Sub Notes

     I don't sub too often anymore. Cara and I are busy enough with work and baby and playstation. Willamette high school was having three math teachers out last Friday and needed an advanced math sub, so I acquiesced. (big word!) A few notes from my recent subbing adventures:

     - I had a conversation with a student that I wish I could have with every student. He doesn't like math - and my guess is he doesn't like school. He said to another student "It's not like the world will end if I fail a math class."
     I was helping another student at this time, but luckily it's part of my job to eavesdrop. I turned around, looked at him, and said "You're right. The world won't end. But the world won't end if you study and pay attention in class either."
     Him: Why? What's the point of learning math? (I love this question).
     Me: What do you want to do when you grow up?
     Him: I don't know.
     Me: Exactly. You don't know what you want to do. So, we teach you everything. That way, when you do start to decide what you want to do, you have options.
     Him: But, what if I just want to work on a boat?
     Me: Great! Go join the Navy and work on a boat.
     Him: (he didn't like this idea) No, I mean like, some old fishing boat.
     Me: Great! If you want to work on an old fishing boat, go for it. But, my guess is you have never actually worked on a boat. What if you do it, and then find out that it's not what you want to do? What if you decide you would really like to design boats or build boats or do something entirely different. Wouldn't it be wise for you to not limit your options now? So that if you want to do something that's hard, or that everyone wants to do, like design video games, you are educated and equipped? You might not care about making a lot of money, and that's fine, but money is very nice to have. It's very nice to be able to take care of yourself and, maybe one day, your wife and family. Well, the best way to make money is with an education, and more specifically with and education in math and science. You're right: the world's not going to end if you fail this math class. But you might end up regretting it later on.
     Him: Huh.

     I don't know if I made any sort of difference in his life or future, but let's pretend like I did, okay? The other boy he was talking to was also paying pretty close attention, so maybe I got a 2-for-1 there.

     - The school wanted me to arrive around 7:30, for classes that start at 8:00. That's no problem. However, the school was on a block schedule and I didn't have a first period, so I didn't start teaching until 9:45. That's over two hours of me sitting by myself in an empty classroom!! Super frustrating. I took my Kindle and Ipod but was still painfully bored. Additionally, their wi-fi was password protected. So I started guessing. Willamette high school is in the Bethel school district, so I guessed variations on "Bethel," "Bethel School District," "BethelSD," "WillametteHigh," "Password" and "Wolverine." A student later told me the password was "B3thel." Dang it!!

     - I sing and hum very often. Especially when I am in the room by myself. Back when I was teaching high school, It was always weird for my TA's, or students that were taking tests, when it would just be me and him or her in a room and I would start singing. I think it's even more weird for students that I am subbing for. I'll walk around handing out papers or checking work, humming whatever song I heard last. I am fine with being the weird-guy-that-likes-math, though.

     - I was just terribly bored while subbing. Students did worksheets, classes were 90 minutes long, I wasn't needed for anything. There was one point where I was spinning on the stool in the front of the room, trying to make myself dizzy. (No, really!) While spinning, I heard a conversation out in the hall. A student had stepped out to talk to a friend. A school employee of some sort - like a campus monitor - was interrupting their conversation. I hear him say "Would you be doing this if your teacher was here?" and promply stopped and looked up. He was giving me a quizzical look. I am a good sub.

March Drives-Me-Mad-Ness

    Clever title!!

Yet another year where I won't need to follow March Madness. Go Ducks.

     I can't bring myself to do March Madness. There are two vital, essential pieces to enjoying and participating in March Madness: You either have a team you care about in the tournament (nope), or you fill out a bracket and enter some sort of a betting pool. I can't bring myself to do the latter.
     You see, I hate being wrong. What I hate even more than being wrong is being factually, demonstrably wrong. Usually if I am wrong about something I can weasel, lie, manipulate, distort and blame to deflect my wrongness. I can't do that with a busted bracket. So I don't make brackets.
     I've tried. I've looked at them and tried to imagine how they will shake down. But I don't watch college basketball and I don't know anything except for who the powerhouse names are. So I'll have a bracket with teams like Duke, North Carolina, Kentucky, Indiana (what?!) Arizona (huh?) UConn of course, and a token SEC/Big East team make their elite eights and go from there. Of course I am wrong somewhere. I am more wrong than most other people. I hate it.
     If a team I care about is in the dance (read: Oregon), than it's almost worse. Should I play favorites and put them in the Elite Eight? Should I understand that they are never that good and knock them out in the first round? These decisions stress me out. There's two main options: 1) I put them deep into the tournament. This means I am either rooting for my team to win, knowing that when they lose I will be doubly let down, or knowing that if they go farther than I predicted I will have some amount of disappointment and can't fully enjoy their success. 2) I have them lose in the first or second round, which results in me just rooting for my favorite team to lose and that's annoying.

Ugh. What about this isn't stressful?!

     Do you see how impossible this is!?
     This is an issue with sports in general. Your team will ultimately lose, unless they win it all, and your team won't win it all. It's a guaranteed disappointment. I guess this is an issue with most of life. Louis CK does a stand-up bit about how marriage, like buying a puppy, is an investment in disappointment. When you bring a puppy home, you are telling your family "Someday soon, maybe in 3 years or 12 years, we are going to be sad!" This is how I feel with March Madness

     It causes me to miss out. I am not drawn to watch the games, because I don't have any vested interest, and the basketball isn't really that good. I didn't get to see Stephan Curry blow up 5 years ago, I generally missed out on the Butler runs, and I just avoid conversations for a couple of weeks. The rest of the world is having fun and excited and has an outlet until we hit spring, and I am just glad that no one will know how little I know about college basketball these days.
     This is honestly part of why I love math. I can do a problem and be right. There's no question as to whether or not I am right, most of the time. Additionally, it has taught me how to argue and use logic to my advantage, so when I am wrong, it is probably due to some gray area of some sort, and I can deflect.
     Also, I am a very healthy, normal person with no social issues.

Seeing God in Your Experiences

     I once heard that people see God in a way that fits with their own experience. Specifically, artists see God as the "great artist in the sky," and architects see Him as "the great builder in the sky." If your job doesn't lend itself to that (i.e., He's not the great data enterer in the sky), then whatever you find important, or find meaning in, is where you might see Him. The great gardener, or great parent, or great grand-parent (ha!). You get the idea.
     This notion has stuck with me since then, probably because I found it to be true for me. I of course see God as "The Great Mathematician in the Sky." The difference between my view of Him and yours is that I think mine is more accurate.
     There are three traits associated with God at His most fundamental levels. He is omnipotent (all powerful), omniscient (all knowing) and omnipresent (all everywhere). Guess what! Mathematics pretty much contains all of those all-everything qualities as well!
     It's hard for me to convince someone that math is all powerful, because in and of itself it has no real power, but mathematics is both what governs all of our physical world and the primary tool that allows us to learn and discover anything and everything about existence. Since it applies to most everything, I call that darn-near omnipotent.
     The rules and logic of mathematics are quite literally inescapable. Wherever you go, 1 + 1 = 2, and whatever time period you are in you can solve an equation for x. This makes mathematics omnipresent.
     All knowing is a bit of a stretch, but I think it is still sensible. In terms of the hard sciences, everything that we think we discover, or have agreed upon to be a scientific truth must first be verified through mathematics. If the math isn't there, than the theory isn't either. The issue with this, and with all the sciences in my opinion, is that we are always learning more and more. We don't know if what we know now is the whole story or just a part of it. So I submit that mathematics are all knowing, we just don't know all mathematics.

     Calculus is tough to sum up as one main idea. If I had to, I would say it is the study of calculating the infinite. Finding the sums and products of numbers as they get infinitely big or infinitely small. I'd like to explain this real briefly, but to insert equations and fractions I have to use a diferent program, so the text size and style will change slightly, which I know is jarring. Please bear with me.

     The idea being, I can't actually calculate two-to-the-infinity. In fact, I can't even calculate two-to-the-really-really-big number (even 2^100 is absurdly huge). But since I can look at the smaller numbers, I can use those smaller numbers and the trends that I see appearing to know that if I could add up every possible term from one to infinity, they would add up to 1. I think that's cool.
     I can't truly comprehend infinity. If I could, it wouldn't really be infinite, now would it. I don't think there is a better word to describe God than the word infinite. This means that I can't truly comprehend God. If I could, He wouldn't really be God, now would He.
     But! When I started to wrap my ahead around this calculus idea - that I could determine what happens as things get infinitely big or infinitely small, I suddenly started to realize that I could understand God in a similar fashion. I can't understand the infinite depth of His power, presence or knowledge, but what I see in little pieces can give me a clue as to the larger trends. This might not be a radical idea (in fact I think it's really the only option), but the fact that I was able to do it with math excited me.
     I began to really dive into my studies. In college I did three things: play super smash brothers, play ultimate, and study math. I felt like the more I learned about math the more I was learning about God. I still do, and I eat it up every chance I get.

     Well, enter some random guy who informs me that everyone sees God in a fashion that fits their experiences. The great whatever in the sky. It turns out what I was doing with mathematics, everyone can do with everything. I'd felt that by learning more about math I was learning more about God because I felt like mathematics had some special insight into God's character. It turns out that everything points to God. The more we learn about science and humanity and art and nature and batman, the more we are learning about God.
     (I still do feel that math has special insight into who God is - more so that most other disciplines - because I think if God is bound by anything it is logic. I was once asked at a Bible study if there was anything God couldn't do. My reply was "Of course there is. He can't do anything that isn't logically consistent. Meaning, He can't simultaneously do something and not do that same thing." I think logic (and by extension mathematics) is an integral part of His character. I know there are smarter people than me that have thought about this, and I think the idea relates to His nature and His administration, but I don't know.)
   
     So whatever you are doing, if you do it to your fullest, you are glorifying God. I find that relieving.

Finding Your Center (Of Your Pizza)

     People walk up to me all the time and ask "tell me a good math trick, something I can use in my everyday life." No, this really happens!! Well, let me tell you, I have some great ones. I know how to multiply by 9's really quickly, I can tell if a number can be divided by 3 really quickly - lots of great stuff.

     If you are like my wife, you are absolutely horrible at cutting pizza slices. I mean, you might be the worst person in the world at finding the middle of your pizza. What drives me crazy is she will cut across the pizza, clearly not hit the middle of the pizza, and then just keep going like her first cut was a good one! Can you imagine? Then, I am stuck with the task of handing out disproportionately cut pieces of pizza to people. Everyone automatically assumes that I am giving them their slice based on how I perceive their eating habits, based on their weight. That's not fair to me! I am doing that, but I wouldn't be so crass as to symbolize it through pizza sizes. Please!

     I needed to solve this problem in my household before my marriage and friendships came crashing down. I needed geometry.

     You might not be aware of a cool property that all circles have. If you pick a point on a circle, and from that point draw a right angle within the circle, the sides of that angle will intersect the circle at a diameter. Here's what I mean:
     First off, it's important to make sure we know that a diameter passes through the center of a circle. Any other line segment through a circle is called a chord:

What we want is to find the center of any random circle. (Or, really, we just want to be able to cut a bunch of diameters). Start by picking any random point (we'll call it "S" for "start." Or "square." No, wait, that's confusing. "Sircle.") Also, grab a right angle:

If I put the corner of my right angle directly on the edge of the sircle circle, the right angle will intersect the circle at two points (we'll call them A and B):

If I connect points A and B, we have a diameter:

And of course, you can repeat this process one more time to cut another diameter, and this will result in finding the center of your circle:

So, this can work for any random picture of a pizza I pull of the interwebs:

As you can see, the center that was cut in this pizza is actually slightly off. Shame.

Now we can find the center of any pizza, and avoid any awkward moments when having guests over. Ha! A great right angle to use might be the box the pizza came in, but I am sure you can find others as well.

It looks as though I have made your life a little easier by answering a question you have never asked and didn't really care about anyways. You are welcome.

(Hint: this method also works on pies, cakes, waffles and well-made pancakes, but not quiche for some weird reason).

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.)

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.

Inheriting Appreciation

    I had a teacher in college named Shlomo. He's a fantastic teacher; his lectures are rife with information, you learn a ton, and he imparts a lot of passion and excitement into the material. Shlomo told a story one time about going to an art museum, just for kicks. He wasn't an art fan, but wanted to see what the fuss was about. He remembers seeing a painting and not being very impressed with it. He went back to the same museum later, and there was a tour taking place, so he tagged along. The tour guide stopped at this same painting and espoused all of the things he loved and respected about this painting. Shlomo left with the same love of the painting as his tour guide, and he learned something too: Sometimes people don't appreciate something until they learn why it should be appreciated.
     A little later (this might have happened in the same lecture, in fact) Shlomo is working through a proof of something. I don't know what the proof was on, but it was in a geometry class. He looks at an equation and says "Maybe you don't like this equation. You don't like it? You change it. Where else can this happen in life but in mathematics? You want something, you get it. Only in mathematics."
     I'd never thought of this. Truly I didn't really stop to think about it until much, much later. His lectures were dense, and I usually left feeling a bit punch-drunk as my brain tried to wrap itself around the new keys to the universe it had discovered. One day later I was solving an equation with a bunch of fractions, and instinctively I got rid of the fractions. No one likes working with fractions.

     It hit me at that moment. I didn't want the fractions, so I got rid of the fractions. I was able to get what I wanted immediately. This doesn't happen in life! Only in mathematics!!

     Let me explain a bit more, but it involves actual math so feel free to skip down to more good stuff. 

The above equation has 4 different fractions with no common denominators. I have to teach my students how to solve these types of equations, and when I show the first one their faces go white. I say "What, you don't like fractions? Fine. I don't either. Let's get rid of them." If we multiply everything by 30 (the lowest common denominator of all 4 fractions), we get the equation 20x + 24 = 15x - 25.  No fractions. Hallelujah.

     I pause here and yell at my class. We didn't like the fractions so we got rid of them. You have that kind of power! You have that freedom! If you don't like something in math, you can change it! You're given rules to follow and you follow the rules, but do whatever you want. Where else in life does this happen? Where else can you want something and get it right away? If you want to get stronger, you have to work hard. If you want to get richer or prettier, you have to be extremely lucky or you have to work hard. If you want to get taller, well you can't. But in math, when you want to get rid of the fractions, you get rid of the fractions. I love that about mathematics. I love many things about mathematics, but that's a big one.


     My hope is this is a turning point for my students, where they go from what is their worst-nightmare-of-a-problem, to a realization that math is potentially wonderful. I try to keep sharing this point, and don't mind it when they laugh at my nerdiness. Just like how Shlomo inherited the love of a painting from someone else, Shlomo had to show me this appreciation for math, and then I inherited it. I hope if I can show this appreciation to my students, they can inherit it as well.



     Well, I've thought more about this, of course - this relationship of rules and freedom in mathematics. As long as you follow the rules you can do what you want. My parents have often said that kids want boundaries. They want to have clear rules of what they can and cannot do. They want to have these rules enforced consistently and fairly. If they have freedom within those rules they will be more or less happy and content. This is just like mathematics. There are extremely clear rules that are very, very consistently enforced, but if you follow those rules you can do as you please.
     So then I think about society. Ideally, society would be full of easy-to-understand, all-encompassing, non-loop-holey rules that are enforced but are fair. This doesn't happen, and that leads to all kinds of problems, but wouldn't it be great of society was more like mathematics in this sense?
     (Heck, isn't this kind of like how God gives us rules to follow, but freedoms within those rules?)
     You are thinking "But Grant, what about art? What about freedom of expression? What about diversity? What if some rules work for some people and not for others?" My reply? Shut up hippie. Seriously, though, that's all true. We won't ever have that idyllic society where a fair and just government makes and enforces laws that are for the equal protection and benefit of all. It won't happen. That sucks.
     But I can have that idyllic world in my small, misunderstood world of mathematics. There's freedom in all of those restrictions and rules that students don't understand. I just need to get them to appreciate it.