Wednesday, May 2, 2018

Calculation was the price we used to have to pay to do mathematics

By Keith Devlin

You can follow me on Twitter @profkeithdevlin


Ever since mathematics got properly underway around 3,000 years ago, there was only one way to achieve access to the field. You had to spend many years developing a fairly extensive calculation skillset. In the first instance, to pass the graduation and entrance examinations to gain initial access to the field. Then, once accepted into the world of mathematics, calculation of one kind or another was what all mathematicians spent the bulk of their mathematical time doing. Arguably, for most of mathematics history, the subject really was, to a large extent, primarily about calculation of one form or another. Newton, Leibniz, Bernoulli (any of them), Fermat, Euler, Riemann, Gauss, and the other greats of times past, were all superb masters of calculation. (We should also include Boole, since his famous Boolean algebra is also a calculation system.)

But whereas most laypersons seem to think that calculation is all there is to mathematics, surely none of the greats did. Calculation was an important tool (more accurately, a set of tools) you needed to do mathematics, they must have realized, but the essence of mathematics is much more, a plateau of knowledge that transcends all the calculation techniques.

In the 19th Century, that somewhat tacit understanding became explicit. The increasing complexity of the problems mathematicians tackled led to a series of results that defied the human intuition. (Several of them were referred to as “paradoxes”.) This led to an intense period of mathematical introspection, where the primary focus was not performing a calculation or computing an answer, but formulating and understanding abstract concepts and relationships. In other words, a shift in emphasis from doing to understanding. What had previously been implicit, became full-on explicit.

Mathematical objects were no longer thought of as given primarily by formulas, but rather as carriers of conceptual properties. Proving something was no longer a matter of transforming terms in accordance with rules, but a process of logical deduction from concepts. Mathematics was reconceptualized as “thinking in concepts” (Denken in Begriffen).

This was, in every sense, a mathematical revolution, with the primary revolutionaries being
leading mathematicians such as Lejeune Dirichlet, Richard Dedekind, Augustin-Louis Cauchy, Karl Weierstrass, and Bernhard Riemann.

To give just one instance of the shift, prior to the nineteenth century, mathematicians were used to the fact that a formula such as  y = x2 + 3x – 5  specifies a function that produces a new number y from any given number x. Then the revolutionary Dirichlet came along and said, forget the formula and concentrate on what the function does in terms of input-output behavior. A function, according to Dirichlet, is any rule that produces new numbers from old. The rule does not have to be specified by an algebraic formula. In fact, there's no reason to restrict your attention to numbers. A function can be any rule that takes objects of one kind and produces new objects from them.

This definition legitimized functions such as the one defined on real numbers by the rule:
If x is rational, set f(x) = 0; if x is irrational, set f(x) = 1.

Of course, you cannot draw a graph of such a monster. Instead, mathematicians began to study the properties of abstract functions, specified not by some formula but by their behavior. For example, you can investigate questions such as, is the function one-one, injective, surjective, continuous, differentiable, etc.?

For any mathematician alive today, mathematics is a subject that studies formally-defined concepts, with a focus on the establishment of truth (based on accepted axioms), with various forms of calculation (numerical, algebraic, set-theoretic, logical, etc.) being tools developed and used in the pursuit of those goals. That’s the only kind of mathematics we have known.

Except, that is, when we were at school. By and large, the 19th Century revolution in mathematics did not permeate the world’s school systems, which remained firmly in the “mathematics is about calculation” mindset. The one attempt to bring the school system into the modern age (in the US, the UK, and a few other countries), was the 1960s “New Math”. Though well-intentioned, its rollout was disastrous, in large part because very few teachers understood what it was about – and hence could not teach it well. The confusion caused to parents (other than mathematician parents) was nicely encapsulated by the satirical songwriter and singer Tom Lehrer (who taught mathematics at Harvard, and did understand New Math), in his hilarious, and pointedly accurate, song New Math.

As a result of the initial chaos, the initiative was quickly dropped, and school math remained largely unchanged while real-world uses of mathematics kept steadily changing, leaving the schools increasingly separated from the way people did math in their jobs. Eventually, the separation blew up into a full-fledged divorce. That occurred in the late 1980s. The divorce was finalized on June 23, 1988. That was the date when Steve Wolfram released his mammoth software package Mathematica. Within a few short years of that release, if not on the release-date, Mathematica (and a similar package released a few months later in Canada, Maple) could answer pretty well any school or university math exam question with at least a grade B+, and very often an A.

The days when calculation (of pretty well any kind, not just numerical) was the price humans had to pay to do mathematics were over.

Given that thirty years have passed since that initial epochal moment, and most of the world has still not woken up to the fact that the entire mathematical world has changed dramatically and forever, let me repeat the core of that statement in caps.

THE DAYS WHEN CALCULATION WAS THE PRICE HUMANS HAD TO PAY TO DO MATHEMATICS ARE OVER.

To be sure, after that symbolic 1988 date, it took a few years for the change to percolate through the system, gain momentum, and eventually reach critical mass. Three further developments were also hugely significant: the birth of the World Wide Web in 1989 and the browser in 1993, and the launch of Wolfram Alpha in 2009. (Others might want to add other factors. I’m being selective here.)

Talking about being selective, I’ve mentioned Wolfram products twice now. Though I was a member of Wolfram’s Mathematica Advisory Board in the first few years, I have no stake in or involvement with the company. While both Mathematica and Alpha were indeed major players in changing the way mathematics is done – particularly in applied settings – I am citing those particular products largely as icons, using two specific products to represent a range of new digital tools that were being developed around the world at that time. While Wolfram’s systems were ones I myself made early use of in my work, other mathematicians were also active in that digital mathematical revolution, using different systems. Still, Mathematica was the system that caught the public attention.

Since the turn of the new Millennium, I doubt if anyone making professional use of mathematics in their job, or indeed any adult using mathematics in their everyday lives, has taken out paper-and-pencil and followed a classical algorithm to add, subtract, multiply or divide numbers in an array of real-life size, or perform complex algebraic reasoning to solve systems of equations, or solve problems using calculus, or any other established mathematical procedure. Not only would it now be a waste of valuable human time and energy doing something a cheap machine can do in far less time with no possibility of error, but many of the problems that people encounter in their careers and lives have simply too much data for the human mind to handle. Those same digital tools that have made the execution of mathematical procedures unnecessary have also come to dominate and drive our world, so many of the problems that require mathematics in their solution are now simply beyond human capacity. That’s why Amazon Web Services has become such a behemoth for data storage and processing.

But that does not mean humans no longer need to have some mathematical skills. On the contrary, they are as crucial as ever – unless you are willing to be totally reliant on others, but personally, I have never felt comfortable doing that with things that are part of my life every day. What has changed are the specific mathematical skills required today. There are plenty of things computers cannot do or do poorly. Genuinely creative thinking and analogical reasoning are two obvious ones – though with today’s massive cloud computing resources we can use systems that provide an approximation often adequate for the purpose, and on occasion can be better than humans.

Mostly, however, where you need humans is going from a real-world challenge situation to formulating one or mathematical tasks that can help you make progress. Sometimes, progress means solving a real-world problem in the sense of getting a specific answer (say, a number), but much more commonly it’s about finding a better method, where “better” can mean faster, cheaper, safer, or whatever other criterion is important, and where the change may involve developing a new method or improving an existing one.

This way of using mathematics was the focus of that mini-course I gave at a California school (Nueva School) in January of this year, that I wrote about in the February, March, and April posts to this column.

Though several mathematicians and mathematics education scholars expressed agreement with what I wrote, my articles brought some critiques from teachers and parents. The critiques all made reference to my asides about the Common Core State Standards in the first two of the posts. Since “Devlin’s Angle” no longer seems to be a target for the CCSS social media trolls (likely because the yield of issues to react to relative to the length and substance of most of my posts makes it less rewarding to them), I made some efforts to find out what exactly it is about the CCSSs that they objected to. As far as I could ascertain, the issue was inevitably (and predictably) to do with particular implementations of the Standards in specific curricula or (and this seems to be the most common occurrence) claims that a particular homework exercise was a “Common Core exercise”, which of course it cannot be since the CCSS are, as the name indicates, purely a set of standards to attain, not in any way a curriculum or curriculum content.

More generally, in fact, pretty well all critiques of the CCSS are due to a complete misunderstanding of what they are, why they are, and what they say. The issue was nicely dealt with in this 2014 article in the Hechinger Report.

My reason for bringing the Common Core into my series of posts was to point out that the standards were developed precisely to help guide school districts, schools, and teachers in the tricky task of updating K-12 mathematics education to adequately prepare tomorrow’s citizens for life and work in a world where calculation is no longer a central pillar of mathematics.

Having said that, I should point out that the above statement in no way implies that we should drop the teaching of basic arithmetic and algebra from the school system. As I discussed in some length in the third of my Nueva-inspired articles, the change that is required in K-12 math education is not so much in the mathematical topics but the reason they are now being taught and, in consequence, the way they should be taught.

Teaching for execution is no longer the primary driver, since no one using mathematics in the real world does that anymore. What is now of crucial importance is teaching for understanding. Digital systems outperform humans to an insane degree when it comes to execution. But they don’t understand; people have to supply that.

I leave you with an image I pulled from one of those Common Core social media rants some time ago. (I no longer remember the exact source.)

Typical social media posts about Common Core mathematics.
I have three comments about the post on the left. First, the mathematics in the bottom left is not some fancy new algorithm, it is what a child wrote down in reasoning (sensibly) about a particular arithmetic problem. Second, if you are unable to follow what the child is doing, you would have trouble making effective use of mathematics in today’s world. It’s pretty basic. (Your kid just did it, right?) Third, if you are a parent and you don’t see why it is important that today’s school students acquire those math reasoning skills, please don’t communicate your skepticism to your children. Doing so would be a great disservice, to your child, to your child’s math teacher, and to society. The mathematical world has changed significantly. That occurred over twenty years ago. It is not going to change back. Sit back, relax, be encouraging, and let the kids take over. They do just fine with it.

REFERENCE: During the period when the computer revolutionized how mathematics is done, I edited the American Mathematical Society’s “Computers and Mathematics” section of their monthly notices publication, sent to all members. I wrote about the column and that period in general in a paper that I submitted to the Proceedings of the Jon Borwein Commemorative Conference, held in 2017. Borwein, who died tragically young in 2016, was a leading pioneer in bringing digital technologies into mathematics. You can access a preprint of the paper HERE.



Wednesday, April 4, 2018

How today’s pros solve math problems: Part 3 (The Nueva School course)

By Keith Devlin

You can follow me on Twitter @profkeithdevlin


NOTE: This article is the final installment of a four-episode mini-series posted here starting in mid-January. In writing it, I have assumed my readers have read those three earlier pieces.

At the end of last month’s post, I left readers with a (seemingly) simple arithmetic problem. I prefaced the problem with the following two instructions:

1. Solve it as quickly as you can, in your head if possible. Let your mind jump to the answer.

2. Then, and only then, reflect on your answer, and how you got it.

The goal here, I said, is not to get the right answer, though a great many of you will. Rather, the issue is how do our minds work, and how can we make our thinking more effective in a world where machines execute all the mathematical procedures for us?

Here is the problem.

PROBLEM: A bat and a ball cost $1.10. The bat costs $1 more than the ball. How much does the ball cost on its own? (There is no special pricing deal.)

What answer did you get? And what did you learn from the subsequent reflection?

Before I continue, I should note that the use of this problem (which you can find in many puzzle books and on countless websites) in the context of trying to maximize the human mind’s innate abilities in order to become good 21st Century mathematical thinkers, is due to Gary Antonick, with whom I co-taught a Stanford Continuing Studies adult education course last fall. It was in that course that I gave the second iteration of the UPS problem I subsequently based my Nueva School course on. The discussion of the bat-and-ball problem that follows is the one Antonick presented in our course.

Now to the problem itself. The most common answer people give instantly to this problem is that the ball costs 10¢. It’s wrong (and many realize that is the case soon after their mind has jumped to that wrong number). What leads many astray is that the problem is carefully worded to run afoul of what under normal circumstances is an excellent strategy. (So if you got it wrong, you probably did so because you are a good thinker with some well-developed problem-solving strategies— problem-solving heuristics is the official term, and I’ll get to those momentarily. So take heart. You are well placed to do just fine in 21st Century mathematical thinking. You simply need to develop your heuristics to the next level.)

Here is, most likely, what your mind did to get to that 10¢ answer. As you read through the problem statement and came to that key phrase “cost more,” your mind said, “I will need to subtract.” You then took note of the data: those two figures $1.10 and $1. So, without hesitation, you subtracted $1 from $1.10 (the smaller from the larger, since you knew the answer has to be positive), getting 10¢.
Notice you did not really perform any calculation. The numbers are particularly simple ones. Almost certainly, you retrieved from memory the fact that if you take a dollar from a dollar-ten, you are left with 10¢. You might even have visualized those amounts of money in your hand.

Notice too that you understood the mathematical concepts involved. Indeed, that was why the wording of the problem led you astray!

What you did is apply a heuristic you have acquired over many financial transactions and most likely a substantial number of arithmetic quiz questions in elementary school. In fact, the timed tests in schools actively encourage such a “pattern recognition” approach. For the simple reason that it is fast and usually works!

We can, therefore, formulate a hypothesis as to why you “solved’ the problem the way you did. You had developed a heuristic (identify the arithmetic operation involved and then plug in the data) that is (a) fast, (b) requires no effort, and (c) usually works. This approach is a smart one in that it uses something the human brain is remarkably good at—pattern recognition—and avoids something our minds find difficult and requiring effort to master (namely, arithmetic calculation).

Of course, primed by the context in which I presented this particular problem, you probably expected there to be a catch. So, after letting your mind jump to the 10¢ answer, you likely took a second stab at it (or, if you were anxious about “getting a wrong answer,” made this your first solution) by applying an algorithm you had learned at school. Namely, you reasoned as follows:

Let x = cost of bat and y = cost of the ball. Then, we can translate the problem into symbolic
form as x + y = 1.10 ,   x = y + 1

Eliminate x from the two equations by algebra, to give
1.10 – y = y + 1

Transform this by algebra to give
0.10 = 2y

Thus, dividing both sides by 2, you conclude that
y = 5¢.

And this time, you get the correct answer.

You may, in fact, have been able to carry out this procedure in your head. When I was at school, I could do algebraic manipulations far more complicated than this in my head, at speed. But, truth be told, since I started outsourcing arithmetic to machines many decades ago, I have lost that skill, and now have to write down the equations and solve them on paper. (This is a confirmation, if any were needed, that arithmetic calculations do not come naturally to the human brain. Over the years, as my mental arithmetic skills have declined, my pattern recognition abilities have not diminished, but on the contrary have dramatically improved, as I learned—automatically, through exposure—to recognize ever more fine-grained distinctions.)

Whether or not you can do the calculation in your head, it is of course entirely formulaic and routine. Unlike the first method I looked at (a heuristic that is fast and usually right), this method is an algorithmic procedure, it is slow (much slower than the first method, even when the algebraic reasoning is carried out in your head), but it always works. It is also an approach that can be executed by a machine. True, for such a simple example, it’s quicker to do it by hand on the back of an envelope, but as a general rule, it makes no sense to waste the time of a human brain following an algorithmic procedure, not least because, even with simple examples it is familiarly easy to make a small error that leads to an incorrect answer.

But there is another way to solve the problem. It’s the way I addressed it, and, according to Antonick, who has given it to many professional mathematicians and asked them to vocalize their solutions, the way many math pros solve it. Like the first method we looked at, it is a heuristic, hence instinctive and fast, but unlike the first heuristic method, it always works.

This third method requires looking beyond the words, and beyond the symbols in the case of a problem presented symbolically, to the quantities represented. Though I (and likely other mathematicians) don’t visualize it quite this way (in my case it is more of a vague sense-of-size), the following image captures what we do.

http://www.stanford.edu/~kdevlin/

As we read the problem, we form a mental sense of the two quantities, the cost of the ball-on-its-own and the cost of the bat-plus-ball, together with the stated relation between them, namely that the latter is $1 more than the former. From that mental image, where we see the $1.10 total consists of three pieces, one of which has size $1 and the other two of which are equal, we simply “read off” the fact that the ball costs 5¢. No calculation, no algorithm. Pure pattern recognition.

This solution is an example of Number Sense, the critical 21st Century arithmetic skill I wrote about in the January 1, 2017 Huffington Post companion piece to the article I published on the same day as my article about all my math skills becoming obsolete, which I referred to in my last post here on Devlin’s Angle.

It is, I suggest, hard to imagine how a computer system could solve the problem that way. (Of course, you could write a program so it can perform that particular pattern recognition, but the essence of number sense is that you can apply it to many numerical problems you come across.)

Those three ways to solve the bat-and-ball problem I just outlined are examples of what the famous Australian (pure) mathematician Terrence Tao has called (in his blog), respectively, pre-rigorous thinking, rigorous thinking, and post-rigorous thinking. You can also listen to him explain these three categories in a short video in the Numberphile series.

Post-rigorous heuristic thinking is how today’s math pros use mathematics to solve real-world problems. In fact, as Tao makes clear, post-rigorous thinking is what the pros use most of the time to solve abstract problems in pure math. The formal, symbolic, rigorous stuff comes primarily at the end, to check that the solution is logically correct, or at various intermediate points to make those checks along the way.

In the case of solving real-world problems, the pros almost always turn to technology to handle any algebraic deductions. In contrast, though pure mathematicians sometimes do use those technology products as well, they often find it much quicker, and perhaps more fruitful in terms of gaining key insights, to do the algebraic work by hand.

So, one of the big question facing math teachers today is, how do we best teach students to be good post-rigorous mathematical thinkers?

In the days when the only way to acquire the ability to use mathematics to solve real-world problems involved mastering a wide range of algorithmic procedures, becoming a mathematical problem solver frequently resulted in becoming a post-rigorous thinker automatically.

But with the range of tools available to us today, there is a good reason to assume that, with the right kinds of educational experiences, we can significantly shorten (though almost certainly not eliminate) the learning path from pre-rigorous, through rigorous thinking, to post-rigorous mathematical thinking. The goal is for learners to acquire enough effective heuristics.

To a considerable extent, those heuristics are not about “doing math” as such. Rather, they are focused on making efficient and effective use of the many sources of information available to us today. But before you throw away your university-level textbooks, you need to be aware that the intermediate step of mastering some degree of rigorous thinking is likely to be essential. Post-rigorous thinking is almost certainly something that emerges from repeated practice at rigorous thinking. Any increased efficiency in the education process will undoubtedly come from teaching the formal methods in a manner optimized for understanding, as opposed to optimized for attaining procedural efficiency, as it was in the days when we had to do everything by hand. See Daniel Willingham’s excellent book Why Don’t Students Like School? for a good, classroom-oriented look at what it takes to achieve mastery in a discipline.

Now to that UPS routing problem that was the focus of my Nueva School course. [You will find it discussed here.] Here are some of the hints and suggestions about solving the problem I made to the students in the three courses where I used it. Whether they followed my advice was entirely up to them. The purpose of the course was not to solve the problem unaided—even an entire semester would not be enough time for that with students who had never approached a problem the way the pros do. Rather, it was to give them an experience of the method.

First, they had to work in teams of three to five. I let them select the teams, but said it would be good if at least one person on each team felt they were “good at math.”

Then, start out by using Google to find out what you can about the problem domain, and any attempts made by others to solve it.

Whenever you come across a reference to a concept, an approach, or a method that you suspect might be relevant, use general Web resources like Wikipedia to get an initial understanding of what they are and what they can do.

Follow any leads your search brings up to solutions of problems that look similar. Note what methods were used to solve them.

If you come across references to others who have worked on the problem, or a similar one, send them a brief email. You may not get a reply, but occasionally you will, and it could be invaluable. (I receive such emails all the time. Mostly I do not have time to respond, but occasionally one lands in my inbox when I have a spare moment, and I happen to know something that might help, so I shoot back a brief reply, often just a reference to a particular source.)

When you get to a point where you need to perform a specific calculation, perhaps because you have found a solution to a very similar problem someone else has obtained and published, but your data is different, use Wolfram Alpha. It is structured so you can use pattern recognition (of formulas) to identify the appropriate technique and then edit the example provided to be the one you want to solve.

Reinforce your use of Wolfram Alpha by using YouTube to find suitable videos that provide you with quick tutorials on the technique.

The resources I just mentioned are all listed on that chart of “Important Mathematical Technology Tools” I published with the first two articles in this series.

As it turns out, with the UPS routing problem, the sequence of steps I have outlined so far quickly leads to identification of a small number of possible solution techniques for which there are many very accessible YouTube videos, and in fact, for this problem there is no need to go much further into my list of tools, if at all.

You should, though, check out the various other resources on my list, to see what they offer. Each new problem has to be approached afresh, in its own terms. Twitter is on my list because it is my list, and I have sufficiently many math-expert Twitter followers that a quick tweet can often yield just the information I need, saving me having to send out a slew of emails to people I think might be able to help. LinkedIn is also idiosyncratic to me, since I have a good network of mathematics and technology professionals I can contact. But the other resources are pretty generic.

Ideally, everything goes much more smoothly if you can avail yourself the services of a math consultant to assist you in negotiating the various resources. (I was that consultant to the teams in the three courses I gave.)

Interestingly, in the final meeting of my Princeton class (which was the fist time I used the UPS problem in a course), after having the student teams present their solutions, I gave the solution I had obtained, at the end of which two individuals came up to me to say they hoped I had not minded their sitting in on the class. (It was an experimental course, and there had been strangers sitting in for one or two sessions throughout the semester, so I had not paid them any attention.) They were, they said, postdocs working with Professor X, who was a math consultant for UPS and had worked on the algorithm the class and I had been trying to reverse engineer. Hence their curiosity-driven attendance on the last day! Unbeknownst to me, my final lecture had been my oral exam!

“How did I do?”, I asked. “You got it pretty well right,” they replied.

Which was nice for me, but it would not have mattered if I had followed a different track. What was important from an educational standpoint was the process.

Something else I suggested to the class was to come up with a solution—any solution—as soon as you can. “Don’t worry if it is optimal or even right,” I said. “Just check it by computation, perhaps in the form of a spreadsheet simulation. Once you have some solution that you can check (in the case of my UPS problem, check against the shipping data I supplied, or any other UPS data you can find on the Web), you can iterate to find a better one. It might turn out that your first solution, or your first three or four, won’t even get you to first base, but in the process of formulating and checking those initial attempts, you will inevitably gain insight into the problem you are trying to solve. Remember, computation is cheap, fast, and essentially limitless.”

If you are not familiar with this way of solving math problems, it may not seem like an approach that will work. But it does. It is, in fact, how all of today’s pros do it!

If you have not already done so, now is a good time to check out the dictionary definition of the word heuristic! Here is Wikipedia’s (at the time of writing):

“A heuristic technique (from the ancient Greek for “find” or “discover”), often called simply a heuristic, is an approach to any problem-solving, learning, or discovery that employs a practical method not guaranteed to be optimal or perfect, but sufficient for the immediate goals. Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision. Examples of this method include using a rule of thumb, an educated guess, an intuitive judgment, guesstimate, stereotyping, profiling, or common sense.”

Without an expert consultant, the heuristics approach to solving real-world problems can work, but it definitely goes a lot faster, and with a far great likelihood of success, if you have a math expert you can call on. Not to “do any math.” Computer systems handle those parts. Rather to help you negotiate the vast array of resources at your disposal and select the most promising one(s) to try next. For that is what using mathematics to solve a real-world problem really boils down to these days: managing resources.

And managing resources is something humans are innately good at. Natural selection always favors those creatures which are best able to manage the available resources. We are here as present-day humans because as a species we are good at doing that. What is new in the case of mathematical problem solving is that pieces of mathematics (formulas, equations, procedures, algorithms, techniques) are now among the “intellectual Lego pieces” (freely accessible on the Web) we can use as we assemble a solution.

As the students in my three courses could, in principle, attest, you don’t need vast expertise in mathematics to work this way. You just need to be a good thinker able to work in a small team. I say, “in principle,” since I think it highly likely most of not all the students felt they did not do much at all by way of using math to solve a problem. But that, I would say, is because they have a conception of “using math to solve a problem” rooted in the Nineteenth Century, if not the Fourth Century BCE. From my perspective, they absolutely were able to do what I just said they did.

Of course, they were not as good at it as I am. I’ve been at this game a lot longer, and, make no mistake about it, experience counts. (I think it is close to being the only thing that counts.)

What does not count, at least to any extent even remotely approaching the prohibitive degree it used to, is the ability to “do the math.” You just need to be able to select (hopefully, with help from someone with experience) the right pieces from the available online mathematical smorgasbord, and stitch them together in an appropriate way.

This kind of problem-solving doesn’t feel like math (as we all came to love or hate), that’s for sure. In fact, it doesn’t even feel like work. Once they got into the swing of it, even the students who declared they were not good at math or did not enjoy it, found they were having a good time, working in teams in a creative, explorative way. For the fact is, properly approached, humans enjoy problem-solving. (That’s another consequence of natural selection— problem-solving, particularly group problem solving, is one of our species’ key survival advantages.)

In fact, another way to look at the recent revolution in how we “use math” to solve real-world problems, is that it has brought “using math” into the mainstream of human group activities we naturally find enjoyable. At heart, mathematical thinking is little more than formalized common sense. It always has been. Which means it is something we can all do. (In my 2000 book The Math Gene, I presented an evolutionary explanation for the human brain’s acquisition of the ability to do mathematics, which implied that mathematical capacity is in the human gene pool, and hence available for all of us to “switch on.”) What caused many people problems over the centuries was that, before we had technologies that could handle the formal symbol-manipulation stuff, the only way to employ our innate capacity for mathematical thinking was to train the brain to do those manipulations. But manipulating algebraic symbols with logical precision is most definitely not something our brains evolved to do. (Our early ancestors’ lives on the savannas did not present much by way of a need for algebra.) So we find it very hard. Only with great effort over several years can we train our brains to do such work. And even then, we are error-prone.

Incidentally, practically everything I have said in this article applies to the way 21st Century coders work. In coding as in mathematics, the days are long gone when it was all about writing thousands of lines of instructions. The modern-day mathematician’s Web resource MathOverflow (on my chart of useful math tools) was modeled on, and named after, the coding world’s StackOverflow. Both groups of professionals use heuristics. In today’s world, highly regarded math problem solvers and good coders have simply acquired a richer and more effective set of heuristics than the ones who are less highly ranked. And for the most part, developing heuristics is a result of reflective experience, not some innate talent.

And there you have it. The primary goal in 21st Century mathematics-education-for-all is the development of a good repertoire of heuristics.

I’ll leave you with a graphical summary of Tao’s categorization of the three kinds of mathematical thinking we can bring to problem-solving. I introduced this categorization above to provide a perspective on the three phases each one of us has to go through to become proficient mathematical (real-world) problem solvers. But it also provides an excellent summary of three historical stages of mathematical thinking as it has evolved over the past ten thousand years or so, from the invention of numbers in Sumeria, where the mathematical thinking of the time was accessible to all, through three millennia of formal mathematics development, where many people were never able to make effective use of it, and now into the third phase, where, because of technology, mathematical thinking can once again be accessible to all.

http://www.stanford.edu/~kdevlin/

To be sure, we do not know the degree to which people have to master rigorous thinking to become good post-rigorous thinkers. As I already noted, I don’t for a second imagine that stage can be by-passed. (See the Willingham book I cited.) But, given today’s technological toolkit, including search, social media, online resources like Wolfram Alpha and Khan Academy, and a wide array of online courses, it is absolutely possible to master most of the rigorous thinking you need “on the job,” in the course of working on meaningful, and hence motivational and rewarding, real-world problems.

This is not to say there is no further need for teachers. Far from it. Very few people are able to become good mathematical thinkers on their own. Newtons and Ramanujans, who achieved great things with just a few books, are extremely rare. The vast majority of us need the guidance and feedback of a good teacher.

What the inevitable transition to 21st Century math learning requires is that mathematics teachers operate very differently than in the past. The days where you need a live person to deliver information are largely over. Today, teaching is much more a matter of being a coach and mentor. To be sure, you can occasionally find such teaching on the Internet, but it works only if you can be one-on-one with that teacher. I expect there will be change, but I don’t expect an economy of scale. If I had to make a guess, I would predict that in due course you will find your (specialist) math teacher by going online to a Math-Teacher-Match.com website, where you will be paired with a practicing 21st Century math professional who spends part of each day coaching and mentoring students.


LABELS: mathematical thinking, problem-solving, rigorous thinking, pre-rigorous thinking, post-rigorous thinking, Terrence Tao, social media in mathematics

Friday, March 9, 2018

How Today’s Pros Solve Math Problems: Part 2

By Keith Devlin

You can follow me on Twitter @profkeithdevlin


CHANGE OF PLAN: When I wrote last month’s post, I said I would conclude the description of my Nueva School Course this time. But when I sat down to write up that concluding piece, I realized it would require not one but two further posts. The course itself was the third iteration of an experiment I had tried out on a university  class of non-science majors and an Adult Education class. This series of articles is my first attempt to try to describe it and articulate the thinking behind it. As is often the case, when you try to describe something new (at least it was new to me), you realize how much background experience and unrecognized tacit knowledge you have drawn upon. In this post, I’ll try to capture those contextual issues. Next month I’ll get back to the course itself.


We all know that mathematics is not always easy. It requires practice, discipline and patience,  as do many other things in life.  And if learning math is not easy, it follows that teaching math is not easy either. But it can help both learner and teacher if they know what the end result is supposed to be.

In my experience, many learners and teachers don’t know that. In both cases, the reason they don’t know it is that no one has bothered to tell them. There is a general but unstated assumption that everyone knows why the teaching and learning of mathematics is obligatory in every education system in the world.  But do they really?

There are two (very different) reasons for teaching and learning mathematics.

One reason is that it is a way of thinking that our species has developed over several thousand years, that provides wonderful exercise for the mind, and yields both challenging intellectual pleasure and rewarding aesthetic beauty to many who can find their way sufficiently far into it. In that respect, it is like music, drama, painting,  philosophy, natural sciences, and many other intellectual human activities. This is a perfectly valid reason to provide everyone with an opportunity to sample it, and make it possible for those who like what they see to pursue it as far as they desire. What it is not, is a valid reason for making learning math obligatory throughout elementary, middle, and high school education.

The argument behind math’s obligatory status in education is that it is useful; more precisely, it is useful in the practical, everyday world. This is the view of mathematics I am adopting in the short series of “Devlin’s Angle” essays of which this is the third. (There will be one more next month. See episode 1 here and episode 2 here.)

Indeed, mathematics is useful in the everyday practical world. In fact, we live in an age where mathematics is more relevant to our lives than at any previous time in human history. 

It is, then, perfectly valid to say that we force each generation of school students to learn math because it is a useful skill in today’s world. True, there are plenty of people who do just fine without having that skill, but they can do so only because there are enough other people around who do have it.

But let’s take that argument a step further. How do you teach mathematics so that it prepares young people to use it in the world? Clearly, you start by looking at the way people currently use math in the world, and figure out how best to get the next generation to that point. (Accepting that by the time those students finish school, the world’s demands may have moved on a bit, so those new graduates may have a bit of catch up and adjustment to make.)

If the way the professionals use math in the world changes, then the way we teach it should change as well.  Don’t you think? That’s certainly what has happened in the past.

For instance, in the ninth century, the Arabic-Persian speaking traders around Baghdad developed a new, and in many instances more efficient, way to do arithmetic calculations at scale, by using logical reasoning rather than arithmetic. Their new system, which quickly became known as al-jabr after one of the techniques they developed to solve equations, soon found its way into their math teaching.

When Hindu-Arabic arithmetic was introduced into Europe in the thirteenth century, the school systems fairly quickly adopted it into their arithmetic teaching as well. (It took a few decades, but knowledge moved no faster than the pace of a packhorse back then. I tell the story of that particular mathematics-led revolution in my 2011 book The Man of Numbers.)

The development of modern methods of accounting and the introduction of financial systems such as banks and insurance companies, which started in Italy around the same time, also led to new techniques being incorporated into the mathematical education of the next generation.

Later, when the sixteenth century French mathematician François Viète introduced symbolic algebra, it too became part of the educational canon.

In each case, those advances in mathematics were introduced to make mathematics more easy to use and to increase its application. There was never any question of “What is this good for?” People eagerly grabbed hold of each new development and made everyday use of it as soon as it became available.

The rise of modern science (starting with Galileo in the seventeenth century) and later the Industrial Revolution in the nineteenth century, led to still more impetus to develop new mathematical concepts and techniques, though some of those developments were geared more toward particular groups of professionals. (Calculus, for example.)

To make it possible for an average student or worker to make use of each new mathematical concept or technique, sets of formal calculating rules (algorithmic procedures) were developed and refined. Once mastered, these made it possible to make use of the new mathematics to handle—in a practical way—the tasks and problems of the everyday world for which those concepts and techniques had been developed to deal with in the first place.

As a result of all those advances, by the time the Baby Boomers came onto the educational scene in the 1950s, the curriculum of mathematical algorithms that were genuinely important in everyday life was fairly large. It was no longer possible for a student to understand all the underlying mathematical concepts and techniques behind the algorithms and procedures they had to learn. The best that they could do was master, by repetitive practice, the algorithmic procedures as quickly as possible and move on. [A few of us had difficulty doing that. We wanted to understand what was going on. By and large, we frustrated our teachers, who seemed to think we were simply troublesome slow learners. Some of us eventually learned to “play the mindless algorithm game” in class to pass the test, but kept struggling on our own to understand what was going on, setting us on a path to becoming mathematics professors in the 1970s.]

It was while that Boomer generation was going through the school system that mathematics underwent the first step of a seismic shift that within a half of a century would completely revolutionize the way mathematics was done. Not the pure mathematics practiced by a few specialists as an art—though that too would be impacted by the revolution to some extent. Rather, it was mathematics-as-used-in-the-world that would be radically transformed.

The first step of that revolution was the introduction of the electronic desktop calculator in 1961. Although, mechanical desktop calculators had been available since the turn of the Twentieth Century, by and large their use was restricted to specialists—often called “computers” in businesses. [I actually had a summer-job with British Petroleum as such a specialist in my last three years at high school, and it was in my final year in that job that the office I worked in acquired its first electronic desktop calculator and the British Petroleum plant bought its first digital computer, both of which I learned to use.] But with the increasing availability of electronic calculators, and in particular the introduction of pocket-sized versions in the early 1970s, their use in the workplace rapidly became ubiquitous. Mathematics underwent a major change. Humans no longer needed to do arithmetic calculations themselves, and professionals using arithmetic in their work no longer did.

It was not too many years later that, one by one, electronic systems were developed that could execute more and more mathematical procedures and techniques, until, by the late 1980s, there were systems that could handle all the mathematical procedures that constituted the bulk of not only the school mathematics curriculum, but the entire undergraduate math curriculum as well. The final nail in the coffin of humans needing to execute mathematical procedures was the release of the mathematics system Mathematica in 1988, followed soon after by the release of Maple.

In the scientific, industrial, engineering, and commercial worlds, each new tool was adopted as soon as it became available, and since the early 1990s, professionals using mathematical techniques to carry out real-world tasks and solve real-world problems have done so using tools like Mathematica, Maple, and a host of others that have been developed.

Simultaneously, colleges and universities quickly incorporated the use of those new tools into their teaching. And while the cost of the more extensive tools put their use beyond most schools, the graphing calculator too was quickly brought into the upper grades of the K-12 system, after its introduction in 1990.

Yet, while the pros in the various workplaces changed over to the new human-machine-symbiotic way of doing math with little hesitation, most educators, exhibiting very wise instincts, proceeded with far more caution. The first wave of humans to adopt the new, machine-aided approach had all learned mathematics in an age when you had to do everything yourself. Back then, “computers” were people. For them, it was easy and safe to switch to executing a few keystrokes to make a computer run a procedure they had carried out by hand many times themselves. But how does a young person growing up in this new, digital-tools-world learn how to use those new tools safely and effectively?

To some extent, the answer is (and was) obvious. You teach not for smooth, proficient, accurate execution of procedures, but for broad, general understanding of the underlying mathematics. The downplay of execution and increased emphasis on understanding are crucial. Computers outperform us to ridiculous degrees (of speed, accuracy, size of dataset,  and information storage and retrieval) when it comes to execution of an algorithm. But they do not understand mathematics. They do not understand the problem you are working on. They do not understand the world. They don't understand anything. 

People, on the other hand, can understand, and have a genetically inherited desire to do so.

But just how do you go about teaching for the kind of understanding and mastery that is required for students to transition into worlds and workplaces dominated by a wide array of new mathematical tools, where they will encounter work practices that involve very little by way of hand execution of algorithms?

We know so little about how people learn (though we do know a whole lot more than we did just a few decades ago), that most of us with a stake in the education business are rightly concerned about making any change that would effectively be a massive experiment on an entire generation. So we can, and should, expect small steps, particularly in systemic education.

In the U.S., the mathematicians who developed the mathematical guidelines for the Common Core State Standards made a good first attempt at such a small step. True, it quickly ran into difficulties when it came to implementing the guidelines in a large and complex public educational system that is answerable to the public. But that is surely a temporary hiccup. Most of the problems at launch came from a lack of effective ways to assess the new kind of learning. Those problems can be and are being fixed. Which is just as well. For, although it’s possible to argue for tinkering with specific details of the Common Core State Standards guidelines, in terms of setting out a broad set of educational goals to aim for, there is no viable alternative first step. The pre-1970s educational approach is no longer an option.

In the meantime, individual teachers at some schools (particularly, but not exclusively, private schools) have been trying different approaches, in some cases sharing their experiences on the MTBOS (Math Twitter Blog-O-Sphere), making use of another technological tool (social media) now widely available. [For a quick overview of one global initiative to support and promote such innovations, the OECD’s Innovative Pedagogies for Powerful Learning project (IPPL), see this recent article from the Brookings Institution.]

The mini-course I gave at Nueva School in the San Francisco Bay Area last January, which I talked about in the first of this short series of essays, is one such experiment in teaching mathematics in a way that best prepares the next generation for the world they will live and work in after graduation. I tested it first with a class of non-science majors in Princeton in the fall of 2015 and then again with an Adult Education class at Stanford in the fall of 2017. The Nueva School class was its third outing.

With the above backstory now established, next month I will describe that course and talk about how today’s pros “do the math”. (Again, let me stress, I am not talking here about “pure math”, the academic discipline carried out by professional mathematicians in universities and a few think tanks. My focus here is on using math in the everyday world.)

In the meantime, I’ll leave you with a simple arithmetic problem that I will discuss in detail next time.

It comes with two instructions:

  1. Solve it as quickly as you can, in your head if possible. Let your mind jump to the answer.
  2. Then, and only then, reflect on your answer, and how you got it.
The goal here is not to get the right answer, though a great many of you will. Rather, the issue is how do our minds work, and how can we make our thinking more effective in a world where machines execute all the mathematical procedures for us?

Ready for the problem? Here it is. 

PROBLEM: A bat and a ball cost $1.10. The bat costs $1 more than the ball. How much does the ball cost on its own? (There is no special pricing deal.)

Wednesday, February 7, 2018

How today’s pros solve math problems: Part 1

Last month, I wrote about my recent experience teaching a three-day mini-course in the Nueva School January electives “Intersession” program. What I left out was a description of the course itself. I ended with the below diagram as a teaser. I said that, when reading in the usual left-right-down reading order, these were the technology tools that I typically turn to when I start to work on solving a new problem.


A number of mathematicians commented on social media that their list would be almost identical to mine. That did not surprise me. My chart simply captures the way today’s pros approach new problems. A number of math teachers expressed puzzlement. That too did not surprise me. The current mathematics curriculum is still rooted in a conception of “doing math” that developed to meet society’s needs in the 19th Century.

Actually, I should point out that the diagram above is not exactly the one I published last month. I have added an icon for a spreadsheet. A mathematician in Austria emailed me to say I should have included it. The two of us had corresponded in the past about the use of spreadsheets in mathematics, both in problem solving and in teaching, and we were (and are) very much on the same page as to their usefulness in a wide variety of circumstances. My excuse for overlooking it the first time round was that it was only the second technological tool I brought into my mathematics arsenal, so far back in my career that I had long ago stopped thinking of it as something new. (The first piece of “new tech” I adopted was the electronic calculator, and that too did not appear in my chart.) I suspect that almost all math teachers, and indeed, pretty well all of society, make frequent use of calculators and spreadsheets, not only in their professional activities but in their social and personal lives as well. Still, the spreadsheet is such a powerful, ubiquitous mathematics tool, I should have included it, and now I have. (Its use definitely figured in the guidance I gave to the Nueva School class.) I have placed it in the position in my list that, on reflection, I find I turn to in order of frequency.

Some of the responses I received from teachers indicated that I need to clarify that, by “solving a mathematical problem”, I mean using mathematics to solve a real-world problem. The problem we worked on at Nueva School was one UPS worked on not long ago: “What is the most efficient way to route packages from place to place?” More on that later. A simpler example in the same vein is when we ask ourselves “Which kind, model, and hardware configuration of mobile phone best meets my needs within my current budget?”—an example where, for most of us, the item’s cost is high enough for us to weigh the (many) options fairly carefully.

This is clearly not the same as “solving a math problem” in a typical math textbook. For example, “What are the roots of the equation x2 + 3x – 5 = 0?” Those kinds of questions are, of course, designed to provide practice in using various specific, sharply focused, mathematics techniques, procedures, formulas, or algorithms.

Those techniques, procedures, etc. are the basic building blocks for using mathematics to solve problems in real life, but they don’t really present much of a problem, in the sense the word is used outside the math class. Indeed, the reason it can be valuable to master those basic techniques, etc. is that being able to use them fluidly means they won’t be a problem (in the sense of an obstacle) that gets in the way of solving what really is a mathematical problem (e.g., which phone to buy). That, of course, is why we call them basic skills. But having mastery of a range of basic skills does not make a person a good problem solver any more than being a master bricklayer makes someone an architect or a construction engineer.

My focus then, is on using math to solve real-world problems. That’s where things are very different from the days when I first learned mathematics. Back in the 1950s and 60s, when I went through the school system, we spent a huge amount of time mastering algorithms and techniques for performing a variety of different kinds of numerical and symbolic calculations, geometric reasoning, and equation solving. We had to. In order to solve any real-world problem, we had to be able to crank the algorithmic and procedural handles.

Not any more. These days, that smartphone in your pocket has access to cost-free cloud resources that can execute any mathematical procedure you need. What’s more, it will do it with a dataset of any real-world size, with total accuracy to whatever degree you demand, and in the majority of cases in a fraction of a second.

To put it another way, all those algorithms, techniques, and procedures I spent years mastering, all the way through to earning my bachelors degree in mathematics, became obsolete within my lifetime, an observation I wrote about in an article in the Huffington Post in January of last year.

So, does that mean all that effort was wasted? Not at all. Discounting the fact that in my case, I was able to make good use of those skills and knowledge for several decades before the march of technology rendered them obsolete, the one thing that I gained as a result of all that procedural learning that is as valuable today as it was back then, was the ability to think mathematically. I wrote about one aspect of that “mathematical thinking” mental ability, number sense, in a simultaneously published follow-up piece to that Huffington Post article.

In today’s world, all the algorithmic, computational, algebraic, geometric, logical, and procedural skills that used to take ten years of effort to master can now be bought for $699. At least, that amount (the price of an iPhone 8, which I chose for illustration) is all it costs to give you access to all those skills. Making effective use of that vast powerhouse of factual knowledge and procedural capacity requires considerable ability. Anyone who mastered mathematics the way I did acquired that ability as an automatic by-product of mastering the basic skills. But what does it take to acquire it in an age when all those new tools are widely available?

The answer, of course (though not everyone involved in the mathematics education system thinks it is obvious, or even true), is that the educational focus has to shift from procedural mastery to understanding. Which is precisely the observation that guided the Common Core initiative in the United States. Yes, I know that the current leadership of the US Department of Education believes that the Common Core is a bad idea, but that is an administration that also believes the future of energy lies in fossil fuels, not renewables, and the highly qualified, career-professional contacts I have in the Department of Education have a very different view.

How do you acquire that high-level skill set? The answer is, the same way people always did: through lots of practice.

But be careful how you interpret that observation. What need to be practiced are the kinds of activities that you would use as a professional—or at least a competent user of mathematics—in the circumstances of the day. In my school days, that meant we had to practice with highly constrained, “toy” problems. But with today’s technologies, we can practice on real-world problems using real-world data.

Almost inevitably, when you do that, you find you frequently need to drop down to suitably chosen “toy problem” variants of your task in order to understand how a particular online tool (say) works and what it can (and cannot) do. But today, the purpose of, say, inverting a few 2x2 or 3x3 matrices is not (as it was in my day) so you can become fluent at doing so, and certainly not because you will actually invert by hand that 100x100 matrix that has just reared its ugly head in your real-world problem. No, you just need to get a good understanding of what it means to invert a matrix, why you might do so, and what problems can arise.

And you know what? That’s rarely a problem. Once you have identified a mathematical technique you need to understand, the chances are high you will find not one but a dozen or more YouTube videos that explain it to you.

These new tools certainly don’t solve the problem for you. [Well, sometimes they may do, but in that case it wasn’t a problem that required the time of a mathematician. Better to move on and put your efforts into a problem that cannot be solved by an app in the Cloud!] All that these fancy new tools have done is change the level at which we humans operate.

At heart, that shift is no different from the level-shift introduced in the 9th Century when traders in and around Baghdad developed techniques for doing routine arithmetic calculations at scale, by performing operations not on specific numbers but on classes of numbers. One of the techniques they developed was called, al-jabr, a term that ended up giving the name we use today to refer to that new kind of calculation procedure: algebra.

Throughout mathematics’ history, mathematicians have calculated and reasoned logically with the basic building blocks of the time. Today’s procedures (that have to be executed) turn into tomorrow’s basic entities (on which you operate). A classic example is differential calculus, where functions are no longer viewed as rules that you execute to yield new numbers from old numbers, but higher-level objects on which you operate to produce new functions from old functions.

So (finally), what exactly did we do in that Nueva School mini-course to illustrate the way today’s pros use math to solve a problem? The problem, remember, was this: Reverse engineer the core algorithm than UPS uses to route packages from origin to destination?

To start the class off—they worked in small teams of three or four—I provided a small amount of information to get them started:
  1. Tracking information for a fairly large, heavy case, including a partially dismantled bicycle, I had shipped from Petaluma, California to Fair Haven, New Jersey, in 2015. See image below.
  2. I told them I sent the case by “three day select.”
  3. I reported that my package went by plane from Louisville, Kentucky, to the UPS facility in Newark, where it was immediately loaded onto a truck, and was delivered to the intended Fair Haven destination with just a few hours to spare within the three-day period guaranteed.
That information, I told the class, was enough to figure out how the routing algorithm worked. [This itself is useful information that I did not have when I first solved the problem, but they had to figure it out by the end of the course, so I was happy to give them additional information.] In solving this problem, they could elicit my help as their “math consultant,” to call on with specific questions when required. But they had to carry out the key steps.

They could, of course, use the various tools in my “modern math tools” chart, and any others they could find. (Since the UPS routing algorithm is an extremely valuable trade secret, they would not find that online, of course.)

Next month, I’ll tell you how they got on. In the meantime, you might like to see how far you can get with it. Happy problem solving! Happy mathematical thinking!

Part 2 will appear next month.

Tuesday, January 23, 2018

Déjà vu, all over again

I gave a short course at a local high school recently. Three days in a row, two hours a day, to fifteen students. To my mind, it was a huge success. By the end of the course, the students had successfully reverse-engineered UPS’s core routing/scheduling algorithm. In fact, they spent the last half hour brainstorming how UPS might improve their efficiency. (My guess is the company had long ago implemented, or at least considered, the ideas the kids came up with, but that simply serves to illustrate how far they had come in just six hours of class-time.)

To be sure, it was not an average class in an average high school. Nueva School, located in the northern reaches of Silicon Valley, is private and expensive (tuition runs at $36,750 for an 8th gader), and caters to students who have already shown themselves to be high achievers. Many Silicon Valley tech luminaries send their children there, and some serve on the board. They have an excellent faculty. Moreover, the fifteen students in my class had elected to be there, as part of their rich, January, electives learning experience called “Intersession”.

I was familiar with the school, having been invited to speak at their annual education conference on a couple of occasions, but this was the first time I had taught a class.

Surprisingly, the experience reminded me of my own high school education, back in the UK in the early 1960s. My high school was a state run, selective school in the working class city of Hull, a major industrial city and large ocean fishing and shipping port. Socially and financially, it was about as far away as you could get from Nueva School on the San Francisco Peninsula, and my fellow students came from very different backgrounds than the students at Nueva.

What made my education so good was a highly unusual set of historical circumstances. Back then, Hull was a fiercely socialist city that, along with the rest of the UK, was clawing its way out of the ravages of the Second World War. For a few short years, the crippling English class system broke down, and an entire generation of baby boomers entered the school system determined to make better lives for themselves—and everyone else. (“Me first” came a generation later.)

We had teachers who had either just returned from fighting the war (the men on the battlefields, the women in the factories or in military support jobs), or were young men and women just starting out on their teaching careers, having received their own school education while the nation was at war. There was a newly established, free National Health Service, an emerging new broadcasting technology (television) run by a public entity, a rapidly growing communications systems (a publicly funded telephone service), and free education, including government-paid- for university education for the 3 percent or so able to pass the challenging entrance exams.

We were the generation that the nation was dependent on to rebuild, making our way through the education system in a social and political environment where the class divisions that had been a part of British life for centuries had been (temporarily, it turned out) cast aside by the need to fight a common enemy across the English Channel. The result was that, starting in the middle of the 1960s, a “British Explosion” of creative scientific, engineering, and artistic talent burst forth onto the world. Within our individual chosen domains, we all felt we could do anything we set our minds to. And a great many of us did just that. About half my high school class became highly successful people. That from a financially impoverished, working class background.

It was short lived, lasting but a single generation. I was simply lucky to be part of it.

What brought it all back to me was finding myself in a very similar educational environment in my three days at Nueva School. The circumstances could hardly be more different, of course. But talking and working with those students, I sensed the same thirst to learn, the same drive to succeed (in terms they set for themselves), and the same readiness to keep trying I had experienced two generations earlier. It felt comfortingly—and encouragingly—familiar.

But I digress. In fact, I’ve done more than digress. I’ve wandered far from my intended path. Or have I? The point I want to get across is that when it comes to learning, success is about 5 percent talent, 35 percent the teachers and students around you, and 60 percent desire and commitment. (I just made up those figures, but they represent more or less how I see the landscape, having been an education professional for half a century.)

It turns out that, in today’s world, given those ingredients, in roughly those proportions, it is possible for a small group of people, in the space of just a few days, to make significant progress in solving a major problem of massive societal importance. (If you can figure out how UPS performs its magic, you can do the same thing with many other large organizations, Walmart, Amazon, United Airlines, and so on.)

How can it be possible to take a small group of students, still in high school, and make solid progress on a major mathematical problem like that? It would not have been possible in my school days. The answer is, in today’s world, everyone has access to the same rich toolset the professionals use. Moreover, most of those tools—or at least, enough of them—are free to anyone with access to a smartphone or a personal computer. You just have to know how to make effective use of them.

Next month, I will describe how my Nueva class went about the UPS project. (I had done it once before, with a non-science majors undergraduate class at Princeton University. Doing it with high school students confirmed my belief that a group with less academic background could achieve the same result, in the process providing me with some major-league ammunition to back up my oft-repeated—and oft-ignored or disputed—claim that K-12 mathematics education is in need of a major (and I mean MAJOR) makeover. (After the invention of the automobile, it made more sense to teach people how to drive than how to look after a horse. I feel the math ed argument should end with that razor-sharp analogy, but it rarely does.)

As I say, that discussion is for next month. But let me leave you with a teaser. Actually, two teasers. One is my January 1, 2017 opinion piece in the Huffington Post, "All The MathematicalMethods I Learned In My University Math Degree Became Obsolete In My Lifetime." The other teaser is the diagram I will end with. It summarizes some of the most useful tools that a professional mathematician today uses when starting to work on a new problem. (Note: I’m talking about using math to solve real-world problems here. Pure mathematics is very different, although all the tools I will mention can be of use to a pure mathematician.)

This is my set of “most useful tools,” I should note, and reading the diagram left-to- right, top to bottom, the tools I list are roughly in the order I have used them in working on various projects over the past fifteen years. Other mathematicians might produce different collections and different orders. But they won’t be that much different, and I’ll bet they all begin with the same first tool.

If you find this diagram in any way surprising, you likely have not worked in today’s world of mathematical problem solving. If you find it surprising and are in mathematics education, I respectfully point out that this is the mathematical toolset that your students will need to master in order to make use of math in the world they will inhabit after graduation. You may or may not like that. If you don’t like it, then that is unfortunate. Mathematical problem solving is simply done differently today. It just is.