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Monica Wijers, Dede de Haan, Pauline Vos,Rijkje Dekker, Harm Jan Smid, Adri Treffers,Marja van den Heuvel-Panhuizen, MarjoleinKool, Mar van Zanten, Koeno Gravemeijer,Wil Oonk, Ronald Keijzer, Joke Daemen, TonKonings, Theo van den Bogaart, PaulDrijvers, Martin Kindt, Kees Hoogland, Irisvan Gulik-Gulikers, Jenneke Krüger, Jan vanMaanen, Michiel Doorman, Aad Goddijn, Edde Moor, Wim Groen, Floor Scheltens, JudithHollenberg, Ger Limpens, Ruud Stolwijk, Jande Lange

David Webb, Frederick Peck, Dirk De Bock,Wim Van Dooren, Lieven Verschaffel, ErichWittmann, Cyril Julie, Faaiz Gierdien, AbrahamArcavi, Berinderjeet Kaur, Wong Lai Fong,Simmi Naresh Govindani, Petra Scherer,Betina Zolkower, Ana María Bressan, SilviaPérez, María Fernanda Gallego, Xiaotian Sun,Wei He, Dirk De Bock, Johan Deprez, DirkJanssens, João Pedro da Ponte, JoanaBrocardo, Christoph Selter, Daniel Walter, DorAbrahamson, Elisa Stone, Kyeong-Hwa Lee,YeongOk Chong, GwiSoo Na, JinHyeong Park,Omar Hernández-Rodríguez, Jorge López-Fernández, Ana Helvia Quintero-Rivera, AileenVelázquez-Estrella, Mogens Niss, Zulkardi,Ratu Ilma Indra Putri, Ariyadi Wijaya, UshaMenon, Paul Dickinson, Frank Eade, SteveGough, Sue Hough, Yvette Solomon

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Driving to Hamburg Thematic Afternoon

European Didactic Traditions: the Netherlands

2

U H

110

O

215

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115

Distance to O

Dist

ance

to B

Driving to Hamburg: schematizing

?

4

Driving to Hamburg: animation

H H

110

O

215

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115

Problem orientation: � Starting point in U: (O, B ) = (215, 330) � End point in H: (O, B ) = (225, 110)

Model: � u = distance to Utrecht (independent variable) � O(u) = distance to Osnabruck = | u – 215 | (dependent) � B(u) = distance to Bremen = | u – 330| (dependent) � P(u) = (O(u), B(u)) -> So we have a parametric curve!

5

Driving to Hamburg: model

H H

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Teaching experience

� 13-14 year old � High achievers � Bilingual stream

6

Teaching experience

7

Teaching experience

8

Conclusion: activity features What makes this a nice problem? � Realistic, meaningful context as point of

departure � Unconventional, non-routine problem

(no time-distance, but distance-distance graph) � Different types and levels of approaches and

solutions � Input from students, interaction between

students and between teacher and students These aspects are core in NL math ed tradition. To design such productive tasks is our challenge!

9

Paul Drijvers, Freudenthal Instituut p.drijvers@uu.nl

2016-07-27 www.uu.nl/staff/PHMDrijvers

Thank you! Thematic Afternoon

European Didactic Traditions

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From proof of conceptto large scale projects

• Whitnall study à Mathematics in Context• de Lange, van Reeuwijk, Burrill et al

• Mathematics in the City• Fosnot, Dolk et al

• Statistical reasoning• Gravemeijer, Cobb et al

• ARISE/COMAP: Assessment Tasks à Curriculum• Garfunkel, van der Kooij et al

• Assessment: RAP à CATCH à BPEME• Abels, Dekker, de Lange, Feijs, Querelle, Webb

US Attraction to RME

• Unique context-first approach

• Preformal models and tools

• Robust approach to assessment

Context first approach

Pre-formal models and tools

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Thema&c  A*ernoon:  European  Didac&c  Tradi&ons  –  Netherlands  Wednesday,  16.00-­‐18.00  

 TWO  DECADES  OF  RME  IN  INDONESIA:  

FROM  ICMI  SHANGHAI  1994  TO  ICME  HAMBURG  2016      

Zulkardi,

Ratu Ilma Indra Putri Sriwijaya University

Aryadi Wijaya Jogyakarta State University,

Indonesia

OUTLINE  

Ø   Mathema@cs  reform  using  RME  in  Indonesia  Ø   The  development  of  PMRI  

Ø   Ini@a@on    Ø   Implementa@on    Ø   Dissemina@on    

Ø   PMRI  growth  beyond  project  Ø   Development  of  a  web  portal  on  PMRI  Ø   SEA-­‐RME  Course  Ø   Founding  an  Indonesian  Journal  on  (R)ME  

Ø   PMRI  con@nues    

Math  Reform  using  RME  in  Indonesia                In  1994,  Jan  de  Lange-­‐  Director  of    Freudenthal  Ins@tute  presented  his  keynote  in  ICMI  Interna@onal  seminar,  in  Shanghai.                    Prof.  Sembiring  saw  Jan’s  presenta@on  about  RME.    As    Dik@’s  team  who  search  what  the  best  math.  educa@on  to  change  math.  modern  in  Indonesia,  Sembiring  invited  Jan  to  come  to  Indonesia  in  order  to  help  Indonesia  reforming    mathema@cs  educa@on.      

The  ‘culture’  decision  of  Indonesia  to  adapt  RME  to  Indonesian  context    

                       In  1998,  Jan  de  Lange  came  to  ITB,  Bandung  visited  pak  Sembiring.  With  Prof.  Tjeerd  Plomp  (UTwente)  and  Annie  Kuipper,  they  conducted  a  workshop  on  RME  with  20  candidates  and  selected  6  Ph.D.                  Indonesian  Government  sent  and  funded  the  six  students  to  do  a  Ph.D.  research  on  RME  in  the  Netherlands.  The  six  followed  the  ‘sandwich    PhD.  Program”    between  UTwente  and  Utrecht  University.  Finally,  they    got  PhD.  on  RME  in  December  2002.    All  of  them  are  professor.  (Prof.  Sutarto  is  also  here  now!)  

THE  DEVELOPMENT  OF  PMRI  Year   Reform  Movement  of  PMRI  1994   Ini@a@on:  Prof.Sembiring    Prof.  Jan  de  Lange  in  ICMI  Shanghai    1998-­‐  2002  

Six  PhD  candidates  sent  to  the  Netherlands  to  learn  RME  

2001   Implementa@on:  PMRI  is  started  and  Small  Project  PMRI  was  started  in  3  ci@es  in  Java  

2006-­‐  2011  

Dissemina@on  of  PMRI  (DO-­‐PMRI)  Project  funded  by  Dutch  Government  about  25  out  of  34  Provinces  and  linked  by  a  Web  Portal  P4MRI.net    

2008   Star@ng  a  Joint  Master  program  on  RME-­‐PMRI  (IMPOME)  among  Unsri  Palembang-­‐Utrecht  University  and  Unesa  Surabaya  

2010   Star@ng  Mathema@cs  Literacy  Contest  (KLM)  and  Journal  on  (R)ME    

2011   Star@ng  SEA-­‐DR  Conference  on  Design  Research  on    

2012   SEA-­‐RME  for  Teacher  from  ASEAN  countries    

2016   Star@ng    PhD  program  on  PMRI  and  Na@onal  Center  in  Unsri  Palembang    

A  decade  of  PMRI  in  Indonesia:    Story  form  some  chapters      

1.! P4MRI UNSRI 2.! IMPOME (MASTER PMRI) 3.! KLM 4.! SEA-DR 5.! P4MRI.NET 6.! JOURNAL (R)ME 7.! INDOME (DR. PMRI)

1.! P4MRI ACEH

1.! P4MRI ULM

1.! P4MRI UNY 1.! P4MRI UNESA

1.

THE ACTUAL INFRASTURUCTURE OF PMRI (Sembiring, et.al ,2011)

Portal  PMRI  www.p4mri.net    

hdps://sites.google.com/site/kondriunsri2013/  

http://seminar.fmipa.unp.ac.id/seadr16/

impomeunsri.wordpress.com  

1. Counting orally during Gasing is moving Example of research using PMRI

2. Students count orally in conjunction with the

movement of clock hand

3. Counting the strip and the space of the strip of second interval on the clock

Scores  PISA  Mathema@cs  of  Indonesian  Students    in  Year  2000  to  2012  

(IDN)  Students’  difficul@es  with  context-­‐based  (PISA)  problems  

(Wijaya,  2015)  Four  types  of  errors  in  solving  context-­‐based  problems:  •  Comprehension:  38%  •  Transforma@on:  42%  •  Mathema@cal  processing:  17%  •  Encoding:  3%  Conclusion:  IDN  students’  mainly  have  difficulty  in  comprehending  a  context-­‐based  problems  and  in  transforming  them  into  mathema@cal  problems.  

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Web blog PISA Indonesia (www.pisaindonesia.wordpress.com)

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FUTURE  OF  PMRI:  CHALLENGES  

•   LOWEST  PISA  SCORE  &  HAPPIEST  IN  THE  SCHOOL  

•   SUSTAINABLITY  OF  PMRI  INFRASTRUCTURE    

•   CENTER  OF  EXCELLENCE  OF  PMRI  •   ….  

MARS  PMRI  

Thank  you  for  coming    

E-­‐mail        :  zulkardi@gmail.com    Blog              :  www.p4mri.net;    

Facebook  :  facebook.com/zulkardiharun;  Mobile  phone:  +62-­‐8127106777  

 

REFLECTIONS FROM ENGLAND AND THE CAYMAN ISLANDS

Sue Hough, Paul Dickinson, Steve Gough, Yvette Solomon, Frank Eade

Manchester Metropolitan University

ICME conferenceJuly 2016

BRIEF OUTLINE

! Dealing with clashing educational ideologies

! RME based projects in the UK

! RME based projects in The Cayman Islands

CLASHING IDEOLOGIES

CLASHING IDEOLOGIES

HOW DO YOU KNOW YOU ARE RIGHT?

‘I know it’s right because that’s how I was taught to do it’

‘I was taught this method and just accepted it’

‘I know they’re right because I’ve been doing it for years and have checked my answers’

HOW DO YOU KNOW YOU ARE RIGHT?

‘I wasn’t 100% sure I was right, this is a regurgitation of a procedural method’

‘I can’t think of why I am right’

‘I have no concept of what these answers mean in terms of the actual question, and no idea if they’re even sensible. If I made a mistake a wouldn’t notice’

! 100% used a procedure to answer

! 71% used a procedure to justify a procedure! 0.5% used estimation! 22.5% said they didn’t know why

they were right.

CLASHING IDEOLOGIES

! Radical differences between RME and English education system in terms of :

! Student and teacher expectations about the nature of mathematics and mathematics classrooms

As seen in ourcurriculumtextbook resourcesclassroom culturesassessment systemsaccountability structures

!

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! Under RME assessment problems should be:

! Accessible and worth solving

! Unfamiliar, giving opportunities for students to formulate their own constructions

! Facilitate ownership and decision making on the part of the student

(Van den Heuvel-Panhuizen, 2005)

‘TEACHING TO THE TEST’

RME TRIALS IN ENGLAND

Trial 1 – 2004 - 2007

Over 400 project students aged 11- 14 across 12 schools

Project teachers taught using the ‘Mathematics in Context’ textbook series developed by the University of Wisconsin in collaboration with the Freudenthal Institute

Project teachers attended 6 days training per year, supported by Manchester Metropolitan University and the Freudenthal Institute

TRIAL 1 - OUTCOMES

! Similar performance for project and control students on traditional examination questions

! Performance on unfamiliar type problems was significantly better for project students compared with control (36% correct v 17% correct in the case of ‘lower attainment’ range)

! ‘Evidence that project pupils’ approach to problem solving changed and this influenced how they understood the mathematics’

(Searle and Barmby, 2012)

TRIAL 1 – A DIFFERENT APPROACH TO SOLVING

PROBLEMS

TRIAL 1 – A DIFFERENT APPROACH TO SOLVING

PROBLEMS

TRIAL 1 – A DIFFERENT NOTION OF PROGRESS

! A shift from ‘doing something with the numbers’ to ‘making sense of the problem

! A recognition of progress in understanding the concept of area as exemplified by the use of a ‘model of’ the situation and how this develops into a ‘model for’

(Streefland 1985, 1993)

! An appreciation of the iceberg model and the ‘landscape of learning’

(Webb et al 2008, Fosnot & Dolk, 2002)

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RME TRIALS IN ENGLAND

Trial 2 – 2007 - 2010Smaller scale, around 240 project students aged 14 – 16, across 10 schools

Project teachers taught using the ‘Making Sense of Maths’ resources developed at Manchester Metropolitan University in collaboration with the Freudenthal Institute

Project teachers attended 6 x 2 hour twilight training per year, on a voluntary basis, supported by Manchester Metropolitan University.

TRIAL 2 - OUTCOMES

! Similar performance for project and control students in National GCSE examinations.

! Performance on unfamiliar type problems was significantly better for project students compared with control in the middle to lower attainment range

! Publication by Hodder Education of ‘Making Sense of Maths’ as a textbook series.

RME TRIALS IN ENGLAND

Trial 3: 2012 - 2016Working with students aged 16 and over who have notgained the National GCSE examination to an acceptable standard.

Success rates for these re-sit students are traditionally very low (In 2012-2013 only 9.3% of students went onto improve their GCSE grade.)

(Department for Education, 2014)Very short intervention, 12 hours on Number, 9 hours on Algebra, out of their 9 month resit course.

TRIAL 3 - OUTCOMES

! Small but significant gains for the project group on Number

! For some students using a variety of contexts which led to use of the bar model helped them to see connections across elements of the curriculum

! Some students used the bar model as an algorithmic strategy, rather than as a ‘model for ‘ making sense of a problem

PRE AND POST-TEST APPROACHES TO FINDING SUSAN’S ORIGINAL

CAR PRICE

PRE AND POST-TEST APPROACHES TO SHARING £140 IN THE RATIO 2 : 5

TEACHER IMPACT

! ‘The students show much more confidence….much less of a correct/ incorrect environment compared to a traditional approach and so the students are not frightened to ‘have a go’ as much

! ‘They stopped asking ‘What is the point in this?’ They stopped saying ‘can we do something different?’ I stopped replying ‘you have to do it because you have an exam in it’ Energy levels were higher in the room, a lot more discussion took place

TEACHER IMPACT

! ‘the underlying understanding being really important, so I’m pulled in two ways….I really like what you do and buy into it, and the other side of me is saying’dam with this group, I’ve still got to this, this and this, and when am I going to do it?’

RME BASED PROJECTS IN THE CAYMAN ISLANDS

! 2011, Frank Eade from MMU becomes Mathematics Advisor for the Cayman Islands

! Primary students in the Cayman Islands were expected to learn rules given to them by their teachers, many were behind, participated little in class and constantly asked for help, afraid to take mathematical risks

! Very little use of context, models or imagery. Even the context of money was not well understood.

RME BASED PROJECTS IN THE CAYMAN ISLANDS

! Introduced ‘Mathematics Recovery’ training (Wright et al, 2014). Extensive use of models such 10-frames, the 100 bead bar and arrays… but little use of contexts

! A group of Primary teachers were trained in use of the empty number line and how to use contexts as a point of entry

CAYMAN ISLANDS -OUTCOMES

! Increased awareness in students of how calculations like 91 – 37 can be represented on a number line

(In 2011, 16% of Year 6 could explain compared with 46% in 2013.)

CAYMAN ISLANDS -OUTCOMES

! Increased use of a range of informal strategies for division problems like 222 ÷ 3, rather than following a procedure

CAYMAN ISLANDS -OUTCOMES

! At Secondary age group, under guidance from Frank, teachers began trialling the ‘Mathematics in Context’ textbooks.! Pre and post-test mean scores for a 13 question

test are shown below

Pre-test mean

Post-test mean

Set 1 44% 45%Set 2 14% 26%Set 3 5% 15%

CAYMAN ISLANDS -OUTCOMES

! Students showed initiative using a range of solution strategies:

CHINKS OF LIGHT

! Where teachers receive substantial training in the use of RME, they experience significant shifts not only in the way they see mathematics but in the way they operate within their classrooms

! The latest version of an English curriculum does stress the need to develop mathematical reasoning and the ability to problem solve as well as procedural fluency

! National public examinations in England to include more problem solving with open ended questions set in real life contexts

(OCR, 2014)

EXAMPLES WE HAVE TRIALLED…

‘Making Sense of Maths’ series published by Hodder

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RealisticMathematicsEducationSomeremarksbyacriticalfriend

CyrilJulieUniversityoftheWesternCape,SouthAfrica

RME has demonstrated itself as a viable approach for incorporating some of its ideasin the state-mandated school mathematics curriculum in South Africa

ThebroaderSouthAfricanschoolmathematicscontext

Mathematics/P1 4 DBE/November 2014 NSC

Copyright reserved Please turn over

QUESTION 3 3.1 Given the quadratic sequence: – 1 ; – 7 ; – 11 ; p ; … 3.1.1 Write down the value of p. (2) 3.1.2 Determine the nth term of the sequence. (4) 3.1.3 The first difference between two consecutive terms of the sequence is 96.

Calculate the values of these two terms.

(4) 3.2 The first three terms of a geometric sequence are: 16 ; 4 ; 1 3.2.1 Calculate the value of the 12th term. (Leave your answer in simplified

exponential form.)

(3) 3.2.2 Calculate the sum of the first 10 terms of the sequence. (2)

3.3 Determine the value of: ...51

141

131

121

1 ¸¹·

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(4) [19]

High-stakesmathematicsexaminationattheendof12-13yearsschooling

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Teacher:WhenItaughtthestufftheycoulddoit.WhenIsawtheresultsIwonderwhetherIwasteachingthem.

Learner:WhenwedidtheworkinclassIunderstoodandcoulddoit,butintheexaminationitwasasifIwentblank

RMEislowonincorporationofstrategiesandtacticstodevelopproceduralandotherfluenciestoaddresswhatIcallthe“forgetproblem”

90

Operating with powers (I)a × a × a × b × b × c = a3 × b2 × c1 = a3b2c

The exponents of a, b, and c are 3, 2 and 1.

With the exponents 3, 2 and 1 and the letters a, b and c also can be madeother products, for example: ab3c2.There are six different products that one can make using a, b, c and the

Write the other four products.Multiply the six products to each other.The result can be written in the form a .... b .... c. ....

If it is known that b = a, you can simplify a3b2c

a3b2c b = a

a3a2c

a5c You can do the same with the other five products.

How many different products do you get? Which ones? If you also know that c = a you can write each of these products as a power of a. Which one?

(note: the exponent 1 is mostly not written)

Which exponents do you get?

exponents 1, 2, 3 .

1

composed by

PositiveAlgebra

a collection of productive exercises

Martin Kindt

PositiveAlgebra,MartinKindt

educationinMathematicsvs mathematicsinEducation

educationinMathematics

Inductionintothewaysmathematiciansworkandthepracticeofdoingmathematics(SeymourPapert’s “Teachingchildrentobemathematicianvsteachingthemmathematics”)

mathematicsinEducation

DevelopmentandnurturingoftheglobalandresponsiblecitizenwhohasasenseofthecontributionsMathematics,aspartofthetotalityofhumanity’sknowledgeheritage,havemadetothecurrenthumancondition

education in Mathematics

mathematics in Education

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Mathematicsasbackdroptocrime(andpossiblyother)novels

Hacking,AI,encryption,decryption,largeprimenumbers,etc

GlobalInformation

Violenceincreasedsincethe50’s

Healthincreasedsincethe50’s

Betterhealthcontributestowardstheincreaseinviolenceorvice-versa

Currentlyafatheris25yearsolderthanhischild.Insevenyearshewillbe5timesasoldashischild.

Whatisthefatherdoingnow?

Years Child Father

Now P P+25

7yearslater P+7 P+32

5(P+7)=P+32P=-¾years

NotstrictlyRMEbutDutch

THANKYOU