Realistic Mathematics Education

1 RRealistic Mathematics EducationMarja Van den Heuvel-Panhuizen1andPaul Drijvers21 Freudenthal Institute for Science andMathematics Education , Faculty of Science &Faculty of Social and Behavioural Sciences,Utrecht University, Utrecht, The Netherlands2 Freudenthal Institute, Utrecht University,Utrecht, The NetherlandsKeywordsDomain-specific teaching theory; Realisticcontexts; Mathematics as a human activity;MathematizationWhat is Realistic MathematicsEducation? Realistic Mathematics Education hereafterabbreviated as RME is a domain-specificinstruction theory for Mathematics , which hasbeen developed in the Netherlands. Characteristicof RME is that rich, Realistic situations are givena prominent position in the learning situations serve as a source for initiatingthe development of mathematical concepts, tools,and procedures and as a context in which studentscan in a later stage apply their mathematicalknowledge, which then gradually has becomemore formal and general and less context Realistic situations in the meaningof real-world situations are important in RME, Realistic has a broader connotation means students are offered problem situationswhich they can imagine.

2 This interpretation of Realistic traces back to the Dutch expression zich REALISEren, meaning to imagine. It is this emphasis on making something real inyour mind that gave RME its name. Therefore, inRME, problems presented to students can comefrom the real world but also from the fantasyworld of fairy tales, or the formal world ofmathematics, as long as the problems areexperientially real in the student s Onset of RMEThe initial start of RME was the founding in 1968of the Wiskobas ( Mathematics in primaryschool ) project initiated by Edu Wijdeveld andFred Goffree and joined not long after by AdriTreffers. In fact, these three mathematicsdidacticians created the basis for RME.

3 In 1971,when the Wiskobas project became part of thenewly established IOWO Institute, with HansFreudenthal as its first director and in 1973when the IOWO was expanded with the Wiskivonproject for secondary Mathematics Education ; thisbasis received a decisive impulse to reform theprevailing approach to Mathematics the 1960s, Mathematics Education in theNetherlands was dominated by a mechanisticteaching approach; Mathematics was taughtS. Lerman (ed.),Encyclopedia of Mathematics Education , DOI ,#Springer Science+Business Media Dordrecht 2014directly at a formal level, in an atomized manner,and the mathematical content was derived fromthe structure of Mathematics as a scientific disci-pline.

4 Students learned procedures step by stepwith the teacher demonstrating how to solveproblems. This led to inflexible and reproduc-tion-based knowledge. As an alternative for thismechanistic approach, the New Math move-ment deemed to flood the Netherlands. AlthoughFreudenthal was a strong proponent of themodernization of Mathematics Education , it washis merit that Dutch Mathematics Education wasnot affected by the formal approach of theNew Math movement and that RME could s Guiding Ideas AboutMathematics and MathematicsEducationHans Freudenthal (1905 1990) was amathematician born in Germany who in 1946became a professor of pure and appliedmathematics and the foundations of mathematicsat Utrecht University in the Netherlands.

5 As amathematician he made substantial contributionsto the domains of geometry and in his career, Freudenthal (1968,1973,1991) became interested in Mathematics educa-tion and argued for teaching Mathematics that isrelevant for students and carrying out thoughtexperiments to investigate how students can beoffered opportunities for guided re-invention addition to empirical sources such as text-books, discussions with teachers, and observa-tions of children, Freudenthal (1983) introducedthe method of the didactical phenomenology. Bydescribing mathematical concepts, structures,and ideas in their relation to the phenomena forwhich they were created, while taking intoaccount students learning process, he came totheoretical reflections on the constitution of men-tal mathematical objects and contributed in thisway to the development of the RME (1973) characterized the thendominant approach to Mathematics Education inwhich scientifically structured curricula wereused and students were confronted with ready-made Mathematics as an anti-didactic inver-sion.

6 Instead, rather than being receivers ofready-made Mathematics , students should beactive participants in the educational process,developing mathematical tools and insights bythemselves. Freudenthal considered mathematicsas a human activity. Therefore, according to him, Mathematics should not be learned as a closedsystem but rather as an activity of mathematizingreality and if possible even that of , Freudenthal (1991) took over Treffers (1987a) distinction of horizontal and verticalmathematization. In horizontal mathematization,the students use mathematical tools to organizeand solve problems situated in real-life involves going from the world of life into that ofsymbols.

7 Vertical mathematization refers to theprocess of reorganization within the mathematicalsystem resulting in shortcuts by using connectionsbetween concepts and strategies. It concerns mov-ing within the abstract world of symbols. The twoforms of mathematization are closely relatedand are considered of equal value. Just stressingRME s real-world perspective too much maylead to neglecting vertical Core Teaching Principles of RMERME is undeniably a product of its time andcannot be isolated from the worldwide reformmovement in Mathematics Education thatoccurred in the last decades. Therefore, RMEhas much in common with current approaches tomathematics Education in other countries.

8 Never-theless, RME involves a number of core princi-ples for teaching Mathematics which areinalienably connected to RME. Most of thesecore teaching principles were articulated origi-nally by Treffers (1978) but were reformulatedover the years, including by Treffers total six principles can be distinguished: Theactivity principlemeans that in RME stu-dents are treated as active participants in thelearning process. It also emphasizes thatR522 Realistic Mathematics Educationmathematics is best learned by doingmathematics, which is strongly reflected inFreudenthal s interpretation of mathematicsas a human activity, as well as in Freudenthal sand Treffers idea of mathematization.

9 Thereality principlecan be recognized inRME in two ways. First, it expresses theimportance that is attached to the goal ofmathematics Education including students ability to apply Mathematics in solving real-life problems. Second, it means thatmathematics Education should start fromproblem situations that are meaningful tostudents, which offers them opportunities toattach meaning to the mathematical constructsthey develop while solving problems. Ratherthan beginning with teaching abstractionsor definitions to be applied later, in RME,teaching starts with problems in rich contextsthat require mathematical organization or,in other words, can be mathematized andput students on the track of informal context-related solution strategies as a first step inthe learning process.

10 Thelevel principleunderlines that learningmathematics means students pass various levelsof understanding: from informal context-relatedsolutions, through creating various levels ofshortcuts and schematizations, to acquiringinsight into how concepts and strategies arerelated. Models are important for bridging thegap between the informal, context-relatedmathematics and the more formal fulfill this bridging function, models haveto shift what Streefland (1985,1993,1996)called from a model of a particularsituation to a model for all kinds of other,but equivalent, situations (see also Gravemeijer1994; Van den Heuvel-Panhuizen2003).