Is it a bad sign that I'm already fantasizing about dropping out and starting a restaurant or bakery? On my bookshelf, waiting to be read cover-to-cover, is a copy of Culinary Artistry, which is about cheffing. Dorenburg and Page distinguish between three kinds of cooking:
- "Cooking as a trade", where your primary goal is sustenance, and with your limited repertoire you're hoping your customers go away thinking "I'm full."
- "Cooking as craft", the style promulgated in the greatest cookbooks, has its main goal enjoyment; a chef should have a wide repertoire of classic dishes, and hope that the customers go away thinking "That was delicious."
- In "cooking as art", on the other hand, a chef prepares her own recipes, and any given night the menu will be very limited; customers should be entertained, and go away thinking "Life is beautiful."
This trade/craft/art distinction is useful in other disciplines. Everyone should be able to (but many can't) do mathematics at the most basic of trade levels — monetary arithmetic, being duly suspicious of newspaper statistics, etc. It bears remembering that tradesmen often have incredible skill: Alice Waters a few years ago left the artistry of Chez Panisse a few years ago to try to reform the public school food; she is now cooking trades food for the masses. I wonder whether an actuary considers her mathematics to be trade, craft, or art.
I feel like the majority of expository mathematics writing, and the entirety of an undergraduate math major curriculum, falls under "mathematics as craft": classic results presented (hopefully) well. There is certainly an artistry to teaching well, but the teacher-as-artist focuses on the delivery, not the mathematics itself. I have not yet tried serious mathematics research; I know I am excited by artistic mathematics, but I also know that I adore teaching, and I have often comforted myself with the reminder that, if research is not for me, I can have a good life teaching at a liberal arts college.
To do original, beautiful research, however, requires the creativity of an artist (and lots of crafts- and tradesmanship). A calculation can prove a new theorem, just like a recipe can create a new meal; a few geniuses create new recipes, new fields, new mathematical insights. If I can learn to be a mathematician-as-artist, then I will stay in research.
An advanced social dancer is an artist, although often not a great trades- or craftsman. An advanced ballerina is a virtuosic tradesman. When I go to Pilates, I am engaging in movement-as-craft.
Chez Panisse has a fixed menu every night: Mondays are $55 + drinks + 17% tip + 8.75% tax / person, and the price increases through the week. Other great restaurants also have small, constantly changing menus. Moosewood offers a limited, changing vegetarian menu every night: your choice of three or four entrées, a couple salads, etc. In both cases, recipes are original, based on seasonal organic ingredients and the tastes of the chef.
If I were to start a restaurant, I'd want it to be like the Moosewood. Cooking as craft does not excite me. I like knowing how to make all sorts of dishes, of course, because I like to understand how food works — the science and history — but I would never want to work in a restaurant where customers pick dishes from an extensive fixed menu, and I make those. As chef, I am the dom: I pick and create a menu that you will eat and hopefully find transcendent.
But the fantasy is not that I would run a restaurant (at best, I'd be a member of a cooperative restaurant like Moosewood). Rather, I'd like a bakery, making gourmet breads, and possibly serving pastries, coffee, sandwiches, and soup. Here it is food-as-craft, but as baker I don't serve anyone. I still make my loaves, and then sell the completed objects to you. Your choice is limited, and what I have available will change: there will be some staples, of course, but each day two soups will be available, and they will change by the week (one on Sundays, one on Wednesdays), and salads will be based on seasonality. If the bakery is successful, I will continue to expand: breads first, then pastries and coffee, then sandwiches and salads, and then dinners with a daily-changing menu. Such an operation is much more work than one person can manage.
In fact, however, I will never own a bakery, although I will continue to bake for myself, friends, and family. What I like best is that I can create everything from scratch, from raw ingredients. And I like sharing my food, and eating what other people have made, also from raw ingredients. I'm exceedingly happy when I am spending my time creating not just food, but also objects: I would love to learn more about woodworking, plumbing, pottery, knitting. However, I do hope to remain an academic. I love thinking about mathematics and communicating it.
The current fantasy, then, does not include selling food, but also does not include purchasing much. I'd rather move away, as much as possible, from the industrialized economy, towards one populated by human craftsmen and artists. Thus, the fairy-tale future involves a university, yes, where I will work six to nine months a year, but also a large house on a lot of land, outside the city but within the public-transportation network (there is no vehicle I enjoy more than the train, except possibly the bicycle). For the three months of summer I will be a full-time farmer, growing, pickling, and canning enough vegetables to live on.
Especially in the North, where the growing season is short but furious, I could avoid too much overlap between Spring planting, Fall harvesting, and Winter teaching. I have grown up in the West, and would like to return to the Pacific Northwest; on the other hand, I have no real desire to stay in the U.S.; perhaps I will live in British Columbia. At such latitudes, in this fairy-tale I will practice mathematics by night and farming by day.
Many of my recipes, some posted here, some e-mailed, some posted elsewhere, are available here.
6 comments:
There is certainly an artistry to teaching well, but the teacher-as-artist focuses on the delivery, not the mathematics itself.
I don't know that I quite agree with this. While it's true that a teacher-as-artist needs to deliver well, I would argue that the best artists also (1) choose subject matter well (2) discerns what to highlight and what to place in the background (3) combines #2 with the appropriate context to display the subject in a way that allows the audience to fully appreciate the content. One could argue that these are about delivery, but I say they are about understanding the math. To me, delivery has more to do with basic oratory technique like rhythm of words and interacting with the audience. In other words, I think that a good orator (who happens to be a mathematical dolt) who has memorized a bad math talk could deliver the bad math talk well -- perhaps even masterfully captivate the audience; but only a good mathematician with good oratory skills could design and execute a good math talk because the teacher-as-artist focuses on both math and delivery.
YL, I was hoping you'd respond. I will try to clarify my point, but I'm not entirely convinced, so perhaps you can say more about teaching.
Perhaps I should have picked a word other than "delivery". I have no doubt that a great math teacher must be a great mathematician-as-craftsman, which is how I understand the kind of deep knowledge of mathematics a teacher must have. But I think that choosing subject matter, correctly highlighting it, and setting it in context are more like the skills of a museum curator than of a painter. I think the teacher is more like Yo Yo Ma, masterfully performing J.S. Bach's pieces, than like Bach, who certainly was not as good a cellist (and, depending on your definition, perhaps wasn't as good a musician).
One of the best math instructors I've had has published many textbooks, but is the first to point out that his only original research was his Ph.D. thesis. Of course, other fantastic math instructors are also phenomenal researchers... I think, though, that these skills overlap only to the extent that both creating and teaching mathematics are centrally focused around how to think about mathematical objects.
I absolutely plan on being an artist after I graduate. If this artistry is based on any one (or combination) of food, mathematics, and teaching mathematics, I will not be unhappy.
But I think that choosing subject matter, correctly highlighting it, and setting it in context are more like the skills of a museum curator than of a painter.
To pick this metaphor to death, I think that the museum curator's role is more akin to the Mathcamp Academic Coordinator's role. As for choice of subject matter and how to highlight it, let's take Rembrandt's
Anatomy Lesson as an example. Rembrandt noticed that the anatomy lesson would be an interesting thing to document, so he chose to paint it. He could have painted like a photographer recording an event, but instead he highlighted various faces in the painting and receded other items into the background. The painting would be less powerful if it had been done like a photograph because irrelevant details would have distracted the viewer from the rest of the composition.
I think the teacher is more like Yo Yo Ma, masterfully performing J.S. Bach's pieces
What distinguishes a genius cellist like Yo Yo Ma and a great teacher is audience expectation. If I go to a Yo Yo Ma concert, I expect a transcendent artistic experience that mesmerizes me and makes me think, "Life is good." If I go to a phenomenal lesson by a great teacher, I also expect transcendence but I also expect to learn. The goal of a teacher is not exclusively to captivate the audience. I would say that the teacher-as-performer focuses on delivery, but that no matter how captivating and entertaining his audience found him, I would not consider him a teacher-as-artist unless he transferred some mathematical knowledge to his students.
I think that the best teachers combine performance (to captivate the audience),
knowledge (so they can interact intellectually with the audience), and artistry (organizing the matter in such a way that it looks both beautiful and clear).
One of the best math instructors I've had has published many textbooks, but is the first to point out that his only original research was his Ph.D. thesis.
Yes, but I like to think that lessons can be original. For example, Sam Vandervelde (of the Mandlebrot competition) recently wrote up a series of lessons plans for an MSRI publication. One of them includes a way to teach Euler characteristic through a game. Clearly, Sam was not the first to discover the topological invariant, but he may be the first to teach it in that particular way, and this required what I would consider artistry.
In my original post, I had cited Dorenburg and Page, who place transcendental experiences squarely in the realm of the artist, as opposed to that of the craftsman.
But then you point out, and I fully agree, that a Yo Yo Ma concert or a terrific math class is also transcendental. Certainly it's not just the Bach or the mathematics that's transcending: the same piece played by someone not Yo Yo Ma would not affect me as much.
Is learning really distinct from being transcended? And is the art of helping someone learn (or even creating a new mathematician) really inseparable from the art of creating new mathematics?
We often recognize great mathematicians who are not great teachers, and great teachers who are not great mathematicians... or perhaps the greatest teachers (of mathematics) must also be great mathematicians, in a profound (but not research-based) way.
You've thought about these issues much more than I have, so I will trust your ideas. On the one hand, I'm arguing a position that's certainly more simplistic and biased than it should be. On the other hand, I do think that this trade/craft/art distinction is a powerful and useful one, that can shed light on the different modes that make up mathematical, or culinary or musical, practices and experiences.
Is learning really distinct from being transcended?
There's overlap, but ultimately they are different. An intellectual epiphany may feel transcendental, but not all transcendental moments are about learning. No matter how beautifully Pete Sampras serves, I will learn more about serving by going to a good tennis instructor than by watching ten-year-old Wimbledon finals.
And is the art of helping someone learn (or even creating a new mathematician) really inseparable from the art of creating new mathematics?
Some educators break down teacher knowledge in the following way: "content knowledge" (how much math do you know?), "pedagogical content knowledge" (what are the best examples to illustrate a particular mathematical concept? what is the best metaphor? what is the best way of organizing an explanation/proof to that a student can understand it?), and "teacher knowledge" (when you look at a student's work, can you tell what's salvageable and what's not?)
All of these skills are useful to the budding (or even veteran) researcher. But the arena in which the professional researcher and the professional instructor have these skills may be different. For example, a researcher may be far more likely to be fluent in these knowledges in his/her research area and the upper-division undergraduate/graduate material that underlies it. A good high school teacher will have these skills applied to high school material. Have you ever tried teaching fractions or negative numbers to an elementary school student who is failing? I have, and it's hard. I am not sure that I ever succeeded. I was absolutely unable to detect where the student's lack of understanding came from. In contrast, when I teach calculus or ODEs, I am usually able to diagnose the root cause of a student's lack of understanding.
All this said, the other skill necessary in the art of teaching/advising that isn't as necessary in the art of research is a fundamental understanding of other humans. It doesn't matter how good these skills are if you consistently demoralize and repel students (even if unintentionally). A good teacher has the skill of conveying information in an honest way that the student ultimately wants to receive.
The distinctions you mention are indeed useful and interesting. I'm not sure exactly how they would fit into the realm of teaching, but here is a stab at it.
1. "Cooking as a trade", where your primary goal is sustenance, and with your limited repertoire you're hoping your customers go away thinking "I'm full."
You have to teach a lot, and you have to teach many sections of the same class, day in, day out, year in, year out. You don't have the time to explore other subject material, but at the end of each year, your students leave satisfied and can pass whatever standardized exam is fashionable at the time and apply their solid knowledge to the classes for which yours was a prerequisite. (Your teaching methods, though, are independent of said standardized exam.)
2. "Cooking as craft", the style promulgated in the greatest cookbooks, has its main goal enjoyment; a chef should have a wide repertoire of classic dishes, and hope that the customers go away thinking "That was delicious."
You teach a variety of classes and have a command of a wide variety of material suitable for the range of students you teach. You have some time to explore material independently, and from time to time, you spice up your classes with material you've just learnt yourself. Your students leave your class happy about the cute examples/demonstrations you put in and able to generalize the skills you taught them to future classes and projects.
3. In "cooking as art", on the
other hand, a chef prepares her own recipes, and any given night the menu will be very limited; customers should be entertained, and go away thinking "Life is beautiful."
You only ever teach classes/workshops of your choice. You have virtually no required teaching assignments, and spend almost all your time designing ways of teaching whatever material catches your fancy. Your audience leaves feeling as though their intellectual horizons have been drastically widened. You have shown them a corner of mathematics that is beautiful and done it in a pristinely clear way. The audience cannot wait for an opportunity to revisit concepts that you explained or play with an example you discussed.
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