Number system and computation with a duo of artefacts: the pascaline and the e-pascaline

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Edito

ors: Xuh

hua Sun

n , Berin

nderjee

et Kaur , Jarmila Novo

otná

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The Twenty-third ICMI Study:

Primary Mathematics Study on Whole Numbers

Macao, China

University of Macau

June 3 - 7, 2015

Proceedings

Edited by Xuhua Sun, Berinderjeet Kaur and Jarmila Novotná

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International Programme Committee

Co-chairs: Maria G. (Mariolina) BARTOLINI BUSSI (Italy), Xuhua SUN (China) Members: Sybilla BECKMANN (USA), Sarah GONZÁLEZ DE LORA (República

Dominicana), Berinderjeet KAUR (Singapore), Maitree INPRASITHA (Thailand), Joanne MULLIGAN (Australia), Jarmila NOVOTNÁ (Czech Republic), Hamsa VENKATAKRISHNAN (South Africa), Lieven VERSCHAFFEL (Belgium); Abraham ARCAVI (Israel - ICMI Secretary General) ex-officio.

ICMI Executive Advisors: Ferdinando ARZARELLO (Italy – ICMI President),

Roger E. HOWE (USA – ICMI liason)

Local Organizing Committee Chair: Xitao FAN

Co-chair: Lai LEONG

Associate Chairs: Rachel Wan Hang SIO, Kin Mou WONG

Members: Alan Wai Lon CHANG, Ping CHAO, Kwok Cheung CHEUNG, Raymond Kuok Vai CHIANG, Vera Si Nga CHOI, Jianxia DU, Chunlian JIANG, Meng Ngai LAO, Brendan Chi Hang LEI, Stephen Kam Chio LEONG, Xiaoyu WEI, Bonnie Hoi Ian WONG, I Lin WONG, Ian Nam WONG, Juan ZHANG

Acknowledgement:

We want to acknowledge the significant work done by all the IPC members, who voluntarily invested time and effort to be part of the IPC on top of their other obligations. They were friendly, and willing to work hard when some speed-up of the process was needed to meet the deadlines (We specially acknowledge Hamsa Venkat for the help in language editing of some parts). We want also to thank the IMU Secretariat and the University of Macau, the Education and Youth Affairs Bureau,

Macau SAR (DSEJ; 澳門特別行政區政府教育暨青年局) and Macao mathematics

education study association for the generous financial support for both the IPC meeting and the Conference. The colleagues and technical staff of university of Macau contributed effectively to preparation of the conference management system, of the conference and of the proceedings (We specially acknowledge Xitao Fan for effective administration support and leadership of ICMI STUDY 23 conference). Last but not least, Lena Koch (IMU Secretariat at WIAS) helped to solve many problems during the whole process.

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ICMI Study 23 Table of contents

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TABLE OF CONTENTS

Xuhua Sun: Preface ... 1

M.G. Bartolini Bussi & Xuhua Sun: The ICMI Study 23: Primary mathematics study on whole numbers ……….. 3

Plenary lectures ………... 9

M.G. Bartolini Bussi: The ICMI Study 23 plenary speakers... 9

H. Bass: Quantities, numbers, number names, and the real number line ...……….. 10

B. Butterworth: Low numeracy: From brain to education………. 21

L. Ma: The theoretical core of whole number arithmetic …….……...………... 34

Theme 1 .………... 39

X. Sun & S. Beckmann: Theme 1: The why and what of whole number arithmetic ... 39

N. Azrou: Spoken and written numbers in a post-colonial country: The case of Algeria ………... 44

Ch. Chambris: Mathematical basis for place value throughout one century of teaching in France ……...………... 52

N. Changsri: First grade students’ mathematical ideas of addition in the context of lesson study and open approach ……….... 60

J. Cooper: Combining mathematical and educational perspectives in professional development ……... 68

J.-L. Dorier: Key issues for teaching numbers within Brousseau’s Theory of didactical situations ….……….. 76

L.R. Ejersbo & M. Misfeldt: The relationship between number names and number concepts …………...……….. 84

S. González & J. Caraballo: Native American cultures tradition to whole number arithmetic .……….. 92

C. Houdement & F. Tempier: Teaching numeration units: Why, how and limits ….. 99

R. Howe: The most important thing for your child to learn about arithmetic ………. 107

L.M. McGarvey & P.J. McFeetors: Reframing perceptions of arithmetic learning: A Canadian perspective ……….……… 115

J. Sayers: Foundational number sense: The basis for whole number arithmetic competence ………..……….. 124

M.-K. Siu: Pedagogical lessons from TONGWEN SUANZHI ( 同文算指 ) – Transmission of BISUAN (筆算 written calculation) in China …….……… 132

X. Sun: Chinese core tradition to whole number arithmetic ……….….………. 140

E. Thanheiser: Leveraging historical number systems to build an understanding of the base 10 place value system ………...………... 149

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D. Zou: Whole number in ancient Chinese civilisation: A survey based on the system of counting-units and the expressions ……….…………..……… 157

Theme 2 ...……….………... 165

L. Verschaffel & J. Mulligan: Theme 2: Whole number thinking, learning and development ………..

A. Baccaglini-Frank: Preventing low achievement in arithmetic through the

didactical materials of the PerContare project ………... 165 169

I. Elia & M. van den Heuvel-Panhuizen: Mapping kindergartners’ number

competence ………... 177 P. Gould: Recalling a number line to identify numerals ……….……... 186 S. He: How do Chinese students solve addition / subtraction problems: A review of

cognitive strategy ……….….. 194 Y. Ma, S. Xie & Y. Wang: Analysis of students’ systematic errors and teaching

strategies for 3-digit multiplication ………..………. 203 J. Milinković: Counting strategies and system of natural number representations in

young children ………..……….

J. Mulligan & G. Woolcott: What lies beneath? Conceptual connectivity underlying

whole number arithmetic ……...……… 212 220 P. Nesher & S. Shaul: On the semantics and syntax of ‘+’ and ‘=’ signs ……... 229 A. Obersteiner, G. Moll, K. Reiss & H.A. Pant: Whole number arithmetic –

competency models and individual development...

N. Roberts: Interpreting children’s representations of whole number additive

relations in the early grades ….……….………. 235 243 N. Sinclair & A. Coles: ‘A trillion is after one hundred’: Early number and the

development of symbolic awareness ………. 251 L. Verschaffel, J. Torbeyns, G. Peters, B. De Smedt & P. Ghesquière: Analysing

subtraction-by-addition in the number domain 20-100 by means of verbal protocol vs reaction time data ………... 260 D.-Ch. Yang: Performance of fourth graders in judging reasonableness of

computational results for whole numbers ………...……... 268

Theme 3…...……….. 277

M.G. Bartolini Bussi & M. Inprasitha: Theme 3: Aspects that affect whole number learning ……….. 277 M. Bakker, M. van den Heuvel-Panhuizen, & A. Robitzsch: Learning multiplicative

reasoning by playing computer games ……….. 282 D.L. Ball & H. Bass: helping students learn to persevere with challenging

mathematics ……….……….. 290 M.G. Bartolini Bussi: The number line: A “western” teaching aid ……….... 298 B.R. Hodgson & C. Lajoie: The preparation of teachers in arithmetic: A

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M. Inprasitha: An open approach incorporating lesson study: An innovation for teaching whole number arithmetic ……… 315 S. Ladel & U. Kortenkamp: Development of conceptual understanding of place

value ……….. 323 A. Mercier & S. Quilio: The efficiency of primary level mathematics teaching in

French-speaking countries: A synthesis ……… 331 Y. Ni: How the Chinese methods produce arithmetic proficiency in children ………

A. Peter-Koop, S. Kollhoff, A. Gervasoni & L. Parish: Comparing the development

of Australian and German children’s whole number knowledge ..………

D. Pimm & N. Sinclair: The ordinal and the fractional: Some remarks on a

trans-linguistic study ………... 339 346 354 T. Rottmann & A. Peter-Koop: Difficulties with whole number learning and

respective teaching strategies .………...

S. Soury-Lavergne & M. Maschietto: Number system and computation with a duo

of artefacts: The pascaline and the e-pascaline ………. 362 371 J. Young-Loveridge & B. Bicknell: Using multiplication and division contexts to

build place-value understanding ………. 379

Theme 4 ...………. 387

J. Novotná & B. Kaur: Theme 4: How to teach and assess whole number arithmetic 387 M., Alafaleq, M., Mailizar, L., Wang & L. Fan: How equality and inequality of

whole numbers are introduced in China, Indonesia and Saudi Arabia primary mathematics textbooks ……….. 392 M. Askew: Seeing through place value: An example of connectionist teaching ……. 399 A. Barry, J. Novotná & B. Sarrazy: Experience and didactical knowledge – the case

of didactical variability in solving problems ...………... 407 A. Brombacher: National intervention research activity for early grade mathematics

in Jordan ...……..………... 415 Y. Cao, X., Li & H. Zuo: Characteristics of multiplication teaching of whole

numbers in China: The application of the nine times table ………... 423 A-L. Ekdahl & U. Runesson: Teachers’ responses to incorrect answers on missing

number problems in South Africa ……….. 431 A. Gervasoni & L. Parish: Insights and implications about the whole number

knowledge of grade 1 to grade 4 children ………. 440 B. Kaur: The model method – A tool for representing and visualising relationships 448 P-J. Lin: Teaching the structure of standard algorithm of multiplication with 2-digit

multipliers via conjecturing ………... 456 C. Pearn: Same year, same school, same curriculum: Different mathematics results 464 G. Sensevy, S. Quilio & A. Mercier: Arithmetic and comprehension at primary

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I.N. Wong, C. Jiang, K-C. Cheung & X. Sun: Primary mathematics education in Macau: Fifteen years of experiences after 1999 handover from Portugal to

mainland China ……….. 480

Q-P. Zhang, K-C. Cheung & K-F. Cheung: An analysis of two-digit numbers subtraction in Hong Kong primary mathematics textbooks ……….. 488

X. Zhao, M. van den Heuvel-Panhuizen, & M. Veldhuis: Classroom assessment techniques to assess Chinese students’ sense of division ……….. 496

Theme 5 ………... 505

H. Venkat & S. Gonzalez: Theme 5: Whole numbers and connections with other parts of mathematics ………. 505

Y. Baldin, M.C. Mandarino, F.R. Mattos & L.C. Guimarães: A Brazilian project for teachers of primary education: Case of whole numbers ……..……… 510

S. Beckmann, A. Izsák & I.B. Ölmez: From multiplication to proportional relationships ...……… 518

L. Chen, J. van Dooren & L. Verschaffel: Effect of learning context on students’ understanding of the multiplication and division rule for rational numbers ……. 526

S. Dole, A. Hilton, G. Hilton & M. Goos: Proportional reasoning: An elusive connector of school mathematics curriculum ………...…... 534

A. Eraky & R. Guberman: Generalisation ability of 5th - 6th graders for numerical and visual-pictorial patterns ………... 542

F. Ferrara & O-L. Ng: A materialist conception of early algebraic thinking ... 550

K. Larsson & K. Pettersson: Discerning multiplicative and additive reasoning in co-variation problems ……… 559

M. Mellone & A. Ramploud: Additive structure: an educational experience of cultural transposition ………..…. 567

L. Venenciano, H. Slovin & F. Zenigami: Learning place value through a measurement context ..………... 575

H. Venkat: Representational approaches to primary teacher development in South Africa ……….……… 583

Y-P. Xin: Conceptual model-based problem solving ……….………. 589

J. Zhang, Y. Meng, B. Hu, S.K. Cheung, N. Yang & C. Jiang: The role of early language abilities on math skills among Chinese children ……… 597

Panels ………... 603

M.G. Bartolini Bussi: The ICMI Study 23 panels ……….………. 603

F. Arzarello: Panel on tradition……….. ………….………... 605

J. Novotná: Panel on teacher education ……….. 613

L. Verschaffel: Panel on special needs ……… 619

Appendix. The Twenty-third ICMI Study: Primary Mathematics Study on Whole Numbers. DISCUSSION DOCUMENT ………. 625

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ICMI Study 23 Theme 3, Saury-Lavergne & Machietto, Duo of Artefacts

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NUMBER SYSTEM AND COMPUTATION WITH A DUO OF ARTEFACTS: THE PASCALINE AND THE E-PASCALINE

Sophie Soury-Lavergne(1), Michela Maschietto(2)

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IFÉ ENS de Lyon, France, (2) UNIMORE, Italy

Abstract

We are using a duo of artefacts, constituted by a mechanical arithmetic machine and its digital counterpart, to enable six-year old French students to learn about numbers. The experiment shows the separate conceptualisation processes involving numbers as sign of a quantity and number sequences on one side and recursive addition, computation and its effect on the decimal code for numbers on the other side. The duo of artefacts enabled the design of situations that required these processes to be connected. We observed how students and teachers used the duo and discuss the results concerning the conceptualisation of number.

Key words: duo of artefacts, e-pascaline, number system, pascaline Teaching number decimal system and computation

One of the aims of the first year of compulsory education for 6-year old French students is to learn the decimal system of writing numbers and to use it to perform computation. In their review of studies about whole numbers, Nunes and Bryant point out the key question: ‘how do children come to understand that any number in the counting sequence is equal to the preceding number plus 1?’ (2007, p. 4). We reformulate this in a more general way by asking how children connect what they know about numbers, number sequences and the manipulation of quantities with what they know about computation, performing addition or subtraction within the number system. The question also concerns the representation of number, taking into account that the number sequence is often learnt as an oral sequence, while the number system is a symbolic written system.

In France, before they are six, students begin to learn the numbers up to 30 at “école maternelle”. For these children, number is used to indicate a specific characteristic of a collection, which is the quantity of its elements, or a position in a list (Margolinas and Wosniack, 2014). Comparing numbers involves going back to the collections and operating on their objects. The number name is a label that signals the number of elements. These names are ranked and can be recited in a given order. Children count by using the oral list of number names, which turns to be an action on the objects of the collection. They may even perform some kind of addition, which is, in fact, the union of two collections of objects. Once the two collections are unified, children can determine the number of its objects. Even if they use digits to write a representation of numbers, they use these as an icon and do not manipulate the decimal number system.

When children start “elementary school”, during the first year of compulsory education, they have to learn the decimal number system used to represent

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numbers up to 100. The aim is that they understand how this system is linked to the collections of objects they have used to count, in a more precise manner than just as different names for different kinds of collection, and that one can operate on numbers by operating on their digits. They have to build the relationship between successive numbers in the number sequence; for instance the fact that the number next in the list is the previous one plus one unit.

We are investigating the use of a duo of artefacts, with a physical machine (the pascaline) and its digital counterpart (the e-pascaline), in the learning of numbers and computation. Following Italian research about mathematical machines (Maschietto and Bartolini Bussi, 2009), we assume that the physical machine enables the action-perception loop linking eyes and hands which is important for mathematical conceptualisation (Edwards, Radford and Arzarello, 2009; Kalenine, Pinet and Gentaz, 2011). However, using only a limited range of physical material may also lock students into procedures that require the presence of the physical artefacts, even when the didactical sequence and the teacher try to take this possibility into account and to facilitate the transfer and generalisation of procedures. Therefore, we have extended the physical artefact by a digital version of the machine that enables students to use their procedures in another context (Maschietto and Soury-Lavergne, 2013).

The aim of this paper is to describe how the duo of artefacts offers students a way to learn about the number system by solving problems that require flexibility in moving between number writing and computation. We have designed the e-pascaline in order to enlarge the mathematical experience of the students and to make complementary activities with the two kinds of artefacts. This paper discusses a French teaching experiment carried out in ordinary classrooms with voluntary teachers (Soury-Lavergne, 2014).

Materials and methods

A duo of artefacts: the pascaline and the e-pascaline

The pascaline is an arithmetic machine composed of gears analogous to the famous machine, called Pascaline, invented by the French mathematician Blaise Pascal in 1642. It is a crucial tool in the history of European mathematics because it represents the first example of addition performed independent of the human intellect. When the pascaline is introduced in the classroom, the reference to Pascal and his motivation for the construction of the machine plays an important role in the vision of mathematics as a cultural product. It provides a symbolic representation of the whole numbers from 0 to 999 and enables arithmetic operations to be performed. Each of the five wheels has ten teeth. The digits from 0 to 9 are written on the lower yellow wheels, which display units, tens and hundreds from the right to the left (Fig. 1 shows the pascaline displaying the number 122). When the units wheel (respectively the tens wheel) turns a complete rotation clockwise, the right upper wheel (respectively the left upper wheel) makes the tens wheel (respectively the hundreds wheel) go one

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clicks on the units wheel, is hence no longer possible. In such a way, the student has to look for another strategy to perform the addition.

The e-book “Counting the e-pascaline clicks” contains a task, which consists in minimising the number of clicks required to write a number on the e-pascaline. This task appears to only concern the writing of numbers, but in fact, it requires the exploration of different ways of reaching a number through combinations of additions and subtractions. Starting with the e-pascaline displaying 0, there are three possible procedures to display a given number on the e-pascaline. Let’s consider an example. The number 17 can be written by iteration (17 clicks) or by decomposition (8 clicks), but the minimum of clicks is obtained by a computation 20-3 (5 clicks). This third strategy requires knowledge about the decomposition of numbers and also for the students to change their point of view on the problem and to move from writing the number to computing it.

Experimental setting

The didactical sequence has been developed within a French project gathering teachers, researchers and teacher educators <http://ife.ens-lyon.fr/sciences21/ressources/sequences-et-outils/pascaline-CP>. It has been experimented during the last school year by a team of eight teachers who did not participate in the initial project. During the twelve week experiment, we were able to directly observe working sessions with two classes of six-year old students. We also collected information from the teachers through interviews and written reports throughout the experiment. The results refer either to direct observations of students’ behaviours (with teachers Stina and Nelly) or to the teachers’ reports and interviews (Stina and Cleo).

Results

In the teaching sequence, students first worked on addition with the pascaline alone and then used the e-book. In almost every teaching experiment with the pascaline, students’ initial strategies to add two terms below ten were: (i) the two terms were written on two separate wheels and the result was expected to appear on the third wheel; (ii) the addition was done by mental calculation and the result was written on the pascaline. With the first strategy, analogous to the use of a calculator, the user transferred the main part of the work to the pascaline. With the second one, the user performed the main part of the work. The iteration procedure appeared only after teacher interventions and discussion about the two previous strategies.

Adding numbers with the pascaline and the e-pascaline

In Cleo’s class, the iteration procedure appeared after she suggested using the units wheel alone. Then, with terms over ten, mistakes did not increase enough to make the students look for another strategy, even when the teacher suggested that they did. Only one of Cleo’s twenty-three students found the decomposition procedure. When Cleo first set the addition e-book to the students as an individual activity, they still had the possibility to use the pascaline to do

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answer the question “Why haven’t we succeeded in obtaining ‘smileys’?”. She also used an intermediate task, with the pascaline, to make pupils find decompositions of numbers. With this intermediate situation, she introduced the solution by asking students to complete a partial decomposition with a subtraction, such as 28 = 30 – … . After this episode, most of her pupils were able to solve the problem.

We directly observed the first use of the e-book in Nelly’s class and we made two important observations. First, students didn’t use the iteration procedure. They directly used decomposition, starting with the tens digit. This meant that they reproduced on the e-pascaline the spatial organisation of the digits in the written number. Moreover, two numbers, 9 and 19, provoked different procedures, although the successful procedure asked for computation. Students failed to write 9 with the minimum of clicks while they succeeded with 19, first writing 1 on the tens wheel and then turning the units wheel one click in the anticlockwise direction, after having observed that the tooth with the digit 9 was close to the red triangle. They finished by adjusting the tens wheel (one click) when they observed that it had returned to 0. Their procedure could be represented by the computation 19 = 10 – 1 + 10. The fact that they didn’t do this for 9 illustrated that they are not in the process of computing but in the process of writing the number (and adjusting the wheels if needed). Their procedure is not equivalent to 19 = 20 – 1, which could have been transferred to 9.

Discussion and conclusion

We have elaborated the duo of artefacts and the e-books to build didactical situations that require the evolution from the iteration strategy to the decomposition one. With the “minimum of clicks” e-book, a third strategy requires computing. It is worth remarking that on one hand, the addition e-book explicitly required computation, but, as in any process of computing, it requires taking into account the way numbers are written with digits and not just adding units one by one (Nunes and Bryant, 2007). On the other hand, the “minimum of clicks” e-book explicitly required writing numbers, but a successful strategy required the students to compute. Both situations were rather difficult for students and successes resulted from teachers’ interventions. They revealed that the concept of number, its properties and the signification of its digits code are not stable. They compel the students to make connections between the number designing a quantity and the number represented by its digits code. This relationship frames the fundamental conceptual understanding of whole numbers. Another example of this relationship is the connection between two successive numbers in the sequence, the operation +1 and its effect on the digits of the numbers codes.

In this learning process, both artefacts of the duo played a crucial role. The pascaline was used to produce a sequence of clicks that can be counted. Meanwhile it displays numbers code and ask for operating on these codes. The e-pascaline provided different constraints and feedback that led students to

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change their strategies and to deal with complementary conceptualisation of numbers.

Further observations are needed to deepen our understanding of the different aspects of numbers that are developed by students while using the duo of artefacts. We have planned to conduct further experiments in France as well as in Italy, to record students’ strategies with the pascaline and the e-pascaline and to address some cultural issues.

Acknowledgements

The study has been conducted with the funding of the Canopé network, in charge of pedagogical support and pedagogical resources dissemination in France.

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