Mathematics and Science Education
19 August - 30 August 2013
Realistic Mathematics
Education
–
A teaching approach
that makes sense
Marja van den Heuvel-Panhuizen
Freudenthal Institute
Faculty of Science
Faculty of Social and Behavioural Sciences
Faculty of
Social and
Behavioural
Sciences
Faculty
of Science
Secundary
Education
Early Childhood
Special Education
Primary Education
Vocational Education
Freudenthal Institute
Realistic
Mathematics
Education
• “real world mathematics”
• to imagine = ZICH REALISEREN
• context-based
real world or fantasy world
formal world of mathematics
Realistic
Mathematics
Education
Freudenthal Institute
1971 - ....
~1968
2013
• still under construction
• over the years
different accentuations
Mechanistic
mathematics
education
Realistic
Mathematics
Education
New Maths
~1968
Grade 3
1969
(9e edition)
Grade 3
2013
% market share RME textbooks
% market share Mechanistic textbooks
1960s
1980s
1987
1992
1997
2004
Mechanistic
Mathematics
Education
Realistic
Mathematics
Education
- bare number calculations
- activity principle
- reality principle
- level principle
- intertwinement principle
- interactivity principle
- guidance principle
- little attention applications
(especially not at start)
- teaching is transmission
* atomized
* step-by-step
Realistic Mathematics Education
transmission approach to learning
applications
target
constructivist approach to learning
applications
applications
source
target
TIMSS 2003 Study - Grade 8
International average: 38% got a full credit
Dutch students: 74% got a full credit
Freudenthal
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.
“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)
Treffers 1987 (Three Dimensions)
mathematizing
“real” world
1
2
mathematics
Realistic
Mathematics
Education
– various levels of
understanding
– progressive schematization
– models as bridges
- activity principle
- reality principle
- level principle
- intertwinement principle
- interactivity principle
- guidance principle
Grade 1
Grade 1
formal
calculation
1
5
1
1
5
1
1
1
1
1
1
1 1
1
1
1
1
1
cross-section
six and six is ...
calculation
by structuring
calculation
by counting
calculation
by structuring
longitudinal-section
formal
calculation
Treffers 1987 (Three Dimensions)
Progressive
schematization
Progressive ‘complexization’
Progressive schematization
12
6394
1200
5194
1200
3994
1200
2794
1200
1594
1200
394
120
274
120
154
120
34
24
10
12
100
100
100
100
100
10
10
10
2
532 r. 10
6394
2400
3994
2400
1594
1200
394
360
34
24
10
12
200
200
100
30
2
532 r. 10
6394
6000
394
360
34
24
10
12
500
30
2
532 r. 10
6394 532
60
39
36
34
24
r. 10
whole-number-based
written calculation
53 5459
5300
159
159
0
100
3
103
digit-based
written calculation
53 5459 103
53
15
0
159
159
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
bus stop
model of
on
31
model for
off
on
39
difference
minimal
or more
off
On which table
do you get more?
model of
or
Which fraction is larger ?
15 or 6
12
8
model for
3
6
4
8
Mathematics in Context
mathematics
textbook series
for grades 5-8
Romberg (Ed.) (1997-1998 ...)
Grade 5 (- 6)
Learning trajectory for percentage
qualitative/informal way
of working
with percentage
percentage as
descriptors of
so-many-out-of-somany 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
Calculating via 1%
Year
Total
Number Percent
of
number
of
of
marathon
of
drop outs drop outs
runners
Describe your strategy
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
Jimenez
20%
Jacobs
25%
Peresini
Fulhouse
30%
15%
10%
Situations of change - prices
Additive way:
‒ 25%
75%
$3.20 ÷4 = $0.80
$3.20 ‒ $0.80 = $2.40
Multiplicative way:
x 0.75
$3.20 x 0.75 = $2.40
Check the sale price by making just
one calculation on you calculator
80%
x 0.8
80%
x 0.8
64%
x (0.8 x 0.8) is x 0.64
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
sale price
75%
100%
96
?
x 0.75
÷ 0.75
original price
original price
Shifts in function of the bar model
representational model
estimation model
Year
of
marathon
calculation model
thought model
Total
number
of
runners
Number
of
drop
outs
Percent
of
drop
outs
Describe your strategy
shifts in
 context
 domain
 function
contextconnected/
informal
level of
understanding
model of
of
for
model for
of
for
and so on
of
for
general/
formal
level of
understanding
model of
Realistic
Mathematics
Education
model for
your teaching
Thank you
m.vandenheuvel@fi.uu.nl
http://www.staff.science.uu.nl/~heuve108/download/summer-school-readings