Mathematics
Problem Solving for Able and Gifted School Students
The ECU program provides year-long after-school courses for students
aged 10 to 15. It has operated since 1992 at the Mount Lawley campus
of the University, and is directed by Dr Norm Hoffman. Each year
more than 200 students are enrolled. They come from primary and secondary
schools, government and non-government, throughout the Perth metropolitan
area.
The high school students participate in a national program, Mathematics Challenge for Young Australians. This program is produced by the Australian Mathematics Trust. It comes as a complete package, with student notes and problems, and teacher guides. ECU offers three levels of the program: Euler (Year 8), Gauss (Year 9), and Noether (Year 10).
At the primary level there are two programs, Primary 1 (mainly Year 6 students) and Primary 2 (mainly Year 7 students). These have been specially developed and refined over a number of years by Dr Hoffman. They focus on systematic approaches to problem solving. Systematic approaches are explored in arithmetic, symbolic and geometric situations. There is also a strand that focuses on chance processes and the nature of randomness. The scope of activities in the primary programs is best illustrated by the following examples. Solutions are given on the bottom of the page.
Arithmetic: List all the 3-digit numbers for which the sum of the digits is 4.
Symbolic: List all the 3-letter arrangements that can be made from the letters of the word “page”.
Geometric: Find the total number of triangles (of all sizes) in
the diagram below.
Chance Processes: Fruit Drop lollies
are made in ten different flavours: Apricot, Banana,
Cherry, Date, Elderberry, Fig, Grape, Kiwi fruit,
Lemon and Mandarin. In each production run, ten million
lollies are made, one million of each flavour. They
are then mixed thoroughly and put into packets of
ten. On average, how many different flavours might
you expect to get in a packet?
In the primary classes, the problems are explored over an extended period, typically 2 or 3 months. First a simple version of the problem is examined, then a simple extension, then more complex extensions. Each problem is explored to sufficient depth for students to appreciate the solution strategy involved, and the process by which it is obtained. The inter-relatedness of many of the problems is drawn out.
At all levels of the program (primary and secondary) there is a consistent focus on clear and effective presentation of solutions. Students not only improve their problem solving skills but they also develop skills in presenting a solution, once it has been obtained. The marking of the solutions also provides a means of monitoring student progress.
Student achievement is recognised through certificates
presented in August and November. Photographs of
Dr Hoffman with the highest achieving students can
be viewed on the Presentation page of the web site:
http://www-chs.ecu.edu.au/courses/mpsp/awards.html
Solutions:
| 0+0+4 = 4 | 3-digit numbers: 400 |
| 0+1+3 = 4 | 3-digit numbers: 103 130 301 310 |
| 0+2+2 = 4 | 3-digit numbers: 202 220 |
| 1+1+2 = 4 | 3-digit numbers: 112 121 211 |
Algebraic problem:
pag pga apg agp gpa gap
pae pea ape aep epa eap
pge peg gpe gep epg egp
age aeg gae gea eag ega
Geometric problem: There are 69 triangles altogether.
Chance processes problem:
By Monte Carlo simulation, the expected number of
different flavours is approximately 6.5