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Who's the target audience for that explanation? It seems like the initial question can be solved much more easily by rewriting x^2 + x = 6 as (x)(x+1) = 6. Seei
by wcarey 14y ago
Who's the target audience for that explanation? It seems like the initial question can be solved much more easily by rewriting x^2 + x = 6 as (x)(x+1) = 6. Seeing that x can be two seems intuitive, and that x can be -3 a bit less so.
I wonder whether talking about algebra as the manipulation of statements to construct consistent sets is more productive than talking about error in systems?
Instead of x hiding a value, we want to know what statements can we make about x that are consistent with x^2 + x = 6.
- pavel_lishin 14y ago> seems intuitive You can't really write "well, it seems intuitive" into a logical proof, or into an algorithm.
- ChuckMcM 14y agoGood point, you write it "by inspection" :-)
- alanctgardner 14y agoBut the point of this site is seemingly "understanding", not an algorithm - see the title of the page ("right, not rote"). However, what they really seem to be doing is breaking down what should be an intuitive process - understanding equalities and how to work with them - into several smaller, vague sub-processes which also aren't really easy to understand. I would argue that explanations like this might hurt someone's chances of understanding concepts like systems of equations later on.
- pavel_lishin 14y agoI thought it was meant to give us an intuitive explanation for why we approach things via the "factoring algorithm" as a method of solving these guys. In any case, it's just a simple example case. x^2 + x = 6 can probably be solved just by staring at it for a little bit, but once the equation gets more complex, trying out a couple of numbers in the [-5,5] range is not gonna work anymore.
- louischatriot 14y agoyour factoring, although correct, is not really useful in the general case. For example with x^2+x=1.
- wcarey 14y agoTo be sure. But it is useful in this particular case, so why wouldn't we use it? Similarly, you can write every multiplication or natural numbers as a summation, but we don't because multiplication notation adequately handles the case where the term of summation doesn't change and is simpler to reason about. The motivation for summation is that sometimes we want to add up things where multiplication notation fails. Similarly, the motivation for factoring is that sometimes we run into an equation that we can't intuitively reason about. His explanation would be more persuasive to me with an example where simple inspection fails. If the target audience for the article is people who understand factoring and are amused by thinking about it in different ways (the bit about systemic error was new to me and intriguing!), then I think it succeeds. If the target audience for the article is people who don't yet understand how to factor, but do understand why to factor, then I think it sort-of succeeds. There are probably people (the author is certainly one?) for whom his explanation will seem intuitive and compelling. If the target audience is people who neither know how to factor or why, then I think it's more problematic. It's symptomatic of a rush to abstract and generalize that often leaves students wondering why the abstractions are more valuable than more intuitive reasoning. An example that requires factoring would speak to that motivation a bit better, I suspect.
- oo 14y agoWhy not? Unless you limit yourself to the domain of the set of rational numbers. Enlarge the domain by adding in the magic number "5^(1/2)", and from an algebra standpoint there is no difference between solving X(X+1)=6 and X(X+1)=1. Well technically there is indeed a slight difference: in the case of X^(X+1)=6 the roots live in the same domain with the coefficients, while in the case of X^(X+1)=1 the roots can only be find in a larger domain where new kinds of numbers are included.