How would you critique math as a game?
I have no idea what you mean by math. Like what people do in school?
By Math, I mean what people do in school.
Low depth, it’s rote. A lot of math that people perform in school have the issue that it’s searching for static solutions to static problems, more like a puzzle than a game. A lot of it involves following exact steps, memorization of things like multiplication tables.
Trouble is that to build math as a skill, you don’t have many clear alternatives. To guarantee understanding, you need to present an exact problem with a clear solution. Trouble with that is that it doesn’t necessarily promote understanding of the material either, it’s just about getting the answers.
The mathematical system as a whole has a tremendous level of depth as a system (though it’s practically a tautology to say this), but individual math problems as presented have basically no depth. Apply algorithm, receive solution. Much like a hand-holding tutorial in many games.
I myself didn’t really enjoy math until I learned to program. I was advanced in mathematics in high school, finishing calculus in junior year, though I never really enjoyed math class. I learned in junior year also how to apply what I had learned in trigonometry (pre-calculus) in programs.
I had been programming in flash and something had finally clicked about how to use variables that never made sense for me before, and I was using it in all different ways, starting with a really basic problem of trying to make an object smoothly move towards the cursor.
In middle school I had tried programming an MMO chatroom thing (way out of my depth, I know, I never got anywhere close to network code or even typing in text) and in the process I had made a little enemy character I wanted to move closer to the main character, so I just programmed it to move +5 on the X axis if the main character’s position is greater than it, and vice versa for Y axis and lesser positions. I expected the result to be the enemy just moving smoothly at the character no matter which direction they went, but instead got a weird diagonal focused movement (which I’d later learn is due to a similar reason as the straferunning bug).
In high school I tried again, this time making it so an object would find the distance on the X and Y axes, and move half the distance. It moved smoothly now, but bizarrely, slowed down as it got closer to the mouse (zeno’s paradox anyone?).
Later in high school I realized I had to use trigonometry, treating the distance and direction I wanted it to move as a polar coordinate vector, then converting that to a Cartesian coordinate. This means first finding the angle between the mouse and object by taking their respective cartesian coordinates and running them through arctangent (cotangent), then running sine and cosine on the resulting angle and multiplying the results by the speed I wanted the object to move at, and adding the results of those two operations to the X and Y position of the object. And it worked, it finally moved in a smooth pattern towards the mouse at a steady speed.
And finding that answer for myself was fun. Of course, once the object got within 5 units of the mouse it started spazzing out, because it would move past the mouse and then back again, because it couldn’t center on the mouse’s position, but that was easily solved.
Finding that solution, thinking through the problem like that, that’s not something you get in math classes. Taking programming classes in my senior year of high school, even those weren’t really the type of exploration I had done on my own, because specific things were expected out of the code, and though I was allowed to reach the solution a number of ways, the actual process of coding wasn’t nearly as open ended.
A lot of games require the use of math in one way or another. RPGs are especially notable. All games involve math in some way, but most of the time it’s hidden internally inside the system. Even in games that explicitly show the numbers at work, performing operations on those numbers generally isn’t necessary for the end user.
For example, in Dark Souls and Demon’s souls, rolling is determined by what percentage of your equip burden your current equipment takes up in weight. So you have a slower roll if you’re over 25%, 50%, or 70%. This means players need to make an estimation of what half their equip burden max is versus the current number. Though there’s an easier method, they can just try on equipment and roll to figure it out brute force. Dark Souls 2 goes further and does away with this by simply displaying the percentage of your equip burden currently consumed on the equip menu directly.
In fighting games, damage scaling is a big concern during combos. Combos typically scale damage either per-hit or per-move. What this means is that adding additional hits/moves to your combo will make later hits weaker, usually meaning you want to stick the harder hitting moves earlier into your combo rather than later, and avoid low damage multi-hitting moves where possible. This can mean that in some cases, you get more damage out of a combo by subtracting moves even though the combo is longer. In some games, particular moves themselves have proration values, like guilty gear, blazblue, and persona 4 arena all have extra proration on their light attacks. This means that the combo punch, kick (1 hit), slash, hard slash with Sol does the same damage as slash, hard slash (68 damage). P K S S H (84 damage) does less than SSH (92 ). Again, the trouble here is that people don’t necessarily need to do math to understand this, they can apply the abstract principle, and try things out brute force style. People don’t seriously sit there adding up all the numbers and doing the math for how much the damage is scaled each hit.
A similar deal goes back to RPGs, sure there’s a lot of math that went into deciding how much EXP is used to level up, and which stats should be gained, but people know they can just level up more and they’ll be strong enough to win.
Smash Bros is probably one of the best example systems for ridiculously complicated mathematical underpinnings that people don’t need to understand to play the game. The amount of knockback inflicted on opponents is determined by a formula that combines the damage of the target with their weight, and multiplies it by the knockback scaling of the move, added to the base knockback.
This knockback value is then multiplied by .4 to determine the number of frames the target remains in hitstun.
The direction of knockback is essentially a polar coordinate vector, that is affected up to 18% by DI, which is when the control stick is held perpendicular to the angle of knockback (arbitrary to the move), proportionally to how perpendicular the stick is. It applies this final polar coordinate as cartesian X and Y velocity values to the character, which steadily degrade due to air control (maybe air friction too) and gravity.
From a player’s perspective, you don’t need to really know any of that. You can play the game perfectly fine without running that formula or understanding any part of it. You only need to know that at 100% or so, a strong hit’s gonna kill the guy. Physics in real life are complicated, but you don’t have to understand the mathematics to understand physics. Knowing the math is important for a NASA engineer, not for guys playing soccer.
Here’s a practical example that I’m stealing from Sirlin’s podcast after having learned it the practical way in StarCraft: Brood War. When you have a bunch of workers on a mineral deposit, and the opponent is harassing your workers, do you pull them all off the mineral deposit to avoid losing any of them, or keep them working so you’re continually earning minerals at the cost of maybe one or two workers? Answer: you pull them all off, because sacrificing your intake of minerals for a brief period of time is better than your actual rate of production going down.
Maybe we could reintegrate this mathematical understanding more into the actual application side of games? I considered this in the past, show the damage calculations more and have the player manipulate them more directly, but I never got around to a practical means of implementing this. Computer games can integrate real-time tests of mathematical skills in ways that homework problems can’t. You have stuff like Brain Age, which I wouldn’t call a good game, but it does the thing it needs to do well. Grading based on it, or something like it is tricky though, because people can suffer on the real-time side even if they understand the math, and we want to hand As to people who are merely competent and understand thoroughly, not who are exceptional and perfect.
How can you make a good game that teaches math too? Something to think on in the future. I fear the goals may be incompatible and a good game would be something too loose to pass a grade on, despite obviously testing the skills in question.
Common Core Controversy
I feel like this is a kind of inevitable side topic. Common Core exists to address a lot of the concerns I have about how math is taught. In the old Math curriculum, kids are expected to perform algorithms on the numbers instead of developing strong understanding of what the numbers are and how they relate to one another, which is why I get shocked, scared, and disappointed when I hear about people in their first year of college only just taking Algebra 2. (or Algebra 1!!) Having gone to an art college, a constant joke whenever anything math related came up was that we were art majors, and not good at math. hahaha. (read as a dry laugh)
So the implicit goal of common core is to attempt to show different ways of solving simple arithmatic problems so that kids can understand the units being manipulated, and thereby be prepared for algebra rather than struggling with variables that they can’t plug into their calculators. Calculators themselves are kind of an issue too. With them being so readily available kids want to take the fast and easy solution to a given math problem.
A lot of the tricks common core attempts to teach are things I picked up incidentally. I don’t use the standard carry methods of addition and subtraction, even though I’m capable of them.
Something I personally disdained in school was the necessity of showing my work, because I hate writing by hand. I’m slow at it and have poor handwriting. Common Core, as well as other homework and class work requires that you show your method, which is painful for students who are good at mental arithmetic (which in my opinion is the skill that actually needs to be established by the end of education, but which is regrettably indistinguishable from cheating)
Common core requires more writing out of process and more steps to achieve an answer, so naturally it’s a bunch of shit that I as a kid didn’t want to deal with. The methods involved make sense and solve clear issues with standard algorithms (like what a pain it would be to use the borrowing method to get 1000-3)
I think ultimately the problem with common core is that a lot of it is communicated poorly, even though it’s trying to do something that’s ultimately the right thing to do. The more fundamental problem with the curriculum is that having kids run through different methods for solving math problems doesn’t necessarily teach them how the numbers work in an abstract way, give them deep understanding of mathematical units. It probably just means they’ll be following more simple algorithms for deriving the answer rather than reaching the deep understanding that they can derive the answer multiple ways, and some methods work better or worse for different problems, then matching those methods to the problems in question.
And that’s hard to teach, kids are like a black box, like a hundred thousand black boxes from a bureaucratic perspective, and we need to inspire a change within the black box and check the change has actually occurred, but the change cannot be systemically tested for.
The end result of this teaching method might be that we’re overloading the kids with a ton of extraneous information and not necessarily teaching them any better for it.
I went to the conference Games for Change in 2014 (got a press badge through a former GYP member, wish I was there for Raph Koster’s closing speech the following year, I got stuck with Leigh Alexander) and attended a couple talks by educational game makers who took the approach of making games with a theme like exploring your inner biology, then info dumping on the side when the game itself didn’t really have anything to do with biology. Making games about math is hard, and will probably require a more formal understanding of game making to really accomplish. I spoke on the side with some people involved in education and they weren’t very happy with it either, claiming it was more a marketing sham. Make the educators very happy, give them clear signs of progress from kids, but not teach them through the material.
Gamification is a popular buzzphrase, but in practice it doesn’t actually deal with making games (or making activities more fun). It’s more often about theming and reward structures than games. It’s not interested in making the process itself fun, rather in making enjoyable things complement the process, or giving purpose to the process. I guess this is a very practical downside to the current non-formal way game development is looked at.
The most sensible method in my opinion to reform math education is to find students that are behind the curve, and investigate what teaching methods are capable of bringing them up to standard or above the standard. Once a method is found that works on those students, see if the results can be replicated with a new group of students (to control for the risk that what helped the first group improve was merely additional tutoring time).
What common core looks like to me is an experiment that wasn’t properly vetted ahead of time. It’s like trying to do what you think is the right thing, rather than finding the right thing and doing it. Not playing to win.
Might be a fun research project to look up every math game and compare methods. Older ones to my recollection used theming and basic math problems, but lacked strategy + depth.
Oh and Zoombinis are on steam, that’s nostalgic.