Here’s a technique that works for most of the activities in the Puzzles section
of this site:
I start by showing the class a completed puzzle and ask them to figure out what
the rules must be. This is a lot of fun – even the youngest students are very
good at hunting for patterns. (Where appropriate in these pages, I’ve included
“demos” with each type of puzzle.)
Then we do at least one puzzle together. I insist that students not only use
mathematical language to make their suggestions, but that they justify their
reasoning. I’ll play dumb if I have to; it usually takes only one or two vague
descriptions or incomplete arguments for students to start to be very precise.
Ideally, every student should get a chance to suggest and explain a step. It can
be difficult to keep the discussion from being dominated by a few quick
thinkers. I’m sure you have your own ways of doing this.
Having the class solve a puzzle or two together enables students to see that
there are often several good places to start a puzzle and often several chains
of reasoning to follow from any one position. Some chains are short and direct,
while some are impressively long and even hard to follow. Since all (correct)
chains of reasoning will ultimately lead to the same result, people can
succeed using very different approaches. Noticing this can reduce the
anxiety many students feel about solving math problems.
Once the class has a good idea of the goal and some of the logical techniques
useful for solving the puzzle, I hand out a page of fairly easy puzzles. As each
student finishes that page, he or she receives a page of slightly harder ones.
It helps everyone keep track if each page is a different colour.
Here is a tricky bit: we want increments in difficulty to be small enough to
keep frustration to a minimum but large enough to let the students know that
they are getting more skilled. You’ll have a few students who can solve the
first puzzles quickly. They can always be individually encouraged to skip a few
– just don’t let them go directly to the hardest ones, as that generally leads
to frustration and a tendency to give up.
We all know that success breeds success and that there’s nothing like the
excitement of saying “I got that one!” to encourage students – and everyone else
– to try something slightly harder.