"Doing Math"

     I will often hear students say that they are solving problems without doing any math. An example: I put an equation on the board and asked my students to make a table of points that solve the equation:

2x + y = 12  was the equation.

Some points that solve the equations are (0, 12), (1, 10), (2, 8), (3, 6), (4, 4), etc. There are an infinite number of points that solve the equation.

When I ask my students how they got the points, a lot of them well say something (sheepishly) like "well, I didn't use any math, I could just look at the equation and see it."

When they say "I didn't use any math," what they mean is they didn't write stuff down and do adding/subtracting multiplying/dividing. Like, at no point did they plug in a 5 for x:

2x + y = 12
2(5) + y = 12
10 + y = 12
-10          -10
y = 2

so the point (5, 2) also solves the equation.

The above work they call "math."

     I try to stress that "doing math" isn't actually math. It's arithmetic. The adding/subtracting multiplying/dividing of numbers is simply arithmetic which is just a part of math.
     But mathematics is really just logic. It's logic applied to a very precise, well defined system. Looking at the equations 2x + y = 10 and just "seeing" points is really an application of logic, and that's math.

    Similarly, some students find more points in the equation above by noticing the pattern (every time the x-value increases by 1, the y-value decreases by two in the (x, y) points). They seem to feel that noticing and using a pattern isn't "math." No, it's not "arithmetic," because math is nothing if not lazy people noticing patterns to make their lives easier.

     I feel like this must be a result of poor teaching, from a young age. We call arithmetic "math" and never really teach straight, plain logic. It's frustrating, and something that CS Lewis mentions often throughout his writings. Teaching logic is challenging, and arithmetic can be easily grasped at a young age, so i understand its emphasis, but this is a huge disservice to our children.

Solving a System Through Combination

Systems of equations is a fundamental algebra topic. If 2x + 3y = 4 and 6x - 9y = 48, what do x and y equal?

One method of solving this system is through combination (also called elimination, and also called addition).

2x + 3y = 4
6x - 9y = 8

multiply the top equation by -3

-3(2x + 3y) = -3(4)
    6x - 10y = 50

which gives

-6x - 9y = -12
6x - 10y = 50

the point of doing this is to have two variables that have opposite coefficients (in this case the x's).

We can then add up the equations


   -6x - 9y = -12
+  6x - 10y = 50
         -19y = 38

if we divide both sides by -19, we get

         -19y = 38
          -19    -19

            y = -2

if y= -2, we can plug that back in to either of the original equations to find x

   2x + 3y = 4
   2x + 3(-2) = 4
   2x + (-6) = 4
           +6   +6
   2x = 10
   2x = 10
    2       2
    x = 5

so x = 5 and y = -2. The point (5, -2) solves both equations.

But, let's look at that step where we added up the equations again:


   -6x - 9y = -12
+  6x - 10y = 50
         -19y = 38

     I always, after a couple days of doing this method, ask my students "did anyone wonder why we are allowed to add up equations? We've never done that before. Was there a moment where anyone thought if that was even allowed?"  Usually one or two students admit that they briefly had that question (and it's usually a girl, truth be told).

     Why is that allowed? It's kind of weird! Well, first things first, we aren't actually adding up the equations, we are adding up expressions, the two halves of the equations (6x - 10 is an expression, 50 is an expression, and they were equal to each other).

     The Addition Property of Equality states that if a = b, then a + c = b + c, which basically means you can add whatever you want to both sides of an equation.

Well, let's look at that new equation we made by multiplying by -3:

 -6x - 9y = -12.



I ask my students, can I add 4 to both sides?

-6x - 9y = -12
           +4      +4      of course I can (some students don't think I can, in fact, because there was no -4
                                anywhere in the equation. I try to clear this up.)

so, could I add 6x, if I wanted to?

-6x - 9y = -12
           +6x    +6x    yes, again, I could do that.

I could also add 3 to one side and 2 + 1 to the other, right?

-6x - 9y = -12
          +3     + (2 + 1)     as long as I am adding the same thing to both sides, then I can do anything.

Well, look at that other equation 6x - 10y = 50. The equals sign means that the two sides are the same. 6x - 10y is the same as 50. So really, if I want to add 50 to both sides of the first equation:

-6x - 9y = -12
          +50    +50
I could change that first "50" into 6x - 10y (because they're the same thing! 6x - 10y = 50)


-6x - 9y = -12
+(6x - 10y)   + 50


and at this point, it is easier to just add up the equations and save some writing.


   -6x - 9y = -12
+  6x - 10y = 50
         -19y = 38

and then solve from there.

     Does anyone care about this? Again, I would have maybe one or two per class who were interested. But understanding little things like this can really pay off in the long run. I was never taught this, not remotely. I was in a van driving to a frisbee tournament talking about math with a couple of friends, and some made the distinction between adding equations and adding expressions. I took some time to put the rest together and a lot of other things in math started to fall into place. I want to give this opportunity to my students as well. So I try to teach my students why.

I Am An Idiot.

     My beloved cousins up in Portland get together and make frozen burritos. Lots of them. I'd be exaggerating if I said they make thousands of them, but they certainly make hundreds. They brought some down to Eugene right after Wyatt was born (I like to pretend that I got the most out of all of the Eugene Gilchrists, due to my favored status, but I don't think it's true).  I've been taking a frozen burrito to work for lunch every day ever since. It's been a lifesaver and the burritos are delicious.

It's proof of my idiocy

     The burritos are individually wrapped in wax paper and are filled with chicken, pork, or beef.
     The kitchen where I am working this fall is pretty bare. No plates or napkins (no sink to wash them), no silverware, no cups and no faucet. There is a fridge, freezer, microwave and coffee maker. So, I microwave the burrito on top of its wax paper. Usually I do this for about three and a half minutes. Sometimes the wax paper gets stuck to the burrito a bit but I can live with it.
     Today, eating THE LAST burrito that I was given, I microwaved it for about four and a half minutes for some reason. That's a long time. When I picked up my wax paper, the burrito fell through it, making a big hole in the middle. Again, for some reason, I didn't think much of this and took it into my office. I looked at the bottom of it. It was kind of slimy, but almost in an appetizing way. I thought to myself, "is there any chance there is wax on this?"
     I took a bite.
     I chewed for a few seconds.
     Something hard started crunching.
     I pulled out a bunch of hard wax from my mouth.
     I am an idiot.
     I took the burrito and ate into just the top of it, trying to avoid the wax, and made a mess.
     It still tasted like wax.
     I am an idiot.

Life at Conception and Initiative 26 - Part I

(A two-parter on when life begins and my views on abortion. Part II is here.)

     I should start by saying that I am pro-life, just so we all know where I am coming from. If anyone says to me that a woman "should have the right to choose" what happens to her body, my response tends to be "I completely agree, and she had the choice to not have sex." I don't even think that's as strong an argument as "well what if that fetus is a girl, why doesn't she get to choose?" I think sex is the root of the argument, when people make it about "life."
     "Life." That's what's up for debate. When does a human life really begin? When does a person become a person? I am out of my league in this conversation. I have half-formed opinions from half-understood facts and statistics that may or may not be accurate. This is me thinking out loud, because I tend to understand things better when I think them out loud in my head (it makes sense). Being misinformed and out of my league has never stopped me before, after all.

      I'm going to start with the most logical approach I can. Let's say life doesn't start at conception, as this law is trying to say it does. The logical question that follows is then, when does it begin? I've heard people say that it begins when the embryos "start to look like a person." This is incredibly dangerous, because not only is it stupid (it really is), it's subjective. But more importantly, if that was the definition of human life, "looking like a person," the true outcome of that line of thinking is that we would have levels of humanity. The deformed would be less human than the non deformed. Those in their prime of life would be more human than infants or the incredibly old. A man who gets in a car wreck and loses his legs is suddenly less human than he was yesterday. This is obviously a notion that few would accept (and we tend to not thing too highly of those that would accept it). But why do we reject it?
     Why are opposed to the idea of levels of humanity? There must be something that unites all of humanity beyond how we look.

When did Wyatt start thinking? I'm pretty sure it was long
before this picture was taken.

     Maybe a person becomes a person when he or she can think. When there are synapses firing in the brain. I don't know when that happens in a human embryo (and am not going to bother to look it up), but I know at some point it goes from not-happening to happening, right? Is it at that moment little Wyatt became a human? Well, what separates human thinking from animal thinking? It can't be problem solving or language, as lots of mammals can do that, just not as well as us. Is it the ability to think abstractly? I don't know if animals can do that (and I don't know if anyone knows). But, suppose there was a human with a defect and he couldn't think in the abstract. Would he no longer be human? I don't think anyone would say that. When a person lies in a vegetable-like state, are they no longer human? I don't know. Some might say he's not. I don't think those people have the plug pulled because they are no longer human, but because their quality and hope of life are so diminished that it's in everyone's best interest. What am I talking about? Point of the paragraph: I don't think that "thinking" is what makes us human. Pretty much all animals have a brain.
     So it's not based on how people look or when they start thinking, in my opinion.
     What's left? A heartbeat? That's actually a pretty good benchmark. Once that heart starts to beat, we have a person. But really, the key isn't the heartbeat. Heartbeats have a special meaning to us because we can feel them and hear them and pretend like that's where emotions come from (despite the fact that we really feel them in our gut, right?). But the heartbeat is just a means to an end: pushing blood around to get stuff to different parts of the body. Picking "heartbeat" as the benchmark is fairly arbitrary. What if someone gets a heart transplant? In those few moments when there's no heart inside him, is he no longer human? Of course not. Or if someone has a machine pumping their blood? Still human. The heart isn't the issue, it's the whole system of bodily functions. Well damn it, those functions are starting as soon as that baby is conceived. They change and morph over the next several months, but they are fundamentally the same, right? (no really, I am asking, I don't know). Those functions are no different than a bear's or a wolf's or a rats. Rats.
     If it's not the brain, the heartbeat, or the physical appearance, what's left? Really, the question becomes "what makes a human a human," or, different from animals. That is a question of soul. Humans have a soul. Define that however you want, I don't actually care for the moment, but let's just agree to call the difference between humans and animals "the soul." (If you think humans are no different from animals, you are deluding yourself.)

     So, pretend we agree that humans are humans because they have a soul. Then, back at our original question, if having a soul makes a person a person, when do they get that soul? Conception makes the most sense. They become a human when that sperm and that egg become best friends and start growing. It just fits. If the soul came any later, you would have to assume that someone or something is giving that person a soul, and that someone or something would really have to be God by some definition, and at that point the morality and ethics of abortion are pretty much lost (if there is a more-powerful, higher being, we shouldn't be undoing its work of creating life). If the soul isn't bestowed at some point, it must happen at conception.

     This is what I do. I grab an argument or statements or assumptions and I take it to its grandest, largest extremes, and I take it to its smallest, most minute extremes. If the arguments hold up in these extremes, they are probably valid, and if the arguments break down, they are probably flawed. I do this with everything, all the time. I can't help it. So, yes, I concede that I am nitpicking notions like "life begins with a heartbeat" and that I am drawing fairly broad, grand conclusions about the existence of a soul or a God. But my logic is pretty sound. I can't prove anything, but doesn't it at least raise enough questions about the alternatives? Don't you have to question the idea that a person becomes a person at any time other that conception?

     Believing that life begins at conception is logically consistent. It is also the simplest, easiest definition. I am not going to try to apply Occam's Razor, because it doesn't really fit here, but the fact that it's the easiest point to start from is a great reason to back it, in my opinion. Get rid of those grey areas if you can.

     Unfortunately, this isn't a very convenient place to start life. It means that some effective, popular forms of contraceptive are actually killing people (because they do their work after conception), and it means that throwing out fertilized eggs when a woman is receiving in vitro fertilization is murder, and it means that early abortions are more horrible than they were before. All three of those results suck, but there you have it. You aren't going to find a very convenient place to say that a human embryo isn't a person, and convenience isn't the goal, anyways.

     If this initiative is passed, a lot of things follow. It is almost surely going to go all the way to the supreme court, which will either undermine or strengthen Roe v. Wade. It will make getting an abortion illegal in Mississippi, and it will make the process of in vitro fertilization much more complicated and less likely to be successful. It will make post-conception forms of contraception illegal. This initiative is a very big deal.

Part II is here

Life at Conception and Initiave 26 -- Part 2

A two-parter on when life begins and my views on abortion.

     I just spent 1300 words explaining why I think life begins at conception. Know what? I don't think that really matters. I don't think that's the real issue here, or the real issue with abortion. I think the real issue is that people want to have sex without consequences. People like having their get-out-of-jail free cards and want to defend it and protect it and keep it legal, so they fight over it.
     As I said last time, I am pro-life. I think the labels "pro-choice" and "pro-life" are stupid, because no one is really "anti-life" and no one is "anti-choice," but they are the political terms that have been solidified. I will use pro-abortion and anti-abortion to call it what it is.
     I don't think there are many people that have the active, thoughtful mindset of "I don't care if I get pregnant (or my girlfriend does) because I can just have an abortion." I really don't. I figure at the very least, abortions are expensive and annoying, so people would rather avoid them. But I do think that in the back of peoples' minds, there is this notion that "it's okay if I (or my girlfriend) get pregnant because it doesn't have to change my life." That thought is there for a lot of people, and you will never convince me otherwise. How could it not be? This comforting thought allows men and women to have sex and if a girl gets pregnant, they can continue their lives essentially unaffected. This is good news, because sex is fun. And sex without consequences is more fun. If we can convince ourselves that sex without consequences is possible, sweet! That just means ignoring the fact that sex has an emotional hook to it, that it tends to produce life, and that it often transfers diseases.
     Well, emotions can be managed, diseases can be treated, and life doesn't start right away anyways, so free sex is still fair game.

     The consequences exist. They just do. So, abortion becomes an option to get rid of one of those consequences. People that are opposed to Initiative 26 have a lot of concerns about it and what it leads to - I am just going to focus on abortions for now.
     There is the legitimate concern that criminalizing abortions will lead to more back-alley, coat hanger abortions and overdose abortions. This may happen. It's truly unfortunate. But there are consequences for actions, and getting an abortion is just a consequence for trying to avoid another consequence. It will still drastically reduce the number of abortions that occur.
     A lot of people argue that this won't fix the problem. That there are too many unwanted pregnancies, that these children will grow up in homes that aren't able to provide, or that it will cause further problems on society. This is also probably true. Here's where I propose a radical idea:
     Don't make criminalizing abortion the only step towards stopping abortions and unwanted pregnancies.

     There's not just one solution to the problem!! Why is that so hard for people to see?! Don't just stop abortions. Educate kids better! Inform them of the consequences of sex, the safety and benefits of abstinence, AND how to use condoms! Have parents start preaching values again! Attach a modicum of morality and ethics to sex!
     And if you are a person or group who is anti-abortion in an active, loud sense, then you damn better be pro-adoption. And I don't mean pro-adoption in the theoretical, supportive sense. I mean you better get off your ass and adopt some kids that need a home. If you care so much about those unborn babies, give them a good home to come to.
     There's not one solution to the problems that ail our country. The financial crisis we are in is not due to a lack of oversight by the government. It is not due to rich people not paying enough taxes. It is not due to massive corporations not paying enough taxes, or banks being dumb with their money, or huge debt by the government, or huge debt by the American people, or defaulted housing or a lack of jobs. It is due to ALL of those things. Fix all of those things. Stop blaming each individual thing and fix all of them.
     Abortion is no different. The problem isn't that people want to have free sex. The problem isn't that kids are uneducated, or that there is a lack of morals, or that men aren't taking responsibility for creating life, or that there aren't enough viable candidates for adoption. It's all of those things. We have to fix all of them.
     I truly believe these problems, all of them, start at home. Families are broken down through divorce, values and morals have been lost, money is no longer saved, and everyone wants a nice house, car and TV. This lack of family values trickles into how we buy things, what we watch, and who we vote for, and here we are. Oversimplification? Yes. Accurate? Probably. Do I have any solutions? Men need to nut up and start raising families and everyone should go start buying into some reasonable system of morality and ethics. I can recommend true Christianity if you don't know where to start (meaning, not Christianity preached by someone trying to profit off of you or their fame, for starters). And start saving your money.

     I don't know what I am talking about. I understand that. I turned from a logically thinking person to a crazy man spewing his ideas of how the world should look. But I just might be right. I think I am right. Sex isn't free, abortion is killing a person, life begins at conception, and work hard to raise a good family.

Stuff That Happened To Me Today

     A student of mine at LCC went deer hunting and brought me back some venison steaks and a stew. Do I have to give him an A now? This is a quandary. It was very thoughtful of him to give me some meat without me prompting him at all. I guess, When he told me he was going to miss class because he was deer hunting, I said "if your freezers full you can always throw some venison my way," but it's not like I asked for some venison.
     More subtly, on the first day of classes I always begin by saying "My name is Grant Gilchrist, I've been teaching here for two years, and I can be bribed." As a government employee I feel like that's the right thing to do.

     I grabbed my frozen burrito and put it in the microwave for a few minutes, and went to my office to check facebook grade. When the microwave beeped that it was finished, I went back to the kitchen and checked the freezer again. My burrito wasn't there (venison still was, luckily). Where did it go?! Did someone seriously steal my frozen burrito? As I was freaking out I smelled it in the microwave and felt like an idiot.

     Did you know that a gallon's size was determined by the size of a goat's stomach? I really hope that's true because I've been telling my students that for years.

     I think I encountered my first brown recluse spider: the very poisonous spider that lives in the northwest and hides in dark, dank corners. I was moving a bag of pellets from a corner of the garage to the pellet stove, and I'm pretty sure that bag had been sitting there for a couple of years. When I picked it up, I thought "I bet there's a brown recluse under this thing. Sure enough!!! There, on the bottom side was a big, really dark spider with a huge butt (that must hold all of its deadly poison I am sure) that started moving around frantically. I ran with the bag into the house, threw it on the stove, and freaked out a little. I did my best to bury the spider underneath the pellets and hope it burns.
     I told all of my friends about my near-death encounter. Then I looked up brown recluses. Turns out they don't look like the thing I've had nightmares of, and don't tend to live up here. It definitely wasn't a recluse on my bag. I don't plan to tell my friends this.

     I was subbing at a high school and a kid just got up, went to the back of the room, sat on the counter and started talking on his phone. I wasn't teaching or anything, it was general work time, so it wasn't offensive or disrespectful. Usually if I see a kid texting, I write down my "phone number" on the board, and if they are asking their friends for help with math they can just text me. This makes them blush like crazy, and depending on the reaction I will start calling them "texter" and have texter work on the problem I was doing. It's lots of fun for everyone except maybe one person.
     Well, this was a new one for me (someone cavalierly on the phone in the back of the room). I grabbed my phone, held it up to my ear, and started talking on the phone right next to him. "Yeah, so I guess this is the place in the room where we just get on the phone and talk. No, there's not any more appropriate place to do this. Yeah, I know, right!?" Then I started basically repeating whatever he was saying on the phone into my phone.

Picture on my phone

     He turned to me, showed me his iphone picture and said "I'm talking to my dad." I showed him my phone with a picture of Wyatt on the desktop and said "I'm talking to my mom!!" He handed me the phone and said "no really."  He was talking to his dad. Not that it's any better. I handed back the phone and started yelling at people (on my phone) to buy low/sell high, move the inventory and stop mismanaging my funds. Things like this went on for a bit longer until I was bored. He outlasted me.

     When subbing, I try to not call of names for attendance if I can help it. It kind of halts the class, people start talking, and it reminds the class that I am not in charge. So if I have a seating chart, I just use it, and if not I take attendance at the end. I work really hard at saying people's names right on the first try. I usually read through the list once or twice to myself before I go live, and try to work through how I think names are pronounced.
     I was doing a great job. I called Juan Juan, Jesus "hey-suess" and Julia "who-leah." But her name was Julia Anderson. Julia with a J. And I'm an idiot.

Awful, Awful Math Problems

     I had a student approach me after class this week. He said there was a problem he was asked in a math class a long time ago that he'd never been able to solve, and it had stuck with him. It's a classic textbook problem:

     "You're at the zoo and look into the arctic exhibit. You see that there are 18 legs and 6 heads. How many polar bears and penguins are in the exhibit?"

     This is one of the dumbest possible problems. It's just so stupid. If you are looking in the exhibit, why wouldn't you just count the animals instead of their legs and heads? If you can see their heads, you can see what kind of animals they are! And why are the polar bears and penguins together in the first place? They live as far apart as possible on the earth. Not to mention that those polar bears might try to eat those penguins (we don't know, BECAUSE THEY DON'T LIVE WITH EACH OTHER). What a crappy zoo! Why am I at that zoo?! I hope I didn't pay to go to that zoo.

     If you were to give this problem to a young child, like an elementary school student, they would ask all of those questions I listed above. If you gave it to a high school student, they would sigh and try to get through the problem. At some point along the way those high school students were conditioned to accept these dumb parameters, where logic is thrown out so we can practice using logic. Is it any wonder a lot of students hate math? They are asked to pointless, not-remotely-applicable things like this all the time!!

    The problem is supposed to be solved like this:
     x = # of polar bears, y = # of penguins

     4x + 2y = 18    (equation for legs, 4 per bear and 2 per penguin)
     x + y = 6          (equation for heads, 1 per bear 1 per penguin, ideally)

     And then you solve using substitution or elimination/combination (there end up being 3 of each, who does that in a zoo?!).  Substitution and elimination are worth learning and practicing. This problem (and almost all the problems that come with this topic) are not worth doing. There aren't a lot of simple, entry level problems that use these ideas in mathematics. These ideas are useful in chemistry and city planning and other areas, but they are fairly complicated to start.

     This guy talks about how awful math problems and textbooks can be (his name is Dan Meyer and he teaches in Santa Cruz), and everything he says is legitimate. I am looking for problems in real life when I might potentially use systems of equations, but haven't found any. I'll work at it, and if you know of any, please send them my way. (grant.gilchrist@gmail.com)