The education of all young people
in the 21st Century requires astrong definition of numeracy
Max Stephens
The University of Melbourne, Australia
Second International Education Cooperation
Symposium, Tokyo 23 January 20051

Outline of presentation• Defining “numeracy”. What does this termmean for school education and for life
beyond school education?
• Strong and weak definitions of numeracy(“old thinking” and “new thinking”). Whatis the reason for this change?
• Some practical illustrations of what itmeans to be numerate in the 21st century.
2Part 1
Defining Numeracy

3Two key themes• A major policy objective of all governments
is to ensure that all students attain soundfoundations in literacy and numeracy• In our increasingly globalized economies,
we cannot have one definition (実り多い) of
numeracy for rich countries and anotherdefinition of numeracy (粗末な) for those inpoor and developing countries
4

Numeracy – a new term
• Numeracy is a recent concept that has
different meanings for different people.
The term “numeracy” first occurred inEngland in the Crowther Report (1959)• It was re-defined in the Cockroft Report(1982) as “the skills and dispositions
needed by ordinary people in work anddaily life(万民が仕事や生活で必要とする技
能と能力)”
5

Mathematical literacy• In the USA, some mathematics educatorsand politicians(行政官) prefer terms like“quantitative literacy(量的識字力)” or“mathematical literacy(数学的識字力)”• The focus is the same: to be able to use
the mathematics one has learned: in life,
at home, in paid work, in further study, andin community and civic life6OECD – PISA (2000)• PISA assessment focuses on young
people’s ability to apply their knowledge
and skills to real-life problems andsituations– Are students able to analyse, reason andcommunicate their ideas effectively?
– Do they have the capacity and strategies tocontinue learning throughout their lives?
7OECD – PISA (2000)• The OECD considers that– mathematics, science and technology are
sufficiently pervasive in modern life that• personal fulfilment, employment and fullparticipation in society increasingly require an
adult population, which is not only able to read andwrite, but also mathematically, scientifically andtechnologically literate• Mathematical literacy = numeracy
8

PISA defines mathematical literacy• as mathematical knowledge and skills forlife …
• as an individual’s capacity– to identify and understand the role thatmathematics plays in the world– to make well-founded judgements and– to use and engage with mathematics in waysthat meet the needs of that individual’s life asa constructive, concerned and reflective
citizen9Numeracy – an Australian definition
Numeracy=everyone’s business (1997), which was
funded by the Australian Government, provides
a definition which underpins current policy:
To be numerate is to use mathematics effectively
to meet the general demands of life at home, in
paid work, and for participation in communityand civic life
数量図形的思考力を備えているとは、自宅での生活、そして職業、コミュニティや市民としての生活する際の、一般的な課題に遭遇した際に数学を、効果的に利用することである。
(Numeracy=everyone’s business, p. 10)
10What is numeracy about?
• The United Kingdom focus of numeracy has been largely
on number and measurement. But now embraces space,and data as well
• In Australia, Numeracy=everyone’s business (1977), forexample, acknowledges that numeracy must encompass
all the key concepts and skills of mathematics (数学のすべての鍵概念・技能).
• Research shows that many students are unable to taketheir mathematics beyond the mathematics classroom– numeracy must pervade (浸透する) all areas of the curriculum

11

Numeracy and success at
school• Students who cannot make connections
between the mathematics they have beentaught and the mathematical knowledge
they need for their other school subjects– are the most likely to ‘drop out’ of high school– are also most ‘at risk’ of being unemployed
after leaving school12Three different emphases• Early years of school - the focus of‘numeracy’ is on building understanding of
key ideas of number and measurement• Middle years of school - the focus of‘numeracy’ expands to include making
links between mathematics and otherschool subjects and its uses beyond
school13

Three different emphases• Senior high school - how well do young
people use and apply their mathematicalknowledge and skills to a range ofcontexts they are likely to meet in theirother school subjects, in further study andtraining, and in community life
• This view of numeracy is used in OECD’s
PISA study (results for 2003 just released)
14

Part 2Strong and weak definitions of
numeracy or mathematical
literacy15

Public discourse• The word “Numeracy” is an artefact (人工的所産)
of public debate (政治的論争) to draw attention
to those aspects of mathematics that areconsidered important for all students regardlessof their intended specialisation• In Japanese, the closest equivalents may be“suryo zuke tekina shikoryoku” (数量図形的思考力)
or Basics ‘Kiso Kihon’ in the words of the
National Course of Study
16

Strong and weak definitions• The word, ‘numeracy’ belongs to the sphere of basic
education and to the compulsory years of schooling• There are strong (実り多い) definitions of numeracy, andweak/narrow (粗末な) definitions• In countries, where the compulsory years of schoolingare limited to the elementary school, there tends to be alimited or weak (粗末な) definition of numeracy• There students require basic primary education andbasic numeracy (number, measurement and money) toperform basic tasks in one’s job and at home (=numeracy: weak definition) (粗末な)17

Do we want two definitions?
• As a society’s expectations of general or basic education
increase, definitions of numeracy become stronger (実り多い定義が求められる)• Two definitions of numeracy divide rich and poor
countries: strong definition (実り多い) for developed
countries and a weak/narrow definition (粗末な) for
developing countries• Two definitions perpetuate (永続させる) inequalities,
especially economic and educational imbalances18Changing economies - Japan• In Japan between 1990 and 2000, thepercentage of labour force engaged in– agriculture and related industries fell by 28%– manufacturing and mining fell by 12%– knowledge and service related industries
increased by 15%• knowledge and service related industries
now constitute 65% of the total labour
force (Labour situation in Japan, 2003)
19

Changing economies - Thailand• In Thailand between 1990 and 2002, thepercentage of labour force engaged in– agriculture fell from 64% to 36%– manufacturing & mining rose from 10% to 16%– knowledge and service related industries increased
from 26% to 48%• knowledge and service related industries now
constitute nearly half of Thailand’s labour force(Asian Development Bank: Key Indicators of Developing Asian andPacific Countries, 2003)
20

Changing economies• In many countries, manufacturing andagriculture are becoming less labour
intensive and more capital intensive• Capital intensive work requires fewer
people and higher mathematical literacy• The growing knowledge and service
related industries also require higher
mathematical literacy
21Changing work patterns• A working life in the 21st century will not be
spent in one job, or with one employer, orwith one company, like Japan in the 1980s(Horio, 1997)• Today’s generation of young workers willneed to be retrained in the skills the needand may change their work several times
in a working life (Morishima, 2000)
22

An ‘old’ world view
• Agricultural and industrial societies could allow
many students to drop out of schoolmathematics.
• There were always plenty of low skilled jobs forthese students to undertake• In these societies, it was sufficient to selectthose who could work in technology, engineeringand other careers which required a strong
mathematical preparation
23

An ‘old’ world view
• The education system was designed to
identify those who were likely to succeedin these “elite” careers• Schools adopted a pyramid approach tomathematics, selecting and rejecting, andexpecting that many students would failalong the way (人々を序列化し少数の成功者と多数の失敗者をその過程で選別する)(see Uzawa, 2000)
24

An ‘old’ world view
• Mathematics was used to select students at theend of primary school for a limited number ofplaces in secondary schools• In these societies, people also entered a stablelabour market (安定した労働市場) where jobs
were well defined and where a person’s entire
working life might well be spent in one job orwith one company• As a result, people could be trained in the skills
required to do their work over the course of aworking life25

Rapid social change• Information and service driven societies require
quite different knowledge bases. Their labourbase is structured quite differently. There arefewer and fewer unskilled jobs.
• Manufacturing and agriculture are both
becoming capital(資本) intensive and less labour
intensive.
• Capital intensive work requires people with highability to process numerical, statistical, andgraphical information (= numeracy: strong
definition) (実り多い).
26

Rapid social change• Today, the labour market is not so stable• Jobs can disappear or can be re-defined interms of the necessary knowledge and skills
required to carry them out• There is a rapid growth in part-time work, “oncall work”, self-employed work and in small
businesses that respond to market forces• The school system needs to prepare young
people for a very different world of work fromthat which their parents knew

27Mathematical literacy• PISA 2003 used a strong framework formathematical literacy in terms of:– mathematical content:• quantity• space and shape• change and relationships, and• uncertainty

28

Mathematical literacy• PISA 2003 used a strong framework formathematical literacy in terms of:– situations or contexts where mathematics isused• personal• educational/occupational• public• scientific29

Mathematical literacy• PISA 2003 used a strong framework formathematical literacy in terms of:– mathematical competencies• thinking and reasoning• argumentation• communication• modelling• representation• problem posing and solving• using symbolic, formal and technical language andoperations
30

Part 3
Some illustrations of mathematical
literacy needed for the 21stCentury31

From the middle school
Investigating Japan’s population
3-4 lessons

32

Population of Japan• From the newspapers and from their social
studies class, students know that Japan’s
population is “ageing”• To investigate what this means thestudents looked at Japan’s population in1925 and 1998
• They used EXCEL to make a graph fromthe numerical data
33

> 85

80 ~ 8575 ~ 80

70 ~ 75

65 ~ 70

60 ~ 65

55 ~ 60

50 ~ 55

45 ~ 50

40 ~ 45

35 ~ 4030 ~ 35

25 ~ 30

20 ~ 25

15 ~ 20

10 ~ 15

5 ~ 100~5

x 1,000 people

Population of Japan in 1925

9,000

8,000

7,000

6,000

5,000

4,000

3,000

2,000

1,000

0

Age Group

34

> 85

80 ~ 8575 ~ 80

70 ~ 75

65 ~ 70

60 ~ 65

55 ~ 60

50 ~ 55

45 ~ 50

40 ~ 45

35 ~ 4030 ~ 35

25 ~ 30

20 ~ 25

15 ~ 20

10 ~ 15

5 ~ 100~5x 1000 people

Population of Japan in 1998

12,00010,000

8,000

6,000

4,000

2,000

0

Age Group

35

Japan’s population• Students noticed very different shape ofthe two graphs• Median age in both cases is quite different• They found population data for Australia(1921 & 1998) and USA (1925 & 1990)• They used technology to create graphs forAustralia and USA
36

Japan’s population• Size of population is different, but students
compared the shapes of the graphs fromthe 1920s and 1990s• Japan, Australia and the USA all have asimilar “population profile” in the 1920s• Students explained what was similar fromthe graphs from the 1920s

37

Japan’s population• In the 1990s the profile of Japan’s
population is very different from Australia
and the USA• Students identified key differences• What problems are likely to arise forJapan in the future?
• What predictions can be made?
38

Japan’s population• Technology helps students to present
complex information in a way that they can
see the problem and make some analysis• At the conclusion of their study– Students provided some feedback to theteacher at the end of this topic– Their teacher gave students some further
questions to consider
39

From the middle/senior highschoolLooking at the environment
2 lessons40Reducing Bluegill population in
Lake Biwa• Bluegill fish have become a problem in
Lake Biwa• Shiga prefecture has a plan to catch acertain number each year to reduce thepopulation• Students in Years 8 and 9 at Tennoji
Junior/senior high school investigated this
topic (ICME-10: Yanagimoto, 2004)
41

Reducing Bluegill population in
Lake Biwa• Students used graphics calculators to do
the mathematical modelling of this theme:– According to the fishery section of Shiga
prefecture in 1999, it is estimated that there
are between 1500 t and 3000 t of bluegills in
Lake Biwa– They plan to catch 300 t each year to halve
the population of blue gills in 10 years– The budget for this is ¥ 55 000 000
42Reducing Bluegill population in
Lake Biwa• Assumptions used by the students in theirmodel:– (1) the present population of bluegills in Lake
Biwa is between 1500 t and 3000 t– (2) the breeding rate of bluegills is constant– (3) the capture of bluegills is 300 t every year
– (4) other conditions are not considered43Reducing Bluegill population in
Lake Biwa• Students used Casio graphic calculator fx9700GE and its table and graph menu
• The students initially– Used 15% as the breeding rate of bluegills,– Input the recurrence formula• An+1 = An + 0.15 An – 300• estimating the present amount of bluegills at 2250 t(A0 ) as the average of 1500 t and 3000 t• Let n be from 0 year to 10 years
44Reducing Bluegill Population in
Lake Biwa
Bluegill Population (t) at 15% breeding rate and removing 300t per year
3500

3000

2500

Tonne

2000

1500

1000

500

0
0

1

2

3

4

5
No of Ye ars

6

7

8

9

10

45

Reducing Bluegill population in
Lake Biwa• The graph shows that with 15% breeding
rate the population of bluegills increases• Next, the students took 5% as thebreeding rate of bluegills and used thesame recurrence formula• In this case, the population of bluegills
decreases immediately and becomes zeroafter nine years
46Reducing Bluegill Population in
Lake Biwa
Bluegill Population (t) at 5% breeding rate and removing 300t per year
2500

2000

Tonne

1500

1000

500

0
0

1

2

3

4

5
No of Ye ars

6

7

8

9

10

47

Reducing Bluegill population in
Lake Biwa• Students then tried various breeding rates• They showed that if the breeding rate is
10.23% the number of bluegills becomes
half in 10 years• Students then did some extensions.
• They used 1500 t and 3000 t separately as
the actual population and combined thesenumbers with different breeding rates
48

Reducing Bluegill Population in
Lake Biwa
Bluegill Population (t) at 10.23% breeding rate and removing 300t per year
2500

2000

Tonne

1500

1000

500

0
0

1

2

3

4

5
No of Ye ars

6

7

8

9

10

49

Reducing Bluegill population in
Lake Biwa• Students could quickly create different
graphical representations using fx-9700GE• Students concluded that– mathematics is useful (essential?) in
investigating environmental problems– validity of the model depends on the accuracyof the breeding rate and population estimate– more detailed modelling may be needed tosolve the problem
50

Conclusions• In these examples of mathematical literacy– Focus is on using mathematics to investigate
a practical problem– Using mathematics to create an image or
representation of reality– Access to technology is important– The most difficult mathematics is not always
required51

Conclusions• In these examples some considerations
become very important:– What is the quality and reliability of the data?
– Does the real situation need to be simplified?
– What results are given?
– What predictions can be made?
– What limitations or restrictions need to beplaced on these predictions?
52

Summary• In Japanese, there may be no exact
equivalent for the term “Numeracy”• “Mathematical literacy” as used by PISA
may be more useful expression• In the past, numeracy focussed on
calculation and simple measurement• This weak definition is not sufficient foryoung people in the 21st century
53

For the 21st century• Representing, interpreting and analysing
quantitative information is central tomathematical literacy (numeracy)• Computer and hand-held technology
makes representation easier and quicker• In an information society, representing andanalysing information is a key element ofmathematical literacy54