Monday, September 17, 2012

Yearning for Some Joy of X

Last year, I decided to take Algebra I in order to get over my math anxiety. I wrote about it a couple of times here and at in the New York Times. In what anyone but myself could have predicted to be a critical error on my part, I began attending class halfway through the year. I missed the first half of the textbook, never learned how to sing the quadratic equation to the tune of some Christmas carol, and I could only attend class three days out of five because of my teaching schedule, so I was doomed to failure. This year, I have no such excuses. I am all in, from day one, four out of five classes a week.

If I was anxious about my performance in math the first time around, imagine the added pressure I am under now. My students are all aware that I am re-taking the class, and while they finished Pre-Algebra last June, I finished it in June of 1982. My teacher, Alison Gorman, calls on me in class, I do the homework every night at the dining room table while my sons do their homework, I hand in review sheets, and I just took my first test. I planned to take the test along with the rest of the class, but an emergency parent meeting took precedence, so I took my test at home. My sons were fighting over some thing or other at the time, so my test-taking environment was not what you might call calm and stress-free.

That said, things are going really well. Alison allows us to hand our review sheets in early for corrections, and I was the first student to earn a perfect score on the first one. Definitely a new Jess Lahey moment, and the review sheet has a place of honor on our refrigerator. I have been doing well on my homework, and was feeling confident going into my first chapter test. I will hand that in tomorrow, with all fingers crossed.

In the meantime, I've been trying to work out precisely what it is about math that freaks me out. Part of it is certainly the habit I've gotten into, the automatic assumption that I can't execute simple math without using my finger to draw the math problem in the air so I can remember to carry the one. Part of it has to do with my frustration that when when my husband or son multiplies 26 * 4, they simplify the whole endeavor to 25 * 4 for an easy 100, then remember to add on those extra four, for 104. I don't naturally think that way.

Then, last week, I isolated the real source of my frustration. I was doing the following problem:

30 - 5 * 3.

Simple, right? Well, here's what irks me. You have to do the multiplication first, because that's the rule according to the order of operations (PEMDAS - parenthesis, exponents, multiplication/division, addition/subtraction). you do 5*3, which equals 15, then remember that the negative sign is there and take 15 away from 30? Or do you multiply a -5 by 3, which automatically gives you a -15, which you then add (or, as it's negative, take away) the 15 from 30?

Turns out, it works either way. Which is my problem. Alison claims that it doesn't matter. Rules are rules.

Rules. Tricky things, those rules. In Latin and English, rules are slippery creatures, best banned from my classroom. As soon as I tell a student they can rely on a grammatical rule, an exception sneaks out of the recesses of the English language to bite me on the butt and prove me wrong. So this whole trusting of rules thing does not lie within my comfort zone. I brought this up with Alison, and she smiled at my discomfort, reassuring me that I can relax. Rules are the rules, and that's the nice thing about math. Certainty.

My certainty lasted until second period Algebra class. Alison was explaining why numbers to the exponent of zero always equal one - it's a rule I did not understand last year, even after a very sweet student explained it to me repeatedly (see her explanation here). This year, I finally got it. In the middle of the explanation, Alison stopped, got a strange look on her face, and looked at me. She sheepishly apologized to me as she began to write the exception to the rule she had articulated not an hour before. Zero to the exponent of zero is not, in fact, one, but...drumroll...undefined. Not definable. Ahem. An exception. See? Bitten right there in the butt.

And then, an angel appeared. My personal math angel, who, as angels are wont to do, managed to make my life simultaneously simpler and much, much more complicated.

My math angel? Steven Strogatz - sorry, Professor Strogatz. He's a kick-ass teacher, bard of all things mathematical, and a lovely person, as I found out last week. He wrote a series of articles a while back for the New York Times that were amazing, and in advance of the release of his new book, The Joy of X (yeah, I great, you know he did a fist-bump with his agent when he came up with that title), he's doing a new series for the Times. The first piece, on singularities, appeared last week, and I wrote to compliment him on it.

I wrote him an email, he wrote me and email, and we got to chatting about math. I explained my whole distrust of rules issue, especially in light of the zero to the zero debacle, and he replied to that email with his thoughts on the subject. My original email is in black, his are in red (I edited out some of his non-math comments):


Sorry to bother you again, but there's a major issue of math you should be made aware of. 

I'm writing right now about the fact that math "rules" make me nervous. [They should make you nervous. There's a beautiful book called John Stillwell called "Yearning for the Impossible" on this very point.   It argues that the great advances in math come when people break the rules. Two simple examples: (1) Rule: you can't divide 5 things evenly among 3 people, because 3 doesn't go into 5. True in the world of whole numbers. But if you "yearn for the impossible" and open your mind, break the rule, and invent fractions, suddenly your system of math becomes more free AND more powerful. (2) Rule: You can't take 6 away from 5. True, in the world of positive numbers. But if you yearn for the impossible and enlarge your concept of number and invent negative numbers, you can now work the math of of debts, which you couldn't even conceive of before. I discuss this sort of thing in the beginning of The Joy of x. It's the great narrative arc that runs through all of early math (and later math too, as Stillwell explains).]    I teach grammar and languages and there, "rules" are always in quotations - they are subject to the inevitable exceptions. So, when I come across the homework problem:


I get nervous. Is that -5*3 which is -15 [yes] that I take away [no. You don't take it away. If you're regarding it as -15, you need to add it, not take it away, from 30. You already used the negative sign to make the 15 into negative 15. You don't get to use it a second time to do a subtraction. I'll avoid the temptation to make a bad linguistic joke here about double negatives. Boo!]  from 30, or is that 5*3, and the negative is a subtraction symbol rather than a negative symbol - I mean heck, I don't know enough to know the difference, I managed to take linguistics as a substitute for college math…

Anyway, the fact that the answer is the same either way, and the fact that my Algebra I teacher can explain that it will always work out the same way, no matter what, does not ease my worry. 

There are always exceptions. My husband explained that this is what is lovely and amazing about math, that there are not exceptions. 

Alison Gorman, my lovely Algebra I teacher, explained that this is what is lovely and amazing about math, that there are not exceptions. 



I went in to math class the next day and learned why 3^0 is 1. Last year, I wrote about this, about how I had to just accept it as a RULE because I could not SEE it, and math people offered some lovely comments that went over my head. 

But Alison explained it so I understood this year, and I was happy. 


She then gasped and realized that she was about to make me very unhappy. Just when I had been mollified, when I realized that I could take solace in the rules of math, I could TRUST the rules of math, she offered up the one exception to that rule. 

Damn that 0^0 and its exceptive qualities. [yes, 0^0 is worthy of a column. Very tricky.]

Back to square one and my distrust of math rules. 

This is what I am writing about this evening. Although, as a side note, I did get an A on my first Algebra I chapter review, and I have decided that my new favorite thing to do is simplify. It's quite satisfying. Although when I was just asking my otherwise brilliant husband what the word I was looking for, "What am I trying to say…I simplify like…terms? variables?" he said, absentmindedly, "You simplify like a 'muthah'" which I assume he meant as some sort of compliment. 

Anyway. That's what I'm writing about, and I expect you to fix that whole "exception" issue in math as it appears to be my main stumbling block and source of distrust and unease. [Hope my comments above help. The biggie, which I did not talk about yet, is infinity. The Greeks after Aristotle said the rule is, you can't think about infinity. Not allowed. Too many paradoxes. Math was stuck with that rule for thousands of years, till the 1600s, when the yearning for the impossible got too overwhelming, and the need too great. And the result was calculus, which tamed the infinite and harnessed it to solve all kinds of problems that ultimately made modernity possible.You'll see what I mean when you read the book...]

Jess [Steven]

So you would think that Algebra I would be a breeze, what with Professor Steven Strogatz willing to personally explain the subtleties of my Algebra homework, but I guess not. And now that my teacher, my students, my students' parents, my blog readers and math stud Steven Strogatz are aware of my adventures in Algebra I, the stakes have been raised. Good lord, I hope I did well on my Chapter One test.

P.S. Professor Strogatz's Radiolab segments are some of their best, and they generally produce the best stuff on the radio. His TED lecture on synchronicity in the natural world rocks, too.

P.P.S. After reading this post, he recommended this page, for its discussion of 0^0.

UPDATE: I just got my test back from Alison, and I got a 97/102! Some stupid addition errors and not fully understanding how to simplify some fractions lost me a couple of points, but I'm pretty darn proud of myself today.

Friday, September 7, 2012

The First Stage of Pip's Expectations

It's time! It's time! I get to start Great Expectations today! Oh, how I adore this book. I love reading it to my students - kids who, like Pip, are on their own journey. They come of age as Pip comes of age, and that's what makes this book a perfect middle school read. This year, I have promised my students that we will read the first third of the book together, and that I will give them everything they need in order to understand the book. Last year, I did the same for A Tale of Two Cities' most challenging chapter, "A Hand at Cards," and that seemed helpful for my 8th graders who were not quite able to untangle the figurative from the literal in that chapter.

To that end, I am posting this "Jess Notes" summary of Book I of Great Expectations. Yep, it's long. So get yourself a cup of tea, maybe a cookie or two, and settle in. Or even better - pick up a copy of the book and read along with us. I promise, it will be worth your time.

The First Stage of Pip’s Great Expectations

My father’s family name being Pirrip, and my Christian name Philip, my infant tongue could make of both names nothing longer or more explicit than Pip. So I called myself Pip, and came to be called Pip. (1)

With these words, Dickens deposits us down in medias res, or right in the middle of things, on Christmas Eve of 1812, in the company of our narrator and main character, Pip. Not Philip, just Pip. Plain old Pip. As we find out in the second paragraph of the book, Pip’s parents are dead, so it’s been up to Pip to not only name himself, but to figure out who he is and what he might become. This should sound familiar - a hero, orphaned, or arising out of unusual origins strikes out on an adventure. It’s one of the oldest stories humankind has. Consider some heroes from literature and film: Harry Potter, Jesus, Luke Skywalker, Percy Jackson, King Arthur, Pip - they are all either orphans, misplaced royalty, or born out of unusual circumstances. We meet Pip, the hero of Great Expectations, as he struggles to interpret the details of his dead parent’s graves, just as he would study photographs. He derives impressions of his parents from the shape and slant of the letters, the form of the words, and the placement of the epitaphs. Pip’s five brothers and sisters lie next to his parents, under five small gravestones. Our guide to this story, “the small bundle of shivers growing afraid of it all and beginning to cry was Pip,” (2) is absolutely alone in the world, in a marshy little village where the Thames River meets the sea.
As we already know from reading the back of the book, Pip will meet a stranger, and that scene gets the plot rolling: 
            A fearful man, all in course gray, with a great iron on his leg. A man with no hat, and with broken shoes, and with an old rag tied round his head. A man who had been soaked in water, and smothered in mud, and lamed by stones, and cut by flints, and stung by nettles and torn by briars; who limped and shivered, and glared, and growled; and whose teeth chattered in his head as he seized me by the chin. (2)

I adore this passage, because it’s quintessential Dickens, all wrapped up in one descriptive paragraph. First of all, it just goes on forever. It’s a beautifully complex sentence - well, technically, it’s a complex-compound sentence, because it has at least two independent clauses and one or more dependent clauses. It also appeals to the rhetorician in me. I get a little worked up over rhetoric, the use of language to persuade, to move, to elevate language above the sum of its parts. It can be as simple as repetition, or as complex as elaborate linguistic patterns. This passage embodies a couple of lovely rhetorical forms. It’s a gorgeous example of polysyndeton, or the repeated use of conjunctions (such as and, but, or) in order to create an effect, such as rhythm, mood, or pace. The effect created here is the rush of Pip’s confusion and panic. By using the conjunction “and” over and over to string together a bunch of descriptive phrases, it evokes Pip’s fear. Pip wants desperately to tell us about this figure, but he’s frightened, and a little out of his head. He’s not sure where this confused list of characteristics might end; he’s making it up as he goes along, and therefore, he’s rambling and imprecise in his language. To add to the loveliness of the paragraph, Dickens also uses parallel structure, or isocolon - the repetition of sentences, phrases, and clauses, all in the same syntax - to create a rhythm and pace. That’s a lot of terms to define in one place, but no worries; syntax is just a fancy term for word order. Look back at that paragraph; Dickens repeats adjective-preposition-noun, adjective-preposition-noun, over and over again. “Soaked in water, and smothered in mud, and lamed by stones…” Beautiful, right? Dickens does not just tell us that Pip’s freaked out and out of sorts, he freaks us out through the very language he uses to describe the very thing that’s causing the freakout. The paragraph sweeps us up in Pip’s anxiety, and barrels along as if it’s never going to stop - until it does, abruptly, when the convict grabs Pip by the chin in mid-repetition, snapping Pip out of it, and breaking the rhythm. Bam. The convict grabs Pip by the chin, and the passage screeches to a halt. 
I mean come on. This is great stuff. Dickens is using language not just to tell - that’s for amateurs - he’s using language to show, to create a mood, and evoke emotions in the reader that mirror his main character’s. This is why we read Dickens. There are plenty of books out there that simply spin a good plot and keep us entertained, but language of this caliber is rare and magical. But back to the story, because Pip’s in dire straits at the moment.
The convict lifts Pip up by his ankles, literally and figuratively turning Pip’s world upside down, and demands that Pip steal food, or “wittles,” for him, as he’s starving. Once the convict finds out that Pip lives with a blacksmith, he demands a file as well, to remove the chains still attached to his leg. He’s escaped from one of the hulks, or ships, that transports criminals to Australia and lies moored offshore, and he can’t re-enter civilization with a leg-iron tethered to his ankle. In order to assure Pip’s obedience, he tells him that he has an accomplice nearby, a fearsome, savage man who wants nothing more than to get at Pip and eat his liver right out. Pip is so afraid that he runs home, without stopping, to gather the requested supplies for the convict.
As it is Christmas Eve, Pip’s sister, Mrs. Joe, and his brother-in-law, Joe, are expecting guests for dinner. Pip is terrified, but he takes a moment out of the narrative to give us some backstory and description regarding his guardians. When Pip’s parents died, Mrs. Joe grudgingly took Pip in, and brought him up “by hand.” In the parlance of Victorian England, this literally means that Pip was bottle-fed rather than fed by his mother or a wet nurse, but in an ironic twist on the term, Pip implies that brought up “by hand” means she beat him on a regular basis. “Having at that that time to find out for myself what the expression meant, and knowing her to have a hard and heavy hand, and to be much in the habit of laying it upon her husband as well as upon me, I supposed that Joe Gargery and I were both brought up by hand.” (6) Dickens is very concerned about the treatment of children in Victorian England, and this motif, or recurring idea, comes up often in his stories, such as David Copperfield, and Oliver Twist. Dickens himself was forced into factory work as a child when his father was imprisoned, and those experiences led to a lifelong interest in the plight of children in the early years of unregulated, industrial England. When we read about how Pip is treated by his sister, it’s hard not to think of the starving orphan Oliver Twist when he tells the ruthless master in the workhouse parish, “Please sir, I want some more [gruel]”
Pip may not be starving, but he is subjected to terrible physical and verbal abuse. His sister beats him with a stick called “the Tickler,” and while Joe tries to run interference for him, Pip usually bears the brunt of his sister’s frustration over her unhappy life. Joe and Pip are more siblings than father and son, and they depend on each other for emotional comfort and comic relief. They play games with their meals, protect each other from Mrs. Joe’s wrath, and clearly adore each other. The evening Pip encounters the criminal for the first time, Joe senses that Pip’s upset, but Pip’s afraid to tell him anything for fear of that accomplice, waiting out in the marshes, eager to make a meal of Pip’s liver. And here’s where we run into another of my favorite quotes: “I was in moral terror of the young man who wanted my heart and liver; I was in mortal terror of my interlocutor with the iron leg; I was in mortal terror of myself, from whom an awful promise had been extracted…” (13) And there’s that rhetoric again. This time, it’s anaphora, or repetition at the start of successive sentences or clauses. As if we had not figured it out already, Pip is in mortal terror, so much so that he repeats it for us a couple of times to hammer the point home. The anaphora lulls the reader into the rhythm, until that last phrase shakes things up - not only because the rhythm gets interrupted, but because it’s an odd thing to be in mortal terror of one’s self. The young man who wants his liver? Sure. The convict with the iron leg? Of course. But himself? Both the incongruity of the idea and the change in the object of the mortal terror trips the reader up just enough to make us stop and think about what we just read. It’s lovely, just lovely.
After a fitful night, Pip wakes up early and steals food from the pantry and a file from Joe’s forge. As he makes his way across the marshes to find the convict, Dickens takes some time to describe the marshes. Description is one of the other linguistic forms, along with rhetoric, and Dickens is a master at it. He could simply tell us that the marshes are cold and foreboding, but instead, he renders the marshes as a character in the novel, more antagonist than mere setting. This mist obscures Pip’s vision, both literally and figuratively. It obscures his vision of what lies ahead in the terrain and what lies ahead in his future. Pip can’t see the horizon line of the marshes any more than he can see what lies ahead in the trajectory of his life. It isn’t until Pip leaves the village, and its mist recedes, that he believes he can finally see his future, clear and bright. But let’s get back to the marshes, because the convict is out there, shivering and waiting for Pip’s delivery.
The one thing that is clear to the reader as Pip traverses the marshes, looking for the convict, is that Pip suffers from a seriously guilty conscience. In his heightened fear and guilt, even the oxen in the fields take on the appearance of judgmental clergymen, ready to hurl accusations over their fences. Finally, Pip comes across a man he believes to be his convict, at least from behind. He startles the man, dressed in the same coarse gray, with the same leg-iron as his convict, but with a different face. The man runs away, and Pip assumes he’s just stumbled across the young man who has been yearning to eat his liver. Pip continues on and finds the convict - his convict, as he’s come to think of him - and gives him the food and file. The convict scarfs the food down like a wild dog, frantic. Pip watches him eat, and tells the convict, “I am glad you enjoy it.” This is an important moment. Despite his incredible fear and guilt, Pip hopes the convict likes the food. Even the convict is surprised by Pip’s sentiment.
“Did you speak?”
“I said I was glad you enjoyed it.”
“Thankee, my boy, I do.” (17)

Pip’s generosity of spirit, even toward a convict who has threatened his life, tells us - and the convict - much about Pip’s character. Again, Dickens does not come right out and tell us that Pip is kind and caring (a technique called direct characterization); he does not even have Joe tell the reader through some heavy-handed attempt at description (“Oh, Pip, how kind and caring you are!”). No, that’s not Dickens’ style. He allows Pip to show us who he is through his own words and actions, a much more subtle trick, called indirect characterization.
Pip leaves his convict, still sawing away at the iron on his leg, and joins the Christmas Eve festivities back at home. Apparently, Pip’s relatives view Pip’s mistreatment as part of the festivities of the holiday, because they berate him mercilessly over dinner. To add to Pip’s agony, he is feeling guilty about having to steal the food and file, and fears he may be found out at any moment. Probably, he believes, when his sister goes into the pantry for the pork pie he’s stolen, or offers the brandy, which has been replaced by tar-water, a vile-tasting liquid she makes him drink as a tonic. At the moment he’s convinced the gig is up, “I could bear no more…I released the leg of the table, and ran for my life. But I ran no further than the house door, for there I can head foremost into a party of soldiers with their muskets, one of whom held out a pair of handcuffs to me, saying, ‘Here you are, look sharp, come on!’” (28)
It’s at this point that readers might start to notice that certain chapters of Great Expectations end in cliffhangers. Dickens sold many of his novels to magazines in monthly serial installments, and the cliffhanger was a great way to keep readers on the hook until the next installment was published. If you read carefully, you can tell which chapters ended the installment - they are usually the chapters with the most tense and exciting endings.
But back to Pip, standing there at the door, positive he’s about to be arrested for the theft of “wittles” and a file. It turns out the soldiers are not there for him, but for Joe’s services. They are in pursuit of an escaped convict, and their handcuffs are in need of repairs. Joe fires up the forge and as they wait for the work to be done, wine is passed around. Pip and Joe are invited to come along on the hunt for the convict, with Pip riding on Joe’s back. The soldiers signal the alert when they come across two convicts fighting in a ditch, and Pip recognizes them as his convict and the liver-craving convict immediately. The two are bloodied and bruised from their struggle, and they each claim the other was intent on murder.
Despite his curiosity, Pip is nervous about getting too close; he’s afraid that his convict will think that Pip alerted the police to his whereabouts. However, when the convict catches sight of Pip, he does something truly surprising. The convict lies. He confesses to stealing the food and file from Joe’s house, apparently as a ruse to keep Pip out of trouble. It’s a wonderful moment, made even more wonderful when Joe, in another moment of Dickens’ graceful characterization, quickly offers that the convict is welcome to whatever food he needed, as he would not have wanted the convict to starve, no matter what he’d done in his past.
The next couple of chapters offer up some exposition, another of the four rhetorical form we’ll encounter (the others are narration, argumentation, and description). Exposition conveys backstory, information the reader needs to understand in order to move the plot forward. Joe and Pip spend some quiet time together, reading, and through their conversation, we learn that Joe is illiterate, that he suffered at the hands of a cruel, alcoholic father, his mother died, and his loneliness drove him into his marriage to Mrs. Joe. Pip and the reader come to understand the source of Joe’s tolerance for Mrs. Joe’s fits of temper. “I see so much in my poor mother, of a woman drudging and slaving and breaking her honest hart and never getting no peace in her mortal days, that I’m dead afeerd of going wrong in the way of not doing what’s right by a woman, and I’d fur rather of the two go wrong the t’other way, and be a little ill-conwenienced myself. I wish it was only me that got put out, Pip; I wish there warn’t no Tickler for you, old chap; I wish I could take it all on myself…” (50) It’s official: Joe’s a really good guy. Despite the abuse of Mrs. Joe, he wants to do right by women, and more than anything else, he wants to protect Pip.
Now that the reader has all of this vital backstory and context, we arrive at a critical moment in the plot and in Pip’s journey. Pip is about to receive an invitation, what the author Joseph Campbell refers to as the “call to adventure.” Some calls are literal invitations - Harry Potter received a living room full of owl-posted letters - but Pip’s invitation much is less literal. His invitation comes in the form of Pumblechook, a disagreeable and pompous relative, who has arranged for Pip to play at Miss Havisham’s mansion, Satis House. Miss Havisham is a rich and noble woman who lives in town, and while Pip does not understand the purpose of the visit, wondering “what on earth I was expected to play at” (53), he is packed off to Miss Havisham’s. The preparations Pip is subjected to before his first visit may seem like a simple humorous scene, but when viewed in terms of the hero’s journey, it’s much more significant.
…She pounced upon me, like an eagle on a lamb, and my face was squeezed into wooden bowls in sinks, and my head was put under taps of water-butts, and I was soaped, and kneaded, and toweled, and thumped, and harrowed, and rasped, until I was really quite beside myself. (53)

Pip refers to this treatment as “ablutions,” because that’s exactly what this washing is. Ablutions, from the Latin abluere is a washing away, a cleansing, used most often in the sense of ritual purification in preparation for some journey or ritual. Through this humorous description (and did you notice the polysyndeton again? The way the repetition of the conjunction “and” makes the torture of being soaped and kneaded and toweled and thumped just sweeps the reader up in Pip’s passive yet violent experience?), Pip is cleansed in preparation for the experiences that lie ahead. He’s about to embark on his adventure, his quest for those great expectations, and like a medieval knight preparing for a religious quest, he is purified before setting out.
Upon his arrival at Satis House, Pip is greeted at the gates by a young girl named Estella, “who was very pretty and seemed very proud.” (56). She allows Pip, but not Pumblechook, who has accompanied him on his journey into town, to pass over the threshold of Satis House. The hero of our story has thus arrived at another significant moment, what Campbell calls “The Crossing of the First Threshold,” the moment when the hero crosses over from known to the unknown world. In mythology, the threshold is guarded by a sentry, such as an ogre or the monster dog Cerberus. Pip’s threshold is guarded by a monster even more fearsome: a pretty and proud teenage girl, and she will change Pip’s view of himself and his world forever. He arrives at Miss Havisham’s thinking of himself as a normal child, maybe even a little smarter than the average child thanks to Joe’s lavish praise of his reading and writing. By end of the chapter, however, Pip has found out that in Estella’s estimation, he is common, coarse, and poor. Worst of all, he has begun to fall in love with Estella and all she represents to him, and he knows in his heart that he is unworthy of her love.
She gave me a triumphant glance in passing me, as if she rejoiced that my hands were so coarse and my boots were to thick, and she opened the gate, and stood holding it. I was passing out without looking at her, when she touched me with a taunting hand.
“Why don’t you cry?”
“Because I don’t want to.”
“You do,” said she. “You have been crying till you are half-blind, and you are near crying again now.”
She laughed contemptuously, pushed me out, and locked the gate upon me. I went straight to Mr. Pumblechook’s, and was immensely relieved to find him not at home. So, leaving word with the shopman on what day I was wanted at Miss Havisham’s again, I set off on the four-mile walk to our forge, pondering, as I went along, on all I had seen, and deeply revolving that I was a common laboring-boy; that my hands were coarse; that my boots were thick; that I had fallen into a more despicable habit of calling knaves jacks; that I was much more ignorant than I had considered myself last night; and generally that I was in a low-lived bad way. (65)
            This is the moment Pip loses his innocence, for Estella’s observation is offered up to Pip just as the apple was offered to Adam. Adam understood he was human, naked, and he was ashamed, and Pip understands that he is low, and coarse, and unworthy. As it’s impossible to go back home, back to the place of childhood innocence, Pip’s only choice is to move forward, away from this new impression of himself and toward the person he will become. In order to make his way in the world Estella has revealed to him, he must embark on a quest for his great expectations - to deserve Estella, to raise his status and become a gentleman, to discover who he wants to be. Pip understands the significance of this moment, and tells us:

That was a memorable day to me, for it made great changes in me. But it is the same with any life. Imagine one selected day struck out of it, and think how different its course would have been. Pause you who read this, and think for a moment of the long chain of iron or gold, of thorns or flowers, that would never have bound you ,but for the formation for the first link on one memorable day. (72)

In order to become a gentleman, Pip will need education and money. On the very first morning of his new life, Pip reaches out to his friend and teacher, Biddy, for an education. As for the money, Pip is in a bit of a predicament. From the day he entered Joe and Mrs. Joe’s home, the course of Pip’s life had always pointed toward the forge, to become Joe’s apprentice, and take over Joe’s business when Joe was ready to retire. But even Pip’s modest proposal costs money. At the time Dickens was writing, an apprenticeship was an official, contractual obligation, an agreement one had to apply for, and pay a fee to obtain. Pip has been looking forward to his apprenticeship with Joe for his entire childhood, but when Miss Havisham offers to pay for it, and end his visits to Satis House, he knows the life of a blacksmith will never be enough for him. “Finally, I remember that when I got into my little bedroom, I was truly wretched, and had a strong conviction on me that I should never like Joe’s trade. I had liked it once, but once was not now.” (107)
Pip has the presence of mind to be ashamed of himself, but he believes that the forge, Joe, his home, his village, are not his destiny. Look for the anaphora in the following lament about the curse of his shame:
It is a most miserable thing to feel ashamed of home. There may be black ingratitude in the thing and the punishment may be retributive and well deserved; but that it is a miserable thing, I can testify.
Home had never been a very pleasant place to me, because of my sister’s temper. But Joe had sanctified it, and I believed in it. I had believed in the best parlor as a most elegant saloon; I had believed in the front door as a mysterious portal of the Temple of State whose solemn opening was attended with a sacrifice of roast fowls; I had believed in the kitchen as a chaste though not magnificent apartment; I had believed in the forge as the glowing road to manhood and independence. Within a single year all this was changed. Now, it was all coarse and common, and I would not have had Miss Havisham and Estella see it on any account. (111)

Despite his dissatisfaction, Pip commits to his interminable tenure in the forge. It is at this point that a journeyman named Orlick takes work at the forge with Pip and Joe. Dickens uses a wonderful bit of biblical allusion, a reference to a story outside the text, to give us all the information we need to know about Orlick’s character. He describes Orlick as “Cain or the Wandering Jew” (118), and, if we know our Old Testament, we understand that Orlick, like Cain, or the Wandering Jew, has been condemned by God to wander the world in retribution for some evil he’s committed in his past. Dickens does not have to go into a lot of narrative about what Orlick may or may not have done, the reference to Cain serves that up in a much more subtle fashion. No good can come of Orlick’s presence in the forge, and he brings a sense of danger and foreboding into the story. Orlick hates Pip because he understands that no matter how hard he works in the forge, he will always have to defer to Pip, Joe’s successor in the family business. Orlick hates Joe because he knows Joe will always prefer Pip. Orlick hates Mrs. Joe because, well, she’s a shrew; what’s not to hate.
Pip tries to stay focused, to stay away from Miss Havisham’s as long as he can, but when her birthday rolls around, he makes the trip into town to thank her once again for putting up the money for his apprenticeship. At least, that’s his ostensible reason. His real reason is, of course, to see Estella. However, Estella is abroad, getting a lady’s education, and Pip knows he’s even further from earning her love than he’s ever been. “Abroad,” said Miss Havisham; “educating for a lady; far out of reach; prettier than ever; admired by all who see her. Do you feel that you have lost her?” (122) Pip sees no way out of this life that’s been forced on him, and as if to hammer home this point, the fog hangs heavy, wet, and thick over the village.
Arriving home, Pip finds that his sister has been savagely attacked by an unknown intruder. She is, from this night, permanently damaged mentally and physically, “destined never to be on the rampage again while she was the wife of Joe.” (125) Pip feels guilty about the attack on his sister, particularly as the weapon is determined to be a “convict’s leg-iron that had been filed asunder.” (126) Pip is does not think it was his convict who committed the act, but he assumes that he is responsible for the weapon, for it is likely the same leg-iron that his convict removed with Joe’s file. Even in her disabled state, Mrs. Joe’s behavior toward Orlick suggests that he was her attacker. She can no longer speak, but she draws pictures of hammers on her slate in a desperate attempt to communicate the identity of her attacker, and is particularly eager to stay on Orlick’s good side whenever he is around.
Save for this one change to Pip’s household, life continues in “a regular routine of apprenticeship life, which was varied, beyond the limits of the village and the marshes, by no more remarkable circumstances than the arrival of my birthday and my paying another visit to Miss Havisham.” (131) Pip is bored, his life is monotonous, he’s depressed, and he can’t even bring himself to recount the details of what appears to be the interval of a few years for us. The only detail he bothers to note is regarding his opinion of Biddy. He’s begun to notice her. Not that she can compete with Estella, of course. “She was not beautiful  - she was common, and could not be like Estella - but she was pleasant and wholesome and sweet-tempered.” (131) Dickens makes a clear comparison, positining these two women as foils, opposites meant to contrast each other, to highlight the differences between them. Estella and Biddy. The difference is apparent from the moment we learn their names. Estella: Spanish for “star,” luminous and beautiful, cold and distant. And Biddy? Well, Biddy is another word for chicken, and slang for a female servant. Their personalities only serve to reinforce their roles; they may as well be from two different species.
Biddy does not understand or encourage Pip’s desire to become a gentleman. She’s perfectly content in her life and wishes the same for Pip. Unfortunately, Pip’s desire for great expectations has begun to change him, and where he used to possess empathy and care for other’s feelings, he manages to insult and alienate Biddy when he confides in her about his regrets.
“If I could have settled down,” I said to Biddy, plucking up the short grass within reach, much as I had once upon a time pulled my feelings out of my hair and kicked them into the brewery wall; “if I could have settled down and been but half as fond of the forge as I was when I was little, I know it would have been much better for me. You and I and Joe would have wanted nothing then, and Joe and I would perhaps have gone partners when I was out of my time, and I might even have grown up to keep company with you, and we might have sat on this very bank on a fine Sunday, quite different people. I should have been good enough for you; shouldn’t I, Biddy?” (134)

Pip’s implication is that he can never be enough for the lovely Estella, but he would be enough for Biddy. That’s low, and certainly not what a woman of marriageable age would want to hear. To her credit, she brushes the comment off and forgives Pip, because Biddy is simply that type of person; kind, gentle, and willing to see the best in Pip. The polar opposite of Estella, who is happiest when Pip is in tears over her.
Four years go by in much the same manner as the years before, and then, like a bolt from heaven, Pip is visited by a stranger who changes his life. Jaggers, a lawyer whom Pip has seen at Miss Havisham’s house, informs him that he has been granted his dream, the promise of great expectations.
“I am instructed to communicate to him,” said Mr. Jaggers, throwing his finger at me sideways, “that he will come in to a handsome property. Further, that it is the desire of the present possessor of that property that he be immediately removed from his present sphere of life and from this place, and be brought up as a gentleman – in a word, as a young fellow of great expectations.” (144)

There are two conditions on these “great expectations,” however. Pip must “always bear the name of Pip,” and “the name of the person who is your liberal benefactor remains a profound secret.” Pip – just plain old Pip – must keep his name as a reminder of his origins. A fortune cannot change the fact that he remains just Pip, and his benefactor wants him to remember this no matter how far he travels from the marshes and the forge. Despite the directive that he must retain his diminutive name, Pip immediately assumes a selfish and pompous persona. He orders clothes for his new life as a gentleman of London, but has them delivered to town rather than to his village, because he believes he will be stared at by the villagers now that he is something of a local celebrity. As he becomes more and more vain, he also becomes more and more dissatisfied with his family and former life. He feels his new station and fortune elevate him above his family and peers, until even his familiar old bed cannot offer the comfort it once did.
Looking towards the open window, I saw light wreaths from Joe’s pipe floating there, and I fancied it was like a blessing from Joe – not obtruded on me or paraded before me, but pervading the air we shared together. I put my light out, and crept into bed; and it was an uneasy bed now, and I never slept the old sound sleep in it anymore. (152)

Before Pip’s departure for London, where he will commence his gentlemen’s education, he pays a last visit to Miss Havisham, whom Pip assumes to be his benefactor. He expresses his gratitude, kisses her hand, and as he leaves, Miss Havisham reminds Pip, “You will always keep the name of Pip, you know.” Despite her rejoinder, it is too late. Pip is already quite changed. He may still answer to Pip, but he’s rapidly forgetting where he came from. He will not even allow Joe to walk with him to the village, and not because he is dreading a teary goodbye. His reasoning that of an increasingly pompous Pip, wearing fancy new clothes and an obnoxious attitude, “I am afraid - sore afraid - that this purpose originated in my sense of the contrast there would be between me and Joe, if we went to the coach together.” (166) He’s ashamed of Joe, and would rather not be seen with him on his first morning of his new life as a gentleman.
He regrets this course of action long enough to consider getting off the coach and walking back to the house in order to have a better parting with Joe. Rather than follow his heart, however, he remains on the coach, “and it was now too late, and too far to go back, and I went on. And the mists had all solemnly risen now, and the world lay spread before me.” (167)