Calculus notes - part e^4

1. The current calculus curriculum I am enrolled in at the U of M is not for tech grads. It is for liberal arts students. It does not use computers or graphing calculators per the Math Department policy, which avoids discriminating against those who can't afford the $100 or so that a Computer Algebra System or HP-83 costs. (I doubt that this policy will stay in place much longer with the $100-semester CLA "technology" fee each student pays with their tuition...the irony is just too great).

2. Calculus is a spatial discipline taught with linear methods. In other words, calculus has to do with rates of change in position, area, volume, speed, etc...all things that have to do with objects that occupy space. The methods used are typically linear symbolic representations of quantities and their relationships. Current teaching methods as far as I can tell emphasize the linear symbolic manipulation as the content of the course, and treat the spatial materials as an illustration of the symbolic principles. I have stated this with prejudice, obviously. (I am using linear in the sense of language skill and sequential processing, not linear algebra.)

3. I am not aware of too many efforts to teach the spatial math subjects with intrinsicly spatial methods. I have a box called the Math Kit that I bought for my son in the early 90's. It has pop-ups and paper engineering devices designed to teach several math concepts physically and spatially, including derivatives and integration. In the late 1960's two professors at the University of Leeds created an interactive computer station that represented work optimization problems as spatial puzzles. In other words, the subject had to arrange colored blocks in an efficient pattern: the blocks represented work crews, job sites, deadlines, and material availability.

Experienced supervisors in the construction field, foremen, laborers, University instructors and students, random townspeople and a few children were the subjects of the experiment. The experienced supervisors did the worst, while the best optimization was performed by an 11 year old girl who simply played with the visual patterns. Apparently the supervisors were inhibited and confused by the representation of a task they already understood in different terms, while the 11 year old girl had the greatest freedom to just deal with the spatial play, color coded objects, and dependencies (rules). The girl was "doing" calculus. Could she have extracted the second derivative of a function in order to establish the unique solution of the absolute minimum value in the function range? Of course not. Could she use her ability in other problem sets? Of course she could. Should 11 year old girls be scheduling work crews for construction companies? No, but they could certainly be showing us something about learning calculus.

The advent of the internet and Java applets should be putting a lot of demonstration programs into the hands of interested students and amateurs. But a Computer Algebra System is not easy to program or learn. You can watch simple illustrations of tangents and other trig and algebra functions if you search hard enough, but there isn't a coherent system available that would work as well for the 11 year old girl and the 40 year old construction contractor and everyone in between. Mathematica and MathLab sure won't do it.

4. Does calculus have to be so hard?
This problem is actually several problems. I will break them down:

a. The University certifies competence. That is its primary product. If it allows the product to be compromised, it goes out of business. It might take 100 years, and it might be happening now, but it is not okay. In order to keep its cred as an institution, the U must make calculus as hard as their competition does. This is a primary concern for the curriculum committees: they don't want to be perceived as shoddy or soft. The up side of this is a pride in craft: if the U said you passed their calculus curriculum, then it means something. The down side of this is a huge inertia. No one wants to both do the work of rethinking existing methods AND take the risk of being seen as shoddy or soft. Bottom line-- it is going to be a long, hard process for the U to insitute meaningful change in its calculus curriclum, absent a real champion with the reckless need to provoke change.

b. Calculus straddles the border between linear, symbolic language and spatial operation. The beauty of this is that it requires, and rewards, a high degree of integration of right and left brain skills. Calculus is one form of math that should be at least as attractive to women as to men. The sad fact is that most of the calculus course is taken up with Algebra -- simplificatons, factoring, etc, and they are pretty much hard core left brain materials, the kind of stuff that inspires the term "math anxiety." The emphasis on algebra could be reduced without compromising the mastery of calculus. Conversely, the algebra skills could be taught with more of a coaching/mastery emphasis, as performance. Just watch "Stand and Deliver" to see what that would look like. Edward James Olmos teaches pre-calculus and calculus to a class of poor kids. The results are amazing. But the process sure doesn't look like a contemporary math class in the U of M. Olmos works like a baseball coach, identifying weaknesses in each players grasp of the fundamentals, then drilling them until they are rock solid.

I guess that in a typical math class, the teacher lectures to the confident upper 30% of the class, and lets the others try to keep up as best they can. The physical drop-out rate in my class is alarming. There were over 40 students at the beginning, and it seems that fewer than 20 come on a regular basis. You couldn't run a business or a ball team with that kind of attrition. The emotional and intellectual drop-out rate is even more disturbing to me. The kids who don't come back will never touch the subject again. The ones who stick it out doggedly and still get a D or F are even worse off...their entire outlook on themselves and their interest in learning is scarred. For what? Who picks up the tab for this casual waste of human resources? In this case, Calculus is hard, not because it is intrinsically hard, but because the politics of the classroom make it hard, and reward the proficient while punishing the student who struggles.

c. Finally, how hard is calculus, really? By using spatial methods, you can teach fifth graders the entire canon of basic principles in three months. I would bet my job on it. Then you can ground them in the basics. You can make them proficient in the underlying trig and algebra in two years. Bet my job on it. I think you could have a curriculum with the right tools and teaching methods that could crank out proficient calculus practitioners with the requisite trig, algebra, geometry by 11th grade with no problem. These should be required of all graduates, with different levels of complexity available. Every high school graduate in the country should be able to follow the basic science in a discussion of the Challenger disaster failure, or follow the basic financial models in a discussion of impact of development on a city tax base using different approaches to wetland conservancy.

If this sounds crazy now, read it again in fifty years.

Do not attempt to adjust your picture....

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