those which leave remainder 2. Are there any others?
Why not? You have explained something! Put a ring
by Joe Watson,
Keele University
around any square numbers you have chosen - and
consider in which column you would place other
square numbers. Do you notice anything? Why does
this happen?
(c) Find two square numbers which add up to 17, to 41,
to 19, to 43. Any conjectures?
In (a) you always get a multiple of 4. But how can we be
sure? Experience - and a cautionary tale or two - shows
that it isn't sufficient to try a few cases and then deduce a
general rule. Analysing special cases is fine for developing
conjectures - indeed, looking at evidence is often the only
way to generate conjectures - but a conjecture is a
wh Y
relatively tentative assertion, not yet justified.
In some simple cases, a diagram provides a convincing
justification:
f?
proo
The diagram for each pair of consecutive odd numbers
contains one "extra" square; this extra square can be fitted
into the gap to give a multiple of 4 diagram. Since a similar
diagram could be drawn for any two consecutive odd
Proving, explaining and justifying are nothing new to
Mathematics -(and there is nothing new in this article!)
- but these are aspects of the subject which are apparently
neglected nowadays. Are they important? The national
curriculum (NC) has little to say about it for the majority
of pupils, and some "opinion-formers" would have us
believe that mathematics is a bag of tricks which our pupils
must know and be able to use in order that, as a nation,
we shall not fall behind others in the economic rat race.
So arithmetic is king, calculators are frowned upon and
the emphasis in on simple skills rather than proving.
Indeed, proving is thought much too hard - and since
hardly any of our pupils are going to become academic
mathematicians (who, it is argued, are the only people who
need concern themselves with proving), we can safely
ignore it.
I don't wish to underestimate the pressures which the
NC has placed upon teachers, who are expected to prepare
their pupils for the appropriate end of key-stage hurdles,
but the NC will not last for ever, and has already undergone
revision. I consider the explanatory aspect of mathematics
an important one, which Mal encourages us to explore an opportunity which I hope will be increased by changes
in emphasis in the future. In ancient Greece, arithmetic
calculations were done by slaves, while educated people
learned geometry, logic and number theory. Today we can
leave arithmetic and much else in terms of "hack" work to
electronic slaves, and let our pupils do more interesting
things.
So to some examples:
(a) Add together two consecutive odd numbers. What do
you get?
(b) Choose a few (whole) numbers and classify them, in
columns, into three types, those which are exactly
divisible by 3, those which leave remainder 1 and
16
Mathematics
in
numbers, the "proof" is perfectly general.
Another, more abstract and arguably more powerful,
method is to use algebra. Part of the justification for using
algebra is that it enables us to succinctly state and prove
general statements.
Let the two odd numbers be 2a + 1 and 2a + 3 (or 2a- 1
and 2a + 1). Then their sum is 2a + 1 + 2a + 3= 4a + 4 a
multiple of 4. Thinking numerically, consecutive odd
numbers will always sum to twice the number in-between
them e.g. 11 + 13= 2 x 12. Since the number between two
odds is even then the result follows.
(b) An algebraic proof is straightforward for those that
can expand quadratic forms: If x is a "3a number" its
square is 9a2 which is of the form 3b; if it is a "3a + 1"
number its square is 9a2+6a+ 1 =3(3a2+2a) + 1, a
"3b + 1" number.
Finally, if x is a "3a+2 number" its square is
9a2 + 12a + 4 = 3(3a2 +4a+ 1) + 1, another "3b+ 1"
number. So, the square of a number can never be of the
form 3b+2.
(Using a diagram rather like that in (a), it is possible,
though not quite so easy, to give a pictorial proof of this
statement. Readers may wish to attempt this with their
pupils.)
(c) Using algebra, one can show that no number of the
form 4N+3 can be expressed as the sum of two square
numbers since each square number is either a "4a" number
(when it is the square of an even number) or a "4a + 1"
number (when it is the square of an odd number). This
can be also demonstrated with a pictorial proof.
Proofand Conjecture
Two problems which will be familiar to many.
(d) Several straight lines are drawn on a sheet of paper.
What is the maximum number of intersections, and the
maximum number of regions which can be formed?
(e) N points are marked on the circumference of a circle.
They are joined to each other in all possible ways. What is
School,
May
1995
the maximum number of intersections, and the maximum
number of regions, which can be formed?
(The maximum number is asked for, since in special
cases points or regions may be "lost" when more than two
lines meet at a point.)
Some experimentation fairly quickly provides a table of
values for each problem:
(d)
Lines
N
1
2
3
4
5
Intersections P 0 1 3 6 10
Regions R 2 4 7 11 16
(e)
Points
N
1
2
3
4
5
Intersections P 0 0 0 1 5
Regions R 1 2 4 8 16
In (d) There is an obvious pattern and we conjecture that
P=N(N- 1)/2 and R=N(N + 1)/2 + 1
Pupils can "spot the pattern" and predict the next number
- they may even be able to "guess the formula", possibly
by examining differences. But without wishing to belittle
this activity, which can form part of exploring the situation,
there is more to do, as the next example illustrates - how
are the results to be justified, so that we may have absolute
faith in them?
In (e) the position seems clear, but although there may
at first appear to be a pattern in the figures for the regions,
the next number of regions is "31" which totally disturbs
this pattern - despite the many attempts at a "recount"
which pupils will often make when faced with this situation!
In fact the formulae are:
P=N(N-1) (N-2) (N-3)/4!
and R = 1 + N(N-1)/2! + P
If further diagrams are drawn, these rules may be verified
- but that still does not prove they are correct!
For me, the value of this example (sometimes referred
to as the "Pancake" problem) lies less in deriving the
correct formula, than in realising that the very plausible
conjecture R=2N-1 actually turns out to be incorrect. It
can be highly instructive for pupils to be faced with patterns
that have a strong association with "known" sequences but
which then break down.
Logical Thought
"If you do your job as Chancellor of the Exchequer
properly, you are bound to be unpopular".
Norman Lamont remarked, "If this is true, I must be
one of the best Chancellors this country has ever had".
Leaving aside his performance as Chancellor, we would
have to award a low grade for logical thinking, for this
statement confuses theorem and converse. What is said is:
proper job > unpopular
This is not the same as:
unpopular = proper job.
Norman has some excuse, since in everyday life two things
are often associated so that they are seen as equivalent.
In mathematics, many converses of true statements are
themselves true; perhaps the most famous is the "Converse
of Pythagoras". But not all converses are true - two
simple examples are:
(j) If the product of two numbers is odd, their sum is
even. (A true statement. Is its converse true? Can you
prove the original statement?)
(k) If a quadrilateral is a rectangle, its diagonals are
equal. (Forwards? Backwards?).
These examples can be used to highlight the directional
nature of some proofs and the role of counter example: a
single non-fitting case is enough to disprove a conjecture
(although conjectures can of course be adapted to exclude
special non-fitting cases), whereas, as we have seen, a
statement cannot be proved true by any number of
confirming instances, no matter how many.
Most of the examples have the advantage of relating to
fairly elementary properties of numbers. Other proofs e.g.
the irrationality of the square root of 2, may be harder to
grasp. It seems important when confirming (or proving the
impossibility of) results that the conclusions are within the
learner's intuitive grasp.
Why should pupils be involved with notions of proof?
Sometimes out of curiosity, "Why is it that no square
number appears in the 3N+2 column?" Sometimes to
avoid error (as in the example (e) above), and sometimes
to disprove things (which means there is no point in trying
to do something which we can show is impossible). An
appreciation of proof arises partly from a desire to examine
Another useful problem in this regard is to examine the
two conjectures:
(f) The values of the function n2 + n 41 increase by 2,
4, 6, 8, ... as n increases by 1 (from 1).
(g) The values of n2 + n + 41 are always prime.
The former is always true (why?) The latter breaks down
(but is true for - 41 < n < 40).
Some interesting mathematical proofs are concerned
how things "hang together" (curiosity), and to communi-
with impossibility:
is formed.
(h) Four jars of marbles have these numbers of marbles:
10, 13, 19 and 20. A "move" consists of taking one marble
out of any of three jars and placing them all in the fourth
jar. Can you end up with all marbles in the same jar? (This
could involve a "parity" proof involving considerations of
odd and even numbers.)
(i) It is impossible to traverse the diagram below, tracing
each path only once and visiting all the points without
taking your pen off the paper. Why?
cate this understanding tidily.
The user of mathematics can take matters of justification
largely on trust "provided the rule works that's good
enough for me". The "pure" mathematician (in the
psychological sense) is concerned to feel the pulse and
know whether or not something is always true. In education
we may need to show both how maths is used and how it
The complex world in which we live - with Jumbo jets,
satellite TV, electronic communication networks, .
depends on having at least a few people who can understand
these complicated devices and the mathematics which lies
behind them. More importantly, we need to develop the
spirit of enquiry that all young children seem to have
(sometimes in irritating abundance ...) so that insight is
given a high priority and an appreciation of structure is
encouraged.
Arithmetic is not enough, we need citizens who can think.
For a proof of the "Pancake" problem see Baxandall et al., "Proof in
Mathematics" KMEP 1978 available from Keele University.
See also:
Beevers, B. (1994) Patterns which Aren't, Mathematics in School, 23, 5.
Johnston Anderson (1995) Patterns which Aren't Are!, Mathematics in
School, 24, 2.
Mathematics in School, May 1995
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