A case of Universititis

Today, I helped grade exams for the first time – as one of twelve teaching assistants in “Linear Algebra I” at the university here. First, let me be clear – we do not assign the letter grades. We receive solutions for the problems and work in groups to correct them. One group corrects problem one, the next problem two etc. Within each group we discuss how many points should be awarded for each part of "our" problem. We can then use that as a rubric to ensure that all members of the group grade "their" problem the same way. Once the professor sees the distribution of points for each problem and for the exam in general, he or she will fix the letter grades.

Now, this exam, like many math exams at this and other German universities, is going to end up having students who have less than 50% of the points who still get passing grades – because if they didn't, far too many people would fail. Out of respect for people's privacy, I won't post the specifics of this class – but I don't know any introductory level class here (Calculus, Linear Algebra, Probability Theory) in the last few years in which more than half of the students would have passed if the passing grade had been left at 50% (some, even after that line was lowered to 35%). 

However, I should give some context. In a math degree in Germany, it's totally normal to fail once or twice. If the exams tend to end up this way, it's only natural. Especially if in your first or second semester you don't really realize what's expected of you. Nobody attaches these big moral weights to failure the way (I think) we sometimes do in the States. And, since it doesn't cost tens of thousands of dollars per year, it's not such a big deal to repeat a class. 

However. There's a wrinkle. If you fail the same class – say, “Calculus I” – three times, you are out. That means that you’re barred from that particular course of study at that university. And you’re not allowed to study that subject anywhere in Germany. Now, for some people, it might not be such a big deal – you had math as one of your two minors. Well, switch your minor and do the rest of your studies and it's all good. But if you were studying Computer Science, Math, Physics, Biology – done. That's why I saw so many German med students when I was studying in Budapest.

Oh, and there are no other grades besides the final exam in each course. The homework you turn in (at least in math) is your way of proving you should be allowed to take the exam in the first place. In this last class, you were required to get 2/3 of the points on the homework assignments (2/3 overall, not per assignment) in order to be allowed to take the exam. But after you're registered for the exam, that doesn't help you. (Footnote: maybe a tiny bit, if the professor is debating between a B and a B+ say, and they see that you always actively participated in the class – it might bump you up a tad.)

One day – two or three hours – and a semester's worth of material. Better hope you have a good day. And, if you can, really understand the material.

Kind of stressful.

However, let's flip the coin. 

Now we’re in a calculus class in, say, a liberal arts college in the US. Homework counts for 1/3 of the grade. Two small midterms count for another third and the final counts for the last third. And someone maybe – MAYBE – gets a C. Or someone gets a B-. And students and parents complain.

Grade inflation is real and serious. Professors, particularly ones without tenure, are extremely vulnerable to course evaluations and there’s a strong correlation of lower grades and worse reviews of professors (I know, I know. It’s hard to separate the feelings of ‘How much I like this professor?’ from ‘How good of a professor this person was in this class?’ but considering someone’s career is very affected by it, we should at least try). And yes, college in the US is extremely expensive – so flunking a class is like flushing lots of money down the drain. But that doesn’t mean professors should let students through who, in every sense of the word, did not make the grade.

But what about the other system, where some people work hard, maybe improve a lot in the course of the semester – and then fail on one day.

Discouraging some students and inducing a panic that makes them want to cram everything for one day (and not learn it all properly, because they feel they can’t in that amount of time) or pampering students and letting some slide so as to avoid angry parents and bad evaluations…

Also, let me add a disclaimer. I’m not a professor. I don’t really have a leg to stand on when I try to spout these views. So, take all of this with a grain of salt. But I am the daughter of a professor and have spent the last seven years in and around university systems. What am I getting at? Well, I’m wondering what the best way is.

As to the German system, I’m not a fan of the ‘one exam, one grade’ method. I think difficult exams are all well and good but I think something – either a midterm or the work on the homework – should also contribute to the grade, if possible. That might also give some people the hint earlier on in the semester that they’re not doing what it takes for the class and make them either drop it or start working harder. As for the American system? That’s a trickier problem, I feel. Do you give honest grades on assignments during the semester to give students the hint, but always give better ones at the end to feed the beast of grade inflation because the students can’t help that they are caught in the middle of it? Or do you stick it to the inflation system and risk your position by grading honestly?

I think there is no easy answer, neither to these questions nor to the predicament as a whole because these issues are very tied up in the two educational systems. The German system, which is also just getting used to the Bachelor/Master thing (they used to just have a ‘Diplom’ degree which kind of lumped both together) and it’s very new to even require grades for early classes instead of just having a handful of large exams toward the end of your studies. The American system – especially the liberal arts wing – is set up so differently: with our "customer-oriented" private colleges, emphasis on teaching and guidance, a student-to-instructor ratio that would be unachievable in public education, the financial strain put on families – it’s hard to turn an enormous ship on a dime. You have to make tiny adjustments and see where they take you.

I believe the answer to this question resists simplicity, if I can paraphrase John Green.

And one of the thoughts I have so frequently is how privileged I am to be able to look at both of these systems from the outside. I think it has allowed me to notice some of the flaws and perks of both. I wonder where each system will be in ten or twenty years' time.

Creatures of Habit

I love having a routine. I love knowing what I'm going to do next and what's going to happen after that. I somehow feel like when there is time boxed out for an activity, there is somehow more time to do that than if the time I had were unlimited. - well, for unpleasant activities, that is.

Sometimes I have wished during my student career that at certain times of day, my books, my notes, or even my laptop would disappear. Maybe they would only be available to me between eight in the morning and four in the afternoon. That means, after that magical time in the afternoon, I cannot work anymore.  Bliss.

Actually, I'm fairly skilled when it comes to setting deadlines for myself and setting aside time to rest, but I am currently learning that this becomes more difficult when there are fewer things to do - and when the deadline is six weeks away, not one week away (like a homework assignment, for example).

At this moment, I have one exam (on March 12th) and then, of course, that Bachelor's thesis. Which is coming along, but slowly. I'm currently trying to work out a proper schedule for myself for this time (now that classes have ended and my days seem to be yawning expanses of time that could be used for sooo many things).

The other thing that has been on my mind lately has been a quote - and of course, I'm not sure exactly who said it the first time, but I believe I heard it first from Stephen Colbert. Sometimes I feel thrilled to sit down to work on my paper, ready to write, full of ideas of how and why and in what order to say things -- and on other days, I will clean the entire apartment before sitting down to do my work because I don't know where to start. I'm not inspired. But, as Stephen or (I'm sure) several other people have said - 'inspiration is wonderful and enough for amateurs - the rest of us show up to work'.

So, it's okay if I don't produce something amazing when I sit down to type or work through a proof. The sitting and the trying and the engaging with the subject matter might be enough - at least if I'm sitting and working and I happen to stumble upon some inspiration, I'll be ready for it.

BUT - in order to get me sitting down, a schedule might be in order. I'll get to it.

Hello again!

It has been simply ages since I have written! I do miss it but at the moment, I'm up to my eyes in research for my bachelor's thesis, which, being in the field of algebra, means that I'm staring at piles and piles of things that look like this:

Algebraic nonsense. No, not nonsense. Just a lot of g's and f's and h's...

Still, I've hit a very productive stride the last few days and I am thoroughly enjoying it.  Much better than the blank panic I felt before!

Also, in my last few days of working at odd hours, I have been struck by something I don't think we hear about a lot - and that is the kindness of the internet. Oh yes, there's also the awful, mean underbelly of the world wide web- demons haunting the comments section, cyber bullying and much much more - but there are some nice corners, too. Some things that just make one realize how thoughtful we homo sapiens can be.

Imagine, lying in a bed and being unable to sleep - plagued by worries. Google for thirty seconds and find instructions on breathing to help you relax, or (a favorite of many people) old, familiar videos that might make you forget your worries - or, what I consider the most thoughtful of all -- someone who has uploaded a youtube video that consists solely of 45 minutes of the sound of rain falling, since apparently I'm not the only one in the world who loves falling asleep to the sound of rain. Also, of course, the typical databases of recipes - for everything from quinoa-banana-chocolate muffins (I've made three batches) to your own shampoo (using rye flour? I tried it this morning. I'll see how my hair is this afternoon!). It's just a cool place, a lot of the time.

Alright.

Back to those g's and f's and h's.

The Little Things

 That title was one of several that almost made it to this blog. The last few days have been filled with moments about which I really have wanted to write – and though each one in itself might not be enough for a full entry, all combined, they make a nice little montage of last week. Plus, like that, I can give them each the title they would have had, had they made it to full entry status.

Work for Lesser Minds, And the Ivory Tower of Mathematics

This week was the first one of our semester here in Germany. That means that after the weeks and weeks of feverishly studying for exams, the strange sad let-down that follows after the exam is over, and after the subsequent weeks of relaxing and visiting family, all of a sudden, I’m on that bus again, heading to the campus every morning.

I am one of those people who tends to work better within the confines of a schedule, so I’m not that upset about the semester starting again, but there are a few things that are intimidating me about this semester (see my other entry about TAing and writing a Bachelor’s thesis!). Also, another plan for this semester for me is to be more engaged in the math department – not just going to the lectures I’m supposed to attend, but also going to talks from visiting professors and seeing what other folks in that area are up to.

So, on Thursday, I went to my first colloquium given by a visiting professor from Smith College, no less. “A female mathematician from a liberal arts women’s college?”, I thought. I just had to go! It had been such a damn mathematical day, too. I had Projective Geometry starting at 8 in the morning, then worked on a Topology homework assignment for about three hours, then prepped my own Linear Algebra workshop for next Wednesday, and then had my “office hour” (I don’t have an office, but I’m available for students if they have math-y questions). In between I also had some food and plenty of coffee. And then it was 5 in the evening and I finally went to this colloquium.

Her talk was on an international quarterly called The Mathematical Intelligencer – which is very much unique. Not specific to one area of mathematics, the Intelligencer encourages experts in various fields to write about what is important and happening in their field – but to write it for other experts, not just ones in their field. (For people of high mathematical literacy, as she explained.) The Intelligencer also prints columns on things like the history of mathematics, looking at what was happening 50 or 100 years ago – and it even includes articles simply on the inherent beauty of certain mathematical concepts, such as space-filling curves (look here!), fractals, or other amazing constructions. They also print articles about math education –  a subject I care very strongly about.

However, this makes them a very strange quarterly. And from the perspective of their publisher, they aren’t doing well as a quarterly. How many times were they cited in mathematical papers in the last year? Not that many. How many new, young professors do they have writing for them? Also not many, since they all are worried that they need to print things in more traditional journals. So we have lots of metrics that don’t put the Intelligencer in the best light. But that doesn’t mean it’s not important. What do to? This was the main question the professor had for us and quite a lively discussion ensued, during which the following quote came up. I warn you, this quote contains the true epitome of mathematical and academic snobbery, from the heart of what I call the Ivory Tower of mathematics:

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.

G.H. Hardy, A Mathematician’s Apology

Oh, it’s quite a thing, that essay! It also contains such nuggets as: “Most people can do nothing at all well”. (I feel like I can just hear Saruman reading it. Okay, obviously in Christopher Lee’s voice.) A professor had brought the essay up to give some context to some of the critiques the Intelligencer has received within the mathematical community.

As soon as this quote came up, it was rebuffed by others. No one really admits to agreeing with that sentiment – and indeed, I think most do not. (Although some do.) At that point many things I have thought before were streaming through my mind:, about how well mathematics needs to be explained; how difficult and simultaneously crucial it is to be able to explain mathematics in prose, or hell, even poetry; to make it accessible and exciting and to make folks want to take part.. And as I heard some of the folks quoting Hardy and talking about the attitude within a lot of pure math circles, I remembered what I heard just a few days ago from a friend: “The trouble with exclusive communities is that they tend to die out over time.”

What was so very exciting is that I have had lots of thoughts about the teaching of mathematics, and about making it accessible – and I’ve held many a minor soap box lecture on this topic among my friends – but I’ve never really spoken to professors or graduate students about it or heard them speak on it – so to suddenly hear some of my own arguments coming from other people’s lips was so very exciting. And I spoke up, too. Sometimes it’s difficult to fathom doing anything except that which is expected of me in my classes, but I do think I might want to start being a bit more active in this whole field. Who knows. Maybe the local math department magazine needs another editor.

This is Why

This story comes right at the end of the one I just told. After the colloquium I was heading into town to meet up with some friends for Pub Quiz, our weekly tradition. And as I was in the bus, I was standing there, holding on to the rail to keep from falling over and all of a sudden, I felt like I had a very specific type of x-ray sense. I looked at my feet on the floor, shifting to keep me from falling over as the bus turned, and I felt like I could see through to the wheel and axles and mechanics of the whole bus – I saw signs flashing past on the road and thought about the machines that made them – my music came through my headphones and I saw the chords in my mind like graphs, the harmonies corresponding with beautiful visual symmetry. And I just thought – this is why I study math. Because the world is math. The language of the universe. Yes, I felt like a nerd – but I also felt like I was flying. I’ll take the one to have the other.

That Small Town Feel

At Pub Quiz, one of the waitresses who usually works Thursday nights came over to our table and was taking our order – and as she brought our food and drinks a bit later, she said to me, “Did you cut your hair?” – I was so surprised I couldn’t stop smiling! I’d been recognized, without ever realizing that I’d been noticed in the first place. And then the next morning, I went to the bouldering hall where I’ve been going climbing for the past several months – and the woman who checked me in said, “You DID cut your hair! I saw you in the bus yesterday but I wasn’t sure if it was you, so I didn’t say anything! It looks great!” I never realized . And it’s not that I think I’m incredibly important – or indeed at all important – to any of these people. But it just enhances the feeling of belonging in a place that is just so, so lovely.

"You're not doing it right!"

One of my favorite comedians for the past few years has been a woman named Maria Bamford. I'm not sure if I've written about her here before or if you already know who she is - but I highly recommend her. The kind of off-the-wall, makes-you-cringe-while-you-are-crying-from-laughter comedy of hers is, I think, simply brilliant. I've rarely seen something so original that not only makes me laugh so much that I can't breathe and also touches on subjects that so desperately need it.

One of my favorite sketches of hers is from a particular show, the 'Maria Bamford's One-Hour Homemade Christmas Stand-Up Special' (in which she is in fact sitting down on a couch with her two pugs). One of the main topics she discusses in her sketches is mental illness – something she's struggled with a lot and thinks deserves some more attention – even comedic attention! n this particular sketch, she discusses some of the woes of going to a therapist. This particular therapist of hers insists on getting Maria to follow her rather irrational worries to their rather crazy conclusions – just to prove a point, to make Maria 'face her fears'. For example, Maria says she is worried about making eye contact with people, so her therapist asks the following questions and Maria responds:

(Keep in mind, everyone, that she is doing both voices – and spectacularly so.)

"Maria, what would happen if you made eye contact with someone?"
--"I dunno, probably genocide 'em."
"And then what would happen?"
--"Uhm...I'd go to genocide jail."
"And then what would happen?"
--"Pff...probably genocide everyone in the jail."
"And then what would happen?"
--"...they'd have to find a super-strong jail..."
"And then what would happen?"
--"I DON'T KNOW- you're not doing it right!"

So, I have to write a bachelor's thesis this semester. I'm writing about a topic that is, at best, just barely barely within my grasp as far as understanding goes. Like, I can feel it brushing the tips of my fingers. But I have not grasped it yet. Not even close. I'm also going to be a teaching assistant for a course this semester -something I did and loved at Mills but am a little nervous about doing in Mainz.  Also I have normal classes to attend. Plus, you know, run-of-the-mill "what am I going to do with my life" thoughts. So, as might be expected, I sometimes catch myself freaking out and spiraling into paralysis about all the things I need to do. And sometimes, when I catch myself going into this, I have to ask myself the same questions – or just, that one particular question over and over.

"And then what would happen?"

And luckily, this merry-go-round of questions, me pestering myself about just what would happen if everything I feared happened–- it usually ends with me realizing that the paper is after all, just a paper – the T.A.-ing job just a semester-long job, and in fact, the classes are just classes and people fail classes and are okay -- and after all, I have a family and friends who would greet me with cookies and hugs no matter what happens. So, I suppose things will be alright. Now, I just need to get started!

P.S.: If you feel like seeing some of her stuff, check out this link right here.

All Shapes and Sizes

Let's try not to be so linear. We tend to think in a very Euclidean sense (and why shouldn't we?) when we think about measuring space and objects – and not only that, we also tend to think rather continuously, as opposed to discretely. However, measuring things can actually be done in a variety of ways. Let me explain what I mean.

First of all, I'm currently studying for a measure theory final, which might explain why any of these things are in my head in the first place.

What is measure theory anyway? Basically, a lot of measure theory deals with the measurement of objects – some of which we already how to measure. I can ask you to find the length of a line (or an interval, like [1,5]) or the area of a square or even the surface of a sphere (though I recently completely forgot the formula for exactly that calculation at a pub quiz a few weeks ago – quite embarrassing). If you've had some calculus, you also know how to measure the area under a curve by using an integral. And we also know how to calculate the volume of things like spheres, cubes, or other shapes.

As you might have noticed (though we don't always say it so directly) all of these measurements have an inherent dimension. 'Length' is always one-dimensional, 'area' two-dimensional etc.  Obviously. We just say 'length of a line' instead of saying 'the default one-dimensional measure of a line' because, since it is the default, we don't think that there could be other ways to measure such a thing. Measure theory looks at some of these other ways. But I'm getting a little ahead of myself.

I mentioned earlier that we also tend to think continuously as opposed to discretely. Things that are discrete are a little bit different. For example, if we consider the interval [0,1], which includes all numbers between zero and one, that's a continuous chunk of numbers. If you were to draw it, intuitively, you would not pick up your pencil or pen – you would just trace along a number line from zero to one, hitting all the elements in that interval. However, if we think about the set {0,1} (notation is very important in math – [0,1], (0,1) and {0,1} all mean different things!) – then, the only things in that set are the number zero and the number one, nothing in between. Two unconnected dots on a number line. A discrete set.

And what measure theory tries to do is to measure things like these sets. But imagine trying to use that intuitive 'length' notion to measure something that is discrete – something whose elements are separated by space, even if it's a very small amount of space. How does that even work? Does it even make sense to try to apply that definition? Not really.

We need a new idea of measurement.

So, that's some of the fun stuff we've been getting up to in the lecture this semester. If you'll let me use as a premise that we do have a tool (several, in fact) that can handle discrete inputs and measure them – then we can come back to the idea of dimension. What if you're trotting down a road in the mathematical world and you come upon a set Ω. Very startled, you want to know some things about it – its measure, for instance. But we know nothing about its contents. They could be related to a function or an algorithm. They could be points, vectors, or even  – gasp! – nxn-dimensional matrices. We really don't know much. So, how do we find out what dimension it has? 

What are some things that could go wrong here? I claim there are several ways to get the wrong impression. What if this set you happen to meet is actually just a kind little square, but you don't know that, and mistakenly, you try to measure it with a three-dimensional measure, like volume. What happens if you try to figure out the volume of a square? Well, you get zero. Just like if I tried to measure the length of a single point. If my object is n-dimensional and I try to measure it with a measure any bigger than n (in this example, n+1), then I get zero.

And what if I make a mistake the other way around. What if the set I encounter happens to actually be a cube and I try to calculate its area. I don't just mean the surface area – the surface is two-dimensional, that wouldn't give us any trouble. But the area of the entire cube? That would have to be infinite.

So, let's get back to that road in the mathematical world and back to Ω. You want to figure out what its measure is – but in order to do that accurately, you have to figure out what dimension it has.

I'm skating over a few technical details here, but basically, in one area of measure theory, we talk about something called the Hausdorff measure. (Hausdorff was, as you may have guessed, some old math guy. He actually did quite a lot of cool things.) The definition of the Hausdorff measure is pretty hairy – it looks a bit like this:

(If I have any math critics reading this, I realize this is the delta-approximative, s-dimensional Hausdorff measure – I thought it was the prettiest formula to show here. The actual formula for the Hausdorff measure hides all the details by taking the limit.)

And just like with our intuitive standard ways of measuring things – like length and area and volume – there are different versions of this measure in different dimensions. And there is also something called the Hausdorff dimension. The Hausdorff dimension of a set is simply the dimension that is 'correct' – which is to say, if I try to measure that set with a dimension that is higher than the Hausdorff dimension, I'll get zero for the measure (like measuring a square's volume) and if I measure it with one that is lower than the Hausdorff dimension, I get infinity (like the area of a cube). So, to figure out what dimension that set has that you see walking towards you on a lonely road, simply measure it with a few different dimensions and see where the jump occurs. That's it's Hausdorff dimension.

Many of you will say: "But when will I ever be walking down a lonely mathematical road? When will I ever need any of this?" My answer: "Yes. But... isn't it cool?"

Ponderings from the Internet Age

I'm not sure whether the thought I am thinking now is well formed enough for me to explain it, but I'm going to give it a try. I just sat down with a nice cup of jasmine tea and was about to do some proofreading for work when, as is so oft the case, I decided to briefly take a glance at my Facebook.

Now, I have had an off-and-on relationship with Facebook for years. I was one of the first people in my age group back in middle school to get a Facebook account, at a time when nearly all of the people on the site were in college -- but back then, I was taking a lot of dance classes and performances with the Allegheny College Dance Department, so many of my friends were that age. Then, when I did my year abroad in high school, it was a great way to keep in touch with folks, and to share parts of my experience with large groups of my friends or acquaintances at once. Then, after I got back, there was about a year in college when I simply deactivated my account, not liking the "timesuck" that it was.

But now, after this and that and oh so many awkward moments of "You don't have a Facebook? I guess I can email you, but I never check my email..." and simply wanting to see the pictures of my friends on vacation and the children of extended family - well, I'm back on it again. But I've noticed a new trend in my news feed of late (that is to say, the past year or so) that I find troubling.

First off, a few caveats. 1. The things I am about to list may be specific to those people I happen to be "friends" with on Facebook. 2. I may be extremely sensitive and pick up on things no one else would think were an issue. 3. Well, I'm sure there are more. I'm just going to tell you what I think anyway.

There seems to be a big wave of negativity these days. I don't just mean that bad things are happening in the world and people are upset about them, but rather there is a desire to post about things that make people upset, hoping to have their outrage validated by others. For example, there are often internet articles that are fads; they go viral and zoom around from one person to another over the course of a few days and get a lot of "buzz". Now, most mainstream "click me" articles, even if they're supposed to "make you cry" or assure you "you won't believe what happens next" -  well, they are frequently problematic. Whether it's racist, sexist, classist, whichever-ist - there are usually some problematic undertones. Sometimes they are obvious, sometimes you really have to dig for them -- and someone will. And I think there's a difference between commenting when someone else posts said article and saying (KINDLY) "I think this article has a few issues because...." and between posting that article (as many of my Facebook seem to acquaintances do) simply out of bile. Just so they can say "Take a look at this disgusting exhibit of the patriarchy" - or whichever the enemy du jour is.

I don't know why this bothers me so much - but I find that every time I read over my news feed, I end up unhappy. And I feel there's a difference between responding truthfully and critically to other people and the world, and bringing up something you are angry about merely out of spite, encouraging more anger, hoping for that validation... I just feel it multiplies. It makes me not want to look at Facebook anymore, but maybe that's not such a bad thing.

A few more caveats at the end - I am not advocating pretending everything is jolly and ignoring all the bad things in the world - absolutely not. Also, it's quite possible that after having attended an extremely liberal women's college in the Bay Area, I have an above average number of vocal, angry activists on my news feed.

If you are at all interested in a discussion of similar topics in the form of a podcast, I highly recommend going to this webpage and clicking on the podcast called 'Our Computers, Ourselves'.  It's a great listen - talks of technology and how we feel about it and what we do to it vs. what it does to us. Go for it. :)