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Alf wrote this article in 2006 in
response to a request to choose a favourite problem.
I tend to think more about 'spaces for exploration' than 'problems
to solve' when I am planning teaching, so I have found deciding on
a favourite problem, as well as how to write about it, hard. But
the problem I imagine I have used with more groups, for longer than
any other, is one I call 'Functions and Graphs'.
The challenge is simple to state:
Given any equation, predict what
its graph would look like.
It is harder to set up and is never-ending to work on. I start by
playing a 'function game'.
I'll write, in silence, as dramatically as possible, something
$5 \to 12 $
$3 \to$ ... and offer the pen (in silence still) to a student to
come and fill in the gap.
I have a rule in mind and will respond to what is written with
trying to maintain
silence for as long as possible.
If a student gets a
then they choose the
next starting number.
I'll use a linear rule to start off year 7; quadratic if it is a
year 8 I have worked with before.
At the point of getting descriptions of the rules I will get as
many as I can from the class:
I'll write $ x \to \cdots$ and may have to explain I want someone
to write the rule they were using.
Typically there will be many different rules:
$ x \to x \times 2 + 2 $
$x \to x + 1\times 2$
$x \to xx + 2$
I will accept these rules exactly as the students write them and
explain how a mathematician would interpret them if there is any
ambiguity (e.g. $xx$ means $x^2$). I offer creating a picture for
each rule as a mechanism for deciding which ones are the same or
different and which fit our game.
Students write each rule across the top of a page and choose a
random selection of 5 numbers (including some negatives) to list
under each one, e.g.:
$x \to xx + 2$
$5 \to $
$3 \to $
$0 \to $
$-1 \to $
$-4 \to $
Having filled in these inputs (many conversations usually
necessary for the negatives - I am deliberately evasive at this
point; one of the powers of the activity for me is that it offers
students a chance to try and make sense of operations with
negatives) we re-write input and output as co-ordinates and plot
them all on the same grid. A discussion of what is the same or
different usually throws up many rich questions for exploration;
e.g. "Which rules give curved lines and which give straight?"; "How
do I get parallel lines?".
The power of the activity, for me, lies in the scope it offers for
students' choice and exploration; the naturalness of the questions
it throws up; the interest it tends to generate and the range of
curriculum areas students will be working on (negatives,
co-ordinates, substituting into formulae, indices, writing and
interpreting algebra, not to mention all the using and applying
issues). This wide range of topic areas allows me to work happily
on this problem with a class for a month. This is a problem I will
work on with classes from year 7 to year 13 Further Mathematicians
(who by then are on to graphing rational functions). Year 8
students can work on gradients of curved lines (having sorted out
$y=mx+c$) and deriving gradient functions in the same room as other
students grappling with how to label axes and add a negative
number. There is also, of course, a natural link in to using ICT
and graphical calculators.
In the end, what I like most about this problem is its familiarity;
I have worked on it with every class I have ever taught. Rather
than this becoming monotonous, it seems to mean that each new class
goes further than the one before, as I become more and more attuned
to what students say and to what possibilities there are for
exploration. And I imagine this is true for everyone's 'favourite
problems' - in the end it does not matter so much what they are as
what use you have made of them.
Alf Coles is the Head of
Mathematics at Kingsfield School in Gloucestershire, UK.
In 2010, Alf Coles and Tracy
Helliwell wrote an article Kingsfield School -
Building on Rich Starting Points which elaborates on these