What is mathematical creativity, how do we develop it, and should we try to measure it? PART 1
By Keith Devlin @profkeithdevlin
In its January 1 Learning Blog, the business networking service LinkedIn published a list of the skills today’s large companies value most in their employees, as obtained from survey data. The report notes that 57% of senior leaders in business value soft (human-centered) skills over hard skills, pointing out that “the rise of AI is only making soft skills increasingly important, as they are precisely the type of skills robots can’t automate.” In the case of mathematics, this aligns with the theme that occupied Devlin’s Angle for most of last year, starting with the January post.
Survey respondents ranked the top five of those soft skills in the following order (most valuable first): creativity, persuasion, collaboration, adaptability, and time management. (For the top 25 hard skills, see the LinkedIn blogpost. I note that mathematical skills do not appear anywhere in that list—at least not under that name, but keep reading.)
As regular Devlin’s Angle readers will know, as a lifelong university scholar and educator, I have never viewed K-16 education as being job training; it’s life preparation. But as I also always add, jobs and careers are part of life, so it would be irresponsible for educators to ignore the realities that will face the students who graduate from our institutions.
For instance, and to pick up the main theme of last year’s posts, until the late 1960s you had to master numerical calculation (ideally fast and accurate) in order to (1) live a successful, productive, and rewarding life, (2) get many jobs, (3) acquire a mathematics education and use mathematics, and (4) acquire an education in a STEM related field and work in a STEM area. So, it was important that schools taught basic arithmetic. Up until the late 1980s, it was likewise important for colleges and universities to ensure their students master other forms of calculation (most notably, algebraic). But with the arrival of electronic calculators in the 1960s and computer packages like Mathematica and Maple in the 1980s (and particularly after the appearance of Wolfram Alpha in 2009 and Desmos in 2011), the need to master any kind of calculation had been eliminated. Since no one in the world (at least the parts of the world with Cloud access) ever needs to do calculation themselves, there is no longer an imperative for schools or colleges to teach it.
At least, there is no need for anyone to teach calculation (of any kind) so their students can execute algorithms by hand. But in place of that now obsolete skill set, there is a new one. In today’s world, we all need to be able to make good use of those new calculation technologies. To achieve that, we need to provide students with a good understanding of the calculations those technologies can perform, and that surely requires that those students achieve some level of mastery. But the goal of calculation instruction today should be understanding, not execution, so the level and nature of the required mastery is different.
With calculation now automated, the creative aspect of mathematics now occupies primary place. But what exactly is the “creative aspect of mathematics”?
Prompted in part by the LinkedIn article, I had a fascinating email exchange about this question recently with a long-time friend in the ed tech industry. A former school teacher (not STEM), he shares my interest in finding ways to make productive use of technology to improve teaching and provide access to quality education in particular to groups currently under-served (for various reasons). Though my friend has spent many years in the tech world, like me he thinks that technologists who approach education simply as another domain in which to find markets for the products they have built are unlikely to create anything of educational value. You need to start with a good understanding of, and some considerable experience in, education and then look for ways technology can help—either an existing technology or one that has to be designed and built.
The goal of our exchange was to answer these specific questions: Can digital technologies, in particular digital mathematics learning games, help develop creativity, and can they measure it?
In using the term “mathematical learning game,” I mean a game explicitly designed to support the learning of specified mathematical skills. All games produce learning, and indeed all games can result in the acquisition and development of skills and attitudes useful in doing mathematics. Anyone who does not see that has not played many video games—or does not really understand what it means to do mathematics. But our focus was games developed specifically to provide learning of specified mathematical skills. Hence my choice of term.
Before we could answer those questions, we had to decide what we meant by “creative mathematics.”
Without question, the first step anyone should make when trying to answer a question in today’s world is do a quick Google search. (That is true even if you have some prior knowledge of the topic, which I did—more on that in Part 2 of this post.) In my case, Google instantly brought up some helpful resources, among them:
A youcubed page from my Stanford colleague Prof. Jo Boaler.
A 1997 book called Creative Mathematics aimed at elementary school teachers.
A website called creativemathematics.com providing teaching resources for elementary school teachers.
A 2017 blog post with the wonderfully provocative title Mathematics must be creative, else it ain’t mathematics, written by a former research mathematician now focusing on teaching, called Junaid Mubeen, who I have interacted with productively on social media from time to time.
But pretty well everything that came up near the top of my search assumed we all know what the term “creative mathematics” means. (Actually, I prefer to use the longer term “creative mathematical thinking,” to emphasize that it is the process of doing—or using—mathematics that we are referring to, not the body of knowledge found in textbooks that the word “mathematics” commonly suggests.) The articles I found seemed to be using the adjective “creative” to evoke a word cloud along the lines of “lively, engaging, fun, enjoyable, experiential, multidimensional, open-ended, exploratory, intriguing, satisfying, …”
None of the words in that word cloud are exclusively connected to mathematics, though all can (and to mathematicians do) apply. What was significant, I found, was the absence of two words that definitely apply to mathematics and one that applies more or less exclusively to mathematics. The first two words are “difficult” and “challenging” and the third is “algorithmic” (or “procedural”, which would be equivalent in this context).
Let me start with “algorithmic.” That’s the one that, of necessity, used to be front, center, and most significantly temporally first, in mathematics education, but which has now been relegated to a teaching tool to be used to develop understanding. “Algorithmic” is the elephant in the room that the terms in that word cloud were trying to distance themselves from. In fact, you can simplify the entire cloud with one word: “non-algorithmic.”
The point is, we humans evolved to understand, and we find the act of achieving understanding rewarding (both psycho-chemically and cognitively). We are meaning-seeking agents. The three traits understanding, planning-based-on-understanding, and communicating-our-understanding-and-planning, are Homo sapiens’ evolved skills to compensate for our lowly position in the “red in tooth and claw” ranking table.
What we did not evolve to do, and until very, very recently in our history had no need for, is execute mathematical algorithms. We started to do it because, a few thousand years ago, we got to a stage where we had to. But it was difficult for the human brain to do and took time and effort to master. The majority of people disliked it from the start, and many never did succeed. In particular, it required suppressing the very thinking processes our brains found natural and enjoyed doing, as we trained our minds to act like mechanical devices. (Hence the use of derogative colloquial terms such as “number crunching”, “grinding away”, and “turning the handle” to refer to computation.)
Not to put too fine a point on it, “algorithmic thinking” is an oxymoron. The trick to being able to master execution of algorithms was to suppress the brain’s instinct to think and force it to slavishly follow the rules. Few of us were able to do that well.
I think those (relatively few) of us that succeeded were able to do so because we took pleasure in understanding how and why those algorithms and procedures worked, and appreciated the human creativity that went into designing them. That was definitely the case for me. As I have written about often, I was the last kid in my school math class to do well on tests in the lower grades, because I was never able to just learn the rules and apply them; I kept trying to make sense of them. (In later grades, I figured out how to play that game, accepting that to succeed in the education system I had to first master the procedures and get good grades, and then try to figure everything out later. I got good at that. So good, in fact, that I was a mathematics graduate student before I really understood the calculus methods I used efficiently to get A’s on tests in high school and as a mathematics undergraduate.)
To get back to my theme: What is the opposite of “machine-like thinking”? “Creative thinking” seems to capture the essence. (A somewhat equivalent phrase would be “the kind of thinking the human brain naturally finds pleasurable.”)
And once you are at that point—when you have discarded “algorithmic thinking”—you can safely throw in those other two missing words “difficult” and “challenging” when talking about “creative mathematics.” For the fact is, the pleasure we get from using our minds the way they evolved is all the greater when we succeed in something we found difficult.
So now we have made some progress in answering the question “What is creative mathematical thinking?” It is non-algorithmic mathematical thinking. We have defined it in terms of what it is not.
But can we turn that into a positive definition? For that is what my friend and I would need in order to pursue our email exchange about how to develop and use technology to help develop creative mathematical thinking, and even more so if we want to measure it (with or without technology).
To be continued next month in PART 2
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