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Me: Algebra is one of the most mistaught subjects that I've seen (Calculus being another).
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Jason H. Smith: Care to explain why you think so? I'm interested.
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No problem. There's this huge chasm between the way mathematics is used and the way that it is taught at the introductory levels. Maybe I am guilty of puffery here, but where else is a subject taught to millions of people, but virtually unused in that form by those who have mastered it.

Let me explain in more detail (much much more) what I'm talking about. I've taught "college algebra" a number of times (like this rather than this) (rather poorly I might add - that's why I don't teach any more :-( ). The emphasis has always been on solving certain types of equations (linear in several variables, quadratic (e.g., ax^2 + bx + c = 0, radicals including the present of square roots, factoring polynomials in a single variable, and so on.

Then you get trigonometry which starts with angles and the basic properties of the sin, cos, tan, etc. functions. Then the average course dwells on identities of these functions. I've seen courses where the principal goal was to memorize the various identity rules and know how to apply them to get the desired end state identity.

Finally, calculus does differentiation, series and sums, and integration (usually finishing with integration in multiple variables). Except for a lucky few, this is the limit of virtually everyone's exposure to mathematics. Often people don't even bother with "algebra" in the first place. In my cynical opinion, the approach can be summed up as "memorize the rules, figure out how they work, and pass the course" or basically that these courses teach a set of algorithms for a small rigid subset of mathematics.

There are two problems with this. First, most (IMHO) of the most useful and interesting mathematics simply is ignored. Second, the true ideas of mathematics and their benefits aren't being developed in these courses or used.

Let's start with the list of stuff that isn't covered. I don't know of any high schools or colleges that require knowledge of discrete mathematics. I'm talking about graph theory, combinatorics, set theory (ok, this gets covered a little, usually), discete dynamical systems, etc. I bet a bunch of this could be more useful than say trigonometry, no harder to understand, and engage students more. number theory and geometry should be more available than they are. These two subjects are among the most interesting subjects in mathematics. True, they don't have much to do with real life, but they get the student thinking. Even probability and statistics (often glanced at in introductory courses) can be taught in interesting ways.

I've heard the comment from many mathematicians that they really didn't use or understand algebra properly until they had taught it. This is interesting in that it highlights that mathematicians rarely **use** what they teach in these early courses.

Then when you get to the **ideas** that students should be exposed to, you see more problems. The emphasis is on following algorithms and making graphs rather than on what should be important. My favorite low level course was business math. The reason was because it discussed one of the fundamental aspects of mathematics - namely making a mathematical model of a problem and using that model (and some mathematics) to deduce properties and solutions for the original problem (and it did a fairly neat job too). That's the sort of thing that most scientists, engineers, business people, etc. do every day.

What do I think is lacking from mathematics teaching in high school and college. First, the principle of abstraction is missing. A problem can be made easier by removing the parts that add complexity, but don't change the solution very much. The second is the ability to make a mathematical model of a problem. Finally, the variety in mathematics is being missed and with this, a lot of stuff to engage people's minds is not being used.

Note I haven't said anything about the quality of teaching. This is a somewhat independent issue. Let's say that many people have ranted on the quality of their teachers whether they be good or poor. But how much better could good teachers be if they were teaching the really interesting stuff, and not just what's acceptable. My general impression is that there's two current faulty modes of thought on teaching mathematics. The first is the "old math" which insists on blackboards, doing work by hand, etc. while the "new math" wants to make use of these shiny boxes (you call them computers :-).

In a way, the problem (IMHO) is that we're (as a global society) are teaching low level math 19th century style. Here's an example. While looking for more information on the history of teaching mathematics, I found this (slow) history of teaching mathematics in Germany. This link describes the broad changes made in German mathematics teaching (in high school) since the 19th century. It is notable for its brevity. I.e., not much has happened to change the high school math course (except for events in the 60's and 80's) since 1905! It may turn out that computers are undesireable in mathematics classes. However, the many advances in mathematics in the last hundred years should be reflected. Also, some of the best mathematics is actually the earliest. Two thousand year old math (and the challenging thought and discourse behind it) also isn't reflected in our faded 19th century classroom.