Transcript PowerPoint Presentation - Professor Paul Dowling

What is mathematics not?
• natural or inevitable
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Yoruba
Aboriginal people
Luria
Bernstein
Heath
Alexander Luria’s interviews
The following syllogism was presented. White bears exist only where it
is very cold and there is snow. Silk cocoons exist only where it is very
hot. Are there places that have both white bears and cocoons?
Subject: Kul, age twenty-six, peasant, almost illiterate.
‘There is a country where there are white bears and white snow.
Can there be such a thing? Can white silk grow there?’ …
‘Where there is white snow, white bears live. Where it is hot ,
there are cocoons. Is this right?’ …
Where there is white snow there are white bears. Where it is hot
there are white silkworms. Can there be such a thing on earth?
What is mathematics not?
• natural or inevitable
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Yoruba
Aboriginal people
Luria
Bernstein
Gee
Heath
What is school mathematics?
• collection of objects and rules for
their combination
• pedagogic ‘theory’ that enables
mathematics to cast a gaze beyond
itself
Interpretive
scheme
Triangle task
The birthday party
What is school mathematics?
• esoteric domain
• pedagogic ‘theory’
• public domain
What else is school mathematics
• horizontal discourse
– tacit principles
– informally acquired
– context-dependent
• vertical discourse
– explicit principles
– formally acquired
– context independent
(Basil Bernstein)
What else is school mathematics
• high discursive saturation (DS+)
– principles explicit
• low discursive saturation (DS-)
– tacit principles
• all practices will incorporate both
– ie no such thing as purely vertical or purely
horizontal discourses
• Mathematics substantially DS+
A teacher interested only in answers might have seen this, from
seven-year old Mary, and just marked it right:
25 - 18 = 7
Fortunately, Mary’s teacher was interested in processes,
encouraging the children to write everything down. This is what
she actually got from Mary.
25 - 18 is 5 take away 8 is 5 steps forward and 8 steps back,
that’s -3. 20 take away 10 is 10. 10 and -3 is 7. 7 is the answer.
From this she learnt that Mary had a good ‘feel’ for numbers and
an understanding of place value and negative numbers (although
we would not necessarily expect the latter at this level, nor does
the National Curriculum require it).
(Nelson Mathematics)
What else is school mathematics
• dominant form of argumentation is
syllogism
– All those born within the sound of Bow Bells are
cockneys
– Dowling was born within the sound of Bow Bells
– Dowling is a cockney
What else is school mathematics
• empiricist
• children need concrete experiences if they
are to acquire sound mathematical concepts;
• like adults, children learn best when they
invent and make discoveries for themselves;
(Nelson Mathematics)
What is a triangle?
What is Mathematics?
• esoteric domain is non-arbitrary,
self-referential content
• abstraction from the ‘real world’ has
taken a very long time
• DS+ privileges professional middle
class
is mathematics necessary?
• for domestic practices such as
shopping?
– settings incoroporate resources
– shoppers develop strategies
does maths help you
understand the economy?
On the news recently it was said that the
annual rate of inflation had fallen from
17.4% to 17.2%. What effect do you think
this will have on prices? (If answer ‘none’)
What do you think ought to happen if it
had fallen to, say, 12%?
does maths help you
understand the economy?
• Government data puts annual food inflation at
6.6%. But in the malls of Britain shoppers
would be quick to say the number-crunchers
are having a laugh. At Asda, a dozen free
range eggs cost £1.75 in May last year. Now
the price tag is £2.58—a rise of 47%. In
Sainsbury’s, 500g of pasta has gone up from
37p to 67p—an 81% increase. Bread is up by
20%, English cheddar by 26% … the list goes
on.
Katie & Arlo
Word Problems with Katie
and Arlo
Lauren can solve math problems very quickly.
She can solve 17 math problems in one
minute.
How many math problems can Lauren solve in
8 minutes?
esoteric domain translation
expressive domain text
• Lauren solves 17 problems in each minute.
• How many problems has she solved after two
minutes?
• How many after three minutes?
• How many after five minutes?
• How many after 20 minutes?
• Suppose we give her some more difficult problems.
She can do only 12 of these in each minute. How
many in ten minutes?
expressive domain text
Domains of practice
descriptive domain
Mathematics is not useful!
jim
zola
G police
Y police
Domains of practice
Adding temperatures
Ghost Blasters
practical strategic space
Institutionalisation
Formal (I+)
Informal (I-)
DS+
discourse
idiolect
DS-
skill
trick
pedagogic
exchange
Practical strategic space 2
Field of practice
Category of
Author
Closed
Open
Open
Closed
charismatic
traditional
liberal
bureaucratic
Authority
Target
Interaction
between
Similars
Disimilars
Closure
Openness
equilibration
exchange of
narratives
pastiche
hegemony
Modes of interaction
Summary 1
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•
•
•
Sociological/linguistic perspective
Constructed nature of theory
Maths is not universal
Abstract nature of school discourse
privileges European professional
middle class
Summary 2
• Maths does not emerge easily from
the ‘real world’
• Maths is not useful
Summary 3
•Domains of practice
•Practical strategic space
•Authority strategies
•Interaction
These interrogate rather than
prescribe practice