Dragonbox and The Philosophy of Mathematics

I recall an anthropologist’s remark that some Australian aboriginal marriage rules required fairly advanced maths to solve.

…. But will it teach them to spork an oppressor in the eye? Or is that the Greek Mythology Wii game?

I should think the key question is, “when the time comes that needs to compute something useful–like the time she needs to leave the house in order to catch a ttain–or useless–like the value of x for 3x+2=31–will she need to relearn from scratch, or shrug and say ‘Yeah, I did that on my iPad.'”

I agree with Danyel. I’ll add that I suspect she will not quite simply shrug and say she already knows it, but will be a flash of insight away from realizing she already knows it.

Chris Armstrong, math doesn’t require numbers, so sudoku is math even though its numerals are inessential.

What Jamie and Danyel said.

http://en.wikipedia.org/wiki/Mathematics_of_Sudoku
“The general problem of solving Sudoku puzzles on n2 × n2 boards of n × n blocks is known to be NP-complete.”

@10

If I may answer in Jamie’s place. Of course you could solve a Soduko by trial and error, but then you could also solve an algebraic equation by trial and error by substitution (admittedly in most “usual” settings this potentially requires an infinite number of trials, but this is a minor point). The whole point of algebra is that for many algebraic equations there is a systematic way to do this — that is an algorithm. Arguably, one of the main mathematical points of algebra is to determine what sorts of algorithms can be used to solve what sorts of algebraic equations — indeed, answering this sort of question is what Galois is (mathematically) famous for.

I don’t know if there is such an algorithm (I’ve never been very interested in it) for Soduko, though Manta’s post suggests that if there is it is quite inefficient, and most casual Soduko solvers probably don’t use it. I wouldn’t be surprised if more dedicated probably have developed heuristics that if systematized could be turned into an algorithm. Tying back to the original post, this is exactly what John’s daughter is doing.

I just realized that I systematically misspelled Sodoku.

@11: I would say that the mathematical part is in the discovery of an algorithm (but then when we do algebra we are not doing math either, but simply applying the rules…).

Is my daughter actually doing algebra, even though she thinks she’s growing a dragon, or clearing a level, or playing a card game?

Hum, very interesting. On Youtube, one can see children playing both the cartoonish version and the usual notation version. As a professional mathematician (albeit one ten orders of magnitude less important than Bourbaki), I would say that your daughter is doing something-something intellectually challenging and apparently fun and enjoyable-but not mathematics (I’ll leave it to you to decide if she’s doing algebra; it has always been a source of puzzlement to me that the name of a difficult, abstract, sophisticated and historically quite recent part of math has been chosen to denote an elementary part of the US curricula).

As far as I can tell from the aforementioned Youtube clips, if you want for instance to “subtract” (or in the cartoonish version to slide a night version on one side of the equation), the game provides by itself small boxes on the other side where you have to slide again the symbol. If you want to “divide”, small boxes appear under all the terms of the other side. The problem, the way I see it, is that the player is thus accustomed to thinking that the rules of algebra are, well, rules. If the game wants me to leave the box alone and if for that reason I have to slide a night symbol here and then another one on the small box there, so be it. If tomorrow (or at the next level) it wants me to slide the night symbol on one side but not on the other, or maybe up on one side and down on the other, then so be it as well. The rules have change. The problem is that the rules of mathematics are not arbitrary, they are the manipulations that guarantee that the latter equation will be true if (and sometimes only if) the former is (for a formal and mathematical accurate statement of the criticism, notice for instance that Dragonbox does not seem to care if you divide by zero or not so that I could easily devise a Dragonbox setting leading to the equation 2=3, or perhaps cloud=star, and your daughters will happily solve it; not that there is anything wrong with it, of course, but a proof that they are not doing math).

Math educator (at least at post high-school level) wage a constant battle against the belief, very common among students, that in mathematics, you have the “right” to do some stuffs and you don’t have the “right” to do others and moreover that the rules change from grades to grades (oh, before we did not have the right to take the square root of -1, now we do, or conversely oh, I had the right to divide until learning about rings and now I don’t anymore). The truth is that you have the right to do whatever you like, but only a small subsets of the things you may do will lead you to the correct answer, not in the sense of the answer that will grant you points at your exam, but in the sense that this answer will have actual predictive power on the outside world.

Case in point, for the mathematically inclined among the CT readership. In a recent exam I gave on combinatorics for an elementary college-level course, a subtle question involving the binomial coefficient (2n,n) was lurking at the very end. During the exam, I notice a student having obtained the (correct) answer (2n,2n/2). I congratulate him briefly but then he gives me worried look. “I have a problem”, he tells me. “What’s that? This looks good.” “My problem is that I don’t know if I have the right to replace 2n/2 by n”. “Well, that’s just usual division”, I tell him. “Yes, but I don’t know if I have the right to do it inside a binomial coefficient.”

So what are your daughters doing if not math? They’re exercising their planning abilities, their abilities to notice patterns, to adapt their strategies and actions depending on contexts (what is usually called “executive functions”) etc. All things which are highly beneficial to learning in general and mathematical learning in particular. So it seems to me that Dragonbox is a recommendable educational game. Playing it might not be doing math, though.

Edward Z. Yang made a nice interactive tutorial of the sequent calculus, and he argues that it’s useful because even though you may not understand what you’re doing, you’re proving something (potentially something useful).

I suppose it could be seen as a sort of DragonBox for adults.

Z @ 17: Actually, it is not possible to divide by zero in Dragonbox. You can only divide (or multiply, add, or subtract) by the symbols which appear in the “menu” at the bottom of the screen. Zero never appears in this menu. Of course, zeros can show up in the playing area, but you can’t do anything with these zeros except click on them to make them vanish.

You are right that Dragonbox does not teach people why the rules of algebra work the way they do. But that is not its job. Its job is to give the player an opportunity to practice using these rules to manipulate symbolic expressions and solving simple problems.

As you say, these rules are presented within the game as being totally arbitrary, like the rules of chess. Thus, no one can “learn algebra” just by playing the game in isolation. The game must be supplemented (either before, during, or after playing) with an explanation of what the symbolic expressions mean, and why the rules work the way they do and not in some other way. Nobody can truly understand algebra (or any other computational technique in mathematics) until they understand that the computational rules are not arbitrary; they are entirely determined by the underlying semantic content.

Another important benefit of the game, which should not be underestimated, is purely psychological. A lot of students have a huge amount of anxiety about mathematics, and the more advanced the mathematics, the greater the anxiety becomes. Dragonbox defuses this anxiety by presenting algebra as nothing more than a slightly silly game with quirky graphics and cheerful music. Of course mathematics is much more than “just a game”. But for students who are paralyzed by fear of the subject, it is psychologically helpful to approach it as “just a game”, until they develop some confidence.

Marcus Pivato,
I find myself in complete agreement with what you wrote, both on the psychological value of an apparently genuinely fun game convincing children that they can do it and on the need to supplement this (eventually) with an explanation of the semantic content.

Regarding division by zero, yes, I noticed on the clips that the kid could do only one thing with zero elements: make them vanish. But that doesn’t change at all my point. As long as a formal system does not distinguish non-arbitrarily between division by zero and division by a non-zero element, this system is not part of math (from a philosophical point of view) and interacting with it is not doing math (from a pedagogical point of view). The implication that most of what is taught in math class up to high-school level is not math is sadly correct.

As an aside, I’m well aware of the history of algebra: the fact that it was developed in the 9th century AD makes it, well, a difficult, abstract, sophisticated and historically recent part of math (compare with geometry and number theory). Indeed, Al-Khwarizimi’s text is not what is usually taught to school children, it would typically be much closer to high-school level. But anyway, I wrote this about algebra by way of apology for not knowing quite exactly what Americans meant by algebra.

“a the game” s/b “at the game” in my previous comment

Z@23 “As long as a formal system does not distinguish non-arbitrarily between division by zero and division by a non-zero element, this system is not part of math (from a philosophical point of view) ”

Uh? Abstract algebra is for the most part working with arbitrary rules.
You can invent some rules and study what happens: of course, to catch the interest of people you have to give some motivation, but once you have the rules the game itself is its often own justification.

When algebra was imposed on me in the 1950s, the desired performance was recognizing patterns in strings of numbers, letters, and other symbols, and selecting the correct algorithm (ritual) to transform one string into some other string that was considered the ‘solution’ on rather arbitrary grounds. Very little was explained, nothing was proved, and no connections were drawn to other views of the world. It was, however, called ‘math’ . Learning it was like crossing a desert. When I actually found out what mathematics was about a few years later, I was astounded. If one is going to question whether this Dragonbox game is math, one might also want to question whether high-school algebra is math on similar grounds. Or maybe things have changed in schools.

Z wrote:

“The problem, the way I see it, is that the player is thus accustomed to thinking that the rules of algebra are, well, rules. ”

I don’t agree with this, at least not in the way you formulated it.

It is in fact perfectly true that the rules of algebra are rules, it is also true that you can change those rules and get different results. The difference between associative algebra and non-associative algebra is that one subject studies objects that obey some rule while the other studies objects that don’t usually obey it.

It is perfectly true that those rules are not arbitrary, but their non-arbitrariness comes from the fact that those rules are the result of abstracting away from some human activity, formulating that abstraction into clear and non-ambiguous rules and then working out their implication.

I see it as perfectly possible that some discoveries in physics can lead to the development of a new mathematical formalism, a formalism that can provide a useful language for the physicists (Of course, this happened historically several times).

Your objection should to be directed rather to the very superficial understanding of those rules and of what the full implications of the rules are.

That is a valid point and also a valid criticism of Dragonbox as doing mathematics: Dragonbox is about the human activity of playing Dragonsbox and you can abstract away from that activity in several ways. One way would be to eliminate all the elements of DragonBox that are different from elementary algebra — another way is to eliminate all the math out of it and simply believe that Dragonbox is about moving pretty pictures around.

Both possibilities can be actualized in the mind of the child, but with the right guidance a child can be taught to abstract away the fluff and remain with the algebra in mind.

When I was a child (9 years old or so) I had a toy that had a 3×3 array of push buttons, each with an associated light. Pressing a button flips some of the lights between on and off. For each button, the set of lights that it causes to change state is always the same.

Now, of course, I know abstract algebra, and I’d just write this as a 9×9 matrix multiplying a 1×9 matrix over GF(2), and invert the matrix.

My 9-year old self, however, had not done those abstract algebra courses: but quickly worked out that the order in which you push buttons doesn’t matter (commutative); that pushing a button twice cancels out (inverses), and proceeded(*) to write down the general method for getting from any state to any desired state. (Which is effectively inverting the matrix).

(*) Took me about 2 days, if I recall correctly. Having done freshman “Algebra I” will save you some time in figuring it out from scratch…

#34 Surely. The “E” is backwards and the “A” is upside down!

I was reading

⊢ ∃x. P(x) → (∀y. P(y))

as

⊢ (∃x. P(x)) → (∀y. P(y))

But I suppose the expressions are supposed to have the fullest extent allowable, so that it should be

⊢ ∃x. (P(x) → (∀y. P(y)))

(and therefore he must indeed mean the second formulation, as the negation of the first is just a statement rather than something which could be proved invalid)

Is that a negation sign, or a turnstile?

All math is thinking (things through to solve problems) and all thinking is math, in my opinion. When you have four errands to do and decide what order is best to do them in, you are doing math. (You may not be doing it well.) Trial and error is, I suspect, the basic technique that our neurons are programmed with, but it can lead to other specialized techniques, such as algebra and calculus, which work faster at certain types of problems.

That is all.

It seems to me it probably develops skills that are easily transferred to “algebra”. So presumably the player will find “algebra” easy to learn/practice. Of course it is also possible (depending on the design of “dragonbox” to create certain intuitions that might get in the way.

Learning the mechanics of the steps, is like learning what are legitimate moves in chess. The trick for non trivial algebra problems, is to have some sort of intuition so that you are more likely to choose manipulations which bring the problem closer to solution. Just randomly applying transformations until (you hope) you recognize a solved problem tends to be pretty inefficient. I can easily imagine this game might help develope those intuitions.

I second all that Marcus Pivato has said. As for the Yes and No, cognitive skills are not as simply categorized as we might like to think. A good example is the Wason task ( https://en.wikipedia.org/wiki/Wason_selection_task ). It stumps the vast majority of people and has stumped even some professors of logic. However, if it is framed as a question of the detection of cheating on social rules, the majority of people have no trouble with it. Context matters.

Piaget gave us a good way of comparing the Dragonbox game and algebra. Playing the game requires concrete operations, while doing algebra requires formal operations. But doing algebra not only requires formal operations and context freedom, it requires applying those formal operations in varying contexts. That is why word problems are one of the most difficult and most important aspects of algebra. If you know algebra, you can apply it to the Dragonbox game, but not the other way around.

This is not to disparage the game. In fact, it is a good idea to start children off with learning concrete operations as a preparation for learning formal operations later on.

Make that – 3 cookies per jar.

Error of typing error, not calculation!

:)

I’ve been reading lots of mathematicians in this thread, but my take as an economist who teaches math in an applied context is different. My students need two types of skills with algebra, manipulation (the rules stuff) and translation, being able to read an algebraic formulation as a statement of logic involving concepts that can be conveyed verbally as well (although usually with great loss of efficiency). This second is crucial: it means being able to translate a narrative story into algebra, and algebra into a narrative story, not getting hung up on the notation.

From what I can tell, Dragon Box excels at the first but doesn’t go after the second.

When I was in 7th grade I had a brilliant math teacher (Bill Steele, a miracle) whose entire course consisted of having teams of students solving brainteasers. (There were no textbooks of any kind.) The puzzles started out being on the easy side and very gradually became harder and harder. By the end of the year, the puzzles were so complex that, to solve them, we had to write down symbols and work out their manipulations. The following year, when I took intro algebra, I took one look at the textbook and thought, Why am I repeating this? We figured out all this stuff last year to solve our puzzles.

I felt as though algebra were a private language that I had invented myself (along with my classmates). Again: brilliant.

Thanks for the response Marcus – nice explanation.

The trick, however, is to translate the insights of lateral thinking into a rigorous, precise, logical argument. This is where the technical skill comes in, and where a lot of people get stuck.

Something that is also at the heart of the social sciences… if only as an aspirational goal :)

I think that learning math, and truly, is about self-motivation. Therefore, I think that games are a bad way to teach math. You’re interested in the act of playing and process goes on in your mental hinterground, far from the lights of your attention. In a sense, this logic flows from my thinking about how games are not art, and it’s very hard to actually make both worthwhile art and worthwhile gaming in the same package. Okay, I won’t respond to any derailing, just laying out how I think. This game, while a better attempt than the other sordid attempts at making math “fun”, still burys the appreciation of the intellectual heart of the project inside the gameplay–even though it *is* the gameplay! Such that it led to the philosophical question heading this blog post. The task was about taking care of a dragon. No matter how nicely procedural or artistic you manage to do it, in order to do it *well*, you must ignore the appreciation of what you did, since you’re focused on the goal. If you are actually interested in *teaching* mathematics, you have to instill an appreciation of the act itself, no matter how insubstantial the pleasure not at knowing how to solve a problem, but knowing how to think a problem.

Generally, I think the best way to interest children is through well written books about math and math history.

Try: http://www.amazon.com/Math-Girls-Hiroshi-Yuki/dp/0983951306/ref=sr_1_1?ie=UTF8&qid=1367555098&sr=8-1&keywords=hiroshi+yuki

for starters. In general, as an adult, I understood math vastly better when I read things like Nahin’s An Imaginary Tale. And right now, I’m working on Flake’s The Computational Beauty of Nature.

The main problem with math education is in part described above…people just don’t get taught mathematical intuition, and they never come into contact with how urgently people might feel a need to solve certain problems. Nor do they come into contact with good writing, either. A dense math book is great in a lot of ways, but it should be more of a resource than quite the center of the pedagogical approach. There needs to be more digressions, math history, integrated math with other subjects, so forth and on.

I hope mathematicians find the fact that several math savvy people had difficulty agreeing over an apparently simple qustion to be antidote for the scorn math savvy people have over the rest of us.

BTW, when I sing, Germans don’t burp, but the eyes of those near me squint. If that can be derived from those squiggles than I’m a double Dutchman.

@John Holbo (54)

There is already an educational/play device for arithmetic with a see-saw such that if the sums on each side of the see-saw are equal, the see-saw balances.

I hope mathematicians find the fact that several math savvy people had difficulty agreeing over an apparently simple qustion to be antidote for the scorn math savvy people have over the rest of us.

Scorn? What scorn? I only am annoyed with people who are *proud* of their innumeracy, as if it were somehow cute.

As to the question of whether playing Dragonbox is “doing math”, I agree that it’s teaching formal manipulation, and not understanding the basis of that manipulation, and not understanding how to take a “word problem” and put it in a formal setting.

However, let’s not understate the importance of formal manipulation: some part of the power of mathematics lies in the fact that the right notation and formal rules allow useful relations to be derived quickly and mechanically. That’s what *makes it* the right notation.

Also, if I understand correctly, while the Dragonbox “game” involves formal manipulation, it is not *rote* manipulation: the player must figure out which transformations to apply and in what order. So there is a creative element, and the player is proving a little theorem: “if , then x=5”, say. Seems like math to me.

Does anyone else remember WFF-‘N-PROOF?

Interesting. I recently started playing some casual games both on the desktop and an Android tablet, and the big surprise has been how educational some of them are. Though the skills you acquire – to do with problem-solving, mathy things as in resolving patterns, and strategy – don’t easily translate into classroom curriculum objectives. The disagreement here between people accomplished with math (from varying angles and at varying levels, I’m sure, between formal-logical and practical-applied) illuminates that the question of “is she doing math” may be not the most productive to ask: a “no” doesn’t meant it’s a useless activity for the development of the brain circuits or thinking patterns required to reason about school math, and a “yes” doesn’t imply that you can throw out entire years of math class and replace them with game playing. Also, purely from personal anecdata, I’m sure how much development of mathematical thinking a kid gets out of the game depends of whether they fall back on being told in the end what next manipulation to do.

In any event, I’ll go looking for it.

Informal learning through play and games has always been encouraged by middle class parents.

The mathematicians’ comments on this thread are fascinating. For example, I wouldn’t have thought it possible for one to be “doing algebra” without at the same time “doing math(s)”. (By analogy, if I’m “doing epistemology”, I’m also at the same time, “doing philosophy”.)

Re Harald K @19: That Yang tutorial is nice, but I’d think that natural deduction is easier to wrap one’s head around than the sequent calculus, no? Not that I know of any good links for the former….

(And OCS’s 35 wins the thread, obviously.)

Math is presented as a form of WORK, an expenditure and that is how it will then be perceived. It is taught that way to acclimate students to working for others. Math can be presented as a form of PLAY however, a pay off and that will be how it will be perceived. It is not taught that way because that does not acclimate students to working for others, rather it encourages them to take charge of their own efforts. Self directed individuals do not help the powers that be to remain in charge so every effort is taken to make math as dreary as possible and to then force students to deal with it.

If someone looks at this picture, is he doing math?
http://www.math.ntnu.no/~hanche/pythagoras/pythag.gif

All the nice things other people and I said about Dragonbox also carry over to looking at these kind of nice proof without words figures. That said, I would answer no to your question, and exactly for the same logical reasons I would say that playing Dragonbox is not doing math. You can play Dragonbox without ever getting that there might be a problem with dividing by zero. You can look as long as you want at the nice picture you linked without ever getting that there might be a problem with the following equally nice one.

http://upload.wikimedia.org/wikipedia/commons/5/5e/Missing_square_puzzle.svg

students required to memorize the names and date of birth and death of 10 important philosophers together with the title of one of their book are doing philosophy

Or like when I try to explain the concept of validity in an intro class, and half the students rote memorize the definition instead (later, invariably getting switching it with the definition of soundness). In any case, point well taken.

The taboo against memorizing dates can sometimes go a bit too far. It is really not a bad idea for people to have a rough idea of chronology, otherwise the past can seem like a chaotic mishmash. Memorizing dates can be a very effective way to quickly obtain this. Knowing key dates can be a key to remembering a lot of other things.