Children 'Buy into' Mathematics through non-routine Learning
By Don Allen, F.C.C.T.

Canadian College of Teachers
Occasional Papers
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" ideas.  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.  Secondly, the 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, 1980's), 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."

Several 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 impressive.   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  "divisor" concept.  Numbers are three-way classified as "abundant," "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, to resolve.

Four elementary number conjectures that have been attractive with young learners, serve to reveal difficulties--and perils--inherent in attempted demonstration.

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.

Secondly, De 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 mathematics discovery.

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 bridge"?  Undoubtedly.  But a senior professor, one who shows great interest in how children best learn, still sentimentally retains the shaved die that he "tossed" in his youth.

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, "sprouts."

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 innovation.

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.


2003 Don Allen

Bright Math Camp.