Principles and practices in arithmetic teaching Innovative approaches for the primary classroom

edited by Julia Anghileri

Open University Press Buckinsham . Philadelphia

48 Margaret Brown

indrvidudl teacher Over tinre these balances are cotrt~riually shift~rig accord- ing to the degree of power held by different ~nterests, and in partiiular depeliding on the clr~rrg~ng degree of centralizat~on

lhe ddlustments needed in what would be a complex task, even tn stable cond~tions, can be potellttally excltlng but can also be potentlafly exhauslng

Q and u~~derrllrll~ng ot confidence and self-esteem. ~ e t t i n g the degree of cen- tral~~dtlon and the pace of change right 15 soniething wh~ch we in England Realistic mathematics education in hdve not yet achieved. the Netherlands

'larja van den Heuvel-Panhuizen

Introduction

This chapter can be seen as a guided tour tlrrorlgii scrnie nrain aspects of tlie Dutch approach to matlrematics education and will focus on the ri~rrrtber straid in primary school mathe~natics. T1.w rilairl questioris that will be con- sidered are:

How do we teach ~rlthrnetic in primary \cfioolr 111 the Netherldrld,? * What does our artthmet~c curr~culurrr contdin?

This guided tour will not take you into classrooms nor will it provide you with a representative sample of Dutch classroo~n practice (this can be seen in the chapter by Julie Menne in this volurne). Instead, it will introduce you to a tireoretical framework of teaching matlrernatics and the teaching activities that are in tune with the ideas this involves. What it will do is show you attain- ment targets identified in the Netherlands regarding mathematics, and give you an idea of the position that we want to reach in the cnd. Do not expect to get complete answers and a thorough overview of the I)utch approach to niathematics education from this trip as our approach to inathenlatics edu- cation is too complex. Moreover, the difficulty is that there is not a unified Ilutch approach. Instead, there are some shared basic ideas about the what- and-how of teaching mathematics.

These ideas have been develcrped over 30 years and the accum~ilatiori arid repeated revision of these ideas has resulted in what is now called Realistic Mathematics Education (KME). Irlherent in KME, with its founding idea of mathematics as a human activity, is the idea that it can never be considered as a fixed and finished theory of mathematics education. We see KME as 'work under construction' (van den Heuvel-Panhuizen 19%). 'The different accentu- ations are the impetus for this contirluir~g development. (For a Inore detailed discussion about tlre RME approach by means of developlrrental research, see Chapter 11, this volume.)

( '

SO Maria van den Heuvel-Panhuizen

'This chapter will outline key characteristics in the teaching approaches now widely adopted in the Netherlands and will identify the thinking and research that has lecl to these ideas. It will show sorne of the work that is done to irnple- rnent the RME approacii in classroom practice.

The Dutch approach to mathematics education

Despite this clear statement about horizontal and vertical mathematization, RME became known as 'real-world mathematics education'. The reason, how- ever, why the Dutch reform of mathematics education was called 'realistic' was not only because of the connection with the real world, but was related to the emphasis that RME puts on offering the students problem situations that they can imagine. The Dutch trailslation of 'to iinagine' is 'zich KEALIS- Eren.' I t is this emphasis on irlaking something real in your nlind, that gave RME its ilarne. Far the problems to be presented to the students this ineans that tlle coiltext can be a real-world context but this is not always necessary. The fantasy wc-rrlri of fairy tales and even the formal world of mathematics can be very suitable contexts for a problem, as long as they are real in the student's mind.

Pri Ne

I"he development of Realistic Mathematics Education started almost 30 years ago with the fouliciatiorls laid by Freudenthal and his colleagues at the former IOWO, the oldest predecessor of the Freudenthal Institute. The impulse for tlie reform rnovenient was the inception, iri 1968, of the Wiskobas project, initi- ated by Wijdweld and Goffree. The present form of RME was mostly detet- ~nined by Freudeiltllal's (1977) view that mathematics must be connected to reality, stay close to children and be relevant to society. Instead of seeing mathenlatics as subject matter that has to be transmitted, Freudenthal stressed the idea of mathematics as a hurnail activity. Education should give students the 'guided' opportunity to 'reinvent' mathematics by doing it within a process of progressive ntathernatization (Freudenthal 1968).

Later on, 'l'reffers (1978, 1987b) explicitly for~nulated the idea of two types of process, distirtguished as 'horizontal' and 'vertical' mathematization. In horizontal mathet~latization tlie stucients come up with mathematical tools that can help to organize and solve a problem located in a real-life situation. Vertical matlienlatization is the process of reorganization within the matbe- matical system itself, for instance, findsng shortcuts and discovering connec- tions between concepts and strategies, and then applying these discoveries. In short, horizontal mathertlatization involves going from the world of life into the world of symbols, while vertical mathematization means moving within the world of symbols (Freudenthal 1991: 41-2).

Realistic mathematics educatior~ in the Netherlands 57

nciples underpinning teaching methods in the Ftherlands

RME reflects a certairt view of ntatlleinatics as a subject, of how children learn mathematics and of how mathe~rlatics slsclul~i be taught. These views can be characterized by the following six principles,' some of which originate more from the perspective of learning and some of which are more cortnected to the teaching perspective.

1 Activity priizciple

The idea of riiatfiematizatio~l clearly refers to the coitcept of rnathen~atics as an activity which, according to Freudenthal (1971, 1973), can best be learned by doing (see also Treffers 1978, 1987b). The students, instead of being the receivers of ready-made mathematics, are treated as active participants i11 the educational process, in which they develop by tlremselves all sorts of ntathe- rnatical tools and insights. According to Freuderrthal (19731, usirig scientifi- cally structured curricula, in which stucierits are confrollted with ready-macie rnatllenlatics, is an 'anti-didactic inversiotl'. It is based on tlie false ass~iriiptio~l that the results of inathernatical thinking, placeci in a subjecl-matter frame- work, can be transferred directly to the students. The consequence of the activity principle is that the studellts are co~lfroslted with problem situations in which they can, for instance, produce fractions arid can grad~lally develop an algorithmic way of nluitiplication and division, based oi l an informal way of working. Related to this principle, students' ow11 procltictioris play an important role in RME.

In RME the overall goal is that the stucleilts triust be able to briilg into actlon mathematical understanding\ drld tools when they have to solve problems This linplies that they must learn 'mathematic\ so 'tc to be urelul' (see I reuclen- thal 1968) The realtty pr~nclple is, however, not only recognizable at the end of the learning process m the area of appltcatioll, r e ~ l ~ t y 1s albo conie~vetl ds d

source for learning rnathem~tlcs Just as ~natheilidt~cs Jtose from the math- enxatlzat~on of reality, so must learning mathciuatics also originate in inathe- rnatiz~ng redhty.

In the early days oi KMC ~t wac already einphari~ed that it childreii ledrn mathematics ~n an ~soldted idshion, divorced frorn exper~c~lced redllty, it wtll be qu~ckly forgotten and the chrldren wit1 not be able to apply IT tbreudenthdl 1971, 1973, 1968) Rather than begirtn~rlg wrth certatll abrtrdctlorls or deb- nrtiong to be appl~ed later, one rtiust start with rich contexts deirraiidrng mdth- ematical orgdn1r;ation or, In other word\, contexts that taxi be rx~atlleiiidtt~ed (Fteudenthal 1968, 1979) While working o11 context problelrls tlle \tndellts can develop mathematlcal tools and under\tandrr~g

52 Marja van den Heuvel-Panhuizen Realistic mathematics education in the Netherlands 53

.3 L e ~ v i pr-it fciple

Learning mdthelnatics rnearis that students pass various levels of understand- ing: from the ability to invent informal context-related solutions to the cre- ation of vario~is levels of short cuts and schelnatization, to the acquisition of insights into the underlyirlg principles and the discer~ln~etlt of even broader relationships. The condition for arriving at the next level is the ability to reflect on the activities conducted. This reflection can be elicited by inter- action.

Models serve as an important device for bridging this gap between informal, context-related mathematics and more formal mathematics. First, the stu- cients develop strategies closely connected to the context. Later on, certain aspects of the context situation can become nlore general which means that the context can assume, more or less, the character of a model, and as such can give support for solving other, but related, problems. Eventually, the nlodels give the students access to more formal mathematical knowledge. In order to fulfil the bridging function betweell the inforrnal and the forrnal level, models have to shift from a 'model of' a particular situation to a 'model for' all kinds of other, but equivalent, situations (see Streefland 1991; Treffers 199la; Grave- rneijer 1994b; van den Heuvel-l'anhuizen 1995). The bus context (van den Brink 1989) is an example of a 'daily life' context which can evolve to a more general anti formal level. In the beginning the situation is more or less pic- tured to describe the changes at the bus stop (see Figure 4.1). Later on the bus context has become a 'model for' u~iderstanding all kinds o f number sen- tences. Tlken the stucients can go far beyond the real bus context. They can even use the model for backwards reasoning (see Figure 4.2).

An important requirenlent for having nlodels functioning in this way is that they are rooted in concrete situations anti thar they are also flexible enough to be useful in higher levels of mathematical activities. This means that the models will provide tile students with a foothold during the process of verti- cal mathe~llatizatiorl, without obstructirlg the path back to the source.

E'igure 4.1 A t the bris stop &oimr: Streetland (1996. 15, 16)

in i out IS I ,

dtfference at least -6 or more

'L (I

t 2 h I

Rwre 4.2 Two ninr~ber sentences Sottrte: Streetland (1996: 17)

t n C t

The strength of the level principle is that it both guides the growth in mathematical urlderstanding and it gives the curriculnn~ a lor~gitudinal coherency. This long-term perspective is very characteristic ct f KME. Ttiere is a strong focus on the conllection between what is learned earlier and w11at will bc learned later. A powerful exaniple c)f sucfi a 'longitudinal' xnodel is the number line (see Figure 4.3). It begins in first grade as (a) a beaded necklace on which the studerits can practice all kinds of courltirtg activities. In higher grades this chain of beads subseq~iently becomes (b) an empty numi2er line

q n F I ~ I ? 19 I8 l o

Fibwre 4.3 Dltfefexlt appearance ut the rlurriber lt~te

54 Marjd van den Heuvel-Panhuizen Realistic mathematics education in the Netherland5 55

for supporting dddltlons dnd subtract~ons (see the chapter by Julie Menne of organlzatron lnstedd of ddaptlng kssons to the different &iirty level' about her Julnplng Ahead programme for underach~evers in the early grades of the students by medns of dbrhty grouping, dtfferencer betweeit studerlts dre

Kuthven, Clldpter 121 (c) a double number line for supporting problem for by provldlng them w ~ t h problerns thdt call be solved ratlo$, and (dl a frdctronlpercentage bar for supporting worhng with frat vels of understandmg.

tlons and percentdges.

Characteristic of RME is the fact that mathematics as a school subject is no. split up illto distinctive learning strands. From a deeper mathematical vieM the chapters within mathematics cannot be separated. Moreover, solving rick context problems often means that you have to apply a broad range of math enlatical tools and understandings. For instance, the rnirror activity in Figurr 4.4 clearly shows how geometry and early arithmetic can go together. Thf strength of the intertwineme~~t principle is that it gives coherency across tht curriculum.

5 Intertzctiorz principle

Within RME, the learning of mathematics is considered as a social activity. Bj listening to what is found by others and discussing these findings, the student: can get ideas for improving their own strategies. Moreover, the interaction car evoke reflection which enables the students to reach a higher level of under. standing. The significance of the interaction principle implies that whole-clasr teaching plays a very important role in the RME approach to mathematic: education. This, however, does not mean that the whole class is proceeding together and that every student is following the same track and reaching the same level of develapn~ent at the same moment. On the contrary, within RME, children are considered as individuals, each following an individual learning path. This view of learning often results in pleas for splitting classes up into small groups of students each following their own learning trajectories. In HME, however, there is a strong preference for keeping the class together as a

Figure 4.4 Mirroring and counting Source: TAL Tea~n (1998)

Grridance pn'tzciple

ne of Freudenthal's (1991) key principles for nlatiiexxlatics educatiol~ is that should give students the 'guided' oppom~lity to 're-invent' matl~einatics. In ME both the teachers and the curriculurrl have a crucial role in steering the amlng process, but not in a fixed way by demonstrating what the students ave to learn. This would be in corlilict with the activity prirrciple and would !ad to pseudo-understanding. Instead, the studc~~ts need roorn to construct ~a&ematical insights and tools by themselves. In order tu reach this yosition le teachers have to provide the studerlts with a learzling environlnent in rhich this constructing process can emerge. A requirclllerlt far this is that rachers must be able to foresee where and how they can anticipate the &dentsf understandings and skills that are just corning into view in the dis- mce (see also Streetland 1985). Educatiorlal programs should contairr scen- rios which have the potential to work as a lever to reach shifts in the students' ~nderstanding. Crucial for these scenarios is that they always keep in view the cmg-term teachingflearning trajectory based on the goals one wants to attain. Vithout this perspective it is not possible to guide the students' learning.

Nhat are the determinants of our mathematics curriculum?

Unlike many other countries, at primary school level the Netherlands does not have centralized decision maliing regarding cxnicutu~~l syllabuses, textbooks and examinations (see Mullis et id, 1997). 'reachers have tlexibility with respect to their teaching and can make niany educational decisions either by thern- selves, or as a school team, including choice of textbooks and even what cur- riculum to teach. To give some exmpfes, teachers are allowed to make changes in their timetable without asking the director of the school (who often teaches a class too), and the teacher's advice at the end of primary school, and not a test score, is the most important criterion for allocating a student to a particular kvel of secondary education, Despite this treedorn in educational decision making, the mathematical topics addressed in primary schools do nor differ a lot between schools. In general, all the schools follow the same curricuium. This takes me to the question: what dcter~tlines this curriculum?

Until recently, there were three important determirlarlts for rnathematics education in primary school:

* the mathematics textbooks series; * the 'Proeve' - a document that describes the nlathematlcal content to be

taught in prlmary school; and

56 Marja van den Heuvel-Panhuizen Realistic mathematjcs ducauon in the Netherlands 57

* the key goals to be reachled by the end oof prinlay school as descr~bed by ~ h r govertinient,

7116) ~letm?ritzirzg role of textbook5

Maiy rehnn nlox7eixents around the world appear to be aimed at gettirig r of textbt2oks. 111 the Netherlaids, however, the contrary is the case. Flere, tlre iniprovemerit of nlathe~liatics education depnds targely on textbooks, which have a &termlr~ing role in mather~iatics education and are the most Impon. ant tools that &wide the teachers9eaching. This is true of both the coatexit and the teaching methods, although regardillg the latter the guidance is not sut- ficient to reach all teachers. Many studies have, for example, provided indi- cations that the irnl~lerrientation of RME in classroo~n practice is not yet fully achieved (Graven1eijl.r rt ul. 1993; van den Heuvel-Panhuizen and Vennwr 1999).

Currently, about 80 per cent ot Dutch pnrnary schools use a ix~atl~ematia textbook series which was inspired, to a greater or lesser degree, by RUE' Com. pared to even 10 or 15 years ago, thtfirs percentage has changed cons~derably; at that time, oniy half of the scliools worked with such a textbook series (De Jong 1986). The development of textbook series is done by coxnmercial publishen tchapter by Kees Buys in this volume), The textbook authors are inctependent developers of matheruatics education, but tlley can also make use ot the ideas for teacliing acTirrihes resulting trom deveiopnlental research at the Freudtm- tlrai Irlstitute (and rts predecessors) and at the Dutch Instiwte for Curriculum Developmmt (the &LO).

Loafing back at our reforin Itlovenlent in mathematics education, it is clear that the refon11 proceeded Sri a very interactive and infornial way. There was I ~ O interference from the government. Instead, developers and researchers, in col- faharation with teacher educaturs, school advisors and teachers, worked out teaching activities and Ieaf~Wg strands. Later on, these were included in text- books. An impomnt aid to the development of textbooks has been the mid- arice wilicll, since the mid-tlOs, comes from a series of publications calkd the 'Proeve van een Nationaal Prograrluna voor het r&en-wisku~ldeonderwijs i)p de basissci~ool' (Design of a natiorral progrmlme for mathenlatics education at prlrnary school) (Tkfiers et al. 1989). It is of note that the title refers to a 'riational prografrune' white in fact, tliere was no interference from the govern- n1e11t.

'Creffers is the mail1 author of the 'Proeve' and work on this series is still going on. The documerlts corrtain descriptions of the various domains within mathematics as a school subject and, although it is written in a very accessibfe style with a lot of examples, it is not writ t~n as a series for teachers. Instead, it is meant to be a support for textbook authors, teacher trainers and scliool advisors, many of whom are significa~it contributors to the series.

1 1 ~ 8 key goals )itr matlzrrtra~c,r etiucc-rticwr

lntll recently there was rio real itlterference from the L)u'lclr goverr1nrc.nt garding the content of the educatlrtrlal progranrmes. A tew year\ dgo, huw- w r , the policy of the goverrrrnerrt ~hanged. 1x1 1993, the Dutch hiltillstv 01 .ducnt~on puhltshed a lsst oi attanmart targets, c~iltid 'Ley go~lt ' [.or edch uhject these goals descrrbe whdt has to be learned by the end of ~~rtlxrary rl~txit. For matilematics t-he hst cotlcrsts ol2.3 goals, rpllt up lx~rto vx dorndlrls

k b l e 4.1). Compared to goal descriptio~ls orld prograrrzmcs Roni other countries it IS

lotable that some widespread mathematical topics are nut trkexltiorted i r l this list, such as, for illstallre, proltlem solving, probabiliit~ corribinalorics and logic. Another striMng fedtnre of: the list is that it i s so iiimitcd, This ixreans that the teachers have a lot of freeitor*. in interpreting tlie goiils. At the same Cirrte, however, such a list does riot give much support to teaclters. As a result the tist actually is a 'dead' document, rnostly put away in a drawer wlilerk it. arrives at K h ~ l :

Nevertheless, this first list ot hey gods was ot tmyortance for llutch ~iratlre- matics ducat~on. The publicatton of the list by the govemrrrent corrtirrried md. in a way, vaMated the recent cfiaxiges 111 our cunlculum. the nictltr clranges have bee11 that:

more attention is paid to ~rzesital arithmetic and estilnatiorr; formal operations with fractions are no lorrger in the core curriculntll; geolnetq is officially included in the curriculuxxr; insightful use of calculators is iacorpouated.

However, riot ail these chd~rges have yet been 1ei1eclted In the textb(tohs or rmplenleiited in our present classroorxr practice 1 h ~ s IS ecl~eclakly trve for geometry and the use of cal~vtators

In the years after 1993 drsxlusstorls emerged dt>out these 23 key goals (see Ile Wit 1997). Almost everybody agreed thdt these could 11ever be sufficient to kqve support kor Impromng cla~sruom practice, nor to dssess the outcctme ot education. The latter IS conceived by the governrrlerrt as a powerful tool tor sdfe~ardtng the quality of education. For both pur1msm, the key godB were judged to fail; slmply stating goals is not enough to improve practice, and the key goals are not formulated prec~sely erlough to provade yardstlchs for test- tng.

Blueprints of longitudinal learninglteaching trajectories

For several years it was unclear which dlrcctlon would be cho5en to irtiprove the key goab: wkxether tor each grade a more ctetailrtil hst of goals expressed ro operational terms would be created, or whether a descrrptiorl that supports teaching rather than pure testing would be develcrp&. 111 1W7, the goverrl- inent chose tentatively the latter axict asked the Freliderrthal ltlst~tute to work

58 Marja van den Heuvel-Panhuizen Realistic gnathematics education ~n the Netllerlands 59

Tikble 4.1 llutch hey godis of priniary school mathematics

~;enex.ii abilirier 1 The student& can count forward and backward with changing ~triits.

2 Ttie students can do addition tables and n~ultiplicatiotl tables L L O to ten.

3 The students can do easy mental-arithmetic probletns in a quick way with insight in the operations.

4 The students can estimate by determining the answer globally, also with fractions and decimals.

5 The students have insight in the structure of whole nu~llhers and the place-value system o f decimals.

h The students can use the calculator with insight. 7 The students can convert into a mathematical problem, si~~iple

probterns which are not presented in a niathematical way. Written 8 The students can apply the standard algorithms, or algorithms variations of these, for the basic operations, addition,

subtraction, rx~ultiplication and division, in easy context situations.

Ratio and 5, The students can compare ratios and percentages. percentage 10 The students can do simple problems on ratio.

11 The studet~ts have understanding of the concept percentage and can carry out practical calculations with percentages presented in simple context sihrations.

12 The students understand the relation berween ratios, fractions, and decitnals.

Fractions 13 The students know that fractions axid decimals car1 stand for sttveral meanings.

14 The students can locate frac%ions and decimals on a rlun~ber line alid can convert fractions into decimals; also with tile help of a calcl~lator.

15 The students can compare, add, subtract, divide, and multiply simple fractioris in simple context situations by means of rnodels.

bleasurernent l b The sttidents can read the time and calculate time intervals; also with the help of a cdlendar.

17 The students can do calculations with money in daily-life context situations.

18 'The students have insight in the relation between the niost important quantities and the corresponding units of measurentent.

1Y The students know the current units of nleasurernent for length, area, voiunre, tirne, speed, weight, and temperature, and can apply these in simple context situations.

20 The students cm read sirnple tables and diagrarri and produce them baxd on own investigations of sinlple context situations.

tiecnl~etry 21 The students have some basic corrcepts with which they can organize and describe a space in a geometrical way.

22 The students can reason geonletrically for which they use buildings of blocks, ground plans, tnaps, pictures, and data about place, direction, distance, and scale.

23 The students a n explarn shadow Images, can conlpound shapes, and can dev~se and tdenttiy cut-outs of regular oblects.

ctn curriculuni develop~zlet~t for rrtatbernatics. "rttir decisiosl resulted in tile start of the Tttssenduclen Annex 1,eerlijnen Cl'iZI,) (Intermediate C;oa!s Ailrrex i,earning/'reacl~ing l'rajcctories) I'sojr.ct (19%) which thc Freudelitlial Insti- :ute is carrying out together with the SLC3 a~icl C811 with tlie ptkrposr of *nharlcing classroor~t practice starting with tile early grades. @LO is the 13utciit Institute for Curricltlum L)eveloptnent. Cell i s the sctrtool ;~clvisc.)ry centre for tile city of Rotterdam. I'rcrbably in the near tuture rtlr National lrlsfitt~le for tiducational Measurement ctr CEO will also officially partici[)ate ill the 'l'A1. Project.)

'I%e first focus of the project has been or1 the development of a rlescril)lio~~ ol a loiigitudirral Iearrli~lg/teaclling trajectory orr wliolc-tlumbrr aritfrnrt3fic. In November 1998 ttte tirst descriptions f o r tlze lower grades (4- to 8-year-<>ids) were publislrted. 'The defixrtitive version followed one year later ('Freffvrs ct cr f .

1999). The project corttinurs with a whole-nutnher trajectory for the higilcr grades. Later on, other strands will be worked out.

In the whole-riumber trajectory, arithrrletic is interpreted in a broad way tt) irlclude lrtulnber krlowledge, rlurrrber sense, rnerital arith~~-tetic, estimatitiri a d algoritirtms. Ac-tually, the description is rl~earlt t o give an overview of how all ttrese elernents of nu~nber are related to each other, botll in a 1o1lgitudin;il diid in a cross-sectiorral way. Crucial to tlic trajectory tr1ue~)rints arc the steppirlg stones which the students will use (Irt one way or anotlter) orr their way to reachirlg the goals at the mid of primary scltool. 'l'ltcse steltpirlg stones car1 be seen as i~ltermediate goals. As erld goals, Itowever, tlsey difler in n-ta~rty rcspccts frorlr the usual erld-of-gradr goill descriptiorls, wlliclt are usually very sigid i r i order tct be apprc)priate its a direct hasis tor testirtg. 'I'lle irttencirci t>lueprints lor the lear~lirlg/teaching trajectories arc, i r l several respects, the opposite to gtral descriptict~rs traditiorially stipgrosed to guide the curriculum. Irrstead of urlartr- biguous goal descriptions in behavioural ternrs, the Iearriir~g trajectories ill provide teacllers with a riarratiw sketch of how t l ~ e learrtirrg process can pro- ceed provicleci that a route is followed.

The trajectory bluei~rints are in ilo way rrlearlt as rtlcipc Itooks, rallier tf1t.y are i r t te~~ded to provide texlrers with a rlletxtal educatioslal Itrap ttlat can lrekp thern, if rlccessary, tct make adjustrrie~its to the textboctk. Arrotlier ditferericc from the traditiortal goal ciescriptiort is tlhrt illere is 1x0 strictly lurmuidted stxncture. 111 addition, the learrli~~g processes are 11ot regafcied as a coiitinuctiis process o f srriall steps, nor arc the itidelween goals ccrrrsidereri as a che~'k1ist ill which the ticks tell you liow far your stiidertts have gone. Suill an apprctach lleglects the discolltiriuities in the learxling process aird does riot takc into account the degree to which ~ i r ~ ~ i ~ ~ r s t i d i g and skill biertornratrce are cLeter- t-tlirled by the context and differ brtwecrl ixttllviduals. Instead ot a cltecklist cii isolated abilities, tlte trajectory l)lueyrir~ts try to irrakt* cclrar horu rclevatlt skills and understandirlg are bttilt up in co~rtt~ectioii wit11 each otller.

60 Marja van den Heuvel-Panhuizen Realistic mathematics education in the Netherlands 61

T'lic bitllfi~zg fi)rr%l of kvefs urid the ciicittcticcrl use oft!ilein

11 is this level characteristic of the learning processes that brings coherence to the learnitig/teaclting trajectory. A crucial implication is that childsc~ can understand sornetlli~lg on different levels and several can be workirlg or the salrie problems without being at the same level of understanding. Tlli distillction of levels in understallding, which call have different appearance for different subcion~ains within the whole rtumber strand, is very fruitful fa working on the progress of children's understandirlg. It offers footholds lo stirxtulating this progress.

As an exatnple one rttigtit consider the levels in counting that we have dis tinguisheci for the early stage of tlle develop~llellt of number concept in tlir early years (see TAL, 'Peam 1998):

context-connected counting; object-connected counting; (towards) a more formal way of counting.

-. l o explairr this level distinctior~ and to give an idea of how it can be used for making problenis accessible to children and for eliciting shifts in levels of understanding, one could thirrk of the ability Lo courit up to ten. What do we have to do if a child does not make any sense of the 'how many' ques- tion (see Figure 4.5)' Does this mean that the child is simply not able to do tfie counting?

'That this is not necessarily the case txay becot~le clear if we move to a con- text-con~iected question. This means that a plain 'lrow rnany' question is 11ot asked, but that a context-connected question is used, such as:

How old is she (while referring to the candles on a birthday cake)? (Figurt 4.6) How far can you move (while referring to the dots on a die)?

Figure 4.5 How ntany . . .?

Fibwre 4.6 How old . . .? Source: 'WL team (1998: 26)

* How high is the tower (while referring to ttre hlocks of wlicll tire tower is built)?

In the context-cormected questions, tile colitext gives rneiillirlg to tiltA co~r- cvpt of nuniber, This coiltext-connrctc'd co~intirlg precedes the level of the cibiect-connected counting in wlricfr tlre clrildretr can handle lire direct 'Itow many' questiorl i ~ r relation to a collection of concrete rtbjects withottt any rrferet~ce to a meaningtzll context. Later on, the presence of tile collcrete objects is also not needed artytnore to answer 'lrtrw many' ciuestio~ls. 'I'hrough synlbolizing, the children have reirclieti a level of u~tderstarrding in wlliclr they are capable of what nright be called 'formal countirtg', whidr mwrrs that tfiey can retlect upon nurnber relaticrns a~i t l that they can 111akc use of this liliour- ledge. Regarding the field of early calculatirlg in Grade 1 (with ntlrrrbess up to 20), the following levels I~ave beexi identified (see 'Preffers rt (81. 19991:

calculatirlg by counting: calculating 7 + 6 by layilxg ~tocvn seven I-guilder coins and six 1-guiltier c o i ~ ~ s arid courtting o ~ t e by ozle; calculating by structuring: calctilatirtg 7 + 6 by laying tlowt~ a 5-guiltier coil1 and 1-guilder coins;

* formal calculating: calculating 7 + 6 witflout usir~g coins ancl by rnahitlg t~sc of 11ulliber knowledge about 6 + 6.

I n the higher grades when students are doing calculations on a for~llal level the above levels can be recogrtized irr the three ditterent calculation strategies for additiorr and subtraction up to 100:

* the 'jumping' strategy, which is related to calculating by coulrtrl?g: it i~rtplies keeping the first number as a whole trurrlber, e.g. 87 - 39 = . . . 87 - 30 = 57 . . .57-7 = 50 . . . 50 - 2 = 48;

* the strategy of splitting ~lurrrbers ill tens artd ortes, wl~icll is related to cal- culatislg by structurilzg: it implies nlakirtg use of the cleci~nal structure, e.g. 8 7 - 3 9 = . . . 8 0 - 3 0 = 5 0 . . . 7 - 7 ~ 0 . . . 5 0 - 2 ~ 4 8 ; flexible couritirrg, which is related t o fortnal calculatirig: it i r ~ ~ p l i ~ s liiakirlg use of knowledge of rru~rtber relations and properties of opcratiuns, e.g. 87 - 3 Y = . . . 8 7 - 4 0 = 4 7 . . . 4 7 + 1 = 4 8

These ideas about counting on a number tine as a base for coullting up to 1f.N are further developed iri the cliapter by Julie Menne in this volurrre.

lrrsight into such didactical levels provides teacl~ers xvitlx a powerful nuinstd)' for getting access to children's uncierstandillg a~rcl tor working urr shitts itr understanditlg. After starti~lg, for ixrstarlce, with context-cormected questioils ('How old is she?') the teacher can grariuatly pusl.1 back the context and ctme to the object-connected questior~s ('flow many carldles are on the birthday cake?'). The level categories for calculatior-is tip to 20 and 100 differ ctlxlsider- ably from, for i~tstance, levels based on prablern types alici levels based on Ibe size of the numbers to be processed. They also cieviate from the ntore gellerai

62 Marla van den Heuvel-Panhuizen

concrete-abstract distinctions in levels of understandinlr and from level dis- " tinctions ranging from material-based operating with numbers to mental pro- cedures; verbalizing is seen as an intermediate state.

As far as some of the main ideas behind the trajectory blueprints are regarded, we are just at the beginning of work on them. We do not yet know how they will function in school practice and whether they can really help teachers. Investigations to date (De Goeij et at. 1998; Groot 1999; Slaveriburg and Kroonernan 1999), howev-r, have given us the feeling that the latter might indeed be the case and that we h& triggered off soGething that can bring not only the children to a higher level but also our mathematics edu- cation. The interesting thing for us was to discover that making a trajectory blueprint was not only a matter of writing down in a popular and accessible way for teachers what was already known, but that the work on the trajectory also resulted in new ideas about teaching mathematics and involved revisiting our current thinkirig about it.

To conclude

In this chapter I have attempted to outline the main characteristics of an approadi to arithnietic teaching that has been, and continues to be developed in the Netherlands. The metaphor of a guided tour through the Dutch land- scape has a special meaning for Dutch mathematics educators as it was Freudenthal(1991) who called the last chapter in his last book, 'The landscape of mathematics education'. This chapter probably inspired Treffers when he took a well-known poem of the famous Dutch poet Marsman to summarize Dutch mathematics education in primary school. Let me conclude with this poem.

Thinking of Holland

Realistic mdthematics education in the Netherlands 63

late

Th~s 11st of principles 1s an adaptation of the five tenets (31 the tranleworh fur the KML tnstructlon theory that have been dist~ngulshed by Irelfers (1987) phmomertukogl- fa) exploration by means of concepts, brldg~ng by vertical tnstruinents, pupils' own constructions and produchons, interactive m2r1.~diort, arid intertwtnlng ot ie~1nlng strands. Out of the ux princtples des~ribelf In the present section, the hrbt three b ~ v e reinarkable consequences for I<ME asbessment (ree van den Meuvel-l'anhui~en 1996)

Thinking of Holland I see wide rivers

winding lazily through endless low countrysides

like rows of empty number lines striping the horizon

I see multibase arithmetic blocks

low and lost in the immense open space

and throughout the land rrlathematics of a realistic brand.

(after H. Marsman's 'Denkend aan Holland'; adapted by A. Treffers 1996)

188 Kenneth Ruthven

Callbridge in ?vldrch 1999; attd the 23rd Annuai Conference of the international Group for the Psychology ot Mcrthernatics Education, held at the Technion - Israel Institute of Technology - in July 1999 (Ruthven 19?9b). I arn grateful for the invitations to partici- pate in these meetings, m c i for the many helpful ideas and suggestions gleaned.

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Name index

Alexander, P , 137 Angltileri, 1 , 2, 4, 5, 7, 8, 65, 73, 74, 79,

81-3, 90, 119, 120, 121, 123, 128-0, 18 1

Ashcratt, M , 180 Ashew, M , 5 , 7, 9, 121, !$I, 134, l(5,

137, 140, 143, 183 Athnson, S , 129

Ball, D., 30, 149 Baroody, A,, 120, 137 Bauersfeld, H., 29, 31 Becker, J., 68,76 Beishuizen, M., 5-1 1, 24, 28, 29, 66, 72,

74, 75, 77, 82, 90, 111, 116, 119, 120-3, 126-9, 183

Bell, A.,134 Bereiter, C., 126 Bibby, T., 7 Bierhofi, H., 9, 11, 69, 74 Bowers, J., 25, 29 Bramald, K., 77 Brown, M., 7, 11, 15, 16, 33-5, 74, 79,

134, 137, 142, 163, 168, 171, 181 Buys, K., 12, 56, 6&, 72, 91, 96, 119-21,

130, 137

Carpenter, T., 77, 119, 127, 128, 129, 137 Case, R., 126 Chaplln, D., 83, 177, 187 Cobb,P., 17, 18,19,21,22,26,29,31,

129, 147, 157, 158

(:ockburrl, A., b Y

Gockcrutt, LV., I2, 43, 45- 7, 70, 7q, 82, I b<>

IJantzig, 'C, 15, l h T)e Goeij, E., 62 L>e Moor, E., 56, 12 1, 12rt E)e Wit, C., 57 Ikri, M.,90 Desforges, C:., n9 i)cutctr, A., 82, 92 L>owling, I!, 142

Elberr, E., 157 P.miro, t i . , 82 Frne\t, I?, 72

Farrell, S., 74 Feijs, E., 56, I:etix, C:., 126, 127 Fertnerria, E., f 19 I;iscl.ibein, E., 90 Fitzgerald, A,, 160 Foxmarl, D., 122, 127 Frarthe, M., 119 Frctrdertt2ta1, H., 4, 34, SO, 51,

55, 52, 73, 148, 151, 154-6, 159

Fuson, K., 24, 119

Garden, K., 42 (ire, J., 137