Real Analysis: Week 2 (and doing things with real numbers)

Since learning about some of the basic properties of the real numbers, about how they form a line that is completely continuous, we’ve begun to talk about what those properties allow us to do with the real numbers. When we think about and talk about math, we tend to think and talk about manipulating numbers, not simply admiring their existence. The idea that “proving calculus” is important rests on the fact that our society requires (and our minds enjoy) this kind of numerical manipulation, and that we want the results of these endeavors to have some real meaning.

The first type of active manipulation of numbers that we talked about was the act of arranging real numbers into a sequence. A sequence of numbers is a subset of real numbers which are arranged in a particular order. Further, each element of the sequence can be constructed from some formula; that is, if you want to find the fifth term in the sequence, you can plug “5” into a given formula and acquire that term. For example, we can create a sequence of all numbers of the form {1/n} where each “n” that we plug in is a natural number (the kind of number that you can count on your hands). In the sequence, the first element is 1, the second element is 1/2, the third element is 1/3, and so on.

This sequence, like all sequences, contains an infinity number of terms. This is because there are an infinite number of natural numbers, and therefore an infinite number of “n”s to plug into the given formula. An infinitely long list of numbers all sharing a common formula is a pretty astounding, but it turns out that many sequences have an even more incredible property. This property is called convergence, and it means that after a certain term in the sequence, there are an infinite number of terms which are all practically equal to each other.

The sequence {1/n} is a sequence with this property. Eventually, as the natural numbers which we plug into the formula get very large, the term as a whole gets very small. And when n gets extremely large, when we start dividing 1 by numbers like 1,000,000,000, the elements of our sequence get very close to each other and to zero. Because of this, we say that the sequence {1/n} converges to zero. Interestingly, the terms are never actually equal to zero, and never actually equal to each other. 1/1,000,000,000 is a different number than 1/1,000,000,001, although they are very, very close together in value.

There are actually a lot of sequences that converge, and which converge to a variety of real numbers. This property of convergence is deeply tied to the structure of the real numbers- that line which we learned about at the very beginning of class. Since that line contains every possible number, it is possible for it to contain an infinite number of numbers which are as close together as we could want, without those numbers ever actually being equal. This allows for convergence: if the real line were missing values, there would be gaps in the line which inhibited the closeness of terms in a given sequence.

Sequences, while fascinating, might seem on the surface to not actually “do” much in terms of manipulating numbers. They hold a lot of power, however. They allow us to not only order information, but a potentially infinite amount of it. We can add terms of these sequences together to get new sequences, or to get sums of numbers that approximate complicated functions. These are all things we’re going to talk about in the next few days of class, as we keep looking at what the line of real numbers allows us to do.

 

~Anna

 

Week 2: The Essay

After a relaxing weekend filled with reading, coloring, exploring the arts, and taking a stroll with nature, our class piled into the room. With our refreshed minds we prepared for our next assignment.

The Essay is our last big assignment that is supposed to encompass our lives. The first day we described our golden memory, which is the memory that really stood out to us. For some, like myself, it was a very emotional and spiritual experience. Finding that golden moment and highlighting some of the key moments was probably one of my favorite parts of the class, because I was able to find that special moment and other moments leading up to it. After we all found our golden moments, we shared with our class. I don’t know about you, but there is something wonderful about sharing something special to you from a composition notebook.

In order to make our golden moment stand out from our essay, we were also asked to write about memories that go with the same theme as the golden one. In addition to that, we had to read essays that embodied voice, imagery, and characterization. The next day we came up with our own list of what these writing techniques meant to us and what it means to evoke that in our pieces. And each day we would share our writing pieces and give each other feedback on how to enhance and further grow our essay. We had a writing day on Friday, to start typing up our pieces and arranging them to our liking. The first draft of our essay is what we will be in Workshop for Third week. I would tell you more about that, but I will save that for another time. Until next time!

Week 1: The Ask Album

The first day of Creative Nonfiction Writing started by climbing 3 flights of stairs in Armstrong and into a packed classroom. Our professor, Felicia Chavez, set the tone waited till everyone got settled. There were 25 students in class room, 9 of them were wait-listed. We all wanted to be in the class. We all wanted to be a part of what was bound to be a spiritual experience. Professor Chavez passed out the syllabus and what was expected of us. She wanted to make it very clear that this class was very different from other creative writing classes, let alone English class, and if this wasn’t the class for us, then we should leave. Nobody left. Professor Chavez continued to talk about the course and then took a 3 minute break so that people could leave if they wanted to. Nobody left. Those who were wait-listed were asked to leave and told they were more than welcome to take her other blocks. The 16 of remaining sat in a circle and learned about the first assignment.

The Ask Album is the big question that you don’t know the answer to. Its the big red door with the brass know in your head that is filled with possibilities but not answers. Our task was to physically make our album revolving around the big question by first Friday. It was a pretty daunting project in my eyes. It seemed less daunting but still imminent when she said that we would be working on our Ask Album all week. Each day, she would give us writing prompts as homework to figure out that question. We would write for at least 30 minutes and then edit for no more than 30 minutes. When we would share our pieces, some of the most beautiful sentences came out. Some were moved to tears, some to laughter, but in the end we were all here for each other.

Friday came and we all piled in quite unsure if our project made the cut. Professor Chavez was so happy to see everyone and our Ask Albums. When we all presented our Albums, I felt closer to my class. It was like we were this little family sharing our feelings every day. We were all emotionally and physically exhausted but all extremely proud of each others work. Professor Chavez gave us our assignment for the weekend and I think a few of us groaned before actually reading it.  Our homework was a Play-List of relaxing and pleasurable things we get to do over the weekend. I thoroughly enjoyed all of it by the end and was ready for week two to star20160911_204531

Real Analysis: Week 1 (and one really important line)

One week ago, I didn’t know what Real Analysis even was. I knew it must be important (it’s a requirement for the major, after all), and that it was probably a theory based course (since we’re “analyzing” how numbers work, rather than learning the methods that these workings allow). Going into the first day that way was exciting- after a summer spent mostly at home, it felt like I was throwing myself into some far off corner of the universe and waiting to find out what it looked like.

As it turns out, Real Analysis is a little bit more like every corner of the universe. Our professor, Molly, introduced the class as a course that would allow us to begin “proving calculus.” And calculus- that’s huge. Calculus may be only one branch of mathematics, but it’s the branch that most of math education in the US most naturally leads to. That makes sense, too, since calculus has so many applications in so many important fields. Proving the validity of calculus implicitly assures the validity of fields in economics, physics, chemistry, and pretty much any natural or social science which uses complicated statistics to understand a data set. Real Analysis, then, seemed like some pretty powerful stuff.

We started our first class with some of that powerful stuff, right away, talking about what the set of “real numbers” even is. It’s hard to wrap my head around it, to be honest, because here’s what the real numbers are- they’re the line of every possible number that exists in either the positive or negative direction of zero, with not a single gap in them. The numbers we deal with most frequently in our day to day life are “natural numbers,” the kinds of numbers you can count on your fingers. They’re every number you can get by starting with 1 and adding 1 to itself as many times as you could possibly want. It’s an infinite number of numbers, but compared to the real numbers, the natural numbers are missing a lot. To get to the real numbers from the natural numbers, you first step through the integers. The integers include the natural numbers, every negative version of a natural number, and zero. But we’re still not even close to the real numbers. Next we step through the rational numbers- these are every number you can get by taking a ratio of two integers. So numbers like 13/167, or 2/9, those are rational numbers. And as you can probably imagine, that’s also a really big set of numbers.

But it turns out that, for as many rational numbers as there are, there are far more numbers that can never be expressed as a ratio of two integers. So if we were to lay all of the rational numbers out in a line, there would be spaces, empty spots of numbers which we couldn’t express using that set. The real numbers, though, fix that problem. They include every rational and irrational number. Essentially, any number that you could find out in the world, only the set of real numbers is guaranteed to include it.

So the real numbers are pretty special. We learned that in the first fifteen minutes of class. And it turns out that because the real numbers are so special, mathematical systems that use the real numbers have some cool properties. We’ve been exploring those kinds of properties since them. For example, any interval of numbers (like the interval from 0 to 1, written as (0,1)) is guaranteed to have a single largest element, one unique number bounding the entire set below it. But if you don’t use the real numbers, you can’t guarantee such a thing. That’s just one example, but the essential thing I’ve learned so far is this: the simple fact that the real numbers exist allows for and dictates the behavior of entire fields like calculus . The rules of most math that I’ve ever seen in my life are themselves governed by a line.

For only the start of a class, that’s a pretty amazing thing to have already learned, and I can’t wait to see where in the universe we end up in the next few days.

 

~Anna

 

Week 4: Project Poetry Runway Show

Hey all,

My final blog post is all about Project Poetry! On Monday, Jane had us all over for a class dinner at her house, which was delicious and lots of fun! We talked about poetry, our papers, and what we were all going to do this summer. Then we moved inside to watch an episode of Project Runway (since some of us, myself included, had never seen the show before). Jane chose an unconventional materials challenge where the contestants had to design and make dresses out of old electronic equipment. It was really fun for all of us to watch and eat dessert together!

Now onto our own competition. We had four judges: a former student of Jane’s and three senior Poetry majors, including our class assistant Bo Malcolm. For each category the contestants would stand one by one and read their poem aloud to the judges, then hand them a paper copy so they could look at the poem on paper. Once everyone finished, all the students and Jane would leave the room while the judges deliberated. Then we would come back in, the judges would announce their top three, give feedback, then announce the winner.

The first category to compete was the Ode to Your Age category. Out of six competitors the judges’ top three were Tara, Jack, and Susannah. After some great feedback for each of the three, the winner was…

Tara!

Here is Tara’s beautiful poem:

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The second category to compete was the Eavesdropping Poem category. We only had four competitors for this one, so the judges gave feedback to the top four instead of the top three. These were Kai, Ethan, Ashley, and Jonathan. All the poems were witty and fascinating, but there had to be a winner. The winner was…

Ethan!

Here is Ethan’s wonderful eavesdropped conversation poem:

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The third and final category was the Untranslatable Word category. This one had a whopping nine competitors, and the judges’ top three were Sam, Mary, and Jonathan. And the winner was…

Mary!

Here is Mary’s poem about an untranslatable word she came up with:

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Each winner was presented with a prize! Tara won a book of Rilke poems, Ethan won a little notebook and some fun mustaches with which to disguise himself while eavesdropping, and Mary won a book of untranslatable words. Everyone wrote so many beautiful poems, so here are some of the others that people sent me to include, along with some pictures!

Here is Jack reading his Ode to his Age:

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Here is Jonathan reading his Ode to his Age:

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Jonathan also wrote for both the other categories. Here is his Eavesdropping poem:

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And here is his Untranslatable Word poem:

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Now for a photo of the top three from the Ode category getting the judges’ feedback:

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Here also are two poems by the lovely Kai:

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And here is a photo of how we sat in the classroom as an audience.

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The final poem we absolutely have to include is Sam’s beautiful Untranslatable Word poem. She wrote two connecting poems, one in Spanish and the other it’s translation in English. Here she is reading it for the judges and the class:

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Thank you to everyone who has been reading this blog throughout this class. Thank you to Jane for being an amazing professor and for encouraging our creativity as well as our love of poetry. Thank you to my classmates who made class one of the best parts of my day this block. I’ll miss you all. Have a great summer everyone!

Over and out,

O

 

 

 

 

Week 3: Ian Williams, Nature, and Politics

Hi all,

So this week we spent some time workshopping and editing our second papers, and also discussed some dramatic poems on Tuesday. The one we talked about the most was Robert Frost’s “Home Burial,” which is a beautiful and tragic poem about a husband and wife who have lost their child. They’re both coping with their grief in different ways, so their marriage is suffering. One of the aspects of the poem I really loved is how the husband and wife have a whole argument without really communicating or listening to each other at all. The way Frost writes the scene makes it feel very real and tangible.

For Wednesday we read Ian Williams’ Personals, and I think I can say without a doubt that we all loved that collection. Ian Williams is brilliant, and writes everything from clever, witty poems like “” to deeply profound poems like the “Rings” series. Even his witty work is profound and his profound work witty, which is an admirable talent. “Hay,” in particular struck all of us in this witty and profound way:

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Then on Thursday we talked about form (not that we haven’t been talking about form this whole time, but today we covered a few specific forms, namely Villanelle, Sestina, and Haiku). We read some great villanelles like Dylan Thomas’ “Do Not Go Gentle Into that Good Night” and Elizabeth Bishop’s “One Art.” Jane’s not really the biggest fan of sestina’s, so we didn’t talk about them too much, which was fine with us because instead we went outside and sat on the grass in the sun and wrote haiku. Here are a couple pictures of us:

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My classmate Jonathan spent a lot of his time making a fun flower crown out of daisies, and also wrote some haiku he was willing to contribute.

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I also wrote some haiku myself:

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Last but not least, Friday we discussed political poetry, looking at poems like Carolyn Forché’s “The Colonel” and Mark Doty’s “Charlie Howard’s Descent.” We talked quite a bit about the interplay of passion and restraint in political poetry, because it’s impossible to write a good political poem about something you have no passion for, but the writer also benefits from exercising restraint and focusing their poetry in a detailed way. Poetry can be used both to work through and explain difficult or traumatic events in a beautiful and haunting way.

That’s all for this week! This weekend we’ll be beginning work on our final papers and finishing up poems for Project Poetry.

Over and out,

O

 

Happy Friday, and Welcome 4th Week!

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Greetings from the Sociology Department… It’s Candy Friday! Both class and campus are buzzing today, for several reasons.

Firstly, the weather. It is sunny and HOT in Colorado Springs, so much so that we had class outside on the quad! Our discussion today was regarding our developing research studies; the first draft is due this Sunday at 5pm. All three groups submitted a piece of published literature that is working to guide our development of a research question and subsequent findings for each of our fields. My group, remember that we observed Penrose Hospital, submitted this article on hierarchies, teams and webs in the medical workplace and how they coincide to ensure efficient patient care and satisfied employees. The second group contributed an article regarding WIFI in community places like cafés, and how it is a factor in social dynamics of the business. They have observed the goings-on at Colorado Springs’ The Wild Goose, a notoriously hip coffee shop (their cinnamon rolls are quite something). The final group is researching gender dynamics and masculinity within men’s lacrosse at CC, and submitted an article that parses out the importance of body and self and meaning within lacrosse.

Going into the weekend, my research team of women has a good idea of how to organize our coded data sets into meaningful, patterned findings. Our paper just hit 25 pages, with a substantial literature review. Excited to see what everyone thinks of it (we’re peer editing on Monday, which is why we read all these different articles today)! Another aspect of the project that is due on Monday is our individual portfolios, filled with our own field notes, analytic memos and a reflection of what it’s like to participate in a team-researched and -written study.

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Another buzzing topic is Llamapalooza! CC’s annual music festival is slated for this Saturday, and it looks like the weather is going to hold for us. Palmer Hall, the picture below, is where Symbolic Interactionism has been held all block. Tomorrow, these sidewalks will be filled with colorful students in sandals and sun hats with face paint to go around.

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On the topic of symbolic interactions, music festivals are excellent examples of the presentation of self! Think about how students will dress differently and behave differently, based solely on their location at a distinguished event that values sunshine and music and good vibes. What is expected in terms of clothing, food and drink, schoolwork and conversation changes uniquely tomorrow on campus. The majority of CC won’t be in Tutt Library, even though 4th week is looming. More skin will be showing than usual and less responsibility will be felt and projected. Fun isn’t a feeling; it’s a performance.

My next post will be after the final work of 4th week but until then, hoping you’ll be enjoying sunshine somewhere, too! We’ve got the social event of the season, and seeing as how it was cancelled last year due to pounding rain, we’re hoping it’ll be twice as great!

Week 2: Imagery, Metaphors, and Starting Project Poetry

Hi all,

This week we’ve continued discussing more awesome poetry from some great authors. We started off the week on Monday with a Meter and Terminology Exam, just to make sure we remembered all the terms and scansion skills we practiced last week. We also talked about some poems by Native American author James Thomas Stevens. Then that evening we attended a reading by James Thomas Stevens and several other Native American poets and prose writers in Gaylord Hall. It was especially cool to listen to them read because one was Colorado Coffee’s very own Byron Aspaas, who read a short piece from his memoir. A lot of us had no idea Byron was a writer, and were entranced by the beautiful and personal stories he shared with us.

On Tuesday we did a cool project with imagery. Jane gave us this poem, “City Limits” by Joseph Hutchison, and asked two people to draw on the board the image they pictured in their heads while reading the poem. It’s a beautiful description of the relationship between cities and the natural world. Here’s the poem:

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Mary and Kai drew on the board for us. This is Mary’s image:

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And this is Kai’s:

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Both images highlight different aspects of the poem and represent the imagery in different ways. Which image do you feel most connected to after reading the poem? Would you draw your own image differently?

We spent the rest of the week focusing on metaphor, and on Thursday had a great discussion about Sharon Olds’ “Sex Without Love.” Olds uses metaphors of dancers, ice-skaters, religion, and runners to describe people who have sex without love. This poem is wonderful because it does what great poems do: it comes close to saying the opposite of what it says. The speaker of the poem seems to disapprove of people having sex without loving their partner, but she also talks about those people in terms of strong and reverent metaphors, using words like “beautiful,” “great,” and “true religious.” Ultimately, however, she suggests that despite the power and pleasure that can be attained by loveless sex, those people, whether they know it or not, will ultimately end up a “single body alone in the universe.”

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Last but not least, on Thursday we also kicked off Project Poetry, a competition we invented inspired by some of my classmates’ love of Project Runway. We brainstormed and came up with several sets of creative constraints, then voted on our three favorites:

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In order to compete, everyone has to write at least one of the three poems. We’ll submit them on 4th Monday and have some senior poets come into class to judge and pick the winners. Jane will be our Tim Gunn to give us advice and help us out as we go, and the winners of each category will get a prize! (Exactly what that will be is TBD.) So look forward to more info on Project Poetry come 4th week! (I’m sure everyone will be quite secretive until the day comes for the competition.)

Happy snowy weekend!

O

A Look at our Readings and Thesis Presentations

The following books have all been assigned over the course of the last week, and we’ve had some interesting follow-up discussions (and an essay to write!) regarding the ethics and efficacy of ethnography and participant observation as a research method. I highly encourage, if you’re interested, that you check these books out. They are written on a spectrum of styles, from first person all the way to third person omniscient and work meaningfully to portray the perspectives of marginalized persons living on the fringes of American life.

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These books are all accessible to the educated reader, but I would argue that all three need to be read with a critical eye. That means thinking about who the researcher is, which stories he/she decides to include in the text and why, and how he/she navigates the connection between micro- and macro-sociological patterns. One of the more clear examples of the effect of human, subjective interaction, is from On the Run; the author’s key to acceptance in the community that she is researching is a Black man named Mike. Upon taking her under his wing, Mike describes to Alice (the author and researcher) unabashedly why she is a poor fit for his own community. “First off, my clothes were all wrong – they didn’t even match. My toenails were bare and uneven, and what was I doing wearing flip-flops in January anyway? […] The way that I spoke was strange, and I could stand to get a little more husky. Plus, I didn’t know how to walk or hold my body right” (2014:223). In scenes like this, I can imagine the young Alice Goffman, in her doe-eyed whiteness, sticking out like a sore thumb from a community that she grows to consider herself part of.

Another part of this week has to do with CC’s Sociology Department at large! Today, Friday the 29th, was Senior Thesis Day. Here’s what the schedule looked like:

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I attended the second session, “Sociology of College Life,” which had a fabulous mixture of qualitative/quantitative and local/national research completed by three women of Sociology that will be graduating this year. From inclusivity to sexual assault, the three presentations investigated social problems that affect college students around the country as well as the CC student body in particular. One of the main takeaways that connected to Symbolic Interactionism is the way in which the publicized image of CC (supposedly, we all ADORE the outdoors and spend all our free time in the woods, which is ironic given my “About Me” at the bottom of this page) unequally affects students of color in an exclusionary way. It’s research like this that will keep Colorado College a changing place, and one that cannot ignore its social issues thanks to students brave enough to confront them.

Theses are required of Sociology majors at CC, whether they be a research project or literature review, on a topic of their choosing. To be completed senior year, each student plans for two blocks to be conducting research, writing, and editing with the help of peers and an advisor of their choice. What follows are a few photos of the women that so successfully completed and presented meaningful research; I can’t wait to see them walk the Commencement stage in just two weeks!

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A final note: I spent my last time in the field at Penrose Hospital observing yesterday. My group has started to more specifically investigate the systematic separation of hospital personnel based on profession (as dictated by uniforms) and how those professions correlate to race. Gender dynamics have also been interesting to observe more closely; the profession of nursing is both largely female and largely social (they chat… a lot…) in the cafeteria.

It’s snowing buckets on campus! Nothing like a late April snowstorm in Colorado Springs!

What are we here to do?

The Queery Is

First and foremost:

Queer Theory is not just for Queer Bodies to participate in. 

As I define it, queerness can be as minute as an identity signifier outside of the normative, but it can be as expansive as deconstructing and reconceptualizing our “natural” worlds. Queerness, through subversion, transgression, and the necessity of survival, exposes the artifice of society and what  has been  assumed to be the essence of humanness. It interrogates the idea of an essence at all.

On the first day,  a classmate of mine said
“There have been certain people who have made sense of the world for others..”
Those people in American society have been white, hetero, and male. Therefore,  the compulsory understanding of natural has aligned itself with their  positioning. A position of power. A position of regulation. And a self-perpetuating, self-sustaining, misconception of naturalness functioning, solely, to reinstate a dynamic power.

 SOCIETY’S ARTIFICE.

“The fallacy of binary opposition” is a phrase I first encountered in Mock Trial. For those of you who have yet to embrace this deliciously nerdy subculture, it is essentially learning a real case, with an assigned position (defense or prosecution), and then competitively reenacting that case in courthouse.  Where “binary opposition” asserts itself is in the construction of our team’s argument. We have to believe that we are right and they are wrong. Us/them, right/wrong, prosecution/defense –those are our functioning binaries.  However, in order to move forward, we have to learn both sides of the story (high-five debate team tactics)– and this is where the fallacy is revealed.  The narrative of opposition between two objects (between us and them) is the one we are creating.

And I think we can all agree that we are surrounded by spectrums and fluidity, not hard lines. Or maybe we don’t agree… but,

LET’S GO FURTHER

Binary structures work as a tool to create power dynamics, founded in an assumed or constructed, irrefutable truth. For example, the gender binary male/female: where male is the dominating force. One of the underlying assumptions is the “truth” that women are weak and men are strong, which gives reason for men to be in a position of power. And (surprise, surprise) these support systems take a binary form: strong/weak, natural/unnatural, sensible/silly.

While homosexuality threatens (or challenges) the power dynamic of the male/female binary, the term was created* in conjunction with its own binary force: homosexual/heterosexual. What this, then, did is reestablish lost power within the heteronormative, where homosexuality is subordinate, unnatural, etc.

​Part of this “truth” is grounded in the assumption that the dominating side is the majority, and it’s power is simply a matter of numerical fact. And in some instances, binarisms do follow this narrative. But can dominance be as basic as having the most bodies? How much of it is really the power of numbers or just the reinforcement, and gravitation towards, a larger binary construction for justification. What binary is that, you ask? Majority/minority. But if we think about colonization, which party had the power? The colonizer or the colonized? The majority or the minority?

Or if we think about the dynamics of a plantation house, where the plantation owners were the minority and the slaves the majority. A different binary was put in place to ensure positions of power, white/black, where scientific “truths” of “natural evidence” of inferiority were created to justify claims of necessary control.

In no way am I trying to invalidate the existence, significance, or consequences of oppositional binaries we are swimming in– male/female, white/black, good/bad, etc– but more so wish to illuminate the destructive pattern of thought (its sole purpose being to establish the dominate/subordinate power dynamic). ​​And for those of us who want to move towards balance and equality, or (at least) consensual inequality, we have to strip down the layers of our reality.

So where do we go from here? Well, 

TO EXIST CONSCIOUSLY WE MUST KNOW THE WAYS IN WHICH WE HAVE BEEN CONDITIONED TO MOVE, THINK, AND BE. ​

Queerness, as I embrace and use it, is a resistance to this conditioning.