Are We Teaching Math All Wrong?

William Whewell once contended that math education should build upon geometry, which focuses on logic and concrete objects, rather than upon analysis (algebra, calculus), which focuses on abstractions and symbol manipulation. Oddly enough, another blogger -- James Chastek at Just Thomism -- seems to have encountered the same post today and had this to say:

We’ve clearly fallen into exactly the fault that Whewell wanted to avoid, and have dumped the geometrical approach almost entirely while dedicating years to teaching analytic methods. The fundamental pedagogical mistake is in this approach is that it teaches abstractions before teaching what they are abstracted from.

Not content with this error, we went in the 1970s to teaching New Math; viz., abstract set theory in grade school! A lot of geometry taught today is actually analytical geometry: The focus is on formulas for calculating lengths, areas, volumes of sundry geometric figures. Few indeed are those fortunate enough to prove geometrical properties with straightedge and compass and reasoning from the postulates, as TOF was, Lo!, these many years ago in geometry class from Sr. Amelia. 

That was sophomore year. Freshman year was given over to Algebra. In my section -- Freshman 7 -- we covered both Algebra I and Algebra II, the latter being normally reserved to Junior year. If I read Whewell, Siris, and Chastek aright, we should do Geometry first and postpone Algebra until later.

Fermat's Last Stand

Okay, this is a hoot. The link gives the summary of the play, which involves such things as the Shroud of Turing and the Mathematical Pirates as well as an encounter with St. Thomas Aquinas.  The tunes I found hard to hear, but the YouTube site has the lyrics printed.  Herewith, three of the immortal songs:
The Mathematical Pirate Shanty
Thomas Aquinas' Song
The Battle Hymn of the Republic of Letters

An Interview with David Berlinski

 

Found upon the Web and reprinted here without comment, but with some formatting. Berlinski is a mathematician and well-known gadfly and works with the infamous Discovery Institute. However, he also has a wicked sense of humor, very much like the late C. Hitchens. TOF does not know when this interview was written or for what outlet. He is not, for all that, a supporter of "intelligent design." He's better described, says Jonathan Witt, as a skeptic toward Darwinism and a friendly critic of Intelligent Design.

An Interview with David Berlinski

Jonathan Witt

Math Is Hard

Teresa Ghilarducci, at the New School for Social Research, describes a trip to North Carolina, which left a colleague "shell shocked." The colleague asked: "How can it be legal to have so much poverty in such a wealthy state?"



Aside from the intriguing notion of ending  poverty by simply passing a law making it illegal -- what an idea!  We could do the same with gun violence -- there is the following observation. 

According to Kids Count, New Hampshire has the lowest rate of child poverty, at 11 percent. Ranked worst is Mississippi, where a third of children are poor. But Mississippi is poor over all; it has the lowest median income in the nation. And New Hampshire is rich; its median income is the third highest. I get that. So the child poverty numbers may say more about income than about the management of the state budgets.
But let's look at North Carolina. It is the 39th richest state, and yet it ranks 12th for the percentage of children living in poverty--only 11 states fare worse.

Pi Day!

Today is Pi Day.  So along with the Alamo and the Maine, remember the pi.

π r²

But as a friend once told me, "Pie are not square; pie are round.  Cornbread are square." 

If we superscribe a circle with squares of side r, it is clear that the area of the circle is less than four of these squares (left).  Four such squares is 4r².  If we inscribe the circle with squares of diagonal r, the area of the circle is clearly greater than four of these squares.  Thanks to Mr. Pythagoras, we know that if the diagonal is r, the side is SQRT(2X), so the area is r²/2.  Four of these is 2r².  Therefore, the area of a circle must lie between 2r² and 4r², and 3r² seems a reasonable guess.  If not for those curvey lines....  So it turns out to be "a little bit more than 3."  To wit:

3.1415926535897932384626433832795028841971693993751058209749445923078
164062862089986280348253421170679821480865132823066470938446095505822
317253594081284811174502841027019385211055596446229489549303819644288
109756659334461284756482337867831652712019091456485669234603486104543
266482133936072602491412737245870066063155881748815209209628292540917
15364367892590360011330530548820466521384146951941511609...

How's that for a pi in the face?  All hail, the mighty PI. 

However, there is seldom need for anything more than 3.14159; or even 3.1416, if we round.  Every really real circular object can be measured to a specific number of decimal places, and so the ratio C/d will always be a rational number, no matter how many decimals are in our instrument.  So what is this "irrational" pi anyway but a pure spirit, not found anywhere in the real world?  Hunh?  Credulous believers in PI even call it "irrational," but "rationalists" know there is no such thing in reality.  A fig for your pi.

There is apparently a movement afoot to replace π with 2π in radial formulae, calling it τ (tau).  There is a certain elegance in notation in trig if you do this.  Unfortunately, the area of the circle becomes τd²/8 (or τ r²/2 if you prefer), which is not so elegant.  The circumference of a circle is 2π r, which would be τ r.  So τ is the ratio of the circumference to the radius (C/r) while π is the ratio of the circumference to the diameter (C/r) or the ratio of the area to the squared radius (A/r²).  Notationally, tau may simplify linear formulae while pi makes more elegance dealing with square formulae.  Perhaps we should define a number for V/r³. 

This Gödel is Killing Me

I really wish I could take credit for that pun; but no...
Some years ago, J.R. Lucas formerly a mathematician and philosopher of Merton College, Oxford, wrote in "Minds, Machines and Gödel" (and later in The Freedom of the Will) that Gödel's Incompleteness Theorems imply that mechanistic theories of the mind are false. 
Gödel's theorem states that any consistent system strong enough to produce simple arithmetic contains unprovable, though perfectly meaningful formulae, some of which we, standing outside the system, can see to be true.

Example: Consider the statement G: {This formula is unprovable-within-the-system}. 

• If G is false, then the formula is provable-in-the-system and G is true.  A contradiction.
• If G is true, then G is unprovable-in-the-system.

• If G is provable-in-the-system, then it is false; but if it is false then it must be provable; again a contradiction. 
• If G is unprovable-in-the-system, then G is true. 

Thus we see that G must be true and must also be unprovable-in-the-system

(It could be argued that G might be true and unprovable in a system S, yet provable in some larger system S*.  But then there must be a Gödel sentence in S*, and so on.)