I have met people who really distrust letters as mathematical
symbols - perhaps because they had a negative experience with
algebra at school. Indeed, this seems to be one of the areas of
maths that many adults are quite happy to confess to never quite
having understood, and for which they could see little use in the
grown up world! In this article we'll talk about what algebra
actually is, why it is important, and how we can make really secure
links from informal algebra at primary stages to the more formal
algebra of the secondary curriculum.
It's probably easier to describe algebraic thinking than algebra
itself. Shelley Kreigler in her article 'Just
what is algebraic thinking?' suggests three main ideas:
Algebra is sometimes referred to as generalized or abstract
arithmetic. This means exploring numbers and how they combine
together, and becoming familiar with ideas such as the inverse.
This starts in primary school and when done well can lay a solid
foundation for formal algebra. Teachers who help children to
understand the underlying principles (for example moving from 'if I
know 7 +3 = 10, what else do I know?' to 'If I know 359 + 763
= 1122, what else do I know?' give them networks of connections
that they can draw upon when they begin the formal study of algebra
'If I know a + b = c, what else do I know?'.
So how do we begin introducing unknown numbers to be worked out
and discovered? How do we begin teaching algebra?
Missing number questions are either based on known facts, or on
known connections. Putting what needs to be found out anywhere
except at the end of an equation often begins orally and is based
on known facts.
'What do you need to add to seven to make ten?' 'What do you
know that might help?'
'What do I need to take from six to leave four?' 'What do you
know that might help?'
A Stage 1 problem, which comes in the form of a simple
calculator game, is Secret Number. Yet
another is Number
Lines, which uses numbered strips rather than calculators. Both
these are simply expressed algebraically in the form 'n + a = b' or
'n - a = b' and lend themselves to representing the unknown by a
symbol or letter.
In Stage 2 there are several questions where shapes represent
Shapes is a simple example. The first three questions could
equally well be written
7+a+17 = 25
17+14+b = 21 and
14+2c+34 = 136.
Shape is possibly more difficult because multiplication is
usually harder than addition, less numerical information is given
and all the examples are interrelated. Nevertheless this is a good
example of where different symbols represent different numbers, but
in each case the same symbol represents the same number.
All of these examples (and there are many more) help children to
become used to representing an unknown by a symbol, whether that is
a box, other shape, or letter. The next step is beginning to
manipulate these shapes or symbols according to algebraic
Algebra is the language of mathematics. Children will eventually
need to be able to read, write, and manipulate both numbers and
symbols in formulae, expressions, equations, and inequalities.
Being fluent in the language of algebra means understanding its
vocabulary (i.e. symbols and variables) and being able to use the
correct grammar ie algebraic rules.
The meaning of the equals sign changes from being interpreted in
the early years as "is" or "makes", or in the case of subtraction
as "leaves", into an understanding of balance. When I did algebra
at school (many years ago!) we were taught methods which took all
unknowns, or all of a certain letter, to one side of the equation.
The equals sign expressed a balance. Do the same thing to both
sides and both sides will still balance. This idea is expressed
neatly using a number balance, sometimes called a "Balance Bar" and
sometimes an "Equalizer".
There are several NRICH problems using this device. A Stage 1
the Balance, explains it all. There are several other Stage 1
problems such as Are You Well
Balanced? and a Stage 2 problem, Balance of Halves
which includes halves. These all use the Equalizer for finding
unknown numbers. As a way into more formal algebra, you could use
the interactivity and record the situations and the subsequent
movements using symbols.
This means solving a real world problem using algebraic
thinking, and later setting up equations or inequalities and
solving them using algebraic rules. At early levels problem may be
best solved using trial and improvement, but some children will
already have more sophisticated (and potentially algebraic) ways of
thinking about problems.
At Stage 1 I first came across Eggs in Baskets
and Lots of
Lollies. These are the kinds of problems which are best done
practically with counters. This is also true of Heads and Feet
which is simple using counters (heads) and sticks (legs). This is a
version of a problem with very simple numbers which comes in many
guises in many different places. A much more complicated variation,
because the legs are not in pairs and less information is given, is
Zios and Zepts.
Some children may use symbols to represent different unknowns and
some may even devise formal equations. Many others however will be
using algebraic thinking even if they don?t record it using
There are plenty of Stage 2 problems that have unknowns that can
be solved algebraically. Two examples are Cherry Buns and Buckets of
Thinking. In the first it is the weight of an egg and in the
second three different, but interrelated, amounts of water.
There is another important way that letters are used in school
mathematics. The letter is not an unknown to be discovered but a
generalisation which covers many different examples. A few months
ago I was working with a Year 5 and 6 class on a problem which
called for a generalisation. It was an ideal time to use, in fact,
it turned out it was to introduce, a letter to sum up what we were
doing. Conventionally, I used "n". The class, including the
teacher, looked bemused. Obviously, they had not met this
"n", I said, "stands for any number." I wrote 'any' on the board
with a much enlarged 'n'. Everyone's faces cleared and the teacher
smiled. I felt rather pleased. It seemed a happy, if somewhat
erroneous, explanation of why I was using "n"!
There are some generalising problems at Stage 1. One example is
At Stage 2 there are several. A simple one is Up and Down
Staircases and slightly more complicated Break it Up! and
even harder Sticky Triangles. All three of these would be suitable
for older learners as an introduction or practice in generalising.
Some of them lend themselves to working with symbols.
Secondary mathematics relies upon children having positive
experiences with algebraic thinking in their earlier schooling. As
primary practitioners, our role is to help children to realise that
algebra is not just about putting letters in place of numbers, but
about thinking in a different way.