaches can be said to begin by looking at how nature works, but the moment you start to create a model, you leave nature and are into the realm of human theorizing. From then on, the only available metrics are (1) degree of fit to observations and measurements, (2) degree of utility, and (3) degree to which we find the model’s assumptions reasonable.

There is lots of slack here.

In (1), what are we observing and measuring? (They are often entities created by those very mathematical theories, e.g. mass, length, volume, velocity, momentum, temperature, etc.)

In (2), how do we define utility? Doing stuff, building stuff, understanding stuff, teaching stuff, or something else? (Each with the various audience/use/purpose caveats I raised earlier.)

Then there is (3). Unless we make some initial assumptions, we cannot get a theory off the ground. And make no mistake about it, we do begin with assumptions. Not arbitrary ones, to be sure—not even close to being arbitrary. For the resulting theory to be fully accepted (as a plausible explanation or model), it has to accord to any and all the available facts, and it has to be falsifiable—it should make claims or imply conclusions that we can attempt to prove wrong.

For instance, a mathematical theory that implied 3 = 4 (as an identity of integers) would be immediately rejected.

What about a theory that implies 0.999… = 1.0, where those three dots indicate that the decimal series continues for ever? According to the widely accepted, standard definitions that mathematicians use to provide meaning to the concept of an infinite sequence of decimal digits, this identity is correct. Indeed, it can be proved to be correct, starting from the reasonable, plausible, and accepted basic principles (axioms) for the real number system.

Most university math students learn about the framework within which 0.999… is indeed equal to 1.0. (Though many of the popular “proofs” you come across are not rigorous.) As a result, many mathematically educated people will state, as if it were an absolute fact of the world, that 0.999 = 1.0. But that is not true. The identity holds because we have made some assumptions about how to handle infinity. It’s easy to overlook that fact. So let me provide a further example where it may be less easy to miss an underlying assumption.

Graduate students of mathematics are introduced to further assumptions (about handling the infinite, and various other issues), equally reasonable and useful, and in accord both with our everyday intuitions (insofar as they are relevant) and with the rest of mainstream mathematics. And on the basis of those assumptions, you can prove that

1 + 2 + 3 + … = –1/12.

That’s right, the sum of all the natural numbers equals –1/12.

This result is so much in-your-face, that people whose mathematics education stopped at the undergraduate level (if they got that far) typically say it is wrong. It’s not. Just as with the 0.999… example, where we had to construct a proper meaning for an infinite decimal expansion before we could determine what its value is, so to we have to define what that infinite sum means.

It turns out that there is an entire branch of mathematics, called analytic continuation theory, that provides us with a “natural” meaning for (in particular) that sum. And when we calculate the value using that meaning, we arrive at the answer -1/12. See this Wikipedia article for a brief account.

Incidentally, just as with the 0.999… example, you will find purported “intuitive proofs” floating around, among them this video that went viral earlier this year, but those arguments too are not rigorous.

Both frameworks, the one that yields a value for 0.999… and the one that produces a value for 1 + 2 + 3 + … , satisfy all the requirements of being reasonable, plausible, consistent with the rest of mainstream mathematics, and useful (in studies of real world phenomena, including physics). If you accept one, you really cannot reasonably deny the other. Rather, you have to accept the implications they yield, even if they at first seem counter to your expectations.

True, neither identity accords with our experiences in the physical world, since those experiences do not involve any infinite quantities or processes. (So there is nothing to accord with!)

One of the things surprising examples involving infinity remind us of is that mathematics is not “the true theory of the real world” (whatever that might mean). Rather, mathematical theories are mental frameworks we construct to help us make sense of the world. They survive or wither according to the degree to which they continue to accord with our real world experiences and to prove useful to us in conducting our individual and collective lives.

To return to geometry. For most people, throughout human history the geometry of the world experienced was planar Euclidean geometry, which accords extremely well with our everyday experiences.

But for the global air traveler (such as long distance airplane pilots), and for the astronauts in the International Space Station, spherical geometry is “the geometry.” In still other circumstances (for the most part, physics and cosmology), hyperbolic and elliptic geometries are the best frameworks.

For the artist trying to represent three dimensions on a two-dimensional canvas (or the movie or video-game animator trying to represent three dimensions on a screen), projective geometry is the best framework.

Picking up on my opening example, when you adopt a geometric perspective to try to understand growth in the natural world, you find that fractal geometry is the most appropriate one to hand.

And, finally, when you adopt a geometric perspective to try to make sense of social life in today’s multi-cultural societies, you may find that higher dimensional Euclidean geometries seem to work best, as I explain in this video (30 minutes) taken from a talk I gave at a conference in New Mexico earlier this year. (The relevant segment starts at 3:20 and ends at 11:00.)

The fact is, there is not just one geometry, and there is no such thing as “the geometry of W,” where W is a real world phenomenon or domain.

Likewise for other branches of mathematics we develop and use to understand our world and to do things in our world.

This means that, whereas, within mathematics there are “right answers,” when you apply mathematics to the world, that certainty and accuracy is only as good as the fit between the mathematics (as a conceptual framework) and the world.

And now we are back, more or less, at the topic of my previous Devlin’s Angle post. It merits a second look. Given the nature of the modern world, with mathematical models playing such a major role, with major consequences (in banking, information storage, communication, transportation, national security, etc.), we should not lose track of the fact that mathematics is not the truth.

Rather, it provides us with useful models of the world. As a result, it is a powerful and useful way of making sense of the world, and doing things in the world.

This distinction was not particularly significant for anyone growing up in the 20th century and earlier. Back then, there was usually no danger in viewing mathematics as if it were the truth. But it is an absolutely critical distinction to keep in mind for those coming of age today.

That New Mexico talk video I referred to a moment ago was in fact from a conference on middle school mathematics education, and was an attempt to raise awareness among middle school math teachers of the need to make their students aware of the way mathematics is used in the world they will live in and help shape, emphasizing not only mathematics’ strengths but also its limitations.

When you think about what is at stake here, much of the current debate (largely uninformed on the opposition side) about the Common Core State Standards resembles nothing more than two elderly bald men arguing over ownership of a comb.

In the case of the UK’s Falkland’s War of 1983, where this analogy originated, both sides appeared equally stupid. The sad aspect to the CCSS debate is that the level of ignorance (or malicious intent) on the “Stop” side forces many well-informed teachers and mathematics learning experts to devote time to the debate, lest ignorance prevail and our kids find themselves unable to survive in the world they inherit. (What the debate should focus on is how to properly implement the Standards. There be dragons, and someone needs to slay them.)

WORTH LISTENING TO: American RadioWorks has just aired an excellent radio documentary about the Common Core, in which we hear from real teachers who have been using it, both in states where it has been implemented according to plan and others where the implementation has been modified.

Friday, August 1, 2014

Most Math Problems Do Not Have a Unique Right Answer

One of the most widely held misconceptions about mathematics is that a math problem has a unique correct answer.

(Some of those who hold that view also think that there is just one correct way to get that answer. A far smaller group, to be sure, but still a worryingly large number. Still, my focus here is on the first false belief.)

Having earned my living as a mathematician for over 40 years, I can assure you that the belief is false. In addition to my university research, I have done mathematical work for the U. S. Intelligence Community, the U.S. Army, private defense contractors, and a number of for-profit companies. In not one of those projects was I paid to find "the right answer." No one thought for one moment that there could be such a thing.

So what is the origin of those false beliefs? It's hardly a mystery. People form that misconception because of their experience at school. In school mathematics, students are only exposed to problems that (a) are well defined, (b) have a unique correct answer, and (c) whose answer can be obtained with a few lines of calculation.

But the only career in which a high school graduate can expect to continue to work on such problems is academic research in pure mathematics—and even then (and again speaking from many years of personal experience), cleanly specified problems that have (obtainable) "right answers" are not as common as you might think.

Since the vast majority of students who go through school math classes do not end up as university research mathematicians, whereas many do find themselves in careers that require some mathematical ability, it's reasonable to ask why their entire school mathematics education focuses exclusively on one tiny fraction of all possible mathematics problems.

The answer can be found by looking at the history of mathematics. Starting with the invention of numbers around 10,000 years ago, people developed mathematical methods to solve problems they faced in the world: arithmetic and algebra to use in trade and engineering, geometry and trigonometry for building and navigation, calculus for scientific research, and so forth.

While some of that mathematics was required only by specialists (e.g. calculus), arithmetic and parts of algebra in particular were essential for everyday living. As a consequence, mathematicians wrote books from which ordinary people could learn how to calculate. From the very earliest textbooks (Babylonian tablets, Indian manuscripts, etc.), two kinds of problems were presented: algorithm ("recipes") problems that showed the steps to be carried out to do a particular kind of computation, presented without any context, and word problems, designed to help people learn how to apply a particular algorithm to solve a real world problem. Ancient and medieval textbooks had many hundreds of such problems, so that a trader (say) could find a problem almost identical in form to the one he (and back then use of mathematics was primarily a male activity) actually wanted to solve in his business. If he were lucky, all he would have to do is substitute his own numbers for those in the book's worked word problem. In other cases, the book might not provide an exact match, but by working through five or six problems that were close in form, the individual could learn how to solve his real problem.

For the majority of people, that was enough. Life simply did not require anything more. The problems they faced in their everyday activities for which mathematics was needed were simple and routine. The mathematical word problems that today seem so unrealistic were by and large remarkably similar to the problems ordinary citizens faced every day.

"When do I need to leave home in order to catch that train?" There wasn't an app to tell you the answer; you had to calculate it yourself. That word problem about trains leaving stations in your math class showed you how.

Arithmetic, in particular, was an essential, basic life skill that remained so until the development of devices that automated the process in the 1960s. I am a member of the last generation for whom the question "What do I need arithmetic for?" simply did not arise. (We asked it about other parts of mathematics.)

But that computer technology that eliminated the need for people to be good calculators led to a world in which there is a huge demand for higher order mathematical skills, starting with algebra. I wrote about this change in this column back in 1998, in a piece titled "Forget 'Back to Basics.' It's Time for 'Forward to (the New) Basics.'" Looking back at what I wrote then, I am amazed at just how much things have changed in the intervening 16 years. In September of that year, Google was founded, and the Web became a dominant force in our lives and our work.

Today, we have instant access to vast amounts of information and to unlimited computing power. Both are now utilities, much like water and electricity. And that has led to a revolution in the mathematics ordinary citizens need in order to lead a fulfilling, productive life. In a world where procedural (i.e., algorithmic) mathematics is available at the push of a button, the need has shifted to what I and others have been calling mathematical thinking.

I wrote about this in my September 2012 Devlin's Angle. Broadly speaking, mathematical thinking is a way of approaching problems that is based on classical mathematics, but takes account of the fact that computation (both numeric and symbolic) can be readily done by machines.

In practical terms, what this means is that people can now focus all their attention on real-world problems in the form they are encountered. Knowing how to solve an equation is no longer a valuable human ability; what matters now is formulating the equation to solve that problem in the first place, and then taking the result of the machine solution to the equation and making use of it.

In the 1960s, we got used to the fact that the arithmetic part of solving a mathematical problem could be done by machines. Now we are in a world where almost all the procedural mathematics can be done by machines.

Of course, this does not mean we should stop teaching procedural mathematics to the next generation, any more than the introduction of pocket calculators meant we should stop teaching arithmetic. But in both cases, the reason for teaching changes, and with it the way we should teach it. The purpose shifts from mastering procedures—something that was necessary only when there were no machines to do that part—to understanding the concepts sufficiently well to make good use of those machines.

Though this change in emphasis has been underway for some years now, it did not garner much attention in the United States until the rollout of the Common Core State Standards, which are very much geared towards the mathematical thinking needs of the 21st century. The degree to which many parents were shortsighted by the shift was made clear when some of them took to social media to complain about the kinds of homework questions their children were being asked to do. While some of those questions were truly, truly awful, others garnering a lot of critical SM comments were actually extremely good.

What was particularly ironic was that many parents, faced with being unable to assist their child with elementary grade arithmetic homework, did not draw the obvious conclusion: "Gee, if I cannot understand something as basic as integer arithmetic—however it is done—there must have been something really lacking in my own education." Instead, they jumped to the totally off-the-wall conclusion that the current educational system must be wrong.

That's like waking up in the morning to find your car won't start and saying, "Oh dear, the laws of physics don't work." The smart person says, "I need to replace the battery."

I'll tell you something. I was taught math the "old-fashioned way" too, and some of those student arithmetic worksheets were new to me when I first saw them. But regardless of any views I might have as to how it is best taught in today's world, it didn't take a lot of effort to figure out what those kids were doing on those worksheets posted on Facebook. It was just whole number arithmetic for heavens sake! Anyone who understands the basic ideas of whole number arithmetic can figure it out.

It was not my training as a professional mathematician that helped me here. It was the simple fact that I understand whole number arithmetic, something that goes back to my early childhood, when I did not even know there was such a thing as a professional mathematician, let alone aspire to be one. Unfortunately, many Americans were never taught to understand arithmetic, they were just trained to execute procedures. It's not their kids who are being short-changed. They—the parents—were!

Breezing into this fray is University of Wisconsin mathematics professor Jordan Ellenberg, with his new book How Not To Be Wrong. I knew I would find a kindred spirit when I read the book's subtitle: “The Power of Mathematical Thinking.” With a Stanford MOOC and an associated textbook both called Introduction to Mathematical Thinking, how could I not?

Ellenberg's title is superb. In one fell swoop, it casts aside that old misconception that mathematics provides "right answers," replacing it with the far more accurate description that it is a great way to stop you being wrong. For, like me, he focuses not on the internal activities of pure mathematics, rather on how mathematics is used in today's real world.

To be sure, also like me, Ellenberg has devoted a lot of his career to working in pure mathematics, so he loves searching for those "right answers," and he enjoys the subject in its own terms. We both know that there are eternal truths within mathematics (a better term would be "tautologies") and have experienced the thrill of going after them. But we both realize that what we do as pure mathematicians is a very specialist pursuit. The society that supports us when we do that does so largely because of the payoff in terms of the benefits that emerge when mathematical thinking is applied to real world problems.

Ellenberg's book is chock full of examples of those benefits, from many walks of life, presented with a delightfully light touch. He grabs the reader's attention with his very first example, taken from the Second World War. The U. S. military chiefs wanted to reduce the number of warplanes that were being shot down. The obvious solution was to add more armor to protect them. But armor adds weight, which limits the distances that can be flown and the duration of the mission, as well as increasing the production cost. So the question was, where is the most effective place to put that extra protection?

To answer this question, the chiefs brought in a team of mathematicians to analyze the evidence and determine what parts of the aircraft were most likely to be hit. They examined the fuselages of all the damaged planes that had flown back after being hit to see where the most damage was. It turned out that the engines had an average of 1.11 bullet holes per square foot, the fuel system had 1.55, the fuselages 1.73, and the rest of the plane 1.8.

So where was the optimal place to add extra armor? According to the data, the fuselages took a lot of hits, while engines suffered the least damage. So an obvious suggestion was to add armor to the fuselages. But that was not what the mathematicians suggested. Their solution was to add the armor to the engines, the part that had fewer hits when the planes got back.

And they were right. I'll leave you to figure out why that is the best solution. It's a great example of mathematical thinking. After you have convinced yourself why adding armor to the engines was the best strategy, you should buy a copy of Ellenberg's book and gain some understanding of just what mathematical thinking is, and why it is a crucial ability in today's world.

(My own book on mathematical thinking is more of a "how to" guide, as is my MOOC. Another, excellent book on mathematical thinking, that is somewhere between Ellenberg's and mine, is Burger and Starbird's The 5 Elements of Effective Thinking.)

Finally, and to some extent switching gears (and definitely switching media), I want to draw your attention to a new video game, DragonBox Elements, by the Norwegian-based educational technology company WeWantToKnow. The company made a splash with its first game, DragonBox (Algebra) a couple of years ago.

Unlike my own work in educational videogames, through my company BrainQuake, which is very strongly focused on real-world mathematical thinking, the DragonBox folks are seeking to enhance and strengthen school mathematics.

When I first played the new Elements game, I was initially confused, since I approached it with a Geometer's Sketchpad expectation. But Elements is not a geometry construction/exploration tool. The focus is on the importance of providing justification for steps in a proof. Knowing why something is true. And that is not only a key feature of GOFM (“Good Old Fashioned Math”), as was taught for two thousand years, it's one of the aspects of mathematics that is characteristic of mathematical thinking (as used in the real world). Euclid, the author of the first Elements (the book), would surely have approved.

The modern world has not made GOFM redundant. What has changed, and drastically, is the way GOFM fits in with the rest of human activities. Unless you are going to make a career for yourself in pure mathematics research, GOFM today is simply an amazingly powerful tool for acquiring one of the most important cognitive capacities in the 21st century: mathematical thinking.

In today's world, most of the important problems are complex and multi-faceted. There are few right answers. As Ellenberg demonstrates, mathematical thinking can help you choose better answers—and avoid being wrong.