The purpose of this group is to gather congress participants who are interested in research and development in the teaching and learning of number systems and arithmetic, including operations in the number systems, ratio and proportion, rational numbers. Any current issue related to the main theme of TSG 10 may be considered in discussion. Examples of such issues are the development of ‘number sense’ in students, the role of contexts and models in teaching and learning about number and arithmetic, and the development of teaching/learning units that connect basic arithmetic skills with higher order thinking skills. From an international perspective, we will study and discuss advances in research and practice, new trends, and the state-of-the-art. The focus is all levels from primary through tertiary level.

TSG participants are expected to stay with their group throughout the four sessions. We find this very important if the group is to be able to exchange experience, built networks as well as form collaborations for future research.

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Tuesday, July 8. 12:00-13:00

Session 1: Key note speakers: “State-of-the-art”. Session Chair: Dirk de Bock

• (1) Dr. Zalman Usiskin, Professor of Education, University of Chicago, USA: The Arithmetic Curriculum and the Real World

Abstract: The relationships between abstract arithmetic and the real world are dealt with inconsistently in most curricula. Each of the common arithmetic operations is a mathematical model for counting and measure situations found in the real world. These models parallel the theoretical properties of the operations and provide the basis for more sophisticated models found in algebra, geometry, analysis, and statistics. The absence of explicit instruction in these models may explain why many children have difficulty applying arithmetic.

• (2) Dr. Darcy Hallett, Assistant Professor, Memorial University, Canada: Effects of Fraction Situations and Individual Differences: A Review of Recent Research Regarding Children’s Understanding of Fractions

Abstract: The goal of this paper is to provide a review of the recent and promising research regarding children’s ability to work with fractions. I have chosen to focus on fractions because many researchers have claimed that children have especial difficulty in learning them. Streefland (1991) has even stated that fractions are “without doubt the most problematic area in mathematics education” (p. 6).

Wednesday, July 9. 12:00-13:30

Session 2: Short oral paper presentations. Session Chair: Bernardo Gómez

Theme: Multiplication, division, fraction

• (1) Susanne Prediger: Discontinuities for Mental Models: A Source for Difficulties with the Multiplication of Fractions

Abstract: Different theoretical approaches offer different ways of explaining students’ welldocumented difficulties with arithmetical operations like multiplication of fractions. The article recalls a conceptual framework that integrates approaches focusing on meanings of operations into conceptual change approaches. It offers first results from an empirical study on discontinuities and continuities of models for the multiplication of fractions.

• (2) Rose Elaine Carbone & Patricia T. Eaton: Prospective Teachers’ Knowledge of Addition and Division of Fractions

Abstract: This study reports the initial findings of two collaborating mathematics educators from the United States and Northern Ireland on their prospective elementary teachers’ understanding of rational numbers. Prospective elementary teachers were evaluated on their ability to create appropriate real life problems illustrating the addition and division of fractions. The similarity of the misunderstandings that these prospective teachers exhibited offers ways for mathematics educators to inform and improve their teaching. This research also expands international collaborations.

• (3) Marta Elena Valdemoros: Planning Fraction Lessons: A Case Study

Abstract: We are doing a case study with three basic education teachers who have joined a master degree focused on a professional strengthening of their teaching experience. In the current research phase we explore how they plan activities for teaching fractions and what kind of difficulties they confront in such planning. In this document we will only make reference to the case of Delia, a fifth grade teacher at an elementary school who has decided to plan her fraction lessons relying on the meaning of measure as the planning’s didactic object. This case study has been done from two fundamental methodological instruments: the observation that took place at a master degree seminar (where the activities proposed by Delia for teaching fractions as a measure were presented) and the interviews, where the aforementioned teacher reflected about the obstacles she experienced through the didactic design process.

• (4) Wim van Dooren; Dirk De Bock; Marleen Evers & Lieven Verschaffel: The Role of Number Structures on Pupils’ Over-Use of Linearity in Missing-Value Problems

Abstract: This study builds on previous research showing that primary school pupils over-rely on proportional methods when solving non-proportional missing-value word problems. It is hypothesized that when the numbers in word problems form integer ratios, this will stimulate pupils to apply proportional methods, even if this is inappropriate. It is furthermore expected that the effect will diminish from grade 4 to 6 (with pupils’ age and proportional reasoning experience). The results confirm both hypotheses.

• (5) Fátima Mendes & Elvira Ferreira: Developing Multiplication

Abstract: The research project “Developing number sense: curricular demands and perspectives” 1 studies the development of number sense in children from 5 to 11 years old. The project team included classroom teachers and researchers that developed and experimented tasks and task chain that intended to foster number sense. This paper focuses on one of the project case studies. This case study analyses the implementation in a 2nd grade class (7-8 years old) of a task chain related with multiplication. We will center the discussion on the strategies used by children in one particular task.

• (6) Issic K. C. Leung; Cho Paul & Regina M. F. Wong: Learning Alternate Division Algorithm in Enhancing the concept of Rates and Density

Abstract: The traditional long division algorithm assumes that users can apply a guess-andmatch type mental process of searching for a maximum that is not greater than the dividend at the initial stage of this algorithm. This optimization process requires heavy cognitive load on metal calculation on applying rules and regulations that does not correlate to life experience of sharing objects. By introducing the special method of learning the concept of division, the Partition of Quotient (POQ), we find that it can enhance the effectiveness of learning the concept of rate in science, in particular, the concept and property of density of an object.

Friday, July 11. 12:30-13:30

Session 3: Short oral paper presentations. Session Chair: Chun Chor Litwin Cheng

Theme: Addition and integers

• (1) Bny Rosmah Hj. Badarudin & Madihah Khalid: Using the Jar Model to Improve Students’ Understanding of Operations on Integers

Abstract: The focus of this paper is to report on a study that assess students’ knowledge and understanding of integers before and after the intervention teaching using the ‘jar model’. The paper will concentrate on the kind of errors students make in learning integers and how the ‘jar model’ was supposed to enhance students’ understanding instead of memorising rules like ‘negative times negative gives positive’ etc. Analyses from interviews and performance data of the pre and postintervention stage revealed that most of the students can understand the jar model and thus improvement can be seen from the result of the post-test.

• (2) Patricia Baggett & Andrzej Ehrenfeucht: A New Algorithm for Column Addition

Abstract: We show a modern version of an old “dot algorithm” for column addition of whole numbers and decimals. This new “dot” algorithm is at least as efficient as the “standard” written algorithm currently taught in schools, but has 2 advantages: It is easier to use, especially for adding more than two numbers; and it is not “mechanical”. Users can develop their own strategies based on patterns of digits in 1 column, making computation faster and easier. This makes the new algorithm more challenging and interesting. Theoretical underpinnings of the algorithm, historical data, and comments of people who have already learned it will be given.

• (3) Mária Slavíčková: Experimental Teaching of the Integers by Using Computers

Abstract: This paper deals with using computer on the mathematical lessons. It is focused on the teaching of the integers by simple program based on the Theory of Constructivism. We provide an experiment in 3 classes and compare their results. There were no significant difference between them, but we find important result – pupils, which use educational software based on the theory of constructivism have better results in context tasks than the pupils, which do not use this kind of software.

• (4) Raisa Guberman: A Framework for Characterizing the Development of Arithmetical Thinking

Abstract: Based on a previous study and the Van Hiele Model about levels of geometrical thinking development, I propose a framework for characterizing the development of arithmetical thinking. The framework is based on the profile of students’ reasoning and explanations of arithmetical activities. Data were collected from 190 questionnaires. The quantitative analysis of the results of the questionnaires included calculations of the relative frequencies of levels of arithmetical thinking in the population surveyed. What is outlined in the present paper may provide a possible tool to be used by mathematics teachers.

Saturday, July 12. 12:00-13:30

Session 4: Short oral presentations by the TSG 10 team members. Session Chair: Bettina Dahl Søndergaard

The members of the organizational team of TSG 10 present their point of view on important issues for further research in the field of TSG 10 or reflections on the issues related to TSG 10. This is followed by a general discussion with the audience.

• (1) Dirk De Bock: Operations in the number systems: Towards a modelling perspective

Abstract: Elementary mathematics education often focuses one-sidedly on the technically correct and fluent execution of basic operations like addition, multiplication, and direct and inverse proportionality. As a drawback, children tend to perform these operations also beyond their proper range of applicability. In this discussion paper we first provide some research-based illustrations of this phenomenon. Second, we formulate some recommendations for the improvement of educational practice by bringing the modelling perspective more to the forefront of mathematics education.

• (2) Bettina Dahl Søndergaard: A Brick in the Wall of Mathematics Education Research in Number Systems and Arithmetic

Abstract: This paper reflects on the state of the research and development of teaching and learning of number systems and arithmetic. This includes a discussion of the descriptive/ explanatory and normative dimensions of mathematics education research in relation to the ten papers in the proceeding. I also refer to an example of a normative statement about standard algorithms in Danish primary education. I conclude that we have reached far but we are not finished yet.

• (3) Bernardo Gómez: Models, Main Problem in TSG10

Abstract: The models are present in several of the contributions from the TSG10, who are concerned about the difficulties presented in teaching and learning fractions. Some of the difficulties are related to the generalisation of multiplying and dividing operations. In this study, the alternative historical approaches to tackle this generalisation are analysed. In one of them, connections are sought between models of operations with natural numbers and those with fractions, in order to facilitate the conceptual change, and in the other this conceptual change is avoided.

• (4) Chun Chor Litwin Cheng: Concepts Acquisition in Addition and Place Value (ICME-11 TSG10)

Abstract: The concept of place value and addition are presented in several of the contributions in TSG10, and the learning difficulties and possible concepts formation in learning addition are addressed. Many researches showed that children’s learning experience in addition is not as easy is we thought. Most of the time, we teach addition according to textbook material and many textbooks are algorithm-teaching based. Algorithm teaching resulted in systematic error and lack of focus on the development of place value, which hindered children to understand algorithm in multi-digits addition. Also, the representation used in carrying in addition does not relate to the concept of place value or addition. In this paper, the process of addition learning and formation of place value is analysed.

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