I believe that word problems can be used for both training students to use some methods, as well as actually solving real world problems. For the former, we can use the example of (a+b)^2. We can definitely show the students the algebraic, or even the geometric derivation like so:
Source: https://i.ytimg.com/vi/n_4t9JnwE9s/hqdefault.jpg
While the practicality of this problem is nonexistent, it does give the students more practice, and builds skill.
Generality, and abstraction are very important aspects of mathematics. Another example that can be brought up here is whether teaching factoring is valuable. Factoring, the way we do it in high school, does not generalize and it is not connected to mathematics in its general picture. Whether we should teach it or not, I am not yet sure.
When it comes to "the idea of 'pure' vs. 'applied' mathematics", from our experience it can be thought of applied mathematics and mathematics that hasn't been applied it. There are several examples, such as number theory (RSA), when mathematics was there for the taking and to be used in real world applications.
The ideas above definitely rely on our familiarity with contemporary algebra, since it's the framework in which we have built the reasoning and general ideas. It could be that some aliens have invented their math in their own way, but the math we have thought out so far would probably match up to isomorphism with theirs.
All of these ideas related to Babylonian and Egyptian mathematics in the sense that humans have been thinking abstractly and making abstract mathematics ages ago, and it's still a continuing process.
Very interesting discussion, and I like the thoroughly impractical (but somehow satisfying) word problem you've used as an illustration. Lots to think about in terms of whether to teach factoring, and whether an alien version of mathematics need necessarily be isomorphic to what we have developed!
ReplyDelete