Dr. Harold Don Allen, F.C.C.T.,
Nunavut Arctic College, Teacher Education Program

Don Allen passed away in 2020.


Youngsters given a wide choice of mathematics-related activities, investigations, and background subject matter tend to choose well in terms of furthering their insights into mathematics and building their mathematical maturity. So claims a Canadian mathematics educator whose practice has been to offer out-of-school students the widest possible options and to allow them to dictate the development of programs of enrichment mathematics.

Children of exceptional ability delight in choosing what they will learn, and my experience is that, freed from a pre-set curriculum and viewing content apart from mainstream mathematics, they tend to choose well, says Harold Don Allen, a Montreal educator currently best known for summer and weekend work with the gifted and talented of the Ottawa area.

Mathematics apart from the mainstream, material which schools, even colleges, have scant time for, includes much of "the poetry of mathematics," Allen asserts. He cites topics from informal geometry, probability, number theory, combinatorics, and non-routine problems solving as being well suited to capturing the imaginations and strengthening the abilities of younger students.

Teens think school mathematics is anything but useful, and there they are wrong. What we teach is exactly what is needed to empower them, to enable them to go on in, and to apply, the subject. It is not, however, what best might capture the curiosity and the imagination of the ablest. Were we to teach English in the spirit in which we teach mathematics, it would be the business letter, to the virtual exclusion of Wordsworth and Shakespeare. Allen was addressing a section meeting of the Tenth World Congress on Gifted and Talented Education, currently being hosted by the University of Toronto.

Able, motivated students as young as ten could choose from such diverse tasks as estimation, enumeration, lottery simulation, map colouring, network tracing, model construction, competitive and solitaire math-related games (nim, sprouts, hex, life), non-routine problem solving, and conjecture investigation. The challenge of the more difficult problems seemed to possess distinct appeal, Allen noted.

Students in such circumstances learned several important lessons: to listen critically to one another, neither to reject wholly nor to accept un-questionably without due reflection, and to build thoughtfully on one another's insights and conjectures.

They also learned that solutions rarely come instantaneously, but rather hunch by hunch, step by step, Allen observed. The quitter rarely makes much headway in real problem solving.

Allen served as visiting professor and demonstration teacher in McGill University's gifted/talented summer programs, a teacher education undertaking, and did out-of-school enrichment programs at Nova Scotia Teachers College and Quebec's Champlain Regional College before focusing his curriculum development on parent-organized Ottawa-area enrichment programs.

Two number activities which received sustained interest from this summer's student groups have been the following.

A search for "hailstone numbers," so-called. Choose a counting number. It 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. Repeat the process with the new number so obtained. Further repetition may cause the sequence to rise to impressive heights, then (like a falling hailstone) crash to 1. Numbers that lead to 1 are hailstone numbers. Thus, 5 goes to 16, to 8, to 4, to 2, to 1. All numbers so far tested end at 1, but some (for example, 27) lead the investigator on a merry chase before their eventual tumble.

A "square of differences." Choose four numbers, 0 or positive whole numbers. Write them at the corners of a square. At the midpoint of each side of the square, write the number you get by subtracting the smaller from the larger of the numbers on that square. Repeat the entire process with the new, rotated square so formed. Repeat until you get four zeros, which you will in surprisingly few steps. Students came up with four numbers which required 16 steps, and looked upon this as their finest accomplishment.

There is material in such investigations for a currently neglected science fair approach to mathematics, Allen has suggested.