When 1 + 1 Doesn't Equal 2

My bike Shadowfax. He's so fast he's blurry.

     I used to go for bike rides along the Willamette River in Eugene at night. When I was in college and wanted to clear my head, get some exercise or mull over a math proof (really), I'd hop on Shadowfax and do the loop. Sometimes at 2:00 or 3:00am if I was up.
     One time I stopped at one of my favorite spots - one of the bridges over the river. I was looking at the moon/street light hitting the water and two guys rode up and stopped to chat. They may have been homeless, they may have been drunk, and they may have been on drugs - sometimes it's hard to tell. One of them asked what I do. I told him I was a math major at the U of O. He scoffed. Mathematicians. Let me tell you about mathematicians. One plus one does not equal two, one plus one equals one plus one and two equals two. He went on in this line of reasoning for a while, talking about all kinds of related topics. He honestly seemed like one of those bums who has all the answers to the questions no one is asking.
     Shadowfax and I rode on, and I didn't give it much thought until later. Until now, really. I've talked about how equality means that one side is the same as the other, (or that each side of an equation is the exact same as the other). 2x + 3 = 5 when x = 1 and 1 + 1 = 2.
     Equality only exists in the abstract, though. 1 + 1 = 2 only if you are dealing with a concept, like "numbers" or "apples." By that I mean, if I were to take a literal, tangible apple and then put another apple right next to it, those two apples are not the exact same. They weigh different and look different and one of them has more worms. No two apples are exactly the same. 1 + 1 does not equal 2, because I don't have two of the exact same thing.
     But If I think of those two apples as the generic, abstract label of "apple," then I go from having one apple to two apples when I add them up. I can "add" the two apples because I can label them as "the same," and this abstract label lets me deal with equality. Does this make any sense? One plus one does not equal two, unless the two things are completely identical. One plus one equals one plus one, when the two things are not identical.
     What does this mean? It means that in the real, tangible world, equality doesn't exist. No two things are exactly the same out here.
     Well, what does that mean. Equality from a social perspective means that all people are given the same opportunity, regardless of gender, race, religion, etc. Is that even possible? No, it's not, because no two people, cultures, backgrounds or beliefs are the same. Equality can't happen in the real world. Maybe that notion is obvious, I don't know. But it's undeniably true.
     So should we even pursue social equality? There will always be rich and poor, healthy and sick, feeble and strong. Can we really grant Equal Opportunity to people applying for jobs, when one guy comes from a wealthier family than the other? To truly grant equal opportunity we'd need to make everyone go to the same school and come from the same family and be born at the same time. That sounds stupid.

Cara had a photo idea and ran with it.

     When I think about the ideals of social equality, from the vantage point that equality doesn't remotely exist in the real world, I start to think it's misguided. Equality: treat all people the same. No! We don't treat men and women the same, nor should we. I don't have breasts to feed my son. Cara doesn't have muscles to move heavy things (and I mean, like, really really big muscles like mine). Men and women are so clearly not equal that trying to treat them the exact same is absurd. (And for the record, there's little doubt that women are better than men, in my opinion. As a whole, the only thing that they aren't capable of doing as well as men are feats of physical strength and agility, which isn't such a big deal in today's society (we're not running from sabretooth tigers anymore. Yes I am putting parentheses within parentheses.), and they can bear and nurture children. Of course I am going to hold a door open for them!).
     This is all very easy for me to say. I am a white, straight, middle-class, educated, attractive, charming, physically imposing Christian male. Some of those attributes are by choice, some are by chance, some are by birth, and some are fabrications, but the point is I am not a person who is historically oppressed in the United States. It's easy for me to say things like "we don't need to try to treat people equal when they are not equal," because it doesn't tend to impact me negatively. But I recognize that those in power view "not equal" as "inferior" and begin to exploit. I get that, and I think things like Affirmative Action and Equal Opportunity hiring are wonderful things (although I think affirmative action should move towards being based more on socio-economic standing and less reliant on race, but that's for another time). So let me go back a step.

     Equality doesn't exist in the real world, only in the abstract. Fine. But if I am going to apply my purely mathematical logic to the real world, I need to apply it to my abstract world as well. Equality only exists in the abstract, but I still use it. I still write and solve equations as if 2x + 3 equaled 5. In my perfect, ideal, non-real world, equality is real and it is useful. Is it such a stretch to think that the abstract concepts of equality in the real world aren't useless either? Math breaks down when you apply it to physics. It turns into a model of the world and not an absolute law, mostly because we can't ever know everything and have all the necessary data. In the same way, equality breaks down in society, it becomes a goal we should strive for. We can't ever have a perfect society, but we can strive for it in the same way we strive for cleaner energy or faster transportation. Really, as a Christian I am called to be perfect, knowing full well that I never will be. I am called to strive for it, just like as a society we should strive for equality.

     People and groups and individuals and genders are not equal. That is a wonderful, necessary thing. If they were truly all the same, the world would be a simpler, more boring-er, less accomplished place. We shouldn't treat everyone the same, that's a disservice to all of our tremendous differences. We should treat everyone with the same respect and the same regard (at least as far as it is truly deserved), but we shouldn't treat them the same. 1 + 1 doesn't equal 2, thank God.

     (One last, tangentially related indulgence: This is why I love math. I like to take things that it teaches me, logic it has instilled in my brain, and see how it applies to other things. This application tends to shed new light on things for me, things that other people have probably had figured out for a long time. I don't expect other people to do this with mathematics, but I think they should do it with whatever they know and love. And I don't expect my students to need to be able to solve quadratic equations in life, but damn it I wish they were all equipped to at least analyze their world with cold, hard logic. That's all math is, logic applied to a specific, well defined system. A system where 1 + 1 = 2 and that idea leads to a lot of other ideas. How can I convince a 14-year-old with an iphone that this is worth caring about? I don't know. Thanks for reading.)

What "The Monster at the end of this Book" Teaches Us About Video Games

     Did you read this book when you were young? It was my favorite, mostly because Grover is hilarious. In this book, Grover reads the title page and gets nervous, because he is scared of the monster at the end of the book. He doesn't want to get to the end of the book and see the monster, so he does everything in his power to stop you from turning pages. He begs and pleads, ties corners of pages together, and freaks out everytime you turn the page. I, the young reader, have god-like powers in this scenario and continue on, until we find that Grover himself was the monster and drugs are what we should really be afraid of.

     I've been reconsidering this book, because I recently found the sequel, Another Monster at the end of this Book, which adds Elmo to the mix. When I say "found," I mean it literally: it was sitting on a projector at work and I took it, along with a couple other children's books.

     Here's what I've reconsidered: What if there was a real monster at the end? I mean, what if we got to the end and Grover was right all along and ends up stuck in a page with a scary creature? I, the almighty page turner, would be a huge jerk to do that to the poor guy. If I knew that there was actually a monster at the end of the book, shouldn't I stop reading pretty early on to keep my lovable, furry old friend Grover safe?

     These sorts of dilemmas come up all the time in video games. I think it's a trope of Japanese storytelling. You'll be progressing through hours of game time, and if it's a good game you become pretty attached to the world and characters filling the game. Video games progress linearly, so for the most part you have to keep moving forward, and at some point you take an action that causes unbelievable devastation. The whole world lies in ruin, forests become wastelands, monsters roam the city, and it's kind of your fault. It's up to you (and maybe your misfit gang of rebels) to make right these awful wrongs. No one in the world ever blames you, because the bad guys were going to do bad things anyways, but dang it!! I was working so hard to get the three spiritual stones so I could protect princess Zelda, and Ganondorf wanted me to do it all along to realize his true power?! I feel so used!

     I have come to the point in a couple of games (Final Fantasy 3 and Zelda: Ocarina of Time come to mind, both truly fantastic games that you should try if you haven't) where I know that progressing in the story means things get real bad before they get good again, and they are bad for a long time. If I enjoy the world before it gets bad (which I do, it's happy and fun and the music is uplifting), why would I progress? If I am truly taken by the story, the best thing I could do for those little people is leave my game on and my character standing in a field, playing some music. This is a tough concept to grapple with when you are 12 years old.

     This method of having the hero unwittingly bring about disaster must be a flawed story-telling method, at least when the reader is an active participant. As I said earlier, in my experience it's a very Japanese approach (based on the video games I've played and cartoons I've watched, and yes I am an adult). It's different from the heroic cycle of western lore. In that heroic cycle, evil is coming towards the protagonist and he/she refuses to take action early, for whatever reason, and this leads to disaster. The refusal to take action is the problem, and the hero has to fix it. Conflict is pushed upon the hero. This is vastly different than the hero unwittingly unlocking the magical powers that end the world. As a powerless observer, watching someone screw up and try to make amends is very entertaining, but as an interactive controller of the story, this method of progression is unsettling and doesn't make me want to continue.

The Fourth Wall

     That's where video games are trying to be different. Video games work hard at being an interactive story, much like Grover and his book. Grover breaks the fourth wall and talks to the audience as easily as I break his brick wall between pages. Video games talk to the main character, but tell me I am that character, which is the same thing as talking directly to the audience. They try to make the player/reader feel like he's involved and has a say in what happens. Instead of watching a movie where the hero saves the world, they let you push a few buttons (or turn a few pages) to help along that protagonist, but in truth you have little control over how the story goes (kind of like life, which is essentially a bigger world with more systems and variables at play, but the ending is going to be the same no matter what choices you make -- you will die). Games work harder and harder to make their worlds a self-sustaining story, but whatever the programmers include in their game is what dictates how the story goes. There really is no freedom (at best there are multiple endings to work towards) in a video game, much like a book or a movie (In truth, video games are fundamentally the same as those choose-your-own-adventure books that you think will be so fun at first and then you realize you'd rather just read a good story). 

Grover and I both break it (we are very strong)

     Part of the problem is just bad story telling. Video games have wonderful visuals and truly beautiful music, but the dialogue and narration are exceedingly poor (Game makers are working on that and getting better, this is after all a young medium.) When a video game does a good job of immersing me in the story, making me feel like I am the one who is going to save princess Zelda, the video game needs to recognize that I don't want to hurt any innocent bystanders, just like how I wouldn't want to make Grover run into the swamp thing either. If it wants me to truly invest in the world and become immersed in its characters and interactions, it either needs to a) not have me be the one to end the world, or b) have that end of the world come about in a way that is very consistent with the established characters and themes.

     I've thought of ways this is similar to real life. How walking down a path will take you to the end of that path. If you don't like where you're heading, turn around (if you don't want to end the world, stop playing). Predestination in life is another obvious discussion that could arise from reading Grover's book. I am sure that discussion happens all the time when little Johnny and his mom realize they actually had no control over Grover's fate, the almighty author did. I think there are lots of little pieces that have application, but nothing worth expanding upon. I suppose the nice thing about these stories is that they end and begin in very good places, you just have to get through the bad stuff in-between to finish. This is a good life lesson. Things will work out and drugs are the real monster.

     

Explaining Division of Fractions

     I find that most of my students aren't terribly concerned with why things are the way they are in mathematics. I am of the opinion that this is because when students ask why questions when they are young, they don't get answers. If they ask "why is a negative times a negative a positive" they get an answer like "just because," or, if they are home-schooled, "because I said so." I'd like to discuss some of those why topics in math, most of which I don't think are asked that often.

     Little known math fact: When you divide by a fraction, you multiply by its reciprocal. I say "little known" because every student ever forgets this fact from one day to the next. Maybe, just maybe, if they understood why you multiply by the reciprocal they'd remember. Putting equations in a blog is pretty hard, so I am going to get a bit tricky:

     If I show this to my students, I let them know before and after showing it that they don’t need to “get it,” but it’s worth seeing. For a couple students it makes sense (and they seem to find it clever, and hopefully deepen their appreciation of the subject). I was never taught this, I was nerdy enough to look it up. Now I know why.

Growing up PK

     Pastor's kids rebel. That's the stereotype. They are subject to so much pressure and scrutiny that they become a rebel without a cause and start dancing all footloose and other things happen. Well, I'm a PK and I didn't rebel. Neither did the PK's around my age at school or at my church. Why is that? I don't really know, but I am going to pretend like I do.
     I think a big part of it is that Eugene, Oregon, is not a "churchy" town. Going to church on Sunday is not the socially correct thing to do, so being a PK doesn't carry as much weight here as it does in a lot of other places. There isn't the pressure of a whole town watching and analyzing your every move, and I think that makes a huge difference. When I went to school, I would guess that most of the people in my class didn't go to church at all, and many more were just holiday church-goers. All of those people, and by extension, all of their families, don't really care about who the pastor's son is.
     Growing up in a non-churchy town has a strange effect as a PK. Whenever someone found out what my dad did, they would change how they acted around me. They'd try to stop cussing, maybe not tell a dirty joke, sometimes even apologize for using the Lord's name in vain. I kind of liked it. I think the idea of religion was really foreign to a lot of my classmates, so they weren't sure how to act around a person like me. This might be the case for all PK's everywhere, and in fact it might be the case for all obviously-Christian people as well, I don't know. But people would definitely change when I was around, and it was a little weird.
     Since the whole town wasn't concerned with their pastor's children's lives, that really just leaves the church. When I was growing up (through college), my dad was a pastor at a really big church, with thousands of people attending on a given weekend. I'd be in sermons, I was leading small groups, I was even asked to represent "the youth of the church" at a special ceremony for the groundbreaking of a new church building. With all of those people and the attention that comes with it, I never felt like I was being terribly scrutinized, and that's a testament to the people in that church. I didn't feel a lot of pressure to act or be a certain way. It's very possible that I didn't feel that pressure because I was already acting in ways that were acceptable, for the most part, but I think it was different than at a lot of other churches.
     My uncle (dad's brother) is also a pastor at a big church, up in Beaverton, Oregon. I remember walking into that church with my cousin Jess on a Christmas Eve service, and feeling everyone looking at me. I just knew it was happening. We as a family would all go sit in the front row of the church and wait for the service to be over so we could get at our presents, but it felt different. It's possible I was aware of the attention because it was new to me (like how my room never smelled to me but always did to the girls that came over -- once, and always asked why it smelled like damp goose and never came back). But I honestly think the attention I received at my uncle's church was fundamentally different from the attention I received at my church.
     But the main place that pressure can come from - the pressure to act or be a certain way - has to be from home. There were a few times when my mom or dad would say something like "what would the church think," or "how would that look to the church," but a few times in 21 years is not that many. I came from a home that allowed me to be who I wanted to be, and that takes away any external pressure that might exist. I think even if I did go to a church, or live in a community, that was very preoccupied with the behavior of its PK's, having parents who didn't care if I was in AWANA or going to youth group every week would have been enough. Again, that might be a chicken-egg sort of conundrum (maybe my parents were laid-back because I was a good kid), but I don't think so.

     There were a couple times in high-school when I was feeling stifled or judged for how I was behaving, or what I was planning on doing. I remember thinking, briefly, that I was tired of having to live up to other people's expectations, or that there was a lot of pressure on me because of who my dad was. I can understand that pressure being overwhelming if it is consistent and strong. But I made a realization at a very young age that has served me well for my life up to this point: no matter how many people are watching me and have expectations for me, it pales compared to the fact that God is watching and has expectations for me. In fact, pleasing Him was more important than pleasing everyone else, which is actually a freeing revelation (as opposed to piling on more pressure, in addition to all the people). It was like the corollary to the cheesy cross-stitch "Me and God is always a majority." Which, in math, is read Me + God > Everyone Else. I understood that if there was any reason to feel pressure from other people, there was much reason to feel "pressure" from God, in a good way.

What if he likes art? What if art is his thing?
We'll have nothing in common.

     I wonder what it'll be like for Wyatt (and his future siblings) to be a pastor's grandkid (PGK). Will he still be treated differently by peers when they find out? Having a dad for a teacher might be similar, if I end up teaching at his school. I guess I'll have to be careful to not put any unnecessary pressure on him to be a model churchgoer, or a fantastic mathematician. Things to think about I suppose.

Making a Difference

Sometimes I really feel like I am getting through to my students:

  When I asked him what time it was, he said "blue."

Stuff That Happened to Me Today: Building a Deck

     True story: A student of mine approached me at Lane Community College after class. She said "Grant, I have a question. I am building a deck and want to know how I can make sure to get a right angle when I am laying it out." I got excited, and asked her what we had to work with. She had the pile of wood from Jerry's, all cut and ready to go, a tape measure and a chalk line.

     Perfect. Absolutely perfect. This was the most validating thing that had ever happened to me in my math career. In geometry you do what's called "constructions," where all you have is a pencil, straight edge, and compass, and you need to make things and prove things and do things and things. It's fun and challenging and I will do it in my spare time (like, at a boring meeting, or when my wife is telling a story).

     I essentially had a compass and straight edge to make this right angle from a given side. Here's what I tried to explain:

Let's start with a picture I took of their house (well, I copied it off google maps)

Now let's focus on the pertinent part.

We want to make a right angle from point A upwards, and want to find the most accurate way to do that (note: not the easiest way, the most accurate way).

Along the same line as the side of the deck you have plotted, mark a point the same distance away from A on each side. (This is easiest shown with a circle)

Points X1 and X2 are the same distance from
A (6 feet) and lie on the same line as the deck

So we end up with

Having never remotely done this in real life, I can safely recommend using a distance of about 6 to 8 feet from A to X1 and X2.  Now we can use the points X1 and X2 to make a perpendicular to A. 

Whatever that distance from A to X1 is (let's say it's 6 feet), choose a substantially larger distance to measure. I'm going to use 10 feet to make the pictures work, but the longer the better (it really depends on how far out you want your deck to go, but we'll work with this). Draw two circles around X1 and X2, each with the same exact radius (10 feet in this case).

Points X1 and X2 are each the center of a circle with a radius of 10 feet.

You don't need to draw the entire circles, just trace enough to find the point where the two circles intersect above Point A.

Using your chalk line, mark the line from Point A to the intersection above Point A (let's call it Point I... for "interesting") and you have a perpendicular line! It's as accurate as you were.

Anywhere along that line you can put the other corner of your deck. Wasn't that fun!?

There are all kinds of tools and tricks construction workers use to make right angles that I will pretend to know, but given just a tape measure, chalk line and wood we can do it (truth be told, you don't even need the tape measure - a chalk line and a place to start is sufficient).

There are a couple of faster ways, from this point, to finish the rectangle. Maybe for another time.

Prayer Without Ceasing

     I used to talk to God all the time. When I woke up, when I went to bed, and throughout my day I would tell Him what was going on, what I was thinking, and what I was thankful for. This was in middle school and early high school. I didn't know it at the time, but I had a really healthy, ideal prayer life. I think this was due to the fact that, at that time in my life, I had few friends and was a weird mix of shy and socially awkward. When I had no one else to talk to throughout my day, and really a lack of confidence in the idea of finding actual people to talk to, prayer was a natural resort for me.
     I can safely say I started high school with three friends, Mac, Josh and Jered. They were great friends and I had a lot of fun with them, but they also had their own things (swimming and football) and I spent most of my non-school time alone. Then things started to change. My funny, confident side grew tired of being stifled by my shy side. My older sister's friends seemed to enjoy me and like me as a peer instead of as a friend's little sibling. I remember the first time I cracked a joke to the class and everyone laughed and I thought "okay, I can be funny out loud."
     Suddenly I was making friends. I would see people and say hi to them, even talk to them, when walking through the halls. This was new to me. I had to work at it and get better at it. I started running conversations and interactions through my mind, analyzing them.
     I think this was when it changed. I started to think about myself and my strengths/weaknesses instead of talking with God throughout my day. I was growing selfish, honestly. With this selfishness I think the main thing I lost was my communication with God. I noticed the change. One time I walked out of a grocery store and had a thought like "I sure am funny," probably after a particularly sarcastic day of mine, and then almost immediately thought "huh, I haven't really talked to God today. In fact, I haven't talked to Him in a while." I missed the conversation and was aware that I missed it.

     Sometimes I will try to trace a line of thought I've had. It's a fun, challenging exercise. You know when you are thinking about something, say, work, and next thing you know you are thinking about ligers (not to be confused with tigons. How did you get from work to ligers? I try and remember what my connections were. Something like "I need to get to work early tomorrow to finish grading. Should I bike or should I drive? I'll drive. Which route is fastest at 7:00 am? Depends on the lights. I wish the city would release an app telling me the timing of the lights, so I could plan a route just as I start driving. Or better yet, let google maps do that. We could have a whole society of efficient, linked transportation. A bunch of batmobiles. Wow, Batman is so cool. Catwoman's in the next Batman movie. I wonder if she'll have a pet tiger or something. Tigers are also pretty cool. Did I bring my Detroit Tigers hat with me? Ligers are real. They seem like they would be fake but they are totally real." All I can remember is I was thinking about getting to work and now I am thinking about an absurd animal. I trace the conversation as best I can (and usually find all the links, you should try this sometime).
     This happens to me when I am praying all the time. I start praying with the intention of thanking God for lots of things, and all of a sudden I am drawing up The Annexation of Puerto Rico in my mind. These tangents weren't a problem when I was "praying without ceasing" because God was right there with me, agreeing every step of the way. "Yes, Grant my son," He'd say in His deep, all-knowing voice, "Boardwalk and Park Place are overrated when compared to the Greens." Now when I pray, I am starting and ending a conversation, and it seems rude or inadequate to get on so many tangents.

Wyatt often puts his arms up, straight in the air,
while sleeping or while awake. We like to say he
is always ready to worship. He's not doing it here.

     I've realized that I replaced my thinking from the one thing that is truly worthy of my praise to a bunch of things I think are cool (myself, friends and batman). In the process of me doing that, I lost my prayer-without-ceasing mindset, and it won't come back easily. Like trying to improve your posture, or trying to brush your teeth every day, it's a hard habit to get back into once it's gone. I've had days where I've told myself "let's try this again" and usually don't do too well. In fact, I feel like I can't do it anymore, as if it was a gift that I threw away.

     Now it's starting to really bother me. I was with my son Wyatt one morning - and I mean one disgustingly early morning - and prayed for him a little bit. Mid-prayer, I realized I hadn't prayed for him, or at all, in a couple of days. This really disturbed me. If there is anything I should be praying for it's the little ball of helplessness in my life. Why hadn't I prayed for him in a while? I think that the fact that I used to pray effortlessly, I didn't need a routine or system. Now that the effortlessness of it is gone, I need to either get it back or buckle down and commit.

     So, I guess there was something gained and something lost when I started to gain self confidence in high school. I wish I could have the best of both worlds, and if I could do it all over again I probably would, but here I am. I guess I just have another things to work towards.