## Sunday, January 29, 2012

### Against Factoring Trees

Nowadays, I'm anti-factoring trees. It's funny, because they're not usually part of the classes I teach, but they've come up a few times recently -- including twice just yesterday (as I write this), when they were included in a new book I received, and then within an hour a student came asking about them (because they were part of a YouTube lecture on reducing radicals she'd been trying to watch).

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.

1. 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.
What do you think: can elementary teachers keep using them, until a HS teacher shows students the "normal" way to factorize?

2. ^ 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.

So 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.

3. I was taught factoring by the sieve of Erastothenes method in elementary school in grade 3. This was in Poland, in a very run-of-the-mill public school on the outskirts of a city of ~500,000, in 1985. Somehow it stuck with me quite well, if such an anecdote is of any use. The former village school was turning 100 years old, and the building grew around the original brick house that was the school in the late 1800s.

We did not use factor trees as a representation, or at least I don't recall it at all. We simply wrote down an ascending list of factors as their product equal to the factored number, each factor explicitly repeated, say 8 = 2 * 2 * 2. The division and keeping of the current result was expected to be done ideally in memory, or, as it's called in the U.S., by mental math.*,** It was acknowledged, of course, that we were not all that good at division nor at mental math, so we were, reluctantly, allowed to keep margin notes of the same, and be tidy and compact in our long division (where called for) -- after all, you wouldn't want to run out of space on the margin!

We were routinely reminded that those marginalia were a crutch, and in the long run it was to be done in memory. I also think that we were expected not to take any note of carries nor borrows in long addition/subtraction in grade 3. It was allowed in grade 2, again, reluctantly. It made the pencils last longer, you see ;)

In retrospect, I think that the K-3 math education I had in elementary school was stellar. It was the most bang for the buck you can imagine. Our teacher (we had just one teacher except for arts, PE and environment) probably subsisted on what was gas money for her U.S. counterpart at that time. We were expected to be tidy and use as little paper as possible, so a 60 page 5mm-on-center-gridded math notebook would usually last at least a semester. Here's what I recall. We were taught basic set theory in grade 1. Then positional representation, with emphasis on the fact that it was a representation. That's why at least I could never be dragged into silly and pointless (cuz they miss the damn point!) discussions about how close to 1/3 0.(3) *really* is.

I don't believe that numbers were referred to as abstract objects before grade 5, but there were certainly hints to that in grade 2 at the latest. We were taught, explicitly using the professional terms and not some kiddyification of the same, about the set of natural numbers, then the set of integers, the set of rational numbers, and about irrational numbers. I mean, come on, I never had to use the term "irrational number" ever in my life, but I knew it at age 8. That's *all* by the end of grade 3, I believe.

I mean, come on, we had to be fluent in simplifying fractions and all four long operations on floating point decimal representation numbers by then. I think we had to understand the roman notation as well, but that was an afterthought.

* Whoever came up with that term was mental, too.

** My daughter is often told, in a decent public school in the U.S., in grade 2, that they are not supposed to do mental math, and instead have to follow some useless canned procedure to arrive at the result. You know, because that's what's "expected of a good student". That's what teaching to the test gives you, I'm afraid.

4. By grade 8 we knew about the polynomials, very basics of properties of functions and simple analysis of functions, basics of trig functions. We'd routinely plot 2nd order polynomials, and were expected to have a parabola stencil. It was then up to one to correctly locate the graph axes and calculate scale for the unit divisions that had to be marked.

By grade 12 we pretty much were done with what would pass for calculus 101 and linear algebra 100, but that was admittedly in a public high school that was less run-of-the-mill.

At that time, elementary school was grades K-8, and high school 9-12. I'm afraid to admit that most of the math I know I have learned by end of grade 12. I have of course learned quite a bit of applied math after that, but it seems like not all that much, somehow.

5. I now recall that we had to have a sine stencil as well. Both were expected to be scaled in 1=1cm in both abscissa and ordinata. Polish being the pretty language that it is, plenty of mathematical terms are repurposed common words, no latin imports here. Integrals are called virgins, using the archaic variant of that word.

1. Great observations, thanks for sharing those!