First stab at actually teaching

(Originally written March 2012)

Still doing the math intervention twice a week.  There are always some stragglers who don't show up on Monday, so this week she had me teach them on Tuesday to try to get caught up.  This worked out well--since I'd sat through the class on Monday I knew what to teach and how to teach it.  Experienced teachers can sometimes "wing it" through new material, leveraging their overall knowledge of teaching techniques to come up with something decent with no real preparation.  I don't have that, but I am able to do the opposite--if I have something like a script to follow I can follow it, and that's pretty much what I did.

We are covering percentages.  The basic idea is that every percentage computation is of this form:

percentage/100 = portion/total

This breaks down into two broad categories:  overall proportion, and change proportion.  The overall proportion is easier, you tell the students to set this up first thing, and look for the "is number", the "of number", and the "percent number".  So, they normally write something like this:

x %                   15      is
-----------  =   ---
100                   60      of

It really helps to have them write out the "%", "is", and "of", which helps them keep track of what they are doing, retracing their steps if necessary.

Normally a word problem will give you two out of the three:  you set a variable for the third value, do a cross multiplication, then divide out to make the "letter a loner".  They can learn to do this mechanically with repeated drills, you need to keep after them to write out their steps.  If a student doesn't understand something about a math problem, 90+% of the time they can figure out what went wrong if they simply write out all the intermediate steps.

The change proportion ones are trickier to do, and much trickier to teach.  The first step is normally to write out the problem template similarly:

percent      change
-------- =  -------------
100            original amount

But there are usually two major steps you have to do:  compute the change amount, and compute the change percentage.  Sometimes you have to do one first, sometimes the other first.  I didn't state that explicitly, perhaps I should have.

One thing I do do that the other teachers don't, as much, is the Socratic method:  when introducing a new concept, I ask a question and then pick out a student to answer it.  This helps keep them subject-focused.

One problem I'm having, though, is keeping *myself* subject-focused when the teacher is teaching!  I spend a lot of time making mental notes of her teaching techniques and the reactions of students, to the point where I lose the thread of the subject at hand.  This will be a particular challenge in math, since the mindset of teaching and the mindset of math are so different.  I kind of got caught when my pants down when she called me up to the board to work through the problem, since I had been watching how she and the students said and did things, to the point of excluding what they were actually saying.

How to Organize the Different Components of Math Learning

This summer I've been studying how to put math concepts across to young minds, and there are a lot of different strands and substrands to it. The five strands model seems to be widely accepted, but it seems to me that there are really only three, with one having many substrands.  The three are conceptual understanding, procedural fluency, and problem solving, as described in
http://www.p12.nysed.gov/ciai/mst/math/standards/revisedlintro.html . In this context "conceptual understanding" basically means Bloom's taxonomy, procedural fluency means doing the basic operations quickly and accurately. Problem solving is stickier--it means taking the concepts you've already learned, recognizing which ones to apply to a given problem, and how to apply them. This isn't easy, especially because many problem have more than one way to solve them, and it's what I need to learn more about.

The five strands, by the way, add "productive disposition" (basically, a positive attitude) to the list, and split "problem solving" into "strategical competency" and "adaptive reasoning". Productive disposition is of course an important component, but not something to be planned or tested for, so you can't make it part of a formal structure. As for splitting up problem solving--I don't see any reason for that, maybe when I work out the concept better I'll see the point. I should think that through while I'm playing chess.

Reaching a class: the first five minutes are crucial

The teacher I'm helping with my volunteer work had a rougher time than usual today.  She had to drill them with jumping jacks, insistent questions, etc., to get their attention.  She eventually got the class back to paying attention, sort of, but it was a lot of work.  The problem was the very first problem, which was not an equation (too many unknowns--unknown number of pizza slices for unknown number of guests) and for which the technique she wanted was not totally clear.  The students intuitively knew the answer, but it was not clear what she was trying to get across with the technique, and she basically lost them for a while because of that.

It appears that, when you are starting a class, you need to boot up with something interactive, comprehensible, and not too challenging, to get a good feeling going around the class.  Even if it doesn't contribute too much to the actual learning, you need to set a friendly, positive "onda".

Reaching the kids: set a few simple rules, lather rinse, repeat, repeat, repeat

Currently volunteering in an intervention class for sixth graders.  The teacher is enthusiastic and smart, I am learning a lot from her.  Right now they are covering basic algebraic techniques for solving equations.  To drive the point home to the kids, she started off with two simple rules:

1)  Letters are loners
2) Anything you do to one side of the equation you do to the other

That really is all there is to it, when teaching the basics of algebra.  You want one side of the equation to have a single instance of a variable with no coefficient (letters are loners), and the other side to have no constant.

Many kids don't get the "what you do to one side you have to do to the other" thing right off the bat but, as with many math concepts, it sticks with them better if you anthropomorphize it.  In this case, she says that if you do something to one side the other side "gets jealous".  I tried this out on my son and at first he took it too literally, saying "why would one side always want what the other side has"?   So I explained that sides of the equation are like his little sister--if she sees you eating liver and mud, then she will say "I want some liver and mud too"!   Then it made more sense to him!