How do we measure mathematical creativity?

In last week’s post I described the BrainQuake Score feature of the new BrainQuake app, using my own progress through the game as an example. Here is what I saw when I brought up my BQ Score after completing 111 puzzles.

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This is the BrainQuake Score I had earned in completing 111 puzzles. I was playing as a strong player, solving all puzzles with 3-star solutions. This was reflected in my Performance score of 100%. Many early puzzles do not require much, if any, creativity, but by puzzle 111, I had solved enough puzzles that my Creativity score had climbed to 59%. Image by Keith for BrainQuake.

How does the app calculate that Creativity score? What does it mean? More generally, what exactly is “mathematical creativity”? And is it something you can assign a number to?

Dictionaries typically define creativity” to mean something along these lines (taken from Dictionary.com): the ability to transcend traditional ideas, rules, patterns, relationships, or the like, and to create meaningful new ideas, forms, methods, interpretations, etc.; originality, progressiveness, or imagination.

Though outsiders sometimes think (erroneously) that mathematics is not creative, that belief (sadly) is a reflection of an impoverished mathematics education, focused purely on learning and practicing rules and procedures for calculation and (prescribed) textbook-problem solving.

But to think that is the essence of mathematics is equivalent to thinking bricklaying is the essence of architecture. Laying bricks is a valuable skill that takes practice to master, and it plays an important part in constructing many new buildings. But in of itself, it is not creative. Architects and builders, on the other hand, make creative use of bricks and other materials (and of the craftspeople who work with those materials) in order to design and construct new buildings.

It’s the same with mathematics. Those rules and procedures are the bricks out of which creative mathematicians make new discoveries and solve problems— in the majority of cases (though often not in the school math class) challenging, real-world problems. Creativity is such an important part of professional mathematics, that every four years the international mathematical community awards a prize regarded as equivalent to a Nobel Prize, called the Fields Medal, for particularly creative work.

Fields Medals represent the pinnacle of mathematical creativity, but the concept itself plays a role in mathematics from the very beginning levels. Indeed, creativity is as important as correctness in mathematical problem solving.

In fact, in today’s world, where we have (and use) computers to carry out all the mathematical procedures (calculating, solving equations, etc.), creativity is arguably more important than correctness. (Because the computer, if used properly and appropriately, is unlikely to make an error.)

Fortunately, mathematical creativity can be developed, assessed, and, in certain circumstances, numerically measured. What makes the latter (the quantitative measurement) possible, in particular, is the special nature of mathematical thinking.

Indeed, the nature of mathematical creativity has been the subject of a considerable amount of academic research. I described some of that research in two articles I wrote for the Mathematical Association of America (MAA) in February and March 2019.

[In the same articles, I reference the degree to which today’s major technology companies, in particular, value mathematical creativity in their employees. I also provide a link to two excellent — and entertaining — TED Talk video presentations on creativity in education by the British educational-change advocator Sir Kenneth Robinson.]

BrainQuake’s learning puzzles are designed to develop, require, and measure mathematical creativity. To do so, we draw on a substantial body of research into mathematical creativity culminating in an article titled “Mathematical Creativity”, by Gontran Ervynck, an educator in the Faculty of Science at the Katholieke Universiteit Leuven, in Belgium, published in 1991 in the academic text Advanced mathematical thinking, edited by David Tall (Dordrecht: Kluwer), pp.42–53.

In the second of my two MAA articles, I explain Ervynck’s definition of mathematical creativity, a definition now widely accepted in the mathematical community. To cut to the chase, mathematical creativity is non-algorithmic decision making.

The precision of this definition is made possible by the heavily rule-based, procedural (bricklaying) aspect of mathematics.

Math educators know how to assess and give numerical grades to how a student peforms a calculation or executes a procedure; that’s what most summative math assessment consists of. But that boils down to measuring how well a person can do what a machine can do perfectly. It was important throughout most of history, but has been far less significant since the 1990s, when computers were developed that could do all the algorithmic work.

An experienced teacher can (qualitatively) assess mathematical creativity by looking at the non-algorithmic steps a student makes to solve a problem. In a BrainQuake puzzle, where every step towards a solution is carried out within the app, the game-engine can do that automatically, and moreover compute a numerical measure of that creativity.

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Part of what is required to solve a Tiles puzzle is mentally calculating how each tile will grow. But the solution also depends upon which tiles you use and where you place those tiles. That is non-algorithmic (i.e., mathematically creative) thinking. The game-engine can measure both kinds of thinking. Image by BrainQuake.

With the Tiles puzzle, the distinction between the algorithmic thinking and the creative thinking is clearly delineated: the former is arithmetical (and to come extent algebraic) reasoning, whereas the latter is spatial reasoning.

With Gears and Tanks, the algorithmic and creative parts are more intertwined, though still fairly easy to identify. In keeping with traditional mathematics education, I’ll leave that identification as an exercise for the reader.

[Don’t panic. I’ll come back to this topic in a future post. Now the new app is out, we will be continually improving and fine-tuning the assessment algorithms as we continue to conduct studies and analyze the (anonymous) data of thousands of players, so I’ll have plenty of reason to return to this topic as I explain more of what is going on under the hood!]

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Solving Gears and Tanks puzzles involves a fair amount of arithmetic calculation (and you have to get it right). But in Gears, picking the order in which you collect items requires creative thinking. And solving a Tanks puzzle requires creativity when you have to use two or more valve settings. Both types of puzzles can require other instances of creative thinking as well. This can all be tracked and measured. Image by BrainQuake.

The two MAA articles I cited discuss the degree to which mathematical creativity relates to the more general notion of creativity that people often refer to. We tend to think of general creativity as something special that few people possess, but as Ken Robinson notes in the two videos linked to in those MAA articles, it’s actually very common, and all young children exhibit impressive creativity.

But as Robinson goes on to observe, in the days when it was important (for the individual and for society) that everyone mastered fast, accurate computational skills, the effort (often tedious and repetitive) required to achieve that had the unintended side-effect of surpressing children’s innate creativity (especially in mathematics).

Fortunately, today’s technology-rich world is different, and we can not only let that creativity blossom, we can actively encourage and develop it. Doing this for mathematics was always one of our goals at BrainQuake.

– Keith

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