Problem solving
Rationale: While a great deal of Ôproblem solvingÕ ought to
happen in regular classrooms, the Internet can theoretically offer some extra
benefits to students, including a regular supply of suitably targeted problems,
opportunities to share solutions and even opportunities to get professional
feedback on their work. [There seems little argument for a set of problems on
the web that could just as easily be written in a textbook, with nothing else
added.]
Here are some examples:
Problem of the Week
(POW) was one of the early initiatives of the Math
Forum site in the USA, a massive site with many resources of different
kinds. Problems are posted at different levels and in different branches of
maths, and students can submit answers to these. Good answers are posted on
the web and comments on various solutions are offered. In recent years, the
cost of doing this has been prohibitive, so that some previously free
services are charged for (such as individual mentoring of students), but it
is still possible to access the site productively without charge. Explore
this page to see how mentoring works and also to look at problems and their
(annotated) student solutions. A substantial archive of problems, solutions
and commentaries of the past several years is also accessible. 
The International
Mathematical Olympiad is a mathematical problem solving competition for
school students that has been held each year since 1959. National teams
(including Australia)
are selected and specially trained for the competition which requires a very
high level of problemsolving ability and access to and experience of a suite
of mathematical techniques not conventionally taught in schools. On this
site, John Scholes of the UK has kindly provided a catalogue of past problems
and their solutions. The solutions are very brief and would often be hard for
students without substantial mathematical experience to follow. However, a
site like this provides access to mathematical work that is rarely available
at a typical school to students. 
The Australian
Mathematics Trust is responsible for the Australian Mathematics Competition,
which focuses on problem solving, but at a lower level than the IMO.
Organised by the Australian Mathematics Trust and sponsored for many years by
Westpac, the AMC is a very large annual competition administered through
schools. Other opportunities to learn about problem solving through the Trust
are also described on this site, including the Mathematics Challenge.
Students who are interested in such activities may learn about them from the
web and then discuss their interests with their school. 
This site offers an
opportunity for students to solve some word problems, and to get some hints
if they get stuck. Unlike many sites concerned with problem solving in
mathematics, this is not a competition site (although some of the problems
have been modified from competition problems.). Indeed the authors are
emphatic: ÒThis is not a test
but a set of carefully selected problems which can help you improve your
problem solving skillsÓ. Problems are graded by level (Canadian Grades 512).
The mathematics covers a wide range of areas and all problems have complete
solutions as well as hints. 
This US website for
middle school students uses Flash to offer detailed algorithmic help with
Ôword problemsÕ, with a new ÔepisodeÕ being produced each week. Students can
see how to think about the stepbystep solution of standard problems, with
opportunities to be engaged along the way (responding to teacher questions,
trying for themselves, etc.) The site features an online calculator and
online drawing area. While it doesnÕt deal with problem solving in the normal
sense of the word, it provides an example of how the web might be used to
help students with mathematical tasks, and may be of help to students. Episodes
are archived and classified. 
The UKbased NRich site
has a rich store of problems which are changed monthly. The problems are
identified as appropriate to different stages and many contain interactive
elements. Problems usually come with notes for a teacher and a hint. The
featured solutions from students from around the world provide a nice touch.
The site has been going for a decade now, so that the archive of past issues
in the left column provides a wealth of interesting problems, wellpresented
and nicely targeted. The link above takes you to all the problems for This
Month, which you can refine to different stages with the links in the left
column. 
The wonderful PLUS
magazine, a companion to the NRich site for more sophisticated students, has
a regular Puzzle Page, with an original puzzle each month and solutions in
later months. There is also an archive of previous puzzles accessible from
the site (Explore the archive at the top of the page). Although one
definition of a puzzle is that it is a mathematical problem that doesnÕt lead
to further mathematical thinking, many of the puzzles on this site will allow
students (and their teachers) to engage in mathematical exploration. 
