By Keith Devlin @profkeithdevlin

“What is creative mathematical thinking?” That’s the question I set out to answer last month. The discussion got this far: Creative mathematical thinking is non-algorithmic mathematical thinking.

The question arose when a long-time friend (and former teacher) from the ed tech world and I had an email exchange, prompted in part by the publication of a LinkedIn survey of industry leaders which ranked creativity as the number one skill they look for in employees.

The online magazine EdSurge picked up on the LinkedIn survey results to conduct its own (informal) survey of various thought leaders in different domains (film, writing, teaching, museums, and technology companies of different sizes), asking, “Is creativity a skill (that can be developed through practice and repetition)?” They published the results in the January 21 issue.

The answers given ranged all over. An associated Twitter poll EdSurge conducted came down slightly in favor of “yes.” None of this is scientific, of course. The relevant takeaway is that professionals in different areas for whom creativity is a relevant notion do not agree as to what it is. (Nor did my ed tech friend and I.)

Moreover, the EdSurge survey was by no means specific to mathematics. Indeed, the only responses that came close having particular relevance to mathematics or mathematics learning were acclaimed teacher and Moonshots author Esther Wojcicki’s view that creativity is not a skill but a mindset, and Google Education Evangelist Jaine Casap’s observation that: 

 “[Creativity is] embedded in problem-solving, for example. You must use creativity to think of new ways to define and solve problems. Creativity also separates us from machines or robots. For example, an algorithm is a prescribed process, a pattern of commands a machine (or technology) follows. A human can look at issues from a variety of angles—in a nonlinear way! Creativity can be the ‘how’ part of problem-solving.”

None of those asked gave a definitive answer to the question as to whether creativity could be objectively measured. For my ed tech friend and I, however, leaving the question unanswered was not a viable option. We wanted to know if it were possible, in principle, to develop digital tools that developed creative mathematical thinking and measured it. We needed a definition. It did not have to be “the correct definition.” That seems out of reach given where we all are today, if indeed there is a definitive, clean, concise answer. But is there a notion of “mathematical creativity” that (1) makes a reasonable claim on being referred to by that name, (2) can be implemented in a digital math learning tool, (3) is developed by engaging with the tool, and (4) permits automated assessment by the tool? As long as the notion is easy to understand and clearly specified, such tools could be built. Everyone would know exactly what skill or ability (or mindset, etc.) is being developed and measured, and researchers could take on the task of determining how the defined notion and its implementation compare with other learning outcomes and metrics.

As it turns out, there is such a notion, which had been doing the rounds since the early 1990s. Before I say what it is, it’s probably a good idea to watch (or, re-watch) two excellent TED talk videos on creativity by Sir Kenneth Robinson: His talk Do schools kill creativity? given in Monterey, CA, in 2006 [SPOILER: The answer is “yes”] and the sequel Bring on the learning revolution!, given at the same venue in 2015.

Most people I have talked to about creativity have already seen those videos, and agree that Robinson is absolutely right in saying that creative thought comes naturally to humans, with young children exhibiting seemingly endless creativity in all manner of domains. Anyone who has spent any time with young children, as parents, teachers, or whatever, has surely observed that. But as Robinson correctly, and eloquently, observes, systemic education tends to drive the creativity out of them. 

In the case of mathematics education, creativity is suppressed by the adoption of an excessive focus on the mastery of basic algorithmic skills. To be sure, mathematics educators could, until recently, defend that emphasis by pointing to the crucial need to master calculation—a need that lasted throughout the three millennia period up until the 1990s, when calculation was a crucial life skill but there were no machines to do it for us. 

ASIDE: While that defense has some merit, I find it hard to accept that the need for calculation “drill” meant the almost total suppression of creative mathematics. “Drill of skill” turned into “drill and kill”—the precious commodity killed being any interest in mathematics as a pleasurable mental activity. There was never an either-or choice; time could have been devoted to engagement with creative mathematical thinking.

Be that as it may, with Robinson’s talks fresh in my mind from an N’th re-watch, I went back and looked at the one notion that, by and large, mathematicians had agreed was a reasonable first definition of mathematical creativity. (At least, the relatively few mathematicians who had spent some time trying to come to grips with the elusive concept so agreed.) 

That notion has a history going back to the 1940s, which seems to be when mathematicians, mathematics educators, and philosophers first started to reflect on the issue, of particular note among them being Henri Poincaré (1948), Jacques Hadamard (1945), and George Pólya (1962).

Mathematical creativity – a definition

The definition mathematicians and mathematics educators settled on is very much along the lines of the

mathematical creativity is non-algorithmic decision making

we eventually arrived at in Part 1 of this post.

Taking that general idea as a starting point, Gontran Ervynck, an educator in the Faculty of Science at the Katholieke Universiteit Leuven, in Belgium, came up with a definition (Ervynck 1991) of mathematical creativity that I personally find productive (as do many others).

I’ll elaborate a bit about the background to Ervynck’s contribution later, but first let me cut to the chase and present his definition. I should, however, preface it by noting that he was trying to define creativity in advanced mathematical thinking. What I find attractive, however, is that his definition distills mathematical creativity to an essence that works equally well for learners of all ability levels, both for learning and assessment. Moreover, that notion could be implemented in digital learning tools. 

Ervynck approached mathematical creativity in terms of three stages of mathematical competence (Ervynck 1991, pp.42-43):

The first stage (Stage 0) is referred to as the preliminary technical stage, which consists of “some kind of technical or practical application of mathematical rules and procedures, without the user having any awareness of the theoretical foundation.” 

The second stage (Stage 1) is that of algorithmic activity, which consists primarily of performing mathematical techniques, such as explicitly applying an algorithm repeatedly. 

The third stage (Stage 2) is referred to as creative (conceptual, constructive) activity. This is the stage in which true mathematical creativity occurs, and consists of non-algorithmic decision making.  Ervynck comments that “The decisions that have to be taken may be of a widely divergent nature and always involve a choice.”

Although Ervynck describes the process by which a mathematician arrives at the creative thinking stage after going through two earlier stages, his description of mathematical creativity nevertheless ends up very similar to those of others who have considered the topic of mathematical creativity, such as Poincaré and Hadamard. 

I should point out that, in accepting Ervynck’s concept as a working definition of mathematical creativity, mathematicians and mathematics educators are really taking the word “creativity” and giving it a specific meaning within mathematics. (Mathematicians do this with everyday words all the time.) In this case, the result is a notion that (1) makes sense within mathematics, (2) makes sense within mathematics education, (3) can be applied to all mathematics learners, regardless of experience or ability, and (4) can be applied to mathematics learners in a graded fashion, based on the nature of the choices they make. In addition, it accords very well with the kind of creativity Ken Robinson talked about in his talks. That’s why I like it so much.

What the definition does not capture, however—at least not directly—is the notion of mathematical creativity that is tacitly assumed when we talk about highly creative people. That kind of population was the focus of Einav Aizikovitsh-Udi’s 2014 study The Extent of Mathematical Creativity and Aesthetics in Solving Problems among Students Attending the Mathematically Talented Youth Program.

While Ervynck’s three-stages concept still applies to exceptional individuals, the essence of creativity that Aizikovitsh-Udi studied involves making highly unusual choices that lead to unusual results that stand out from most others. The mathematics community as a whole has very little difficulty recognizing that kind of creativity when we see it, just as is the case for exceptional creativity in all other domains. But do we understand it? Do we know how to develop it? Do we know how to measure it?

Regardless of any progress we may one day obtain on those questions, the Aizikovitsh-Udi is interesting as it stands as a study of exceptional mathematical creativity as it exists. Certainly, the goal of the study was not to figure out if that kind of creativity could be effectively assessed algorithmically, by technology or by hand. To do so would presumably require analyzing the sequences of choices that lead to the desired result, but such an approach seems highly unlikely to be successful. Algorithms can identify unusual sequences of steps, but as any research mathematician knows from long and frustrating experience, the vast majority of those unusual sequences don’t work—even if they seem like wise choices at the time.

In contrast, the thought experiment my ed tech friend and I were having was the degree to which technology could develop and measure the (mathematical) creativity in regular children that Ken Robinson was talking about. Such a technology, it one were possible, would clearly be a significant benefit to the mathematics education community. I don’t think that is necessarily out of reach. In fact, starting with the Ervynck notion of mathematical creativity, I see real potential to make useful progress. But time alone will tell.

Finally, I promised I’d say something about the history of studies of mathematical creativity that led to the Ervynck definition.

The earliest attempt I am aware of to study mathematical creativity was a fairly extensive questionnaire published in the French periodical L’Enseigement Mathematique in 1902. This questionnaire, and a lecture on creativity by Henri Poincaré to the Societé de Psychologie, inspired his colleague Jacques Hadamard to investigate the psychology of mathematical creativity (Hadamard, 1945). Hadamard based his study on informal inquiries among prominent mathematicians and scientists in America, including George Birkhoff, George Pólya, and Albert Einstein, about the mental images they used in doing mathematics.

Hadamard’s study was influenced by the Gestalt psychology popular at the time. He hypothesized that mathematicians’ creative process followed the four-stage Gestalt model of preparation–incubation–illumination–verification (Wallas, 1926). That model provides a characterization of the mathematician’s creative process, but it does not define creativity per se. 

Many years later, in 1976, a number of scholars interested in the notion of mathematical creativity came together to form the International Group for the Psychology of Mathematics (PME), which began to meet annually at different venues around the world to share research ideas. In 1985, a Working Group of PME was formed to look at creativity in advanced mathematical thinking. The volume Advanced Mathematical Thinking, edited by mathematics educator David Tall at the University of Warwick in the UK (Tall 1991), resulted from the work of that group. In Chapter 3 of that book, Ervynck presents his analysis of mathematical creativity.

The PME volume is a mammoth, comprehensive work, full of powerful insights, that I have done no more than delve into from time to time. From what I’ve read (and from what Tall says in his Preface), at the end of the day, we really don’t know how the logically-sequenced solutions and proofs mathematicians write out relate to the mental processes by which they arrive at those arguments. Tall writes (p.xiv):

“[T]here is a huge gulf between the way in which ideas are built cognitively and the way in which they are arranged and presented in deductive order. This warns us that simply presenting a mathematical theory as a sequence of definitions, theorems and proofs (as happens in a typical university course) may show the logical structure of the mathematics, but it fails to allow for the psychological growth of the developing human mind.”

Salutary advice for teachers and students alike.

Final thoughts

My take-home conclusions from my discussion with my ed tech friend? With today’s technologies having eliminated the need for humans to master computation (of any kind), learning and assessment have to focus on creative mathematics (as defined above). 

Teaching computational skills was relatively easy—albeit too often done in a way that turned people off the subject—and assessment could be done with automation. In contrast, developing and assessing creative mathematics are much more problematic. 

Technology may help for the early school grades, say through to middle school, but even then it is likely to be a challenging task to develop systems that work really well, and in my view it’s highly likely that if they do work well it will as supplementary tools dispensed as and when appropriate by an experienced teacher.

As to higher grade levels, I’d look to the considered opinions of experienced mathematics educators and developmental clinical psychologists. They, perhaps informed by conclusions generated by machine-learning algorithms, can certainly have (some) value in terms of identifying creative mathematical talent. Such an approach could be useful in deciding who should be given the benefit of a focused mathematical education and when to conduct an educational intervention for a particular student. Decisions about resources allocation have to be made, and it’s always better to make them with as much information as possible. And from society’s perspective, technology can surely help develop creativity and provide useful measurements of an individual’s creative potential. But at the end of the day, each individual decision is at best an educated bet.

In particular, the most dramatic forms of creativity are often missed as such at the time. Georg Cantor’s theory of infinite sets was initially regarded as the wild mental ramblings of a deranged mind; only later was it recognized as a work of creative genius. In earth science, it took fifty years before the scientific community recognized that Alfred Wegener’s theory that the surface of the earth consisted of separate plates, whose drifting led to the formation of today’s continents and were the cause of earthquakes, was a creative explanation having scientific validity–supported by evidence not available in Wegener’s time. And in music, Stravinsky’s Right of Spring met a similar fate. Etc.

Leaving creative genius aside, however, I should conclude by acknowledging that these Final Thoughts about the potential for ed tech in the development and assessment of creative mathematical ability, are at present no more than a considered (and somewhat informed) opinion from an experienced mathematics educator. Pass the salt.

References

Aizikovitsh-Udi, E. (2014). The Extent of Mathematical Creativity and Aesthetics in Solving Problems among Students Attending the Mathematically Talented Youth Program. In Creative Education 5, pp.228-241

Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), Advanced mathematical thinking (pp. 42-53). Dordrecht: Kluwer.

Hadamard, J. (1945). Essay on the psychology of invention in the mathematical field. Princeton, NJ: Princeton University Press.

Poincaré, H. (1948). Science and method. New York: Dover.

Pólya, G. (1962) Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving.New York: Wiley

D. Tall (Ed.) (1991). Advanced mathematical thinking. Dordrecht: Kluwer (2002 edition available on Google Books)

Wallas, G. (1926). The art of thought. New York: Harcourt, Brace & Jovanovich.