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Here is one potential example from a number of years back. Maybe it counts as "back then." My daughter's algebra teacher was teaching the laws of exponents. When faced with "what is x^0," the teacher gave axiomatic reasoning of "x^0 is always 1. Nobody knows why, you just have to memorize it." When I confronted him about it, his response was along the lines of "we don't have time to get into concepts and teach reasons; we need students to memorize and move on to the next thing for the state tests." More and more, educators in the US are realizing that memorizing things just wont work. You get students who can't recall if x^0 is 0, 1, or x, and they lack the mental model to figure out the answer. Math nowadays strives to give students the mental models to derive answers. I would hope today's students to go "I know x^3 is xxx, and x^2 is xx, so x^1 is x, so the pattern is divide by x, so x^0 must by x/x or 1". This applies to even more common things. Many of us who did well in math did so because we saw patterns and relationships. Now educators are striving to help students see those patterns and relationships. In similar fashion, and to address a concern a few comments up, many older US students (and adults for that matter) can't figure out percents. They were taught a formula of [base * percent / 100] and then they have to unpack the term "percent," where they remember there was something divided by 100, and then they find themselves all mixed up because of failed memorization methods. I can recall in grade school and high school, many kids did not realize they could find 20% of something by multiplying it by 0.2. They first would multiply by 20, then divide by 100, and when I told them they could just combine those two steps, their mental model would break. It got even worse if you tried to tell them "what is 1/10th of the item? now double that, and you'll have 20%". To summarize, "outdated" vs "nowadays," the students would try to add algorithmic steps to their mental tool box that work in very specific situations, and now, many educators are trying to help children internalize concepts. [edit, formatting. I wanted the xxx and xx to originally have an asterisk between them to show multiplication, but then everything went italic. Why does formatting have to be such a pain?] |
While that's clearly a bad answer, it's not clear that an answer to the effect of: "That's a great question but answering it requires getting into a number of somewhat advanced topics. But if you're interested, here's a good place to start." would be.
Personally, I actually could not tell you the answer myself--although I could look it up.
I agree that, where practical, understanding "why" is preferable. But, even in more advanced subjects, you sometimes just don't have the basics to derive everything from first principles. High school physics is an obvious case when most students haven't had any calculus yet. But even intro courses at a lot of good colleges often ask you to accept certain things as true because understanding why they're true requires significantly more background in the subject.