Blueprint for a Revolution in the Reproduction of Practical Knowledge

Introduction #

1 #

It’s a long story, but basically Super Practica will be the optimal method of reproducing mathematical knowledge, if it can be completed. (Reproducing means learning by the way.)

Super Practica is a game that you play. This game is not frustrating. You see yourself improving at a rapid pace. When you’re done you become an expert in the topic of the game, and you’re able to apply outside the game what you have learned in the game. With very few exceptions, every player can play this game and see the same effects, and without great difficulty. This game can be completed in a small fraction of the time of a typical educational program for the same topic, with a small fraction of the energy, and with a small fraction of the resources.

Learning mathematics this way is not only possible, but optimal. This follows from the nature of learning and from the nature of mathematics.

2 #

The design of Super Practica is the solution to the problem of optimally reproducing mathematical knowledge, and perhaps more generally practical knowledge. It is not yet complete, but whatever a complete solution would look like, it must follow the same outline of solution as I give in this blueprint.

My methods of design are applicable more generally, but I will focus on arithmetic. What functions for arithmetic will function for most of mathematics; and what functions for mathematical knowledge will function, to some extent, for practical knowledge in general.

This blueprint contains an outline of the design of Super Practica and an explanation of the theory behind it. It and the proof-of-concept demo I release with it are the results of 7 years of design, research, philosophical criticism, theoretical development, prototyping, and redesign.

The demo showcases some of the designs in this blueprint, especially those given in chapters [2] and [7]. I call the demo “version 0.6” in reference to prior, unreleased prototypes.

I write this to share my work, to make clear what Super Practica is, and to persuade people to help me.—Because I can’t make Super Practica alone.

3 #

No traditional instruction or testing is necessary or efficient for learning mathematics. Gameplay and gameplay alone is efficient and sufficient. This is not a stretch of wishful thinking, but rather follows from a consistent application of empiricist and materialist methods of analysis and reasoning, opposed to the mysticism and confusion prevalent in mathematics education.

You learn to play the game by playing the game, and insofar as the game is similar to a mathematical practice, you learn the mathematical practice by playing the game.—I often repeated this logical statement to keep myself on the right track. Now I think it’s oversimplified and ambiguous, but it still gives a fine introduction to the thinking behind Super Practica.

One problem with this summary statement is it treats games and mathematical practices as being distinct from each other, the game being only a means to the end of learning mathematics. But it’s more fruitful to think of mathematics as already being a game and then to ask “What kind of game is mathematics?”

1: What Kind of Game is Mathematics? #

1.1: Mathematics is a game #

It is commonly understood that mathematics has rules.—If someone fails at doing mathematics, we say that they don’t know the rules or that they haven’t followed the rules correctly. Mathematics also has goals.—Mathematical problems have solutions that are obtained by doing mathematics. These two elements, rules and goals, are the elements that games are made of. In this basic sense of sharing elements, mathematics is a game.

The association of mathematics with games isn’t new. Plans to teach mathematics using games range from Plato attributing the practice to Egyptians [D.1], to talk of “gamifying mathematics education” today. But these various plans make the same crucial mistake: They see games as something outside of mathematics, introduced into mathematics to make its education more engaging. Mathematics itself is left a mysterious activity consisting of some kind of incomprehensible genius and insight. [A.5] So long as the practice of mathematics is a mystery, educators will fail to teach it well, games or no games.

The value of identifying mathematics with games lies in demystifying it, not in making it fun. By applying our understanding of games to mathematics, we can understand mathematics. What we know of learning to play games we can thus apply to learning to play mathematics; and what we know of designing games we can thus apply to designing mathematics. This is why I say mathematics is already a game and doesn’t need to be “gamified”.

So mathematics is a game, but what kind of game is it? Mathematics has problems and solutions, so it looks like a puzzle game, but it’s more than a simple puzzle game. We must look closely at the gameplay of solving a problem of mathematics to tell precisely what kind of puzzle game it is. [A.1] [A.2]

1.2: Gameplay 1 - Setup #

Suppose you have a mathematical problem that you set out to solve, but don’t immediately know how to solve. Take a problem from life rather than an artificial one from a book. You think you should be able to figure it out but can’t think of the method for it. So you sit down at a desk with pencil and paper and try to work it out.

To solve the problem you must act on the paper with your pencil. Each step toward solution is an action in the game. But what are the rules of these actions? Looking at a blank piece of paper, there don’t appear to be any rules. Here it’s easy to be lost and have no idea where to start.

Skilled practitioners of mathematics quickly overcome this first hurdle by “setting up the problem”. To do this, you start writing down all the relevant measurements that might be useful for solving the problem. Then you write down formulas you can remember that you might need to use. You try drawing a picture, which takes a few attempts.

If this still doesn’t kickstart your mind into solving the problem, you can ask yourself “What exactly is it that I want?” Then you write down a statement of the problem. What you want is a number that answers a question—how many packages you need to buy, how long a trip would take, or some such thing. Having done algebra before, you call this number x. Then you can write a problem-statement: “I want to find x, which is …”

1.3: Gameplay 2 - Solution #

You now have a piece of paper in front of you with all these things scribbled down on it. Where do you go from here? You look over the data, the formulas, and the problem-statement, hoping something will come to you.

You think of something to do, though you’re not sure how it will help, if it will help at all. You think to plug in these two pieces of data into this formula. So you do this calculation and produce a number as the result. Then you think adding up some of the numbers you wrote down might help, so you do this and get another number out of it.

The words in the problem-statement help you to figure out how to proceed. The word “total” suggests adding things up to a sum, the word “each” suggests multiplying, and other words suggest other calculations. These calculations are a few preliminary actions that may or may not make progress to a solution. You try them out and hope they help.

It could happen that all this is still not enough to solve the problem and you have to think of some new ways to proceed, but let’s consider the happy ending instead.

After these attempts, while you’re looking at the picture you drew, you suddenly see the path forward! Now you know which calculations to do and proceed without hesitation to the solution. You perform a sequence of actions—calculating, manipulating the equation, and such things. At the end of this sequence you have solved for x.

With the answer in hand, you can get up and do whatever it is that you needed this answer to do.

1.4: The nature of insight #

So we find that there is a flash of insight at the core of the gameplay. Much of the mystery of mathematics stems from the inexplicability of this insight and the feeling of elation that comes with it. It is typical confusion to call it “genius” or a “vision” or “inspiration”—meaning spiritual influence. These intuitive explanations are lousy because they cannot predict or replicate the insight. We will only understand the phenomenon of insight when we can make an empirical claim that reliably predicts it.

The empirical explanation has been found by the mathematician George Polya: The moment of your insight is precisely the moment at which the problem you are looking at starts to look very similar to a problem you already know how to solve. [A.1] Setting up the problem and the preliminary actions attempting to solve it serve to transform the problem from vague ideas to concrete figures on paper that look like a problem you have already solved before.

The sudden knowledge of how to solve the problem derives from the sudden applicability of your prior experience of solving similar problems. This explains the confidence of the final sequence of actions that lead to the solution.—It is the confidence of past success.

1.5: Rules 1 - Actions and states #

What are the rules of the game of mathematics? [B.1]

Mathematical symbols and diagrams are pictures that suggest rules to follow. If you perform an action on a picture, drawing on top of it or redrawing it with a minor change, you will obtain a new picture. [A.1] The new picture then suggests other things you can do with it—other actions you can perform on it.

For example, suppose you see the picture 2 + 4 on your paper. What can you do with it? You could solve it:

2 + 4 = 6

Or you could rewrite it to be equal to the same expression:

2 + (2 + 2)

Or you could change the expression to be 5 times what you have:

5 × (2 + 4)

Or you could equate it to another expression:

2 + 4 = 4 + 2

Or you could define a variable by it:

x = 2 + 4

There are many more things you could think to do. These are all valid actions of the game that yield valid states of the game. And each new picture has the same breadth of valid actions as the last.

1.6: Rules 2 - Puzzles and procedures #

When a game has many valid actions but only one solution, then that game is a puzzle to find the right actions to do. For example, consider the problem of solving for x in this equation:

x + 5 = 2x − 1

Here it is necessary to isolate the variable x, which can be done by subtracting x from both sides. It’s a simple solution, but if you perform valid actions without a strategy, then it’s unlikely that you’re going to come to a solution by chance.

Other pictures always imply following the same algorithm to change it. Then setting up the picture by itself implies the result, and there are no choices to make in gameplay. For example, such is the procedure of deduction on a Venn diagram.

1.7: Rules 3 - Interconnection #

There are many things you might do in solving a problem. When you calculate a sum, you follow the rules of addition. When you do multiplication, you follow other rules. When you manipulate an algebraic equation, you follow still other rules. The rules change with every calculation and with every picture you draw.

Yet all the numbers—the inputs and outputs of calculations—are still connected on the same piece of paper. So the game has many different rules, all interconnected together, and the connection-points between them are the symbols and pictures on the paper.

Most commonly it is numbers that move—repeat—from calculation to calculation, but the gameplay of mathematics has little to do with the particular symbols that are numbers and everything to do with the use made of these symbols. We can therefore in passing demystify the nature of numbers: They are a common interface for connecting different mathematical practices to each other.

For example, consider these two pictures which set up and record a calculation:

3 + 8 = ?

3 + 8 = 11

They fit into a solution method outside of themselves. The 3 and 8 in the first picture came from somewhere else before being put together to set up the calculation. Then the 11 goes into another picture with different rules, and so on.

This is how you play the game of mathematics. You move from picture to picture, applying different rules, and carry forward elements such as numbers from one to the next. To play the game of mathematics is to play many interconnected games.

1.8: Goals #

What are the goals of the game of mathematics?

The goals vary with every problem, and each problem has goals on multiple levels. Despite the variety of particular goals, the gameplay of mathematics commonly follows the same sequence of 4 generic goals [A.1]:

1. Set up the problem. This goal is to determine which rules you should follow to start playing the game of mathematics such that it ends in giving you a solution that you can successfully apply. There may be false starts in doing this, and it’s often not clear if the problem has been fully set up, so you might come back to this after you thought you were done.

2. Solve the problem identified by the problem-statement. This is typically an answer to a question of mathematics in mathematical terms. For example, in the most common problems of algebra the goal is to solve for x, or in other words to answer “What is x?” The “answer” to a problem-statement may be a picture as well as a statement.

3. Verify your solution. Unlike in most puzzle games, there is often doubt in mathematics whether your solution is correct. To properly do mathematics, you must verify your solution. For example, once you have solved for x in an equation, you can verify your solution by substituting it for x and then evaluating to check that both sides of the equation are equal.

4. Apply your solution. This goal is not strictly part of the game of mathematics, but mathematics is only useful due to its ability to inform a plan of action in non-mathematical terms.

1.9: Verification 1 - Reliability #

The place of verification in mathematics is commonly confused. A skilled player of mathematics can blaze through solving a set of similar problems because they have enough rational confidence in their methods to not stop and verify. But a beginning player of mathematics would be foolish to take this shortcut. Mathematics is nothing without the ability to use it reliably, and to solve mathematical problems reliably generally means to stop and verify.

Cases of experts reasonably cutting their verifications short or apparently skipping them mystify this necessity. The confusion is compounded by how schools commonly alienate answering from verifying. The way a student knows they have the right answer in a school is either by drilling enough that they do not make mistakes, or by consulting authorities—the teacher or the answer key. But outside of schools you don’t tell that your answer is the right answer by either of these methods.

1.10: Verification 2 - Structure #

An example should hopefully make clear the verificational structure of mathematics. [A.1] Suppose we want to verify the result of evaluating 5 × 4. You will likely be able to call the answer 20 with full confidence. Your prior experience makes verification unnecessary for you. But suppose we wanted to make our verification complete, such as to explain it to someone who does not share our experience and confidence. How would we do this?

We may rewrite it as (5 + 5 + 5 + 5). We count 4 fives and transform the multiplication problem into an addition problem. Now we can add the fives and get 20 again. This verifies our memorized answer of 20 by the method of rewriting a multiplication problem as an addition problem and solving it.

But what if even our evaluation of (5 + 5 + 5 + 5) was in question and we wanted to verify this too? We may draw out 4 sets of 5 dots and count the dots we have drawn. This verifies the addition problem by drawing a picture of dots and counting the dots. We may also verify the multiplication problem by drawing out dots straightaway, without the intermediate addition.

Now suppose someone questioned even this count of dots! Could we verify this too?—No. If someone questioned the correctness of our count, we could only recount or try to teach the questioner how to count.

Most mathematical practices are built on top of other mathematical practices, so we can verify the higher practice by the lower. But some mathematical practices are self-sufficient. They include counting, recognizing symmetry in a picture, and noticing the identicality of signs.—These are the foundations of the verificational structure of mathematics.

1.11: Verification 3 - Confidence #

Verifications can run all the way down the verificational structure, but they typically stop short of reaching a foundation because the verifier has enough reason to be confident in their answer before then. A memorized answer is a fine place to stop, so is a proof, and so is “a risk you’re willing to take”. What works to verify—to acquire enough confidence to proceed—depends on the context of the problem and on personal reasons for confidence, such as one’s familiarity with a branch of mathematics.

It follows that there are many ways to verify a solution, that different people will find different value in different verifications, and that what look like verifications may be functionally useless for the purpose. Whatever works to give rational confidence works for verification, and when rational confidence is achieved there is no value in further verification.

1.12: The game of mathematics #

Now we can tell what kind of game mathematics is.

To start playing you translate your problem into mathematics, and to finish you have to translate your solution from mathematical to real terms. This is to select the right game to play, like selecting the right tool for a job. There are many wrong answers but also many correct answers, which makes this part of the game a creative puzzle.

Once the game is selected and set up, it consists of the definite tasks of obtaining and verifying the solution. Many methods of solution and many methods of verification may be effective, and the methods of each may consist of many sub-methods. At the higher level of analysis there is much room for choice in how to do things—in how to connect sub-methods together. At the lower level, gameplay consists of strict rule-following activities that transform pictures and symbols by one action at a time.

Mathematics is a big game made up of smaller games, which are made of even smaller games, which are made of yet smaller games. This is not like a game that is a collection of minigames.—The smaller games of mathematics interconnect together in a wide variety of ways, and the player must figure out which path to follow.

With a particular problem to solve, the many small games have to be sequenced together. Each small game ends in an intermediate victory, but it is only in reference to the original problem that it can be recognized as a victory. Completing these small games in sequence together yields an answer. Then the player must perform the anticlimactic and often boring task of verifying that the answer is correct.

If no particular problem is attended to, then the player can play the many small games and follow the many interconnections between them in many interesting directions. Mathematics is then an exploration game. Otherwise it is a puzzle game with a combination of creative invention, logical planning, unexpected discovery, and boring routine.

1.13: Introducing Super Practica A #

Now that the kind of game that mathematics is has been specified, we can reap the fruits of the analogy. Mathematics is a game, but it is a game that is frustrating and difficult to play. We can now apply our knowledge of making games easy to learn and play to making mathematics easy to learn and play.

This is a project of redesigning the game of mathematics. But there are no very mechanically similar games to draw on for guidance. I have therefore had to make innovations to effect this redesign myself.

I have named this project Game Formation for its design of having a united formation of many small, interconnected games. But then I decided Super Practica would be a better name instead.

Super Practica has become more than this, but it begins with the project of redesigning mathematics. Call this first part Super Practica A.

2: Redesign of Mathematics to be Easy to Play #

2.1: Mechanization 1 - Introduction #

The project of redesigning mathematics to be easy to play should be centered on the design and development of a computer-game. There are many benefits that computer-games have over other kinds of games:

• There is no problem of the rules being too vaguely specified such that they are misunderstood or not followed exactly.
• They can be easily and cheaply reproduced as software that can run on compatible hardware around the world.
• They allow automation of processes that are otherwise inconvenient or labor-intensive.
• It is easier to iterate their design over time and release improved versions.

It is not the rules or the tasks of mathematics that make it difficult to learn or to play. They are rather what make mathematics valuable and applicable to life, so we should keep them the same in our redesign unless we have very good reason to change them. What requires redesign are the means of applying actions, keeping track of the game-state, and so on. These are all the things that make mathematics confusing and hard to follow. To do this, we can draw on the history and expertise of designing games to be easy to play and devices to be easy to use.

Because mathematics is already a game and its rules are kept largely intact in our redesign, I call this a mechanization of the game of mathematics. Our purpose is to reconstruct its rules of practice as the mechanics of a computer-game. [B.3]

2.2: Mechanization 2 - Requirements #

How can we reconstruct rules of practice as mechanics? For talking about this I find useful the interaction-design vocabulary of affordances, constraints, and signifiers. [B.2]

A game has valid actions and invalid actions. If a player performs invalid actions, like moving a rook diagonally in chess, then we say that the player stops playing the game or stops playing it correctly. Mechanization requires that all valid actions of the game should be afforded.—The computer interface should provide a means of performing each action. A good mechanization should have distinct signifiers for each affordance—a visible and unambiguous interface. And invalid actions should be constrained.—The computer interface should disallow performing invalid actions.

A game also has state. In computer-games it is customary to hide some of the game-state to better surprise and excite the player. But self-directed play requires visible state so that one can judge how the game can legally progress. A mechanization should therefore not have hidden state, though this is unconventional for computer-games.

A player who plays such a mechanized game cannot help but play the game that has been mechanized. We may say that they must follow the rules of the game because the computer program does not permit otherwise.

2.3: Usability 1 #

Before attempting any design, it is a good idea to decide how to evaluate the success or failure of the design. The test of whether a computer program or any other product is easy to use is known as a usability test. The success of our redesign should be judged by means of usability tests.

That is, we should systematically observe users with no prior experience play (or “use”) Super Practica A and note their stumbles and frustrations. These observations will inform further redesign to make the game easier to play. Then the redesign should be tested again, effecting an iterative cycle that produces a minimally frustrating design. [B.6]

“Usability testing” implies the question “testing usability to do what?"—We test that the game (the computer program) is usable to play the game (the mathematical practice). The different senses of the word “game” make this confusing [B.4], so I will call usability “playability” in the rest of this blueprint.

2.4: Playability 2 #

Some benefits for playability follow directly from the mechanization of mathematics.

One is the immediacy of feedback. The natural process of learning mathematics is to attempt, fail, and be corrected—because mathematics is a game, and this is the natural process of learning to play any game. Its teaching fails when the attempts are too cautious, or when the corrections are too slow or vague or unpleasant. Slowness, vagueness, and unpleasantness in feedback can all be eliminated by design.

Another is the opportunity for animation. Playing on paper or a board leaves a record of the game-state step by step, but this makes ambiguous the nature of the transformation at each step. In showing examples of mathematics, students often ask such questions as “Where did the 5 come from?” The answer to this question on a board is to draw an arrow from one number to another. This serves to disambiguate the transformation of the step, but drawing such arrows everywhere tends to clutter the board. Animating the step-transitions, such as having a number move from one place to another, is a better solution. Such animations can be programmed at every step all across mathematics once it is mechanized.

2.5: The pimnet system #

Such are some of the benefits if we can make a mechanization. The difficult part is actually making one, and doing so without losing what makes mathematics worth learning in the process—that is, without “dumbing down” mathematics.

I have made the innovations necessary to do this. The system of Super Practica A can be broken up into these parts:

• Pims: “Pim” is short for “programmable interactive module”. These are the smallest of sub-games corresponding to the combination of pictures, symbols, and rules of a particular mathematical method.

• Pimnet: Pims connect together in a pimnet, which is short for “programmable interactive module network”. This corresponds to the work that is done on the same paper with repeating symbols and pictures. The move from one pim to another within the pimnet corresponds to the move from doing one calculation to doing a different kind of calculation on the same paper. [1.7]

• Levels: A level corresponds to a task of mathematics and ends in the verification of the task. [1.8] The gameplay of each level is determined by the layout of its pimnet.

2.6: Interaction #

The gameplay of a mathematical solution method is play on a picture that has the effect of transforming the picture. To play on paper, the player must remember what actions can be performed on the picture and how to perform them. This design is open to improvement.

For a maximally intuitive interface, let the picture be an interface for its own transformation. In interaction-design terms, give the picture affordances for the player to transform it by. The picture thereby becomes not just a picture, but a dynamic environment for the player to act in via mouse cursor.

For example, imagine the “×” in the expression 3 × 8 as being a button. Upon this button being pressed, the whole expression gives the new picture 24.

This design gives an affordance for evaluating the expression and a signifier for the affordance. This is a general method of adding affordances and signifiers to the pictures of mathematics: Make elements of the picture into buttons that trigger transformations of the picture.

2.7: Pims #

There are many other things we can do with the picture 3 × 8 besides evaluating it to 24, as described in section [1.5]. Every valid action should have a mechanical affordance, but how can we fit so many affordances to a single picture? We should have separate signifiers for each affordance, but having so many affordances and signifiers in the same picture will proliferate buttons or context-menu options, making the design complicated and hard to use.

There are two general solutions to this design problem, and they can be used together.

First, let the pictures of mathematics, such as 3 × 8, be reused across many small games that look similar but that have different mechanics. What can be done with a picture—what game is playable on it—will be determined by the programmed mechanics that come with it. I call such a bundle of programmable mechanics on an interactive picture a “pim”.

Second, let the particular affordance be consequent of the combination of the affording element and the state of the manipulating object.—In other words, let the mouse be settable to different tools that do different things.

2.8: Pims together #

Next we need a system for viewing the different pims together on the same screen like we can view different mathematical pictures together on the same paper. This can be done by copying the windowing system from common computer desktop environments. Let pims be contained in such windows.

When pims are placed close together, one pim can act as a guide for what to do on another pim.

2.9: Interconnections 1 - Objects #

There should be affordances for the interconnections between pims described in section [1.7]. A drag-and-drop interface will suffice for most of these interconnections, so let the player be able to drag objects from one pim to another. A simple case is when a non-symbolic object like a counter can be moved between two pims of the same type.

2.10: Interconnections 2 - Symbols #

Symbolic objects like numbers are typically constrained in their arrangement on a picture, such as to fit a syntax. Manipulating these would be more intuitive if they were fitted into slots, rather than being “standing” objects in a field. Therefore let some pims have slots that afford dragging to duplicate the symbol and place it in other slots of other pims.

This lets symbols be dragged from one slot to another, copying symbols from pim to pim, but the symbols must be initially generated somehow. Let there be pims specifically for this purpose, which I call selector pims. For example, let a selector pim have buttons to match a common keyboard’s keypad, and let the player be able to generate a number in a slot by pressing these buttons.

2.11: Interconnections 3 - Translation #

Other interconnections between pims require translations from one pim to another. A simple solution is for a pim to convert dropped objects into a form native to it. For example, let a dropped symbolic number be converted into a number of counters equal to that number.

Such translations can go in many ways. From object to symbol, from symbol to object, from one object to another object, and from one symbol to another symbol. These interconnections mechanize the reinterpretation of pictures in terms of other pictures.

2.12: Level-start #

The gameplay of the middle part of each level of the game consists of using this pimnet system of interconnected pims to obtain a solution. But the beginning and ending parts of each level still require plans of mechanization.

The beginning of a level corresponds to setting up a problem. For this, let there be an interface such as in a sandbox game which enables the player to select from the whole range of pims and set them up with initial objects and symbols. This way, the player can construct the layout of a level before going on to play the level.—But then the wide range of options would likely make the player lost and frustrated, just as in beginning a mathematical problem on paper. We can improve upon this design.

A good way to avoid getting lost in setting up a problem on paper is to copy an example you have seen before. So for our design, let there be templates for different layouts of the pimnet, called solution templates, that correspond to plans of solution for problems. It will be more convenient and less frustrating for the player to select the correct template from a set of relevant ones than to reconstruct the whole solution method from scratch.

2.13: Level-end #

The ending of a level corresponds to its verification, but its structure described in section [1.10] makes it difficult to mechanize. Following verifications down to a foundational verification at the end of each level could be difficult to show clearly on a single screen, and would waste the player’s time. The verifications at level-end should therefore be abbreviated and stop short.

The natural time to stop verifying is when one is rationally confident [1.11], but different players would become confident at different times and by different means. If we left it up to the player to determine how to verify and when to stop, it would likely make for a frustrating game. It takes care to maintain honesty in mathematics and not pass off results one is unsure about as being certainly correct. These standards should be mechanized on the side of the game instead of being left to the player to fulfill.

To do this, we can copy the design of many puzzle games and finish the level with an automated and animated test. The Incredible Machine and Opus Magnum end levels with such a test of the player’s solution, and usually make clear why the solution works or what went wrong for it to fail.

So let there be a “Verify” button that the player can press to run an automated and animated test of the solution. There should be visual effects to show what goes right and wrong in the test. And the player should have an option to speed it up, because the player could be confident enough without it, and so would find the test tiresome and uninformative.

2.14: Design problems #

Let these designs serve as an outline of what is necessary for the mechanization of arithmetic. They cover some of the most common and universal elements of mathematical practices. They also serve to illustrate the design problems involved in mechanization.

The requirements of Super Practica are for it to accurately mechanize the original practice, to make it be easily playable, and to do this within the limits of available hardware. Its design problems arise from the difficulty of doing these things together, and from maximizing playability on top of them. Some design problems might be unsolvable, but the designs given here solve the most worrisome problems in at least the core of arithmetic.

This is enough to begin the development of Super Practica, but it is just the beginning. The mechanical redesign of mathematics will require solutions to many more design problems. Every additional mathematical practice will come with a new set of design problems to solve. Even the designs I have given here are open to improvement and redesign.—I have chosen to keep things simple rather than to elaborate the best solutions I could find to these problems.

2.15: Introducing the theory #

But what exactly is the benefit of this mechanical redesign? It “makes the game of mathematics easy to learn and play”, but what does this mean for the rest of mathematics and for mathematics education?

Super Practica could function as an educational program or as a more visually appealing calculator, but to call it these things would be to misunderstand its true value. Super Practica is ultimately a project of developing the optimal method of reproducing mathematical knowledge. This has required a theory which I call the theory of practical knowledge. [C.2]

3: Fundamentals of Practical Knowledge #

3.1: The nature of imagination #

A common belief is that mathematics is essentially something that goes on in imagination. If this is its essence, then mechanizing mathematics would destroy it. This belief is founded on confusion about imagination, so we should demystify imagination. All mystical and spiritual explanations of imagination are worthless because they cannot predict or replicate the phenomenon of imagination. As in section [1.4], we need an empirical claim that can reliably predict it.

First, let’s note that we can observe imagination to only follow perception. [C.1] The things one can imagine while thinking, reading books, hearing stories, planning what to do, and dreaming derive from one’s prior experience. The novelty of what one can see in a dream is at its greatest only a mutation of what one has already seen in life.

“The Tetris effect” is a popular term for the common experience of involuntarily imagining the visuals of Tetris and thinking in terms of the mechanics of Tetris, which happens after playing Tetris for a long time. This is also found to happen with other games and activities. [C.1] On the basis of this evidence I propose that gameplay produces imagination. The Tetris effect is an effect of not just Tetris but of games in general, including mathematics.

3.2: Ecological habituation #

For precise design, it will help to put this theory into precise terms: The affordances, constraints, and signifiers of one’s environment are internalized if they are constant and engaged with for a long period of time. [B.2] That is, if a person has a constant relation to their environment, they continue assuming, expecting, acting upon, and imagining that relation even as they move into different environments. Such a person habitually relates to new environments in the same way as the old, so I call it ecological habituation. [D.4]

According to this theory, we expect to see the greatest ecological habituation with computer-games, and especially ones you can play without breaks in the action like Tetris. We expect to see less ecological habituation with non-mechanized games like mathematics and repetitive activities like driving a car. And we expect to see much less ecological habituation when a person does not do any particular activity for long.

If this theory is correct, then playing Super Practica should be the best method of developing mathematical imagination—better than playing mathematics with pencil and paper. This follows from the accuracy of its mechanization, if its mechanization is accurate.

3.3: Practical knowledge #

So the player is ecologically habituated to mathematics by playing mathematics. What exactly does this have to do with mathematical knowledge?—Everything! Observation of the Tetris effect is observation of a natural ability seen apart from its regular function. Its regular function is knowledge.

The ecologically habituated player is not surprised by the various events of the game and does not need to be reminded of its rules. Such a player has developed habits—which may be dysfunctional but are usually effective—for completing the various tasks of the game. And if a player is very highly habituated to a game, then they can “take the game with” themself.—A highly habituated chess player can play chess without looking at a chessboard. [C.1]

These are clearly aspects of knowledge. But it is not book-knowledge, or memory, or understanding, or theoretical knowledge. It is rather game-knowledge, or as I prefer to call it practical knowledge, to include practices that have not been so codified as you expect a game to be.

Mathematical knowledge consists mostly, if not completely, of practical knowledge. After all, to know mathematics is to follow its rules and achieve its goals. [1.1]

3.4: Specification of practical knowledge #

Practical knowledge is always gained in an environment, but an environment alone does not determine the habits that one gains by living in it. Different people can learn different things in the same place. One person can develop functional habits while another can develop dysfunctional habits. [C.4] So what determines the particular habituation?

There are many causes: The person’s prior habits, their goals, their emotional motivations, and many more. [C.1] All these I encompass in the general term that is the person’s ecological attitude. To continuously have the same ecological attitude in the same environment is what I have previously called “a constant relation to the environment”. I accordingly derive this formula: A person’s ecological habituation is a function of the person’s environment and ecological attitude.

And a specific attitude to a specific environment is identical to a game! A game of The Floor Is Made Of Lava is specified [B.3] by the room—the environment—the player plays in, and by the rules and goals—the attitude—which a player follows. To play a game is to have the necessary abilities to start playing, the right emotional state to continue playing, interest in not failing the goal, and everything else encompassed by ecological attitude. [C.6]

The conclusion is thus: The specific practical knowledge that one learns is specified by the game that one learns it by playing. [B.4]

3.5: How you learn to play a game #

There are two parts to learning to play a game that are worth distinguishing: How one learns to play a game, and how one learns to play a game well. Ecological habituation denotes only familiarity with a game, but we are also interested in how one gains enough competence in playing a game to win the game.

So how does a player gain competence?—There are so many different games, so many skills that may be valuable, so many approaches that may be effective, so much variety, that we can hardly say anything about what it takes to become competent in any game in general. We ask how experts got good at playing any game, and the response we can generalize from them all is only “By playing it.” [C.1]

We can therefore summarize learning to play a game and learning to play it well in one general statement: You learn to play the game by playing the game. Everything else is specific to the game.

This is not a trivial conclusion. It implies that our general advice to aspiring mathematical practitioners should be “Do math.” In other words: Disregard the distractions and supposed shortcuts that stop you from doing mathematics—playing the game—yourself.

3.6: Contrast with educational theories 1 #

The theory of practical knowledge developed thus far can be summarized by these fundamental propositions:

• Practical knowledge is gained by playing the game it is practical knowledge of.
• A game consists of its environment and the attitude and abilities required to play it.
• You learn to play the game by playing the game. [C.7]
• Playing a game requires the player’s participation—the choice to play. [C.6]

And mathematics is a game—more precisely, a collection of interconnected games—so this can also be considered a theory of mathematics education. Let’s contrast it with common educational theories. These can only be valuable insofar as they give definite suggestions for how to design educational programs, so to cut the fluff and nonsense let’s consider the implicit theory of most educational programs:

• Knowledge of mathematics is composed of many concepts.
• A concept can be learned by being instructed in it, or by using it, or by some other method.
• There are many different ways to learn a concept, though some ways are better than others.
• However they learn it, a student must do many exercises so that the concept sticks with them.
• Teaching is a hit-or-miss affair. A good teacher will fail to teach if the student is untalented or disabled, or if the teaching method does not fit the student’s learning style.

This may be called a theory of “concept acquisition”. Most educational theories are only variations on this core. It is the same core whether they emphasize instruction or activity as the best means of learning concepts, or make more or less of the differences between students. The various psychological theories of learning are mostly speculations on the structure of concepts and rarely give definite-enough suggestions to be usable by educators. [D.2] [D.3]

3.7: Contrast with educational theories 2 #

So is mathematics a bunch of games or a bunch of concepts, if either of these? The reason for deciding in favor of my theory, in short, is that my theory can be a scientific theory, whereas concept acquisition theories must necessarily be unscientific.

Games, gameplay, and the phenomenon of learning to play can be directly observed, so we can develop a scientific theory about them. In contrast, “concepts” are abstractions that signify nothing in particular, so we could not possibly make a scientific theory about them. [D.4] Concepts do not have real existence; they are only vague words that help us talk about the phenomena of knowledge and learning without understanding them.—If a teacher says that they have taught their student a concept, then I say that this teacher does not understand what has actually happened.

The claims of “learning disabilities” and “learning styles” in theories of concept acquisition are also incompatible with empiricism. Empirical evaluation of the success of a program requires not jumping to disability as an explanation.—If we can always blame the student (or “discover” a new “learning style”), then we can always avoid admitting the defects of an educational program. In contrast, my theory of practical knowledge makes falsifiable claims about the conditions in which a person will learn to play a game.

I will now explain what actually happens in the course of instruction. By examining effective teaching methods and explaining what precisely makes them effective, we can begin to develop a reliable method of reproducing practical knowledge—a method not so hit-or-miss as schools and educational programs almost universally settle for.

3.8: The true value of rules #

It is a common assumption that a person must first learn the rules of a game before they can play the game.—“To play chess you must first know its rules.” But we see that beginning players can start playing a game without reading or hearing its rules at all. [C.1] Rules are therefore not necessary to play a game, so what is their value?

What is a rule?—It is an imperative statement. An imperative statement is spoken by one person to another to command or suggest that they do something. And playing a game requires participation, not obedience to commands, so the rules of a game are valuable for being suggestions for how to play.

3.9: Continuous correction #

We rarely see beginning players read all the suggestions of a rulebook and then play the game confidently without mistakes. More commonly, we see beginning players switch between playing the game and reading its rules. Over time, they consult the rules less and less until they can play with confidence and without error. [C.1]

The statements of a book are an abstraction of speech. When a player consults a rulebook for a suggestion, it has the same effect as being given the suggestion by a person. So if the rules of a rulebook are abstracted suggestions, what is the repeated consultation of a rulebook an abstraction of?—It is an abstraction of continuous correction during gameplay!

Sure enough, another way a beginning player can start playing a game is by being continuously corrected by a player who already knows how to play the game. [C.1] This is the most common case of effective instruction by a teacher.—The instruction is a suggestion, the teacher is a player who already knows how to play, and it is effective only as a correction in response to a player already trying to play the game. [C.7]

3.10: Suggestive signaling #

Computer-games have developed methods of guiding a player to play correctly that are more effective than reminding the player of the game’s rules. [C.1] Rules take time to listen to and cannot be very precise without being very long, so correction can be done better by suggestive signaling.

This is to give the player different signals in reaction to their gameplay. A teacher can give them by interjections and gestures, and a computer-game designer can give them by visual and audio effects. The player can understand these signals in relation to their immediately preceding gameplay, and can therefore take them as suggestions for how to think about their own gameplay and how to play differently.

It takes time to learn the meaning of every new sign, and using more signs presents more danger of misunderstanding. It therefore seems best to limit suggestive signaling to these 3 most basic, universally valuable, and universally understood signals:

1. Affirmation: For example, say “good” or draw a checkmark.
2. Rejection: For example, say “no” or make an abrupt motion.
3. Attention: For example, point at something or make something shake.

I have already used affirmation and rejection signals in my design of automatic verifications in section [2.13]. For another example, consider how these signals could be applied to guide a player to start playing a racing game correctly:

• Direct the player’s attention to the next checkpoint by periodically having the player’s avatar emit a ball of light that moves to the checkpoint.
• Signal rejection of going backwards by saying “wrong way” when the player does so.—The tone of voice makes it function as a rejection even if the player cannot speak the language.
• Signal affirmation of clearing checkpoints by progressively filling a checklist of all the checkpoints that have to be cleared to finish the race.

It is a common confusion to speak about such signals in terms of incentives, but these signals rather suggest and communicate, not “incentivize”. [B.7]

Signs acquire their meaning by consistent usage, so consistency is more important than the sign itself. But not any sign can function as any signal.—An annoying sound will not function as an affirmation, and a subtle motion will not function to attract attention.

3.11: Modeling #

Another method of suggesting how to play is modeling, which is to play the game as an example for the player to imitate.

Solving mathematical problems requires recognizing the similarity of your problem to one you have solved before [1.4], and the memory of past solution can come from observing someone else’s solution as well as by solving a problem yourself. As this implies, we find that modeling and imitation are extremely effective for teaching and learning to play mathematics, as well as games more generally. [C.1]

But it is often not clear when you follow a model whether you follow the model correctly.—It is not clear what parts of the shown gameplay are meant to be suggested and what parts are incidental. Modeling is therefore best used together with suggestive signaling.

3.12: The true value of instruction #

The greatest value of verbal instructions comes from their ability to serve as suggestions for how to play. This is at its most streamlined and efficient when the suggestions are given by intonation and gestures rather than words—when they stop being instructions. But instructions have value beyond being inefficient suggestions.

Consider the instruction: “The rook moves along the ranks and files of the board.” A beginner who has never before heard these words in the context of chess will not be able to understand the sentence at all.—It is useless for the purpose of instructing.

But what happens when the beginner sees a demonstration together with the instruction, seeing the teacher move the carved figure back and forth with the word “file” and from side to side with the word “rank”?—The instruction gives names to the demonstration! The true value of instruction is to connect speech to gameplay.

A player who has heard this game-speech can go on to use the new vocabulary and grammar by imitation. They can begin to speak about the game. And what does this specialized speech do for the player?—It makes the not-always-present states, objects, and patterns of the game be always immediately available by speaking their names. It affords communication with other players who have learned the specialized language, and it affords planning ahead.

3.13: Game-plans #

A game has a metagame, which is the transformation of the game by the social context of its players. This includes their game-speech and the game-plans they share. Games that have been played for longer and by more people typically have more developed metagames; and the more developed the metagame, the more valuable are its game-plans to new players. To hear and follow the communally developed, time-tested game-plan is a shortcut to victory for the new player over the experienced but merely intuitive player. [C.1]

This shortcut makes instruction appear more effective than it really is. The instructor gives a game-plan: “Develop your pieces.” The player follows the plan and sees improvement in their gameplay. Both the player and the instructor feel that this teaching has been valuable, but this shortcut is a trap.

The effective game-plan is a product of the metagame, not of the instructor. Every player is capable of judging the plans of the metagame, and this judgment is a skill that gets better only with the experience of playing the game and of judging game-plans. An instructor jumps ahead, giving a game-plan that the player cannot effectively judge the value of. This encourages reliance on the instructor’s authority over personal judgment and therefore stultifies the player, creating a barrier to their personal progress. [C.2]

Stultification is a constant danger of authoritative communication, but this does not imply that authoritative guides to gameplay are always bad. On the contrary, all the methods of suggestion described here depend on the player recognizing the authority of the teacher for their effectiveness.

3.14: The true value of teaching #

So while a player can only learn to play a game by playing it, learning alone is often less efficient than learning with a teacher’s guidance. There are many valuable things that a teacher can do to help reproduce the practical knowledge of a game in a player:

1. Specify the game by setting rules and tasks.
2. Set up the game’s environment.
3. Be a model for how to play.
4. Be another player to play with.
5. Correct the player’s failure to play the game.
6. Give suggestions for how to play better.
7. Converse about the game in its specialized vocabulary.
8. Introduce the metagame and its game-plans.

If we dissipate the confusions of concept acquisition theories and observe what is effective in reproducing practical knowledge, then we find that the true value of teaching reduces to these things.

This explains why many methods of teaching mathematics have been effective even when they did not value gameplay—the only means of learning to play mathematics. Successful students of mathematics must have learned to play mathematics on their own by playing it on their own, and the instruction must have done some of these valuable things for them.

3.15: The optimal method 1 - A start #

The theory developed so far gives this partial answer for how to optimize the reproduction of practical knowledge:

1. We must specify the practical knowledge to reproduce by specifying a game. This means to specify an environment and the “rules” for play in it.
2. We must make the game easily playable in this environment.
3. We must guide the player in their play using the effective methods of suggestion.

This is only a start, but on this basis we can conclude that the optimal method of reproducing practical knowledge must be a well-designed computer-game:

• To exist in an environment is not the same as to play a game in it, but these two distinct things unite in a computer-game.—A computer-game can embody the ecological attitude of a game as well as its environment. [2.2] [B.5]
• A computer-game can be optimized for playability. [2.3]
• Most of the things a teacher can do in the way of communicating ecological attitude—specifying games and suggesting how to play—a computer-game can do as well, and often much better. [2.1] [2.4]

The value and promise of a game like Super Practica should therefore be clear: Here is a way to get prospective players playing the game of mathematics as efficiently and easily as possible. Here is a way to include in the game exactly what is valuable for the player’s improvement and to eliminate all frustrations and stultifications. All that is necessary is to develop the game and let interested players play it.

4: Measurement of Practical Knowledge #

4.1: The value of measurement #

The theory of chapter [3] clarifies the value of Super Practica A outlined in chapter [2]: It’s a start toward optimally reproducing mathematical knowledge, but only a start.

To find and develop the optimal method, we must find a way to measure mathematical knowledge. Then we could judge the success of Super Practica in reproducing mathematical knowledge, and we could compare different programs and designs to see which is best. An empirical measure would afford optimization.—We could propose a redesign and confirm its success by observing an improvement in our measurements.

4.2: The true measure of mathematical knowledge #

The mathematics testing common in most schools ostensibly measures mathematical knowledge, but on close examination this is found to be only wishful thinking. Such a test puts the student into the artificial and arbitrary situation of sitting at a desk and guessing what answers and answer-methods are expected for decontextualized problems. It is always questionable if the successful completion of such a test will imply the ability to do mathematics.—It might or it might not.

I call this “schooltesting” because it’s primarily related to the operation of schools and only tangentially related to honest empirical testing. [D.5]

To find the true measure of mathematical knowledge, we need to observantly and honestly answer how we actually test mathematical knowledge. We do not look at a person’s mind, or brain, or behaviors to do this. We look for the person successfully using mathematics. If we see that a person can do mathematics, then we say that this person knows mathematics. The measure of mathematical knowledge is successful mathematical practice.—More generally, the measure of practical knowledge is successful practice.

This is ultimately how we test the schooltests. If a schooltest says that its taker does not know mathematics but we see that they can do mathematics, then we say that the schooltest is flawed. And if a schooltest says that its taker knows mathematics but we see that they cannot do mathematics, then we say again that the schooltest is flawed. We test the schooltest by the real test, and the real test is the test of using mathematics in its natural situation.

4.3: The optimal test #

To efficiently observe if a student has gained competence in some mathematical practice, put them into a situation where performing this practice is necessary for success, and then observe if they succeed or fail. For example, let the student build a bridge and see if it collapses. If the student succeeds in building a usable bridge, then you can deduce that the student knows how to build a usable bridge.

This is efficient both for evaluating the student’s abilities and for providing the student with opportunities for further learning: If the bridge collapses and kills hundreds of people then you can be sure that the student will learn from their mistake. But such tests can be dangerous and expensive.

An optimal test would combine all the features we want in a test.—It would combine the natural situation’s contextual detail and its facilitation of learning from mistakes, with efficient reproduction and safety. What combines these together is a computer simulation, so the optimal test of mathematical knowledge must be a well-designed simulational computer-game.

4.4: How not to well-design a simulation #

The problem remains to well-design a simulation to be the optimal test. How could this be done?

A good place to begin is to note that mathematical word problems amount to badly designed simulations. They rely on the unpredictable workings of imagination suggested by text to construct an environment and a context for the problem. Another limitation of textual form is that describing more contextual detail requires more text, which takes more time to read. This makes text less efficient for communicating contextual detail than a picture or a graphical game.

So we could try to go beyond the limits of word problems and simulate the environment and context of a natural situation in full:

• Natural situations are rich in different things you can do, so we could simulate all the tools scattered about the workplace and all the things you can do with them.
• Natural situations have a social context, so we could simulate the friendships and tensions that people have with each other.
• Natural situations are rich in the perceptible details of their environment, so while we’re at it we could simulate every tree swaying in the wind.

This would clearly be a mistake, so when does simulating a situation in greater detail stop being valuable?

4.5: Transferability #

To design a simulation well does not require accuracy per se, but a specific kind of accuracy. The primary feature of a good simulation is the ability of the player who passes the simulational test to pass the natural-situational test. In theoretical terms, the measure of the value of a simulation is the transferability of practical knowledge from the simulation to the situation it simulates.

Maximum transferability can be found in the trivial case that the successful performance of a practice is a sure and certain measure of the ability to perform that same practice.—If we observe all players who successfully build a bridge in a simulation to successfully build a bridge in the natural situation, then we verify that we have a good simulation. The value of a simulation is determined by how well it approximates this ideal. We have a well-designed simulation when its successful completion approximates to being as certain a predictor of ability as the successful completion of the original situation which it simulates.

Accordingly, a simulation needs only to reproduce the essential and important details of a situation, not all of its details. Call this the essential accuracy of the simulation. This is a relative term, so no simulation can be essentially accurate without reference to a situation it is essentially accurate to.

4.6: How to well-design a simulation 1 #

This empirical measure affords us the ability to design experimentally. Suppose that a player has successfully performed a mathematical practice in the game but then was unsuccessful in performing it outside of the game. Then we should say that the game was not similar enough to what it was supposed to simulate. Then we may redesign the game to be more similar and then test it again. And if that fails, then we may redesign it to be more similar still and then test it again, and so on, repeating this process until it finally works and we say it is hence similar enough.

However, as in the case of building a bridge, it can be dangerous and resource-intensive to apply the natural-situational test. The whole point of designing simulations is to replace the natural tests with more efficient and safe tests, so this trial-and-error method of design is generally not worth following.

We should rather rely on theory and the good judgment of designers. The determination of what is essential in a simulation amounts to a set of theoretical questions and design problems. We should apply the natural-situational tests as a means of verification to ensure that we are on the right track, but most of the time we can only apply them sparingly.

4.7: How to well-design a simulation 2 #

The theory I propose for designing simulations is my theory of ecological habituation. [3.2] The details of a situation’s environment to simulate are those which a participant must interact with to be competent in the situation. The simulation should include what experts look at to make their decisions—the situation’s signifiers—and what means experts have to implement these decisions—its affordances. [B.2]

The simulation should include only these signifiers and affordances.—It should not include any means of divining what to do or of enacting one’s will that is not present in the natural situation.

But if we just drop the player into the situation, they would lack the context that experts have as they set out to do something. We should therefore simulate this context by clearly suggesting goals. The specific goals to give should be determined according to which goals we judge it is worth becoming expert in achieving. These are not all the various and idiosyncratic goals that the original experts might have, such as surreptitiously carving their name into the product of their work, but only a selection of their goals according to our design.

4.8: The sufficiency of simulation #

Suppose we thus make a well-designed simulation of a situation of expert mathematical practice. This would be a game and a test, and you learn to play the game by playing the game [3.5], so would you become an expert at the game-test just by taking the game-test to see if you’re an expert at the game-test?—Yes, sometimes.

Consider a well-designed simulation of piloting an airplane from one airport to another. As a new player you might have no idea what to do or how to do it. You might repeatedly crash. But after every crash you can develop new plans for what you could do differently to do better next time. Eventually, without any outside help and by trial and error alone, you can become an expert at winning in the simulation. [C.1] It may be impractically difficult and time-consuming, but resources outside the simulation are not strictly necessary.

This is to say that efficiency aside, the game itself is sufficient for becoming an expert player of the game. Therefore also, a well-designed simulation is sufficient for becoming an expert in the situation that is simulated. The specification of practical knowledge in the simulation is sufficient for its own reproduction. [3.4]

4.9: Introducing Super Practica B #

A well-designed simulation is in general not easily playable, but a playable mechanization like Super Practica A is. [2.3] Meanwhile, a playable mechanization is in general not sufficient for reproducing practical knowledge, but a well-designed simulation is. These must be distinct pieces of the optimal method of reproducing mathematical knowledge.

So let’s call Super Practica B the project of developing many computer-games that simulate the various situations of expert mathematical practice. These simulations will render all other tests of mathematical knowledge obsolete because they will be the optimal tests. [D.5] Super Practica A and Super Practica B are worth making separately, but their combination has added value:

• Super Practica B provides the empirical framework to ensure that Super Practica A functions as intended.
• Super Practica B benefits from incorporating elements of Super Practica A into its own design and for its own purposes. [5.5] [5.8] [5.17]
• Because you can become an expert at the game-test by playing the game-test, the combination eliminates the inefficiency of playing game-tests separately from games that reproduce practical knowledge.

Super Practica A is designed to optimize playability, and Super Practica B is designed to specify the practical knowledge of mathematics and optimize its transferability to situations where mathematical expertise is required. Combined together, they make a game that’s easy to start playing and that ensures expertise upon its completion.

5: Simulation of the Situations of Expert Mathematical Practice #

5.1: The nature of application 1 - Problems and solutions #

We must understand the situations of expert mathematical practice before we can simulate them. The most important of these is the application of mathematics to solve problems with some immediate practical purpose—that is, “real-world” problems. After all, we wouldn’t value mathematics as we do if we couldn’t apply it.

When you apply mathematics to solve a problem, you first understand the problem in non-mathematical terms and then you reinterpret it in mathematical terms. [1.8] For example, if your problem is to cut a circular cake equally for everyone, you can solve it by this method:

1. Count the number of people who will get a piece.
2. Divide 360 by the count to determine the angle from the center between each cut.
3. Cut the cake such that each slice is separated by this angle.

This solution method is distinct from the problem, but connects to the problem as its solution. You could solve the problem of cutting a cake approximately equally by a different mathematical or even non-mathematical method. You could also reapply the solution of equally dividing a circle to other problems that have nothing to do with cutting cakes.

The vocabulary of problem and solution-domains is helpful for analyzing situations in which mathematical knowledge is applied. In the example, the solution-domain is geometry, and the problem-domain is food preparation. These identify different contexts and expectations of knowledge.—The shift from one context to another implies a shift in what is the relevant information and expertise. When a solution of a solution-domain is applied to a problem of a problem-domain, I call it an application.

5.2: The nature of application 2 - Sequence #

The full sequence of an application of mathematics is thus [A.1]:

1. You work in the problem-domain and find a problem. For example, suppose you’re refurbishing a room and want to replace the carpet, but you’re not sure which dimensions of carpet to buy.

2. You determine what mathematics is applicable and set up the problem. For example, you make measurements of the walls and draw a picture of the floor of the room with the measurements on it. Then you find information for the shapes of carpet you can buy and write down what may be acceptable solutions.

3. You solve the problem mathematically and obtain the result. For example, suppose the suitable carpet comes in rectangular strips and you want to approximately minimize the area of unused carpet. To do this you try out different arrangements of the strips on the picture and select the one that appears to be best. This could be done by drawing on the picture and calculating lengths or by making to-scale cutouts.

4. You interpret your solution in the context of the problem-domain and enact the solution. For example, you identify your selected arrangement with the type and number of carpet-strips to buy, then you buy them, and then you cut and arrange them according to your selected arrangement.

5.3: The nature of application 3 - Complications #

The solution of the solution-domain is complicated by the contextual details of the problem in the problem-domain. For example, cutting a carpet into many tiny pieces can help to maximize the use of its area from a mathematical perspective, but will ruin the solution in context. These details may be precisely translated to additional conditions for an acceptable solution in mathematical terms, but this makes the mathematics less elegant. So it is generally preferred to speak of “common sense” while not being as precise as one could be in defining what makes an acceptable solution.

Divisions of labor are another complication. These are often arbitrary, and different divisions will imply different points at which one problem-domain ends and another problem-domain begins. For example, the precise division of labor between the manufacture and installment of a carpet will determine what is the job of each.—The carpet could be cut into shape in the factory or in the room where it is installed. The particular problems and their appropriate solutions will vary according to the division of labor.

5.4: Simulating application 1 - Common mechanics #

What we must simulate in Super Practica B are the problem-domains of mathematics. Super Practica A deals largely with its solution-domains [2], and Super Practica B deals largely with its problem-domains.

But mathematics has very many problem-domains, and each of these has very many relevant details that complicate its solutions. There’s no way around this difficulty: To simulate real-world problems, we must simulate the real world—but only to the point of essential accuracy. [4.5]

Each problem-domain will require some unique mechanics for its simulation, but we can simulate much of what is essential in all problem-domains by reusing just a few mechanics:

• Let the player be able to control a character and move around in the setting of the problem-domain. This would tend to be either a workplace or the “field” of fieldwork.
• Let the player be able to examine things, write things down, and make measurements. These mechanics will be used for determining and setting up problems.
• Let the player be able to perform actions that interact with and change the environment. These mechanics will be used to enact the solutions to problems.

5.5: Simulating application 2 - Modes #

We should somehow connect the problem-domain to the solution-domain. So let the player be able to switch between “situation mode”, in which the player controls a character in the problem-domain, and “math mode”, in which the player solves problems in mathematical terms using the mechanizations of Super Practica A. [E.3]

There should be some kind of input and output between the situation and math modes. For example, let measurements insert into the math mode as symbols on a pim. [2.10]

5.6: Simulating application 3 - Similar games #

So what kind of game is the situation mode?—Its mechanical features [5.4] make it similar to the genre of puzzle-adventure games. These are often called “point-and-click” games, but this interface is not essential to their design. The genre’s essential mechanics are that a player character can navigate an environment, obtain things from it to use as tools, and interact with it by applying various verbs and inventory items to its objects. This is a good place to start for our design.

In abstraction, the gameplay of such games consists of finding and using so many keys on so many locks, and the designer makes only a few combinations of key and lock yield progress in the game. Applying the correct key to the correct lock triggers a scene that changes the game’s environment to progress the game. But precisely what using the chicken grease (a key in the player’s inventory) with the cannon balls (a lock in the environment) will do is unpredictable. There is an arbitrary connection between the lock-and-key mechanism and its function of changing the game’s environment.

This arbitrariness enables more creative puzzles and stories by letting developers fill in what happens. But we have no use for such arbitrary connections in our simulations. We should replace them with more direct mechanical connections between what can be done and its effect on the environment.

5.7: Simulating application 4 - Initial design #

It would take very much time to develop unique mechanics for each problem-domain and for each action the player can take within each problem-domain. So to the greatest possible extent, we want the mechanical connections between the player and the environment to follow a reusable pattern for many different ways of interacting with the environment and to enable the essential accuracy of each simulated interaction.

I propose this design: Let the abstract key be a minigame that the player can initiate by using a tool, and the abstract lock be the place to initiate it. Then let the mechanics of the minigame connect to the mechanics of changing the environment. The gameplay of the minigame thus constitutes a second puzzle on top of the first puzzle that is to find the place to initiate it.

For example, the minigame-key of using a screwdriver can be triggered anywhere and can change the environment wherever there is a screw, but it is only valuable to use it to change the environment in only some places for only some screws.

The purpose of this design is only to function as a first approximation to a solution. It will be easier to see what redesign will be valuable by observing it in a prototype, and to make this prototype is my plan for continuing development on this front. But there are more situations of expert mathematical practice beyond application to consider.

5.8: Time constraint #

Doing mathematics under time constraint is another set of situations of expert mathematical practice. Knowing why you have to do something quickly is irrelevant to successfully doing it quickly, so it will suffice for the essential accuracy of the simulation to only simulate the time constraint without other context.

This is easy to do: If we value a player’s ability to perform a mathematical practice quickly, we may take the level which mechanizes that practice in Super Practica A, and add to it the goal of completing it within a time limit. [E.6]

5.9: Resource constraint #

As with time constraint, it will suffice for the simulation of resource constraint to simulate only the resource constraint without other context.

There are many kinds of resource constraints. The traditionally most valued are the constraints to paper and pencil, to straightedge and compass, and to nothing apart from one’s own body—typically called “mental calculation”. But there are many possible combinations of resources we can simulate constraint to: A calculator, an abacus, a standard computer program, and more. The choice of which of these to simulate should be determined by which resource-constrained situations we value expertise in.

There is some value in simulating paper-and-pencil gameplay on a computer even though one can just play with paper and pencil instead.—It can make use of the benefits of mechanization, including those described in sections [2.4] and [3.10]. But the design of mechanized simulations of resource constraint tends to run into apparently unsolvable problems: You cannot accurately simulate the act of using a pencil and paper by having the player use a mouse and keyboard, for example. In general, simulation of resource constraint is limited by computer hardware.

It is therefore an unavoidable imperfection that Super Practica should lack accurate simulations of many resource-constrained situations—but this is true only in the short term. Advances in computer hardware could make these simulations practicable.

5.10: The practical nature of proof 1 #

An important but often confused expert mathematical practice is that of making mathematical proofs. This is a preferred topic of mystics, so I must first demystify it. [A.5]

Consider what applying the Pythagorean theorem can do for you. If you can make two measurements of the sides of a right triangle, then the Pythagorean theorem tells you how to deduce the third length without having to measure it. This is useful if two measures on a right triangle are much easier to find than a third that you want to find. For example, suppose you want to measure a length between two points that runs across a pond. Rather than getting wet in the pond, you can measure the lengths of two lines that extend from those points and run over solid ground to connect at a right angle. Then you can calculate the length of the hypotenuse from these two other lengths.

Such a calculation is only a guess until it is verified. [1.9] The natural way to verify it is to measure the length of the hypotenuse directly, but that would get you wet. Instead, you could examine the proof of the Pythagorean theorem and become rationally confident of its applicability to the same extent as you would be if you had measured the hypotenuse. So we see that a proof can serve to verify a calculation by more efficient means.

5.11: The practical nature of proof 2 #

Mathematical proofs function to make fit for reliable use general statements about mathematics called theorems. Their primary use is to substitute for other verifications, but their benefits turn out to be greater than only making existing verification more efficient. Some things are not practically verifiable at all by direct measurement, such as astronomical distances. Proofs therefore supercharge the capacity for verifying proposed solutions to mathematical problems, and thereby yield a great increase in the quantity and reliability of mathematical methods. [A.1]

Though they have this great value, it is also true that it may be proof enough, in some context, of the reliability of a method that it has worked 100 times out of the last 100 times it has been applied. And it is also true that you can often perform mathematics well enough without anyone proving the validity of your methods. But such mathematical practice is fast and loose, and a more reliable way of doing things is often preferable. The practice of mathematical proof yields this valuable but often unnecessary reliability.

5.12: The social nature of proof 1 - Persuasion #

One does not “prove a theorem” simply, though this is a natural way of speaking. One only proves a theorem to a person. Mathematical proofs are made by people and to people, and mathematical theorems are accepted and used by people.—They are not universal truths that stand apart from humanity. Recognizing this is necessary to understand the place of proof in society and to sweep away related mystical confusions. [A.6]

Proofs verify the reliability of mathematical methods, and they are themselves verified by surviving criticism and persuading honest and intelligent critics.

The means of persuasion is argument. The general form of these arguments is to begin with more certain—accepted—premises and derive what are otherwise less certain—not yet accepted—conclusions. These arguments may be formalized, but whether they are made in formal or informal terms, their efficacy depends on the presence of critics who allow themselves to be persuaded by good arguments and who do not allow themselves to be persuaded by bad arguments.

5.13: The social nature of proof 2 - Consensus #

The critic of the proof could be the same person as the user of the method that the proof verifies, but more commonly there is a division of labor between the calculators and appliers on one side and the provers and critics on the other, such as between engineers and mathematicians. (Let’s call the provers and critics mathematicians from now on, though one may do these things as an amateur.) Mathematicians as a community set the standards for what constitutes a sound proof, though following their particular standards is not necessary for using and finding value in proofs.

Theorems are commonly organized in axiomatic systems.—Some statements are proclaimed certain, called axioms, and by arguing from these one can proclaim just as certain the conclusions of the argument. One proves a theorem in the context of some axiomatic system.—The proof of a theorem varies according to what axioms and other theorems are accepted prior to it. [A.1]

So what are the axioms and theorems of an axiomatic system, if not universal truths?—They are attempts at rational consensus by the community of mathematicians who make them and use them. [A.3]

Such, then, is the situation of mathematical proof in its most basic form: Proofs are made within a community of critics, they give rational confidence in theorems, and these theorems find use in mathematical methods of solving problems and verifying solutions. Divisions of labor between pure and applied mathematicians, scientists, and other experts only complicate this core situation.

5.14: Existing simulation of proof #

The situation of mathematical proof is socially complex, but we only need to simulate what is essential to its simulation, and that is the rational, critical dialogue between mathematicians. We must simulate precisely the rules of argument and persuasion that they use when they argue and are persuaded rationally—rationally according both to them and to us. The result is a game of how to converse and how to change your beliefs.

Such a project of making explicit the implicit rules of rational argument in a community has been carried on by many philosophers already, such as under the name “rational reconstruction”. [A.2] This body of work is relevant for our simulations, but we can draw on a much more developed tradition of what amounts to rational reconstruction: This is the tradition of making proofs in axiomatic systems.

While a game of dialogue can simulate an argument with one mathematician, a game of axiomatic proof simulates argument with a community of mathematicians. The axioms and theorems simulate the consensus, and the proofs simulate the response to criticism. Axiomatic systems are already simulations of rational, critical dialogue between mathematicians. [A.4]

So the simulation that we seek to make for Super Practica B already approximately exists as the game of axiomatic proof. And conveniently, much of the work of specifying the rules of axiomatic systems has been done for us by mathematicians already. What’s left is to mechanize the specification [B.3] and to improve upon its design. [2.14]

5.15: Simulating proof 1 - Two games #

Axiomatic systems efficiently survey the range of accepted arguments, but they are not perfect for this task. They typically do not contain all the arguments that survive criticism and gain assent.

We find that mathematicians have historically engaged in inconclusive arguments over how many axioms an axiomatic system should have. [A.1] It therefore appears that there is an irresolvable conflict between reducing these systems to fewer axioms to make them more elegant, and adding more axioms to fit in unusual arguments to make them more complete. So should we prefer the more elegant or the more complete design of axiomatic systems?

I propose that we should make two types of simulations for proof in Super Practica B:

1. Mechanizations of the more elegant and reduced axiomatic systems, which, while incomplete, would still cover a large part of accepted arguments in their topic, such as 95% or more.
2. Non-axiomatic, rational reconstructions of dialogue between mathematicians, which would cover the remainder of accepted arguments.

The game of proof in dialogue is a secondary and difficult addition to Super Practica, so my focus in this outline will be on mechanizing and redesigning the game of axiomatic proof.

5.16: Simulating proof 2 - Levels #

This game is a puzzle game. The task of each of its levels is to prove a theorem, so the puzzle is how to generate the correct conclusion from available premises. Let the player start with a collection of premises, and let the level be won when a statement matching the theorem is obtained.

To do this, first let the player perform logical-structural actions that set the form of the argument. For example, the form could be to argue exhaustively for the 3 cases of x>y, x<y, and x=y. Then let the player fill in the logical structures with statements to generate conclusions.

What premises a player can use in a level depends on the place of the level in the axiomatic system:

• Let axioms be usable anywhere as premises.
• When theorems have the form “If A then B,” then let the condition A of the theorem be assumed as a premise in the level of proving that theorem.
• To avoid circular arguments in which a theorem could be proven by itself, let levels be arranged in a sequence with only the theorems proven in prior levels of the sequence being usable as premises in each level.

5.17: Simulating proof 3 - Interface #

The pimnet system described in section [2.5], with a few modifications, will be an effective interface for this game.

Let the slot interface of section [2.10] handle statements as well as symbols, so that the player can move premises and conclusions from slot to slot. The player can use this to act on both logical structures and on theorem pims. Since theorems typically have the grammatical form of “If A then B,” such as “If two circles intersect then they do not have the same center,” filling them in can generate conclusions just like for logical structures.

5.18: Outstanding design problems #

The language of premises and conclusions in proofs is a language of general descriptions, and the language of mathematical pictures and their applications is a language of specific descriptions. We can intuitively relate general and specific descriptions to each other, but mechanizing this relation is a tricky problem. For example, we need to mechanize the intuitive logic connecting general statements about isosceles triangles in arguments, to specific statements about pictures of isosceles triangles.

This problem and the one I proposed a solution to in section [5.7] are the outstanding design problems it will be necessary to solve to make progress in the design of Super Practica B.

This is enough for an outline of how Super Practica A and Super Practica B can be developed. Super Practica is valuable enough with these two pieces, but one more piece will bring it to logical perfection.

6: Reliable Reproduction of Practical Knowledge #

6.1: The value of reliable progression #

Super Practica A and Super Practica B are two different projects that should ideally be combined together in the same game. [4.9] The first should be beginner-friendly, and the second should only be completable by experts. The problem with their combination is that while simulation is sufficient for becoming an expert, it’s not necessarily efficient. [4.8] The progression of gameplay would be thus: The player would have an easy time starting with Super Practica A, and then would have to bang their head against a wall trying and failing repeatedly before they might, unreliably, get good at Super Practica B. This design is open to improvement.

The task is to move the player’s level of expertise—their practical knowledge—efficiently and reliably from beginner-level to expert-level. The beginner-friendly mechanizations of Super Practica A and the simulations of Super Practica B won’t achieve this by themselves.

6.2: Existing reliable progression #

Games that reliably move a player’s abilities from beginner-level to expert-level already exist. Consider one of the many computer-games in which the final level is difficult enough that someone who has only completed the first level (A) will not be able to complete the final level (B). In such games it is common that there are many intermediate levels which the player plays sequentially between A and B. In the course of doing this, the player can build their skill such that they can complete the final level.

If the game is well-designed for the purpose, then its player does not get stuck or have to do special practice outside the game to complete it.—Playing through the level-progression builds the player’s skill to the point of completing each level in the level-progression and therefore the whole game. [C.1]

We can therefore make the following claim: For some cases of beginner-completable level A and exclusively expert-completable level B, there is a connecting sequence C of intermediate levels C1, C2, C3, and so on, which the player can sequentially play through to reliably complete level B after starting at level A. In other words, there can be a game where A is the first level and B is the final level with a reliable level-progression in between.

So we would like to find a connecting sequence of levels between the levels of Super Practica A and those of Super Practica B that will make a reliable level-progression between them. But to do this we must first understand how such reliable level-progressions work.

6.3: The nature of difficulty #

An intuitive method of designing a level-progression is to first design a bunch of levels and then to order them along a “difficulty curve”, which means in order of gradually rising difficulty. This is appropriate for simple games, but is a bad idea for more complicated games with many elements.

To speak of a curve implies that each level has an absolute and one-dimensional difficulty. But the experienced difficulty—the only kind of difficulty that matters—depends on the player’s abilities and habits. [C.1] These are different between players and variegated in a single player, so a level’s difficulty is multi-dimensional and relative to its player.

The 1994 Donkey Kong is exemplary of how to make a good level-progression. It switches between introducing new mechanical elements, developing previously introduced mechanical elements, and synthesizing them in new combinations. The player thus habituates to each of these mechanical developments separately and sequentially. Such is a level-progression founded on the understanding that expertise develops along many dimensions. Try laying this game’s levels out along a plain difficulty curve and you’ll understand what a terrible mistake it would be.—It would make the game less fun and more difficult.

6.4: The nature of difficulty spikes #

“Difficulty spikes” are valuable to find on a difficulty curve. These are points of the game where players commonly get stuck or are unable to progress for much longer than they expect. A reliable level-progression amounts to a level-progression without difficulty spikes.

But difficulty spikes are better understood without reference to a difficulty curve, as high relative difficulty for players. And high relative difficulty depends on the player as much as on the level.—What is difficult for a beginner may be easy for an expert. [C.1] The difference between them is their past experience, so high relative difficulty amounts to low relative preparation.—It is to have insufficient past experience to make a task easy to complete.

We find that players get past difficulty spikes either by retrying many times or by practicing to get better outside of the sequence of the game’s level-progression—by habituation and self-guided preparation. [C.1] So what is a difficulty spike but a failure of the game to guide the player’s habituation and prepare them for its own levels?

What games with reliable level-progressions do is make each level of the progression prepare the player for its future levels. The levels of such a game do not only present new and interesting challenges to the player, but also function as guided practice to the game’s completion.

6.5: Playthruability 1 - Measure #

If we can empirically measure the reliability of a level-progression, then we can optimize it. This requires making precise our judgments of what reliability and unreliability are. In intuitive terms, a reliable level-progression is one that any player can play through without getting stuck. We need to put this into precise and measurable terms.

A level-progression is made up of a sequence of levels from level 1 to level N, and to get stuck on some level X is to not be able to advance reliably from X to X+1. To ensure that a player can advance from 1 to N, it suffices to ensure that the player can advance from each level of the progression X to each next level X+1. In theoretical terms, the practical knowledge of the reliable level-progression 1 to X must be transferable to the level X+1 for the level-progression 1 to X+1 to be reliable. [4.5]

The empirical measure is thus: For a sample of players who are known to have completed levels 1 to X and not to have previously completed level X+1, what proportion of them go on to complete level X+1 within desired conditions? The conditions here are what we require to call the progress successful, such as not taking too long. Call this measure the playthrough rate.

The term “playthrough rate” only describes our observations. What we want is not the measured rate but an appraisal of reliability according to it. Call this the playthruability of the game and of its level-progression. So we determine the playthruability of a game by observing its playthrough rate in a sample of players. And a game (or a level-progression) is playthruable if its playthruability is close enough to 100% for some purpose.

6.6: Playthruability 2 - Structure #

Now we can explain how a computer-game with a reliable level-progression works:

• The software and hardware produce a sequence of environments for the player to inhabit. [B.5] The levels are little sub-games within the greater super-game.
• Each level embodies an ecological attitude for the player to habituate to. A single level can be very effective in reproducing the practical knowledge required to play itself. [3.15]
• The game is designed to be fun to play, so that a player is naturally inclined to take on the correct ecological attitude.
• The player’s habituation is ensured by constraining progress until the player completes the goal of the level.
• The levels progressively prepare the player for future levels. A reliable level-progression will only have levels which success in past levels implies a player can complete.

This is the general structure of a playthruable game. The effect of playing through such a game is to become an expert in the tasks and gameplay of its levels—expert enough to complete the game. The expertise is practical knowledge, and the practical knowledge which this game produces is specified by its levels in combination. [3.4]

6.7: The optimal method 2 - Conclusion #

The efficiency of a playthruable game to produce its practical knowledge is maximizable by varying the mechanics of each level individually and the arrangement of its levels in sequence. So because it can be both optimally reliable and optimally efficient, a playable and playthruable computer-game can optimize the production of practical knowledge.

Such a computer-game optimally produces the practical knowledge of playing itself, but if it also simulates the situations of practical knowledge, then it will optimally reproduce that practical knowledge. We must therefore conclude that the optimal method of reproducing some practical knowledge is to play a playable and playthruable computer-game that simulates the situations of that practical knowledge. [C.8]

6.8: The optimal method 3 - Design #

In summary, the general method of designing a game to optimally reproduce practical knowledge is thus:

1. Design a set of final levels B to specify the practical knowledge you wish to reproduce. These should be simulations of situations in which you value expertise. [4.8]
2. Design a set of beginning levels A to be maximally playable by any beginning player. [2.3] These should approximate the simulations within the constraint of playability, and are best mechanized. [2.2]
3. Design a playthruable level-progression of intermediate levels C connecting levels A to levels B.
4. Constrain progress through the intermediate levels C by requiring the player to complete necessary tasks to advance.
5. Optimize the efficiency of the progression within the constraint of playthruability, such as by cutting out unnecessary levels and streamlining others.

Super Practica as outlined so far is close to being such a game, but how can we design a playthruable level-progression for it?

6.9: How to design a playthruable progression #

Being able to test for playthruability affords experimentation in design. Given some first and last levels A and B, we could design many intermediate levels that seem like they would be effective for playthruability, sequence them together, and see what happens. But if we find players getting stuck on a level it will only tell us that the whole game is unplaythruable, not what is wrong with its progression or how to fix it.—It may be best to make a small change or it may be best to throw the design away and start over. And good playthruability tests will not be especially quick or easy to conduct.

So like for designing simulations [4.6], we should only use playthruability tests to keep us on the right track, and we must rely on theory and good judgment to make our designs. Designing a reliable level-progression between some first and last levels again requires solving a set of theoretical questions and design problems. These are problems of what levels to make and how to sequence them together.

It will help to get a store of generally useful methods for solving these problems. I have collected and generalized many such methods from existing games, along with some of my own inventions.

6.10: Quantitative intermediation #

A general method for solving the problem of the player getting stuck on a level is intermediation: If the player has completed a level X but gets stuck on level X+1, then we can try adding an intermediate level between the two that is also intermediate in its design.

A simple case of this is quantitative intermediation. For example, suppose that a level X has 2 enemies and level X+1 which the player gets stuck on has 4 enemies, with no other difference. Then a potential solution is to make a new intermediate level in between them that has 3 enemies. Similarly in mathematics, if the player gets stuck doing a task in which a number is 1000, but has completed a similar task with the only difference that the number is 10, then we can try making a new task in between them in which the number is 100.

6.11: Elemental intermediation 1 - Isolation #

But quantitative intermediation does not work when successive levels have a qualitative rather than a quantitative difference. A method of intermediation that better prepares the player for qualitative differences is elemental isolation, which is to isolate elements of a difficult level into distinct levels.

If we can decompose a level X+1 which the player gets stuck on into the two distinct elements A and B, then we can isolate these elements in intermediate levels before their synthesis in level X+1. The level-progression would thus be:

X → A → B → X+1 (which is A&B)

For example, consider one of the many games that have levels which consist of a stage (A) followed by a boss battle (B), such as Mega Man. For such games, it can be a good idea to add a checkpoint before the boss or to make the boss battle a separate, standalone level. These solutions decrease the likelihood that a player would get stuck and be unable to defeat the boss.—And we find that a player who has completed these levels A and B separately is then more likely to complete both levels in succession and without retrying if they play the game through again. [C.1] This design therefore increases the playthruability from playing the game as a beginner to completing the game as an expert—without retries.

(Stage) → (Boss battle) → (Stage)&(Boss battle)

See the game VVVVVV for an effective and systematic use of checkpoints to prepare the player for completing the game without relying on checkpoints. This goal of expert completion is explicitly suggested in the game.

6.12: Elemental intermediation 2 - Gradual synthesis #

Elemental isolation works for any number of distinct elements. For example, with a level X+1 having 4 elements:

X → A → B → A&B → C → A&C → B&C → A&B&C → D → A&D → B&D → C&D → A&B&D → A&C&D → B&C&D → X+1 (which is A&B&C&D)

But playing through all of these intermediate levels can be tedious and inefficient, so we should not design formulaic expansions like this.

A level X or its previous levels may contain elements of level X+1 which we can assume the player is already familiar with, so we should only decompose and isolate the novel elements of level X+1. And the value of isolating elements is to prevent getting stuck, so we can skip isolating elements that do not present any difficulty. And even for difficult and novel elements, it might be sufficient to gradually synthesize them without first isolating them, like so:

X (A&B) → A&B&C → X+1 (A&B&C&D)

6.13: Elemental intermediation 3 - Variation #

Analyzing a stage and its boss battle as distinct elements is intuitive because they are distinct in chronological order, but other elemental analyses can be valuable too. We can analyze the same game in many different ways, which suggest many different intermediate levels to design.

For example, the boss battle that has been newly isolated in its own level can be further split into a few boss battles that isolate the different attacks that the boss has, or that isolate different features of the environment of the boss, or that gradually increase the speed of what the boss does. These follow from analyses which keep other elements constant and vary “attacks”, “environmental features”, and “speed”.

6.14: Funnel constraint #

So novelty can be decomposed into elements and isolated, but what if these decomposed elements themselves make the player get stuck? This often happens in the initial levels of a game where the player-control mechanics and the level-completion mechanics must be introduced together. Modeling and suggestive signals [3.10–11] can help, but I believe the most efficient solution to this problem is the funnel constraint. [B.2]

To apply a funnel constraint to a level, design the level to limit both the possibility of retrogression and the player’s possible actions. When done right, the effect is to make even completely lost and confused players inevitably complete the level, so long as they continue trying things out.—Imagine the player as doing random actions. Because retrogression is blocked, every action takes them either around the funnel or deeper into the funnel. Because their possible actions are limited, it does not take long for them to do a progressing action deeper into the funnel. The player will therefore before long progress through the funnel to its narrow exit and complete the level.

What does the player gain by this experience?—They see themself become their own model, or by another view, the funnel guides their trial-and-error attempts at completing the level to be maximally efficient. After the funnel constraint is lifted, the player will only need to do what they have done before to complete the level again.

For example, take the first level of Wario Land 4, the Hall of Hieroglyphs, which divides the level into sections that make retrogression to each previous one impossible. The player is therefore more likely to progress to the end instead of getting stuck wandering around. A more constraining example is the first tutorial levels of Opus Magnum. These highly constrain the player’s options relative to later levels.

6.15: Branching progression #

A common design in computer-games is to arrange the game’s levels or challenges in a non-linear order. This is like a tree that branches off into multiple paths, but then the paths rejoin together. The purpose of this design is mostly to keep the player from getting bored.—It lets a player bored with one level along the progression play another level instead.

But there is another benefit to this design: We find that a player who was stuck on one level sometimes gets immediately unstuck when they come back to it after playing levels in a different branch. [C.1] There are two explanations for this:

1. If it is a puzzle, then the player has time to think of how to solve the puzzle while away from the place of its solution. This can also be true of games which are not identified as puzzles, because many tasks are made easier by having a better plan for their completion.

2. The player gains relevant expertise in a level of one branch that is applicable to the level in the other branch. If the direction of preparation between levels goes the other way as well, then we may say that the levels are complementary to each other. The player is then likely to learn more efficiently by choosing to do first whichever they prefer. [C.6]

It follows from these explanations that splitting a linear progression into multiple branches can make it more efficient and playthruable.

Measuring the playthruability of branching progressions will be more complicated, but it can be done. To be playthruable, a branching progression should ensure that either a player could not get stuck on any branch, or that a player who gets stuck will get unstuck by playing through other, suggested branches.

6.16: Introducing Super Practica C #

The design of Super Practica follows the method of section [6.8] to design a game to optimally reproduce practical knowledge:

• Super Practica A is the set of easily playable, beginning levels which are mechanized approximations of the uses of mathematics. [2]
• Super Practica B is the set of final levels which are simulations of the situations of expert mathematical practice. [5]
• Super Practica C is the playthruable progression of intermediate levels which connect the levels of Super Practica A to those of Super Practica B.

When these three pieces of Super Practica A, B, and C are combined together, they make a game that holds unfathomable power: Super Practica.—A game that anyone can play through from beginning to end without difficulty, and completing which guarantees expertise.—The optimal method of reproducing mathematical knowledge.—But why stop at mathematical knowledge?—It could be a game that you can play to get good at anything and everything most efficiently, reliably, and easily.—The ultimate game!

But I’m getting ahead of myself. Is Super Practica C even possible to make?—The only way to determine its possibility is to set out to make it and try solving the design problems involved.

7: Design of a Reliable Level-Progression for Mathematics #

7.1: Progress in many dimensions #

The practical structure of mathematics is formed by the necessity of knowing how to do some mathematics in order to do other mathematics. This structure primarily consists of the verificational structure of mathematics [1.10], but also includes every other aspect of mathematical expertise, which develops along many more dimensions. [6.3]

The design of Super Practica C—of a reliable level-progression for mathematics—should reflect this practical structure, and so should progress along many dimensions. The progression from one mathematical topic or sub-topic to another, such as from multiplication to exponentiation, is only the first of these dimensions.

7.2: Level-selection map #

It will be best to let the player survey the map of their own progress. This will give them greater control over how they pace their play, and it will make it easier to recall the levels they completed.

For this, we can repurpose the design of level-selection that computer-games such as Donkey Kong Country have perfected for the purpose of evoking adventure. That is a layered map with multiple macroscopic and microscopic views, and which connects levels as nodes to each other in each view. Many different dimensions of progress can thus fit to the different connections between nodes at different scopes on such a map.

Let this design be a first approximation: At the outermost scope is a selection of major topics, such as arithmetic, and these progress to other major topics. In the scope of arithmetic is a selection of minor topics, such as fractions, and these progress to other minor topics. In the scope of fractions are level-sets that progress to other level-sets. In each level-set are levels that progress to other levels. And in each level is a selection of level-variants that progress to other level-variants.

Each scope should have branching paths because their levels and level-variants are often puzzles and are often complementary to each other. [6.15] This is unlike educational courses which follow a curriculum. A player finds the most value in a game when they can play it freely and according to their own interests [C.6], so we should let the player follow whichever branching path they like.

7.3: Decreasing guidance 1 - Soft constraint #

Much of the playability of Super Practica A will be achieved by having many constraints [2.2] [6.14] [B.2] and suggestions. [3.9–11] But the simulations of Super Practica B require having comparatively few of these. [4.7] This gives us a general design problem of removing constraints over the course of Super Practica C. A second dimension of level-progression should therefore be for the player to complete mathematical tasks with less and less guidance.

For an example of the problem, suppose a level X lets the player drag the number 5 to the number 6 to generate the number 30. Somehow, the player should advance to completing the level Y where they must select the specific number 30 out of any number at all, given the numbers 5 and 6. The point is to eventually require the player to perform the multiplication manually—without constraint.

A generally effective solution is to add a special intermediate level between levels X and Y: If you want to make a playthruable progression from a level X to a level Y which are completed by performing the same task, and level X has additional constraints on level Y, then it will generally help to make an intermediate level U with undo-capabilities, the mechanics of level Y, and rejection signals in place of the constraints of level X. [3.10] [6.5] [6.13]

Rejection signals communicate to the player their mistakes, so upon seeing a rejection signal, the player can immediately undo their mistake. The result is that rather than being stopped from doing something as in level X, the player can do it as in level Y, but only has to follow suggestive signals to perform the task like in level X.

I call level X the hard-constrained level, the intermediate level U the soft-constrained level, and level Y the manual level. These terms are relative to what they constrain and afford the player to do, such as completing a task.

The progression along these levels is to replace constraints with suggestions and then to remove these suggestions. When effective, the player will find the pattern of the suggestions and will therefore gain the ability to constrain themselves to performing the task correctly. [E.5]

7.4: Decreasing guidance 2 - Full progression #

The progression from hard-constrained levels to manual levels fits into a larger progression of introducing a task and guiding the player’s practice in it. These other parts of the progression might be valuable or skipped according to the particular task:

• Modeling is often valuable [3.11], so let there first be a demonstration of the task.
• Funnel constraint is often valuable [6.14], so let funnel constraints precede regular hard constraints.
• After the player can reliably perform a task manually, its performance becomes no longer valuable to test. So let it afterwards be automated, as my design of mechanizing verification [2.13] calls for.

This second dimension of level-progression therefore runs from end to end by varying a task in these ways:

1. The task is automated (to give a demonstration).
2. The task is funnel-constrained.
3. The task is hard-constrained (without funnel constraints).
4. The task is soft-constrained.
5. The task is manual.
6. The task is automated (to animate verification).

7.5: Exploring variations #

A third dimension of level-progression is the exploration of the different variations of a task.

The same task may have different variations. For example, a multiplication problem may have as its initial picture 4 × 3, or 42 × 7, or 13 × 79, or 525 × 982, or 2733 × 7954, or 0 × 4563, or −852 × −32. While these may all be solved by applying the same algorithm, that the player can solve one variation does not necessarily imply that a player could solve the others.

For a player to get good at completing a task, they must complete it reliably in all its variations. But what constitutes a variation that is worth presenting to the player as a level-variant to play and complete?—We cannot practically present every combination of numbers. We should rather identify every difference that can confuse a player to no longer complete the task, and then we should specify level-variants according to these differences.

7.6: Increasing speed #

A fourth dimension of level-progression is for the player to perform the same task more quickly. This progression is to give the player more and more exacting time constraints. [5.8] [E.6]

Progress along these second, third, and fourth dimensions constitutes getting good at an individual task. These dimensions map to the microscopic relations between levels and between level-variants. Note that there’s no progression for increasing accuracy. [E.7]

7.7: Composing tasks #

After a player gets good at a task they can move on to getting good at another task. A solution method for one problem is typically made of smaller solution methods for smaller problems [1.12], so a fifth dimension of level-progression is the composition of tasks.

The levels should be arranged to put more basic tasks prior to more complex tasks which combine the basic tasks together. The subtasks of a task should first be isolated in their own levels before being synthesized together in a level for the composite task. [6.11]

7.8: Climbing the verificational structure #

A sixth dimension of level-progression is to move up the verificational structure. [1.10] Let the progression run thus:

1. The player solves a problem, and it is verified using a foundational verification method.
2. The player solves a problem, and it is verified using a verification method that the player has used before as a solution method.

In other words, let the solution methods which the player has successfully applied be reused as automatic verification methods later. [2.13] Because no verification method will be used before the player acquires expertise in applying it as a solution method, the player will not be alienated from the verification of their own solutions as criticized in section [1.9]. [E.4]

7.9: Symbolizing pictures 1 - Relation between symbols and pictures #

Throughout mathematics we find interchangeable pictures and symbols. For example, the fraction 1/4 carries the same information as a picture of a pie with 1 region shaded out of 4 equally sized regions. To use pictures is more intuitive and beginner-friendly, but to use symbols is more efficient and flexible. There should therefore be a progression from pictures to symbols. This is a seventh dimension of level-progression.

Consider the task of determining the equality or inequality of the fractions 1/2 and 3/6. It can be completed using this pictorial method:

1. Draw two pictures of circles.
2. Draw lines and shade regions on the circles to shade them in the same ratio as the fractions.
3. Determine that the fractions are equal if the shaded area is the same in both circles.

The same task can also be completed using this symbolic method:

1. Multiply or divide the fractions without changing their value to make them have the same denominator.
2. Determine that the fractions are equal if both their numerators are then equal.

So what precisely is the correspondence between fractions and divided circles that we see here?—Their correspondence consists in them both being functional for solving the same problems! This is the general relation between corresponding symbols and pictures. [A.1]

7.10: Symbolizing pictures 2 - Progression #

Because they can solve the same problems, it follows that symbol-manipulating methods may be verified by means of corresponding picture-manipulating methods.

But it is common that pictorial methods are impractical for verifying tasks solved by symbolic methods.—Comparing 1/2 and 3/6 can be effectively verified by drawing a picture of pies with shaded regions, but comparing 1583/58926 and 11081/412482 cannot be. In general, complicated-enough variations of a task are not solvable by pictorial methods that can solve simpler variations of the same task. So how can these more complicated variations be verified?—By other symbolic methods! For example, multiplication can be verified by addition, as shown in section [1.10].

So let the progression run thus:

1. A task must be solved by applying a pictorial method. It is verified by a foundational verification.
2. The task must be solved by applying a symbolic method. It is verified by the previously used pictorial method.
3. A more complicated variation of the task must be solved by applying the same or a similar symbolic method. It is verified by a simpler symbolic method.

7.11: Collecting methods #

Every mathematical problem has a set of methods that are functional to its solution. [1.12] As the player completes more mathematical tasks by more solution methods, it becomes a new problem to recognize which solution method is functional to complete which task. Dealing with this increase of game-plans [3.13] to choose from is an eighth dimension of level-progression.

The design of solution templates in section [2.12] mechanizes setting up a problem according to a plan. We should first introduce a solution template as being suitable for solving a particular problem, before tasking the player with finding it as the right solution template for a problem. So let the player have a collection of solution templates they can select from to set up their solution. This will be a “solution toolbox” that has more solution templates added to it as more are introduced.

This design of an expanding toolbox might also be valuable for a toolbox of verifications [E.4], and for a toolbox of theorems in the game of axiomatic proof. [5.17]

The player’s progress along these fifth, sixth, seventh, and eighth dimensions is reflected in the player’s ability to complete more mathematical tasks. These dimensions map to the mesoscopic relations between levels and between level-sets.

7.12: Progression to simulations 1 - Direction #

The ultimate end of the progressions of Super Practica C is to prepare the player for the simulations of Super Practica B. This is a ninth, most macroscopic dimension of level-progression.

For a simple design, let proofs [5.15–17], problem-domain applications [5.4–7], and specially constrained practices [5.8–9] have their own level-selection maps, distinct from the one I have outlined for the solution-domain in this chapter. (Though in section [7.6] I called calculation speed a dimension of progression in the solution-domain map. Both of these designs will be functional.)

In each case of simulation, the progression should be for the player to complete levels in the topical, solution-domain map of Super Practica C before advancing to simulational levels in the maps of Super Practica B:

• We should let the player first get good at doing something reliably before tasking them to do it quickly, or with other special constraints.
• Applying mathematics requires using mathematics, such as in the math mode [5.5], so the player should first use it outside the context of application. [E.8]
• Proofs require connection with other mathematical practices to function [5.18], so the progression should go from using mathematical methods to proving the reliability of mathematical methods. This will also be more efficient to introduce things than will always requiring the player to understand a proof before using its conclusions.

7.13: Progression to simulations 2 - Unlocking #

It would not be a good idea to drop the player into a problem-domain without ensuring that the player is able to complete at least most of its problems. (See the open-world puzzle game The Witness for an example of this inadvisable design and its consequence of unplaythruability.) But it would be worse if each problem-domain level were unlocked immediately upon the player’s completion of its requisite solution-domain level. This would ruin the simulation by giving the player an unnatural hint as to how to solve the problem. [4.7]

The best design is therefore likely to be thus: First, let each problem-domain be consolidated in a unified level-set. Then let such a problem-domain level-set be “unlocked”—newly suggested—after the player completes all the solution-domain topics or sub-topics which are necessary for its completion. Consolidating the relations between solution and problem-domains this way should minimize both undesired hinting and the frustration of getting lost.

For the same reasons, the axiomatic systems in the simulations of proof should likewise be contained in unified level-sets that are unlocked in the same way.

7.14: Continuous revision #

Other topics than arithmetic will come with their own dimensions which I haven’t outlined, such as the construction of progressively more ridiculous geometrical figures. Even within arithmetic, I expect that each sub-topic will come with unique details that will necessitate an individualized design, so that no template can be followed to solve all the design problems involved. The system of dimensions given here is only an outline for beginning to construct a reliable level-progression for mathematics.

Such a project as this has never been done before, so I expect that I have missed some important dimensions [E.3] and that I’m mistaken on some things. The measure of playthruability [6.5] is the ultimate means to confirm or reject the reliability of any proposed progression. In response to measuring insufficient playthruability we can criticize, redesign, and improve the progression. This continuous revision makes the design of Super Practica a process that corrects its own mistakes.

8: From Arithmetic to the Moon #

8.1: The empirical framework #

It is not wishful thinking to believe that this, the ultimate game can exist, but only empiricism on the basis of materialist philosophy. If we eliminate all the magical and mystical attributions made to mathematics and to knowledge of it, what is left to observe is a bunch of games that the player gets better at by playing. Super Practica is a project to optimally redesign these games. [6.8]

That Super Practica in a completed form will be the optimal method of reproducing mathematical knowledge follows from its empirical framework:

• The success of Super Practica A is confirmed by playability tests. [2.3]
• The success of Super Practica B is confirmed by natural-situational transferability tests. [4.5]
• The success of Super Practica C is confirmed by playthruability and efficiency tests. [6.5] [6.7]

If these values can be measured, then they can be optimized. By continuously improving its design, Super Practica will tend to optimality in the long run. [7.14]

8.2: Empirical design problems #

The empirical framework makes the design problems of Super Practica into empirical design problems. I have noted the process of solving empirical design problems across this blueprint:

• We first make an initial design to be a good-enough solution that accommodates each requirement with the rest. [2.14]
• We confirm our solutions by measuring the values which prove their effectiveness. [4.1]
• Once we have an initial working design, we can optimize it by proposing new designs and confirming that they are improvements by measurement. [4.1]
• Design problems will not necessarily be solvable. [2.14]
• Each prospective feature—each new topic—comes with a new set of design problems. [2.14]
• These sets of design problems will contain outstanding design problems which are especially valuable but also especially difficult to solve, and which therefore demand the greatest attention. [5.18]
• General outlines of solution can be given for design problems that we expect to repeat across topics, but each particular solution should be individualized to its topic. [7.14]

8.3: Growth and coverage #

The designs I have given in this blueprint are intended to function as an outline for a first, good-enough design for arithmetic. This is to make the game playable enough, mechanized correctly enough [2.2], essentially accurate enough, playthruable enough, and efficient enough.

From here, we can fill in the details of the design for arithmetic and we can progress beyond arithmetic. This will require solving a set of empirical design problems corresponding to each topic.—Super Practica will grow and cover more topics by finding solutions to the empirical design problems of each topic.

This reconceptualization in terms of its design problems lets us look ahead and foresee what Super Practica might look like in an approximately complete form.

8.4: Begin with arithmetic #

Super Practica must begin with arithmetic because it is nearest to the foundational verifications [1.10] and is itself the foundation for most other mathematics. It is at the root of the practical structure of mathematics. [7.1]

Arithmetic is founded on counting, but we cannot begin with counting. Counting is a physical, bodily process. In learning to count you learn to speak the number-words, to raise and lower your fingers, to point at what you count, and to follow a rhythm. [A.1] Such basic bodily actions cannot be simulated by input to a computer via mouse, keyboard, or controller. [5.9] The ability to count and the ability to use a computer must therefore be the necessary prior knowledge to begin playing Super Practica.

Level 1—the first sub-topic—of Super Practica is hence to verify counts. From there we can move on to addition and the rest of arithmetic. This includes topics outside of common curricula, such as numeral systems, geometric numbers, and many more that go under the name of “number theory”.

8.5: Growth along branches #

The major topics following arithmetic are algebra and geometry. Algebra is, like arithmetic again, the foundation of most other mathematics. Its practices are ubiquitous. For example, while propositional logic does not deal with numbers, it does deal with variables, constants, equalities, and so on.

The practices of arithmetic, algebra, and geometry are reused over and over again in more advanced mathematics.—The tree of mathematical practice is a tree of many branches which spring from these topics. The branches of the macroscopic level-progression of Super Practica will reflect the branches of this tree. [7.2]

Continued development along a branch will require solutions to design problems on that branch. There will thus emerge an expanding frontier of unsolved design problems, the solution of which will extend Super Practica to cover more of mathematics. When we reach an apparently unsolvable problem, Super Practica would stop growing along that branch, but could still grow along its other branches.

8.6: Holes in coverage #

What would happen if the outstanding design problems of some topic are unsolvable? That is, how would Super Practica look like if even initially good-enough solutions could not be found?—This would make a hole in the coverage of Super Practica that would have to be filled in by less effective methods.

I can only guess how many holes Super Practica will have. Utter failure would mean that Super Practica must necessarily be full of holes, but my experience of designing it so far tells me that its design problems are generally solvable, even when they are difficult. [E.1] If my estimation is correct, then the completed form of Super Practica will have a theoretically ideal structure in most of its parts, yet would unavoidably have some substantial holes. I expect that the theoretical ideal is not quite possible but can be approximated closely.

8.7: Unstoppable growth #

Being at the trunk, the topics of arithmetic, algebra, and geometry are vital. If their design problems can be solved, it will open up the rest of mathematics for inclusion into Super Practica. If these can be covered by Super Practica without substantial holes, then the tree of Super Practica will spring into so many branches that only a very high rate of holes in these branches could stop its growth.—And the problems of the trunk being cleared would imply that there would not be such a high rate of holes in its branches.

In other words, if Arithmetic, Algebra, and Geometry can be covered by Super Practica, then the large majority of mathematics can very probably be covered by it as well.

8.8: The value of unification #

But why develop this as one computer-game—Super Practica—instead of many distinct computer-games—Super Arithmetic, Super Algebra, and so on—that follow the same empirical framework?

The design of a playthruable progression for mathematics requires the player to play through many intermediate levels. [6.10–13] Because of the great quantity of these intermediate levels, it is essential to the efficiency of a playthruable progression that the time a player has to spend learning to use new interfaces in each level should be minimized. This can be done by reusing interfaces across topics and optimizing their playability.

And the composite practical structure of mathematics [7.7–8] ensures that topics cannot be self-contained.—Super Algebra would require interfaces and mechanics to perform arithmetic, and so would the whole tree of mathematics that grows from the root of arithmetic. Hence, to maximize the reusability of the game’s interfaces, all or most mathematical topics should be unified with arithmetic in one computer-game.

This is not the only effective organization we could consider, but it is a simple and good-enough one. So for now let’s proceed with developing the project of Super Practica as a single, unified computer-game.

8.9: Beyond mathematics 1 - Suitability #

But why stop at mathematics? Many empirical sciences use arithmetic and algebra in some way, so it again makes sense to unify them in the same computer-game. We can apply the optimal method of reproducing practical knowledge [6.8] to reproduce the practical knowledge of any topic. So let’s consider what Super Practica can cover beyond mathematics and what this might involve.

We would have to do for each topic what I have done in this blueprint:

1. Analyze the practices of the topic as a game or as many games. [1]
2. Redesign the game to be mechanized and playable. [2]
3. Identify and simulate the natural situations where expert practice of the topic is valuable. [5]
4. Design a playthruable progression that connects the mechanizations to the simulations. [7]

This method is applicable to topics which have practices and situations of application that are already codified [B.3] and game-like.—The most suitable topics are those which consist of practices with clear rules, tasks, and applications. Mathematics most especially fits these criteria, and other topics fit to a lesser extent.

8.10: Beyond mathematics 2 - Topics #

I grade the suitability of various topics for inclusion in Super Practica thus:

• Mathematical-empirical sciences such as physics and chemistry are very similar to mathematics and are hence very promising topics. But a computer-game must be limited to reproducing the expertise of applying readymade theories to predict what happens in readymade simulations. This is a valuable skill, but it cannot replace truly empirical observation and experiment. [C.5]

• Highly codified topics such as grammar, linguistics, and geography are only somewhat promising. The problem with these is that it is hard to find natural situations of their application that can make for good simulations.

• Some topics such as music and drawing have the opposite problem: They can have clear tasks and applications, but their practice is more freeform and uncodified. These are promising to the extent that they can be relatively codified, such as in the task of transcribing music by sequencing notes.

• Other topics seem to not consist of practical knowledge at all, such as history. There is no reason for optimism about these, but a thorough analysis might yield surprisingly fruitful results.

• Job-specific practices and the various topics of technical tutorials may appear to be promising for inclusion in Super Practica, but this is likely to be overvalued.—The best solution to the problem of needing expertise is in general a usable technology that obviates the need for expertise. Perennial topics such as mathematics are an exception only because their practices have a wide variety of useful and unpredictable applications.

8.11: Summary #

To summarize, Super Practica could probably cover most of mathematics, and its level-progression would look like one unified tree with arithmetic at its root. Beyond mathematics we cannot expect so much coverage, but we can look forward to a substantial patchwork of more limited and disconnected trees where this optimal method is applicable, some of which will connect back to the tree of mathematics.

If we treat problems of human learning as empirical design problems of mass-producing optimally effective environments, then we can envision a computer-driven system of progressively arranged games that will constitute a revolution in humanity’s ability to efficiently and reliably reproduce practical knowledge. This is Super Practica.—The ultimate game!

8.12: To the moon? #

I have tried to look ahead, but I can only speculate. The best way to discover how high this tree (Super Practica) can grow (reproduce practical knowledge) will be to water (develop) it until it reaches its limits (unsolvable design problems). Starting with arithmetic, it might grow as far as the moon!

9: Let’s Develop Super Practica #

9.1: Open-source development 1 #

These are the conclusions I have come to upon asking the question “What would be the optimal method to learn mathematics?” and then answering it. It was clear to me that the answer would be to make a game, and I wanted to design this game. But after having spent so long researching the problem and making an outline of the game’s design, I find that it’s a large enough project that I can’t possibly make it alone. It requires funding and collaborative effort.

But common methods of funding game development are antithetical to the principles with which I have designed Super Practica. If I took money from investors, then I would have to make generating profits—not reproducing practical knowledge—the first priority. If I tried to build a business on the basis of copyright restrictions, then it would disempower rather than empower the game’s players and therefore make a worse game. [B.8] These would not be optimal methods and would ruin what I have taken so much care and trouble to get right.

It therefore seems that Super Practica must be an open-source, free software project to have any chance of being completed without compromises. A copyleft license will be effective to better compete with freedom-restricting imitators, so I distribute the code of Super Practica under the copyleft AGPLv3 license. [E.9]

9.2: Open-source development 2 #

You can find the project of Super Practica, its newest release, its code, and instructions for contribution on the website: [superpractica.org]

Super Practica will require contributions in code, funds, and other things that you can read about on the website. If you are interested and able, then please consider working together with me to make this theoretically optimal method of learning mathematics a reality.

9.3: Proposition for funding 1 #

People already spend a lot of money to buy various inferior and abusive games and educational programs, so the case for me to make is only that people should act rationally and fund Super Practica instead.

Funding development of free and open-source games [E.9] gives you many great benefits that you wouldn’t receive from paying for proprietary games:

1. The developer can’t restrict you from playing the game however you like for any reason.
2. The developer can’t sneak anything malicious into the code.
3. You can study the game’s code to learn more about the game or to learn how to better program your own games.
4. You can modify the game’s code to better fit your preferences. Even non-programmers can achieve a lot by changing a few numbers in the code.
5. You can make as many copies of the game as you want, which you can share with whomever you want.
6. Because it’s so easy to make copies, you can be sure that you won’t be locked out of playing the game. Compare this to games that you can only play as long as a record of your payment can be found, or as long as the company that developed the game is still in business.
7. You don’t have to bother with annoying and unreasonable Terms of Service agreements.
8. Developers must prioritize the needs of funders to ensure continued funding. That means your priorities become the developer’s priorities.
9. The developer doesn’t have to rely on means of funding and profit that are antagonistic to you. [B.8]
10. Because the money doesn’t go into marketing or into giving large payouts to investors and executives, the money is more efficiently used to develop the game.
11. Everyone can play it, so the game is more beneficial to society.
12. The game can be developed continuously by anyone, instead of being unimprovable when the legally sanctioned developer stops developing it for any reason. This makes funding the game before its completion less risky.
13. If you disagree with how the game is being developed, then you can pay other developers to continue development along a different course.

With all these great benefits, it should be clear how irrational it is to pay for proprietary games when you can pay for “free” games instead. To fund free and open-source game development is to pay less for more and better games.

9.4: Proposition for funding 2 #

A game designer is responsible to whoever funds the game, and if you want to design a good game, then the best people to be responsible to are your players. So I especially ask for funding from prospective players.

But the first topic of Super Practica will be arithmetic, and arithmetic is taught mostly to children, and children aren’t a good source of funding. It will therefore be a priority for Super Practica to more quickly cover topics that aren’t commonly taught in schools, so that prospective players can begin to benefit from funding it as soon as possible. There are many interesting and little-known sub-topics of arithmetic that will be valuable for this, including those mentioned in section [8.4].

9.5: Collaborative design 1 - Logical method #

So far this project has been my own invention, but for Super Practica to grow and fulfill its promise it must become more than just mine. I have designed it from the start using a logical method, and this method is well-suited to collaboration among designers. My method has been thus:

1. Identify a design problem. [8.2]
2. Sketch out some designs to attempt to solve the design problem.
3. Make arguments for why some proposed solution will fail, why some solution is a good-enough solution, or why some solution is the best possible solution.
4. Develop and iteratively improve upon the designs. Also criticize and iteratively improve upon the arguments.

Super Practica is thus far a product of the many sketches of designs I have made and of the many arguments I have made about them to myself. I now invite other designers to sketch out designs with me and argue about them with me, using this logical method.

9.6: Collaborative design 2 - Iteration #

Testing according to the empirical framework [8.1] will drive the iterations of Super Practica.—And it is iteration, not validation, that we should seek.

Many designers seek to systematically validate their designs, meaning that they treat the value of a design as being suspect if no tests confirm it. But this can be confused.—Validation can be good practice in cases such as when safety is a concern, but it is generally not worth the expense. For us, it will not be worth the expense of greatly slowing down iteration.

The value of the empirical framework is to have a method of testing to answer challenges and questions that could be made about Super Practica’s design, not to prove that every part of it works or could not be improved. This enables iterative improvement that tends to optimization in the long run.—The point is not to prove present optimality but to iterate toward future optimality. This is a fallibilist understanding of empiricism.

9.7: Collaborative design 3 - Scientific method #

While many tests could be devised for Super Practica according to its empirical framework, only relatively few, high-quality tests should actually be run. It is a common confusion of empiricism to collect a lot of data and then to try to answer every question that seems answerable by it. We should rather collect the precise data we need in order to answer the precise questions we need answers to.

The order of the scientific method is to first make a hypothesis, then to collect appropriate data, and then to decide on the truth of the hypothesis. Anything else enables common fallacies like cherry-picking, wishful thinking about the true relevance of the data to the question, and low-quality, inconclusive tests. (See for example, note [D.5].)

High-quality tests can be expensive to run, so the design of Super Practica should generally proceed according to theories. [4.6] [6.9] Then we can make the most of our tests by testing our theories on particular points which we have good reason to suspect are false. [E.2] The theory of practical knowledge should accordingly be revised in the course of developing Super Practica.

9.8: Collaborative design 4 - Shared premises #

Collaborative argument following the logical method of design will require a basis of shared premises for it to lead to consensus. The premises I have used in my arguments have originated in these different ways:

• Some premises I have systematized over time into my analyses of mathematics [1] [5] and my theory of practical knowledge. [3] [4] [6]
• Others have come from relevant scientific theories, such as on human emotion, navigation, and perception.
• Others are philosophical and design principles. I have either begun with them or have recognized their value in the course of developing my theories and designs.

We should begin with these premises—theories, analyses, and principles—which have made Super Practica into what it is now. So I will finish this blueprint by introducing some of the most important principles of Super Practica besides empiricism and fallibilism, which I have already introduced.

9.9: Design for the player #

The central principle is to design Super Practica for its players and for no one else.

A game that is effective for the purpose of mathematics education will naturally attract the attention of interested third parties including governments, businesses, and parents. But their objectives are not the player’s objectives, and therefore should not be our objectives. So grades, certificates, and curricula have no place in Super Practica.

It seems clear to me that there are inherent conflicts between efficiently reproducing practical knowledge and the interests of the parties which build and design school-bureaucracies. [7.2] [C.6] [D.5] But even if there were no conflicts, it is still best to design for a clear set of users and to fulfill a clear set of requirements, rather than to design for everyone and to do everything. Super Practica is designed for players who voluntarily decide to play it—whether for fun, for self-improvement, or for solving their mathematical problems—so we should not accept contributions or suggestions that do not clearly advance player interests.

The goals of Super Practica are distinct from the goals of educational institutions. To make this point strongly, I say that Super Practica is not an educational game. [E.10] But this does not imply that educators cannot use Super Practica as a resource.

9.10: Respect and affirm the player’s freedom #

It follows immediately from designing only for the player that we should value, respect, and affirm the player’s freedom in playing the game however they like. If players want to play the game in a way that some people call “cheating”, then they’re welcome to do so and we won’t develop any mechanical constraints against it. Likewise, the “unlocking” mechanism of level-progressions [6.8] should be overridable by players.

This principle informs the design of the game, but the problems in following it extend to the organization of the game’s development and distribution. This is why Super Practica must be free and open source.

9.11: Aim for optimal design #

The point of Super Practica is to find the optimal method to reproduce practical knowledge and then to develop it. The empirical framework makes this an optimization problem, so it is an essential principle that we should aim for optimal design—in other words, that we should not take shortcuts.

Super Practica calls for the design and development of a very large quantity of games following very high standards of quality. [6.16] Upon seeing this, the typical designer of a sub-optimal educational program would decide that the project is unviable for requiring too much time and labor to make, and so would start looking for shortcuts. [D.6] But if we decide that a difficult and time-consuming design will be optimal, then that is the design we should gradually work to develop. We should not cut corners or settle for sub-optimality to meet deadlines.

But in the process of optimal design, we should first make playable approximations to ideal designs. We can rely on iteration to eventually lead us to optimality, but we should prioritize goals and improvements to roughly approximate the ideal quickly. This will let players benefit from our work sooner, and it will help us notice the flaws in our designs sooner.

9.12: Design for humans universally #

According to my theory of practical knowledge and my analysis of mathematics, there is nothing in doing mathematics that is essentially different from playing games that consist of moving around, manipulating objects, and conversing with other people. It follows that anyone who can do these things and play such games can learn to play mathematics at an advanced level.—Humans have a universal ability to play mathematics, barring rare debilitations which extend beyond mathematical ability in their effects. So to develop the optimal method of reproducing mathematical knowledge, we should design for humans universally.

Instead of identifying players as having “learning” or other disabilities [3.7], we should assume that all apparent disabilities are only relative, ecological disabilities.—An observation of disability is as much an observation about the situation and the environment as about the person. So if someone who wants to play mathematics has unusual difficulty doing so, we should treat it as a solvable design problem until proven otherwise.

It follows from this principle that Super Practica should support multiple interface devices, and especially inexpensive devices, because lack of access to technology is itself an ecological disability. This can be done by first designing around the most limited devices, and then adapting other devices to this core interface. My designs in this blueprint accordingly assume nothing beyond a small screen with a single-button mouse.

9.13: Minimize frustration, do not maximize fun #

If someone isn’t interested in playing Super Practica, then they can choose to not play it. We have no intention to manipulate anyone into learning anything, hence there is no need for rewards [B.7], distracting effects, or other such mechanics. Neither should any part of the game be “dumbed down” or made inaccurate to keep players engaged. [2.1] Making mathematics more entertaining is beside the point of Super Practica.

Optimizing playability, playthruability, and efficiency implies systematically removing frustrations. [B.6] As Super Practica is developed, we can expect players to have fun playing mathematics, and this will be because mathematics is already fun to play.—It just currently happens to also be frustrating to play. Fun is good, and Super Practica should be fun, but its fun should derive from the fun of mathematics, not from unrelated additions to the game.

9.14: Materialism #

A core underlying philosophy of Super Practica is materialism. This of course implies atheism, which is the systematic rejection of all religion. Relevant to Super Practica is the impossibility of miracle learning, also known as revelation.

Ontological reduction is closely associated with materialism, and is often misunderstood and undervalued. It means to reduce some phenomena to a limited set of observable elements. This is necessary to construct an empirical theory, so calling it valuable is an understatement. Without a reduction, there would be too many things outside the theory that could explain away observations unexpected by the theory.—This would render the theory unfalsifiable.

The theory of practical knowledge is a product of reduction. The point is to reduce the reproduction of practical knowledge to its only cause and ontological basis. [3.4] [C.8]

9.15: Pragmatism/Contextualism #

Another core underlying philosophy of Super Practica is contextualism, which is more commonly known as pragmatism. My concept of essential accuracy [4.5], among other things, is derived from systematically applying the pragmatic method of argument:

If expertise in a simulation lets you do the same things as expertise in what it simulates, then they amount to the same knowledge and expertise. And if what you supposedly learned in a class doesn’t help you do anything, then it’s not real knowledge. Be pragmatic and identify knowledge with what you can do with it. [C.9]

The “context” in contextualism refers to the ontological unit that is assumed in applying the pragmatic method, which is the context of the person whom the argument addresses.—This context includes the person’s situation and their self-understanding of their own interests in the situation. Using “be pragmatic” as a premise in an argument implies argument from the context of someone’s problem to a contextually appropriate solution serving their interests.

9.16: Contextual materialism #

Combining contextualism with materialism makes contextual materialism, which is to say that a situation is as material as an organism or an atom. Always taking a neutral god’s eye view, as materialists have usually done, must have the effect of decontextualizing matter and confusing its relation to us. The existence of matter for us can only ever be the existence of matter with us and in relation to us. We can talk about matter-with-us using the publicly observable, context-specifying words “situation”, “practice”, “game”, and “gameplay”.

What I introduced as an apparent analogy of mathematics to a game in section [1.1] was really the reduction of mathematics to a game. The theory of practical knowledge and the design of Super Practica based on it are the fruits of this reduction.—They wouldn’t follow from a mere analogy.

Accordingly, the theory of practical knowledge is an “ecological” and not a psychological theory. [C.3] [D.3] [D.4] We speak of a person learning to play a game and not about what happens in the mind as a person learns to play a game.—What happens in the world is habituation and gameplay. [3.3] [4.2] In developing Super Practica, we seek to reproduce competent gameplay in games that accurately simulate valuable activities.—We do not seek to reproduce behaviors, cognition, or any other mind-stuff.

9.17: Conclusion #

In conclusion, Super Practica is too good not to make. This blueprint should hopefully give interested co-developers a basis for collaboration beyond trust in my personal vision, so now let’s develop Super Practica!

I call upon all interested programmers, good designers, gameplay scientists, freedom-loving people, math people, people who want to give money to a good cause but aren’t sure what cause to give it to, fans of intellectual progress, gamers, wannabe polymaths, parents who love their children, atheists, slackers, communists, “lifelong learners”, and whatever. Together, we can create the ultimate game, we can show them, we can make lousy mathematics textbooks a thing of the past, and we can grow a tree to the moon!

With patience and teamwork, the optimal method of reproducing practical knowledge can be realized. (Or something like that, probably.) [E.1] Because common methods of teaching mathematics tend closer to worthlessness than optimality, the benefit of this is so extreme that it’s difficult to imagine. The investment of time, effort, and resources will pay itself off by multiple times. [D.6]

Such a game can and will be made whether it’s called Super Practica or something else, whether with me or without me, and whether it respects player freedom [9.3] [9.10] or makes lots of money at the expense of its players. [9.1] [B.8] To help me make Super Practica in its most beneficent, optimal form, see again section [9.2].

A: Notes on Mathematics #

A.1: Empirical sources for my analysis of the structure of mathematics #

Linked from: [1.1], [1.4], [1.5], [1.8], [1.10], [5.2], [5.11], [5.13], [5.15], [7.9], [8.4], [A.3]

Most of my analysis of mathematics is founded on commonplace observations.—Anyone should be able to examine what goes on as they do mathematics, observe the same things as I do, and come to the same conclusions.

I also rely on empirical observations of what professional practitioners of mathematics do, such as engineers and mathematicians, and of how they fit into the rest of society. There are plenty of discussions on the internet that give this information in a non-systematic way. I have also found this information in books and articles variously categorized under “philosophy of mathematics”, “history of mathematics”, “anthropology”, “education of mathematics” using “problem-solving” methods, and other names.

For my understanding of what mathematicians do I have relied on the descriptive writing of especially George Polya and also Imre Lakatos. [A.2] I’m more personally acquainted with what engineers do, so I didn’t need similar guidance for that.

A.2: Theoretical influences on my analysis of the structure of mathematics #

Linked from: [1.1], [5.14], [A.1], [C.2]

My analysis of the structure of mathematics is formed mostly by my methods of game analysis. [B.1] Besides personal researches, some authors have helped me to understand the structure of mathematics.

• Ludwig Wittgenstein’s body of work (after his lousy first book) was the source of many inspirations. It inspired the basic idea of analyzing mathematics in terms of small, interconnected games. [1.7] [1.12] It also inspired my analysis of the verificational structure of mathematics. [1.10] Philosophical Investigations and Lectures on the Foundations of Mathematics are the most relevant books.

• George Polya supplied me with crucial insights, especially about the precise interface between mathematics and mathematical practitioners. The basic story of how the game of mathematics is played [1.2–3] [1.8] is adapted from Polya. My theory of the nature of insight [1.4] is a materialist reinterpretation of Polya’s explanations. My understanding of the nature of application [5.1–3] and of proof [5.10–13] also owe a lot to Polya. How to Solve It is the most relevant book.

• I have also found Imre Lakatos’s work helpful for understanding the nature of proof. The most relevant book is Proofs and Refutations. The dialogue in this book approaches what I have in mind for games that rationally reconstruct dialogue between mathematicians. [5.15]

Various other writing on the philosophy of mathematics has been helpful for me to engage with, but not to nearly the same extent as these works.

A.3: Corroboration of the social nature of proof #

Linked from: [5.13]

The history of mathematics corroborates the social nature of proof: The method of giving example-cases in Euclid’s Elements was later criticized for not being rigorous enough. It was argued that these examples succeeded in proving the theorems only for some cases and not for all cases. What was understood as sound mathematics—and what was sound enough for architects and other practitioners up to that time—was in the 19th century criticized by mathematicians as unrigorous and unsound. They then developed more rigorous axiomatic systems of geometry. [A.1]

There is no better place to locate the change in axiomatic systems and proofs at this time than in the social standard for persuasion in the community of mathematicians.

A.4: Proof that axiomatic proof is simulation #

Linked from: [5.14]

My claim that the practice of axiomatic proof is a simulation of mathematical argument can be verified. I define an essentially similar simulation by the ability of its expert players to move without difficulty to being expert in the same practices that were simulated. [4.5] So ask, observe, and verify whether students who acquire expertise in making proofs in axiomatic systems go on to easily use that expertise in argument with mathematicians.

I would value a thorough test of this being made since for me it’s only a conjecture, but I consider the conclusion to be predictable enough to design according to it without fear of wasted work.

A.5: Mathematical mysticism #

Linked from: [1.1], [5.10]

I criticize “mathematical mysticism”, but what exactly is it and where does it come from?—A typical case is to say that mathematics exists in some world other than the one we regularly perceive. It usually derives from Platonist and Aristotelian philosophies, and these have been very influential, especially in academic and educational settings. The same confusions continue in modern “analytic” and “continental” philosophy, which can be represented by the works of Bertrand Russell and Alain Badiou.

Mysticism is in general assertive, speculative, confused, and nonsensical speech. It is a pretense of knowledge to disguise ignorance, and so is informative of nothing but the mystic’s own feelings, prejudices, and desires.

Recognizing that mathematics is a human activity tends to make clear the fundamental ignorance of mystical doctrines about it. [A.6] But it would be a waste of time to refute every lousy theory and every lousy argument. Developing a productive program like Super Practica will tend to render mystical speculations irrelevant and ridiculous.

A.6: Reintroducing self-consciousness #

Linked from: [5.12], [A.5], [B.5]

One generally effective method of materialist criticism [9.14] is to find the source of mystification in the “lack of self-consciousness” of a person in his or her own activities.

When we do something very often, it becomes redundant to keep saying “I do X with Y to …,” so we tend to speak instead of Y without reference to ourselves or to X. This strips Y of its contextual relations in speech.—We simply say “The Y is …” The decontextualized object Y can therefore begin to seem conscious when it is inert and magical when it is mundane. For example, we say “Numbers exist that are greater than 100,” instead of “People can write more than 100 distinct numbers.” And we say “As x increases, y decreases,” instead of “As we substitute greater numbers for the variable x in the equation, we must substitute lesser numbers for the variable y to observe that the two expressions on either side of the equation will evaluate to the same number.”

Thus I make these formulaic materialist criticisms:

• The mystical theory of the reality of mathematical objects is consequent of the reality of the imagination of the person who is not self-conscious of being a person who can imagine things.
• The mystical theory of the universal truth of mathematical theorems is consequent of the reality of the certainty of the person who is not self-conscious of being a person who is persuaded to certainty.
• The mystical theory of __________ is consequent of the reality of __________ of the person who is not self-conscious of being a person who __________. (You can fill in the blanks to make your own materialist criticism!)

This method is helpful for rediscovering the human organism and human society in human activities, which are not commonly mentioned in language. Empirically speaking, this method is useful for generating likely hypotheses.

B: Notes on Games #

B.1: Methods of game analysis #

Linked from: [1.5], [A.2], [C.1], [C.2]

This blueprint is founded on applying my methods of game analysis to mathematics, in addition to more standard interaction design analysis. [B.2] But my methods are not yet systematic and I change the vocabulary I use with each analysis, so I apply them in only simplified and intuitive terms for this blueprint.

I hope to give a more complete account of my methods of analysis in the future.

B.2: Interaction design vocabulary #

Linked from: [2.2], [3.2], [4.7], [6.14], [7.3], [B.1]

• An affordance is the possibility of action that any object has for a person.
• A signifier is the sign by which a person can recognize this possibility.
• A constraint is the opposite impossibility of action, which is only worth pointing out when an affordance is rather expected.

The vocabulary of affordances, constraints, and signifiers is taken from the book The Design of Everyday Things by Donald Norman, specifically the 2013 edition. Norman developed these from James Gibson’s theory of affordances, as defined in The Ecological Approach to Visual Perception, but minimized their ecological significance. I haven’t been consistent in keeping [3.2] or dropping [2.6] the ecological significance in this blueprint.

I have expanded this vocabulary to include more types of constraints. [6.14] [7.3] The theory and name of the “funnel” constraint is derived from remarks by the game designer Zach Barth in some talk or other.

B.3: Gamification, mechanization, and specification #

Linked from: [2.1], [3.4], [5.14], [8.9], [C.8], [E.10]

The word “gamification” has generally been used to repackage the behaviorist theory of operant conditioning with new vocabulary for use in design outside of games, without this continuity being perceived. Game designers have often applied the theory of operant conditioning, so this is presumably the cause of the misunderstanding. [B.7] Given this history, I consider the term “gamification” to be hopelessly confused and never worth using.

I call making a computer-game out of an existing practice “mechanizing the rules” of the practice. [2.1] This is not “putting rules into code”.—The code is only ever a means to an end, which is the systematic manipulation of computer hardware. Mechanization is rather programming the computer to have mechanics such that interaction with it implies following the rules of the game. But the word “mechanization” makes this ambiguous with the other kind of mechanization that is automation. I therefore prefer to call it “mechanical manumation” to be more precise, but it wasn’t necessary for this blueprint.

The only thing you have to do to make a game out of any existing practice is specify the practice, such as by stating its rules and goals. [3.4] I have called this variously “specification” [5.14] and “codification” [8.9] in this blueprint.

B.4: Ambiguity in the word “game” #

Linked from: [2.3], [3.4], [C.6], [C.8]

My definition of a game is unusual and conflicts with what we intuitively call games. A game is any practice that can be done—“played”—for its own sake. It is reproducible and self-containable practice. And a practice is not distinct from the person who practices it, so a game is only observable and sensical as an ecological object—as a perceptible human-environment relation. [9.16]

This clashes with the common way of speaking of games as commodities or as computer programs—as objects that may be bought at a store or run on a computer. I rather distinguish the game, which implies a player to be perceived, from the materials with which the game is played and which help to play the game. These materials include boards, pieces, rules [3.9], and computer programs. I say “computer-game” to distinguish the computer program from the game it suggests to play or functions to reproduce. [2.3] [3.15] But I haven’t consistently followed this disambiguation in this blueprint.

These distinctions are valuable to precisely describe the empirical observation that people can use the same game-materials to play different games:

• People can play different chess-variants on the same board and with the same pieces.
• People can play different levels in the same computer-game and with different player characters that have different abilities.
• People can play a computer-game competitively or casually—to win or to have fun.

B.5: The nature of virtual worlds #

Linked from: [3.15], [6.6]

When I say that computer-games provide environments, I don’t mean that a game creates any new world beside the real one, but only modifies its own environment by its mechanics and output devices. The environment which a computer-game provides to a player is that of a computer with consistent patterns of interaction. For example, the environment of the game of Tetris may consist of an updating screen with identifiable objects in it and controls for changing the state of the game as shown on the screen. The environments of computer-games can be specified to a very high degree of granularity by programming them, but they are always founded on making consistently patterned use of the variability of input and output devices.

All mystical delusions affirming the non-virtual existence of virtual worlds in computer-games are dispelled if one only sees how clunky and ridiculous the implementing code and its interface to hardware can be. See for example the “Television Interface Adaptor” of the Atari 2600 and how to use it to draw to a display. The apparent worlds of computer-games are like the apparent magic of magic shows that hide action behind a curtain to effect their illusions.

Humans have the natural ability that they can pretend to live in a world unlike the real one, and to use this ability is commonly called “to imagine” and “to play a game”. A virtual reality is achieved whenever anyone uses this ability, and there is nothing essentially different from this in “virtual reality” technology. The promise of “VR” games is rather to extend the technology of computer-games to effect a greater extreme of immersiveness by means of improvements in input and output devices. The mystical theory of virtual worlds is consequent of a lack of self-consciousness by a player who is immersed in gameplay. [A.6]

But I still think the word “virtual” is useful. I use it to refer to the world as perceived, or as apparent, or as acted upon in particular ways—in other words, as played. It is valuable to speak of the world in terms of its human relation. [9.16] Therefore all games, not only “virtual reality” games, effect virtual worlds. This includes even the most unimmersive games that have no characters to relate to and that are played with pencil and paper.—Such is the virtual world of mathematics.

B.6: The nature of frustration #

Linked from: [2.3], [9.13]

Frustration is personally felt as being similar to pain and is antithetical to fun when present in a high degree. We can impersonally observe it as inefficiency in the player’s progress in learning and doing things [2.3] [6.5], and as a common precursor to the player giving up. It is the same phenomenon that is both the pain and the inefficiency, so it doesn’t matter which way we think of it. We design to minimize it either way. [9.13]

For this purpose, it will be valuable to analyze different kinds of frustrations and the situations that cause them. Here are a few to get started:

• One type is that of feeling lost. This is consequent of the player having a lack of familiar things in their environment, or having a lack of direction as to where to go.
• Another is that of feeling powerless, for lack of a better word. This is consequent of a mismatch between what the player expects that they should be able to do, and their ability to do it.
• Another is that of feeling cheated. This is consequent of a game giving suggestions and hints about itself that are contrary to its mechanics.

B.7: Operant conditioning in game design #

Linked from: [3.10], [9.13], [B.3], [B.8], [D.4]

The theory of operant conditioning is a scientific theory of how to manipulate organisms by means of rewards and punishments. It is aptly applied in animal training, but is often confusedly applied in game design and teaching. [D.4]

You can find the theory being applied in many games today, purposefully and inadvertently. Conditioning is especially valuable to keep a player playing despite a limited supply of new content and interesting situations, and despite fundamentally non-skill-based gameplay that precludes improvement by practice. In other words, conditioning is an effective crutch to get a player to keep playing a bad game.

Even game designers who are not specifically interested in this end often use the theory to understand games in confused terms. Consider this progression from the game Baldur’s Gate:

1. The player character gains experience points when the player completes quests.
2. With enough experience points, the player character gains a level, which increases the character’s power and gives them new abilities.
3. With enough power, the player character can more safely explore new areas.

A confused designer might call the experience points “incentives” and the new abilities “rewards”, but incentives and rewards are irrelevant to this design. This is a progression which suggests tasks (quests) to the player to complete, and links their completion (through experience points) to variations of the game’s mechanics (the new abilities and areas). This design functions to keep the game varied, balanced, and following a planned progression, while leaving much room for the player to choose how to play.

It’s unfortunate that most game designers lack a vocabulary to clearly distinguish manipulation from suggestion because manipulation often turns to abuse. The easiest way to tell if a game is abusive to its players is to see what players do when given the option to effect its redesign, such as by “cheat codes”. A newly freed player would choose to gain all of an abusive game’s rewards and nullify all of its punishments, like a newly freed donkey that was driven by carrot and stick would eat the carrot and kick away the stick.

B.8: Methods of abusing players for money #

Linked from: [9.1], [9.3], [9.17], [E.9]

I can think of some commonly used methods of funding development that abuse players. If applied to Super Practica, they would make it a worse, sub-optimal game:

• Copying the game could be restricted so it could be sold to players or schools. This would establish paid tiers of education that would ultimately be enforced by state violence.
• Advertising space could be sold, effectively selling the attention of players. This would be at best a distraction and at worst malicious manipulation, depending on the advertisements.
• The private data of players could be collected and sold, which would violate their privacy.
• Valuable content could be locked behind paywalls. A common design is for this content to be unlockable without paying, with payment unlocking it immediately. This would at best waste the player’s time, and it must waste very much time to make payment appear to be a good idea.

These are just a few of the various means by which a player can be abused for profit. A skilled and unethical game designer can do very much to manipulate players to do things they would otherwise rather not do [B.7], but I don’t want to give anyone any ideas beyond what’s already common.

Don’t you think it would be bad if someone were to make a game based on the designs of this blueprint, but then added to it these abusive methods of funding? The extra funding that can come from using these methods could even give it a competitive advantage! If you think this is worth preventing, then you should consider helping me make Super Practica so that it prevails instead. [9.2]

C: Notes on Practical Knowledge #

C.1: Empirical sources for my theory of practical knowledge #

Linked from: [3.1], [3.3], [3.4], [3.5], [3.8], [3.9], [3.10], [3.11], [3.13], [4.8], [6.2], [6.3], [6.4], [6.11], [6.15], [C.6], [E.1]

The evidence I draw on for my theory of practical knowledge mostly comes from personal research into how people learn to play different games, both as beginners and as experts. The empirical sources for this are publicly available:

• In the many computer-games that have been made is a trove of experimental methods for helping a player start playing and start playing better. Different games with similar but crucially different mechanics can be compared to each other to tell the effect of the mechanics on gameplay.
• Many players post videos and other records of their gameplay on the internet.
• I have made many personal experiments in trying different methods of playing and learning to play, and noting their effects on my gameplay.
• Many educational courses and resources can be found online, and I have analyzed these as games [B.1] so that the subject of study between the computer-games and the educational courses is the same. Practical knowledge is specified by the game that it is knowledge of [3.4], so what is effective for gaining practical knowledge must be found in both computer-games and educational courses.

Rather than compiling a large list of sources, I have treated the evidence as being commonplace in the phenomenon of learning to play games. My method is to attempt to describe what I see clearly and precisely, and then to let anyone contradict me if they can find a counterexample. But this method obscures my reasoning and continuing to follow it will make collaborative development of the theory more difficult [9.5], so I intend to compile particular evidence for particular conclusions in the future.

Some other projects with similar theories and designs to mine have made empirical studies and produce empirical results. The most relevant such project that I’m aware of is Jump Math, founded by John Mighton. Also relevant are the many programs of language acquisition associated with the name of Stephen Krashen, who popularized the research and theories behind them. I haven’t thoroughly studied their research and results, but I consider them to be fairly successful. Their success tends to confirm my theory because they make similar claims as I do—though not the same claims.

Thoroughly verifying my theory of practical knowledge before developing Super Practica wouldn’t be practical because Super Practica is itself the best means of the verification and falsification of the theories behind it. [9.7] This is a pragmatic method of testing a theory: Try using the theory and conclude its truth or falsehood according to whether or not it works. [9.15]

C.2: Theoretical influences on my theory of practical knowledge #

Linked from: [2.15], [3.13]

In developing the theory of practical knowledge I have found it valuable to engage with many works, especially George Polya’s How To Solve It, Stephen Krashen’s Principles and Practice in Second Language Acquisition, Jacques Ranciere’s The Ignorant Schoolmaster, Ludwig Wittgenstein’s Philosophical Investigations, and James Gibson’s The Ecological Approach to Visual Perception. Of existing methods for learning mathematics I analyzed, the most helpful to me was John Mighton’s design and explanation of the Jump Math program, such as given in his book Myth of Ability.

Joseph Jacotot, whose ideas are summarized in The Ignorant Schoolmaster, was the earliest educator I know of who systematically propounded a theory of education that minimizes the involvement of a teacher in learning, as I do. [3.14] My theory of stultification [3.13] is derived and mutated from the theory of stultification developed by either him or Jacques Ranciere. (Ranciere doesn’t make clear how much of his book is summarization and how much is revision.) The observation that setting tasks is sufficient for a student to learn translates to the sufficiency of computer-games and simulations. [3.15] [4.8]

Ludwig Wittgenstein’s work was helpful in providing the intuitive contextualist basis [9.15] of the theory and the triple-analogy between mathematics, games, and language. Language might seem to be irrelevant to mathematics, but learning a language is the case where theories of concept acquisition [3.6] are most clearly false, as explained in Principles and Practice in Second Language Acquisition, so analogies to language were helpful for suggesting fruitful hypotheses about how people learn to play mathematics. [A.2]

My theory does not derive from apparently relevant traditions of study such as pedagogy, psychology, or epistemology. [C.3] I was rather concerned with analyzing effective game design [B.1], and then more recently I found a solid theoretical basis in The Ecological Approach to Visual Perception for the ideas I made sense of as a game designer.

C.3: The structure and subject of my theory of practical knowledge #

Linked from: [9.16], [C.2]

Most theories of knowledge either consider knowledge as being separate from the people who have it, as in epistemological theories, or they consider knowledge as being personal, as in educational and psychological theories. My theory of practical knowledge rather removes conventional boundaries between education and epistemology, between beginners and experts, and between students, teachers, researchers, and theorists. I understand these as occupying different social positions of the same metagame. [3.13] This functions to avoid dualist confusions such as the intuitive and misleading dichotomy between “subjective” and “objective” knowledge.

This unusual structure follows from the way I developed the theory. I didn’t ask anything like “How does knowledge work?” but rather the practical question “What would be the best way to learn mathematics?” Then I determined the best empirical measures by which to judge my answers, and from these measures developed the theory. [9.8] These empirical measures only measure the person and environment together, never apart from each other. [9.16]

So what kind of theory is the theory of practical knowledge, and what branch of science does it belong in?—I follow James Gibson’s use of the term in The Ecological Approach to Visual Perception and call it an “ecological theory”. If the ecologists don’t like it then we can call it “game theory” instead. And if the game theorists don’t like it then they can take it up with the ecologists.

C.4: Ambiguity in the word “knowledge” #

Linked from: [3.4], [C.5], [D.4]

Ambiguity in the words “knowledge” and “learn” can be found in the theory of practical knowledge, as in most theories of knowledge: I use the word “knowledge” to neutrally describe a person’s habits and abilities, as well as to judge which habits and abilities are valuable and worth reproducing. Accordingly, we may call knowledge either all the habits which a player develops by playing a game, or only the ones we judge to be functional and valuable.

I haven’t been precise in this introductory outline, but a distinction between these different uses of the word “knowledge” will be valuable for distinguishing the design problems of Super Practica from theoretical questions about practical knowledge. The theory of practical knowledge should be value-neutral, but the design of Super Practica should be value-interested.

On the value-neutral side, precise theoretical terms include “ecological habituation” and “playability”. On the value-interested side I suggest distinguishing between “knowledge” and “pseudo-knowledge”, and between “functional” and “dysfunctional” habits.

C.5: Epistemic ethics of reproducing practical knowledge #

Linked from: [8.10]

The value of extending Super Practica to cover empirical sciences is to reproduce the practical knowledge of applying their theories. But being able to make a playthruable game to reproduce the application of a theory does not imply that the theory is useful or correct. The methods of Super Practica can therefore be used to reproduce practical pseudo-knowledge [C.4]—competence in making consistently false predictions, such as economists are famous for having.

Similar ethical problems are involved in reproducing mathematical knowledge, but they are not so worrisome. At worst, bad science is destructive, while bad mathematics is merely useless.—Though we shouldn’t waste a player’s time with bad mathematics.

According to the theory of practical knowledge developed in this blueprint, the most effective method to guard against reproducing practical pseudo-knowledge should be thus:

1. Do not attempt to make games which reproduce practical knowledge without particular applications. [5.1]
2. Make persuasive arguments for these applications being valuable.
3. Test and verify the essential accuracy of the game to these applications. [4.5]

But this could be unreasonably strict in some contexts. It might help to consider an analogy to reproducing books: How would a responsible publisher go about reprinting either pseudo-scientific, or outdated and falsified scientific books?—They would attach a notice informing the reader of this relevant context. Such a book would be valuable to someone curious about past ideas, but not so much to a student of current scientific ideas. This book wouldn’t belong in a curriculum, but would belong in a library.

It accordingly seems to me that the ethical obligations are thus:

1. Do not advertise benefits you cannot support by evidence.—The “most effective method” previously mentioned is the method of obtaining this evidence.
2. Contextualize things that may be harmful and easily misunderstood.

It will be worth continuing this ethical discussion as the implications of Super Practica and its methods become more clear in the future.

C.6: The necessity of voluntary play #

Linked from: [3.4], [3.6], [6.15], [7.2], [9.9]

It seems to be an empirical fact that people who are forced to play a game are less likely to learn to play that game well than people who play the game due to their own personal interest and choice. [C.1] My formula of ecological habituation [3.4] incorporates this observation. It immediately follows from it that the player who learns to play a game most efficiently is the player who plays it freely.

The appropriate attitude and emotion in playing a game are essential parts of the game itself.—They are part of its ecological attitude. To only halfway participate in playing a game is to play an altogether different game, such as to play with some rules relaxed, or with the goal being not to win but only to appear to be trying to win. [B.4] Likewise, to play a game in the context of being threatened or having an external reward is to play a different game; and because they are different games, there is necessarily imperfect transferability between them. [4.5] The more coercion is involved, the more different would be the games, so the smaller would be the transferability.

I therefore expect a coerced player to develop some practical knowledge that can be applied in the freely played game, but less efficiently than they would if they had played the game freely. It follows that coercion has no place in the optimal method of reproducing mathematical knowledge.

C.7: You learn to play the game as you play the game #

Linked from: [3.6], [3.9], [C.8]

Anyone who tries to explain how someone learns to play a game eventually runs into a chicken-and-egg problem: A person learns to play a game by playing it, but in order to play it they must first learn to play it. Much confusion derives from this apparent paradox.

My description of continuous correction in section [3.9] resolves this problem. Correcting a player’s play requires the player to first attempt starting to play. A player does not learn to play before they play, but they only learn to play as they play. You must play somehow before you can play correctly. A beginning player makes a switch, gradually or immediately, from doing stuff to playing the game. This shift can be facilitated only by suggestion [3.8–11] and constraint. [2.2]

The complete slogan of my theory of practical knowledge should therefore be: You learn to play the game only as you play the game, and only by playing the game.—But this doesn’t roll off the tongue as well as “You learn to play the game by playing the game.” [Intro.3] [3.5]

C.8: The only method of reproducing practical knowledge #

Linked from: [6.7], [9.14]

The optimal method of reproducing practical knowledge is optimal because it optimizes the only method of reproducing practical knowledge. The levels of a reliable level-progression embody a sequence of ecological habituations [3.15], but practical knowledge can only be gained by some sequence of ecological habituations. [3.2] [3.5] [9.16] [C.7] This is simply the continuous process of learning. So a reliable level-progression in a computer-game is a particular optimized mechanization [B.3] of the process of learning practical knowledge in general.

It can therefore be useful to analyze any method or event of learning practical knowledge by specifying as games the sequence of environments and ecological attitudes which a person “plays” through. [B.4] For example, the sufficiency of simulation for reproducing practical knowledge [4.8] can be explained by the player constructing their own intermediate levels to sequentially adapt to. The player’s identification of something to improve upon can be analyzed as the goal of an intermediate game, and the concomitant “new way of looking at things” can be analyzed as the ecological attitude or rules of the intermediate game. [3.4]

C.9: Is all knowledge practical knowledge? #

Linked from: [9.15]

The ultimate result of adopting the pragmatist way of reasoning in section [9.15] is to identify all knowledge with the practice it enables, so the apparent conclusion is that all knowledge is practical knowledge. Consequently, perhaps the theory of practical knowledge should be understood as a general theory of knowledge, rather than as a theory of only a certain kind of knowledge. [3.3]

This makes for interesting philosophical speculation, but arguing this stronger claim would not help me design Super Practica. It would rather make the theory of practical knowledge more cumbersome to develop and use, since it would have to answer and explain many more cases of apparent knowledge. I hence pragmatically limit my pragmatism and say we should not say that all knowledge is practical knowledge.—This is how you know I’m a true pragmatist!

D: Notes on Education and Psychology #

D.1: The antiquity of the idea of teaching mathematics by games #

Linked from: [1.1]

Plato mentions the practice of teaching mathematics using games in Laws, Book 7 (translated by Benjamin Jowett):

All freemen, I conceive, should learn as much of these branches of knowledge as every child in Egypt is taught when he learns the alphabet. In that country arithmetical games have been invented for the use of mere children, which they learn as a pleasure and amusement.

This dates the idea at least as far back as the 4th century BCE.

D.2: Educational theories versus psychological theories #

Linked from: [3.6]

An example of a theory of concept acquisition can be found in Principles to Actions by the writing team representing the “National Council of Teachers of Mathematics”. Like most practically oriented educational theories, it combines many apparently useful insights into a mishmash of ideas and claims that do not neatly fit together. It puts “conceptual understanding” at the core of its theory, but also affirms “building procedural fluency” and a variety of other things as being valuable in teaching and learning mathematics.

I have much more respect for something “down and dirty” like this than I do for abstract psychological and metaphysical theories like Experiential Learning by David Kolb. The latter are so detached from application and sound empirical methods as to be worthless.

So while the theory I reconstruct in section [3.6] hasn’t been propounded by anyone, it helps to specify the beliefs and methods of teachers of mathematics for productive examination and critique. Concept acquisition theories are vague and various, but unlike neater psychological theories, they’re actually usable.

D.3: Contrast with constructivist educational theories #

Linked from: [3.6], [9.16]

It might appear that Super Practica is an instance of a constructivist educational program, or that my theory of practical knowledge is a constructivist theory. This would be a false impression.

As I understand them, constructivist theories are a subcategory of concept acquisition theories. [3.6] It’s the same idea whether the term for their mystical knowledge-objects is “concept”, “schema”, “mental structure”, “cognitive tool”, or anything else. The difference in constructivist theories is only that concepts are understood to be internal to humans rather than external to them, and therefore must be newly constructed to be learned, rather than being communicated to be taught. My criticisms of concept acquisition theories in general therefore apply to constructivist theories more specifically. [3.7]

A constructivist could find very much to agree with in my theory of practical knowledge, and so could consider games as an exclusive means of “practical concept” acquisition.—Each game could correspond to the set of mental structures that could be constructed only by playing the game. But talk of mental structures here would be superfluous and misleading. The difference between such a psychological theory and my ecological theory of knowledge is between understanding games as efficient means of learning something else—concepts—and understanding games as situations in which knowledge is gained and used. [3.4] At best, a constructivist theory would introduce a worthless vocabulary to describe an inconsequential and illusory mental world of knowledge-objects between players and the games they play.

I attribute the general ineffectiveness of educational computer-games to their common adherence to constructivist educational theories. The labels for these, “educational games” and “learning games”, indicate the confusion.—Learning is an integral part of gameplay, so all games must be learning games! The designers of these games mistakenly make gameplay a mere means to the vaguely defined ends of “education” and “learning”. In contrast, gameplay is both the means and the end of Super Practica. [9.16] [E.10]

D.4: Contrast with behaviorist educational theories #

Linked from: [3.2], [3.7], [9.16], [B.7]

Many of my criticisms are the same as those made by behaviorists, which may lead one to confuse my theory of practical knowledge for a behaviorist theory. This would be incorrect. My theory of ecological habituation [3.2] also has no relation to behaviorist theories of habituation.

I disagree on the following points which appear to be common to all behaviorist approaches to the problems I deal with in this blueprint, according to my limited study of them:

1. They reduce their observations to what they call “behaviors”.—But an adaptation can only be sensically described together with its environment and as being functional to some purpose! [9.16] The reduction to behavior is therefore an over-reduction. [9.14]
2. They consistently miss the value judgment that is inherent in the word “knowledge”. [C.4]
3. Probably as a result of (1) and (2), they confuse learning with manipulation. [B.7] This is a disastrous consequence.

D.5: The pseudo-empiricism and obsolescence of schooltesting #

Linked from: [4.2], [4.9], [9.7], [9.9], [D.6]

“Schooltests” are in general a mere pretense of empirical testing. [4.2] My criticism is based on the 10th edition of Educational Testing and Measurement by Tom Kubiszyn, which gives a fine account of the practice.

The typical schooltest is a pseudo-empirical test because it only makes a show of measuring things that are apparently, but not really, related to the conclusions it is used to justify. There is a logically unsound jump from seeing a student successfully complete a schooltest to deducing that the student has learned the concept that the schooltest is associated with [3.6], and therefore that the student will be able to do things associated with that concept. Why should we assume that knowledge can be expressed by or deduced from answering prompts on paper? Such a claim requires evidence, but evidence is rarely given to support it.

Still, the empirical framework provided by the theory of “educational testing” also allows truly empirical testing. The problem is the theory calls “valid” empirical and unempirical tests alike, and this hole in validation makes its validation process worthless. But let’s consider the best possible tests we could make following this theory:

1. Well-designing a schooltest begins with precisely defining “instructional objectives”. This at best amounts to partially specifying the practical knowledge that is tested—the game that the student learns to play. [3.4]

2. A schooltest is at best a simulation. [4.3] The paper-and-pencil gameplay of schooltesting lends itself to word problems, which can amount to functional simulations. [4.4]

3. At best there would be empirical testing to verify that successful completion of the schooltest implies achieving the instructional objectives. This would be to measure a high rate of transferability of practical knowledge. [4.5]

4. Rather than failing the student who fails the schooltest, at best the student would get feedback and would be able to retry the schooltest. This amounts to giving continuous corrections [3.9], which enables the student to learn by trial and error. [4.8]

So we see that even the best possible case of schooltesting is inferior to Super Practica, which will have better simulations [4.7] and better feedback [2.4], among other things. Hence, Super Practica renders schooltesting obsolete.

D.6: The efficiency of optimal design #

Linked from: [9.11], [9.17]

The attitude of contentment with sub-optimal design is self-defeating.

Optimally designing Super Practica includes making it work reliably to reproduce practical knowledge. [6.7] Educational programs that are not reliable are in fact not functional.—We wouldn’t praise a screwdriver if it only works “much of the time”. I only need to call reliability “optimization” for popular understanding in the context of pseudo-empirical educational theories that blame their failures on students [3.7] and that validate non-functional educational tests which then validate non-functional educational programs. [D.5]

And new educational programs are designed and redesigned with almost every teacher, even though a good program only needs to be designed once. The result of such redundancy and low standards is that a great deal more time and effort is spent to make non-functional programs than would be required to develop Super Practica.

E: Notes on the Development of Super Practica #

E.1: On confidence and failure #

Linked from: [8.6], [9.17]

My confidence that Super Practica is possible to make derives from these things:

• Prototypes of the solutions of chapters [2] and [7] working well enough for the most worrisome problems.
• Materialist heuristics that reduce most problems of reproducing mathematical knowledge to problems of designing computer-games. [9.14–16]
• Empirical investigations that corroborate my theories. [C.1]

I am very confident that arithmetic can be covered by Super Practica, and this has been the focus of my research and design so far. Beyond arithmetic I am less confident, but I think success is probable. The problems of algebra and geometry are the ones to worry about, but if these can be solved then high coverage is nearly guaranteed. [8.7]

“Failure” would mean having to scale back the vision of Super Practica, perhaps admitting that the optimal method [6.8] is “purely theoretical” or unachievable in most cases. Even in the worst case of failure, Super Practica would be worth developing to better understand what is possible and to help develop a better method in the future. The empirical framework [8.1] ensures that it will at least generate valuable information to this end. In this way, Super Practica is failure-proof.

E.2: Noticing potential improvements #

Linked from: [9.7]

Thorough empirical testing [8.1] shouldn’t be done to validate our designs before release [9.6], but only in response to noticing problems in design after release. [9.7] So how could we systematically notice such problems?—A few methods will be effective:

• Ask for and listen to feedback from players. Players especially enjoy complaining about their frustrations and difficulties in public internet forums.
• Watch players play the game. We should encourage players to post videos of first-time playthroughs of any section of the game on the internet.
• Ask experts to judge the expertise of players who have completed Super Practica.

It follows from the effectiveness of these methods and a clear understanding of empiricism that telemetry won’t be useful for us, even if it didn’t conflict with the principles of section [9.10].

The value of telemetry for a designer is to provide an exploratory survey of how people are using a computer program, and this can help the designer get ideas for how the program could be improved. It is not useful for gaining more precise understanding than this.—The data gathered by telemetry is decontextualized, so we should expect it to be systematically biased according to the precise methods of data collection, and this bias cannot easily be corrected.

So empirically speaking, telemetry is valuable for generating hypotheses, but not for testing them. But the listed methods are at least as effective for generating hypotheses, and telemetry comes at greater expense. Hence telemetry would be a mistake.

E.3: The place of Super Practica in its simulations #

Linked from: [5.5], [7.14]

Does the design of connecting a mechanical math mode to a simulation [5.5] ruin the simulation’s essential accuracy? [4.5] It could be argued that it does:

In the real world, you don’t enter a mechanized math mode, but rather pull out papers and a pencil to do mathematics with.

—But this real world is the same real world in which you can play Super Practica! Instead of taking out papers and a pencil, you can turn on a computer and run Super Practica, or some other software that’s helpful for doing mathematics. If we simulate the situation of using a computer to solve mathematical problems, then Super Practica itself should be part of the simulation.—The mechanical math mode simulates this.

But there are also good reasons to value expertise in solving problems without relying on a computer, so we should support the option to disable the ability to enter math mode from situation mode in Super Practica B. And Super Practica C should have an optional progression to prepare the player for these simulations. This would be a tenth dimension of its level-progression. [7.1]

E.4: Simulating verification #

Linked from: [7.8], [7.11]

Gaining expertise in applying a verification method after having applied it as a solution method only requires imitating the automatic verification as a model. [2.13] [3.11] But perhaps players will have difficulty doing this, which would render the dimension of progression in section [7.8] unplaythruable. [6.5]

In that case, my backup plan is to make intermediate levels which will have the goal of manually verifying given solutions. For an example, consider the picture showing an automatic verification in section [2.13], but imagine that the player applies the verification manually. Perhaps these manual verification levels could also be valuable as simulations in Super Practica B, if they amount to simulating the verification of other people’s solutions.

E.5: Problems in decreasing guidance #

Linked from: [7.3]

The soft-constrained level U eases the progression from the hard-constrained level X to the manual level Y, but the result is not necessarily playthruable. It leaves us with the two design problems of making playthruable progressions from level X to level U and from level U to level Y.

1. Can the player easily advance from level U to level Y? This will require the player to be able to complete level U without triggering any rejection signals.—This is simply the method of continuous correction by suggestive signals! [3.9–10]

2. Can the player easily advance from level X to level U? This will require the player to quickly learn the new interface of level U and to find correct answers without taking too long in trial and error.—This is effectively the method of funnel constraint! [6.14]

But though funnel constraint and continuous correction by suggestive signals are often effective, they are not always effective, so this design is not a complete solution to the problem of lifting constraints.

The most common problem for arithmetic is when the trial and error of funnel constraint involves trying any number to find a unique solution. For example, suppose the player must guess the correct 6-digit number to advance.—The possible actions in this funnel constraint wouldn’t be limited enough for playthruability.

E.6: How to design time constraints #

Linked from: [5.8], [7.6]

There are two variables that can be changed to set the time constraint of a simulation: The player must complete a task X number of times within Y seconds.

The difference between increasing X and decreasing Y is important for determining what situation these levels simulate. Setting X = 1 while decreasing Y will simulate the task to “do this as fast as you can;” and keeping Y constant while increasing X will simulate the task to “do this as many times as you can.”

E.7: There is no progression for increasing accuracy #

Linked from: [7.6]

There are many existing computer-games that attempt to develop a player’s mathematical skills by giving them math exercises to solve. It is a common mistake in these to measure and test accuracy as well as speed. This seems to be done in imitation of typing games, but what is good design for developing skill in typing is not good design for developing skill in mathematics.—A typo doesn’t ruin a sentence nearly so often as a mathematical mistake ruins a calculation.

Unless the task is specifically to approximate, anything less than 100% accuracy is a failure of mathematics, and so should be considered a failure in the levels of Super Practica.

E.8: Direction of progression between solution-domains and problem-domains #

Linked from: [7.12]

Some teachers may benefit from first introducing a practical and concrete problem, and then introducing mathematical methods for solving it, but this is not a generally effective progression. It works well only when the player is interested in the given problem, so it implies making more assumptions about the player than we would like. [9.12]

Hence, the progression we design for Super Practica should be from the solution-domain to the problem-domain, not the other way around.

E.9: Free software vocabulary #

Linked from: [9.1], [9.3]

• Free software is software that empowers its users and respects their freedom. It is distinguished from “proprietary”, “nonfree”, and only “gratis” software.—A common clarification is to “think free speech, not free beer.” The standard criteria for determining what is free software and a more complete explanation may be found on the [GNU website].

• I use the phrase open-source development for an effective method of developing free software in public. Because the terms “free” and “open source” clarify each other, I say that Super Practica is a free and open-source game.

• Free-and-open-source-software (FOSS) licenses renounce some otherwise exclusive copyright privileges which enable a wide range of abuse against users of the software. [9.3] [B.8]

• Proprietary software is software that is released without a FOSS license and which therefore restricts the freedom of its users, due to copyright law.

• A copyleft license is a FOSS license with conditions that prevent other developers from developing proprietary software using the source code of the FOSS-licensed software.—In this way, copyleft “uses copyright law against itself”.

The free software movement has focused on counteracting the social destructiveness of copyright law applied to software, but the ideal of free software requires more than that. Beyond the legal problem of counteracting copyright is the design problem of developing software that empowers its users without restricting or manipulating them, and which doesn’t require the user to make special accommodations or sacrifices. I therefore like to speak of freedom-respecting and freedom-affirming design [9.10], which is to me an essential part of truly free software.

E.10: What kind of game is Super Practica? #

Linked from: [9.9], [D.3]

I say that Super Practica isn’t an educational game, so what kind of game is it? An analogy to history should help clarify what place Super Practica has in relation to education and academic knowledge.

Every now and then, a historian takes it upon themself to write a “readable narrative history book” to summarize the findings and conclusions of historians to interested readers.—And what am I doing but developing a “playable simulative mathematics computer-game”? Super Practica is to mathematics what scholarly narrative history is to history.

Narrative history books aren’t “educational resources” but simply “history”. They aren’t placed in a genre of “educational books” but in the genre of “history books”. To call “educational” a genre implies confusion.—We value education because it educates to some end, not because it simply educates and to no end. So the word “educational” is only used to indicate value when value is expected but not identified!—If the true value were clearly understood and agreed upon, then we would refer directly to it and not to “education”.

Hence it would be better to place Super Practica in the genre of “mathematics games” than in the genre of “educational games”.—But this would be redundant because mathematics is already a game [1.1], and calling it simply “mathematics” would also be ambiguous. So we can note the fact that it’s a computer-game and call it “computerized mathematics”, or “mechanized mathematics”, or “manumatic mathematics”. [B.3] Personally, I call it by its structure: It is the composition of many little sub-practica into a collection of larger super-practica.