Teaching 9th graders operations research

Our two-week project with Upward Bound came to a close yesterday. I've been working with Soumia Ichoua on her NSF-supported project to try to interest high schoolers in the fun side of math.

More to come after we've analyzed the assessment results and put the paper together.

Acknowledgements

Financial support for this work was provided by the National Science Foundation (NSF) through grant number 0927129 (Ichoua). This support is gratefully acknowledged. 

Getting to Expression

 Barbara Fister's "Why the 'Research Paper' Isn't Working" has some interesting observations about the teaching and assessment of composition, and I made the connection to the deductive/inductive divide I've been going on about lately. Let me reframe the latter as the "language/expression" divide as a preface:

• A language is a set of knowledge that usually comprises vocabulary, methods, reference points of common knowledge, and a web of connections between concepts. Understanding a language is always a prerequisite to being able to produce it intelligently.
  
• Expression is the illumination of new ideas, new connections, creation of new parts of the language to contribute to the existing corpus. It is realized with different styles in varying degrees of fluency, and allows the display of insights or brilliance.

This is the analytical/deductive vs. creative/inductive divide that I've blogged about before, for example in the previous post. We see this division everywhere. Learning how to use paint versus expressing yourself in the medium. 

A "life-long" learner must either become used to learning new languages all the time or else not plan to live long. My 'stack' of languages to learn currently includes R programming, German, and photography. I think of it in over-generalized terms as a progression from confusion to understanding to expression. I'll come back to that idea.

The "photography language" is one I dabbled in when it meant smelly chemicals and a long time between when you shot a photo and when you got to look at it. Nowadays digital photography obviates many of the skills that one needed, and it does something else very important.  It's not any easier to do digital photography--you have to master software instead of stop bath--but it's so much quicker to get from the snap to the view that you can learn from trial and error in real time. This is a huge advantage. Conservatively, the gap between taking a shot on film and holding a print in your hands is at minimum several hours (leaving aside Poloroids or other quickie formats). With digital it's a matter of seconds. So learning the language through sheer trial and error has been accelerated by a factor of, say 2 hrs/2 seconds = 3600. 

Photo: David Eubanks, some rights reserved

Different learners approach learning language in different ways. Some people like to read all the manuals first, and others start pushing buttons. My wife (laughing at the end of a long work day, above) learned Italian by working all the exercises in two textbooks and then spending a month in Italy. I struggle along with German because I don't have the patience to memorize vocabulary. I try to bridge the confusion/understanding divide by reading novels translated into German (it makes the language much simpler), and look up words that come up frequently enough. Her way is much more efficient than mine.

So, in this epistemological vivisection of learning, the challenge for faculty is to teach and assess the crossing of two metaphorical bridges:

Land o' confusion -> Understanding -> Expression

In "Complexity as Pedagogy" I showed how it's possible to take a very narrow road straight to Expression. That is, one can encapsulate a small part of the language and use it to get right to the fun part. Because, let's face it, creating is fun! And if anything distinguishes humans from the rest of the biological kingdom, it's our blabbing--we like to talk.

An art professor once told me how to learn to draw. He said, just draw your hand over and over again in different positions. After about 500 times, you should be pretty good at it. I don't know if he was joking or not, but this is an example of simplifying the language to the point where you can quickly become expressive.

The practice of assessment should be very different across this divide. Testing language fluency can take many forms, but it's always about correctness, speed, conformity to convention, and so on. One is not supposed to be creative on a spelling test. Otherwise I would have gotten better grades in grade school. Similarly, we're not suppose to invent better names for state capitals for that test, or help the Germans organize the genders of their nouns better. 

Assessing language seems easy because of this necessary emphasis on mastering form. Vocabulary tests, concept inventories, and the like are easily administered, and even testing understanding of subtle connections through the use of the language itself is straightforward. 

Example: In teaching logic, it's simple to write down a logical argument and ask students to justify each step with an axiom or theorem, or even let them find errors with the proof. The only way students can be successful is if they have a good understanding of the language.

This ease of assessment is a bane, and a great peril to learning. Let me finally get to the points I liked about the article I cited way back at the beginning of this piece. Starting with the idea of forcing students to master arcane rules of correct citations, the author notes more broadly that 

I have long agreed with Richard Larson who wrote way back in 1982 that the research paper as taught in college is an artificial genre, one that works at cross-purposes to actually developing respect for evidence-based reasoning, a measured appreciation for negotiating ideas that are in conflict, or original thought.

An artificial genre that is at cross-purposes with original thought. That's pretty damning. But it's these very mechanics of any language that are easily defined, easy to get agreement on, and easy to assess. It's a quick slide down the slope to standardization of a form that becomes inimical to the actual intent of the enterprise! This happens all over the place. Whole subjects taught in school exist only because of such inertia, like Geometry in high school--there's no reason kids should be learning plane geometry with rulers and protractors in this day and age, but it's been so deeply standardized that it's become part of the culture. But I digress.

Barbara goes on to illustrate the point with a fascinating example of how students react to the low-complexity standard we've set institutionally:

I hate it when students who have hit on a novel and interesting way of looking at an issue tell me they have to change their topic because they can’t find sources that say exactly what they plan to say. I try to persuade them otherwise, but they believe that original ideas are not allowed in “research.” How messed up is that? The other and, sadly, more frequent reference desk winch-making moment involves a student needing help finding sources for a paper he’s already written. Most commonly, students pull together a bunch of sources, many of which they barely understand on a topic they know little about, and do their best to mash the contents up into the required number of pages.

Does this sound like a road map from Confusion to Expression? It doesn't to me--it sounds like a Skinner Box: mash the button to get the food pellet.

It shouldn't be hard to fix this problem. That's the good news. The recipe is simple:

1. Focus the language to a small useful subset. In terms of composition, it would mean picking a topic that's narrow enough to actually learn something about quickly.
   
2. Demonstrate and assess--with feedback!--fluency in this new language. Have conversations about what confusion levels, what you know you know, and what you know you don't know. There are lots of creative ways to organize this with mind maps and such, and it also can be fodder for oral presentations, or other engagement activities. Develop fluency in real time.
   
3. Emphasize expression and creativity over form as far as it can be pushed. This isn't always possible, e.g. in logic--you have to be 100% correct--which is why the focus is so important. If you have to absolutely master some topic in order to be creative, make it a small one. 

Note that I am not advocating "free form creativity" devoid of any content or ultimate value. This might be fun for the students, but I don't see how it accomplishes any useful learning objectives. But there's a lot of road between many of our current practices and goofing off in the name of creativity. 

Barbara's ending paragraph is apposite:

But if you want first year college students to understand what sources are for and why they matter, if you want them to develop curiosity and respect for evidence, your best bet is to start by tossing that generic research paper. As for those who will complain that students should have learned how to paraphrase and cite sources in their first semester – we’ve tried to do that for decades, and it hasn’t worked yet. Isn’t it time to try something else?

Yes. 

Creativity as the Pinnacle of Learning

In a recent post on the topic of memory, I noted that this skill was at the base of the revised Bloom's Taxonomy (or taxidermy?). This morning I woke up thinking about the other end: creativity. I've mused about the role of creativity before, and how to teach and it (here and here).

It's easy to make the unwarranted leap from creativity to aesthetics, because we associate art justifiably with a creative process. But I prefer to think of creativity as the production of new knowledge in any context. Let me give a pedantic example:

All men are mortal.
Socrates is a man.
------------------
Socrates is mortal.

The conclusion follows deductively from the two statements above it, so it is not the production of new knowledge. This is the hammer that fell on Bertrand Russell in his quest for ultimate true by means of logic. Logical, rigorous deductive thinking is an essential skill, but it's not creative. In contrast, Aristotle's encoding of logic into language was creative, but I have a more interesting example.

The other day I saw an interesting problem posted on the math subreddit. The diagram below shows a laser beam coming from the right and striking a perfect mirror (the thick black line at the bottom) at angle b, bouncing off, and then striking another mirror placed at angle a with respect to the first one:

The gray line is imaginary here--it extends the line of the bottom mirror to illustrate angle b, but the light hits the very end of the mirror itself. The light will continue to bounce off the mirrors in some way. Where does the beam end up going? I will put the solution at the end, in case you want to think about it first.

The point is that I suspected there should be an elegant way to think about the problem, where the solution--all solutions--would be obvious. So I cast about, looking for it. This is rather like trying to find the light switch in a dark room, as Andrew Wiles put it (see my other posts on creativity for the link). I gave up before I found it. I found an inelegant solution, which was correct, but wasn't creative enough to be called elegant. It was sort of a plodding, "add up the cumulative effect" solution, where you sort of crush a problem with the weight of logical facts until it leaks out its secrets like a garlic clove exudes oil.

My lack of blazing imagination does, however, illustrate that the creative process itself deserves its own "taxonomy." In other words, there are qualitative differences to creative enterprise. Let's take a look.

Creativity as producing new information can start with sheer randomness. Flipping a coin and writing down the results is creative. This sounds too trivial to be counted, but it's not. In fact, it's the singular most important spark of novelty in history. I have two examples. First, physicists have wondered how galaxies formed. If the big bang started from a single point, for example, why wasn't everything thereafter perfectly uniform? Where did all the novelty come from? One proposal is that tiny differences in the primordial universe were seeded by quantum events, which we know to be deeply random. So the largest structures in the universe may have started with infinitesimal randomness. Cool, right?

The second example is the evolution of life, which explores via an ecology a vast space of possible designs for living things. This exploration proceeds by random mutation of genes, and other ways in which genetic material may get mixed around (like parasitism), or a bacteria's lascivious lifestyle with regard to DNA. This is not the deeply puzzling randomness of quantum mechanics, but the sort that emerges from complex systems that is sometimes called chaos.

Randomness is a great entry point into creative thinking. The casting about for novelty is a skill in itself. It requires courage to be wrong, a good idea of how to recognize your intellectual quarry when you've found it, and determination--because it takes a long time for randomness to hit the right target. Louis Pasteur's "Chance favors the prepared mind." has two parts: chance, and preparation. The latter is a formed in the laborious mastering of some discipline or subject.

The whole idea of serendiptipy is based on these two elements, and our culture has benefited handsomely from it: rubber, penicillin, radioactivity, and many more are on the list. Wikipedia has many examples here.

In the last post, I showed an example of a game designed for high schoolers that is aimed at creative thinking in a mathematical context. The essential skills are being able to understand the problem and do basic math (easy), and cast about for creative solutions (fun, I hope). These solutions will start with guessing.

Guessing is a step up from randomness. Humans aren't very good at true randomness--we have to depend on the world around us for that, like moldy bread crumbs falling accidentally into a Petri dish. I suspect that good guessing is an art unto itself, and that it can be taught and practiced. There's an MIT course on the art of making educated guesses with regard to estimation (how many gas stations are in the US, do you think?). Here's the course description from the Open Courseware site (it's free!).

This course teaches the art of guessing results and solving problems without doing a proof or an exact calculation. Techniques include extreme-cases reasoning, dimensional analysis, successive approximation, discretization, generalization, and pictorial analysis. Applications include mental calculation, solid geometry, musical intervals, logarithms, integration, infinite series, solitaire, and differential equations.

This is targeted at students with a good math foundation (everybody at MIT, I guess), but I find it exciting because it shows how to teach a whole course on guessing in the context of a discipline. There's no reason that this couldn't be done in other subjects just as well. Guess-and-check is a fundamental human skill that reinforced our knowledge of the world. Think about kids and the funny way they conjugate verbs at first because they are guessing based on simple rules (e.g "I eated my peas, daddy"). The guess is close enough to communicate, and as an additional reward, they glean information about new complexities of language, if someone is kind enough to point out the right way of saying it.

Problem-Solving might be the next step in the creative chain of being. This is a natural continuation of randomness and guessing, which results in the production of new knowledge in some applied context. This works in art as well as math, I think. It's the evolution from random doodles to purposeful artistic creation. Problem solving weds the analytical/deductive process, discipline-specific skills and knowledge with the trial-and error process that I've described in prior posts on creativity. This is the nuts and bolts of creative production.

Inspiration may or may not be teachable. If we help students to be good seekers of randomness, good guessers, and good problem-solvers, can we help them elevate themselves to inspired thought? I don't know, but I guess that we can provide a fertile environment for this, and foster it in individuals who might otherwise have not reached their potential. I don't really believe that we can take every math student and produce another Gauss or Euler, but we can ameliorate one of the great hidden human tragedies--the many, many inspired thinkers who never got the intellectual cultivation they needed to allow their talents to flower.

This is all first-draft thinking. An interested group of discipline experts could turn these rough ideas into something applicable to a curriculum or institution. To include ways to assess creativity at each step along the way. Disciplines can learn from each other and share approaches, opening up the possibility of interdisciplinary learning. I often though that the math students could benefit from watching art students critically review each others' work.

---

Here's the solution to the problem. I have redrawn it, but I saw it first here. The original problem was in terms of a tiny billiard ball, but I changed it to a laser beam. The key insight is that reflections preserve angles, so that instead of imagining the beam bouncing off at the same angle (incidence = reflection), imagine it passing through the mirror as if it were a pane of glass. Then add another pane of glass where the return bounce would have occurred, so that copies of the mirrors look like spokes on a wheel separated by angle a. This illustrates clearly that the beam will swiftly exit the mirrors and go on its way in most arrangements. The whole process is laid bare. I've illustrated it with a=45 degrees below. This is an inspired and elegant solution, unlike my workable but problem-solving brute force approach (not shown).

Education and Creativity

Assessing Creativity

A piece in the New York Times addresses the assessment of creativity, and has some interesting bits.  Quoting "Rex Jung, a research scientist at the Mind Research Network in Albuquerque:"

One study of 65 subjects suggests that creativity prefers to take a slower, more meandering path than intelligence.  “The brain appears to be an efficient superhighway that gets you from Point A to Point B” when it comes to intelligence, Dr. Jung explained. “But in the regions of the brain related to creativity, there appears to be lots of little side roads with interesting detours, and meandering little byways.” 

 This sounds like the diagram I use with students, to explain how to do proofs or other creative math work.The task is to go from some question (?) to a resolution (!) through structured inquiry.  Which is a fancy way to say trial and error.  Read the diagram from left to right.

Each of the straight segments is an attempt to solve the problem.  The way I've drawn it, the first five attempts don't work, but give some hint about the real solution.  It might not be so straightforward, of course.  Real problems might look more like this:

Here, multiple lines of inquiry finally meet up in an Ah-Ha! moment to provide the solution. Andrew Wiles gives a nice exposition of this process in The Proof, the Nova show on how he proved Fermat's Last Theorem. His analogy about creative exploration is of exploring a darkened room.  You can see a minute of it on YouTube here.  The whole show is highly recommended.  It's powerful.

Note that in order to do the creative exercise, one has to be able do the deductive part of the work.  This entails knowing how to write and think in the target discipline, and knowing the tools available.  Without this background, it's impossible to make even reasonable guesses, or to check your work for logic errors.  Checking solutions is generally much easier than finding them to begin with. 

There are other points of view expressed in the article.  For example:

According to Kenneth Heilman, a neurologist at the University of Florida and the author of “Creativity and the Brain” (2005), creativity not only involves coming up with something new, but also with shutting down the brain’s habitual response, or letting go of conventional solutions. 

This resonates as well.  Trying the same thing over and over and failing isn't productive, and is part of the reason math proofs are hard for students to learn.  Initially they have a limited number of ways of looking at a problem, so are constrained in what they can attempt.  Maybe the lesson there is to train them early to ask the question "what's another way to look at this?"  That may be the heart of creativity.

How could we assess this in practice?  In my last post I mentioned that the visual and performing arts faculty are interested not only in the final product of a creative work, but also the evolution of thinking that went into making it.  This may include a portfolio of intermediate work, for example, something like the lines on the schematic above.  "How many ways can you look at this?" could be an interesting question to routinely ask.  There are established theories for many areas, and these can be used as lenses.  In the end, creativity may be more about the process of creation than the final, visible result.  This attitude could have an impact on the way we teach technical subjects too: given a problem to address, list all the approaches you can think of before trying to solve it. This meta-inquiry leads one to seek out new sources of inspiration.  One time when I was stuck on a math problem I went to an art show and found what I was looking for.  This sort of determined curiosity may not be valued as much as it should be because it may be under the radar, so to speak.  In my experience, we tend to expect curiosity and creativity to just bloom like spring flowers, but paying more attention to the details could help a great deal.

As evidence that the "many views" approach is useful, I submit the Polymath Project by Tim Gowers. You can read a great account of it in the Science News article "Mathematics by Collaboration."  Here's the setup:

Late last January, University of Cambridge mathematician Tim Gowers decided to run a little experiment. Was it possible, he wondered, for a large number of mathematicians to collaborate openly on the Internet, pooling their ideas around a single problem? If it were possible, would it be easier, more efficient, more fun? Could the mathematicians together solve a problem they might not be able to solve individually?

The answer is yes!  This hints at another possible way to look at creativity--if multiple views are important, then collaboration can be a good thing.  I've experimented with this by allowing students to work together on exams.  It depends on the class, but it can work very well.  They tend to sort themselves into groups by ability level, which is interesting.  It also might lead us to question the whole idea of assessing skills.  What if a person has the skill of being a really good collaborator, taking others' ideas and stitching them together in novel ways?  In the usual way courses are run, this could easily be called cheating.  And since assessments of intellectual ability are focused on students as individuals, such traits wouldn't be assessed, and probably wouldn't be valued.  Interesting questions.  Why don't we focus more on group problem-solving or creation, develop techniques for organizing and documenting it, and find ways to assess it?  Maybe this is already being done somewhere?

Grading and Assessment

The topic of how grades and assessments could be aligned has been mentioned here before (see "Assessing away Grades"). Nils Peterson pointed me to an active discussion on the topic on HASTAC called "Grading 2.0: Evaluation in the Digital Age." There are some good links there, topic questions, and several comments. One of the discussion points asks:

3. Can everything be graded?
- How important is creativity, and how do we deal with subjective concepts in an objective way, in evaluation?

Here, I think we run smack into the problem. It goes like this:

1. Grades have economic consequences for both students and teachers.
2. Because of this, grades have to be defensible during a challenge and review process.
   
3. Because they have to be defensible, grades have to have at least the appearance of objectivity.
4. However: the best assessments should be free from economic influence, and may be subjective (see the whole Assessing the Elephant thing).

This problem only rears its head for complex learning outcomes. If you're teaching multiplication tables, it's not a problem to create objective (maybe even valid and reliable) instruments. What about creativity, however, as posed in the question above? Can we really slice up that concept into "dimensions" and rubrics that capture the essence of what creative genius produces?

It's instructive to see how grades are assigned in the fine arts and performance arts. There simply is a lot of subjectivity. I'm generalizing from limited experience, but I think that the key is the attitudes and methods the assessor uses more than the actual assessment. For example, if I say "your work is all derivative and boring," it's very different from saying "to my taste, this doesn't excite me." The former sounds like an objective statement, and the later is clearly subjective. Students aren't stupid, and they know that there's a difference: dressing up subjectivity as objectivity only irritates them. What I've seen from successful art-type assessments is that the effort put into the work counts for a great deal. Creativity is a kind of exploration, perhaps, requiring trial and error, and therefore time invested. Art profs want to see portfolios, sketchbooks, incomplete works, anything that shows that the student is engaged. And the judgment of how much work one has done can be fairly objective; it's something you can talk to the student about and reach agreement on. Of course, the quality of the engagement counts as well, but to some extent I think that comes out also, if one can review all the cars in the whole train of thought. This is certainly true of teaching math at the upper levels. It's a thrill when a student comes to you with "I thought of this problem and tried to solve it. Will you look at it?" Whereupon you're presented with scruffy bits of paper (mathematicians will write on anything) with formulas all over them. Most of them are wrong--false starts. It's like what one of my art colleagues described looking at sketchbooks is like: it's raw and unprocessed, and more powerful than a finished work.

So it may be that there is a natural division between objective and subjective assessments and grades. The former are relatively easy. But maybe for more complex outcomes we need an approach more like that of art: look at not just a finished product on a test or paper, but demand to see the corpus of work, mistakes and all, that led to it. Technology can obviously help with this because information is cheap to store "forever." Portfolio systems as they are generally currently conceived are not really the right tool for this--what you'd want is a virtual space for storing documents and imposing a bit of structure on them. Perhaps a mind-map hyperlinked to documents and meta-data tags on the whole thing, so it can be sorted and presented by different facets. Add the ability for an instructor to freely annotate these nodes and artifacts with hidable notes, and it starts to sound attractive. At any rate, this is not the kind of problem that another scoopful of rubrics can solve.

Evolutionary Thinking

Any reader of my blog knows how I advertise the evolutionary approach to solving hard problems, but what does that mean? I have a prime example to trot out, something I came across on Reddit yesterday. The article is "The story of the Gömböc," which may not sound promising, but trust me. I dare not spoil this tale of discovery by condensing it into some trivial didactic--it's worth reading it its entirety. Commentary after the break.

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This story has potential for all kinds of reflection on the practice of discovery. One could mention the noncognitives--trying every darned thing one could think of without giving up, for example, the willingness to be wrong and learn from it (accurate self-reflection). Or we could talk about the concept of critical thinking as iterating loops of analysis (testing a theory or evidence) versus creativity (finding theory or evidence). It's a marvelous example of critical thinking.

But I'd rather muse about what practical lessons can be drawn about the evolutionary approach. How can you actually do it? Suppose you have a tough problem at hand. How to assess higher order thinking skills in a discipline or how to get better evaluations of teaching effectiveness, or how to redesign general education? What evolutionary principles can potentially help find a solution? I have a few to suggest, and I'm sure you can think of your own (drop a comment if you do!).

First, maybe it's already been solved. If the problem has been around for a while, it's a good bet someone else has thought of it. I'm working on a research problem using a computational model for living things, and ran into a communications problem that looks like the sort of thing NASA would have to deal with. So I emailed yesterday to see if someone has already solved it. The internet is obviously good for that sort of thing. I don't know how many times I've wished I had some simple software tool, and two minutes later discovered that someone had created it and released it into public domain.

Well, if it hasn't been solved satisfactorily, what next? There is no magic evolutionary wand that makes the work go away. Think of picking over 10,000 stones on the beach looking for the right one. But the process can be described very easily.

First, you have to know what you want because you have to be able to identify it when you find it. That is, you don't need to know the form of the solution to your problem (otherwise you already have the answer), but you do need to be able to recognize it as a solution. You may not know off the top of your head what the prime factors of 1,010,299 are, but you can easily verify them once you're told (911 x 1109). This is the analytical part. In natural selection this is done by survival and reproduction. Critters that don't pass their genes on aren't solutions to the problem. For something fuzzy like identifying a good method of evaluating teaching, this is going to be hard to come to grips with. But setting out looking for a solution without knowing how to tell if you find one is a waste of time.

So you could, for example, decide that faculty confidence is the most important thing for teaching evaluations. Then the challenge would be to come up with some way to assess that--probably involving asking them what they thought of this or that method. If this sounds political, it's because it's a problem laced with politics.

With a tool for evolutionary fitness in hand, now you have to find a way to generate potential solutions. This is the creative part. My recent post on brainstorming quotes research saying this step is best not done in groups. But a group is good at weeding through the ideas. In the scenario I described (evaluating teaching), if faculty confidence is the key then having a test group to try out ideas against would be useful--to weed out ones they don't like. I think it's obvious that this weeding group should be different from the idea-production group; people are often too attached to their own ideas to be objective. So two groups: a creative one for idea production. They may or may not have to meet as a group. Then an analytical group for weeding out non-solutions. They have to be very clear as to what a solution looks like. Add a bit of salt and iterate and you have an evolutionary process.

In a nutshell, it sounds very simple. Generate lots of ideas and be ruthless about weeding them out according to your clearly-defined measure of success. Putting that into practice is complicated, however, and some level of formalization could help. Good leadership is a must.

Think about all the committee meetings you've sat in that wrestled with tough problems like these, but without a plan or clear notion of success, and almost certainly without a wall between idea production and idea deletion. How much time was wasted?

No system is perfect, and there are game-theory short circuits to the method I've described too. Without good leadership, the group responsible for judging success (killing bad ideas, keeping good ones) can have too great an influence on guiding where the investigation leads. The conversation can go like this (C = creative group, A = analytic (weeding) group):

C: Here's our ideas so far.
A: Nope. They all stink.
C: (Frustrated) Then what DO you want?
A: Well if you come back with this or that, we might be interested.

This becomes the equivalent of "teaching to the test," and subverts the process. On the other hand, constructive feedback is good:

C: What did you think of idea X?
A: It comes close--but the committee saw Y as a problem. Is it possible to create some variations?

The difference is subtle.

The evolutionary process for decision-making I've described above is theoretical and idealized. In practice it will always involve compromises. Deadlines will pressure incomplete solutions. There may not BE a solution, or it may be impractical. So treat this as a tool in your tool bag and not a panecea for all big problems. After all, natural selection got some things wrong too--I don't have the topological properties needed to stay upright when I fall asleep in long useless meetings, a useful trait indeed (gives me an idea for a comic strip).