Some underlying senses we need before
begin to understand and use mathematics
on our route from infancy to science-see:
. . . in mathematics
|Jerry Manheim tells this story from his course
in Differential Equations:
"I asked him to help me work through a problem (since I firmly believe Math is not a spectator sport). No matter what question I asked he was silent.
"Finally, in desperation, I said: "Look, this is basically subtraction; what if the temp is 4 degrees F outside (O.K., so it 's never 4 degrees in CA....pretend you're in Alaska), and then the temperature drops 7 degrees, what is the new temperature?
I decided to wait him out on this one and eventually he told me that the temperature is now zero.
B's and C's in College Alg, Calc and Analyt (3 terms) and doesn't know about neg numbers! Who in hell's name are we doing a favor for....surely not this ELECTRICAL ENGINEERING student.
|The young budding scientist needs to understand reversing an action--an action such as adding. The reverse of adding is subtracting. And this means he must understand negative numbers. We often can subtract more than we had to start with.||
Negative number -- inverting an operation
as moving up (right) on the line.
just the beginning.
Yet some learners need more work on it.
A single action is the simplest possibility, but In the real world, many actions work at once.
|We might undo some change we make. For example, consider that
we have placed something on a balance pan that upsets the balance.
We can undo our action simply by removing the thing we just put on.
We can also restore the balance by putting something on the other pan which exactly compensates for the weight we put on at first.
just one result
Mother Nature doesn't care
is what children begin to see:
An action can be reversed in different ways:
restore by undoing
or by rebalancing
|Look again. We can restore the balance in a number of other ways, too. We might move the balance pan on which we added the weight a little closer to the fulcrum. Or we might move the other pan farther from the fulcrum. Or we might use several combinations of adding or subracting weight and moving pans.||
When two simple influences can each balance each other, we can see a complete set of actions:
-- the orignial action
Many different factors can add new complexities to real-world events.
There's always another way
The "INRC group" is one of the central logic themes in the pioneering work of Jean Piaget. Another is the set of all possibloe logical relationships of two or more propositions. Our abilities to properly identify and sort through multiple, interacting influences and variables in a problem are based on our abilities to "see" these relationships.
Click on the diagram to see where this lies in the development of mathematical thinking
|We might be able to see logic that we can't explain or describe -- just as we can identify people whom we know when we see them but can tell someone else only a little bit of what is is that lets us know we are right. Most of our mental processes are unknown to us.|
Similarly, "He could even care less," and "He couldn't care less."
“It got to minus ten degrees below zero,” and "It got to ten degrees below zero."
"Do unto others as they would do unto you." and "Do unto others as you would have them do unto you."
All of these share a common error. That error involves a simple abstraction—of the kind that is the heart and soul of modern science and mathematics.
...What is that error?
error in common to the first three statements is the failure to see the
the negative by another negative—a
Negation is a slightly tricky concept. Negation of negation is even trickier, tricky enough to frequently trip up a lot of people. And a lot of science requires a good sense of negation of negation.
Mutual reciprocity (exemplified by the Golden Rule) is a symmetry concept which conceptually stems from a system of potential negations and negations of negations. In physics, the concept of irreversibility is usually misunderstood, demonstating the difficulty of understanding negation of negation. The usually misunderstood concept of Newton's third law of motion (action and reaction) demonstrates the difficulty of understanding mutual reciprocity.
|Some common (mis)uses of words demonstrate that negation is a subtle (abstract) concept at the edges of human comprehension.|
We have ways to use language to express negation.
(not sensing what a multiplicative effect is in such a use of words)
|He hasn't got food on his plate
Our confidence will not easily be restored.
Only over my dead body will they raise your taxes.
|He hasn't got no food
on his plate.
Our lack of confidence will not easily be restored.
Not over my dead body will they raise your taxes.*
...Sometimes we want to say that the negation is not.
||Here the speaker doesn't realize he wants negation of negation.|
|A second one? This guy couldn't
care less about the first.
||A second? This guy could even care less about the first.*|
And language can express negation in mathematics, too.
|Mathematics is careful to mean just what it says.||The negative of a negative is a positive.|
|It got to ten degrees below zero.||It got to minus ten degrees below zero.|