Milan's concern is that in schools we tend to focus on textbook
tasks that can be solved by most children using methods that are
prescribed by the teacher. I think that this is true in British
schools as much as in Czech schools. He set out to devise some
geometric tasks in which the student does not know the solving
method but can discover it for him/herself or in which he/she knows
the method but does not realise that it is suitable for the new
problem. At NRICH, these are just the kind of problems that we are
interested in and write for the website.
Looking at Milan's ideas for problems, and arenas in which
problems could be devised, I was struck by the richness of the
scenarios that he had chosen and also the quality of mathematical
thinking that some simple contexts were capable of provoking. Milan
stresses his desire to devise problems that will motivate children
and engage them as well as extend their mathematical thinking and
develop their imaginations. The idea is to present settings in
which there is no obvious method but in which children can be
encouraged to think deeply about the context and extend their
ability to think mathematically, especially geometrically. So what
do these scenarios look like? The list is extensive and I do not
have the space in this article to show you all Milan's ideas but
here are a few to give you a taster.
Milan suggests placing a die on a square grid and recording the
top number. The die is then turned onto one of the other faces.
Then the top number is recorded again. We developed this idea into
a game called Roll the Dice. Try it yourselves!
It is quite surprising and counter intuitive. Can you explain
why it works as it does? The children we have used the activity
with, are fascinated by the idea and, as well as asking for
reasoning and justification, the task helps to develop recording
skills and systematic approaches to possibilities. I think we
are possibly getting better at encouraging these kinds of
thinking in the context of number work but in shape and space we
do tend to be prescriptive. This problem offers us a geometric
context in which to develop mathematical thinking.
Other suggestions include looking at routes through
squared grids using colour, say red for right and blue for up, so
that different paths can be described by colours simulating the
real movement of an object, a person through streets for example.
This could be extended to include forward and backward movement and
possibly linked with number so that a move to the right
horizontally involves adding one and a move vertically upwards
involves adding five. This idea was developed into a problem called
Routes 1 and 5.
Milan Trch and Eva Zapotilova, (1995), Non-standard Problems:
The Means of Development of Thinking and Geometric Imagination in
The Lowest School Age. Proceedings of the International
Symposium on Elementary Maths Teaching 1995. Charles
University, Praha, Czech Republic.
Milan Trch and Eva Zapotilova, (1997), Non-Traditional
Mathematical Tasks as a Means of Developing Mathematical Thinking
of Younger Children and Problems with their Evaluation.
Proceedings of the International Symposium on Elementary Maths
Teaching 1997 . Charles University, Praha, Czech Republic.
This article appeared in Primary Mathematics,
a journal published by the Mathematics Association , in Summer