I passed by an algebra class and they were working on exponential functions, they never see it in geometry and then we expect them to remember it in algebra 2. It never happens and we just have to reteach the whole topic from the beginning!!!
The above is quoted from a message submitted by a high school math teacher to a mathematics newsgroup last week. The comment addressed the three fundamentals of education: what do we teach? why do we teach it? and when do we teach it? It also reminded me of the distinction between learning and educating. It seems at times like the "education" fan club forgets how often learning takes place outside of the world of education, and the "self taught" crowd loses sight of how helpful schools can be in guiding an individual's learning.
Mathematics is not unique insofar as how it is learned or how it is taught. Mathematics is unique in how it is perceived. For some crazy reason society places the burden of initial education in mathematics in the hands of elementary teachers. That is not to speak down on elementary teachers. On the contrary.
Consider, for a moment, foreign language education. In most places in out society foreign language instruction is held off until the post-elementary years. Why is that? Most likely it is because of the impression that the typical elementary teacher is not skilled (trained?) adequately to teach a foreign language. We hold off French classes, don't teach Spanish, delay Chinese instruction, and so on because individuals skilled (trained?) to teach them are assigned only to older students in post-elementary years. In some parts of the country there is a push to get these teachers into the elementary schools, where, in isolated instances it was done, and then undone because of budgetary constraints.
Take a moment to recognize that musical instrument instruction has acquired such a respect that most students are taught individually or in small groups by a special instructor. Ever come across an elementary teacher required to teach students how to play the clarinet?
Yet, society operates on the supposition that elementary teachers will have the skills (training?) to instruct in mathematics. The same subject which people disdain with the "I hate math" and "I am no good at math" is being introduced by elementary teachers with no more training in mathematics than they have in any other subject.
Suppose we had a "race to the moon" approach in mathematics education. Imagine placing enough math specialists in elementary schools and adjusting priorities so that students would all leave elementary schools with the solid grounding necessary for handling secondary math. Imagine elementary schools leaving elementary school with a number sense so strong and a spatial perception so good that middle and high school algebra and geometry classes become common-sense subjects to them.
Some students slide through mathematics in school with ease and it can be easy to fall into the trap of saying that all should be able to do it if only they would try harder. Quite often academic success is aided by support the student gets outside of school, and a good learner with some motivation and a good mentor (or sometimes all alone) can succeed even in the absence of a school or teacher. But those are the exceptions, and we should not use them to support a less than adequate system.
Here is part of an article by John A. Dossey from The Arithmetic Teacher from 1984:
Please take note that the issue of improving the teaching of mathematics in elementary schools has been around for a long time. The vast majority of those who were teaching when the article referred to here was published have retired by now. yet the issue still exists.
We have spent at least 35 years recognizing this need and nothing has really changed. Maybe it is time to create a grass-roots movement making the case that any and all students in our society deserve the absolute best we can give them and not accept anything less.
If we were giving the students the absolute best, the scene described in the opening quote here would probably not take place: solid fundamentals create a mind where mathematics makes sense, and if something makes sense, it will be remembered. Plus, if all students were more solidly prepared, the geometry teachers would have the time to incorporate exponential functions into their courses.