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About six years ago on this blog, I started thinking about trying to make a web-based game based around high-school algebra, and ended up using Scratch to write a short but playable puzzle game in which one solves linear equations for an unknown ${x}$ using a restricted set of moves. (At almost the same time, there were a number of more professionally made games released along similar lines, most notably Dragonbox.)

Since then, I have thought a couple times about whether there were other parts of mathematics which could be gamified in a similar fashion. Shortly after my first blog posts on this topic, I experimented with a similar gamification of Lewis Carroll’s classic list of logic puzzles, but the results were quite clunky, and I was never satisfied with the results.

Over the last few weeks I returned to this topic though, thinking in particular about how to gamify the rules of inference of propositional logic, in a manner that at least vaguely resembles how mathematicians actually go about making logical arguments (e.g., splitting into cases, arguing by contradiction, using previous result as lemmas to help with subsequent ones, and so forth). The rules of inference are a list of a dozen or so deductive rules concerning propositional sentences (things like “(${A}$ AND ${B}$) OR (NOT ${C}$)”, where ${A,B,C}$ are some formulas). A typical such rule is Modus Ponens: if the sentence ${A}$ is known to be true, and the implication “${A}$ IMPLIES ${B}$” is also known to be true, then one can deduce that ${B}$ is also true. Furthermore, in this deductive calculus it is possible to temporarily introduce some unproven statements as an assumption, only to discharge them later. In particular, we have the deduction theorem: if, after making an assumption ${A}$, one is able to derive the statement ${B}$, then one can conclude that the implication “${A}$ IMPLIES ${B}$” is true without any further assumption.

It took a while for me to come up with a workable game-like graphical interface for all of this, but I finally managed to set one up, now using Javascript instead of Scratch (which would be hopelessly inadequate for this task); indeed, part of the motivation of this project was to finally learn how to program in Javascript, which turned out to be not as formidable as I had feared (certainly having experience with other C-like languages like C++, Java, or lua, as well as some prior knowledge of HTML, was very helpful). The main code for this project is available here. Using this code, I have created an interactive textbook in the style of a computer game, which I have titled “QED”. This text contains thirty-odd exercises arranged in twelve sections that function as game “levels”, in which one has to use a given set of rules of inference, together with a given set of hypotheses, to reach a desired conclusion. The set of available rules increases as one advances through the text; in particular, each new section gives one or more rules, and additionally each exercise one solves automatically becomes a new deduction rule one can exploit in later levels, much as lemmas and propositions are used in actual mathematics to prove more difficult theorems. The text automatically tries to match available deduction rules to the sentences one clicks on or drags, to try to minimise the amount of manual input one needs to actually make a deduction.

Most of one’s proof activity takes place in a “root environment” of statements that are known to be true (under the given hypothesis), but for more advanced exercises one has to also work in sub-environments in which additional assumptions are made. I found the graphical metaphor of nested boxes to be useful to depict this tree of sub-environments, and it seems to combine well with the drag-and-drop interface.

The text also logs one’s moves in a more traditional proof format, which shows how the mechanics of the game correspond to a traditional mathematical argument. My hope is that this will give students a way to understand the underlying concept of forming a proof in a manner that is more difficult to achieve using traditional, non-interactive textbooks.

I have tried to organise the exercises in a game-like progression in which one first works with easy levels that train the player on a small number of moves, and then introduce more advanced moves one at a time. As such, the order in which the rules of inference are introduced is a little idiosyncratic. The most powerful rule (the law of the excluded middle, which is what separates classical logic from intuitionistic logic) is saved for the final section of the text.

Anyway, I am now satisfied enough with the state of the code and the interactive text that I am willing to make both available (and open source; I selected a CC-BY licence for both), and would be happy to receive feedback on any aspect of the either. In principle one could extend the game mechanics to other mathematical topics than the propositional calculus – the rules of inference for first-order logic being an obvious next candidate – but it seems to make sense to focus just on propositional logic for now.

I had another long plane flight recently, so I decided to try making another game, to explore exactly what types of mathematical reasoning might be amenable to gamification.  I decided to start with one of the simplest types of logical argument (and one of the few that avoids the disjunction problem mentioned in the previous post), namely the Aristotelian logic of syllogistic reasoning, most famously exemplified by the classic syllogism:

• Major premise: All men are mortal.
• Minor premise: Socrates is a man.
• Conclusion: Socrates is a mortal.

There is a classic collection of logic puzzles of Lewis Carroll (from his book on symbolic logic), in which he presents a set of premises and asks to derive a conclusion using all of the premises.  Here are four examples of such sets:

• Babies are illogical;
• Nobody is despised who can manage a crocodile;
• Illogical persons are despised.
• My saucepans are the only things that I have that are made of tin;
• I find all your presents very useful;
• None of my saucepans are of the slightest use.
• No  potatoes of mine, that are new, have been boiled;
• All of my potatoes in this dish are fit to eat;
• No unboiled potatoes of mine are fit to eat.
• No ducks waltz;
• No officers ever decline to waltz;
• All my poultry are ducks.

After a certain amount of effort, I was able to gamify the solution process to these sort of puzzles in a Scratch game, although I am not fully satisfied with the results (in part due to the inherent technical limitations of the Scratch software, but also because I have not yet found a smooth user interface for this process).   In order to not have to build a natural language parser, I modified Lewis Carroll’s sentences somewhat in order to be machine-readable.  Here is a typical screenshot:

Unfortunately, the gameplay is somewhat clunkier than in the algebra game, basically because one needs three or four clicks and a keyboard press in order to make a move, whereas in the algebra game each click corresponded to one move. This is in part due to Scratch not having an easy way to have drag-and-drop or right-click commands, but even with a fully featured GUI, I am unsure how to make an interface that would make the process of performing a deduction easy; one may need a “smart” interface that is able to guess some possible intended moves from a minimal amount of input from the user, and then suggest these choices (somewhat similarly to the “auto-complete” feature in a search box).   This would require more effort than I could expend on a plane trip, though (as well as the use of a more powerful language than Scratch).

There are of course several existing proof assistants one could try to use as a model (Coq, Isabelle, etc.), but my impression is that the syntax for such assistants would only be easily mastered by someone who already is quite experienced with computer languages as well as proof writing, which would defeat the purpose of the games I have in mind.  But perhaps it is possible to create a proof assistant for a very restricted logic (such as one without disjunction) that can be easily used by non-experts…

Just a quick update on my previous post on gamifying the problem-solving process in high school algebra. I had a little time to spare (on an airplane flight, of all things), so I decided to rework the mockup version of the algebra game into something a bit more structured, namely as 12 progressively difficult levels of solving a linear equation in one unknown.  (Requires either Java or Flash.)   Somewhat to my surprise, I found that one could create fairly challenging puzzles out of this simple algebra problem by carefully restricting the moves available at each level. Here is a screenshot of a typical level:

After completing each level, an icon appears which one can click on to proceed to the next level.  (There is no particular rationale, by the way, behind my choice of icons; these are basically selected arbitrarily from the default collection of icons (or more precisely, “costumes”) available in Scratch.)

The restriction of moves made the puzzles significantly more artificial in nature, but I think that this may end up ultimately being a good thing, as to solve some of the harder puzzles one is forced to really start thinking about how the process of solving for an unknown actually works. (One could imagine that if one decided to make a fully fledged game out of this, one could have several modes of play, ranging from a puzzle mode in which one solves some carefully constructed, but artificial, puzzles, to a free-form mode in which one can solve arbitrary equations (including ones that you input yourself) using the full set of available algebraic moves.)

One advantage to gamifying linear algebra, as opposed to other types of algebra, is that there is no need for disjunction (i.e. splitting into cases). In contrast, if one has to solve a problem which involves at least one quadratic equation, then at some point one may be forced to divide the analysis into two disjoint cases, depending on which branch of a square root one is taking. I am not sure how to gamify this sort of branching in a civilised manner, and would be interested to hear of any suggestions in this regard. (A similar problem also arises in proving propositions in Euclidean geometry, which I had thought would be another good test case for gamification, because of the need to branch depending on the order of various points on a line, or rays through a point, or whether two lines are parallel or intersect.)

High school algebra marks a key transition point in one’s early mathematical education, and is a common point at which students feel that mathematics becomes really difficult. One of the reasons for this is that the problem solving process for a high school algebra question is significantly more free-form than the mechanical algorithms one is taught for elementary arithmetic, and a certain amount of planning and strategy now comes into play. For instance, if one wants to, say, write ${\frac{1,572,342}{4,124}}$ as a mixed fraction, there is a clear (albeit lengthy) algorithm to do this: one simply sets up the long division problem, extracts the quotient and remainder, and organises these numbers into the desired mixed fraction. After a suitable amount of drill, this is a task that can be accomplished by a large fraction of students at the middle school level. But if, for instance, one has to solve a system of equations such as

$\displaystyle a^2 + bc = 7$

$\displaystyle 2b - c = 1$

$\displaystyle a + 2 = c$

for ${a,b,c}$, there is no similarly mechanical procedure that can be easily taught to a high school student in order to solve such a problem “mindlessly”. (I doubt, for instance, that any attempt to teach Buchberger’s algorithm to such students will be all that successful.) Instead, one is taught the basic “moves” (e.g. multiplying both sides of an equation by some factor, subtracting one equation from another, substituting an expression for a variable into another equation, and so forth), and some basic principles (e.g. simplifying an expression whenever possible, for instance by gathering terms, or solving for one variable in terms of others in order to eliminate it from the system). It is then up to the student to find a suitable combination of moves that isolates each of the variables in turn, to reveal their identities.

Once one is sufficiently expert in algebraic manipulation, this is not all that difficult to do, but when one is just starting to learn algebra, this type of problem can be quite daunting, in part because of an “embarrasment of riches”; there are several possible “moves” one could try to apply to the equations given, and to the novice it is not always clear in advance which moves will simplify the problem and which ones will make it more complicated. Often, such a student may simply try moves at random, which can lead to a dishearteningly large amount of effort expended without getting any closer to a solution. What is worse, each move introduces the possibility of an arithmetic error (such as a sign error), the effect of which is usually not apparent until the student finally arrives at his or her solution and either checks it against the original equation, or submits the answer to be graded. (My own son can get quite frustrated after performing a lengthy series of computations to solve an algebra problem, only to be told that the answer was wrong due to an arithmetic error; I am sure this experience is common to many other schoolchildren.)

It occurred to me recently, though, that the set of problem-solving skills needed to solve algebra problems (and, to some extent, calculus problems also) is somewhat similar to the set of skills needed to solve puzzle-type computer games, in which a certain limited set of moves must be applied in a certain order to achieve a desired result. (There are countless games of this general type; a typical example is “Factory balls“.) Given that the computer game format is already quite familiar to many schoolchildren, one could then try to teach the strategy component of algebraic problem-solving via such a game, which could automate mechanical tasks such as gathering terms and performing arithmetic in order to reduce some of the more frustrating aspects of algebra. (Note that this is distinct from the type of maths games one often sees on educational web sites, which are usually based on asking the player to correctly answer some maths problem in order to advance within the game, making the game essentially a disguised version of a maths quiz. Here, the focus is not so much on being able to supply the correct answer, but on being able to select an effective problem-solving strategy.)

It is difficult to explain in words exactly what type of game I have in mind, so I decided to create a quick mockup of what a sample “level” would look like here (note: requires Java). I didn’t want to spend too much time to make this mockup, so I wrote it in Scratch, which is a somewhat limited programming language intended for use by children, but which has the benefit of being able to churn out simple but functional apps very quickly (the mockup took less than half an hour to write). (I would certainly not attempt to write a full game in this language, though.) In this mockup level, the objective is to solve a single linear equation in one variable, such as ${2x+7=11}$, with only two “moves” available: the ability to subtract ${1}$ from both sides of the equation, and the ability to divide both sides of the equation by ${2}$, which one performs by clicking on an appropriate icon.

It seems to me that one could actually teach a fair amount of algebra through a game such as this, with a progressively difficult sequence of levels that gradually introduce more and more types of “moves” that can handle increasingly difficult problems (e.g. simultaneous linear equations in several unknowns, quadratic equations in one or more variables, inequalities, etc.). Even within a single class of problem, such as solving linear equations, one could create additional challenge by placing some restriction on the available moves, for instance by limiting the number of available moves (as was done in the mockup), or requiring that each move cost some amount of game currency (which might possibly be waived if one is willing to perform the move “by hand”, i.e. by entering the transformed equation manually). And of course one could also make the graphics, sound, and gameplay fancier (e.g. by allowing for various competitive gameplay modes). One could also imagine some other types of high-school and lower-division undergraduate mathematics being amenable to this sort of gamification (calculus and ODE comes to mind, and maybe propositional logic), though I doubt that one could use it effectively for advanced undergraduate or graduate topics. (Though I have sometimes wished for an “integrate by parts” or “use Sobolev embedding” button when trying to control solutions to a PDE…)

This would however be a fair amount of work to actually implement, and is not something I could do by myself with the time I have available these days. But perhaps it may be possible to develop such a game (or platform for such a game) collaboratively, somewhat in the spirit of the polymath projects? I have almost no experience in modern software development (other than a summer programming job I had as a teenager, which hardly counts), so I would be curious to know how projects such as this actually get started in practice.