'Buy into' Mathematics through non-routine Learning
By Don Allen, F.C.C.T.
Canadian College of Teachers
2003 Number 1
Two great sources of parental delight we've come to recognize. One is the child's spontaneous response to the pleasure of being "read to." The other is the child's related thrill in discovering reading for himself or herself.
No quarrel there!
Societies do tend to accrue a rich heritage of stories, legends, games
and rituals that serve to nurture the growing child through the spoken or
written word. Similar initiation
into practices and processes of quantitative and logical thinking, into broadly
mathematical aspects of culture, also have been near-universal.
When you think about it, much that I've shared with children and with
prospective and practicing teachers, over 43 years and more, has been
mathematical in nature. I deem it significant that my greater successes, as I see
them, and my more satisfying moments, have been when I ventured beyond
prescribed syllabus or authorized text, to join children and young adults in
the exploration and extension of "neat"
Step back from textbooks, syllabus requirements, external examinations,
and you sense school mathematics, so defined, as knowledge and skills, taught to
be tested; subject matter primarily chosen as prerequisite to whatever comes
next. That it can be, and quite
defensibly, but students and I have found pleasure and satisfaction on "buying
into" topics and approaches that reveal mathematics as being much more.
Years of honest pedagogical trial and error have enabled students to
guide me on two essential matters. Firstly,
the "content," actually quite diverse, that would lend itself to
satisfying, open-ended group and individual investigation.
circumstances, also diverse, under which such efforts might best be carried out.
This retrospect, prepared for interested teachers and parents, will
touch upon both topic areas and organizational arrangements.
Further, one particular topic area, that of counting numbers and some
related number sequences, will be "opened up" at greater length,
illustrating types of properties and conjectures that math-prone students have
been eager to explore.
Approaches successfully employed in providing such curricular enrichment
have included added topics and activities in a senior mathematics classroom
(High School of Montreal.1959), after-school math clubs (Northmount High School,
Montreal, and Saguenay Valley High School, Arvida, 1960's), "hands-on"
Saturday mornings for upper elementaries (Nova Scotia,
gifted/talented demonstration programs (McGill University,
summers, 1980's), community-level
weekend sessions (Ottawa, 1980's), university-based summer "math
camps" (Ottawa, 1990's and continuing), and invitational evenings for
children and parents (Iqaluit, 1990's). Topics
that have been traditional favourites include recreational topology (map
colouring and route tracing), curve stitching (residue figures), lattice-point
or geo-dot configurations, polyominoes and such numeration and number
sequences, mathematical structure, "magic" shapes,
codes and ciphers,
and conjecture testing. Sessions
have been considerably livened by such "get acquainted" mathematical
games as "nim," "sprouts," and "hex."
such areas will, as we proceed, serve to provide practical illustrations.
The counting numbers; that is, the positive integers, 1, 2, 3, . . . ,
readily give rise to non-trivial questions in which young students may find
interest and real challenge. Consideration
of such numbers and their properties is of great antiquity.
Greek mathematicians looked upon such numbers as numbers of objects,
rather than as abstractions or, say, points on a line.
They called such a number as 14 "rectangular," since 14
objects could be arranged as a rectangle, 2 by 7 or 7 by 2.
The Greeks and their contemporaries also knew numbers as,
say, "square" (a special case of rectangular) or "triangular."
Further, much as astrology had been integral to astronomy, so numerology
had developed within number study. Today,
in different societies, some numbers still are deemed "lucky" or
"unlucky," and newspapers and magazines may perpetuate both
astrological and numerological beliefs. In
a predominantly "rational" society , children find it intriguing that this
should be so.
Counting numbers have come to have a useful three-fold classification. Rectangular numbers, now identified with rectangular patterns
having each "dimension" at least 2, are called composite numbers.
Numbers identifiable with the "lining up" of two or more
objects, but not rectangular, are called prime numbers.
The number 1, the unit, is deemed unique.
A more senior approach to all this is in terms of divisors.
A divisor exactly divides another number; that is, leaves no remainder.
As an instance, 1, 3, 7, and 21 are divisors of 21.
A number having exactly two divisors, 1 and itself and no others, is
taken to be prime. Thus, 19, 97,
and (believe me) 2003 are prime numbers. A
number having more than two divisors is composite.
For example, 12, having six divisors (1, 2, 3, 4, 6, and 12) is
composite. The number 1 itself, the
unit, most conveniently (definitions being arbitrary) now is relegated to a
class of its own.
Alert students spot or seek strategies for identifying counting numbers
as unit, prime, or composite. Twenty-five primes in the first 100 counting
numbers is an instructive opening challenge, for the usual first effort usually
results in one or two "extras"--typically the likes of 91 (which, being 13 x 7, clearly is composite, not prime).
The age-old sequence of prime numbers (2, 3, 5, 7, 11, 13, 17,
. . .)
is carefully defined, but its behaviour can be surprisingly hard to predict. My students like the topic of gaps between consecutive
primes, as 17 and 19, 23 and 29, 113 and . . . what does come next?
Knowing that, say, 2971 is prime (it is) is of little or no help in
determining what prime number follows, they observe.
Prime numbers and such as enrichment?
My youngsters have tended to "buy into" this quite readily. Summer groups have opted for prime-reciting competitions.
Some contestants would try to memorize.
Others would seek to compute on their feet.
Results could be most
Children who enjoy it
can be quite good at this kind of thing.
Mathematicians regard primes as "multiplicative building blocks." A related notion that
students have enjoyed arises from the
concept. Numbers are three-way
"deficient," and "perfect".
Choose a counting number. List
its "proper divisors", the numbers that divide it, not necessarily prime
numbers but less than the number itself. For
12, proper divisors are 1, 2, 3, 4, and 6.
Sum the proper divisors: 1 + 2 + 3 + 4 + 6 = 16.
Since 16 is greater than 12, 12 is said to be an abundant number.
For 15, proper divisors are 1, 3, 5.
Summing, 1 + 3 + 5 = 9.
Since 9 is less than 15, 15 is said to be a deficient number.
In the third instance, equality, where proper divisors sum to the number,
we have, in some sense, perfection. That
is, 6 = 1 + 2 + 3, and 28 = 14 + 7 + 4 + 2 + 1, are called perfect numbers.
Our computer age accrues a growing list of higher "perfects,"
but even the third and fourth , children find, call for a significant
pencil-and-paper or hand-held-calculator search.
Choice of counting numbers (and of prime and related special numbers) as
an initial area for enrichment-related consideration does make sense. The topics lend themselves to interesting and instructive
conjectures, and to open-ended investigation.
Children delight in such "educated guessing," with subsequent
sharing of "discoveries." Other
topics mentioned share such attributes. One consequence can be a group sense of true adventure,
finding a new path en route to worthwhile discovery. Such an approach casts the teacher not as expert as much as
co-learner, a role which I find refreshing in the extreme.
A conjecture, you understand, is something more than a mere guess. The
implication is that the claim may well be true, but in any event should merit
investigation. The bottom line? Such investigation may (i) confirm the conjecture, by proof
or by exhaustive consideration. ("The
sum of two odd numbers is always even."
That we can prove to be true.) The
investigation may (ii) reject it, as by counterexample. ("The sum of three
odd numbers always is prime.":
No. 3 + 7 + 11 = 21, and 21 is not
prime.) The investigation may (iii)
leave the conjecture an open question. (Pairs of consecutive odd numbers which both are prime are
'twin primes.' Examples of twin
primes are 17 and 19, 59 and 61, and 101 and 103."
Open question: Is there an end to the list of twin primes?)
"Open questions" about prime numbers are plentiful. Some are easy to understand, but tough, perhaps impossible,
Four elementary number conjectures that have been
attractive with young learners, serve to reveal difficulties--and perils--inherent in attempted
You may wish to reflect upon each in turn.
Firstly, "Tartaglia," to cite the nickname of an early Italian
mathematician, conjectured in 1556 that the sums
1 + 2 + 4,
1 + 2 + 4 + 8,
1 + 2 + 4 + 8 + 16, and so on, were alternately prime and composite.
Investigate this claim.
Bouvelles, in France, conjectured as follows in 1509:
Choose a counting number. Multiply
it by 6. Then, obtain two new
numbers by adding 1 to, and subtracting 1 from, your result.
Thus, for 5, a counting number, 6 x 5 = 30, and your new numbers are 31
and 29. As it happens, both new
numbers are prime.
De Bouvelles conjectured that with this procedure at least one of the two
numbers produced would have to be prime. Always.
Attempt to check this out.
Thirdly, Christian Goldbach, a mathematical amateur, in 1742 posed in
correspondence the prime number conjecture for which he is best remembered:
Every even number greater than 4 can be written as the sum of two odd primes. The primes need not be different.
Thus, 6 = 3 + 3; 10 = 5 + 5 or 7 + 3; 36 = 19 + 17 or 23 + 13 or 29 + 7
or 31 + 5.
Reference to the classic Goldbach conjecture is found frequently in
upper elementary enrichment materials.
Fourthly, by the time of Euclid, this conjecture known to have received
consideration: There is no last
prime number; that is, the list of prime numbers never ends.
Prime numbers become notably scarcer in higher number ranges.
An eventual end to prime numbers? I've
had young "Philadelphia lawyers" more than ready to argue both sides
on that point.
Students, I find, may be intrigued by whole-number sequences, especially
those which fail to yield the expected result.
Two such whole-group activities merit mention.
"Bracelets of numbers," that's the term I've heard for a
simple activity in which many younger children delight.
The numbers, in this instance, are 0 and the digits 1 to 9.
To start a bracelet, choose
two such numbers, not necessarily different, say 4 and 9. Obtain the next bracelet number by adding your two numbers,
recording only the one's digit of the result.
Thus, 4 + 9 gives 3. Your bracelet, as it stands, reads 4, 9, 3.
Continue the process, using the two final digits.
Thus, 9 + 3 gives 2, and your bracelet reads 4, 9, 3,2. Again, continue the
process. How soon will the bracelet
of numbers "repeat"?
But . . . try again, this time starting with, say, 8 and 4.
Can this start yield the same bracelet, though with a different entry
point? Not really, because adding
of even digits necessarily gives an even result.
Thus, 8, 4, 2, 6, 8, 4 . . . , observe repetition setting in.
Your bracelet in this instance comprises four even numbers, 8, 4, 2, 6.
Creativity suggests some change in the fundamental rule.
Try, say, adding twice the first number and three times the second.
Or, perhaps, try three starting numbers instead of two.
"What might happen if . . ." allows children to open doors to
Especially popular with somewhat older students is the time-tested
Collatz conjecture. Simple rules
applied to counting numbers lead to identification of, and a conjecture
concerning, so-called hailstone numbers.
You proceed as follows: Choose
a counting number. Necessarily, it
is even or it is odd. If the number
is even, then divide it by 2. If it
is odd, multiply it by 3, then add 1. Repeat
the process. Thus, choosing 52,
which is even, obtain 26. Then, 26
also being even, obtain 13. Then,
13 being odd, obtain 3 x 13 + 1, or 40. Continue,
revealing the sequence 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. In eleven steps, rising and falling, 52 proceeds to 1.
Any number reaching 1 is deemed a hailstone number.
Luther Collatz, when a graduate student, conjectured that all numbers
are hailstone numbers. No one has
proven this. No one has discovered
a counterexample, a starting number for which the conjecture doesn't hold.
The conjecture remains an open question.
Twenty-seven, an instance popular with my students, reaches great
heights, and takes many steps before ultimately falling to 1.
Student effort to write out the sequence on adding-machine tape have
spanned three walls of a room.
Changing the rules, as in switching the odd-number directive to
"multiply by 5, then add 1," can open up a whole new investigation.
Children, in my experience, like the idea that a rule that they have
invented may not have been tried before.
Older students also delight in such expressions as 2 n + n + 41 which,
for successive values of n (a counting number), generate far more prime numbers
than might be expected by "chance."
Topics that have "taken off" in such beyond-the-curriculum
contexts I'd not always have anticipated. High
school seniors of 45 years ago were intrigued by chance, far beyond the
permutations and combinations provincial prescription.
"Nickels" were being flipped each recess, accruing evidence
that Canada's 5 cents of those days was slightly biased--as I recall, in favour
of "heads." A
"shaved" die still came up all six faces, but with probabilities that
deviated significantly from 1/6. So
much "water under the
Saturday mornings for upper elementaries arose during a commitment to
pre-service teacher education. I
much enjoyed it, and the children responded enthusiastically to mathematics as
fun. They especially liked the
topological side, from map colouring and route tracing, to draping themselves in
Mobius strip configurations. The
get-acquainted activity that became a feature of all sessions was a friendly
round of English mathematician John Conway's ingenious topological game,
One type of number activity which children like as group effort
involves--I'd call it--number representation.
We choose, typically, a four-digit number, say a dog tag or bicycle
license. We then make use of those four digits, with agreed rules, to
write expressions for counting numbers, starting with 1.
An outstanding experience of this type was many years ago, when duties
took me to the classroom of a master practitioner.
"Show us something interesting," she spontaneously had
directed . I had hesitated, then chalked "3692" on the board.
It had been the number of a boxcar by the train station, I had told the
children, and I had suggested that it could be an "interesting" number.
I challenged the Grade VI class to represent numbers, using a 3, a
6, a 9., and a 2, one of each, in any order, with any
needed signs and rules that they used in everyday mathematics.
Thus, 1 = 9 x 3- 26, 2 = (9 + 2) - (6 + 3), they were fascinated to see.
A week passed.
Ninety-nine of the first l00 counting numbers were posted on the
classroom wall on my return visit. We
collectively rolled up our sleeves and completed the l00, to the children's
delight. That Grade VI teacher
evidently had been drilling fractions. The
children had responded by viewing, say, 4 as 9/2- 3/6.
That activity I can trace to eighteenth century sources.
It's still fun, and "math campers" can pick a number, then
spent a week extending and improving (simplifying) a wall of
representations. Mental math, notational conventions, order of operations, and
cooperative learning, all are elegantly involved.
Week, even two-week, summer "camps," university-based, have
from the start rated high in student and parent satisfaction.
Three factors I identify as contributing to this degree of success.
First, candidate selection. Helene
Desrosiers Gregoire, M.A. Psych., an Ottawa parent, works with other parents and
school personnel to identify children able and motivated to learn and grow in a
less-structured, cooperative environment. Such
children may or may not be doing well in school.
Second, camps ensure an abundance of
"proven" topics and explorations, most of them considered, day
by day, in increasing depth. Each
child, and the group, can exercise choice as to where emphasis is placed.
Third, a supportive atmosphere is highly valued.
Risk taking is strongly encouraged.
"Wrong" conjectures are not laughed at, but built upon in a
positive and constructive way.
Some of the good-natured rivalry of such a group is well exemplified in
collective response to the week-long "estimation" theme.
An imposing jar of jelly beans, gumballs, chocolate "loonies,"
or other such motivating entities remains "up front" throughout the
week. Children post estimates.
Reflection and revision are encouraged.
It's "winner takes all" at week's end, usually with laudable
readiness to share. One important
insight with skills such as estimation is that "majority" (or average)
needn't necessarily be right.
Week-long investigations also frequently include polyominoes and other
composites of "pattern block" figures, geodot polygons on the 3 x 3
(or 5 x 5) array, and similar combinatoric considerations. These allow for significant variation . . . and creative
Many of these youngsters respond well to logical and mathematical
aspects of recreational cryptography, I've found.
Not unusual is spontaneous formation of a group that heads for an
unoccupied classroom to tackle, on chalkboards, a challenge "crypt."
They, or the supervising parent, soon are back to summon the group to
view their accomplishment. Others
are less enthusiastic. Their
hesitancy, I suspect, stems from initial discomfort in a situation where a
solution, or line of attack, may not immediately appear.
There's something near-universal in what these Ottawa experiences have
to tell us, I like to think. Four recent years with Eastern Arctic teachers and
children let me witness delight with such investigative topics . . . and with
learning on one's own terms. In
remote Clyde River, I enjoyed a full morning in elementary classrooms. Afternoon was to be "free"--for parent interviews.
A cluster of upper-elementaries urged that we commandeer the school
library for afternoon mathematics. Believe
me, we did. The Science Institute
of the Northwest Territories arranged "math evenings" and similar
events in Iqaluit and Pangnirtung. Children
and parents learned together at Iqaluit sessions.
What most struck me was how much those children had retained when, months
later, we next met.
Math for fun? My final
Eastern Arctic "math" evening brought a sense of closure to it all.
I had been called upon to perform a truly mathematical function--to call
the numbers at community bingo, a charity event. The Iqaluit Curling Rink was wall-to-wall players, the great
majority Inuit, reflecting the demographics of the town.
At one stage I became increasingly apprehensive, having drawn three,
four, five I-numbers in a row. Such
can happen by pure chance, I well knew. Fortunately,
the players' probabilistic insights, if less formalized, were as sound.
Thousand dollar prizes were duly awarded--and, I sensed, a great time was
had by all.
We do well, I suggest, in whatever context, to "work in", to explore, to share, and to enjoy the creative, ofttimes little considered, "fun" side of school mathematics.
Allen, Harold Don. "Extracurricular Mathematics: Incentive for the Talented." The Canadian College of Teachers, Occasional Papers, No.7 (1983), pp. 12.
Allen, Harold Don. "Math
for the Joy of It: Exploring Challenges and Pleasures of Recreational
Mathematics." Iqaluit: Science Institute of the Northwest Territories,
Public Lecture Series, Unikkaarvik Visitor Centre, 7 April 1994, pp. 22.
Allen, Harold Don. "Mathematics
Curriculum Plus," Nova Scotia Journal of Education, 21 (Winter
1971-72), pp. 26-30.
Collier, C. Patrick. Geometry for Teachers. Boston:
Houghton Mifflin, 1976. Pp. xiv + 331. Treatment
of informal geometry and topology and of motion geometry offer significant
background, insights, and experiences.
Dudley, Underwood. Elementary Number Theory. 2nd ed. San
Francisco: W. H. Freeman, 1978. Pp. ix + 249.
Conjectures of "Tartaglia" and De
Bouvelles (both blatantly
false) are presented as exercises (page 19).
The Goldbach conjecture, among the better known of elementary
mathematics, remains (2003) an open question, and is discussed as such (page
Gardner, Martin. Knotted Doughnuts and Other Mathematical
Entertainments. New York: W. H. Freeman, 1986. Pp. xiii + 278.
A representative collection (Gardner's eleventh) of diverse mathematical
topics based upon his "Mathematical Games" columns in Scientific American.
Ore, Oystein. Invitation to Number Theory. New Mathematical
Library 20. New York: Random House, for School Mathematics Study Group, 1967.
Pp. viii + 129.
"Experiences with numbers accumulated over the centuries with
compound interest, so to speak," Ore observes. A work intended "to make some important mathematical
ideas interesting and understandable to high school students and laymen."
Richards, Stephen P. Numbers at Work and at Play. New
Providence, NJ: By the author, 1987, pp. 213.
A highly varied offering of favourite topics for investigation.
For instructive insights into "hailstone" and related
sequences, see "Rising and Falling of the Hailstone Numbers," pages
Stein, Sherman K. Mathematics: The Man-Made Universe. 3d ed. San Francisco: W. H. Freeman, 1976. Pp. xv + 573. Offers a lucid consideration of the infinitude of prime numbers (pages 13-17), introducing intended readers (teachers) to indirect proof (proof by contradiction). Also develops the less-known conjecture that odd numbers can be represented as a prime plus twice a square, a conjecture which first fails for 5777 (pages 481-82). Also, a remarkable divisor-related conjecture by twentieth-century mathematician and teacher educator George Polya (pages 483-84).
© 2003 Don Allen
Bright Math Camp.