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Transcript of Math Effect
MATH EFFECTDesigning games to teach higher-level mathematics
By Trevor Burnham
I. Prologue
Game designers are under an insidious illusion: They believe that setting matters. Kids,
especially boys, love games in which they explore the depths of space, hunting for
ancient artifacts and vaporizing anything foolish enough to get in their way. Those kids
rarely feel the same passion for solving equations. So, the conventional wisdom goes,
let’s set those math problems in space!
! Math Blaster and, more recently, Dimension M have taken this approach, tweaking
the shoot-’em-up format by overlaying simple math problems on the targets. The goal is
to shoot the right ones.
! Is this attractive to kids? Undoubtably. Engaging? For a while. But it’s also a dead
end. Training kids to blast multiples of 2 as fast as they can will teach them to
distinguish even numbers from odd ones, but it will never answer the eternal question
of “Why?” The opportunity to instill an intrinsic love of mathematics is lost.
! That these games pale by comparison to non-educational action games is a well-
trod truism (the “spinach sundae” problem), but I would offer a different benchmark:
the thousands of whimsical puzzles churned out by Martin Gardner over half a century.
As Ronald Graham of UC San Diego famously said, “Martin has turned thousands of
children into mathematicians, and thousands of mathematicians into children.” These
puzzles are open-ended, with perhaps a hint or two, requiring the solver to carefully
consider possible approaches and probe them for promising avenues. This is how
mathematics is done at a professional level, though you certainly wouldn’t know it
from the endless drills that make up the entirety of today’s education game market.
Why can’t the experience of a Martin Gardner puzzle be encapsulated in a video game?
! I’ll explain how such a feat might be pulled off in part IV. But first, an intermezzo
on a subject near to my heart: What makes a game compelling?
II. Immersion
Part of the problem with using arcade button-mashers as the chassis for electronic math
tutors is that these games rarely hold anyone’s attention for long enough to teach a
complicated subject. Dimension M offers a modicum of storyline, but even a child can
tell that it’s no Lord of the Rings.
! We live in a world filled with sophisticated games offering fleshed-out characters
and epic plots. Education games are rightly consigned to collect dust on a less
glamorous store shelf. The crudeness of education games is an obvious consequence of
their niche status (and concomitantly smaller budgets), but I contend that their niche
status is a product of their crudeness. This cycle must be broken if education games are
to offer players an experience that will keep them attentive for more time than it takes
them to exhaust their trigger fingers. Education games must aspire to be immersive.
! One of the most immersive games of recent years is Mass Effect. Its technical
brilliance—cinematic visuals, orchestral sound, galactic scope—are matched by an
awesome amount of imagination. A staff of four writers painstakingly laid out a game
universe with several alien races, each with their own history and culture, and scripted
hundreds of unique characters to converse with. The game gives you an extraordinary
amount of freedom within the classic save-humanity-from-extinction framework: You
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can customize your character’s appearance (including gender), develop new skills that
determine the paths available to you, and make important ethical decisions.
! For instance, midway through the game, a loyal friend, a Krogan named Wrex,
confronts you when you realize that you have to destroy an enemy research facility. The
Krogans, unable to reproduce after the Turians attacked them with a biological weapon,
are slowly going extinct. Your arch-nemesis, Saren, has figured out a way to breed
Krogans in this facility so that he can use them as soldier-slaves. Wrex holds you at
gunpoint, insisting that you save the facility—an impossible demand. If you’ve focused
entirely on developing your character’s combat skills, then the encounter can only end
in bloodshed. If, however, you’ve picked up some diplomatic ability, you can reason
with him, persuading Wrex that he has to accept the tragic fate of his people for the
greater good of the galaxy.
! A naïve game designer would interpret a player’s love of Mass Effect as a love of
games that involve blasting aliens. I contend that the space operatic setting matters far
less than the sheer amount of imagination that went into bringing it to life.
! Consider the Fallout trilogy, with its darkly humorous take on a post-apocalyptic,
mutant-filled wasteland. Or the Grand Theft Auto games, each of which is a violent
Horatio Alger story of rags to crimelord riches told in a modern American city. Which
setting is “better”? Would grafting math problems onto mutants or rival gang members
make them more appealing? These games are not successful because they give players
new and different things to shoot. These games are successful because their backdrops
are so detailed, so believable. They reward the most urgent drive a child has: curiosity.
! One recent math game, Lure of the Labyrinth1, does offer a compelling storyline
with a vast, whimsical world for the player to explore, including dozens of characters.
Interaction is somewhat superficial, but sufficient to motivate exploration. The puzzles
in the game introduce concepts drawn straight from the curriculum, like measuring the
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1 Link: http://labyrinth.thinkport.org/
area of rectangles and solving linear equations, by analogy. Other than numbers, no
mathematical notation is ever used. The player is tricked into learning math by learning
to play games with mathematical rules. This subterfuge makes for a compelling play
experience, at least for a while. But the game’s content cuts off abruptly at the 8th-grade
level. Why?
III. Abstractness
The foundations of calculus eluded mankind for many centuries after arithmetic had
become essential to human endeavors. It’s not hard to see why. While arithmetic
consists of operations for manipulating numbers—addition, subtraction, multiplication,
division—calculus is about manipulating functions. And functions are a purely abstract
concept, one that children are not exposed to until after they’ve demonstrated a mastery
of arithmetic, the most tangible part of mathematics.
From The Manga Guide to Calculus by Hiroyuki Kojima et al., published in Japan in 2005 and translated into English in 2009. The Manga Guide series has been wildly successful by combining whimsical elements with rigorous explanations and practical examples on a par with conventional textbooks.
At first, a child can only imagine a function as something totally immutable, a black box
that takes one value in and spits another out. This is a crippling notion, because
manipulating functions is a key part of higher mathematics.
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From The Manga Guide to Calculus.
The child quickly hits an abstractness barrier when the core concepts of calculus, the
derivative and the integral, are introduced: These are functions of functions. How can
such a thing be?
! The drills of Math Blaster and the numerical mini-games of Lure of the Labyrinth
are inadequate to bring about such a cognitive shift. No longer can quick thinking be
favored over deep problem-solving. Nor can the dreaded notation be hidden from view.
From high school on through PhD programs, students solving calculus problems are
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implored to show their work. Games have never even allowed any notion of “show your
work.” This is unfortunate. Mathematicians have used arcane notation like
sin2 x dx∫
for centuries, and with good reason: Once learned, this is an elegant and compact way
of expressing difficult-to-understand concepts.
! Games could actually prove to be a fantastically intuitive way of introducing
functions, and their corresponding notation, by analogy to something familiar:
machines. A function is, simply put, a machine for turning one thing into another. That
a machine exists that can turn one machine into another, according to certain rules,
should not be at all surprising.
! Under this analogy, the notation above is equivalent to the following: Take your
sin x machine, and feed it into your x 2 machine. This churns out a sin2 x machine. Then
drop that machine in the funnel of your integration machine, the one with the big S-like
squiggle on it, and you get a machine corresponding to the expression shown above,
one that you could feed particular values of x into, if you so desired.
! This is not to say that I’m proposing a new incarnation of The Incredible Machine
(one of my childhood favorites) with “educational” labels grafted onto the Rube
Goldberg contraptions. What I’m proposing is closer to StarLogo TNG2, but with the
constraint of solving endogenous puzzles with your logical bric-a-brac rather than
facing the daunting task of creating an entire world from the ground up.
IV. Interfacing
The central goal game designers should keep in mind when constructing a true math
game, as opposed to a computation game like Math Blaster or Number Crunchers, is to
make the interface as close as possible to “enhanced pencil and paper.”
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2 A graphical programming environment aimed at children: http://education.mit.edu/drupal/starlogo-tng
! By that I mean, the game should support the same operations that a
mathematician would use a pencil and paper for, while taking advantage of the
computer’s ability to handle the more mindless aspects of problem-solving (such as
computation and graphing, the bread and butter of early mathematics classes). This,
after all, is the way that real mathematicians work. Children believe that math is boring
because they mistakenly associate the word “math” with “computation,” when in fact
mathematics would be more appropriately defined as the art of solving problems by
reducing them into smaller problems that we already know the solutions to.
! I hope I can elucidate the peculiar interface I’m describing with a few examples.
Tutorial: Apples to Apples
The player is dropped into a mysterious world in which people rely on wizards to carry
out their mathematics. These wizards have little power over physical objects, so they
usually create “abstractions” based on those objects, ethereal floating things that the
wizards can then manipulate to solve problems. So there are two kinds of spells: Spells
that can be applied to physical objects, and spells that can be applied to abstractions.
! You, the player, begin with two spells: count and add. A villager approaches you
with a simple problem: He has two buckets full of apples, but he doesn’t know how
many apples he has in total. You can solve this problem in two ways: You can either add
the two buckets of apples, which collide into one overflowing heap (to the villager’s
annoyance—again, your magic only works clumsily on physical objects, when it works
at all), then count the result; or you can count each of the two buckets, causing numbers
and units (“7 apples” and “9 apples,” say) to float above them, then add the two counts.
In either case, you and the villager can both clearly see the wispy, glowing result: “16
apples.” The villager thanks you profusely and collects the ethereal result in a glass jar.
He then asks if you might be interested in a job calculating his orchard’s output. You
accept, and after you carry out some repeated addition, you earn the multiply spell.
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! As you prepare to leave the village, the villager asks for one more thing: He
needs help computing the amounts of the ingredients for a stew he’ll be cooking for a
gathering after you leave. He’d ask you to just compute it for him, but he doesn’t yet
know how many people will attend. What he really needs is a machine that would let
him multiply the amount of each of the ingredients by the number of people attending,
when the number of people is fed into the machine.
! As he’s saying this, a strange thing happens: The phrase “x people” appears in
front of you, crystallized from his words. This bit of involuntary magic indicates that
you’ve earned a most important spell: denote. You’ve leapt from being a mere
arithmetician to a true algebramancer!
From The Manga Guide to Calculus.
With this new magic, the task at hand is easy: Cast a count on a serving of stew he made
for himself, causing a lengthy list of ingredients to appear in the air above it; use your
multiply spell on the ingredient list and the phrase “x people,” and you get a new list
with an x after each of the amounts.3 Storing this result in a spare machine allows the
villager to use it on his own: All the marvelous device requires is for him to state a
number of people, and it will emit the proper recipe. A useful function indeed!
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3 One niggling detail: There should also be a spell for separating “x” from “people,” that is, of making x unitless. Otherwise, the resulting recipe would say things like “1/2x cups of flour-people”—and what is a flour-person? Teaching the proper use of units is important and should be dealt with early in the game. I’ve heard many elementary math teachers abuse units by posing such problems as, “What is five dollars times four dollars?” To which the only correct answer is: “Twenty square dollars.”
The Length of the Diagonal
A short time later, the player is asked to help a group of allies calibrate their catapult.
They have a particular target in mind, which they know to be 3 miles north and 4 miles
east, but they need to know the exact distance. They show you this map, which is to
scale, but an enemy curse has made the distance line immune to magic, so a simple
count won’t work:
The player can use a square spell on one of the lines to turn it into a square with sides of
corresponding length (or, equivalently, multiply the line by itself). The player could,
alternatively, count the line to abstract its length, then square the resulting lengths. After
performing an add, the player can use the square root spell (which is simply the inverse
of the square spell, just as divide is the inverse of multiply and subtract is the inverse of
add) to get the desired distance.
! Of course, this is all a lot of work, so surely some sort of diagonal spell is in order?
In order to earn that, the player will have to prove that the Pythagorean Theorem holds
in general, not just in this one special case.4
3
4?
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4 I won’t go into the details here, but a good starting point for implementing this would be the proof proposed by then-congressman James A. Garfield in 1876. Four years later, he would be elected president. The proof can be found at: http://jwilson.coe.uga.edu/EMT668/emt668.student.folders/HeadAngela/essay1/Pythagorean.html
Crossing the Bridge
This final example illustrates the power of the “enhanced pencil and paper” metaphor:
! Suppose you have a lion, a lettuce, and a llama, and you have to get all of them
across the bridge. Each of these items is enormous, so you’ll have to take them one at a
time. There are only two problems: If you leave the lettuce alone with the llama, you’ll
be left with one very happy llama, but no lettuce. Likewise, if you leave the lion with
the llama, there’ll be not a trace of llama to be found on your return.
! This retelling of the classic wolf-goat-cabbage puzzle is presented by Ian Stewart,
master puzzler Martin Gardner’s most prominent successor, in Another Fine Math You’ve
Got Me Into... (1992). While the problem is not originally his (in fact, it’s attributed to the
8th-century mathematician Alcuin), the solution he presents is an ingenious use of
mapping from one domain to another to shed light on a problem. In this case, we must
transform the dilemma from the world of logic to that of geometry:
From Another Fine Math Youʼve Got Me Into...
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! We’ve done three things in creating this illustration: We’ve taken the eight
possible states of the problem, encoded those states as three-dimensional vectors, and
drawn connections between them representing possible moves, dashing the ones that
would cause a painful end for our llama or lettuce. Now it’s clear: As long as we move
along the undashed lines from our start state (0, 0, 0) to the goal state (1, 1, 1), we can’t
lose! The two direct paths immediately leap out at us.5
! The question is: How do you take this amazing pencil and paper approach, and
make it possible to express through a game interface? What we need is a graph spell that
can show us the possible states and connections of a problem. The result would be a
tangled web, but we can use a vectorize spell to make it into the nice cube that Stewart
gives us. The real trick is to make this combination non-obvious by devising problems
in which each, in itself or in combination with other spells, is useful. Fortunately, the
branches of mathematics have a way of colliding in the face of interesting problems.
! In particular, all roads lead to—and from—calculus, with its two awesome spells,
differentiate and integrate. To find a plethora of puzzles that these spells can solve, simply
walk into any academic department in the sciences or social sciences.
From The Manga Guide to Calculus.
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5 Later in the same chapter, Stewart presents a similarly elegant geometric solution for the Tower of Hanoi puzzle. Interestingly, that classic was featured in Mass Effect.
V. Closing thoughts: “First, do no harm”
I’ve attempted to lay out a set of idioms for shifting mathematical video games away
from rigid computation and toward practical problem-solving, exposition and proof.
The first step is to embed the problems in a believable, immersive universe. Judging by
the accolades Lure of the Labyrinth has received, there is already some movement in this
direction. The next step is to give players an interface that allows them to transform and
reduce problems in creative ways. If a problem can be solved just by staring at it, then
it’s probably not very useful.
! Education games must reverse the damage wrought by games like Math Blaster
and, for that matter, many elementary school teachers, who present “math” as a
hopeless race against the calculator. It would be better for children to learn no math at
all than to learn only its most tedious elements. In the words of Bertrand Russell:
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty.”
From The Manga Guide to Calculus.
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Addendum: Managing complexity in mathematical interfaces
The astute reader may have noticed that I did not describe exactly how the player
selects a particular spell, and may have rightly inferred that the game I envision would
require dozens, even hundreds of possible choices at any particular point. If the game
starts to look like a programming language (even a visual one like StarLogo TNG), can it
still be accessible to people who aren’t already master mathematicians?
! I confess: This is a difficult problem that I have not addressed. As I see it, there
are two ways to tackle it.
! The first is the way that most RPGs, and indeed most software, have organized
their options: nested menus. Today’s children have no difficulty navigating through the
toolbar to locate the most obscure features of Microsoft Word. And as the spells would
be introduced one at a time, with one or two immediate applications, the learning curve
would be somewhat mitigated. However, this would sorely crimp the pace of the game.
! The second and more compelling way is through gestures. This, after all, is how
“real” wizards do it. (They also use chants, which, in light of Tom Clancy’s EndWar,
might be a viable input mechanism as well.) A handful of games, most notably Black &
White, have used mouse gestures for input. Gestures allow menus to be abolished
altogether, freeing the player from the distraction of seeing anything but the game
content. There is also something tremendously satisfying about using gestures for input,
as anyone who’s drawn a swirl to unleash a slew of fireballs at their enemies in Black &
White can attest. The pleasure of gestures is further enhanced by the use of
touchscreens, which are now ubiquitous on high-end mobile devices and becoming
increasingly common on personal computers. As a further advantage, gestures for a
math game could be made to closely parallel mathematical notation.
! While gesture input is not a prerequisite for a satisfying higher math game, it
would be a tremendous enhancement and a significant step toward bridging the gap
between computers and the as-yet-undefeated combination of pencil and paper.
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