These puzzles have been chosen because
they illustrate difficulty caused by
the simplest of mathematical relationships.
...here we have an opportunity we should take advantage of...
Here we have doors into some of
simple, but seldom seen, superstructure of science.
These puzzles can be solved
with the simplest of math
(but math that's seldom seen)
Clicking on a picture leads to a place in Knowledge for Use web site where the puzzle appears.
||Pack the blocks (18) in the box.||
||Arrange the ellipses so that the closer they are, the more similar--and the more similar, the closer.|
|This one is spatial visualization, not abstract math. The insights into the mathematical abstractions generally develop at the beginning of adolescence. Children may see it when their parents don't.||Counting squares to discover that 62 = 62 is only a beginning here. Noticing that a parity relationship exists is a very small step, but it's a step toward the more abstract...seldom seen. It is, however, very easy to see.||This puzzle is virtually unsolvable by trial and error. The arithmetical principle of the smaller puzzle (to the right) is a virtual necessity for this puzzle.||This puzzle yields easily to trial and error. It can, however, be solved by a simple arithimetical principle of odd and even numbers of square spaces--in the 9 planes of the 27 little spaces of the box, and in the numbers of those spaces provided by each of the blocks.||This puzzle has, in our opinion, as great a potential toward more effective understanding of the role of mathematics and science in human life as can be found. For this reason, we expand on some of its implications below.|
and people by size, and colors,
and people by intelligence, and people by "worth,"
and things by "value," and things by cost...
and things that people think can't be measured (even though they can be)
because people don't see the principle that orders the ellipses.
depends on what is meant by "size." The basketball coach wants height; the football coach wants weight (actually he wants mass, and the distinction is important to anyone who understands Newton's discoveries of three centuries agoand which touched off the "scientific revolutions" we have today). There are many different aspects to "size." When we do the usual ranking (linear) we are confusing those different aspects. We usually don't know we are confusing them. We thinkwe feelthat unique rank order is the meaning of measure, the meaning of putting things into mathematical language.
That is profoundly wrong.
Graphically, we can put it this way:
that's the simplest way we can put it.
a little hard to see where this should lead, so we use color as an alternative
Here the shades of blue might indicate "intelligence." Most people think of intelligence as capacity (but sometimes speed) of memory and recall. Math prof Jerry Manheim sees the oversimplification of this viewpoint when he says, "A measure of how good one is in math is not what one can remember but what one can afford to forget."
memory, recall, and even linguistic fluency, are important, but in science
and math there's something else that's far more important. It's a
It definitely isn't the rote and ritual that the vast majority of science students see science as being. It's a set of intelligence components that are orthogonal to that simpleminded notion of "g factor" and which is exemplarized by our comparison of two different physics faculty (from two different universities) in our intriguing observations of Post Modernism.
We're taking only a first step when we replace those blue dots that represent g-factor of intelligence with dots having a second component of color. J. P. Guilford finds, with his factor analysis (one 19th century development of the math of multiple components), seventeen different orthogonal components within what we call "intelligence."
If you take a magnified look at the tiny color-CRT dots on the screen of your computer monitor, you will see that those "BI-CHROMATIC" circles (in the middle of the column to the right) have the blue CRT-dots turned off. Only the red and green dots are in use. They are the two dimensions of the ordering. Two "degrees of freedom." Two "components" of a vector measure. Two of something that is almost universally seen as always one. By psychologists looking at human intelligence. By economists looking at value, benefit, cost... By bigots looking at superficial qualities of "other" people. By "authoritarian personalities" looking at social structures, such as in the miltary, in prisons, in corporations, in families, in neighborhoods...
Scalar measure is rare in the real world. Very, very rare! Oversimplification is the rule.
is one of the best exemplars for understanding
the development of formal operational reasoning.
pair of Paradox Box
Spread eyes on left pair Cross eyes on right pair