2472
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2011-02-01T00:00:01
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These drawings were found in La Pileta cave in Spain and could be up to $25 000$ years old.<br></br>
What do you think these drawings represent?<br></br>
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Could they be about counting things?<br></br>
What might 'stone-age' people count?<br></br>
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Might they be the earliest known number patterns?<br></br>
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You could try and make up your own way of recording counting.<br></br>
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<a href="http://nrich.maths.org/7213&part=">Click here for a poster of this problem</a>.<br></br></mdoxml>
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<p class="editorial">There are no right and wrong answers to this
problem; the drawings could be up to $25.000$ years old, so we
cannot ask the people who drew them! What it does definitely show
us, however, is that the people living all of those years ago had
some sort of system for encoding information. This is what we need
to communicate - we use codes all of the time. Numbers are codes,
and so are letters; put together in a meaningful way, we can then
decode this, and work out what the message is. Symbols can also be
used to transmit information. For example, a smiley face shows that
you are happy, a plus sign means that you add numbers together and
so on.</p>
<p><span class="editorial">Some great suggestions for the "meaning"
of the drawings were submitted. If we can understand the code the
cave men were using, we can then de-code this and read the message
...</span></p>
<p class="editorial">Although this is posed as a mathematical
problem, we must also bear in mind that the drawings may not be
directly maths-related. Lauren, from the Princess Elizabeth School,
suggested that the drawings could be a map of an area, possibly
with directions. For example, the line on top of the vertical
lines, joining them all together, could represent the direction you
should walk in. The number of vertical lines, dipping down below,
represent the number of steps to take in that direction. The
"curvy" lines show a more complicated, curved path.</p>
<p class="editorial">Lauren also made another suggestion:</p>
<div>It could also represent different musical notes: the longer
the line the lower the note is (or maybe the other way around). The
length of the line could also be related to the length of the time
that the note is held for: the longer the note, the longer you hold
it for.</div>
<br></br>
<div><span class="editorial">Emma, from Village Elementary,
suggested that the drawings could represent a family tree, or a
collection of family trees. The long horizontal line that joins the
shorter vertical lines could represent the mother or father. The
vertical lines projecting down from this could represent the
children. Single, unjoined lines could represent unmarried people
with no children, and where lines meet, joining together different
sections, this could represent relations joining the different
family units.</span></div>
<br></br>
<p><span class="editorial">Other people suggested that the drawings
could be some form of early counting system. Joe, from Meridian
Primary, and Lauren from Princess Elizabeth, thought that the lines
could represent a certain quantity. In the simplest case, each line
could mean "one". Then, to work out the total number, you would
just count the number of lines. As Ewan, from Sofrydd Primary
wrote, this a bit like the "tallying" that we use nowadays.
Although it is simple, writing large numbers using simply "ones"
may take a long time, and is very inefficent. So, perhaps
combinations of these lines could mean different things? Perhaps
some of the different symbols represent larger numbers? Read on
...!</span></p>
<p class="editorial"><span class="editorial">Th</span>e code could
be more complicated; each line could mean two, or ten, or $100$.
Or, as Grant from the Village School suggested, the lines could
mean different quantities in different contexts. For example, when
there are six lines joined together by a horizontal line across the
top, this could represent $30$ or another number. A single line on
its own could still mean "one", but when it is grouped together
with other lines, it can mean something different.</p>
<p class="editorial">This is actually similar to the way that the
numbers, which we use today, work. A number $2$ on its own means
"two". However, if the "$2$" is accompanied by other numbers (like
having more lines in the cave drawings), its place (i.e. its
context) in the number is important as it can mean different things
in different places/contexts. In the number $236$, the "$2$" now
means "two hundred". In the number $56721$, the "$2$" now means
"twenty".</p>
<div><span class="editorial">The numbers that we use most of the
time today are known as</span> <span style="font-weight: bold;" class="editorial">Arabic numerals</span><span class="editorial">.
The Arabic numerals are the ten digits: $0, 1, 2, 3, 4, 5, 6, 7, 8,
9$. These digits are used either on their own, or in combinations,
to make up any number that is desired.</span></div>
<br></br>
<div><span class="editorial">There are many other numeral systems,
and they each have their pros and cons. You may have come across
Roman numerals before; these made up the numeral system in Ancient
Rome. Today, in the UK, we use letters for words and separate
digits for numbers. However, the Roman numeral system is based
on</span> <span style="font-weight: bold;" class="editorial">letters</span> <span class="editorial">of the
Roman alphabet. This sort of system is still used today in
languages such as Hebrew and Greek. In these systems, each letter
also represents a number. The way that numbers are "built up"
varies between the languages, as does the methods used to do
calculations.</span></div>
<p class="editorial">Sam, from Village Elementary, thought that the
drawings in the cave may be some form of numeral system:</p>
<div>At first, I thought it may have been a way of tallies (four
lines, then a slash through them represents "five"). But then I
thought that it was more likely for them to be "cave man" numerals,
like Roman numerals. That would better explain the drawings that
don't look like the others. They could be numbers such as $10$,
like the roman numeral for $10$ is X.</div>
<br></br>
<p class="editorial">Different numeral systems have different
advantages and disadvantages. As mentioned above, the Arabic
numeral system uses the position of a number to encode different
meanings. This is good because we need fewer numbers for our code
(less to remember!). Also, it is quicker to write (and read) these
numbers than, let's say, Roman numerals. Compare "$1998$" (Arabic)
to MCMXCVIII (Roman)!</p>
<p class="editorial">As well as being easier to write and read, the
Arabic numeral system also means that calculations are easier to
do. The "changeover" from the Roman to Arabic numeral system meant
that scientists and mathematicians could do more complicated sums
and come up with more complex theories.</p>
<p class="editorial">Think about the advantages and disadvantages
of using the system seen in the cave (if the drawings do indeed
show some form of counting!).</p>
<p class="editorial">What could the cave men be counting? Sam, from
Village Elementary, had a suggestion:</p>
<div>I think it could be used to count how many spears you had, how
much food you have (or need), or it may have even been to play a
certain type of game.</div>
<br></br>
<p class="editorial">Maia, from Culford was also thinking this way.
She thought that the lines may be used to count sheep. Ethan
thought that the lines may represent years, as part of a
calendar.</p>
<p class="editorial">Overall, there is no definite answer for what
the drawings mean. People have different suggestions, or different
versions of a similar suggestion. As long as the reasoning is good,
then this is great! It is fantastic to have lots of different
theories to choose from!</p>
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<h3><span style="font-weight: bold;">Why do this
problem?</span></h3>
<div>The lovely thing about <a href="http://nrich.maths.org/2472">this problem</a> is that it does
not have a 'right' answer as such and therefore it is a great
context in which children can be creative and uninhibited. It is a
fantastic opportunity for you to encourage learners to justify
their hypotheses.</div>
<br></br>
<h3>Possible approach</h3>
<div>Project <a href="/content/id/2472/stone_age2.jpg">the
picture</a> on the screen for the whole group to see. Try to do
this in such a way as to obscure any text so that there are no
clues as to the meaning of the images. Without saying anything else
at all, invite pairs of children to talk about what they think the
picture may show. You could give each pair <a href="/content/id/2472/StoneAgeCounting.pdf">this sheet</a>, which
has a copy of the picture on it.</div>
<br></br>
<div>Share learners' ideas and, depending on what they say, focus
on, or steer them towards, the idea of counting and representing
numbers. Once this has been established, you can tell them where
the symbols were found and encourage them to work in their pairs to
come up with some more detailed explanations of the drawings.
Children could annotate the sheet to record their thinking.</div>
<br></br>
<div>In the plenary, discuss pairs' interpretations of the
pictures. The important thing is to encourage children to justify
their thinking. You may find that some pairs have different ideas
about the same symbol which may provoke a healthy argument! The aim
would be to try and come up with suggestions for each symbol that
don't contradict each other. You could then challenge children to
record other numbers in the same way, for example on mini
whiteboards.</div>
<br></br>
<h3>Key questions</h3>
<div>Do you think the length of the lines is important?</div>
<div>Why do you think some of the drawings have curved lines?</div>
<div>What are the similarities and differences between the
drawings?</div>
<div>How might you write other numbers?</div>
<br></br>
<h3>Possible extension</h3>
Children could create their own way of representing numbers and/or
they could find out about the La Pileta cave.<br></br>
<br></br>
<h3>Possible support</h3>
<div>Some learners may find this task a little overwhelming because
there isn't a 'right' answer. They may well need frequent verbal
reassurance.</div>
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Do you think the lengths of the lines mean something?<br></br>
Why do you think some of the drawings have curved lines?<br></br>
What are the similarities and differences between the
drawings?<br></br>
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<mdoxml version="1.0">Theodore from Trent School wrote to say: <br></br>
I think it is used for counting the days and weeks. <br></br>
What a good idea. If you have any other suggestions, let us
know!<br></br></mdoxml>
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Stone Age Counting
What could these drawings, found in a cave in Spain, represent?
History and Philosophy of Mathematics
History of number systems
Numbers and the Number System
Reading and writing numbers
Numbers and the Number System
Counting