Both doing math and playing chess are frequently touted as beneficial to developing good critical thinking skills and problem solving ability. And on the face of it, it seems that they self-evidently will have that effect. Yet, despite being a mathematician (though not a chess player), I always had my doubts. I long harbored a suspicion that a course on, say, history or economics would (if suitably taught) serve better in that regard. It turns out my suspicions were well founded. Read on.
The point is, mathematics and chess are highly constrained domains defined by formal rules. In both domains, the problems you have to solve are what are sometimes 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 find you can 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.
For example, is it a good idea to install solar panels to power your home? Most of us initially compare several mental images, one of a bank of solar panels on a roof, another of a smoke-emitting, coal-fired, power plant, another of a nuclear power plant, and perhaps one of a wind-turbine. We can quickly list pluses and minuses for each one.
Given how aware we are today of the massive dangers of climate change resulting from the emission of greenhouse gases, we probably dismiss the coal-fired power plant right away.
But for the other three, you really need to look at some data. For example, solar panels seem to be clean, they make no noise, they require very little maintenance, and unlike wind turbines they don’t kill birds. But what is the cost of manufacturing them (including the mining and processing of the materials from which they are made), both monetarily and in terms of impact on the environment? What about the cost of disposing of them when they fail or become too old to function properly? Without some hard data, it’s impossible to say whether they are the slam-dunk best choice we might initially see them as.
In fact, as soon as you set aside an hour or so to think about this problem, you start to realize you are being drawn into a seemingly endless series of “What if?” and “What about?” questions, each requiring data before you can begin to try to answer it. For example, what if a house with a solar-paneled roof is burned in a wildfire, a possibility that residents in many parts of the western United States now face every year? Do those solar panels spew dangerous chemicals into the atmosphere when they burn at very high temperatures? How big a problem would that be? What if, as increasingly happens these days, an entire community burns? How many homes need to burn for the concentration of chemicals released into the atmosphere to constitute a serious danger to human life?
You are clearly going to have to use mathematics as a tool to collect and analyze the data you need to make some reliable comparisons. But it’s also clear that “doing the math” is the easy part—or rather, the easier part. Particularly when there are digital tools available to do all the calculations and execute all the procedures. (See below.) But what numbers to you collect? Which factors do you consider? Which of them do you decide to include in your comparison dataset and which to ignore?
Part of making these decisions will likely involve applying number sense. For instance, for some factors, the numbers may be too small (compared to those associated with other factors) to make it worthwhile including those factors in your quantitative analysis.
Or maybe—if you are very, very lucky—the numbers for one factor dominate all the others, in which case the problem is essentially a kind one, and you can get the answer by old-fashioned “doing the math.” But that kind of outcome is extremely rare.
Usually you have to make trade-offs and compare the numbers you have against other, less quantifiable considerations. This means that problems like this cannot be solved using mathematics alone. And that in turn means they have to be tackled by diverse (!) teams, with each team member bringing different expertise.
For sure, one of the team definitely needs to be mathematically able. But, while just one mathematician may be enough, the others should, ideally, know enough about mathematics to work effectively with the math expert (or experts) on the team.
This is, of course, a very different scenario from the notion of a “mathematical problem solver” that everyone had in mind when I learned mathematics in the 1960s. Back when I was working toward my high school leaving certificate and then my mathematics bachelors degree, with a view to a career as a mathematician, I imagined myself spending most of my professional time working alone. And indeed, for several years, that was the case. But then things changed. Keep reading.
I began this essay with a question: does learning math or playing chess make you a better reasoner—a better problem solver? I hope by now that the answer is clear. For kind problems, almost certainly it does. The largely linear, step-by-step process you need to solve a kind problem involves the same kind of mental processes as math and chess.
But for wicked problems, the above short discussion of selecting among alternative energy sources should indicate that the kind of thinking required is very different. And in an era when machines can beat humans at chess and can do all the heavy lifting for solving a kind math problem (see below), it’s the wicked problems that require humans to solve them.
In other words, in the world our students will inhabit, skill at solving wicked problem is what is needed. And that requires training in mathematics that is geared towards that goal.
So, what does this all mean for us mathematics educators?
The educational preparation for being able to solve wicked problems clearly (see above) has to be 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, as I noted already, the vast majority of problems people face in their lives or their jobs today are wicked problems.
This state of affairs is new, at least for mathematicians. (Not for social scientists.) Until the early 1990s, mathematics educators did not have to face accusations that they were not preparing their students adequately 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 quickly and accurately was important in many professions (and occasionally in everyday life).
But the 1960s brought the electronic calculator, that could outperform humans at arithmetic, and the late 1980s saw the introduction of digital technologies that can execute pretty well any mathematical procedure—faster, with way more accuracy, and for far greater datasets, than any human could do. Once those technological aids became available, it was only a matter of time until they became sufficiently ubiquitous to render obsolete, human skill at performing calculations and executing procedures.
It did not take long. By the start of the Twenty-First Century, we were at that point of obsolescence (of those human skills).
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.
The switch by the professionals in STEM fields, from executing procedures by hand to using digital tools to do it, happened very fast, and with remarkably little publicity. Consequently, few people outside the professional STEM communities realized it had occurred. 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 not only happened fast, it was total. The way mathematics is used in the professional STEM world today is totally different how it was used for the previous three thousand years. And it’s been that way for thirty years now.
As a consequence of this revolution in mathematical praxis—and it really was a revolution—the mathematical ability people need in today’s world is not calculation or the execution of procedures, as it had been for thousands of years, but being able to use mathematics (or more generally mathematical thinking) to help solve wicked problems. [Note that digital tools won’t solve a wicked problem for you. The most they can do is help out by handling the procedural math parts for you.]
Today, to solve a kind mathematical problem, there is almost certainly a digital tool that will handle it, most likely Wolfram Alpha. To be sure, some knowledge is required to be able to do that. And we must make sure our students graduate with those skills. But what they no longer need is the high skills mastery that requires years of deliberate practice. Adjusting to this new reality is straightforward, and many teachers have already made that change. You teach—and assess—the same concepts and methods as before, but with the goal being understanding rather than performance.
When it comes to solving wicked problems, however, what people need is an ability to use mathematics in conjunction with other ways of thinking, other methodologies, and other kinds of knowledge. And that really is a people thing. No current digital device is remotely close to being able to solve a wicked problem, and maybe never will be.
So what exactly do those of us in mathematics education have to do to ensure that our students acquire the knowledge and skills they will require in today’s world?
Well, the biggest impact in terms of changing course content and structure is at the college and university level. Major changes are required there, and indeed are already well underway. In particular, tertiary-level students are going to have to learn, through repeated classroom experiences, how to tackle wicked problems. Project-based teamwork is going to have to play a big role. (See below.)
In terms of K-12, however, there is a good argument to be made for continuing to focus on highly constrained, kind problems that highlight individual concepts and techniques. A solid grounding in basic mathematical concepts and techniques is absolutely necessary for any mathematical work that will come later.
That’s certainly an argument I would make—though as always when discussing K-12 education issues, I hold back from providing specific advice to classroom teachers, particularly K-10, since that is not my domain. Check out youcubed.org for that!
But I do have many years of first-hand experience of how mathematics is used in the world, and based on that background I can add my voice to the chorus who are urging a shift in K-12 mathematics education, away from basic computation and procedural skills mastery, to preparing students for a world in which using mathematics involves utilizing the available tools for performing calculations and executing procedures.
It’s definitely not a question of new content being needed (at the K-12 level). The goals of the Common Core State Standards for Mathematics already cover the main concepts and topics required. Today’s world does not run on some brand new kind of mathematics—though there is some of that, with new techniques being developed and introduced all the time. The familiar concepts developed and used over the centuries are still required.
Rather, the change has been in mathematical praxis: how math is done and how it is used. The main impact of the 1990s mathematical praxis revolution on K-12 is that there is no longer any need for repetitive, deliberate practice to develop fast, fluent skills at calculation and the execution of procedures—since those skills have been outsourced to machines. [Yes, I keep repeating this point. It’s important.]
Insofar as students engage in calculation and executing procedures—and they certainly should— the goal is not smooth, accurate, or fast execution, but understanding. For that is what they need (in spades) to make use of all those shiny new digital math technologies. (Actually, many of them are hardly new or shining, being forty years of age and older. It just took a while before they found their way outside the professional STEM community.)
So, while leaving it to experienced K-12 educators and academic colleagues such as Jo Boaler to figure out how best to teach school math for life in the 21st Century, let me finish by giving some indication of how tertiary education (the world I am familiar with) is changing to meet the new need. That, after all, is what many high school graduates will face in the next phase of their education, so the more their K-12 experience prepares them for that, the better will be their subsequent progress.
In contrast to K-12, when it comes to tertiary education, other than in the mathematics major (a special case I’ll come back to in a future post), the focus should 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 highlighting some key elements of the skillset required to tackle a wicked problem.
Let me stress that this list is one drawn up for college level students. In fact, this post is extracted and adapted from a longer one I just wrote for the Mathematical Association of America, a professional organization for college-level math instructors. Neither I nor (I believe) anyone else is advocating doing this kind of thing at levels K-10. (Though maybe for grades 11 and 12. I have tried this out with high school juniors and seniors, and it has gone well.)
[Incidentally, the 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.]
To solve a wicked problem, you should:
- 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 member, 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.
So there you have it. If you are teaching math in K-12 and you can ensure that when your students graduate they can thrive—and enjoy—working in that fashion, you will have set them up for life.
That’s the wicked truth.
NOTE: A longer, overlapping essay discussing kind versus wicked problems, but aimed at college and university research and education professionals, can be found in my November 1 post on the Devlin’s Angle page on the Mathematical Association of America’s MATHVALUES website.
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