realistic mathematics education a teaching approach … · realistic mathematics education – a...

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

m.vandenheuvel@fi.uu.nl