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 Learner Determination of Content:  Experiences in Mathematics Education
              Outside and Apart from the School Curriculum

                         Dr. Harold Don Allen
                        6150, avenue Bienville
                   Brossard, Quebec, Canada J4Z 1W8

A Gifted Globe:  Tenth World Congress on Gifted and Talented Education
                    Toronto, Canada, 9 August 1993

Selected conjectures, problems, and areas of exploration commonly selected for
individual, small-group, and class investigation.

SQUARES OF DIFFERENCES.  Four numbers (non-negative integers) are chosen and
written at corners of a square.  Pairs of adjacent numbers are subtracted,
smaller from larger, and differences are entered at mid-points of corresponding
sides of the square.  Joined these mid-points yields a new, rotated square.
The process is repeated:  "0, 0, 0, 0" is reached in a finite, and surprisingly
small, number of steps.  Call "Type n" a quadruple which reaches "0, 0, 0, 0"
in exactly n steps:  thus, "3, 4, 3, 4" is Type 2; "5, 3, 7, 5" is Type 3;
"2, 3, 5, 9" is Type 7.  Students have found quadruples through Type 7 with
little difficulty; higher orders call for luck, intuition, or ingenuity:
"1, 21, 58, 126" has been reported to us as Type 15; "60, 181, 402, 810" as
Type 16.  Experimentation with triples, quintuples, sextuples (and corresponding
polygonal configurations) leads to perhaps unexpected results.  Students have
conjectured that there could be a technique for "reversing" the process,
extending to a srequence of "outer squares" and therefore to quadruples of
any Type.

HAILSTONE NUMBERS.  The fanciful name for numbers entering into the rising-
falling pattern postulated in the "3x + 1" or Collatz conjecture.  Choose a
number (a positive integer).  Necessarily, the number is even or it is odd.
If it is even, divide it by 2.  If it is odd, multiply it by 3, then add 1.
Using your result, repeat the process.  The sequence so obtained tends to rise
(due to odd terms), then fall dramatically (due to a multiple factor of 2).
Five, for example, rises to 16, then falls:  8, 4, 2, 1.  Any number which
reaches 1 (that is, in the "hailstone" metaphore, comes down to the ground) is
a hailstone number.  Which numbers are, and which are not, hailstone numbers?
Some numbers rise to astonishing heights, some have long paths of ups and downs.
A student of mine chose to investigate "73":  three walls of a classroom were
covered before "1" was reached. "27" also makes for a rewarding exploration.
Which numbers are hailstone numbers?  Every number that has been tried--but
there is no proof that such must be the case.  "Plausible arguments" can be
offered on both sides of the question.  Any variation on the "hailstone"
procedure, for example a "5x + 1" rule, can yield quite different results.
An ingenious student version noted that numbers greater than 1 are prime or
composite:  if prime, double, then add 1; if composite, divide by the smallest
prime factor, then subtract 1.  Progress of such sequences can be graphed
instructively.

REVERSALS TO PALINDROMES.  "Madam, I'm Adam," "Was it a cat I saw?":
palindromes exist for all major languages, and children know them well.
Correspondingly, a number palindrome has digits which read the same
left-to-right or right-to-left.  For this activity, take a number, a positive
integer of two or more digits.  It is a palindrome (like 30504) or it isn't.
If it isn't, then reverse it (write its digits in the opposite order, initial
zeros being allowed).  Add the number and its reversal.  The result may very
well be a palindrome (643 + 346 = 989).  If it is not, repeat the process.
You obtain a palindrome, in most instances, in a very few steps.  Other
numbers take somewhat longer, and a few (such as 196) offer a challenge to
those who wish to be first with a conclusive result.  Variations?  For one
thing, why limit ourselves to our customary number base?

THE POLE OF A NUMBER.  A concept that rewards thoughtful investigation is that
of the pole of a number.  By number, understand a multi-digit positive integer,
say 6283.  Consider its digits, 6, 2, 8, and 3.  Write the largest number that
can be formed from those digits, 8632, and the smallest such number, 2368.  From
the largest, subtract the smallest:  8632 - 2368 = 6264.  Repeat the process:
6642 - 2466 = 4176.  Again:  7641 - 1467 = 6174.  Satisfy yourself that 6174
now will endlessly recur.  For a four-digit number (digits not all being the
same), this inevitable result, 6174, is called the pole of the number.  Is
there a pole for five-digit numbers, six-digit numbers?  Might there be some
instructive way to alter the rules?

A RACE BETWEEN SETS.  Polya defined a "number of the even type" (not necessarily
what we think of as an even number) as a number (positive integer greater than
1) which, when written as a product of prime factors, has an even number of such
factors.  Thus, 441 (7x7x3x3, 4 prime factors) and 1500 (5x5x5x3x2x2, 6 prime
factors) are numbers of the even type.  A "number of the odd type," correspond-
ingly, when written as a product of prime factors, has an odd number of such
factors.  Thus, 48 (3x2x2x2x2, 5 prime factors), 127 (127, 1 prime factor), and
51975 (11x7x5x5x3x3x3, 7 prime factors) are numbers of the odd type.  Note that
2, 3, 5, 7, and 8 are numbers of the odd type, while 4, 6, 9, and 10 are numbers
of the even type.  Through 10, numbers of the odd type lead numbers of the even
type 5 to 4.  Polya's 1919 conjecture:  starting with 2, counting to any number,
however high, "odd type" numbers always exceed "even type" numbers.  Through 96,
"odds" lead "evens" only by 48 to 47 (but 97, 98, 99 all are "odd type"), so the
race can be close, and instructive to follow.  "Intuition" somehow favours "odds"
as natural winners:  no doubt to the delight of Polya, whose love was heuristics,
the conjecture has been demonstrated to be false.

GEOBOARD SOPHISTICATION.  Polygons having lattice-point vertices are what we
usually investigate on the geoboard, but we tend to back off when the going gets
tough--and the challenge worthwhile.  Limit oneself, initially, to the 3x3
geoboard, nine points (or nails) in the usual square array.  Define polygon--to
rule out non-simple configurations.  Define congruency (rule out look-alikes
obtainable by translation, rotation, reflection).  List obtainable polygons:
the 8 incongruent triangles, 16 quadrilaterals, other polygons.  Define units of
length, area.  Classify the figures by area, perimeter.  Extend to 4x4 and
larger grids.  Extend to grids other than rectangular.  Reconsider your defini-
tion of polygon, your criteria for classification.

STRUCTURE IN MATHEMATICS.  A rich source of important mathematical insights is
any finite system whose properties parallel those which we associate with
numbers and number operations.  I like "braids," exotic combinatoric objects
which are easy to draw, instructive to list, and straightforward to combine
under a simple, binary operation.  "Braids" may turn up in college as a one-
term exercise in group theory--but they can be enjoyed at a much earlier age.
Three-line braids, the simplest useful form, are drawn by placing three dots
in a row above, three below, and making one-to-one linear connections between.
The connections identify the braid.  Six permutations make for six different
three-line braids, and a combining operation yields the 6x6 table of a closed,
noncommutative system.  Systematic investigation yields the 24 corresponding
four-line braids.  Identity elements stand out, but inverses call for some
thought.  The order of elements is a worthwhile concept.  Other "hands-on"
structures include polygonal isometries and clock arithmetics.

POLYOMINO SOPHISTICATION.  The usual polyomino is a plane geometric shape
compounded of congruent squares compounded by linking along wholly shared sides.
Resulting shades are classified by numbers of squares.  Polyominoes are deemed
different if one cannot be obtained from another by translation, rotation or
reflection (essentially, "turning over").  Five different 4-square polyominoes,
12 different 5-square polyominoes, and 35 different 6-square polyominoes exist,
and complete sets often can be sketched on the chalkboard in a single session.
Quadrille paper suffices in the planning stage.  Pieces can be cut out of
heavier stock to permit their properties to be investigated.  A double set of
the 4-squares (10 pieces) will make an 8x5 rectangle, a relatively simple task.
One set of 5-squares will make a 10x6 rectangle, or an 8x8 square with a 2x2
centre "hole."  Much space is devoted to polyominoes in the literature of
recreational mathematics.  Instructive variations on the multiple squares are
multiple triangles (equilateral) aand multiple hexagons (regular); also
multiples of the isosceles right triangle.  A logical step into another
dimension gives "polycubes," multiple cubes sharing a common face.  Classical
puzzles exist of this type.

RESIDUE FIGURES AND PATTERN EXPLORATION.  Patterns of straight lines (which are
tangent lines enveloping mathematical curves) yield an attractive union of
number sequences and geometric representations.  The student begins with a
circle, already divided by n equally spaced, numbered points, 1 to n.  Attract-
ive results are obtained with n = 60, 72, or 96.  The student chooses a multi-
plier, an integer between 2 and n - 1.  The multiplier determines the curve.
The number of points determines the resolution, the detail.  Numbered points
are joined, in pairs.  If the multiplier is 2, then 1 is joined to 2x1, or 2;
2 is joined to 2x2, or 4; 3 is joined to 3x2, or 6; and so on.  When the
obtained product exceeds n, one subtracts n + l as often as is necessary to
obtain a result between 1 and n:  thus, with n = 72, 36 is joined to 72, but
37 is joined to 74 - 73, or 1.  The completed pattern consists of n segments,
which (for a given multiplier, m) envelope a characteristic curve.  A Valentine
heart, a cardioid, results for m = 2; a kidney-shaped nephroid for n = 3; a
three-leaf clover (curve of three cusps) for n = 4.  Rules can be varied, and
the results can be attractive.  This becomes "curve stitching" when theory is
combined with craftsmanship.  Place black velvet on soft pine.  Lay the
completed diagram on the velvet.  Hammer brass finishing nails through each of
the n points.  Wind coloured cord around the nails, one continuous path, along
a route determined by the lines.  Remove and dicard the drawing.  The stitched
curve, with a single knot, now should be ready for framing.  Small values of n
such as 12, 16, or 18 will not define the curve as attractively, but may yield
interesting patterns for colouring of regions.  In every instance, n should be
so selected that n + 1, the number subtracted, is prime.

NUMBER TYPES, DIVISORS, AND SOME RELATED CONJECTURES.  Incredibly challenging
conjectures can be expressed in extremely simple terms in some areas of number
theory, and intuitive exploration has much to commend it.  We restrict our-
selves to positive integers.  Prime numbers have exactly two divisors, 1 and
themselves:  2, 3, 5, 7, 11, ..., accordingly, are prime.  1, the unit of this
system, is unique.  Other numbers have more than two divisors, and are termed
composite:  4, 6, 8, 9, 10, ..., are composite.  Pairs of consecutive odd
integers which both are prime are called twin primes:  11, 13; 59, 61; 71, 73
are twin primes.  Is there a largest, therefore last, prime number?  The Greeks
asked this ... and answered it.  Is there a last pair of twin primes?  We don't
know.  Goldbach asked:  Can every even number, starting with 4, be written
in at least one way as the sum of two primes?  We don't know.  De Bouvelles
noted:  Take any multiple of 6.  Add 1 to it; subtract 1 from it.  The pair
of numbers that you get includes at least one prime.  Thus, 7x6 = 42, a mul-
tiple of 6, and 41 and 43 both are prime.  Is De Bouvelles' conjecture true?
How would you feel if you showed a three century old conjecture to be false?
Divisors of a number which are less than the number itself are said to be
proper divisors.  Early mathematicians, who were heavily into lucky and un-
lucky numbers which implied numerology, spoke of deficient, abundant, and
perfect numbers, according to whether proper divisors summed to less than,
more than, or equal to the number:  6, 28, and 496 were among the first recog-
nized perfect numbers.  Prime numbers can be listed readily by an elementary
but clever approach, the Sieve of Eratosthenes.  As the sequence develops,
remarkable properties are likely to be observed.  Slight variation in the
sifting process will produce not primes but quite possibly an equally interest-
ing set.

OTHER PRODUCTIVE AREAS.  Random walks, intuitive probability, lottery simula-
tion, St. Petersburg paradox, dice-based chaos games.  Recreational topology,
including classic problems, network tracing, map colouring.  Rational and ir-
rational numbers, decimal representations, repetend lengths, other bases.
Prime-rich expressions, representation on Ulam's "square spiral."  Construction
of skeletal polyhedra, from first principles.  Solitaire and competitive math-
related games:  Sprouts, Hex, Conway's Life.  Estimations and other group com-
petitions.

Harold Don Allen, Ed.D., F.C.C.T.,
Toronto, Canada, 9 August 1993.