You cannot get the most you can get out of this unless you've read the beginning, Part One, and the middle, Part Two, because this is The End, my friend.
The elementary school day most usually begins - after brief administrative details - with language arts - reading and writing and word study. But disciplines and curricula overlap, and so some of the mathematics I saw in the earliest grades were the most amusing - and I don't mean that meanly. On a work sheet, one little girl correctly identified all the geometric figures as required, and answered all the questions correctly, deploying every phonic talent she'd acquired: cirkel, sfere, cilinder... and so on.
The first time I sat in to observe explicit arithmetic instruction, I recognized everything I saw: The kids took turns going to the Smart Board and solving equations, inequalities, and simple "algebra" problems - the "x + 12 = 32" kind of stuff, where the task is to solve for "x". But nobody calls that algebra, and instead of variables there are little boxes to fill with a number. None of that was strange and most the kids did just fine. When adding, they carried their 10's forward and marked the acquiring columns accordingly; when doing subtraction, they explicitly carried back their 10's when needed, crossing out the column entry borrowed from, reducing the entry by 1, and adding to the column needing the loan. Nothing peculiar.
Yet there were the two or three kids who simply had no interest, who neither followed along in the text book nor copied down in their notebooks what was being solved on the Smart Board for their benefit: Copy these steps. Do it like this.
Now, somebody might immediately say: THAT'S exactly why conceptual, over mechanical methods - those algorithms - have to be taught. But if not parsimonious, we can find ourselves mired in a plethora of "strategies," yielding at the extreme the kid who knows at least two ways of figuring what is equal to 5 times 6, but who cannot tell you it is 30 without working through one of them. Those stories about High School Freshmen who do not know their multiplication tables to 12 are not apocrypha - I've met those kids, I knew their names....
Every strategy once routinized, one should know, IS an algorithm: A list of instructions directing actions in an order such that by following them, a correct answer to a mathematical question can be given in a finite amount of time. We are, perhaps even as an entire culture, beside ourselves which of the methods is best, if there is only one best, etc.
My overview observation about all the arithmetic (and it all was arithmetic - whether solving equations and inequalities; determining perimeters, areas, and volumes; adding and subtracting fractions, etc.) comes to this: The teachers who kept the best attention of the students were those who did not bog down in the plethora of solution strategies. A problem would be started. A student would be invited to propose a first step. If useful, the teacher would proceed; if not, the teacher would give
maybe one more opportunity, or simply say "Let's start with this step first instead." Then, another student would be invited to propose the next step; if useful, accepted; if not, the teacher would GIVE the next step. The result was to work through as many examples of the kind of problem to be solved as the time given allowed. One should distinguish that from doing the SAME problem over and over more than one way.
So, for example, a class is adding fractions, reducing improper fractions, etc. They come to a problem with different denominators. Somebody gives a solution, adding first the numerators and then the denominators. The teacher then asks the class: Do you ever add the denominators? NO says the class. Do you ever add the numerators when the denominators are different? NO says the class. What is the first step when denominators are different? Somebody ventures: find a common denominator. "Correct" says the teacher. How do you find the common denominator? asks the teacher. No answer. The teacher: You multiply the denominator of the first fraction by the denominator of the second, and THAT number
is your common denominator. Somebody then says (I am not making this up) Why?
And the teacher says: Because if you do it this way, you will get the right answer. If you do it this way, you will not get the wrong answer. I will show YOU why if you want to know why later, but for now, I want you to follow these rules...
Now, I know full well that to some people, some teachers, what I've just described is
horrible teaching. Yet consider this: When it comes to behavior, we expect the kids to do as asked EVEN if they do not know why. At some point, there is the explanation - so you can learn better, so you can be safe, so you can respect others - but the bottom line is, they are EXPECTED to comply with behavioral rules regardless of their comprehending their purpose or effect.
Yet if the matter is a concrete applicable skill required by the curriculum, the explanation: Because it works and so I ask that you follow it - is somehow improper....
And the teacher I just described had control of the class, and people were paying attention.
Similarly with another teacher whose class was computing the volumes of irregular rectilinear solids - think of a smaller cube immediately atop, or adjacent to, a larger cube except that it is one object, as the typical case (or the volume beneath a flight of steps, etc.) Some of the dimensions are given, and the rest can be computed from the explicit measures.
The teacher asks: Who has an idea where to start? Without pictures, this is a bit harder to describe, but generally, the teacher guides the kids into partitioning the larger object into smaller parts where the formula "Volume = H x W x L" can be applied, and then the sum of each of those smaller solutions is the total solution. Whenever a kid suggests a move getting to there, the teacher accepts it; and when the move suggested is wrong, she just says, very delicately "I'm not sure about that - anybody else?" and
there is ONE more chance for somebody to give the next step; if that isn't right, she says "No, let's do this, because we already know ...." and explains why. THAT step done, she asks for the next one...
Otherwise, quite frankly, the kid trying to figure out the problem in front of everybody else becomes the teacher, and the other kids - either because they already get it, or because they haven't got it till now, just drift away. My point here is NOT that the kid figuring out the problem gets it wrong - but that
THEIR exploration is NOT the same as teaching. The following example will, no doubt, push just about every button anyone has about mathematics instruction in the schools:
A class is comparing fractions, converting decimals, ordering on number lines, etc. They are reviewing a worksheet/test returned to them. A typical problem is: Which is the larger fraction: 4/7 or 5/8? A student volunteers to show how she got it right. Up to the Smart Board she goes; she draws a rectangle; she divides the rectangle in half with a line parallel to the base; she then subdivides the top half into 8 pieces with lines perpendicular to the mid-line; then subdivides the bottom half into 7 pieces with lines perpendicular to the base from the mid-line; colors in 5 of the top 8; colors in 4 of the bottom 7, and
voila, demonstrates that 5/8 is larger than 4/7.
Well now, these are some facts about this:
1) There is, in fact, an algorithm for her method. I just described. It can be written with variables, and the procedure is iterative. In a finite amount of time, following it will let you compare any finite number - no matter how large - of fractions.
2) It is laborious. This took some time - several minutes, but the teacher never intervened, and I saw kids just quit watching her work, and fiddle....
3) It works with larger grosser measures, in an ample space, but more complex comparisons would require a ruler - another metric - and space might be limited.
4) The student showing her work
had every problem on her worksheet correct. EVERY single problem, she'd got the right answer.
5) On a MAP test, her method is acceptable (or so I was told just today.)
6) When a boy solved the same kind of comparison problem, he found the common denominator, etc. Making the conversion simply went faster than drawing the picture....
This of course goes to Ines Segert's fundamental objection to the mathematics curriculum in the schools. I know my citing her is perhaps to my detriment as I seek a Board seat, as she is clearly not the most popular of future ex-board members. But I neither betray nor repudiate intellect. I do not know that I agree with her on all points; rather, because we share a common
lingua franca, I understand what she means when she put her point this way (in paraphrase): There is a well established, well known canon of solution methods and algorithms which mathematicians have established through the course of the past several centuries, and the District does not teach this canon.
I do not know if that is true about all the District, or about some levels of mathematics in the district, or only about some teachers in the district; I only know I understand her criticism.
So consider this episode as a more clear demonstration of what the absence of the canon might mean: A class is computing areas. One of the areas is an isosceles triangle. A method in the canon would be: complete the rectangle on the triangle, compute that area, and divide that area by 2. That is not what I saw. Again, the method was to (without pictures, again requiring your imagination) break the triangle into unit squares, and then re-assemble the half-units together to make full units, and then count them. THIS DOES WORK. So do not misunderstand my point. The success depends upon the exactness of the selection of the unit square, such that the slope of the sides bi-sects the unit squares. ...
The issue then for me, as I understand Dr. Segert's concern, is the generalization of any strategy. Clearly, some developing minds will benefit from the kind of visual-manipulation of space the above exercise employs. The question is: When in a student's cognitive development has the EXAMPLE been conflated with the universal principle, such that, her reliance upon the example has become an impediment, rather than a step in advance?
... Well, I could go on like this,
ad nauseum for some readers I am sure. I once self-diagnosed my logorrhea. So I should impose my halt, and leave you with this:
I did like most of what I saw on my visits. I have omitted some particular things that did bother me which go less to curriculum than to discipline. Still, the fact is: Most kids I watched and spoke with are getting an education, though to varying degrees of perfection. That is good. But though one often errs to make the perfect the enemy of the good, I am not yet going to claim that the Columbia Public Schools are good enough....
Word.