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Realistic Mathematics
Education
–
A teaching approach
that makes sense
Marja van den Heuvel-Panhuizen
Faculty of Social and Behavioural Sciences
Faculty of Science
Freudenthal Institute
Mathematics and Science Education
19 August - 30 August 2013
Faculty of
Social and
Behavioural
Sciences
Faculty
of Science
Freudenthal Institute
Secundary
Education
Early Childhood
Special Education
Primary Education
Vocational Education
• “real world mathematics” • to imagine = ZICH REALISEREN
• context-based real world or fantasy world formal world of mathematics
Realistic Mathematics Education
~1968 2013
Realistic Mathematics Education
• still under construction
• over the years different accentuations
Freudenthal Institute 1971 - ....
~1968
New Maths
Mechanistic mathematics education
Realistic Mathematics Education
1969 (9e edition) 2013
Grade 3 Grade 3
1960s 1980s
% market share RME textbooks
% market share Mechanistic textbooks
1987 1992 1997 2004
Realistic Mathematics Education
- bare number calculations
- little attention applications (especially not at start)
- teaching is transmission * atomized * step-by-step
- activity principle
- reality principle
- level principle
- intertwinement principle
- interactivity principle
- guidance principle
Mechanistic Mathematics Education
Realistic Mathematics Education
constructivist approach to learning
transmission approach to learning
target
applications
applications
source target
applications
International average: 38% got a full credit
TIMSS 2003 Study - Grade 8
Dutch students: 74% got a full credit
Rather than beginning with abstractions or definitions to be applied later, one must start with rich contexts that ask for mathematical organization; or, in other words, one must start with contexts that can be mathematized.
Freudenthal
“What humans have to learn is not mathematics as a closed system, but rather as an activity, the process of mathematizing reality and if possible even that of mathematizing mathematics.” (1968)
mathematizing
“real” world mathematics
Treffers 1987 (Three Dimensions)
1 2
Realistic Mathematics Education
- activity principle
- reality principle
- level principle
- intertwinement principle
- interactivity principle
- guidance principle
– various levels of understanding
– progressive schematization – models as bridges
Grade 1
calculation by counting
calculation by structuring
formal calculation
calculation by structuring
formal calculation
Grade 1
cro
ss-s
ecti
on
longitudinal-section
1 1
1
1
1
1
1
1 1
1 1
1 1
1 5 1 1 5
six and six is ...
Treffers 1987 (Three Dimensions)
Progressive
schematization
Progressive ‘complexization’
Progressive schematization
63946000 500394360 303424 210
12
532 r. 10
63942400 20039942400 20015941200 100394360 303424 210
12
532 r. 10
63941200 10051941200 10039941200 10027941200 10015941200 100394120 10274120 10154120 103424 210
12
532 r. 10
6394 5326039363424
r. 10
12
159 0
5300
159
100
3
103
53 5459
whole-number-based written calculation
digit-based written calculation
53 5459 103
53 15
159 159
0
0
Streefland
1985 (Wiskunde als activiteit en de realiteit als bron) 1996 (Learning from history for teaching in the future)
Models as bridges: Model of → Model for
model of bus stop
on off
39 31
on off
model for difference
minimal or more
On which table do you get more?
3
4
6
8 or
Which fraction is larger ?
15
12
6
8 or
model for
model of
Romberg (Ed.) (1997-1998 ...)
mathematics textbook series for grades 5-8
Mathematics in Context
Grade 5 (- 6)
Learning trajectory for percentage
qualitative/informal way of working with percentage
percentage as descriptors of so-many-out-of-so-many situations
quantitative/formal way of working with percentage
percentage as operators
Informal knowledge
How busy will the school theater be? Color the part that will be occupied and write down the percentage of occupied seats
Emergence of the bar model
Emergence of the bar model
Occupation meter
60 out of 80
50 out of 85
36 out of 40
Bar as an estimation model
Poll about favorite baseball souvenir
Giants fans (310): 123 vote for cap Dodgers fans (198): 99 vote for cap Which fans like the cap the best?
Introduction of 1% benchmark
Year of
marathon
Total number
of runners
Describe your strategy Number
of drop outs
Percent of
drop outs
Calculating via 1%
10% 20%
Jimenez
25%
Jacobs
30%
Peresini
15%
Fulhouse
Directly dividing by the whole number
1% is 600÷100 = 6
121 ÷ 6 ≈ 20 121 ÷ 6 = 20.166666 ≈ 20
121÷600 = 0.20166666 ≈ 0.20
Situations of change - prices
Check the sale price by making just one calculation on you calculator
$3.20 ÷4 = $0.80 $3.20 ‒ $0.80 = $2.40
75% Additive way: ‒ 25%
Multiplicative way: x 0.75
$3.20 x 0.75 = $2.40
x 0.8
80%
x 0.8
80%
x (0.8 x 0.8) is x 0.64
64%
Situations of change – interest-bearing account
$447.71
after 1 year: $250 x 1.06
after 2 years: $250 x 1.06 x 1.06
after 3 years: $250 x 1.06 x 1.06 x 1.06
after 4 years: $250 x 1.06 x 1.06 x 1.06 x 1.06
after 5 years: $250 x 1.06 x 1.06 x 1.06 x 1.06 x 1.06 = $354.63
sale price ÷ 0.75
original price
sale price x 0.75
original price
75%
96 ?
100%
representational model
estimation model
Year
of
marathon
Total
number
of
runners
Describe your strategy
Number
of
drop
outs
Percent
of
drop
outs
calculation model
thought model
Shifts in function of the bar model
context- connected/ informal level of understanding
general/ formal level of understanding
model of
shifts in context domain function
model for
and so on
of for
of for
of for
Realistic Mathematics Education
model of model for
your teaching
http://www.staff.science.uu.nl/~heuve108/download/summer-school-readings
Thank you