I’m an artist by trade, and I love creating things. I grew up wanting to be a Disney animator, and my BFA degree is in computer animation. I also happen to love math, and have spent a fair bit of time as a math tutor. I grew up loving math and the intersection that it has with art in things like Origami, the Golden Ratio and Fibonacci numbers (there is a TON of math in art). I love being able to take “right brain” and “left brain” notions and use them to reinforce each other. It saddens me to see students that I tutor fear negative numbers or fractions. Math and art are both deeply inquisitive ways to look at the world around us, and both have a great deal to offer to those trying to understand life and make their cognitive functions more effective, and to each other as disciplines.
Raph Koster brought “Lockhart’s Lament” to my attention, and it resonates with my experience. I managed to find a deep fascination with math early on, and I persisted with it despite my deep disgust with memorization and busywork. Of course, so-called “Investigations Math” is worse, as it doesn’t bother actually teaching anything, leaving students to figure things out on their own. The truth is somewhere in between; students need to learn how math works, but more important, they need to learn why, and how to extrapolate the critical thinking required for mathematical analysis into other aspects of life. Students need to learn how to think, not how to regurgitate.
Of course this has game design applications, since that’s what I talk about around here. Game designers need to give players tools and show them how they work, then stand back and let players play. Good math is playful, good art is playful. It’s the experimentation and discovery that makes them both fun. Games are very similar; the exploration of the game functions and artistic content is a significant part of the fun that can be derived.
To be fair, that’s not the only way to play (or design) games, or the only reason to do so, but it always bothers me when games quickly devolve into reflex checks or memorization hurdles. Likewise, tightly straightjacketed games with little room to explore and experiment don’t hold my interest for long. This is why level-gated games like WoW bother me; I’ve got to jump through the highly repetitive hurdles of leveling (with very repetitive combat) to see more content and get on with exploring and experimenting.
I think it’s no mystery why The Incredible Machine is one of my all time favorite games, and more recently, why Boom Blox and Crayon Physics are high on my list.
I wish people wouldn’t be afraid of math, or dismiss art as frivolous luxury.
I suppose there’s a tangent to be run exploring linguistics and how writing and wordsmithing is similarly creative and playful while being fascinatingly structured. I do lean on alliteration and creative use of words around here, after all. Perhaps that’s best saved for another article, though.
Tish Tosh Tesh 
Have you read Godel Escher and Bach? I think you would really like it. These are the themes is basically talks about but the book is so comprehensive it’s hard to pinpoint an actual subject of the book. But basically, there are common threads in the work of all three, and these also extend to life itself.
I had a history class once where the professor assigned a list of topics, and all he wanted was 3 quick sentences or paragraphs, what is it, why is it important, and an anecdote. I found it depressing how many students around me were having trouble just understanding that they had been assigned storytelling not copying.
I rather tend to agree with you, good design is playful. It’s not about what you want the players to do, rather about the many fascinating things players do when they are empowered to do so. Someone, a modernists of some note I remember, once said that the role of a designer is to cure the disease of bad design around us. In many ways I agree with this, because it’s not a disease that attacks our body directly, but attacks our creativity and imagination. Bad design can make people fear to pick something up, or fear to play with it and turn it on it’s head. It makes us boring.
Not sure if I’m still on topic, but thought I’d add my .02.
In general, there’s a focus on knowing instead of figuring out in most areas, including math and MMOs. A while ago I wrote about programming tests and how writing program snippets with pen and paper and no reference was kind of silly. Real game programming is about tackling a problem and figuring out a solution that fits within the parameters you are given. This often requires a lot of investigation. Sure, if you don’t know how to use common data structures you are probably not going too be as efficient, but you can still be a top-notch problem-solver despite that and accomplish your tasks.
We see the same thing in games. I’ve read a few complaints about “quest helper” type plugins and features in games have taken away a lot of the mystery and fun of exploration in the game. Quests become more like a checklist instead of something fun to do. Again, the focus is more on knowing and completing instead of exploring for reasons of efficiency. The sad part here is this is what many players want, a path laid out before them to follow. Even if you love exploration, your friends and/or competition to get to higher levels may mean that you use the quest hints anyway (even if indirectly from friends who use it too lead the party around).
Interesting to think about the issue, though.
Godel Escher Bach is one of those things I haven’t gotten around to, but really should. Thanks for reminding me about it!
Sara, you’re definitely on topic. Thanks for the comments. I definitely have some education anecdotes that I’d love to correct, and indeed, a big part of that would be to foster experimentation. I feel like Ms. Frizzle sometimes. Learning should be fun, not drudgery.
Brian, I understand the cultural biases in that direction, but I lament them. They are to society’s detriment. I remember a Java class I took where all of the tests asked us to handwrite code. I thought it was lame then, and it was a significant part of why I didn’t take many CS courses in college. I can learn more on my own (faster and cheaper), tinkering with books and experimentation.
Oh, and interestingly, I tend to think that those who are at the top of their game in any competitive venture are those who manage to explore the cutting edge and develop those things that aren’t part of the rote “way things are”. Like Michael Jordan or Michaelangelo, their ability comes with plenty of practice of relatively mundane tasks, but those who truly excel tend to go way beyond the normal, accepted way of doing things.
The correlation, of course, is that those who merely perfect what *is* don’t tend to be among those who redefine what *may be*.
“or dismiss art as frivolous luxury.”
I think you said it best right there. Too often there is a section of society that thinks anything that doesn’t have a practical function has no place in society, but thinking of what the world would be like without the small pieces of art is really what makes life worth living, to resort to a cliche. Everyone indulges in some sort of hobby to some extent or another, some personal pleasure, and at base, I would be willing to say that is only possible through art in some form.
Agreed, Beej. Thanks for stopping by!
I could wax philosophical about art and creativity and how I think they are absolutely necessary to those seeking to understand what it is to be human… but I’d probably risk overexposure to pure nerdiness. Suffice it to say for the moment that I consider art to be essential to sanity and spiritual health.
I agree with you up to a point about “reflex checks” Tesh. On the one hand I find games much more interesting when I have several tactical options available…. on the other hand though, many games would be much less interesting if there were no dexterity to go along with that. Striking a good balance is important for many games.
True, Melf, which is why there are different types of games out there. Some like a Dance Dance Revolution, others like a Final Fantasy Tactics. I’m more of a tactical sort of guy, but I also like Kingdom Hearts 2, which has some Quicktime reflex checks. *shrug*
Perhaps I’m overly sensitive to reflex checks because I have a friend who loves games who happens to be a “high function” quadriplegic, and I have a marked preference for turn-based tactical games that allow me to step away from the game at a moment’s notice to play with my kids.
Ultimately, I’m glad that there are different types of games for different audiences, or for my different moods. I just see pure reflex checks and memorization schemes as being a sort of gaming that doesn’t offer as much rich potential for storytelling that games can offer. It’s still a fun style of gaming, and as you note, a nicely balanced mix can be a great thing. No arguments there. 🙂
I studied a lot of linguistics and English in College and loved it.
Today it’s approached mostly as a science.
Because my students have such a hard time with negative numbers (ie: solve for y in y + 25x = 3x + 7), I started thinking about what the problem was. I would get answers like “y = -28x + 7” or “y = 22x + 7” so it was obvious there was a lack of understanding of negatives.
For my thesis, I began looking into when negative numbers are taught- 7th grade! What?? That’s too late in my opinion. Then I began to look into HOW they are taught- with a number line. But at the very beginning of the first lesson in 7th grade, there is a picture of a boy with a caption above his head reading “I owe my dad $4. I have -$4”
So this idea of owing is tied directly into negatives. So I thought about owing someone some money, paying some back, and figuring out how much more I owed.
If I borrowed $12 and paid you back $7, the problem would look like “-12 + 7” but I would solve the problem, in my head, by counting from 7 to 12. This is not the way we are taught in school. The way we are taught in school is to “find -12 on the number line, count 7 to the right, see what number you land on.” But this isn’t what we do in real life!
Absolute value is the answer. Although “take the difference between the absolute values of the two numbers” is a bit of a mouthful, it is the way to go. This way both numbers, -12 and 7, are treated as real numbers instead of -12 being treated as a number and 7 being treated as a movement. I really think that if we teach kids this way they will begin to see the relationship between positives and negatives and no longer make mistakes when they get to me!
Shana, thanks for stopping by. I agree that negative numbers are taught far too late. Ditto for fractions. I’ve seen *high school* students who are afraid of both. I’m not sure about calling absolute value a solution, though. Yes, that will get the difference between certain pairs of numbers, but if you don’t understand the relationship between those two numbers, you won’t get the correct answer. The difference between -12 and 7 is 19, for example, so taken out of context, cherry picking numbers and just using the absolute value won’t give proper answers. It is definitely worth noting the difference between real numbers and “movements” as you note, you just have to make sure that you’re keeping numbers in context. Absolute value can lose that sometimes if not parsed correctly.
Hi Tesh, thanks for the response. I missed a step in my explanation… First, and most importantly, a student needs to recognize which number is farther from zero so that he will know what sign the answer will take. For a problem lile -12+7, -12 is obviously farther from zero than 7. If the student then markes that number’s sign doen (-) and then takes the difference of the absolute values, he will get the answer. It’s almost like it has to be a 2-part problem. First, find the sign the result will take then find the difference.
If you want, you can stop by my site. I made video of what I mean. Admitdedly saying, “what is the sign of the number farther from zero. Ork now what is the difference of the two numbers’ absolute values” is a real mouthful! But I am convinced looking at numbers this way will begin to help kids see the relationship between positives and negatives.
Shana, that still breaks with something like -12 – 9, though. Going through that two step process will give you -3, not -21 like it should be, since the difference (subtraction) between (absolute) 12 and (absolute) 9 is 3. You’d have to do a check to see if the operation is addition or subtraction and behave differently, or remember that you can see -12 – 9 as -12 + (-9) and change everything to an addition problem that way (keeping in mind to throw that negative sign out in front where necessary). In the latter, the absolute values work since you’re just adding two negative numbers and the logic tracks better.
Either way, it’s a bit more involved than the (to my mind) relatively simpler notion that negative numbers go leftward and positive numbers go rightward, and adding maintains that motion, while subtracting reverses it. (Subtracting a negative number then goes rightward, as it should.)
I’d like to check out your site to see what I’m missing from the explanation… what is your site address?
I think you’re missing one point… This is just a strategy to add two numbers of opposite signs. I hope that kids will realize that -12 – 9 is the combination of two negatives (-12 + -9) and therefore just add and keep the sign. In my experience as a math teacher, it’s the mixed sign problems that give the most trouble.
In any case, on the back of my ruler I give reminders on how to deal with the (-) + (-) and the (+) + (+) situations.
OK, yes, I can see where you’re going, then. Thanks for sticking with it and explaining; I mean no offense. 🙂
I guess it’s just the (small) programmer in me that would rather have a gestalt understanding of negative numbers and positive numbers that can be applied to all situations rather than have an absolute value trick to apply to the mixed sign problems. That just seems like too much of a shift in thinking to really give a clear concept of what the numbers are and why they behave the way they do.
The absolute value trick works to give the correct answer in those problems, but how does it help teach the underlying nature of negative numbers? (It just doesn’t seem like it does by itself, so what framework would you put it in to make the puzzle pieces fit together?)
Have you seen Stand And Deliver? “Fill the hole”? If you’re negative already and subtract more, you stay left of zero. Same with right of the numberline. it’s important to see that in problems if you cross over zero, you see how much you have to “fill the hole” made by the negatives you have. If you have -6 and add 8, you have 2 “outside the hole”. It’s important for kids to see that one negative will cancel with one positive.
I’ve not seen that movie. More the pity, me, I suppose.
So you’re using the absolute value to point out those “matter and antimatter” canceling paired units?
Yes! Matter vs. Antimatter! That’s it!
Definitely rent it. It’s a must see for any math teacher. Think Freedon Writers with less drama and more [all] math.
This is a really random discussion. So random that I will actually comment.
“If I borrowed $12 and paid you back $7, the problem would look like “-12 + 7″ but I would solve the problem, in my head, by counting from 7 to 12. This is not the way we are taught in school. The way we are taught in school is to “find -12 on the number line, count 7 to the right, see what number you land on.” But this isn’t what we do in real life!”
I don’t think this problem is meant to teach kids how to manage debt. It’s just meant to illustrate negative numbers with something tangible so they can understand it. Converting the numbers into absolute values is fine in this example, but will mess the kids up when they have to multiply negative numbers together, or when there are a large number of + and – numbers to add up. I’m not quite sure how this approach to teaching is a problem.
Where I see the problem most is in problems like “solve for y: 3y + 25x = 4x + 7”. You can probably imagine the answers get. The “4x – 25x” is the problem. It’s hard to solve for y when you can’t see that the 25 in “4 – 25” is a negative number, overpowers the 4, and therefore makes the outcome negative.
There are no manipulatives out there for kids to deal with negative numbers. This tool gives them one.
The problem with any addition or subtraction problem is that the first number in the number sentence is treated as a “noun” and the second is treated as a “verb”. In 10 + 7, we’re told to find the real number 10 and then move 7 spaces. Kids should know that both 10 and 7 are real, stationary numbers, not movements.
The movement approach is necessary when learning maths because kids learn to count before they learn to add. After they have done enough addition they will be able to ‘auto-pilot’ a 10 + 7 answer, but before this they will need to count spaces from 10, or 10 from 7. Still don’t really see the problem.
The problem is that 17 year olds aren’t confident that -10+ 7 = -3 even though they have been using the counting method since 3rd grade. If I owed you $10 and paid you back 7, would you count “-9, -8, -7” to figure out how much I still owed you?
When I rephrase a problem like “-10 + 7” into “If I owed you $10 and paid you back $7, how much would I owe you?” my kids get it right away [usually with a “duh” at the end].
There’s a disconnect betwen how these problems are taught with “owing” and “paying” and how they are taught to be solved on a number line. This is the gap my ruler fills.
Either approach works fine when determining the result of something simple like 10-7. If you have a lot of numbers to add/subtract and, god forbid, some multiplication/division, I would much rather a firm understanding of what a negative number actually is.
I would also argue that if you have 17 year olds who have difficulty evaluating -10 + 7, these kids suffer from a learning disability which should have been picked up at a much earlier age. There are very few kids that age who would have any issue with this in my experience. Negative numbers are taught in elementary school.
If you really want to prove your point, conduct a study. Have a workmate who teaches the same part of the curriculum adopt one method while you adopt the other. You will want to compare a few years worth of cohorts, or recruit a large number of teachers. Compare the kid’s grades in a test that evaluates performance on a variety of negative number tasks. In other words, use some science instead of making unqualified statements.
I’ve been holding off on asking, but are you a teacher?
It would probably be easiest to claim that I teach high school maths right? 🙂
I’m a research scientist and teach optics at an early university level. This is perhaps the angle I come from when I say that I prefer kids to have a consistent foundation that allows them to handle more complex problems.
I also tutored several kids in high school maths in my younger days, as my father teaches high school maths and I was somewhat of a maths whiz.
Maybe you should ask your dad how his kids do with negative numbers and report back. It’s hard to have a valid opinion on something you have no idea about.
Maturity, nice.
I already justified my position – if you’re going to completely ignore my arguments then there’s not much use pursuing this further.