Dr.Traveler
Mathematician
- Aug 31, 2009
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Ah, we hear from an NEA sponsor.
Trite response. Makes for a promising first line.
Have you ever taught a class of 22 first graders? (22 being the class size limit in Texas for elementary education the last time I checked) A third or more have an older sibling and enter knowing they can slack and the teacher cannot complain. Four or five have serious learning disabilities (generally at least 1 is a crack baby) and at least two (Perhaps from among the ones with Learning problems perhaps not) are chronic troublemakers. You are not allowed to fail more than two. You are not allowed to get rid of the troublemakers or send the truly brain damaged ones to special education.
So, your solution for this is....
The NCTM is trying to address the failings in the teaching of mathematics. If these are the problems teachers are facing, then the school systems and parents need to address them.
The point is, the bar has to be set high so the students rise to meet the challenge. Your paragraph above makes it sound like the appropriate response is to bring the bar lower and let the kids crawl under it.
Bottom line is this: Everyone knows there are problems in the classroom. The NCTM is trying to address the issues related to mathematics.
Now sure, if you start by selecting only the best candidates out of an entire county, then maybe you can begin working on something more than arithmetic at first grade.
So do so. Pull the ones that can handle the more advanced topics and teach them.
Maybe.
But if you have a set of typical students all the "extra" distractions will be just that - distractions from learning ANYTHING related to math.
Those students will never learn Algebra. They will never get beyond the ability to punch numbers in a calculator and hope for the right answer. They won't be able to manage a tight budget well, nor plan for their retirement properly. They will purchase the large economy size even when the price per unit is more than the smaller size because they can't do that much basic math.
More drivel along the lines of the first paragraph.
As for the Euclidean method - the teaching of Geometry in the classical era was mostly done for the already capable elite who had mastered the art of Arithmetic at an earlier age.
Those students that "mastered" Arithmetic probably learned from Nichomachus Introducto Arithmeticae, which like many textbooks of the day consisted of lists of problems with either no solution, or no indication of how the solution was derived.
Outside of Nichomachus' book or Diophantus book, it was only much later, basically the late 1400's or early 1500's, that mathematicians like Vieta really made Arithmetic a widespread study. Prior to that Arithmetic was done with geometric methods.
Children can learn to tell time because they have a concrete example how it works - the watch on their hand. It is a huge conceptual step from that to modular arithmetic as anyone who has worked with 6 year old children knows.
And with manipulatives you can teach children about symmetric groups or other modular math. The question is should we?
Cumulative tests? Of course tests are cumulative, this is math, where everything they learn build upon what they already know.
We say that in Mathematics, but if you've taught that you know it isn't 100% true once you're past the very basic concepts of Arithmetic. As I said before, when students reach me in Calculus they can't apply the power rule to a simple cube root.
I know for a FACT they were taught exponent rules. They've simply forgotten them as soon as it didn't show up on a test.
However, the students that inherently understood the logic behind how the exponent rules work recover faster than those that simply memorized to get through.
Subtraction (2nd grade) builds on addition. So does multiplication (3rd grade), as does Long Division (4th grade) - in fact starting in 4th grade everything is used for long division.
Trying for too much means the students have an excuse to fail, and justifying an approach because Enrico Fermi could do it that way is asinine.
So again, you're in favor of lowering the bar?
Yes, an advanced approach won't work for every student. Not every student can sum up the numbers from 1 to 100 in elementary school by developing the formula. However, it doesn't hurt to try, and if it sticks good.
Maybe the best way will turn out to be memorization and drilling instead of thinking and reasoning. I can guarantee you though, a student that gets by in life by simply memorizing algorithms will be dead in the water when faced with Integral Calculus, Differential Equations, Analysis, etc. If the goal is to produce more students that can succeed in advanced mathematics course, then more than memorization is required.