Mathematical problem solving requires several important capacities, among them: knowledge of mathematical concepts, rules and methods; mastery of mathematical procedures; adequate understanding of mathematical concepts; mathematical creativity; and, as the teacher in the above video noted, persistence.
I’ll look at persistence in a moment. First though, I note that of the capacities I just listed, only the first two can be developed by direct instruction (where a teacher introduces and explains concepts, rules and methods and then shows students how to use them to solve problems, providing guidance as they try to execute what they have been shown). As a consequence, it’s not surprising that in an educational system built (and budgeted) around a teacher providing instruction, with direct instruction and subsequent tutorial guidance driving most of the activity in the mathematics classrooms, what the students spend most of their class-time doing is learning facts and practicing procedures.
But conceptual understanding, creativity, and persistence are all essential in order to make good use of mathematics in the world. Acquiring those non-factual, non-procedural abilities also requires the assistance of a teacher, but in those three cases the teacher’s role is of necessity more of a coach. Unfortunately, since coaching mostly comes down to one-on-one interaction, systemic education provides limited opportunities for a teacher to help develop those three key capacities.
It’s not that no one tries. Good teachers repeatedly say they value all three, and seek to help their students acquire them. But student assessment within systemic education focuses predominately on execution of standard procedures, and given, as I noted above, the time it takes to ensure students develop the procedural mastery they will be assessed on, coupled with the difficulties of providing one-on-one coaching in a typical school, it’s no surprise that conceptual understanding, creativity, and persistence are left largely to chance.
Fortunately, the development of all three of those capacities can be assisted by technology, including learning games.
For instance, BrainQuake’s products assist in developing conceptual understanding by providing access to some key concepts in a way that breaks the symbol barrier. Note the use of the word “assist” in that last sentence. We make no claims that solving puzzles in the BrainQuake app will, on its own, result is conceptual understanding. But as is well known among the math-learning research community, conceptual understanding comes about only by approaching each concept in different ways. In general, no one approach or representation on its own will suffice. (Lots has been written on this. See, for example, this article published by the Mathematical Association of America.)
Likewise, as I discussed in a previous post on this blog, we also encourage and develop mathematical creativity. In fact, by their nature, most puzzle games require and develop creative thinking. If the puzzles are built on mathematics and the game is designed appropriately, they can also require and develop mathematical creativity. The key is to make sure that the creativity required by the game is (or at least includes) mathematical creativity. Many math learning games on the market do not do this; the mathematical content is simply a product delivered by the game — an approach known derisively as “chocolate coated brocolli”.
Persistence, my main focus here, is largely a freebie that comes automatically with game-based learning. Good games make players want to keep going. Indeed, as all game players (and their family members) know, with a good game the difficulty is stopping play to do something else, such as eat a meal or complete (or grade, if you are a teacher) some math homework.
As with creativity, the key requirement for a learning game, where one of the goals is to encourage kids to want to persist in mathematics, is to make sure that the mathematics is sufficiently well integrated into the game action. In our case, the game action is the mathematics. As I described in a previous post, we take some part of mathematics, represent it in an interactive, visual way, and build a game around that representation. So we use the natural desire to keep playing the game directly in order to encourage persistence in mathematics.
Does that approach work? Those two early users of our first app in the video above certainly thought so.
But why is the development of persistence important in learning–and doing–mathematics? Because learning occurs only when we fail, and being a successful user of mathematics is all about trying repeatedly until you eventually get it right.
Listen to Stanford mathematics education specialist Prof. Jo Boaler talk about the role of failure in math learning in this short video:
Turning to the need for persistence in using mathematics, though traditional, summative assessments of mathematics require “correct answers” obtained in a short period of time, and hence encourage repetitive practice to acquire fluidity at selecting and executing standard procedures to solve highly stylized exam problems at speed, in practical, real-world settings (including the real world of research mathematics), the successful solution to a mathematical problem usually takes considerable time (sometimes days, weeks, or months), with several, and maybe many, failed attempts along the way. Persistence in the face of failure is then a critical requirement for the effective use of mathematics.
When you develop an interactive tool to promote creativity and persistence that has the subject-integration property I just alluded to, you automatically get something else as well: a device that can measure and assess creativity and persistence. (Well, it’s not entirely automatic; you have to develop algorithms that track and process the students’ solutions. We did that. But the data is generated automatically by the students themselves, simply by solving the puzzles.) That’s something new in the math-game learning space.