The squeaky wheel gets the spin.

I met with the director of undergraduate math instruction this morning. He wanted to know what specific problem in my homework was giving me trouble. I said it was getting to the tree and bypassing the forest a little too quickly. He insisted he wanted to get to the tree immediately.

In the interest of getting along with the guy, I tried to give an example of the onerous work-outs that characterized the vast majority of homework. He wanted to know what page and what problem I was having problems with. I tried to explain that wasn't the point of my visit. He insisted on seeing a specific problem. I found one that was similar to a problem that sent me scurring through my algebra reviews and flash cards on Saturday. This seemed to satisfy him.

Once he had connected with me at that level, I tried to raise the level of the conversation to the differences in expectation between the liberal arts student and the IT student. He told me I was taking a CLA math class, and that IT students took a different curriculum. That was scary. I suggested there could be a less draconian presentation for, say, 30 urban studies students that were interested in developing their modelling skills or understanding without the equivalent of a military boot camp. I suggested that making eye contact with the students and making sure that at least the average student in the class completely understood the progression of symbolic processes on the blackboard might enhance the experience. This seemed to be loaded territory, as it evinced a defense of the specific professors in question.

Somewhere along the line of the discussion he began to sense that there was no point to our meeting, as far as he could see. He seemed greatly relieved to see me go.

I was half an hour early for my Urban Studies class, so I got a coffee and stared at the sky, wondering if the Two Cultures of C.P. Snow had in fact become a kind of preemptive rationalization for the vasty gulf between the technicians and the policy makers in our society. Doug is a nice guy with a tough job. But the gap in teaching this kind of material to the people who need it is not narrowing because of the advent of personal computers. It needs attention.

I hope I can contribute to that attention in a positive way.

f''(dialog) lim x-->a = ?

After re-reading the entry below I decided I should do something about this situation. So I sent the following email to Professor Larry Gray, the head of the math department at the U.: (Larry writes publicly about the k-12 math education dilemma, which prompted the digression about my son Sam in this message)

Hi. I am a 56 year old undergraduate at the U taking Calculus 1271. I am an Urban Studies major who is interested in quasi-formal and formal models for solving urban problems. In fact, I am interested in the entire spectrum of modeling issues in public policy and planning. My background is enriched by having worked for 24 years as an information writer for Hennepin County, a full time position I continue to hold.

I am writing you for two reasons. One, my recent experience with intensive precalculus and calculus instruction at the U motivates me to make a few observations about this process. Two, I would like to open a dialog with someone interested in these things in an exploratory mode. Having read some of your writing, I believe you are both committed to the main issues and open minded enough to explore the problems without being defensive or one-dimensional.

I do not represent any camp or ideological point in the spectrum of math education. I am not a member of any group. I have seen two sons go through the gyrations of math curriculum over the last 15 years, and have experienced some of the painful conundrums of the shifting environment. Last year my son was put in Algebra II without experiencing Algebra I. This was justified because South High used a different curriculum than Field School, where Sam was taught until 8th Grade, and the counselor felt that Sam was able to go right into Algebra II. His distress was interpreted as a behavior problem by the teacher and counselor. After an intensive month of investigation and dialog, we got Sam transferred to an Algebra I class, where he proceeded to earn A+ and was considered an exemplary student. The deep human and organizational complexity of these situations humbles me, and despite some of the obvious absurdities that might arise I always proceed from a position of great respect for both the discipline and the persons involved in any conflict situation.

To put it as simply as possible, I believe we could benefit from a return to mastery approaches to basic skills that include drills, memorization, emphasis on mental calculations, development of estimation and judgment, etc, but I also believe that the disparity created by gender and social status in classroom settings is real and can be addressed by curriculum design. I believe that as a society we suffer greatly from the mathematical illiteracy of the voting members of society, but I also believe that there is a culture of technical education that needs to be integrated with an increasingly complicated society. I don't believe that forcing IT people to take dance classes will make them more sensitive to real social problems, and I don't believe that forcing CLA students to sit through what I am experiencing in Calc 1271 is going to be the resolution of a widespread cultural innumeracy that will plague this nation in the coming decades.

If you are interested in corresponding on these issues, I wrote a short essay about my calculus experience which is on my blog at http://www.zeitguy.blogspot.com. I invite a reply. If you prefer a briefer statement I would be happy to condense this for you. I don't expect to change the rules or requirements for getting through this class this semester, but I believe I have something to say and something to offer in effecting change in this process that will benefit others over time.

I was surprised and pleased to see Larry responded immediately:


Dear Jeff,
I read your essay about your experience in calculus class(es). I am
sorry to hear that it is such a struggle, and I am not sure what the
solution might be. In my years of experience working with students, I
find that there are many different reasons why some students struggle
with the class, ranging from a lack of preparation on their part to bad
teaching on the part of the professor or TA. The best way to get to the
bottom of this would be for you to come into our office and talk with
someone so we can evaluate the situation. The best person to talk to
would be our Director of Undergraduate Studies, Prof. David Frank. He
is in charge of all undergraduate math classes. Call his secretary
to make an appoint with him. I'll notify him about your
experiences so he will be expecting to hear from you.
Best wishes,
Larry Gray

Well, I wasn't very clear about my point in writing him, apparently. But it was nice to hear from him in such a friendly manner. Very impressive. Stay tuned for further adventures.

When Sara heard about this, her first question was "Are you going to get your teacher in trouble?" I hope the hell not. That isn't what this is about.

It's about me.

Weapons of Math Destruction

I am studying Calculus in a classroom. It is taught by a qualified University of Minnesota Professor.

Most people I know expect such an experience to be hell.

It surprises me that it is hell.

When I first encountered calculus in the 1960's, it was a mysterious, almost mystical unknown. Only Certain Geeks could approach the temple and few became qualified calculusers. Because of my interest in Science Fiction and scientific approaches to solving social problems, Calculus was never far from any topic I was really interested in. But I always assumed it was, like playing classical piano at the concert level, a skill reserved for a small, deserving minority of people.

Over time, this image of exclusive Olympian existance began to break down. Clues came here and there in articles, stories, conversations. By and large the reputation of Calculus as really hard and unpleasant, reserved for "souless iron-butt repressed drudges" -- the reputation remained intact.

Finally about fifteen years ago I was developing some graphic software, and had learned to code curves and fill areas. By coincidence I came across a short article on calculus that explained it in terms of computer graphics coding. It was an amazing "aha" moment. I knew about slopes of lines and approximating irregular areas with lots of tiny regular areas. I had enough basic geometry and algebra to dimly get the rest of what I was reading.

Calculus was about changing tanget lines along a curve, and approximating the area under the curve with rectangles, in its operational essence. I won't go any deeper into it than that.

After reading the article it felt like a great weight had been lifted from me. I had carried the burden of "not qualified for calculus" around for years as a subtle sabotage of my interest in science. I had attacked the finite maths, such as statistics, probability, set theory, boolean logic, etc, with gusto, because I was qualified for that...I had done well in basic classes and was not impressed with any mystiques.

From time to time after that experience, I would come across something that required thinking a bit in calculus concepts, and I wouldn't be daunted. But I didn't actually tackle the curriculum. I just felt more comfortable, and when the time came to ask questions, I felt as though I could ask them without making a fool of myself.

Recently I decided to bite the bullet. I need operational knowledge of calculus at a much deeper level than before. So I registered for pre-calculus last semester. It consists of trigonometry and the algebra of polynomial and rational functions, for the most part. There are other things. But holy damn, it was hard. I thought my computer experience would serve me in two respects: one, I had written code that created graphic demonstrations of most of the concepts involved, from quadratic equation solutions to animation of sine functions. So I assumed that I would get the concepts easily and could anchor my work on exercises in a secure conceptual understanding.

Not so. The presentation of the detailed aspect of the ideas was never connected to the more abstract level. We marched through horrendous black-board filling exercises in expanding and factoring ugly algebraic expressions. I felt as though I was being dragged behind someone's pick up truck on a chain, and they didn't care if my head was still on when they came to stop five miles down the road.

My romantic notions of the beauty and power of math encountered the ugly truth of math pedagogy. It was taught as a method to separate the class into clear groups: the minority who can function in the hostile environment of rapid, forced manipulation; the majority who can struggle along without raising their heads; and the "unqualified for calculus" who would drop out during the first 10 weeks or so, unable to make any sense or headway in the typhoon of symbols.

After the first two weeks my fascination and infatuation with math was completely scoured out of my little pan of math capacity. All that was left was room enough to fight my way through the increasing pain of homework. Each morning and evening I sat at the dining room table, with index cards, summary sheets, page markers, algebra review books, sheets and sheets of paper scribbled with the scars of a thousand little wounds called "problem solving" by the books and teachers but called torture and indoctrination by the insecure student.

My pride in my programming prowess turned neutral, then to a kind of oblique shame. The class was taught with pencil and paper and, if necessary, cheap math calculators: no graphing or algebraic automation allowed. No computers. We were doing the same drill that students of pre-calculus had done in the 1960's, even the 1930's. No concession was to be made to any social or intellectual shift in attitudes toward the role of math in society. This was meant to be hard. That message was underscored by the stunning rigidity of the class rules. No homework was accepted late, period. No make up tests allowed, period. No excuses were accepted for anything, period.

Since I was struggling so hard to get the problems done, and especially the midterm test problems within the time limits that energetic 19-year olds found daunting, I began to wonder if I would even pass. After my experience of hysterical blindness during the first midterm, and subsequent resignation to handing in half the problems, I was sure I would fail.

But the curve saved me. I had just enough on the ball to be one of the stragglers in the middle group of head-downers. I never quite fell out of the herd, and ended up in fact with a B- despite barely scoring over the mid point of each major test.

The curve saved my participation in math, and allowed me to proceed to the next level of calculus. But the system failed me significantly in not recognizing and responding to my basic problems with algebra.

I have been in calculus for three weeks. The calculus is easy.

The algebra is hell, and there is no relenting from it. It marches like cloned extraterrestrial insect soldiers from Starship Trooper, filling the horizon with its black homicidal whirr of horizontal asymptotes and drone of rationalizing denominators.

Despite my alarmed remarks, the Professor smiles when I say I really need review in Algebra to keep up. "The Calculus is easy, " he grins, "The algebra is hard."

And that is the essence of the situation. Whomever designed this curriculum did not do it from the assumption that a mathematically literate citizen is a better citizen. They designed it to keep the rabble out of the hallowed temples of Math. And as they succeed, and frighten the reasonable, terrorize the faint, and only inspire the unbalanced, they do Math and civilization a great disservice. Keeping an abyss between the math literate and the average educated citizen is not longer an intriguing by-product of our rapid technological ascent in the West, a subject for essays on C.P. Snow's "two cultures". It is a disease. It can be cured.

It might kill me first.

The yoga of fear

Yoga means "yoke." The idea is that the postures, the asanas, integrate the body and not-body parts of ourselves. I say not-body, instead of spirit, mind, or soul, because we don't know what any of the not-body existence really is. We just know it is there. And that it needs to be yoked to the body.

When we get what we want, the body goes one way and the not-body goes the other way, into their own rooms, as it were, to eat from dinner trays and watch their own choice of cable fare.

When we are afraid, suddenly the body and not-body are very interested in each other, both for what they can do for each other, and for more sentimental reasons.

When I sit in Calculus class and can anticipate where the professor is going with the demonstration of, for instance, instantaneous rate of change, my mind wanders off. My body sits in the miserable little desk, and tries to communicate with the outside world through gurgles and twitches.

If the professor suddenly is writing rational functions, i.e. f(x) = 1-(1/x) on the board, fear seizes me. I missed many points on that section in pre-calc. It makes me sweat and tremble. My mind goes blank. I want my mother.

At the moment of greatest fear, my body and not-body are perfectly superimposed, like siblings separated for years and suddenly running into each other in the produce section of the grocers. "Hey!" "Hey, you are looking good!" "Are you still living in Bloomington?" "Looks like we went to the same barbershop." The chitchat masks the great relief, curiosity, and uneasiness occasioned by such chance but profound meetings.

From the back of the mind, the supervisor tells the mind to start taking notes and the body to quit interrupting. The brief moment of rapproachment between body and not-body is over, but it has done its job. For some time afterwards, a melody runs through the chambers of the self, through the shallow blood streams and across the fields of wild identity flowers in bloom. The melody is simple and heartbreaking. When you hear it, you are one. When you don't hear it, you don't know you ever heard it.

No one chooses fear. It is easier to suffer.

Urban studies - the surreal version.

In the actinic light of September, pretense is stripped from the face of the city, and a grim truth is revealed: The city is failing. It is failing its first class of the new school year.

It is late for class, running breathless through a quadrangle among the inimical classicism of a hubris-laden patronage. Its pants fall off. It is wearing Mickey Mouse underwear. Micky and Minnie are doing it. Grim grim grim. The caretakers in their black Mao fashions and lidless eyes stare balefully at the city's Mickey Mouse underwear. They make mental notes on their mental clipboards, and their spit becomes even more acidic.

The city is still healthy enough to have a histamine level, to have a pulse and to create by-products. But it is running, in its shorts, without change for the vending machines.

Overhead, giant blimps move in lazy figure-8 patterns over the freeways, each blimp reflecting the traffic plexus below. In the sky, they make a three dimensional map of the freeway interchanges, like giant flaccid push pins indicating the distribution of gridlock among the withering neighborhoods. The conversation drifts to the blimps, with the assumption that the real "meaning" is about the traffic.

Grim people, who may be women, submit to surgery which removes their eyelids. In exchange for the tearless, blinkless condition they are given unlimited shopping rights and many, many lottery chances to win jobs.

Children ignore their expensive toys, and improvise the history of piracy and the opium trade with paper bags. Some of the paper bags have handles, some are shiney, all become punctured in the active play, but none ever are discarded.

Years later the same children are grown, but not committed to surgery. They bring their paper bags with them into the offices and classrooms, and force the powerless employees and students to write summaries of the bag's stories. To react to the bags, the employees and students must get drunk and violently ill. In the aftermath of the drinking bouts, the Great Fear seizes them, and thoughts of rebellion or impudence are banished. Later, they turn in their papers, and hope for positive attention, good grades, and/or the right to reproduce. The lucky few are allowed to reproduce without deformity. Some die.

Across the river, a green plain rolls forever toward a soft horizon. On the plain, bison stand motionless in the sunlight. They might be plaster. When it rains, the color is washed off the bison, and thier serial numbers can be read with the right optical equipment.

Giant paper plates are rolled out when it rains. They are dragged across the parts of the city not covered by the Mickey Mouse underwear. Inside the city's brain, the effort to form words is a constant source of conflict and dissatisfaction among the many neurons. They are not well trained for their jobs. Their supervisors resort to harassment and open derision to keep them in line. The effort to form words intensifies.

But the city is failing.

It is fall, and school is born from its own ashes again.

The people, some might be women, dress in lidless eyes and never stop swallowing things handed to them on small trays. The trays are everywhere, popping out of tree trunks, kiosks on the sidewalks, windows and delivery trucks. No one refuses the morsels. They keep lifting them with dainty fingers, tasting, chewing, constantly swallowing and never digesting. When they go to each other's homes for dinner, they cannot stop taking morsels off the importuning platters.

The real dinners are prepared by alarmed immigrants, who must make appointments to have their ears sewn shut, their fingers sewn together, their thighs sewn together, and their shoulder blades broken and reset at right angles. They must submit to these surgeries if they want to become citizens of the city. They are alarmed by the rites of citizenship. Some try to erase their features with dilute acid from the sap of plants that grow beside the city industries.

The real dinners sit untouched on the long beautiful tables. Eventually mold and flowers grow from them. Only the children, vexed by their bags and bored with the shallow perversity of piracy, enjoy the flowers.

Isn't it always like that?

What do the men do here?