By Keith Devlin @profkeithdevlin

Does taking a logic course at college make you a better reasoner? How about algebra? Or calculus? Or playing chess?

You hear all of these touted as being beneficial to developing good critical thinking skills and being a better problem solver. And on the face of it, it seems that they self-evidently will have that effect. Yet, despite being a mathematician who concentrated on mathematical logic for my Ph.D. studies and many years of research thereafter, I always had my doubts. I harbored a suspicion that a course on, say, history or economics would (if suitably taught) serve better in that regard. Turns out my doubts were well founded.

During the early 1980s, while still proving theorems and churning out research papers, I began to think more deeply about mathematics education (at that stage, for university students and the general adult population), eventually coming to the conclusion that learning mathematics was ideal preparation for, well, doing mathematics; but not, on its own, particularly conducive to success in solving problems in other domains. (Certainly, pointing to mathematicians who achieved success in other fields does not establish any causality. Successful people come from all over the academic map, including high-school and college dropouts, and career success most likely depends on many factors beyond choice of major.)

The point is, intellectual activities like (pure) mathematics and chess are highly constrained domains defined by formal rules. In both domains, the problems you have to solve are what would many years later be referred to as “kind problems,” a classification introduced in 2015 to contrast them to “wicked problems,” a term introduced in the social sciences in the late 1960s to refer to problems that cannot be solved by the selection and application of a rule-based procedure.

Actually, that is not a good definition of a wicked problem; for the simple reason that there is no good, concise definition. But once you get the idea (check out the linked Wikipedia entry above), you will, to take our cue from Supreme Court Justice Potter Stewart, recognize a wicked problem when you see one. In fact, pretty well any problem that arises in the social sciences, or in business, or just in life in general, is a wicked problem. Mathematics may be (and often is) able to contribute to the solution of such a problem, but rarely yields the solution on its own.

The educational preparation for being able to solve wicked problems is very different from what is required to develop the ability to solve kind problems. In domains like mathematics and chess, once you have mastered the underlying rules, repeated, deliberate practice will, in time, make you an expert. The more you practice (that is, deliberate practice—this is a technical term; google “Anders Ericsson deliberate practice”), the better you become. In this regard, chess and mathematics are like playing a musical instrument and many sports, where repeated, deliberate practice is the road to success. This is where the famous “10,000 hours” meme is applicable, a somewhat imprecise but nevertheless suggestive way to capture the empirical observation that true experts in such domains typically spent a great many hours engaged in deliberate practice in order to achieve their success.

But deliberate practice does not prepare people to engage with wicked problems. And that is a major problem for educators, because the vast majority of problems people face in their lives or their jobs today are wicked problems.

Until the early 1990s, mathematics educators could justifiably claim that having their students engage in many hours of deliberate practice to master a range of basic mathematical skills prepared those students for the lives they would lead. Because being able to calculate (fast and accurately) was an essential life skill, and being able to execute mathematical procedures was important in many professions (and occasionally in everyday life). But the late 1980s saw the introduction of technologies that can execute pretty well any mathematical procedure—faster, with way more accuracy, and for far greater datasets, than any human could do. When that happened, it was only a matter of time before those technologies became ubiquitous, thereby rendering obsolete human skill at performing those calculations and executing those procedures. By the start of the Twenty-First Century, we were at that point of obsolescence.

To be sure, there remains a need for students to learn how to calculate and execute procedures, in order to understand the underlying concepts and the methods so they can make good, safe, effective use of the phalanx of digital mathematics tools currently available. But what has gone is the need for many hours of deliberate practice to achieve skills mastery.

That change happened very fast, and with remarkably little publicity, and in consequence few people outside the professional STEM communities realized it had occurred. [See this book chapter, to be published by Springer Verlag in 2020, for an account of that rapid transition, which was to some extent accelerated by the American Mathematical Society.]

Certainly, few mathematics teachers were aware of the scope of the change, including college instructors at non-research institutions. But in the professional STEM communities, the change happened fast and was total. The way mathematics is used in the professional STEM world today is totally different from the way it had been used for the previous three thousand years. And it has now been that way for thirty years.

As a consequence of this revolution—for so it was—the mathematical ability people need in today’s world is to be able to use mathematics (or more generally mathematical thinking) to help solve wicked problems. (If it is a kind problem, there is a digital tool that will solve it, most likely Wolfram Alpha. Sure, some knowledge is required to be able to do that. But not the high skills mastery that requires years of deliberate practice.)

In my own case, my early 1980s recognition of the limits of traditional mathematics came about just before the dam broke; and by what I now look back on and recognize as great fortune, it gave me a front row seat for the revolution in mathematical praxis, and even a small supporting role (see the book chapter referenced above) in ushering it in.

Specifically, it was my attempt to apply my expertise in mathematical logical to human communication using natural language (wicked problems abound) that quickly brought home to me the limitations of what mathematics can achieve once you step outside the closed word of mathematics itself. In 1987, I finally took the plunge to break out of mainstream research in pure mathematics, and cast my lot in with a small group of like-minded scholars—from other disciplines as well as mathematics and logic—at Stanford (and some surrounding technology companies like Hewlett-Packard, Xerox PARC, and SRI), who had recently created a new, interdisciplinary Stanford research center (the Center for the Study of Language and Information, CSLI), and were in the process of launching a new interdisciplinary undergraduate major and a masters program called Symbolic Systems (SymSys). 

Around 1983, I had started to interact with one of the co-founders of CSLI and the SymSys Program, the (mathematical) logician Jon Barwise, who sadly passed away in 2000. In 1987, Barwise, then the director of CSLI, invited me to spend a year at Stanford. That year turned into two and then morphed into a decade-long collaboration with CSLI from afar, followed by my return full-time to Stanford in 2001 as Executive Director of CSLI. 

My 1987-9 position as a Visiting Professor in Mathematics at Stanford was funded in large part by CSLI, where my research was focused. My teaching duties were to provide graduate courses in model theory in the Mathematics Department and undergraduate courses in logic in the new SymSys program. 

SymSys was a CSLI creation, designed to provide an educational program in the intellectual milieu that formed the core of CSLI research. I took to CSLI and SymSys like a duck to water. This was the intellectual combo I had been looking for. A way to broaden the scope of mathematical thinking from mathematics itself to encompass other frameworks for solving problems and designing things.

The early development of both the center and the educational program reflected the range of people, and their disciplines, who had planned it over many months breakfast planning meetings at various cafes within a 35 miles radius of Palo Alto. [The 35 miles takes you as far as Duartes Tavern in Pescadero, a locally-famous, quirky establishment that figured significantly in the process, and these days is a popular weekend haunt of Silicon Valley professionals on bicycles and motor bikes.] Those researchers came from mathematics, computer science, linguistics, philosophy, and psychology, and from both academia (Stanford) and industry (H-P, Xerox, and SRI). 

As its name suggests, CSLI was created to try to make sense of information, the creation, storage, processing, and transmission of which was making fortunes for an increasing number of new (and a few old) Silicon Valley companies, yet for which there was no guiding theory akin to the way physics guides engineering, chemistry guides biology, and biology guides medicine.

[My own, physics-inspired attempts to describe information while still in the UK involved postulating a hypothetical “particle of information” called an “infon” and trying to develop something like a gravitational field to explain its behavior. It never played out the way I originally envisaged it, but it did eventually prove useful in a number of domains, and likely that early attempt played a role in my being invited to Stanford. I wrote a book summarizing the main results of the CSLI information project in 1991, though by then I was pursuing what was to be a more productive offshoot (at least so far) in socio-linguistics and ethnography.]

The guiding principle behind the Symbolic Systems program was to provide an educational experience that would prepare its graduates for taking a leading role in the technology revolution that was already well underway in Silicon Valley—a revolution the beginnings of which gave rise to the popular name “Silicon Valley” in 1971.

Talking with some of the CSLI and SymSys co-founders soon after my arrival—Barwise, the philosophy professor John Perry, industry computer scientist Brian Smith (Xerox PARC), industry philosopher and AI specialist David Israel (SRI), and others—I learned about the thinking behind both the center and the program. The phrase that remains in my head to this day to describe what they had come up with is “New Liberal Arts”. What I don’t recall is whether one or more of them used that term (each one certainly could have done), or whether that was my description (which it certainly could have been). In any event, the term seems very appropriate. And just as the historically-grounded “liberal arts” has proved to be a powerful vehicle for producing creative people who have successful careers in leadership roles in traditional businesses, so too SymSys has had great success in producing Silicon Valley leaders.

What was that you said? You did not know a good liberal arts education was more likely to produce a business leader than a disciplinary major in STEM, or even business? Well, take a look at the figures. Every year, inc.com publishes a list of America’s top entrepreneurs, ranked according to the percentage revenue growth of the companies they’ve led over the most recent 4 years. To qualify for the 2018 ranking, each company had to have at least $100,000 in revenue in 2014, and at least $2 million in revenue in 2017. Looking at the top 250 entrepreneurs on Inc’s list, how many of them would you expect to have liberal arts degrees?

For some background, around 12% of all graduates major in liberal arts, though only 11% of them go into business. So liberal arts majors who go into business make up about 1.3% of all graduates. In contrast, business majors make up about 19% of all graduates.

Going purely by the numbers, therefore, you’d expect no more than 12 liberal arts majors in the top 250 entrepreneurs (roughly 1 for every 19 business majors). The actual figure is more than twice that many: 29 out of the top 250 entrepreneurs have liberal arts degrees! Extrapolating from that data, you can expect 11.6% of top CEOs have liberal arts degrees, even though only 1.3% of all graduates who go into business are liberal arts majors. And that’s just the top CEOs.

As for SymSys, while it has always been one of the most popular majors at Stanford, the total number of graduates is a tiny percentage of all college and university graduates, so running the numbers is not going to yield a reliable answer. But if you just list the actual SymSys graduates who have made it big, you find it way exceeds anything you’d find from comparing percentages: Reid Hoffman (who was a student of mine in 1987-89) went on to be cofounder and chairman of LinkedIn; Marissa Mayer was a very early employee at Google and went on to be Yahoo's CEO; Mike Krieger is the cofounder of Instagram, the photo sharing app acquired by Facebook for $1 billion; Scott Forstall is a former Apple executive best known for his work creating the iOS software; Chris Cox is Facebooks' chief product officer and chief of staff to CEO Mark Zuckerberg; Yul Kwon is Facebook's deputy chief privacy officer (as well as being the winner of CBS "Survivor" in 2006); Srinija Srinivasan was Yahoo's fifth employee and helped grow the company's search capabilities; Gentry Underwood is the cofounder of Mailbox, the email app Dropbox bought for $100 million; Brian Rakowski is Google's VP of product management for Android; and Elaine Wherry cofounded Meebo, the social media platform Google bought for roughly $100 million.

That’s quite a list. True, you don’t see Google co-founders Sergey Brin and Larry Page in the list. They graduated in Computer Science. However, they took SymSys courses, their advisor was Stanford Prof Terry Winograd, who was a leading figure in the creation and running of SymSys, and from the very start Google has exhibited a strong influence from the program, reminiscent of Steve Jobs’ observation at his 2005 Stanford Commencement address, that he credited much of the success of the Apple Macintosh to his liberal arts education at Reed College in Oregon.

What the successes of SymSys graduates demonstrate is the huge advantage that results from preparing students to tackle wicked problems. Mathematics instruction (specifically, logic) was an important component of the SymSys program from the start (indeed, the need for a mathematician to teach those courses is what led to my job offer in 1987, when the program was launching), and it remains so today, though the program overall has evolved over the years. But mathematics is just a part. SymSys majors learn how to use mathematics in conjunction with other ways of thinking, other methodologies, and other kinds of knowledge, in order to solve the wicked problems for which no current digital device is remotely close to being able to solve (and maybe never will be).

The success of programs such as SymSys surely has implications for K-12 mathematics education. As always when discussing K-12 education issues, I hold back concluding with providing specific advice to classroom teachers, since that is not my domain. (Check out youcubed.org for that.) But based on many years of first-hand experience of how mathematics is used in the world, plus my experience with the Stanford SymSys program, I can add my voice to the chorus who are urging a shift in K-12 mathematics education away from skills mastery to preparing students for a world in which using mathematics involves utilizing the available tools for performing calculations and executing procedures. That is not to say the classroom focus should be on solving wicked problems. You need a good grounding in basic skills to do that, and those take time to develop. (Maybe by upper high school there is scope for some project-based work on wicked problems.) So any change at K-12 level should probably be in manner of teaching, not content.

In contrast, when it comes to tertiary education (with which I am familiar), other than in the mathematics major (a special case I’ll come back to in a future post), the focus should definitely be on developing students’ ability to tackle wicked problems.

How best to do that is an ongoing question, but a lot is already known. That’s also something I’ll pursue in a future post. As a teaser, however, let me end by listing some of the key skills required to tackle a wicked problem. If you are a college or university mathematics educator and this does not sound like the activities your students engage in during a (wicked-) problem solving session, maybe you need to rethink.

• Work in a diverse team. The more diverse the better.

• Recognize that you don’t know how to solve it.

• If you think you do, be prepared for others on the team to quickly correct you. (And be a good, productive “other on the team” and correct another member when required.)

• OTOH, you might even not be sure what the heart of the problem really is; or maybe you do but it turns out that other team members think it’s something else. Answering that question is now part of the “solution” you are looking for.

• Be collegial at all times (even when you think you need to be forceful), but remember that if you are the only expert on discipline X, the others do need to hear your input when you think it is required.

• The other team members may not recognize that your expertise is required at a particular point. Persuade them otherwise.

• Listen to the other team members. Constantly remind yourself that they each bring valuable expertise unique to them.

• It’s all about communication. That has two parts: speaking and listening. If the team has at least three members, you should be listening more than you are speaking. (Do the math as to how frequently you “should” be speaking, depending on the size of the team.)

• The onus is on you to explain your input to the others. They do not have your background and context for what you say. With the best will in the world—which you can reasonably expect from the team—they depend on you to explain what you are advocating or suggesting.

• If the group agrees that one of you needs to give a short lesson to the others, fine. Telling people things and showing them how to do things are useful ways of getting them to learn things.

• These are not rules; they are guidelines.

• Guidelines can be broken. Sometimes they should be.

Incidentally, this list is not just something that someone dreamt up sitting in an armchair. Well, maybe it started out that way. But there is plenty of research into what it takes to produce good teamwork that achieves results. I get a lot of my information from colleagues at Stanford who work on these issues. But there are many good sources on the Web.

NOTE: A shorter, overlapping essay discussing kind versus wicked problems, but aimed at K-12 educators, can be found in my November 1 post on the Stanford Mathematics Outreach Project website, sumop.org