"Tall, Dark, and Mysterious" has a post about the state of college students where she is teaching. She is trying to teach Calculus to students who don't have a good background in the basics, such as algebra. I have reason to believe that this is not isolated to British Columbia, or even Canada.
She writes:
See also: mathematics by pattern matching, word problems, et al. So much of this problem can be traced to the fact that students do not understand that an equation is a relationship among quantities. Each equation they see is a concept unto itself, to be memorized and applied to the word problem on the test that looks like the word problem that I did on the blackboard and that used a similar equation. If the word problem on the test doesn’t look like any of the problems I did in class, then that question is “totally unfair”. Most students will leave the question blank, or solve a completely different problem (one with a ready-made equation) in the space provided.She also writes:
An equation is a relationship among quantities. How many times have I tried, and failed, to get this idea across? Imagine getting students to realize it for themselves! Unfortunately, many students lack the intuitive ideas about math needed to know, even subconsciously and with the help of leading questions, that equations are anything other than a jumble of letters, numbers, and symbols. (Even though I spent twenty minutes showing how we could use the definition of a circle and the Pythagorean Theorem to derive the equation of a circle in the Cartesian plane, nearly all of my students were angry that I wouldn’t provide the formula on the test. I mentioned, incorrectly, that they could derive the formula themselves, on the test, if they needed to; of course, I was wrong.) But I would gladly sacrifice 80% of the poorly-learned content in a first-year college math course if I could instead effect a solid understanding of what equations really meant - and how to get one from a sentence or two of information. I wonder if Fish’s lesson could be adapted to the first-year math classroom.




