Repeated failure is the royal road to learning math

The three puzzles currently included in the BrainQuake app provide natural, manipulable representations of fundamental mathematical concepts. Instead of writing down symbols and manipulating them with paper and pencil, the student manipulates the virtual mechanical device; to obtain a correct solution, the mechanical manipulations necessarily correspond to the symbolic manipulation steps. This provides math learning that Breaks the Symbol Barrier. The game is designed so puzzles can be attempted many times, and replayed to obtain better scores, which science shows leads to better learning. Image by BrainQuake.

The title to this blog is misleading. Repeated failure is effectively the only way to learn mathematics — assuming you understand learning mathematics to include understanding it.

It’s relatively easy to learn a collection of techniques to calculate answers or compute functions, or to master procedures to solve problems, without ever really understanding what any of it really means. You just put in a sufficient amount of repetitive practice. Most of us have used that approach to pass an exam at some stage. As a result of those experiences, we know first-hand that the techniques or procedures that we can execute fluently during the test rapidly fade away afterwards, and without any reinforcement, two months later we would be totally unable to pass the same test.

If you really want to learn something new — truly learn it — you have to be willing to experience repeated failure. That’s a direct consequence of the way the human brain works. If you want to know more about this, check out the book Limitless Mind — Learn, Lead, And Live Without Barriers, by Stanford mathematics education professor Jo Boaler, or view her youcubed blog post Mistakes Grow Your Brain.

[For another take on this, from the perspective of cognitive science, see also my December 1, 2019 blog post for the Mathematical Association of America, and for an instructional perspective see my October 1, 2019 MAA post.]

With so much traditional, systemic education spent in classrooms with an instructor, we tend to overlook that fact that true learning, where what we learn is internalized and never forgotten, is actually quite common outside of the classroom. Learning skills like being able to swim, riding a bike, skiing, playing chess, playing a good game of tennis, and playing a musical instrument, results in skills we never lose or forget. That’s because the only way to master any of those activities is through repeated failure. None of them allow you to achieve temporary mastery to “pass the test”!

A major reason why video games provide an effective medium for (lasting) mathematics learning (with understanding) is that they fundamentally involve repeated failure, but present that failure in a way that, far from being a turn-off (as failure is experienced by many students in a traditional math class), is actually an incentive to keep trying. This is the point James Paul Gee makes in his seminal book on video games, What Video Games Have to Teach Us About Learning and Literacy, and I echoed in my book Mathematics Education for a New Era: Video Games as a Medium for Learning.

When we designed the BrainQuake app, we made sure we both encouraged and supported the player to repeat each puzzle. To do this, we made sure that the majority of the puzzles admit multiple solutions, allowing for improvement with repeated attempts, so finding one solution was not necessarily the end of that particular puzzle. (That’s not possible for some of the more elementary puzzles in the app, but for the most part it is.)

The app encourages players to replay a puzzle as often as they want by providing various ways for players to improve their score.

Puzzle replay mechanisms. Immediately after completing a puzzle, a player is given the option of replaying it (left image, button on left) or moving to the next puzzle (right button) or going to the progress map to find a different puzzle to replay (center button). If this third option is selected, the player is taken to the World Map (right image) and can scroll down looking for other completed puzzles where improved solutions are possible. (“Keith Strong” is me, playing as a strong player. I test the game in different ways; another of my identities being “Keith Weak”.) Image by Keith Devlin for BrainQuake.

First, there is a replay option offered immediately after they have completed a puzzle, prior to selecting the next puzzle. (Left image above.)

Second, the player can scroll through the path they have followed though the game world so far and select any puzzle they see where the score is not optimal and immediately repeat it. (Right image above.)

And third, when they view their BrainQuake Score, they can select any puzzle and elect to replay it on the spot (perhaps to improve one or more factors in their Score). See image below.

The BrainQuake Score. When a player selects the PUZZLE SUMMARY tab on the opening Score summary page (left image), they see a puzzle-by-puzzle breakdown of their Score (right image). If they select a specific puzzle (Puzzle 99 in the image) and think they can get better solution, there is a button giving an option to replay it right away. Image by Keith Devlin (aka “Keith Strong”) for BrainQuake.

To encourage players to make use of the repeat option, the BrainQuake app provides several login hints to create awareness that (many of) the puzzles have multiple correct answers, and can admit improved solutions, and all can be repeated multiple times, using different replay mechanisms.

Hints. Each time a player logs in to the game, they see a hint screen. (They can also elect to receive hints as daily notifications on their device.) Some of the hints inform and encourage them about replay options. Image by Keith Devlin (“Keith Strong”) for BrainQuake.

Not only does the freedom to try–fail–try-again result in better (i.e., real, lasting) learning, it also makes the mathematical experience we provide much more like real-world, professional mathematics. Any mathematics pro will tell you that much of their time is spent trying to correct mistakes and failed attempts, or to improve a previous solution. (Software engineers tell a similar story regarding finding and correcting bugs, and upgrading their code.) Doing math is any many ways all about making, identifying, and fixing mistakes and doing things better!

The possibility to replay to do better is not the only benefit of learning-game design that provides challenges admitting multiple solutions. Such design also provides students with the freedom to approach the problems in their own way. They can learn from their mistakes, and come to realize that wrong answers or failed methods are mere steps toward a successful solution. This can fuel student motivation, mindset, self efficacy and agency, all important components of (far-too-long-neglected) social and emotional learning, a topic this blog will look at in a future post.

So my message to leave you with today is: Keep failing. It’s good for you. Embrace it! (Provided you reflect on the failure and understand what went wrong.)

– Keith