Answer by Jessica Su, computer science Ph.D. student at Stanford:
Lack of math skills from high school.
My father is a professor of engineering and routinely sees people mess up fractions and the distributive law. I tutor a high school junior with engineering aspirations, and he doesn’t understand why increasing and then decreasing a price by 20 percent doesn’t keep the price the same. Of course these people have a hard time in harder classes and have to give up their pursuit of science and engineering.
If I had to guess why students were like this, I would cite at least three reasons:
Algebra is hard. It’s very easy once you understand that the concepts are supposed to make sense and you’re not just supposed to memorize how to do problems. But that usually takes longer than the year students are given to learn algebra. Especially since:
- Most of their courses prior to algebra are memorization-based and not conceptual.
- Most homework problems in textbooks are exactly the same as the examples, which encourages people to memorize specific cases instead of grasping the general concepts.
- The concepts are not always emphasized or explicitly explained.
Because algebra is hard, many teachers shy away from teaching concepts and proofs, and instead they teach mnemonics so students will pass the tests. My precalculus teacher taught us limits by having us memorize rules for “big top” and “big bottom.” This memorization teaches nothing but is the easiest thing for teachers who don’t understand the concepts themselves or students who are so far behind that their teachers have given up on teaching.
Students do not do enough practice. It’s really hard to understand higher-level classes without sufficient practice with lower-level material. My high school precalculus class assigned maybe 10 problems per section. That’s just enough for students to learn how to do the problems and then forget once they’ve passed the test.
I am fortunate that my father had me go through math textbooks and do all the problems in the book, thus ensuring I would never forget high school math. Without this, I wouldn’t see a lot of tricks necessary for more difficult problems, such as completing the square when convenient, multiplying and dividing a fraction by the same number, or expanding and collapsing telescoping sums. I wouldn’t be able to integrate cosine squared because I wouldn’t remember my half-angle formulas. In my physics classes, these were absolutely critical.
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