radical thoughts

What is it about some topics that just never seem to leave the curriculum? People, do we realize what a waste of time it is to spend a week or two simplifying arcane radical expressions?

The purpose of rationalizing denominators predates four function calculators. In those days, one could benefit from transforming an irrational number from the denominator to the numerator. Root 2/2 could be more easily be calculated by hand because you could divide a whole number into 1.414, as opposed to dividing 1.414 into 1.

I also see a lot of algebra II students spending loads of time simplifying the nth root of x to the m, where n does not divide into m evenly. What can be gained from simplifying the fourth root of 32x^6y^9?

Now, I am not above doing "useless" math in class. I love doing mental math, calculation tricks made obsolete by our technology. I love number patterns, e.g. perfect squares can't end in any digit other than 0, 1, 4, 5, 6, 9. I love the Ramanujan story about the number 1729.* And you never know, "useless" math almost always proves to be useful at some later date. We should learn math that is either useful or enjoyable. I can't say that simplifying radicals qualifies on either ground.

Does anyone else have a topic they think is worthless to teach?

*Ramanujan, dying of TB, was paid a visit by his friend and benefactor Hardy, who complained that the number on the cab he took over was most uninteresting--1729. Ramanujan countered that it is the smallest number that could be expressed as the sum of two cubes in two different ways!

to geom or not to geom

The inevitable question arises:

algebra, then geometry, then algebra II, OR

algebra, algebra II, then geometry


As a math department head long ago I liked the former route for accelerated kids. They could learn a little geometry before taking alg II/trig, which seems to make pedagogical sense, and I liked teaching pre-calc kids who had just taken algebra II. Smart ninth graders could handle geometry, and it was a nice break from all the equation solving they had been doing.

Less able kids, I thought, could do four years of math in five: alg I, alg II/no trig (and we always got a lot of entering ninth graders who entered with shaky algebra credentials; see the "my life as a math dog" post), then geom as 10th graders. If for some reason we changed our minds about these kids, they could take trig during the summer, then precalc and calculus. Otherwise after geometry they could take algebra II, which was a rehash of algebra II, then precalc. Almost everyone could manage this sequence, although we always had one or two for which we had to devise individual programs.

This is almost like arguing about which is better, chocolate or vanilla ice cream (or Barack or Hillary?), but we do want your opinions on this one!

please seduce this teacher . . .

. . . so he commits a felony and gets fired. He teaches calculus in DCPS, at the school where the smartest kids go. He does not fully understand the order of operations, and he does not know what is on the College Board syllabus. He makes the students do long anti-differentiation problems by hand but does not look at the steps toward solution. He is just plain lazy, but he has taught for years and has tenure. There is no moving this teacher to some other career.

As a citizen I am outraged the girl I tutor has this teacher. I hope Michelle Rhee can eventually get some qualified young grads to come in and teach math adequately, and I hope we can have lots more accountability for these deadweight teachers.

my life as a math dog

We are creating thousands of math refugees, kids who don't fit in anywhere. They are told they are bright throughout elementary school. In sixth grade, to prove to itself that these kids go to a great school or are part of a great school district, the kids' school tells the kids they are qualified to take algebra I in seventh grade. Never mind that there are only a handful of kids city- or county-wide who are so qualified. The kids and their parents believe it. And perhaps algebra goes OK, and they have a teacher who is not going to really demand all the rigors of a full-fledged algebra I class. After all, the teachers are part of the same school/district who believe in this system. The same thing occurs in eighth grade. The kids take a watered down geometry class and are given a B. Then one of two things usually happens:

1. They change schools, and their new school tests them for algebra and geometry proficiency. There isn't much there. They should get placed down in algebra I, but they bitch and moan (or their parents do) and end up in algebra II and struggle. They hire a tutor. They lose their self-esteem in math.

2. They stay in the same school that started them down this treacherous road in the first place, and they get B's but they don't learn very much. Eventually, they get to pre-calc or calculus and the feces hits the fan. They can't do anything, and they either hire a tutor or take AP statistics. They lose their self-esteem in math.

When I started teaching 30 years ago, we used to complain about too many eighth graders taking algebra. Now it is truly insane.