Dragonbox and The Philosophy of Mathematics

by John Holbo on May 2, 2013

Educational apps for kids are supposed to be fun. The Holy Grail is getting your kid hooked on something that is basically their homework. Via BoingBoing, I found the Holy Grail: Dragonbox. (You can get it through iTunes and from other sources, I’m sure.)

dragonbox

It’s just algebra, rewritten as a genuinely addictive solitaire-ish card game. You have to isolate the ‘box’ (or card) so your dragon will grow. You have ‘powers’ to transform and move and eliminate boxes in various ways (analogs of algebra stuff); and, periodically, you gain new ‘powers’ as you clear levels and your dragons grow. The first thing to say is: gosh, my daughters now fight with each other about who gets to do algebra after breakfast, on the iPad, before school.

The second thing to say is that it raises kind of a funny issue in the philosophy of mathematics. Everything gets introduced in a cartoony, pictorial, non-mathematical way. The x you are solving for is the box with its shy, peeking dragon that won’t grow until it’s ‘alone’ (everything needs to be moved to the other half of the screen). Parentheses are bubbles that a few boxes may be suspended in. Negative numbers are ‘night cards’. A black-and-purple ‘night’ version of a picture card will cancel the regular version out, removing both from play. Numbers are dice pips. Gradually the cartoony stuff is pared back until kids are zipping through screens of stuff that looks a lot like plain old algebra (although, as far as we’ve gotten, the cartoon stuff has not fully disappeared). In the screencap above you see some ‘night cards’ but also something that looks like a negated c. It gets more like that. Belle was amazed to see our 9-year old ‘solve for x’, from a pretty snarly and complex starting point, in a dozen-or-so rapid and confident steps. Now: the philosophy. Algebra turned into a card game is a nice case-in-point for the debate about formalism in mathematics. Is my daughter actually doing algebra, even though she thinks she’s growing a dragon, or clearing a level, or playing a card game? She’s a bit like those tribespeople Wittgenstein writes about, whose stamps and shouts can be related, in a regular way, to moves in chess, but who have no conception that they are playing chess. Very ‘wax on, wax off’.

{ 78 comments }

1

Peter T 05.02.13 at 5:29 am

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

2

John Holbo 05.02.13 at 6:04 am

I suppose I should give my opinion – which I’m not very wedded to: doing algebra requires an intention to do algebra, which my daughter lacks. But derived intentions will do. The makers of the game intended that, by playing a card game, she should do algebra. So, by playing a card game, she’s doing algebra.

3

clew 05.02.13 at 7:56 am

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

4

John Holbo 05.02.13 at 8:02 am

“will it teach them to spork an oppressor in the eye?”

I should think not. Is there an oppressor sporking Wii game?

5

Danyel 05.02.13 at 8:16 am

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.'”

6

Chris Armstrong 05.02.13 at 8:46 am

I think she’s doing algebra. I don’t think not knowing what that means is an issue. On the other hand, people sometimes claim people are doing maths when it seems clear that they are not. I saw a letter to the Guardian once which praised Sudoku for raising interest in maths. Well it’s true that Sudoku uses numbers – but the game would be the same game if it used Pokemon characters instead, wouldn’t it? You just need to be able to recognise whether the same character appears twice in a given row. Doesn’t seem to me there’s any maths involved in that. Might try and tempt my kids with Dragonbox, though!

7

Jamie 05.02.13 at 9:06 am

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.

8

bourbaki 05.02.13 at 9:12 am

I agree with Chris that your daughter is doing algebra. If I understand things correctly she’s just doing algebraic manipulations using a different set of notation (and with the added crutch that presumably j correct manipulations are not allowed). I would say however that she is not however doing math. When she makes the connection between an algebraic problem as usually stated in schoolwork and the game that would be a mathematical insight.

As a caveat I should say I’m a professional mathematician so may have a slightly skewed prespective — in particular I’m totally unqualified to say anything non-anecdotal about childhood mathematical development.

9

Manta 05.02.13 at 9:19 am

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

Chris Armstrong 05.02.13 at 9:39 am

Jamie, I appreciate that maths needn’t involve numbers. My question is whether the process involved in solving a puzzle is really maths. If I set you the following task (which is all Sudoku is):

‘Go plant these three types of plant for me, in three rows, but make sure they don’t repeat within a row’

is that maths? I don’t see that it requires access to any kind of reasoning beyond trial and error. But perhaps it’s still maths – I’m prepared to admit I’m wrong!

11

bourbaki 05.02.13 at 9:57 am

@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.

12

Chris Armstrong 05.02.13 at 10:00 am

That sounds right. I guess, then, that there are more and less mathematical ways of doing Sudoku. I’m afraid I am on the non-mathematical side of the continuum!

13

bourbaki 05.02.13 at 10:04 am

I just realized that I systematically misspelled Sodoku.

14

Niall McAuley 05.02.13 at 10:28 am

As a non-mathemetician, I’d be inclined to agree with bourbaki. I think it makes sense to say your daughter is doing algebra but not maths, in the same way I’d say a computer is generally doing computation but not maths.

15

Manta 05.02.13 at 10:50 am

@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…).

16

Marcus Pivato 05.02.13 at 11:49 am

First, let me say that I think Dragonbox is a brilliant educational game, and I highly recommend it (as both a mathematician and a parent of two school-aged children). Most of the “educational” computer games I have seen are terrible; either they are tedious pencil-and-paper exercises thinly disguised with a shoot-‘em-up video-game facade (usually barely playable, and with terrible graphics), or they are addictive time-wasters with plenty of tasty eye-candy, but just barely enough educational content so that you can rationalize letting your kids play them (they are “educational” in the same way Froot Loops is “part of this complete breakfast”).

In contrast, Dragonbox is both genuinely fun to play (and beautifully designed), and it packs a real educational payload. As John Holbo suggests, the game seems to be inspired by the formalist interpretation of mathematics. Someone obviously said, “Well, if mathematics is “just a game of manipulating symbols”, then why don’t we teach it to kids by turning it into a fun game of manipulating symbols?”

Turning to the question of whether people who play this game are “really” learning algebra, the answer is both yes and no. Yes, someone who masters Dragonbox will develop a good understanding of the formal techniques of manipulating algebraic equations so they can “solve for x“. However, they may not have any understanding of what this means, or how it can be applied to real problems. Consider the following very simple problem: “There were five jars of cookies on the table, each with the same number of cookies. Someone ate two of the cookies. Now there are thirteen cookies left in total. How many cookies were originally in each jar?” Someone could be an expert Dragonbox player, but they might still not have any idea how to answer this question, because they don’t understand how “cookies in jars” is related to “solve for x“. (This is not hypothetical; I have seen this happen.)

This is part of a much broader phenomenon. “Learning” any concept in mathematics is about understanding how it connects to all of the other concepts. So you can develop mastery of one aspect of a mathematical concept (say, formal manipulation) while not grasping another aspect of the same concept (say, geometric meaning, or applications to real-world problems). This fact is, of course, appreciated by most good mathematical educators. Nevertheless, it seems that we still end up with many students graduating the school system who are judged “good at math” because they can play formal games, but have no conceptual understanding of what they are doing.

Some people here have drawn a distinction between “doing algebra” (i.e. a purely formal or computational process) and “doing math”. This is an important distinction, since most of what mathematicians call “math” is more about abstract logical reasoning and creative problem solving, rather than tedious, mechanical computations. However, tedious mechanical computations are still an important background skill, without which it is not possible to do a lot of the abstract logical reasoning. To offer an (imperfect) analogy: we think of writing fiction as the creation of an engrossing story with realistic, three-dimensional characters, sparkling prose, etc. —not as the tedious mechanics of spelling and grammar, much less the physical skill of pushing buttons on a keyboard. However, until someone masters these tedious mechanical skills, they will be extremely handicapped in their ability to produce good literature. In the same way, without mastering “computational” skills such as algebra, it is very difficult to move on to the abstract logical reasoning and creative problem solving which is the soul of mathematics.

17

Z 05.02.13 at 12:35 pm

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.

18

Z 05.02.13 at 1:01 pm

Now that I’ve weighted in on the psycho-pedagogical aspects of the game, let me say a word about its philosophical implications: it has none, and interestingly, this is so for essentially the same reasons that playing the game is not doing math. Again, I could easily devise a Dragonbox setting which leads to a logical contradiction (as far as I can tell from the clips of kids playing on Youtube). Or in elaborate terms, the rules of Dragonbox appear to be inconsistent. From the (maybe not so) well-known logical fact that wrong statements imply anything, the formal content of Dragonbox is empty (any sentence expressed in the Dragonbox formal language is both true and false). Both hard core platonists and hard-core formalists agree that such systems have no bearing on the philosophical status of mathematics (or anything else really).

19

Harald K 05.02.13 at 1:07 pm

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.

20

Armando 05.02.13 at 1:21 pm

Marcus, above, has it right. Moreover, I think a good argument can be made that not only are the mechanical aspects (of things like algebra) necessary in order to do mathematics, but familiarity with them are the best way one has to develop actual insight and intuition into mathematics.

21

Marcus Pivato 05.02.13 at 1:31 pm

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.

22

Marcus Pivato 05.02.13 at 1:41 pm

Z @ 17: By the way, you wrote:

…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.

Actually “algebra” (the kind taught in schools) was developed in the 9th century AD by a Persian mathematician named Al-Khwarizmi. The word “algebra” is a corruption of the Arabic “al jabr“, which in Arabic apparently means something like “restoration” or “completion” —Al-Khwarizmi used this term to refer to the operation of cancelling a term from both sides of an equation. (Incidentally, Al-Khwarizmi’s name is also the origin of the modern word “algorithm”.)

Al-Khwarizmi developed algebra to solve the sort of simple practical problems that would confront a medieval merchant or accountant. It is true that the theory of groups, rings, and categories which is taught in modern university “algebra” courses bears precious little resemblance to what he devised. But it seems that it is historically accurate to use the term “algebra” for what you called “an elementary part of the US curricula”.

23

Z 05.02.13 at 2:11 pm

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.

24

marcel 05.02.13 at 2:20 pm

Let me stipulate that playing Dragonbox is not doing math…

Are the skills developed by becoming good a the game Dragonbox transferable to doing algebra? If one becomes good at Dragonbox, does that help one become good at algebra?

If so, give the kids Dragonbox to play for awhile, then introduce them algebra, drawing parallels between algebraic operations/actions and Dragonboxian ones, no?

25

marcel 05.02.13 at 2:20 pm

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

26

marcel 05.02.13 at 2:21 pm

And “introduce them algebra” s/b “introduce them to algebra”

27

Manta 05.02.13 at 3:11 pm

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.

28

Manta 05.02.13 at 3:14 pm

To be more precise: people study a structure for all possible kind of reasons, but this reason is not part of mathematics per se: you can give no reason and no justification for your choice, and still be doing mathematics.

29

Anarcissie 05.02.13 at 3:37 pm

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.

30

Adrian Kelleher 05.02.13 at 4:10 pm

There’s a slightly older freeware version that runs on an inkPad. What will be the outcome of all this experimentation?

31

Alex K. 05.02.13 at 4:43 pm

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.

32

SusanC 05.02.13 at 5:43 pm

Explaining algebra as a card game is fairly standard stuff. Well, maybe not at the kindergarten level where you first teach algebra, but when you eventually get on to stuff like linear logic, game semantics is the way to go.

And even at the elementary level, algebra feels like some kind of game – a sliding block puzzle perhaps, or Tetris.

33

SusanC 05.02.13 at 6:07 pm

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

ben wolfson 05.02.13 at 6:39 pm

“Edward Z. Yang made a nice interactive tutorial of the sequent calculus,”

In which one of the exercises asks one to prove that

⊢ ∃x. P(x) → (∀y. P(y))

But surely this is false?

35

OCS 05.02.13 at 6:51 pm

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

36

Brendan 05.02.13 at 7:03 pm

And even at the elementary level, algebra feels like some kind of game – a sliding block puzzle perhaps, or Tetris.

I think it feels that way to people who get it. The people I know who struggled with algebra don’t have an intuitive feeling for the rules; instead (when forced to) they painstakingly think through all the steps of subtracting/multiplying/dividing both sides of the equation.

The question is whether developing the intuition first can help produce understanding. I think Jamie (#7) is right, and once you recognize the analogy between the dragon and x everything afterward will come more easily.

The idea (and apparent success) of disguising the mathematical notation is interesting. Maybe it offers a way to distinguish people who think they’re bad at math from those who actually have some impediment.

37

ben wolfson 05.02.13 at 7:29 pm

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)))

38

dsquared 05.02.13 at 8:14 pm

34: don’t see how that’s false – using the official Oxford University schema invented by the late Bob Harwood:

“it’s not the case that if there is any Frenchman who sings, then all Germans burp” (your first expansion)

or

“There does not exist any Frenchman such that if he sings, all Germans burp”

39

dsquared 05.02.13 at 8:17 pm

(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)

40

ben wolfson 05.02.13 at 8:19 pm

38 see 37, but I’m not sure why you’re changing both the predicate and the domain of discourse in your translation.

“If there exists a Frenchman s.t. he sings, then all Frenchmen sing” is false; “There exists a Frenchman s.t. if he sings, then all Frenchmen sing” is true.

41

phosphorious 05.02.13 at 8:21 pm

Is that a negation sign, or a turnstile?

42

ben wolfson 05.02.13 at 8:33 pm

“(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)”

I’m not sure what you mean by this. It is simple to show that the first is invalid: “(∃x. P(x)) → (∀y. P(y))” is false if P is “is prime” and values range over the set {3, 4}. The conclusion does not follow even if we affirm everything in the (empty) set of premises. The negation of the first is, I suppose,

⊬ (∃x. P(x)) → (∀y. P(y))

and as far as I can tell that’s true.

43

JimV 05.02.13 at 8:36 pm

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.

44

RSA 05.02.13 at 9:09 pm

I think a good argument can be made that not only are the mechanical aspects (of things like algebra) necessary in order to do mathematics, but familiarity with them are the best way one has to develop actual insight and intuition into mathematics.

I see the same thing in computer science. I’ve often heard from students, after the semester is over, that such-and-such an exercise to write a program to implement a given algorithm gave them insight into how the algorithm worked.

I think that Marcus’s example of a cookie jar is a good one, and I’ve seen this as well. One of the hardest but most important things to learn in my field is how to formulize a problem so that some solution procedure can be applied. Knowing only the mechanics of the procedure doesn’t go very far.

45

Omega Centauri 05.02.13 at 10:51 pm

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.

46

John Holbo 05.03.13 at 1:21 am

My one complaint about Dragonbox, as a sheer drill exercise, is that it is a bit too quick to prompt you that, if you multiply one group on one side by tomatoface card (for example), you have to multiply everything on both both sides by tomatoface card. The kids may end up relying on the machine to tell them what to do, when it’s a matter of remembering to complete an operation.

There’s a good essay by Russell in which he talk about how the value of studying mathematics consists in realizing that these aren’t just rules that teacher has laid down. That is, it isn’t just political authoritarianism of the math classroom. It’s a glorious, Spinozist kind of freedom through recognition of necessity. Can’t remember which essay. The most obvious disanalogy between what my kids are doing and what ‘real’ algebra involves does indeed come down to this sense that the rules ‘have to be that way’ for non-arbitary reasons. They aren’t so much rules at all as expressions of the nature of the situation you are dealing with. You have to add to both sides because otherwise it wouldn’t still be equal. That most basic of thoughts.

47

Billikin 05.03.13 at 1:49 am

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.

48

floopmeister 05.03.13 at 2:23 am

“There were five jars of cookies on the table, each with the same number of cookies. Someone ate two of the cookies. Now there are thirteen cookies left in total. How many cookies were originally in each jar?”

Marcus: I just add the two back onto the 13 cookies, making 15, and then divide by the number of jars to get 5 cookies per jar.

Presumably this is correct :) but was what I did algebra? Or was it maths, or some other system of solving a real world problem? (Note – not rhetorical questions!)

Never took maths beyond year 10 and I still distinctly remember my teacher stating on my maths report card from that year “XX persists in resorting to lateral thinking to solve set problems”.

Could never understand why that was a problem – but then I guess that’s why my whole academic life has been in the humanities…

49

floopmeister 05.03.13 at 2:43 am

Make that – 3 cookies per jar.

Error of typing error, not calculation!

:)

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Marcus Pivato 05.03.13 at 3:28 am

Floopmeister @ 48:

I just add the two back onto the 13 cookies, making 15, and then divide by the number of jars to get 3 cookies per jar. Presumably this is correct :) but was what I did algebra?

Yes, you just did algebra, because this is precisely the way a trained “algebraist” would have solved the problem. You had 5x -2 = 13. You added 2 to both sides to get 5x = 15. Then you divided by 5 to get x=3.

You seem to be saying, “But I did none of these things! I didn’t write any equations. I just applied common sense.” But here’s the secret: a lot of mathematics is just common sense, formalized. First formalized, and then, through the power and flexibility of that formalism, ramified into possibilities far beyond the ken of “common sense”.

You have just shown how this is true for basic algebra. And the same is true for logic, Euclidean geometry, and even much of differential calculus and real analysis. I often think that one obstacle people have in learning mathematics is not realizing that much of it is simply the art of taking “common sense”, rendering it crystal clear and unambiguous through symbolic expressions and formal rules (which are both less error-prone and more efficient than informal common-sense reasoning) and then applying it to novel situations.

Billikin @47 mentioned the Wason task. This is a good example of how an apparently “hard” logic problem turns into “common sense” when it is given the right semantic interpretation. SusanC @32 mentioned “game semantics”; this is a nice way to make symbolic logic accessible by describing it as a “game” between two players, where each existential/universal quantifier becomes a position in the game where one of the players can make a move. Then subtle points of logic sometimes become “common sense” observations about who will win the game.

Indeed, one of the tricks of a good mathematician is to take an apparently abstract problem and find some concrete “common sense” way to think about it.

I still distinctly remember my teacher stating on my maths report card from that year “XX persists in resorting to lateral thinking to solve set problems”.

I’m sorry to hear you had such a teacher (but not terribly surprised). Higher mathematics is all about creative problem solving, and so-called “lateral thinking” is often the source of the eureka moments. 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.

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Peter Dorman 05.03.13 at 3:36 am

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.

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JSE 05.03.13 at 3:36 am

John, is the essay “The Study Of Mathematics?” Here Russell says

“One of the chief ends served by mathematics, when rightly taught, is to awaken the learner’s belief in reason, his confidence in the truth of what has been demonstrated, and in the value of demonstration. This purpose is not served by existing instruction; but it is easy to see ways in which it might be served. At present, in what concerns arithmetic, the boy or girl is given a set of rules, which present themselves as neither true nor false, but as merely the will of the teacher, the way in which, for some unfathomable reason, the teacher prefers to have the game played. To some degree, in a study of such definite practical utility, this is no doubt unavoidable; but as soon as possible, the reasons of rules should be set forth by whatever means most readily appeal to the childish mind.”

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floopmeister 05.03.13 at 3:41 am

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 :)

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John Holbo 05.03.13 at 4:18 am

Yes, that’s the essay I had in mind, JSE.

It occurs to me that a good way the dragonbox folks might have added a visual metaphor, that would have helped even more, would be introduce a see-saw. The box is on one end of a see-saw, and there is stuff balanced on both end of the see-saw, keeping it level. You have to isolate the box without unbalancing the see-saw.

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shah8 05.03.13 at 4:32 am

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.

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Z 05.03.13 at 9:05 am

@Manta

Uh? Abstract algebra is for the most part working with arbitrary rules.
You can invent some rules and study what happens:

Yes (though this is rarely what real algebraists do or how algebraic structures historically came under scrutiny). Nevertheless, if you wish to have something like a division in your system, you also need to to be able to distinguish zero and non-zero element, otherwise any and all statements about your set of rules will be true, so there is not much study to do (there is one exception that I will leave to mathematicians). Personally, I wouldn’t say the statement Everything is true is part of mathematics (but I guess that YMMV).

Note incidentally that the formalist position in mathematics philosophy is not math is a formal game, it is formal rules are the only thing needed to maintain consistency. Or in other words, it is when you realize that arbitrary rules might be inconsistent and thus that there is a need to ensure consistency in some way that you can become a formalist (or a platonist, or an intuitionist, or a pragmatist or whatever kind of philosopher of mathematics you want to be). Note also that the problem of distinguishing formally zero and non-zero elements is actually extremely hard and has direct consequences on “the real world”. For instance, you could ask your favorite math software to compute the integral from 0 to t of 1/(2+cos(x)) if you doubt me. The problem is easy enough that a good math undergraduate should be able to tackle it but CAS will frequently return a discontinuous output whereas the correct answer is of course continuous; the reason being that CAS are unable to understand a priori where the variable t is “allowed” to go according to the algorithmic formal rules they use to compute antiderivatives and so frequently chose incorrect integration constant at branching points. So not only is the problem of consistency of formal rules a real problem, it is a hard one with immediate real consequences.

@Ben Wolfson

Isn’t this the famous drinker’s theorem (hence true, and famously provable purely syntactically, but not quite trivially)?

@JohnHolbo

The most obvious disanalogy between what my kids are doing and what ‘real’ algebra involves does indeed come down to this sense that the rules ‘have to be that way’ for non-arbitary reasons. They aren’t so much rules at all as expressions of the nature of the situation you are dealing with. You have to add to both sides because otherwise it wouldn’t still be equal. That most basic of thoughts.

Exactly.

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Jeffrey Davis 05.03.13 at 8:47 pm

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.

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SusanC 05.03.13 at 8:53 pm

A further thought: the idea of soundnessmight capture the idea that although the rules of algebra are in some sense arbitrary, they can’t be anything at all. (e.g. we can’t do away with the prohibition on dividing by zero so long as 0.a = 0.b for some a not equal to b. But if you’re willing to change what multiplication means as well, you might be able to get away with it).

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Billikin 05.03.13 at 11:26 pm

@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.

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Billikin 05.03.13 at 11:39 pm

floopmeister: “I still distinctly remember my teacher stating on my maths report card from that year ‘XX persists in resorting to lateral thinking to solve set problems’.”

Marcus Pivato: “I’m sorry to hear you had such a teacher (but not terribly surprised). Higher mathematics is all about creative problem solving, and so-called “lateral thinking” is often the source of the eureka moments.”

I have a friend who did well in algebra but still has blocks in arithmetic. “I can’t carry the one,” she says. In arithmetic class, when faced with multiplying 4 times 48, she multiplied 4 time 50 and subtracted 8. Her idiot teacher scolded her for doing it the wrong way.

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Kiwanda 05.04.13 at 2:03 am

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.

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Anand Manikutty 05.04.13 at 12:59 pm

> 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.
There is a mathiness to it (mathiness is to math as truthiness is to truth), but it is not quite mathematics. If the player actually proves a theorem going through the formal steps of theorem proving, that would be one thing, but unless the exact steps are shown, it is not possible to know if all the various facets involved in the proof are being understood as they are meant to be understood.

Mathematics remains the only way that a number of different facets of a system can be systematically and precisely interacted, and for this reason, it is important, as a philosopher, for me, at least, to see theorem proving being shown explicitly as a series of steps as opposed as to an implicit thought sequence in the mind of the game player.

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Ken 05.04.13 at 2:15 pm

Does anyone else remember WFF-‘N-PROOF?

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John Holbo 05.04.13 at 2:52 pm

“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.”

One nice feature of Dragonbox is that you get an extra star for clearing the level in as few moves as possible. (It tells you how few it can be done in. You see in the image in the post that it’s 12, for that particular screen.) This encourages good thinking about strategies. Although, oddly, I’ve twice cleared a level with a ‘better than possible’ bonus rating, which strikes me as a bit over the top.

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merian 05.04.13 at 4:48 pm

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.

66

bigcitylib 05.04.13 at 9:06 pm

Reminds me a bit of a high-tech version of Wff n’ Proof, which I played back in the 70s (though not very well). Why is author so surprised that we can use games or other exercises to teach useful stuff. Didn’t he watch Karate Kid yonks ago?

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Harold 05.04.13 at 10:12 pm

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

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John Holbo 05.05.13 at 2:21 am

“Why is author so surprised that we can use games or other exercises to teach useful stuff. Didn’t he watch Karate Kid yonks ago?”

What did you think I meant by “Very ‘wax on, wax off’”? Haven’t you seen Karate Kid? I thought everyone watched Karate Kid yonks ago.

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js. 05.05.13 at 6:41 am

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.)

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Z 05.05.13 at 7:33 am

For example, I wouldn’t have thought it possible for one to be “doing algebra” without at the same time “doing math(s)”

I think it is abundantly clear that I suggested one could do “algebra” without at the same time doing “math”, or for that matter doing “math” without doing math, only in the limited sense that what is named “algebra” and “math” in elementary and junior high schools is sometimes so different from the real deal that it becomes its polar opposite. See the moving testimony of Anarcissie at 29 for a prime example.

In fact, I find your analogy quite good. Students (say in junior high-school) who learn to “solve for x” by reflexively performing a series of operations which have inexplicably been declared licit by the instructor are doing algebra exactly in the same sense that 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. If it were a well-established cultural norm in your country that such courses go under the name “epistemology”, wouldn’t you consider the possibility that some students are doing “epistemology” but not philosophy?

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william casey wesley 05.06.13 at 7:51 pm

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.

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Manta 05.07.13 at 7:58 am

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

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Z 05.07.13 at 12:17 pm

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

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Noah 05.07.13 at 3:19 pm

I’d phrase the point that several people have made above somewhat differently.

When your daughter plays this game she’s doing the same thing that most students do when they do algebra, but she’s not doing the same thing that someone who actually understands algebra is doing.

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js. 05.07.13 at 3:59 pm

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.

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Kiwanda 05.07.13 at 8:40 pm

Students (say in junior high-school) who learn to “solve for x” by reflexively performing a series of operations which have inexplicably been declared licit by the instructor are doing algebra exactly in the same sense that 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.

This is a little strong. The problem “Given a condition on x, solve for x” is often a far cry in difficulty and creativity from rote recitation of dates. And how much more is needed for the students to be “doing math”, in the sense that they understand why the operations they are doing are “licit”, than just noting that equality is preserved?

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Harold 05.07.13 at 9:03 pm

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.

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Z 05.07.13 at 9:11 pm

Kiwanda,
I didn’t want to sound too harsh: knowing (in some pragmatic sense) how to solve for x is not valueless, and already an important milestone if we are indeed speaking about junior high-school students. If they are able to explain why the operations they are performing preserve equality, then they are unquestionably doing math (and I would say they are showing some talent towards it). If moreover they know when they need to check that the answer they have found is indeed a solution (a chain of algebraic operations is typically a chain of implications, one has to check the reverse implication in the end), then I would go further and say that they are mathematicians (as a friend of mine recently joked, a mathematician is someone who checks her answer once she has completed a Sudoku puzzle; until that, all she has proved is unicity of the solution).

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