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RealisticMathematicsEducation
Netherlandsdidactictraditioninmathematicseducation
Mathematicseducationsystem
• howlearningandteachingofmathematicsisresearched
• whatistaughttostudents• howteachersteach• whatisinthetextbooks• howstudentsareassessed• howteachersareeducated• howinstructionalmaterialisdesigned
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Freedom of Education
NL Constitution 1848
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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
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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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Paul Drijvers, Freudenthal Institute [email protected]
Driving to Hamburg Thematic Afternoon
European Didactic Traditions: the Netherlands
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2
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U H
110
O
215
B
115
Distance to O
Dist
ance
to B
Driving to Hamburg: schematizing
?
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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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115
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Teaching experience
� 13-14 year old � High achievers � Bilingual stream
6
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Teaching experience
7
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Teaching experience
8
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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
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Paul Drijvers, Freudenthal Instituut [email protected]
2016-07-27 www.uu.nl/staff/PHMDrijvers
Thank you! Thematic Afternoon
European Didactic Traditions
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Thomas A. Romberg Jan de Lange
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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
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US Attraction to RME
• Unique context-first approach
• Preformal models and tools
• Robust approach to assessment
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Context first approach
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Pre-formal models and tools
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Comprehensive assessment
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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
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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
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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.
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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!)
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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
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A decade of PMRI in Indonesia: Story form some chapters
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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)
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Portal PMRI www.p4mri.net
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hdps://sites.google.com/site/kondriunsri2013/
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http://seminar.fmipa.unp.ac.id/seadr16/
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impomeunsri.wordpress.com
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1. Counting orally during Gasing is moving Example of research using PMRI
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2. Students count orally in conjunction with the
movement of clock hand
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3. Counting the strip and the space of the strip of second interval on the clock
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Scores PISA Mathema@cs of Indonesian Students in Year 2000 to 2012
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(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 • ….
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MARS PMRI
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Thank you for coming
E-‐mail : [email protected] Blog : www.p4mri.net;
Facebook : facebook.com/zulkardiharun; Mobile phone: +62-‐8127106777
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REFLECTIONS FROM ENGLAND AND THE CAYMAN ISLANDS
Sue Hough, Paul Dickinson, Steve Gough, Yvette Solomon, Frank Eade
Manchester Metropolitan University
ICME conferenceJuly 2016
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BRIEF OUTLINE
! Dealing with clashing educational ideologies
! RME based projects in the UK
! RME based projects in The Cayman Islands
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CLASHING IDEOLOGIES
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CLASHING IDEOLOGIES
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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’
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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’
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! 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.
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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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!
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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’
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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
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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)
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TRIAL 1 – A DIFFERENT APPROACH TO SOLVING
PROBLEMS
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TRIAL 1 – A DIFFERENT APPROACH TO SOLVING
PROBLEMS
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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.
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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.
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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.
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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
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PRE AND POST-TEST APPROACHES TO FINDING SUSAN’S ORIGINAL
CAR PRICE
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PRE AND POST-TEST APPROACHES TO SHARING £140 IN THE RATIO 2 : 5
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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
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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?’
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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.
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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
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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.)
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CAYMAN ISLANDS -OUTCOMES
! Increased use of a range of informal strategies for division problems like 222 ÷ 3, rather than following a procedure
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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%
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CAYMAN ISLANDS -OUTCOMES
! Students showed initiative using a range of solution strategies:
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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)
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EXAMPLES WE HAVE TRIALLED…
‘Making Sense of Maths’ series published by Hodder
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RealisticMathematicsEducationSomeremarksbyacriticalfriend
CyrilJulieUniversityoftheWesternCape,SouthAfrica
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RME has demonstrated itself as a viable approach for incorporating some of its ideasin the state-mandated school mathematics curriculum in South Africa
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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·
¨©§ �¸¹·
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¨©§ �¸¹·
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(4) [19]
High-stakesmathematicsexaminationattheendof12-13yearsschooling
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Teacher:WhenItaughtthestufftheycoulddoit.WhenIsawtheresultsIwonderwhetherIwasteachingthem.
Learner:WhenwedidtheworkinclassIunderstoodandcoulddoit,butintheexaminationitwasasifIwentblank
RMEislowonincorporationofstrategiesandtacticstodevelopproceduralandotherfluenciestoaddresswhatIcallthe“forgetproblem”
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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
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educationinMathematicsvs mathematicsinEducation
educationinMathematics
Inductionintothewaysmathematiciansworkandthepracticeofdoingmathematics(SeymourPapert’s “Teachingchildrentobemathematicianvsteachingthemmathematics”)
mathematicsinEducation
DevelopmentandnurturingoftheglobalandresponsiblecitizenwhohasasenseofthecontributionsMathematics,aspartofthetotalityofhumanity’sknowledgeheritage,havemadetothecurrenthumancondition
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education in Mathematics
mathematics in Education
!*-)$0K)01)M,#'+",#'%(&)%1)-./(#'%01N
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*#'+",#'%(&)%1)@"1"3#$)3"(3"#'%01#$),#''"3&)$0K
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Mathematicsasbackdroptocrime(andpossiblyother)novels
Hacking,AI,encryption,decryption,largeprimenumbers,etc
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GlobalInformation
Violenceincreasedsincethe50’s
Healthincreasedsincethe50’s
Betterhealthcontributestowardstheincreaseinviolenceorvice-versa
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Currentlyafatheris25yearsolderthanhischild.Insevenyearshewillbe5timesasoldashischild.
Whatisthefatherdoingnow?
Years Child Father
Now P P+25
7yearslater P+7 P+32
5(P+7)=P+32P=-¾years
NotstrictlyRMEbutDutch
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THANKYOU