The Calcul8r

[thecalcul8r]

1.5 Expressions from word statements, 1.6 Introduction to Equations, 1.7 Solving Equations

This was the first time since starting this project that I felt that frustrated feeling again- that 'I-hate-this-I-give-up-I-hate-how-it-makes-me-feel-why-is-my-brain-my-enemy feeling. Basically, the 'math' feeling. And what was so striking about it was that the feeling didn't surface when I hit a hard point in the book. This section was easy. Too easy. I was blowing through it. I was bored the first time I did it, and yet I forced myself to do all of the problems a second time, and yet I still made 3 mistakes that I didn't catch. Each one of them was like a joke-individually, you would say - sheesh, what a one-in-a-million error. Not a single one was a mistake in understanding: they were mistakes in arithmetic, or of reading a sign wrong, or whatever. They seemed random, and yet what isn't random is the absolute, invariable consistency with which I make these errors. I instituted a system to keep myself from making random errors, and I made random errors enforcing that system: once, I had the answer 156 in one set and 56 in the other, and I convinced myself that the first answer wasn't really 156, what looked like a 1 was actually part of the box I drew to make the answer more easily visible on the page. Yeah, no, the right answer was 156, so I got that problem wrong.

And then I asked Schuyler how often he made these sorts of mistakes, trying to figure out what the, like, natural rate of error is for other people, and he thought about it for a while, and he was basically like, never. I make mistakes when I don't understand things, he said, but in Algebra or whatever when I was doing my homework I would always get 100s.

I am telling you now that I never, ever, ever got 100s on my math homework. And what is most infuriating is the feeling that my rate of error has no correlation with how well, or how poorly, I understand something. In retrospect, what made math hard as I moved through eighth grade and beyond was not the difficulty of the concepts but the fact that it grew increasingly unforgiving of my static, high rate of error. In these sets of 10 or 20 problems, which only require one or two steps each, I make 5 or 6 errors, catch 4 or 5 of them, and wind up getting one or two questions wrong in the end - I'd be getting in the 90s, if these were graded. But as we move forward, each problem will require 10 or 20 steps, and making an error anywhere along the line will result in the problem becoming unsolvable, and so, even though I understand the concepts, suddenly I can't get any of the answers right, and I'm failing.

[thecalcul8r]

1.4 Order of Operations

PEMDAS

aka

Please Excuse My Dear Aunt Sally

aka

Parentheses, Exponents, Multiplication, Division, Addition, Substraction

I'm pretty sure this is actually 7th grade math. You can see the way they're building up towards factoring, though: x^2+2x+1, (x+1)^2...(sorry about the gross looking exponents). I still associate those expressions with a sense of relief - it's like, if you end up with one of those when you're done working the problem, you know you did it right.

I should have gotten a perfect score on this one. I should have. I understood it perfectly. I did the entire exercise twice. But I still missed one.

(5x2)^3

Twice, I misread the multiplication sign as an addition sign and did 7^3 instead of 10^3. TWICE. In eighth grade, when these kinds of tiny, twitchy mistakes really mattered, I remember the feeling of being tortured by brain demons, little wayward synapses that would sit in the back of my brain and cackle as they messed with me. And seriously, you can see the joke here, right? By misreading the sign, I turned this problem into 7 cubed - the exact same problem I got wrong last time.

I see you, brain demons. I see you. And I am going to make friends with you, and then invite you, very kindly and mindfully, to get lost.

[thecalcul8r]

1.3 Powers and Exponents

Easy peasy. It's basically just multiplication. I'm operating at a disadvantage anyway because I'm pretty sure your average eighth grader would be using a calculator and I still haven't tracked one down. I enjoy having a reason to be away from my computer screen so I'm just doing the problems by hand.

The method of doing the problems separately, twice, is really working for me. I think if I'd figured that out back in eighth grade, my grade would have gone up half a letter grade at least. I had 3 or 4 sloppy errors and I caught all but one of them.. Funnily enough, all my mistakes happened during the first round - you'd think if they were truly random brain-farts (which they seemed to be on the surface; just bozo mistakes) then they'd happen with equal likelihood both times. And it's not like I realized I had make a mistake the first time when I went back around again - I just sped along happily both times, thinking I'd gotten everything perfect, but NOPE. The one that got through was inexplicable- I said 7 cubed was 147 twice, and both times, I for some reason multiplied 7*3*7 instead of 7*7*7. Brain, you are truly a mystery.

I also had a joyful moment where I got to the first word problem in the book and the character was named Calvin Butterball. CALVIN BUTTERBALL I REMEMBER YOU. We're going to be seeing a lot of him over the next 300 pages. 

[thecalcul8r]

1.2 Variables

Well, this was interesting.

After 1.1, I came up with what I thought was a foolproof plan. When I was done, instead of going over and double-checking the answers (which always makes my eyes glaze over) or re-writing them (which makes me super-focused on neatness to the exclusion of everything else) I would do them, separately, a second time, and then go through and see if the two sets of answers matched. This was actually more appealing to me, even though it's technically more work (or more mental work) because the other two alternatives are too dull.

I was fully expecting to get a perfect score. Instead, I made two mistakes.

The first problem was to solve for 30/x, where x is 45. Really, really basic stuff. (30/45 --> 6/9--> 2/3) Instead, I wrote 3/4. Why? Well, this is a sloppiness error, but to dig into that a little further, I screwed up in two ways. First, this is one of three problems I actually didn't do twice. I skipped over it the first time by accident (it was one part of a 3 part problem, and I'm only doing the odd problems, so I ended up jumping around a bit by accident) and so I told myself I had "caught" the mistake by recognizing that I missed it. That meant I only did it once instead of twice. Then, while I wrote down 30/45, I didn't write down the work that it took to reduce it (30/5 and 45/5, 6/9, 2/3, basically because it seemed too easy...but it wasn't, and I screwed up.

I think a teacher would say I did this because I was "rushed," but that's only true in a very specific way - no one was making me do this; I wasn't hurrying through it so I could watch television; I thought I was being careful until I went back and looked at all the steps I fudged. There's a real danger spot for me where I think things are easy, but they're complicated enough that I still make mistakes on them. Maybe if I can start being more aware of what that sort of slide-y, speed-y sensation feels like, I can be cognizant of when I'm about to make these kinds of errors.

The second mistake was more complicated. It was kind of in-between a word problem and a regular problem; I'll see if I can replicate it here:

I____________12___________I

_____7____I_y__I___?_______I

And we were supposed to write an expression that equaled the value of ?.

I wrote 5-y (twice). The answer is 12 - (7 + y).

A couple of things:

I knew this was hard as I was doing it. In retrospect, I probably thought it was a lot harder than it actually was, and that maybe led me down the wrong path. I was vaguely remembering factoring, and I tried to move the numbers around on either side of the equals sign for a while, and then I backed up and tried to use common sense, and kind of jumped to the answer. My answer still seems right to me, actually.

[five minutes after writing the above, after I contemplated the two answers and substituted in a bunch of numbers and couldn't figure out where I'd gone wrong, I sent Schuyler the problem and my answer and the book's answer, and this conversation happened...]

Schuyler: did you mean "i wrote 7-y"

me:  no

i meant 5-y

 Schuyler:  oh you did 12-7

 me:  yeah

 Schuyler:  no you're wrong

 me:  why?

every time i substitute numbers in it works out

 Schuyler:  ? doesn't = 5 - y

hold on

nevermind you're right

 me:  hahahahahaha

BWAHAHAHAHAHAHAA

MY WHOLE LIFE IS JUSTIFIED

 Schuyler:  but good

but you should have written one of their answers first

 me:  i mean their answers are obviously also right

but i couldn't tell if mine was actually wrong

 Schuyler:  the first thing you write is: 12 = 7 - y - ?

and then start moving things around

 me:  yeah i sort of did that and then i got confused so then i just did it in my head

 Schuyler:  whoops wrong signs

12 = 7 + y + ?

 me:  riiiiight

i remember that

from later algebra

 Schuyler:  that's the first thing that you should have written down

 Sent at 10:26 PM on Wednesday

 me:  yeah, i know, i sort of did, but then i wasn't sure if my next step was wrong

now i know it wasn't

sweet

 Schuyler:  cool good job

  I'm a gennnnnnnnnnius!

No, I'm not. I need to do better at showing my work; I also need to be more confident.

Lessons learned.

[thecalcul8r]

1.1 Operations with numbers

Is there anything more pointless than memorizing math-related vocabulary words? Although I did learn one cool trivia-esque fact: did you know that the bar in between the numbers in a fraction is called a "vinculum"?

Other than that, the work was pretty straightforward. Just do the things in the parentheses first. Yeah, I got it.

I did get one problem wrong, though. I stopped doing the problems halfway through, and when I picked them up again, I was sort of in a rush and I mistook what I'd scribbled down in my notes for the actual problem, and I didn't double check it. My first careless mistake.

Something I did notice: I don't have an eraser on my pencil, and so I couldn't erase when I mis-wrote, and here's a thing: it is very hard for me to write things down perfectly on my first try. I've noticed this before when it comes to filling out forms, or handwriting my comments on student essays. I'm going to make a lot of mistakes; if I want something to look flawless, I'll have to rewrite it 3 or 4 times. That's what I used to do with my math homework - I would be obsessed with making it neat, but because I was paying so much attention to neatness, a lot of times I would replicate the same mistakes 3 or 4 times over.  In retrospect, I don't think that was a good use of my concentration and time. So here's a plan:

-I'm going to make an effort to be neat, but I'm not going to redo things just to make them neater. I'll concentrate my re-do energy on actual mistakes.

-I'm going to try and figure out some strategies for get better at checking my work. Double-checking is so boring. I literally couldn't make myself go back over those problems, and it was my first set! I'm pretty sure this was a common thing back in math class: I would chirpily speed through the problems - because I usually did understand the basic concepts - and then double-checking would be set around re-doing the problems to make them look neater, since thinking through them all again would be dull.

[thecalcul8r]

Foreword to the Teacher

The first part of the book is carefully sequenced to lead to the Quadratic Formula by mid-year. This departure from the more traditional sequence of topics is made possible by technology, specifically, the use of calculators to evaluate radicals.

I remember vividly sitting in eighth grade math and chanting out the Quadratic Formula, and later writing it on my hand in pen to help me memorize it.

"The opposite of b plus or minus the square root of something something all divided by 2ac."

I do remember feeling pretty cool once I finally got the thing memorized; it seemed kind of badass and genius-y, like the secret formula you'd see written on the blackboard in a TV show when a mad scientist discovers cold fusion. I still have a lingering fondness for it and how mechanical and straightforward it is; the sound of it doesn't scare me the way certain other terms do.

I also really love the description of calculators as "technology." Somehow I'm guessing this isn't going to be the last time this book makes me feel old.

Yup, here we go, two paragraphs down:

Students can also learn that the computer is appropriate for repetitive evaluation of expressions, while the calculator or the brain is more appropriate for single calculations...

and later,

this chapter on BASIC programming has been deleted.

Cripes.

...

By Chapter 10, students enter the 'no-holds-barred' phase. Here, clever short-cuts and trial-and-error techniques are encouraged...

Here is something I never did in six years of upper-level math instruction: come up with a "clever short cut." In fact, I have a memory of sad self-awareness from I think ninth grade, when I realized that while I could force myself to do the things I'd been taught, I never "leaped ahead" or figured out new things on my own. My teacher used to include some problems from the next stage of the book as extra credit to see if we could get there without help, and I never could.

When I think about my students in English classes, there are definitely some people that make those leaps and drag everyone else along with them. Often, I try to structure the class in that way, so that it's not just me explaining shit; it's the students having these moments of insight. Usually, when that happens, I feel good and like I've accomplished something as a teacher, but it's also true that this is almost always an example of the 'good' students getting better (and being rewarded for it), which I certainly had mixed feelings about when I was on the other side.

[thecalcul8r]

Foreword to the Student

The letters used will be called variables, because they can stand in for different numbers at different times. Expressions containing these variables can represent quantities in the real world, such as how high a ball is above the ground at various times after it was thrown. By learning how to operate with such expressions, you will be able to predict such things as how high the ball is at a given time, how long it takes to reach a given height, and when it will be back down.

So much of Algebra I is going to be drawing curves on x/y axes, isn't it? It's about two things being tied to together -you turn the handle to move a point on the x axis forward and back, and the point on the y axis moves up and down along with it, tracing out a line. The function is the rule that explains where the line will go. I think I must have seen an internet widget that does that at some point; I tried to search it but I can't find it. Anyway, having that image in my head is giving me a (possibly false) sense of confidence that I understand the basic concept behind all of algebra, and I will basically be fine.

Neatness is an asset in algebra. Clear writing will help you avoid mistakes...Developing the habit of neatness and of saying exactly what you mean will help you in other subjects as well as mathematics.

 And there's the rub. I am not neat. I never have been. I have terrible handwriting. My feelings of helplessness and lack of control when it comes to being neat exactly parallel my feelings about math. They're tied together in a complicated but predictable way, kinda like the x and y axis (groan).

I do not, luckily, have any difficulty saying exactly what I mean.

[thecalcul8r]

A moment of zen

So I have the book and I'm eager to get started. A funny memory came back to me today - how eager I always was to start doing math at the beginning of the year. Every September, I thought "This time it's is going to be different," and the joy and promise of my clean, unopened notebook, freshly sharpened pencils, and an unfamiliar textbook to help perpetuate that delusion. I feel exactly the same way right now: my brain has had twenty years to mature, and I'm not doing this for a grade, and I have the internet and Schuyler to help me, so this time it's going to be different, it's all going to flow smoothly and I won't get sucked down into the messy, ugly reality of myself. I think part of me believes that if I could just master math, I could master a lot of other problems in my life: I would be neater and calmer and more organized and in less of a rush. I would be, basically, a better and less failure-prone person. I have a lot of good reasons for doing this project, but that isn't one of them. I need to let that daydream of perfection go and be okay with being bad at this. Otherwise, I won't learn anything. Making mistakes is the point.

[thecalcul8r]

The Plan

Although the point of this project is to learn calculus, one of my theories about my difficulties with math is that I never really recovered from my first academic implosion in eighth grade. After that, I always felt like I was on shaky ground - like I never really understood what was going on quite as well as I was supposed to. So my plan is to go back to the point where it started to go wrong - Algebra I - and work my way up from there.

I got the textbook in the mail yesterday. Here it is: Algebra I: Expressions, Equations, and Applications by Paul A. Foerster. It's the exact edition I used back in 1994.

The sight of it gave me chills.

This will probably evolve as I go along, but right now my plan is to try and replicate my school experience as closely as possible, starting by sticking to the textbook and working the problems on my own. Once I get stuck, I'll ask Schuyler for help explaining the basics (the equivalent of asking my teacher) and if that doesn't work out, I'll turn to the internet. I'm going to judge my progress mostly by my performance on practice problems (the answer to all the old questions are at the back of the book) and by my general sense that I understand the concepts well enough to move forward. When I'm done with the book (or feel like I've gotten enough out of it to move on) I'll probably take one of the tests available on Khan Academy to see if I've mastered enough Algebra I to move onto Algebra II.

The one exception to the internet-as-last-resort rule is that right now I don't have a graphing calculator (the search for one is in the works) so I'm going to use online programs in the meantime. Schuyler suggested I use Wolfram Alpha, but I'm a little nervous that it'll do too much & make things too easy - that it won't be a "fair" substitution for a calculator. But we'll see.

BTW, last night Schuyler was flipping through my Algebra I book and says, apropos of nothing, "I wonder why I was always so good at math." I almost punched him. But the truth is, that's the question, right? Why was he good, why was I bad, and how does being 'good' or 'bad' at a subject change the way you learn? I have a *lot* of thoughts about that, but I'll share them sporadically as we go through instead of dumping them all here.

Onwards!

[thecalcul8r]

Teaching

One of the reasons I want to learn calculus is because I'm doing a lot of teaching these days. I have been thinking about how strange it is that the subject I teach (English literature) is the subject I've always done well in - so well, in fact, that my own learning process is basically opaque to me. In some ways, it feels like the subject I'm least qualified to teach.

So far,  learning to teach English has consisted primarily of trying to turn a whole lot of vague, intuitive knowledge about how to write into easily summarizable 'facts': put your thesis at the top of the essay; organize your evidence around ideas, rather than chronologically; those ideas should be put in order from the simplest to the most complex; don't end paragraphs with quotations when you can help it, etc., etc.

Except there aren't many more facts trailing along after that "etc.," because this way of "teaching" has almost nothing to do with the way I learned to write. I learned, I guess, mostly by doing - by reading and writing and getting feedback on my writing and writing some more, but to be honest, I forgot 99% of that feedback, since all through high school and college, I got A's. So I just zoomed along, assuming I was doing everything right, and I guess I basically was, because then I got into grad school, and they put me in charge of teaching a bunch of kids to whom writing doesn't come naturally at all. And in consequence, 99% of the time when my students ask me a question for the first time, my instinct is just to throw my hands in the air and say, "I don't know how you do it, you just do."

In contrast, my memories of math are all memories of learning, by which I mean: struggling, thinking hard, pushing past boredom, doing practice problems, memorizing things, searching for alternative explanations, and trying really, really hard to understand. My own experience of the subject matter I'm teaching is entirely irrelevant to the students who actually need my help ("What? It's easy! You just do it! Lucky for you, this subject is intrinsically fascinating!" - not something students want to hear).

[thecalcul8r]

Learning

As you can see from the post below, I never took calculus in high school. I did sign up for it my freshman year in college, but I lost my nerve and switched out before the first class met.

That was fifteen years ago now. I’m 32, and I finally want to learn calculus. I’m going to teach it to myself, with a little bit of help from the internet and my fiancé, who is a graduate student in engineering. This tumblr is the place I’m going to write about it. Mostly, of course, I’m writing for myself – adult learns calculus isn’t exactly the kind of tagline that goes viral – but I’m also directing my writing to a younger version of myself, the girl who was sure she couldn’t do math but couldn’t figure out why.

It seems obvious, in retrospect, that I could do math. Maybe it didn’t come easily, but I’m pretty sure I had the capacity to learn calculus, just because most people do.  I dragged myself miserably and messily through Algebra, Algebra II, and pre-calculus, and I’m 100% sure I could have dragged myself through calculus, too.  This is blog not about proving to myself, or to anyone else, that I can do it, because I don’t have any doubt that’s the case. (Whether or not I will have the willpower to complete this project, of course, is another story).

What I want to do here, now, is learn math differently. I want to start over. I want to understand what math can be when there’s no pressure, no rush, and no shame. More importantly, by revisiting my largest academic challenge as an adult, I want to learn more about how I learn -- to return to that teenage sensation of frustration and flailing and incompetence with an adult's capacity for self-awareness, and see if I can bring myself to a new and better place. I think the odds are that I will still be "bad" at math, but instead of feeling guilty about that, I'm curious about it. I want to know where that feeling comes from, and why, and how to replace it with healthier and more productive ways of knowing. So I would say this isn't a blog about calculus as much as it is a blog about learning. It's a place where I can watch myself learn.

[thecalcul8r]

Bad At Math

I remember the exact moment I went off the rails, math-wise. I was thirteen years old and in the eighth grade. My friend Melissa had invited me to go down to Florida with her grandmother over winter break. I asked my Dad for permission, sure he was going to say yes. Instead, he said, “You’re getting a C in Algebra. I don’t think you should be rewarded for that, do you?”

What followed was an epic, shrieking, sobbing meltdown. My dad was blandly confident that I had gotten a bad grade because I wasn’t working hard, that I could do better if only I tried. This was, theoretically,reasonable; I’d always gotten good grades, and math had never been a problem before. Also, if we’re being honest, I probably wasn’t trying that hard, given that the only other memory I have of eighth grade math is of surfacing at the end of a class period and realizing I’d spent the entire time daydreaming about the Vampire Lestat (it was a different era). But even so, I must have intuited the truth, because the panic that had been building up all year came flooding out: “I can’t do better, I can’t, I can’t, I don’t know why, but I can’t.” 

And I couldn’t. In the months and years afterward, I was proven right: effort and math seemed to have an inverse relationship: the harder I tried, the more intently I tried to focus, the more the numbers swam. At my school, eighth grade was the year in which you were tracked: depending on how you did in Algebra 1, you either went onto the Smart Math track:

Algebra II 

Geometry 

 Pre-Calculus

Calculus

Or one of two Dumb Math tracks. Because of my C, I was put on the middle Dumb Track, and it went like this:

Algebra II 

Geometry 

Statistics and Functions 

Pre-Calculus.

Eighth grade was the year I became Bad At Math: not only in my head, but officially, on paper.  I never recovered. My grades got (slightly) better, but that feeling – that panicky, slippery, I should be able to do this but I can’t feeling— never went away.

[thecalcul8r]

Get ready for some laughs. and some calculus.