Origins of number
MDPT
Culture and Nature
• How are numbers used and conceptualised by
people?
– What differences and similarities are there
between peoples from various communities?
• In what ways do numbers use people?
– Is it the case that humans
• cannot not count?
• cannot not notice ‘more’ versus ‘less’?
Main theme
• As humans we are predisposed to use number
for/as properties.
• In human societies, this predisposition has
flourished in many different ways.
• This short presentation will refer to
– historical snapshots,
– biological results,
– cultural positions.
Historical snapshots
• An ancient artefact from Africa
• Representations of number from
– Middle East
– South America
• A method of calculating from China
Mathematical artefacts: engraved bones
There are now several
archaeological findings
of bones with tallymark engravings.
A famous example is the
Ishango bone,
discovered in 1960 on
the border of modernday Congo & Uganda, it
is ~20,000 years old.
What does this artefact tell us about
the mathematics of that community?
There are different
interpretations of the marks
on the bone.
Historians of mathematics (who
interpret the marks) use them
as evidence to make claims
about the cultural practices of
the people that produced the
marks.
These include the practices of :
• doubling/halving;
• adding and subtracting 1 from
10 or from 20;
• lunar/periodic tracking.
Babylonian civilisation from ~2000600BC
Babylonian number system
• a positional system with a base of 60 rather than the
system with base 10 in widespread use today.
• Babylonians constructed tables to aid calculation:
– Two tablets found at Senkerah on the Euphrates in 1854
date from 2000 BC give squares of the numbers up to 59
and cubes of the numbers up to 32.
– The table gives 82 = 1,4 which stands for
1*60 + 4 = 64
– and 592 = 58, 1 (= 58*60 +1 = 3481).
Why base 60?
• The Babylonians divided the day into 24 hours, each hour
into 60 minutes, each minute into 60 seconds.
• This has survived for 4000 years!
– Notations for sexagesimal numbers, e.g., 5 hours, 25 minutes,
30 seconds include
• 5h 25' 30"
• the ‘sexagesimal fraction’ 5 25/60 30/3600
• 5; 25, 30.
– the number 5; 25, 30 - in sexagesimal form - can be
expressed as a base 10 fraction: 5 4/10 2/100 5/1000
– i.e. 5.425 in decimal notation.
Mayan civilisation 2000BC –900AD
Mayan positional number system
Chinese calculating rods
• These are calculating devices that are precursors of the modern Chinese abacus.
– Evidence of use dates from the first century BCE.
• The procedure for calculating and the
procedure for representing are very close.
– This is unlike the current world-wide standard of
representing number using ‘1’, ‘2’, ‘3’ etc. and
calculating with number using algorithms like
‘column arithmetic’ for addition.
Counting rods in vertical and
horizontal layouts
Place value represented by layout style
Place value is read left to right (as in ‘HTU’)
• ‘Units’ (a.k.a. ‘Ones’) are represented VERTICALLY
• ‘Tens’ are represented HORIZONTALLY
• ‘Hundreds’ are represented VERTICALLY
• ‘Thousands’ are represented HORIZONTALLY
• and so on- gaps left for zero in that order of magnitude
For example:
Use matchsticks for rods
There is difference in opinion as to whether they
worked from the highest order of magnitude down
to the units or from the units up as we do
Adding practice
1.
2.
3.
4.
5.
567 and 312
567 and 352
567 and 358
567 and 338
Your choice.
Neuroscience and mathematics
• This is a developing area of research with new
results being published regularly
• Brain imaging, gives information about location
of brain activity when a person is performing
certain tasks
– techniques include fMRI (functional Magnetic
Resonance Imaging), PET scans (Positron Emission
Tomography).
• Eye movement tracking is used to attribute
attention.
We can’t help seeing quantities!
Although material objects are not directly
associated with a unique number
– e.g., a single apple is not just ‘one’ but can be
associated with many molecules, several pips and a
range of colours,
babies (just a few days old) can distinguish between
visual presentations of one, two or three dots
– perceptual grasp of number is pre-linguistic;
– humans and some animals have this number
perception ability;
– ‘subitizing’ is the term used for seeing a small number
at a glance.
Exact numbers and estimates
• Examples of research into how people recognise,
represent and process number:
–
Recognising small numbers activates a different part
of the brain from estimating two quantities.
•
–
This ‘functional independence’, which is found in other
aspect of mathematics, suggests that there is no such
thing as being ‘globally bad at mathematics’ (Ann Dowker)
Language skills areas of the brain and mathematical
skills areas overlap but are not the same
•
You can be good at mathematics without high overall
intelligence.
Mathematics common to all cultures
• Counting
• Locating
• Designing
• Playing
• Explaining
• Measuring
From: Alan Bishop ‘Mathematical Enculturation’ 1988
A few reference links
• History
– http://www-history.mcs.st-andrews.ac.uk/history/
• Roman numerals
– http://www.novaroma.org/via_romana/numbers.html
• Neuroscience
– http://faculty.washington.edu/chudler/image.html
– http://www.educ.cam.ac.uk/centres/neuroscience/