Tag Archives: math

‘You must keep track of inventory’: Story-problem stories

brandi_math_2014

A friend recently posted this photo to Facebook, and, as I am a Matt, I was drawn to it. But I found this story problem to not accurately represent my desires. I don’t really want any flag at all, but given the choice of the two, I’d much prefer the Mondrian-esque flag on the right rather than the one with “4 equal parts.”

But then, story problems are all too often mere fiction. Students are expected to apply some math processes to situations that, while plausible (like realistic fiction) are not actually, you know, real.

I see some problems here with story problems that have fictional set-ups. One, if the situations are fictional, doesn’t that also suggest that the math is fictional (see also here), too? And if so, why should I spend any more time learning math than I do learning the names of all the dwarves in The Hobbit? And, two, if I’m an imaginative person, I might get so interested in the fictional situation and/or the fictional text that actually doing the math might seem damn boring in contrast, or even besides the point. Who cares about an equally sectioned red flag, if green better matches my living room decor?

Below are some examples (found here, questions that are in the style of the exam given to high school juniors in Illinois) of fictional math situations that seem strange and wonderful.

4. A customer in the music shop where you work purchases 3 cassette tapes. One costs $8.99, one costs $7.99, and one is on sale for $3.99. Excluding taxes, how much does the customer owe?

First, let me ask why the test writers either, A., know preternaturally much about my afterschool employment, or B., are asking me to pretend a whole lot here: I work in a “music shop” — what is this, 1998? Nope, wait, a customer is purchasing “cassette tapes,” so this is 1985. These prices certainly suggest those of the ’80s, but even then, who is selling a cassette for $3.99? Is this a good deal? I mean, is this cheap cassette Led Zeppelin IV or is it the Bullet Boys? If it’s the latter, should I harass the customer for his terrible taste?

So many questions: Why would I exclude taxes — isn’t that included in what the customer has to pay? I mean, am I gonna offer the customer numerous nonfinal tallies, say, giving a subtotal after every item? And why doesn’t this 1985 “music shop” have a cash register to do this work for me? Is the power out? It’s store policy to ask patrons to leave and we lock the store until the lights are back on.

The word “owe” — is the customer taking out a loan to buy these cassettes? Should I, as a math student of the future, advise the buyer not to buy cassettes at all, as the vinyl that now seems inconvenient will be cool again in just a few years, and the CDs that are coming soon will be usable long after cassette players become scarce, but the cassettes he buys will just be stored in some box in a closet in his mother’s house, to be thrown away when she moves out? Would the customer find my future-perspective frightening, as he did not know he would be buying items from a test-taker who was born long after 1985?

6. You must keep track of inventory in an office supply warehouse. This week, 8 computers of a particular model have been shipped out of the warehouse to a local store, while 4 more computers of the same model have been received by the warehouse from the factory. What is the overall change in the number of these computers in inventory this week?

I “must.” Ha, such imperatives. Am I a slave in this office supply warehouse? Or is it a family obligation, like maybe my manager is my wife’s uncle, who was kind enough to give me a job when I got fired (for asking too many questions and frightening the customers) from the “music shop” in question 4 above — and I had to spend some purgatory time in Question 5,  where I was told to Calculate the missing values so as to complete the chart — and so now I feel like I owe the guy, even though our warehousing business isn’t doing so well. It’s these damn Kaypros. I keep telling him that everybody wants IBM or Apple computers, but he won’t listen, and half the computers we send out come back. He just tells me to stick with the strategic plan, but I really think he’s just jealous of my intellect. My wife tells me to bide my time, that the resumes I’m sending out to get a job in my field will eventually pay off, but until then, I’m stuck in the logistics biz. No kid ever grows up and says he wants to be in “logistics” — but here I am. Ah, well, at least I got a place to go and a paycheck to keep my home warm. And thanks to all these Kaypros we keep getting returned, I don’t have to do the inventory myself. Lotus 1-2-3, take it away.

After Question 7 — Calculate the missing values so as to complete the chart — ignores the spreadsheet the Kaypro made, I’m apparently moving to the health care field. Such career whiplash:

8. In the hospital where you work, one of your duties is to take pulse counts. One patient has a pulse count of 21 beats in 15 seconds. At this rate, what should this patient’s pulse count be for 60 seconds?

Man, oh, man, do I hate taking pulse counts. Having to hold the bony wrists while keeping a finger on the sagging flesh of these old arms, it’s the worst. Plus, are we really sure that it should be my “duty”? I mean, I have no experience except selling cassettes and inventorying computers. But around this place, man, you never know. But that’s why sometimes, just to keep myself amused, I’ll count a pulse beat for 10 seconds, and then scream, “OH MY GOD, WHAT’S THAT ON YOUR HEAD?” at the patients just to see how much faster their heart rates can get. I go for unpredictable. But even if I don’t scare them on purpose, how do we know that their heart will beat at an even rate for the next 45 seconds? I mean, dead people’s hearts once worked, too, until they didn’t. And once a person’s dead, at least they don’t have to do these stupid story problems.

Links: Free college for all, crap jobs, math, etc.

1. What college would cost taxpayers if it were free for students. I’m starting to think lately that maybe no one should expect to profit from teaching people or healing people.

2. School of Rock actors, plus 10 years.

3. One explanation for middle-class decline: Even crap jobs paid better 50 years ago.

4. “Would math exist without us?,” continued.

5. How some people follow the Bible literally, but selectively.

6. “Surprising benefits” of smog: A parody and/or a display of rhetorical exercise?

7. SNL’s “I wish it was Christmas today” (aka “Christmas time is here”).

8. “A Comprehensive History of the ‘Cups’ Phenomenon.”

9. Sesame Street clips of the ’70s.

10. Scraps by Emily Dickinson.

11. “The Poem as ‘Thing‘”

12. From Brain Pickings: A list of the best psychology and philosophy books of ’13.

13. Andrew Sullivan says Fox News is anti-Christian.

Math isn’t real, but neither are ‘atoms’

This short video is a little manic, but generally does a nice job of summarizing the philosophical (specifically, metaphysical and epistemological) discussion as to whether math exists somewhere in the universe (though it can’t be, you know, physically detected) or whether math is just a human construct, a set of ideas, or as the video describes it, a “fiction.”

The video’s narrator describes the “math realist” position as believing that math is discovered, like new species are discovered, “out there” in the world. However, since math’s ideas are not physical and thus can never be directly observed by our senses, then believing in “objective” existence of math ideas requires, well, faith.

The video contrasts math to physics and other sciences, which take as their objects of study things that happen or exist in nature. I’d point out, however, that while we can sense things, like a round-shaped, sticky, sweet-smelling pastry we label “pie,” we cannot sense the measurements of the circumference and diameter of said pie placed in a ratio we call “pi”  (also, it’s hard to see how any of the terms just used — circumference, diameter, measurements, ratio, pi, and for that matter, pie — exist except as concepts.

I sometimes tell my students that one way of defining real is by determining whether one would mind being struck by that thing. I don’t want to get hit by a pie, but I don’t see how throwing pi at me could ever hurt.

And so, of course, while physics, chemistry, and biology deal with sensible things, as “ologies,” they are sets of ideas, and so these science concepts, models, theories, and laws may not exist physically any more than math ideas do.

In “Zen and the Art of Motorcycle Maintenance,” Robert Pirsig describes science theories as our contemporary technological culture’s equivalent of earlier cultures’ ghosts — things we think are real but, of course, can’t be seen, etc.

And, of course, none of our words are real, either — depending on how you define ‘real’! If real describes things available to our senses, then words are only interpretations of sounds heard or shapes seen.

So it can get complicated to figure out what’s real when one sees something like IBM’s movie of atoms:

The words “These are atoms. Magnified over 100 million times.” are needed to explain what we are seeing in the video, because what I seem to be seeing is something that looks like little metallic spheres — ball bearings, perhaps. I see ball bearings on a gray background that seems to have shading variations in a gray background.

But we are informed that these are atoms — defined in the press release as “one of the tiniest elements in the universe” — but then, these are not quite atoms. They are “molecules” of carbon monoxide, one atom of carbon and one of oxygen — but this molecule doesn’t look in the movie to have two components.

But “look” and “see” also become problem terms here. The images in the movie aren’t a result of light directly striking a light-sensor (as in a digital camera or in our eyes), but are more like graphs made by information received by the probe in the scanning tunneling microscope. According to the Wikipedia page:

The STM is based on the concept of quantum tunneling. When a conducting tip is brought very near to the surface to be examined, a bias (voltage difference) applied between the two can allow electrons to tunnel through the vacuum between them. The resulting tunneling current is a function of tip position, applied voltage, and the local density of states (LDOS) of the sample.[4] Information is acquired by monitoring the current as the tip’s position scans across the surface, and is usually displayed in image form.

So we’re not seeing atoms. In all likelihood, we can never see atoms. But, OK, let’s call what is detected by the scanning tunneling microscope an atom. But an atom, as we’ve all learned, has constituent parts, which are vastly smaller: protons, neutrons, and electrons. So “atom” just means an organizational level, like, say, a “class” is a bunch of students who gather at the same time in the same room, but there’s no class as a particular physical entity — just particular students.

However, the picture gets more complex, as protons and neutrons are comprised of smaller-yet pieces called “quarks,” and these quarks and other fundamental particles may be made of smaller items yet.

Of course, these definitions are part of an elaborate, technical model of the most basic components and interactions of physical reality, which itself is not seen as a complete model.

But of course, any explanations for things we see in the physical world is going to be an idea, an interpretation, which means it’s not the same as the physical world itself. Even these ideas I’m using now are merely ideas, by which I mean they are arbitrary, subject to replacement or revision as we see fit.

In talking about the arbitrariness of ideas with my high school students, some asked why schools teach “fake ideas,” to which I responded, “It’s better to learn fake things than nothing at all.” I’m not sure I would stand by that position all the time, but it’s worth considering from a philosophical and educational perspective.

Link: Teaching reading is crap-detecting

This New York Times article, saying it’s harder to raise students’ test scores in reading than it is in math, indirectly raises some cultural/epistemological questions about the differences between math and reading.

The article opens with this anecdote from someone who has apparently taught both:

David Javsicas, a popular seventh-grade reading teacher known for urging students to act out dialogue in the books they read in class, sometimes feels wistful for the days when he taught math.

A quiz, he recalls, could quickly determine which concepts students had not yet learned. Then, “you teach the kids how to do it, and within a week or two you can usually fix it,” he said.

Helping students to puzzle through different narrative perspectives or subtext or character motivation, though, can be much more challenging. “It could take months to see if what I’m teaching is effective,” he said.

I have taught high school science and English, and I’m not sure I’d say it’s easier to teach science, because of what it means to “teach science.” The expectation (as I was informed after I pursued a different goal) is for students to learn and apply the set of science ideas (theory of evolution, atomic theory, Newton’s Laws, etc.) that are provided in the textbooks. The discipline of science observes and tries to explain the physical world, but most science classes don’t allow this. Students take notes, do equations, take tests. Real research is not done by most students (though some high school science competitions, such as this one, show that students are capable of doing impressive work).

Science classes, then, just teach a set of ideas — let’s call it a story, Science Stories — and so do math classes. Math classes could be theoretical explorations of these abstract ideas, but many high school math classes simply teach procedures (algorithms) for doing things: to find the area of a rectangle, multiply the base times the height. Sure, that’s useful information, but hardly intellectually all that challenging. Math, as taught to high school students, is a tidy system of right and wrong methods for arriving at an answer. What mathematicians do is far more abstract and creative, of course, but we don’t generally let students see that.

In English, however, we’re actually asking students to do the same things (though obviously adapted to younger minds) that English professors do — read and analyze texts and write about them. What I love about teaching writing is that students are truly CREATING texts. Students in our science and math classes are not making anything — they are just taking in the ideas that others have made.

Of course, creating something is more intellectually demanding than just memorizing and applying an idea (even Bloom’s Taxonomy, that education cliche, says so). So we writing and literature teachers give our students guidelines and models to help them “scaffold” (in the teachers’ vernacular) their way to completed projects.

But of course, there are very few right or wrong essays or literature interpretations — there are worse ones and better ones, and judging which is which is highly subjective. The student essays I like best are those that go beyond what is merely stated in a text to make connections that are not obvious. In other words, I like essays that are interesting, that say things I hadn’t read or thought of before.

Lately I’ve been suspecting that maybe the best way to teach this kind of creative thinking and individual judgment is to model it for my students. As a teacher, I have my own biases and peculiarities, and so I’m not an ideal (Platonic?) model — but maybe learning to be analytical and/or creative is really more of an apprenticeship anyway, rather than something that has set standards for students to adhere to.

And here’s where teaching the study of literature gets interesting and/or controversial. The recent Common Core State Standards for teaching literature include statements such as the following:

CCSS.ELA-Literacy.RL.11-12.2 Determine two or more themes or central ideas of a text and analyze their development over the course of the text, including how they interact and build on one another to produce a complex account; provide an objective summary of the text.

The very use of the the word “objective” in relation to a text is nearly an absurd statement to someone familiar with the critical theories used to interpret literature that arose in the last few decades. Taking undergrad literature classes in the mid-1990s, I gained just a limited understanding of some of these approaches, but judging by the enthusiasms of the younger professors and by the resistance of the older ones, I understood these ideas to be important.

But the Common Core standards seem written in ignorance of these developments in interpretation, as if the standards writers were just gonna elide the last 50+ years of criticism. And though the standards are careful to call their lists of texts for use in classes “illustrative” rather than “recommended,” this listing shows 14 texts, only two of which were published in the last 50 years.

So the act of reading and interpreting texts is something that, in addition to necessitating word-processing skills, also “requires background knowledge of cultural, historical and social references” (as the Times article states), and from these basic skills and resources, we ask students to make coherent, logical statements of analysis. That’s asking a lot of anybody. But then, all too often, standardized reading tests ask students to select an interpretation from multiple choices, which requires students to also analyze the test to see which of the many possible interpretations of a text is the one that the test will honor as the “right” answer. The student has to match minds both with the text-writer and with the test-writer. In the Times article,

But when [the teacher] asked [students] to select which of two descriptions fit Terabithia, the magic kingdom created by the two main characters, the class stumbled to draw inferences from the text.

Uh, yeah. Why only two descriptions? We ask students to make this complex, creative, personal interpretation, and then ask them to compare theirs to an adult’s?

This might all be despair-inducing, except for the fact that when we teach interpretation skills to students, we also empower them to see the tests and the standards as the bullsh*t they so often are. This reminds me of the Hemingway quote featuring his one standard of education:

“Every man should have a built-in automatic crap detector operating inside him.”

(One, too, might employ such a crap-detector while reading Hemingway.)

 

The ‘true story’ paradox: Toward a defensible nonfiction

Many writers tell stories about real people and then claim these stories to be nonfiction, that is, “not false,” but this is a problem, because all stories are false (or are so distant from reality as to be false).

A nonfiction story is a sequence of descriptions (of things, people, etc.). These descriptions are the inferences and interpretations of the colors, sounds, touches, smells, etc., that we take in. As we can’t know the world except through our senses, these interpretations may be flawed, incomplete, or speculative.

As such, anything we say about the real world is at best inadequate and at worst, complete fiction. Any nonfiction story that does not acknowledge this is lying.

The problem lies in conflating the label of “nonfiction” with the idea of truth. Truth — the correspondence of any idea with reality — is not only unknowable; there is no “reality text” or “reality idea” against which to compare a human text or a human idea (that is, all ideas we could have, since we know only our own minds and none other). That which we label physical reality — real, touchable things and sensible energies (light and sound) — are not ideas and do not seem to need our labeling. For me, “real things” are those that I do not wish to strike against my head, as they cause me pain. Ideas may confuse me, but they do not make me bleed.

So a nonfiction text may avoid the true/false dichotomy (itself an idea, of course) altogether if nonfiction is defined narrowly as “the expressions of a particular mind speaking as itself” or perhaps “the expressions of a self” (as Kirsch points out).  The truth test isn’t necessary.

If we stopped wondering whether any claimed-nonfiction story was true or false, we could see it just as an idea, an abstraction, a possibility. We could cease the futile struggle of trying to find reality and we could avoid the delusion of thinking that we had found reality. We could understand that all histories, being stories, are not entitled to a claim to truth. We could see that math and all logic systems are merely ideas, as are all philosophical positions and as are all religious beliefs. We’d be left with our own particular experiences and, perhaps, we’d feel a freedom to interpret these experiences anew for ourselves rather than applying others’ concepts to them, but when we did want to borrow others’ ideas, we’d be aware of our borrowing.

Perhaps realizing that our ideas are not reality would allow us to more clearly see our experiences without filtering these through our own, or others’, previous concepts.

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This short piece was written after the much longer text below. If the tone above seems oracular, that may be a reaction to my feeling in the earlier piece that I was getting too particular, too bogged down, too chatty. The above is the gist of what I was intending to say below, and what I said below came to mind as I wrote in my journal this morning. I sensed that those ideas could be worth posting on ye olde blogge here. I’m posting the lower piece as perhaps a model in the exercise of writing, and perhaps as an antidote to the lordly-and-or-priestly-simplisitc tone above.

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This essay’s point about how humorists like David Sedaris and Sloane Crosley aren’t really writing essays but something else:

The self, then, has always been at the heart of the literary essay. But the new essay is exclusively about the self, with the world serving only as a foil and an accessory, as a mere staging ground for the projection of the self. Formally, one might describe the work of Sedaris, Crosley, Rothbart, and company as autobiographical comic narrative: short, chatty, funny stories about things that happened to me—weird things, or ordinary things that are made weird in the telling. What we now call an essayist used to be called a humorist. Sedaris’s books are sold as essays, but he is plainly trying to be Thurber, not Addison.

This is a particular kind of humor, rooted in the creation of a fictional alter ego who shares the author’s name. This device allows the essayist to claim the authenticity of non-fiction while indulging, with the reader’s tacit permission, in the invention and shaping of fiction.

Sedaris’s essays that I’ve read or heard (as he performed his writings on public radio before he began publishing) are well-crafted, entertaining pieces. By now, I expect that when I’m about to encounter a new Sedaris text, I can relax and prepare to laugh (not that I always laugh — some writings are just a bit peculiar, but I have confidence in a Sedaris piece that I do not in a David Rakoff piece or a Sloane Crosley piece. Somehow those other writers don’t quite feel as entertaining. There’s a fine line Sedaris walks between seeming endearing and empathetic — being Everyman while also displaying his peculiarities ( I recall a “Fresh Air” show a few years back where Sedaris told Terry Gross how he finds bugs to feed the spiders living in his house, and I wondered if Sedaris was putting on a persona in order to make himself seem intriguing (which wouldn’t be intriguing) or if he really were that weird (which would be intriguing, but he was telling this as an anecdote on national media, and I doubt he’d be as successful as he is if he were that socially self-unaware), and somehow Rakoff didn’t strike me as being quite as amusing, and Crosley seemed to not be aware that her story-character, her persona, was not always one I could identify with — particularly in the story in “Cake” where she seemed condescendingly critical of a friend’s wedding, and I sorta cringed to read the story, wondering what this real-life friend would think.

And it matters what one writes about other real people. My brother’s book contained an anecdote about me that was not only “frank and humorous” (as the review at the link says) but also, well, false. The anecdote is that, while camping with my brothers and my uncle, I took our flimsy folding saw into my tent at night so as to defend myself from a bear attack. Amusing, sure, but I have no recollection of that (and nor did my brother, it turned out; he got the story from my uncle). If I took the saw to bed, which I may have, it would have been for the purpose of keeping it dry and/or in a known location. I had no delusions of fighting off bears, but it’s amusing to think that a person would. Clearly, I have a smaller complaint than do many other real people (here’s just one) who’ve been the subjects of works claimed to be nonfiction, but it wasn’t fun to see myself portrayed as a doofus in the seemingly long-lasting medium of print. And because my brother claimed to be writing based on his memories, I felt there was no good way to challenge the story, especially since I hadn’t even remembered this event — and I wouldn’t remember it if it seemed like a non-event to me at the time.  (And of course, in my defense, and for my own publishing purpose, I’m guilty here of writing about my brother.)

To write nonfiction stories, anecdotes, or memoirs is to fictionalize. Stories do not exist objectively in the world and so they must be created by a person’s mind, and even if that mind is not intending to lie, that mind can only tell what sense data it took in (which must be incomplete — our brains are constantly choosing which “inputs” to attend to and which to discard) and that mind can only tell what it interpreted that sense-data to mean. A story can only come from a perspective, but every perspective is limited to the point of being untrustworthy. Where one person sees a threatening man, someone standing to that person’s side might see only a villain on a movie poster, and unless these discrepancies are ironed out while there’s recourse to rechecking the evidence, both interpretations exist and both may be no more true than “that’s what I remember,” which is both inadequate and unassailable.

Sedaris’s stories are unassailable. He tells stories about himself and other real people (he’s pretty hard on his father, for instance, in “Me Talk Pretty One Day,” which  Wikipedia article contains a reference to these troubles, how Sedaris called off a movie of the stories “after a conversation with his sister aroused concerns as to how his family might be portrayed on screen”) while these other people aren’t around to defend themselves. But the stories are entertaining. Sedaris’s essays are well-crafted so as to be entertaining, so as to make them easily digestible for the audience.

An article by Nathan Heller in the 25 Feb. New Yorker (mostly behind paywall) contains this distinction:

“Artists are pulled these days between two warring camps. On one side lie what might be called the Experientialists: those who believe that the point of art is to have the audience undergo a particular experience in time — and that the audience’s responsibility is to submit as fully as possible. (Think of Antonioni, who kept his cameras open to the unexpected.) On the other are the Arrangers: people who think that the role of art is to order, burnish, perform, and engage desire. (Think of Hitchcock.) Experientialism honors the artist’s sensibility: ‘A la Recherche du Temps Perdu’ may be dilated and slow, but it’s only by giving in to the author’s method that we can experience its genius. Arranging, by contrast, defers to the audience: what makes ‘The Great Gatsby’ better than any of a hundred novels with comparable cultural freight is that it’s economically written and smartly plotted, seducing us without special conditions. Diehard Experientialists accuse Arrangers of pandering with ‘easy art’ and cliche. Arrangers mock Experientialists for self-indulgence, tedious abtruseness, and bad faith. (The lousy Experientialist claims that his disjointed, boring novel is supposed to be that way.) The ablest artists are those who inhabit a middle landscape, mastering the art of special attention while meeting the challenges of effortless appeal.”

Clearly, if one’s goal is to make money by appealing to a wide audience, one would make one’s art as effortless as possible, and one would stick to familiar forms. But these familiar forms can sometimes become part of the fabrication that an artist wants to expose.

Sedaris’s stories do not include caveats or doubts that would at least bring in the possibility of an unknowable reality — Sedaris relates reality in his stories. Well, not quite: there is a sense, as Kirsch writes, in Sedaris’s work of an irony that calls “the whole story into question.” But in a sense, this is OK because, I suggest, Sedaris’s essays are never essays (particularly if essays are defined as “attempts” (per the French etymology) in which doubt plays a primary role) — instead, they are scripts for his performances. As a performer, Sedaris is directly in front of an audience, which can boo or leave and so must be entertained. He is able to charge significant prices ($40 to $65 for a ticket in Salina, Kansas) for his public performances — these are not just readings. In giving a performance, there’s not a lot of room to invite philosophical debate — people wanna laugh, and if Sedaris were improvising onstage, or merely telling off-the-cuff anecdotes, people may not be amused. It’s better to go with your best — prepared — stuff in that case.

In contrast, David Foster Wallace’s reportorial essays are not pared down for performance but are stuffed with self-conscious attempts to explain and overexplain — thus, the footnotes. Wallace seems aware, obsessively so, of the problem of writing nonfiction about reality. Yet, he still (as here) does characterize — turn real people into descriptions that might as well be fictional for all the reality about a person that they could possibly capture.

His digressions, while they may reveal the associative nature of his thinking, may also give a sense that he is actually trying to paint a picture of reality through sheer thoroughness.

What I want to suggest is a method of nonfiction that does not attempt to tell stories. (Even when newspapers tell stories, they try to avoid the problem of perspective by simply retelling, by quoting, the stories of those who were there — thus, news gets abstract — and unassailable — by not even trying to discover reality but by mainly retelling others’ stories. Reporters defend themselves by saying “I can only report what sources tell me,” which info could be fact-checked against other sources, but that’s just a matter of comparing stories to stories. I suppose it would be possible to report what one actually sees, but then one is limited by one’s own perspective and interpretation.

[And at this point, I was going to write the ideas that went into the shorter piece at the top.]