A Rationale for The Western Australian Mathematics Problem Solving Program:
Dr Nathan Hoffman, 2013
The following article is in two parts: The first part is a statement of principles based on my experiences, reading and study. The second is a description, with illustrative examples, of the mathematics problem solving program initiated in 1992 and developed since then.
Part 1: Principles that should underpin provisions for mathematically gifted children
It would clearly be thoughtless and inappropriate to provide for mathematical giftedness by having the students participate in weightlifting courses, because weightlifting does not relate to nor build on the students’ special abilities. A curriculum for mathematically gifted students should be based on considerations of the nature of mathematics and the nature of mathematical giftedness.
What can be said about the nature of mathematics?
- Mathematics is a form of intellectual activity. Engaging in mathematical activity is both stimulating and satisfying for many of those who have the intellectual ability to do so.
- Mathematics is a science. It has been described as the Queen of the sciences, and also as a servant of the sciences. It is a powerful tool with which to increase our understanding of the real world. As a consequence, school mathematics programs should bring out the interrelationships between mathematics and the real world.
- Mathematics is the natural language of number, order, and form.
- Mathematics is concerned with pattern and structure, and the use of these to bring order to the study of objects and systems. It is a powerful problem solving tool.
- Mathematics is concerned with patterns of inference and deduction.
- Mathematics is a part of our culture and has contributed to the development of Western civilised society.
What can be said about the characteristics of mathematically gifted children? Typically they have:
- A capacity and willingness to work with abstract notions, e.g. randomness.
- An appreciation of the significance of generalisations, e.g. proofs in geometry.
- A capacity to perceive patterns and relationships.
- A capacity to understand patterns of logic and inference, e.g. mathematical proofs.
- A capacity to solve multi-stage problems.
These observations lead me to recommend that mathematics programs for mathematically gifted children should embody the following features:
Part 2: Description and Example
The programs should provide investigative activities. The students should be given the opportunity to explore problems and develop their own solutions. Often it is possible to structure a sequence of problems, of increasing complexity, leading to a main problem and its solution. Exploration of simple cases is a useful general technique. An example might be the problem of finding the number of 4-member subsets of a 9-member set.
The programs should focus on topics that are mathematically significant. Examples include systematic counting (combinatorics), chance (random) processes, prime factorisation, and methods of proof.
The programs should give preference to topics which are not merely extensions of what is in the standard school program.
The programs should give preference to topics that synthesise two or more areas of mathematics.
The programs should provide ample time for the exploration of a problem. By this I mean weeks and months, not just one or two hours. (Any problem that I can solve in 10 minutes, is not much of a problem for me!)
The programs should place considerable emphasis on the presentation of solutions. Being able to present a solution in a logical and lucid way is a most useful mathematical, vocational and life skill.
What else do we know able intellectually-able young students?
We know that at age 10 or 11, or thereabouts, intellectually-able children enter a new and different phase of intellectual development. Some developmental psychologists call this the Formal Operational phase. Within this phase a mathematically-able child typically becomes able to think and reason abstractly and often shows an interest in strategy games, such as chess, and logic puzzles, such as Sudoku.
Here is an example of an investigative activity that I have found stimulates and challenges 11 year olds, and meets many of the objectives I outlined earlier (3 pages long).