By factoring trees, I mean the method for producing the unique prime factorization of a number that looks like this:
By which we can conclude (re-sorting the leaf nodes) that 48 = 2^4 * 3. Of course, this is pretty customary, and it's how pretty much everyone I know (including myself) learned how to do it.But now my primary complaint against them is that they're a nonstandard method of writing mathematical relationships, and most of all, they're a lost opportunity to practice writing equality statements. With all the problems that students have using, writing, and understanding the equality sign, why not use this as an opening to reinforce their meaning -- especially so in a context just like this, where we do not intend to simplify (evaluate) on the right hand side? Why not instead write in a more standard format like this:
(Or whatever your preference is for use of parentheses or exact number of steps.) It highlights all these issues with the meaning of equality signs that we struggle with later on students' behalf, and it avoids using a special one-off writing technique for the singular task of factoring a number. It's likely easier to read for some students (who may have trouble identifying where the leaves of the tree are). It even saves on lines of paper, and is easier to type out in an email or website if you have to do that. This is actually what I do in class when it comes up now. The more mental connections we can make to the "correct" way of writing math, the better.
I agree with you about getting students used to doing "real math" as soon as they are ready. However, I feel that factor trees are not a bad bit of scaffolding for younger students, until they are ready for fully standard notation.
ReplyDeleteWhat do you think: can elementary teachers keep using them, until a HS teacher shows students the "normal" way to factorize?
^ Good question: I think from where I stand I would actually recommend "no". For what it's worth, I'm dealing with classrooms full of community-college students, none of whom say they've ever seen it before when it comes up in algebra, etc. Then they turn to other resources (like one of my students last week looking at YouTube videos) and wind up confused from the one-off notation.
ReplyDeleteSo if it is taught in elementary school, (a) I'm not seeing any evidence that it "sticks" for my students, and (b) it causes a bifurcation when older students go to read up on it for (what appears to be) the first time.
Better to synchronize it. And I don't see why proper status and usage of the equality symbol is out-of-bounds for elementary students; in fact, it's probably most critical to be taught at that stage.