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.

Criticism

     "Can I just say, you can be a real ass."

     I haven't been the best teacher this term. I've noticed that my energy and patience are much lower than usual. I respond less frequently to emails. I am finishing my grading more slowly, and have made a lot more mistakes in grading (and while teaching) than I am comfortable with. This hasn't sat well with me, as the term has wore on, but I haven't really dealt with it either.
     The day I gave my first test in one of my classes, about three weeks ago now, I got a note from a student (which stated it was written on behalf of at least a couple of students). These students felt that I moved too quickly when teaching, and that I needed to give more time on the test I had them take. I considered their arguments (I feel that I often do move too quickly), but in that scenario there were extenuating circumstances.

     Today, however, I was approached in back-to-back classes. In my math 65 course, a student told me about how he was struggling. He was a very frank, honest and mature person. He told me he thought I was a fine teacher and that if I taught anything but math he'd be "all about" my class, but that he needs more hand-holding and coddling in math (his words). I get this from students a lot. I think all teachers of entry-level math do. Students come in with such a fear of the subject, and a history of frustration, that we are both losing the battle before class starts.
     He explained that he didn't find me approachable. That it took him a lot of nerve to talk to me after class. And that he didn't think he was the only one that felt that way. Okay.

     A couple of weeks ago, in another class, a student did very poorly on her test. She talked to me after I returned the graded tests and asked if she could retake it. I asked her why she thought she would do better the next time, and she told me she met with a tutor (after the test) and understands things better now. I told her she could re-work the test, and if she did improve I'd give her a retake. She didn't improve. I think she felt that I was being dismissive of her (which, in all honesty, at that point I was). I asked her what she was in school for, and she wants to be an elementary school teacher.
    About a week later I tried to talk with her again about her plans and her future. I was trying to get a read on the sort of help she needed, or if it was worth an investment of my time. (It's kind of hard to view someone as "worthy" of extra help, but it happens). She told me she was going to have to quit school after this term because of her baby that was coming soon.
     Well, today in her class I gave a quiz. She finished last, and as she finished she said, to no one in particular, "Math takes me a while, but I'm not stupid at math." I had made no suggestion of impatience or frustration that I was aware of, so I said "Oh yeah, I understand." She followed with "Can I just say, you can be a real ass."
     I wish I could say I was caught off guard.
     "I apologize, that's not my intention. Can you help me? I don't want to be an ass to you or anyone else, so can you tell me what I did that made you feel that way?"
     "When we had that conversation, you made me feel retarded."
     "I'm sorry, I would never try to make you feel retarded, and I don't think you are retarded. Which conversation was it? Was it the one about the test?"
     "No, the one after that."
     I assume she is referring to the one I described above, where I was trying to determine if I should be helping her more. I've run it through my mind several times, and I thought I was being understanding and genuine and showing concern. Apparently I was not coming across that way.

     In the span of about two hours, from two extremely different personalities and backgrounds, I had been told I was an unapproachable ass. Not the qualities of great teachers.

     I have been less approachable this term. I am a bit more - how do I say this - myself this term than I normally let myself be in front of the class. It's not like I am trying to be less approachable, but I am not putting forth the effort that I used to to make it clear that I am happy to help people and take any question. Like I said, I haven't been the best teacher this term. So I guess these two students have confirmed what I have been feeling for a while.
     But here's where I struggle: those tests that my students complained about? This class averages were 74% (good) and 80% (great!) - and I don't give easy tests. I have lots of students in all my classes that ask lots of questions. I crack jokes and they laugh, they crack jokes and I laugh. I am connecting with a strong majority of my students. That's a good thing. I used to work to connect with the others, and this term I am not. That's a problem.
   
     The term is now half over. Five weeks down, five to go. I am going to work at being more energetic, approachable, and not-assy. I think I can do better.

Stuff That Happened To Me Today

     I subbed at the high school today for the first time in like two months.I've learned a couple of tricks in my last couple subbing adventures that I'd like to share with you.

     If the class has no homework assigned, I will lie to them. I should point out, I lie to students all the time. It's one of my main teaching strategies. I tell students I like them when I don't or that I don't like them when I do, or that their test is going to be hard when it's easy, or that I once hit a squirrel in the face with a rock. Things like that. Well, if the class doesn't have any homework, it's because they have a test that day. So I tell the students, "After you finish your test I'll put the homework on the board." When they all finish the test, I say, "You know, you worked real hard on that test and were a good class, so I am just not going to assign the homework. If your teacher has any issue with that, she can take it up with me."

     You should see the smiles on these kids' faces. They call me a hero and applaud me and give me high-fives and handshakes. It's wonderful.

     I wanted to feel that feeling more often. The feeling of being more-than-tolerated. So now, when classes do have homework, I lie. Say they are supposed to do problems 25 - 45 odd. I will write on the board "Page 312, problems 25 - 45 odd, 53 - 61 odd. Then, after a few moments, say, "no, that's too long," and erase the made-up stuff. They get much more confused and stressed, and aren't really relieved or proud of me at all, but I am still having fun. I think that's what matters most.

     I've always been kind of fascinated that I can just say stuff and people do it. Not even in teaching, but in life in general. I have a pathological addiction to lying, I think, kind of like the janitor. Everything written here is totally true, though.

     Another sub trick, although it is really more like a life trick, happens when reading names. I'll read, out loud, about 130 names a day when I sub, and I always feel bad for the kids that always have their names pronounced wrong. I've learned that you are much more likely to say a name right if you say it as fast as you possibly can. This works especially well with Asian names, if I am allowed to say that. Also, I will somewhat randomly alternate between first and last names, so that if a person has one name I am comfortable with I can say it and not look like I am singling anybody out. Still, I really butchered like 5 names today...

     I was walking Wyatt around the loop trying to get him to fall asleep. After a few laps, I burped, then he burped, then he farted and I farted before he was done farting. Then he peed.

     

Tutoring Terrifying Children

Do girls really like this show? Do they want to talk
about it? I have no idea.

     I am tutoring quite a few students in math right now. Most of them are in pre-calculus or calculus, but I cover pretty much anything. I've been tutoring high school students for a few years and am getting a pretty good system going. What I'm doing now for the first time is tutoring an elementary school student. It's a totally different ballgame, and what that I am not really trained for. What's more, it's with a fifth-grade, ten-year-old girl.

     I used to be really good with kids. I counseled at Camp Harlow for years and worked at an elementary school when I was a senior in high school. Every high schooler is good with kids if they want to be, because they are already everything a kid wants to be: older. When you are 17, you're already cool. By wearing a hat backwards and driving your own car and talking to other high schoolers, kids love you.

I didn't draw this, but I wish I could

     Beyond that, however, I am essentially a kid myself. I honestly like the cartoons Spongebob Squarepants and Phineas and Ferb. if Power Rangers was on I would probably watch it if they were air-fighting and not trying to develop plot. I know lots about Pokemon. I can talk and play sports with anyone but also like video games and nerdy things. I have a lot of random, probably-true animal facts at my disposal from working at a zoo. (Did you know that a kangaroo's nasal passage runs around it's brain before going to the lungs, in order to ventilate its brain? That might be true.) I think that farts are very funny if timed correctly.
     But my main two tricks with kids are this: 1) talk to them like they are an adult, and 2) call them "dude." I've found that kids really respond to being talked to like adults, because it doesn't happen hardly ever. I don't raise the pitch ofmy voice or talk super slow or generally baby-talk them, and we can have a normal conversation about how awesome the Dragonzord is (answer: super awesome). I don't talk to their parents about them, I talk to them. They appreciate this. And when you call them "dude," they just assume both you and they are cool, and everyone wins.
     As you've probably noticed, all of my strategies for relating to kids really pertain to boy-kids. I always babysat or camp-counselled boys. I don't know what the equivalent show to "Power Rangers" for girls is, and I am not a Bronie who knows about My Little Ponies. I can't talk fashion, don't want to talk about cute boys, and am not crafty. Girls are (still) terrifying.
     (additionally, at "teacher school" when I got my master's degree and teaching license, pretty much every lesson ended the same way: "gentlemen, DON'T EVER BE ALONE WITH A FEMALE STUDENT EVER EVER OR YOU WILL DIE." We could be talking about returning tests promptly and the instructor would have a statement like "when you are grading your tests make sure you are consistent, and guys, if a girl walks into your room while you are grading by yourself make sure you grab someone to join you immediately." You think I'm kidding, but I'm just exaggerating.)

     I was thinking over all of these things before my first session with the totally intimidating fifth-grade girl. Between me no longer being a cool-by-default 17-year-old, her not being a boy, and me not being around kids for about eight years, I was nervous.

I recognize this is my best weapon for
combating little girls.

     So, before our first tutor session, I was running through all of my conversation options. I had to win her over from day one - not so much for my sake, but because she needs a tutor and I want her to enjoy math, which means she needs to enjoy her time with her math tutor. When she walked in I shook her hand and said hello and had her come sit at the table with me while her dad sat in a chair nearby and read. I told her I liked the hat she was wearing, and the sparkles on her jacket (then felt stupid, was "sparkles" the right term to use there?) and I grabbed a bunch of colored pens for us to do our math with. Colors and pens was a big part of my plan. I tried to have some crafty-things that we did, and I complimented her on her handwriting. I even talked about how hard it is for me to make my "twos" look pretty when I write them. She met baby Wyatt and answered my questions about her family and class. She even giggled when I asked towards the end of the session "so, are there any boys in your class you think are cute?" and promptly explained that boys don't get cute for a long, long time. Getting a giggle was huge.
    I pulled out all the stops. I did everything I could. In fact, I used Letterman's go-to questioning when he interviews a kid on the Late Show: "Are you married? Do you have a house? Do you have a car? Can you drive?" When the kid says no to all these things, Letterman says "sounds pretty boring." The first session went well. Now what? I'm out of ideas already. If you have any suggestions, send them my way. We've had a few sessions and we don't have a common ground yet, meaning nothing to have a conversation about, just me asking questions and her answering them. I am working towards conversation, which I know doesn't necessarily happen with grade-school kids. At the same time, when I counselled boys we would talk about things like Charizard and Dragonball (not what it sounds like). I am sure we can get to the point where she and I can talk about things too, I just might need to get invested in Adelle or something, I don't know (seriously, if you haven't gathered, I have no idea).
     In the meantime I am still nervous before she comes over. Here I am, a math teacher that doesn't blink when a student wants to come over and get calculus help on stuff I haven't seen in years, and I am sweating out working with an adorable, sweet little ten-year-old girl on her long division. Some things never change I suppose.


(PS: if you didn't click that "bronie" link above, it's pretty fascinating/sad. There's a whole bunch of 20-30-something aged men who watch and blog about the new My Little Ponies TV show, and the call themselved Bronies. Look into it: http://www.wired.com/underwire/2011/06/bronies-my-little-ponys/)

"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.

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)

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.)

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.

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.

Explaining Inequalities

     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.

"Why do you switch the inequality when you multiply or divide both sides by a negative number?"

I'm glad you asked. (You're still reading this, right?)

Suppose we have the inequality -2x + 3 < 11. It is solved as follows:

-2x + 3 < 11
        -3   -3                         Subtract 3 from both sides
-2x < 8
-2x  >  8                              Divide both sides by -2 and switch the sign

 -2      -2
x > -4

The solution x > -4 works if you check numbers larger than -4 (like 0).

The inequality, much like the equals sign, is a statement of fact. The inequality 2 > 5 is stating the fact that two is greater than five, and the inequality -2x + 3 < 11 is stating that the left side is less than the right half.

The reason that inequalities switch when you divide by negative is, honestly, because that's just the way it is. Here's what I do with my students:

I put up two numbers and ask which sign goes in-between them.

4          8

and they tell me that a "less than" squeezes in there.

4    <    8

I start doing things to both sides, and check to see if the inequality is still true

4    <    8
+12      +12

16   <   20                   still true, 16 is less than 20, so we subtract 30
-30       -30

-14   <   -10                still true, so we multiply by 5
x5            x5


-70   <   -50                still true, so we divide by 10
/10          /10


-7     <     -5                still true, so we multiply by -1
x(-1)          x(-1)

7       <     5                 this is no longer true. Seven is, in fact, not less than 5. The sign needs to switch.
                                   
The sign didn't need switching with multipling or dividing positive numbers, and it didn't matter if we added or subtracted when positive or negative. It only mattered when we multiplied by a negative number. The same would be true if we divided by a negative number (like negative one).

Another way to describe this is with a number line. I have tried like a trillion ways to create a number line in this space that I can work with, but I am having almost no luck. Here's the best I can do:

-5  <  -3                      but                      5   >  3   

     Imagine having two points on that number line, a point at -3 and a point at -5.

     If we multiply both of those points by -1, we get the new points 3 and 5.

     Well, for -3 to travel to three, it has to travel a total of 6 spaces, but for -5 to get to 5, it has to travel 10 spaces. In that farther distance to travel, the -5 "passes" the -3 and goes from less than to greater than.

That is why the sign switches. Most people don't care. I never had it explained to me, and when I was teaching it one day I thought to myself "why does that happen?" So I went home, figured it out, and started teaching it. I know there are a few students in every class that appreciate seeing that. Hopefully it helps.

I know that no one is still reading this, so I would like to take this opportunity to let the world know that today I wiped my face with a towel that was on the carpet, and my wife told me we had placed that towel under the baby when changing him. Thanks, wife, for waiting until I had finished.

Mathematical Equality

I feel like the meaning of the equals sign is misunderstood by students everywhere. Each year, I start every math class the same way. I draw an equals sign on the board and ask students what it means. Sometimes I get the response I am looking for (it means the two sides are the same), and sometimes students tell me that the equals sign separates the “problem” from the “answer.”

                See, we go into kindergarten and are given a bunch of worksheets that have problems like “2 + 3 = _____” and most of us put a 5 in the blank. “Two plus three equals blank” conditions us to think that the left side of the equals sign is the problem, the right side is the answer, and the equals sign just separates the two of them. Problem = Answer. Really, the equals sign is stating that two plus three is the same as five. We use that blank space to represent the notion of an unknown, as if we were saying 2 + 3 = x and then x = 5.

                This misconception leads to some common errors when solving equations, like 3x + 5 = 17.

   3x + 5 = 17

= 3x + 5 - 5 = 17 - 5

= 3x = 12

= 3x/3 = 12/3

= x = 4

The above work states that every first expression is the same as all the following expressions: 3x + 5 = 17 = 3x + 5 + (-5) = 17+ (-5) = …  At some point there is a statement that 17 = 12 = 4, which of course is not true. This stems from the fact that students view the equals sign as the bridge from the beginning of the problem to the end, not a statement of fact. (Instead, when solving, all four of those equals signs on the left end of the solution should be removed. There’s no “bridge” between the equations, each line simply represents independent statements).

                There are other issues I've found with the notion that the equals signs splits up the problem and the answer. Students who are able to solve 5x + 8 = 23 struggle to solve 23 = 5x + 8. Or, when solving the equation 9x + 5 = 17x - 4, students will subtract the 17x from both sides so that the x's are on the left side, even though I think subtracting the 9x is much easier. 

                I’ve found that clarifying this fact early in a classroom pays dividends long term. It makes the steps involved in solving equations more logical, it allows students to understand the relationship between inputs and outputs in function notation, and, most importantly, it gets students to think about the mathematical statements they are making while writing their work. These are all little things that make life easier in the long run, for them and for me.

                I wasn't great at math when I was in school. I was never told, explicitly, what the equals sign meant, and I would put the equals sign in-between every line when solving equations, like I did above. The first math class I took in college corrected that mistake, and I felt foolish. However, when it was clarified to me, other things began to fall into place as well. I hope to do that for my students a bit earlier.

Life Lessons

                There are life lessons I would try to impart with my high schools students when I had a classroom. Lessons like: go to college, don’t major in English or History unless you either absolutely love it or want to be a teacher, save your money, and don’t get into credit card debt. Teaching math is very challenging and very important, and opportunities to share these sorts of lessons don’t always come up naturally.

                I currently don’t have a high school class. I teach math at Lane Community College, where I would feel a bit odd sharing a lot of the above lessons to adults, many of whom are older than myself, and I substitute teach about twice a week. Tomorrow, if I get the opportunity I am going to share one of my favorite life lessons with my classes at South Eugene High School. I want to see what it’s like for a sub to do more than just hand out a worksheet and take attendance. When I had my own classroom, I had about 120ish students. What if I didn’t have to limit my personal ideals to just the 120ish students I had in a given year? What if I am able to reach more students while subbing? This is a new and exciting thought for me.

                I’ll give you the gist of the lesson, if you are curious (if not, you might as well go to some more entertaining website I suppose). It’s about the balance of freedom and responsibility in our lives. I feel that having more freedom requires one take more responsibility, which might be a new, possibly illogical thought for a 15-year-old, and when freedom and responsibility aren’t balanced there tend to be consequences.

hours-old Wyatt

                When we are born, we have absolutely no responsibilities, whatsoever. And when we are born, we have absolutely no freedom, either. We can’t do anything but cry. It’s a struggle to move our head, we can’t get food for ourselves unless a nipple is in reach, and we certainly can’t go anywhere. No responsibilities (sweet), no freedom (lame).

                When we are very young, we have very few responsibilities (clean your room, eat your vegetables) and just a bit more freedom (physically able to do simple things, freedom to play in most rooms of the house or designated parts of the neighborhood). A bit of responsibility and a bit of freedom, more or less in balance.

                The responsibilities and the freedom continue to grow together for a long time, and a few significant jumps occur. Maybe you get a car, or have your choice of classes in school (or a free period). When those freedoms start to occur, ideally responsibilities come with them. Get yourself to school on time, get a job to pay for gas, maintain a certain GPA or get into college. Still, this balance between freedom and responsibility remains.

                What happens to young men and women, from about age 15 to 25 (maybe 35 these days), in my opinion, is they will often try to keep increasing their freedoms without gaining the appropriate responsibilities. Get your car and go to parties or the beach. Get your free class and skip a few others. Go to college and go wild with experimentation. These actions, which are glamorized and encouraged throughout almost all media, can have consequences which rarely are mentioned. In 2008, the CDC stated that 1 in 4 American girls has an STD, and there are over a million abortions per year in the US, to cite two examples for women. For men, this pursuit of freedom without responsibility has led to less men in college (women outnumber them for the first time), men earning comparatively less than women and possessing fewer jobs than women ten years ago. (I compare men’s jobs to women’s jobs, because obviously men would make less and have fewer jobs now than 10 years ago due to recessions and high unemployment, but in that same time span women are still gaining.)

                I tell my students that I had had my most freedom when I had a full-time job and a full-time marriage: I had enough money to have a place to live, food to eat and a car to drive. When I wanted to go to the beach, I had a friend to go with and didn’t need to ask anyone. I can afford the video games I want and the food I like. They usually counter by saying “it seems like money is the key to freedom.” Well of course it is, at least in the ways I listed above, but you need a job to get money. At least, I did, and most of them do too.

Doesn't look like I've had much freedom lately, does it

                Now, here’s where my theory breaks down, and I used to tell my students this hypothetically, and now I can do it factually: Having a kid gives you much more responsibility and much less freedom. They no longer balance. That’s okay though, because I am happy with the other benefits and joys the new responsibility brings that aren’t freedom.

                So, we’ll see if I can spit this out tomorrow. I am subbing for a teacher I have already subbed for this year, so the students know me a bit. I expect to be mostly ignored or snickered at, but I am used to those things as a math teacher. 

Kinda like this guy

There are so many things wrong with how we teach and do mathematics. I could probably list a dozen problems, but I want to focus a bit (at least temporarily, amiright?)

Math isn't stale or boring. At least, it doesn't have to be, all the time. You can come across something as fun as this and treat it like a trick, or use some math and get underneath it. I often choose the latter, but I've been trained to do so. My students haven't (and when I was their age, I hadn't been either).

I wasn't sure how to get around this frustrating fact. I love math because of how powerful and yet sensible it is. It has an elegance and simplicity that students don't get to see, and don't really want to see. The agonizingly frustrating question of "Where am I ever going to use this" should naturally be replaced with "Seriously?! That's how that works? Why?!" But when I am trying to impart this love to 15-year-olds who have been trained to find the right answer to a homework problems and move on, I typically falter.

Well, this guy articulated my thoughts and gave a solution in his TED talk. (TED talks are a bunch of smart people getting together and sharing ideas in a way that make you simultaneously excited, overwhelmed, and aware of your comparative inferiorities). Mr. Meyer isn't fond of textbooks in math education. He finds their problems contrived and unimaginative and their methods dated in an unnecessary way. He tries to take problems and put them on their head: He'll film himself eating jellybeans and ask how long until they are all gone. He won't state how many jellybeans there are in the first place. He won't give a rate at which he is eating the aforementioned beans of jelly. He will take so long to eat them, on screen, that his students would rather solve the problem than wait to see it end.

He doesn't care about the answer near as much as he does the process, and what's learned along the way towards that process.

This thrills and terrifies me. I love the idea of mathematics without a set solution (because in life, specifically in science, that rarely happens, right?). I like making problems at least interactive and more interesting. I love what he does and want to start trying it. I'll need things like a good computer, camcorder, projector and (ideally) a smartboard in my room to be consistently effective.

I hate the idea of needing to come up with my own complicated, engaging problems. I'm scared that my students won't learn the stuff they need to learn to pass their SAT's. I'm scared they won't learn enough math to pass their next class, which will be taught by a teacher that won't grab a bunch of videos and put them in front of the class. But I can't wait to try it a couple times, here and there.