The brain is a computer uniquely programmed to manipulate concepts and the symbols we (it) invented to represent those concepts. We develop concepts by abstraction from concrete examples; for example, from observing two hands, two arms, two legs, then two apples or two dots on the page, we arrive at the concept of TWO without reference to any object. We learn that the quantity is called “two” (in English), its symbolic representations are the letters T, W and O; the numeral “2” which is more convenient than other representations, such as “II;” and vocalizing the sound of “too.”
But you still have to MEMORIZE the concept’s symbolic representations in the different contexts, which, with the exception of the internationally accepted use of the numerals, is different in every language. What IS the same in every language, and the reason math is called the universal language, is that its rules (its “grammar”) are the same in any language (** + ** = ****; that is, 2 plus zwei is always quatro).
And the reason we used to and still have to memorize multiplication tables, for example, is to learn both the symbolic representations and the rules. So, for example, 3 x 3 = 9 actually means adding 3 to itself 3 times (the definition of “multiplication” is “repeated addition), which with marbles in the sand or dots on the page you can show as *** *** *** and then you rely on your knowledge of the names of the quantities to start counting 1,2,3, 4,5,6, 7,8,9… or you count by threes 3 6 9… or you simply recite what you memorized, three times three is nine.
The point is, your brain can’t help memorizing facts and relationships. And it can’t help manipulating concepts it has formed by observation and memorization. That is what it is built for. But it needs to memorize facts and relationships before it can form and manipulate concepts. That is how it’s built.
But the error in her disapproval of CCSS is very disturbing:
“It’s too big a jump for too many kids,” she said. “It’s a serious problem at the younger level where children are struggling and could get left behind.”
WHAT??? CCSS is a step UP??? Where is her head at?
OK, so she is an art teacher. For her art to make sense,. you have to know which King she is depicting.
These examples show the noble sentiment behind CCSS math. They want all kids to develop the mental agility that the nerds are born with. Too many kids for too many generations had the attitude, don’t bother me, don’t try to make me understand, i never will and I’ll never want to, just tell me the formula to get the right answer. Or as the Russian spy says in the movies, the FOHrrr moo law.
The trouble is, it takes a lot of repetitive practice before you become knowledgeable and comfortable enough to see how many different ways you can represent a problem and find all the possible solutions. Remember counting on fingers? Some people never got past that stage.
And how did our technological civilization respond?
Our nerds built cash registers with pictures on the buttons of a vast keyboard. one button for each kind of burger, fries and coke on the menu. To gain speed, the clerk memorizes the locations of the keys, not the prices on the menu. It also helps management to automate inventory and price changes. No one in the place has to understand or care about any of the details. Just hit the right button.
The same technology enabled nerds to build the infernally complicated accounting rules, tax codes, insurance options and the subsidies under 0bamacare. Why even try to understand? Let the computer handle the details.
So the practical question remains, just what need is there for everyone to understand the minute details, and just how many nerds are needed to advance (or screw up) civilization itself? We the users only have to know how to navigate the user interface. And as Steve Jobs has shown, it takes very few nerds to set the world on a new course. Everyone else has the choice of trying to hang on, or figure out as much of it as they can.
Sounds unfair? Since the dawn of civilization it has always been this way, and it will remain so forever more. The trick always was how NOT to stifle the exceptional few who could make a difference. And we see it here with these CCSS math examples. The evil is not in the desire to raise the kids to excellence by challenging them with higher standards; study after study shows that kids will respond to that. The evil is in the doubt whether the CCSS are higher standards, and the evil definitely is in how the CCSS are implemented. If kids end up being stifled worse than before by being forced to conform to nonsense, then civilization itself will come to a standstill.
Some people are OK with that. R U?