Empty Number Line Research Articles

Pre-service teachers’ use of the jump strategy on the empty number line for mental computation

Background: South African teachers’ mathematical knowledge for teaching and number sense are markedly low, evidenced in national and international research. There is a call for supporting teachers in developing their knowledge and confidence in teaching mathematics.Aim: The aim of this article is to share how pre-service teachers (PSTs) used the empty number line (ENL) to demonstrate the jump strategy (JS) for efficient mental computation, related to the national Mental Starters Assessment Project (MSAP).Setting: This study was conducted at a tertiary institution in the Eastern Cape province of South Africa, with 40 first-year intermediate phase PSTs, during their mathematics didactics course.Methods: The data comprised 40 micro-teaching videos, analysed thematically, following an analytical model for video-recorded data. Videos were categorised according to the way they faithfully followed or departed from the MSAP teacher resources.Results: Findings suggest that more than 70% of the PSTs faithfully followed the teacher resources to teach the JS on the ENL, with regards to use of at least three of the four key elements, namely: ENL, jumps, gesturing, and key phrases and terminology.Conclusion: We suggest PSTs receive explicit instruction on the above-mentioned key elements for teaching mental calculation strategies with the ENL and opportunities to practise teaching these to their peers.Contribution: Most PSTs successfully used the ENL to demonstrate the JS for efficient mental computation, enabled by the opportunity to engage in developing (and reflecting on their) micro-teaching videos.

Mental processing strategies in primary school students: the empty number line

The aim of this research is to determine the mental processing strategies used by 3rd grade primary school students in addition and subtraction using the empty number lines. The model of the research is a case study, one of the qualitative research methods. The participants consist of 143 students; 75 girls and 68 boys, attending the third grade (8-years-old). Content analysis method was used in data analysis. Considering the results of the research, the mental strategies used by primary school 3rd-grade students were gathered in four categories: counting, balancing, first process step and partitioning. It was concluded that the students mostly used the mental processing strategy of using "multiples of 10", which is in the first process category, and this strategy was followed by "counting all" and "partitioning the second number”, the least used mental processing strategy by the students was "using equivalent numbers".

Task beliefs and the voluntary use of the empty number line in third-grade subtraction and addition

This study explored the relationships between task beliefs about the empty number line (ENL), mathematical ability, gender, and voluntary ENL use in multi-digit subtraction and addition. One hundred twenty-three Dutch third-grade students and nine teachers from six schools participated in this study. The multilevel path analysis showed that task beliefs about the ENL mediated the relationship between students’ mathematical ability and their voluntary ENL use. No gender differences were found in the multilevel path analysis. Finally, the results show that task beliefs about the ENL and voluntary ENL use differed across classrooms. The discussion focuses on the implications of the results for using the ENL in diagnostic assessment.

THE NEUTRALIZATION ON AN EMPTY NUMBER LINE MODEL FOR INTEGER ADDITIONS AND SUBTRACTIONS: IS IT HELPFUL?

The number line and the neutralization model have been used very extensively in teaching integer additions and subtractions for decades. Despite their advantages, issues concerning subtractions on these models are still debatable. Therefore, the neutralization on an empty number line (NNL) model is proposed in this research to help students better understand the meaning of integer subtractions as well as additions. This report is a part of a design research study conducted in a classroom of 28 elementary school students at the fifth grade. Data were gathered by collecting students’ written work, conducting interviews and observations during the teaching experiment. This paper focuses on students’ perceptions of the NNL model and what factors that might contribute to students’ achievements in understanding integer additions and subtractions. The analysis revealed that most students overemphasized on the use of the NNL model as a procedural method instead of as a model for thinking. Moreover, students’ mathematical beliefs and conceptions on the use of the column strategy and the absence of a discussion on the need of using the model are found to be some factors that could cause students’ misunderstandings. However, with a thorough planning, the NNL model has a potential to help students developing a meaning of integer additions and subtractions.

Geometric Aspects of Number Line Estimations

Number lines are implicitly embedded in nature. Yet researchers use them for measuring number sense as if they are processed purely through numerical reasoning. We argue that number line estimation tasks are done both by numerical and geometric reasoning. The purpose of this research was to investigate the relationships among mathematics achievement, geometry achievement, spatial skills, and number line estimations. A total of 142 fourth graders were administered 5 different tests: 2 curriculum-based math achievement tests, a spatial visualization test, a number line estimation test, and the Raven Standard Progresive Matrices test. Results showed; estimation accuracy of the relative magnitude of numbers on an empty number line has more to do with geometry achievement and diagrammatic reasoning rather than with numerical or arithmetic reasoning. It seems that number line estimation tasks may constitute an interplay between number and shape. Therefore, we conclude that using multiple external representations of numbers, such as spatial, symbolic, and verbal could be useful in developing a more robust number sense.

Evaluating the Effect of Labeled Benchmarks on Children's Number Line Estimation Performance and Strategy Use.

Some authors argue that age-related improvements in number line estimation (NLE) performance result from changes in strategy use. More specifically, children’s strategy use develops from only using the origin of the number line, to using the origin and the endpoint, to eventually also relying on the midpoint of the number line. Recently, Peeters et al. (unpublished) investigated whether the provision of additional unlabeled benchmarks at 25, 50, and 75% of the number line, positively affects third and fifth graders’ NLE performance and benchmark-based strategy use. It was found that only the older children benefitted from the presence of these benchmarks at the quartiles of the number line (i.e., 25 and 75%), as they made more use of these benchmarks, leading to more accurate estimates. A possible explanation for this lack of improvement in third graders might be their inability to correctly link the presented benchmarks with their corresponding numerical values. In the present study, we investigated whether labeling these benchmarks with their corresponding numerical values, would have a positive effect on younger children’s NLE performance and quartile-based strategy use as well. Third and sixth graders were assigned to one of three conditions: (a) a control condition with an empty number line bounded by 0 at the origin and 1,000 at the endpoint, (b) an unlabeled condition with three additional external benchmarks without numerical labels at 25, 50, and 75% of the number line, and (c) a labeled condition in which these benchmarks were labeled with 250, 500, and 750, respectively. Results indicated that labeling the benchmarks has a positive effect on third graders’ NLE performance and quartile-based strategy use, whereas sixth graders already benefited from the mere provision of unlabeled benchmarks. These findings imply that children’s benchmark-based strategy use can be stimulated by adding additional externally provided benchmarks on the number line, but that, depending on children’s age and familiarity with the number range, these additional external benchmarks might need to be labeled.

The empty number line: A good idea for primary school mathematics?

The empty number line: A good idea for primary school mathematics?

Modelling operations: Counting-based and collection-based approaches to computation

This paper focuses on different ways of modelling operations in mathematics; namely, counting- based and collections-based models. The two models are examined in relation to the literature on diagramme literacy, as well as New Zealand's mathematics curriculum document. Implications for the Numeracy Development Project and the New Zealand Number Framework are also considered. The value of presenting collections- based models in addition to the (counting-based) empty number line models in materials for teachers is discussed. Splitting diagrammes are presented as a collections-based alternative to empty number line models.

초등학생의 수직선 이해와 사용의 어려움

본 연구는 초등학생들이 수 개념과 관련하여 수직선을 어떻게 이해하고 사용하는지, 또 그 학습의 어려움은 무엇인지 파악하고자 하였다. 이를 위하여 수직선 은유가 수 개념과 어떻게 관련되는지 살펴보았고, 프로이덴탈의 수 개념지도론에서 수직선의 역할에 대하여 고찰하였다. 실제 초등학생들의 수직선에 대한 이해와 사용의 어려움을 파악하기 위해 실시한 검사는 수직선에 주어진 위치에서 적절한 수를 대응시키는 문항과 학년별로 수직선이 활용되는 관련 단원 내용을 묻는 문항으로 이루어졌다. 같은 내용과 구조의 문항이지만 수직선으로 표현된 것은 해결하지 못하면서 수 트랙이나 다른 그림으로 표현된 것은 해결하는 학생들이 다수 관찰되었고, 본 연구에서는 이러한 현상의 의미를 해석하고자 하였다. 또한 다양한 교수-학습 자료(수 트랙, 그림, 빈 수직선, 이중 수직선등)를 활용하여 수직선 이해의 어려움을 보완하고 관련 수 개념 학습을 돕는 방안을 제안하였다. The purpose of this study is to investigate how elementary school students understand and use the number line relating number concept and what is the main problem in the learning process. For the efficient achievement of this purpose, we investigated how the number line metaphor is related to the number concept and considered the role of the number line on Freudenthal's number concept teaching theory. The test conducted to find the degree of understanding and difficulty on using the number line by actual elementary school students consisted of two questions ; to find appropriate number corresponding to the given number on the number line and to identify contents of chapters about the use of number line on each grade. It was found that many students couldn't solve the problem represented by the number line though they could solve the problem represented by other ways such as number track and pictures. The only difference between the two problems was the way of representation, and they had same contents and structure. This study tried to figure out the meaning of this phenomenon. Also, by using various teaching-learning method (number track, pictures, empty number line, and double number line etc.), this study was aimed to provide the way to help learning 'related number concept' and to solve the difficulty on understanding the number line.

Developing a Model to Support Students in Solving Subtraction

Subtraction has two meanings and each meaning leads to the different strategies. The meaning of “taking away something” suggests a direct subtraction, while the meaning of “determining the difference between two numbers” is more likely to be modeled as indirect addition. Many prior researches found that the second meaning and second strategy rarely appeared in the mathematical textbooks and teacher explanations, including in Indonesia. Therefore, this study was conducted to contribute to the development of a local instruction theory for subtraction by designing instructional activities that can facilitate first grade of primary school students to develop a model in solving two digit numbers subtraction. Consequently, design research was chosen as an appropriate approach for achieving the research aim and Realistic Mathematics Education (RME) was used as a guide to design the lesson. This study involved 6 students in the pilot experiment, 31 students in the teaching experiment, and a first grade teacher of SDN 179 Palembang. The result of this study shows that the beads string could bridge students from the contextual problems (taking ginger candies and making grains bracelets) to the use of the empty number line. It also shows that the empty number line could promote students to use different strategies (direct subtraction, indirect addition, and indirect subtraction) in solving subtraction problems. Based on these findings, it is recommended to apply RME in the teaching learning process to make it more meaningful for students.

Developing a Model to Support Students in Solving Subtraction

Subtraction has two meanings and each meaning leads to the different strategies. The meaning of “taking away something” suggests a direct subtraction, while the meaning of “determining the difference between two numbers” is more likely to be modeled as indirect addition. Many prior researches found that the second meaning and second strategy rarely appeared in the mathematical textbooks and teacher explanations, including in Indonesia. Therefore, this study was conducted to contribute to the development of a local instruction theory for subtraction by designing instructional activities that can facilitate first grade of primary school students to develop a model in solving two digit numbers subtraction. Consequently, design research was chosen as an appropriate approach for achieving the research aim and Realistic Mathematics Education (RME) was used as a guide to design the lesson. This study involved 6 students in the pilot experiment, 31 students in the teaching experiment, and a first grade teacher of SDN 179 Palembang. The result of this study shows that the beads string could bridge students from the contextual problems (taking ginger candies and making grains bracelets) to the use of the empty number line. It also shows that the empty number line could promote students to use different strategies (direct subtraction, indirect addition, and indirect subtraction) in solving subtraction problems. Based on these findings, it is recommended to apply RME in the teaching learning process to make it more meaningful for students.

Comparing the use of the empty number line in England and the Netherlands

Efforts to meet the needs of children's learning in arithmetic has led to an increased emphasis on the teaching of mental calculation strategies in England. This has included the adoption of didactical tools such as the empty number line (ENL) that was developed as part of the realistic mathematics movement in the Netherlands. It has been claimed that the English use of the ENL differs from that of the Dutch. In this study curriculum guidance materials are examined to investigate these differences. Two examples of teaching sequences are provided to illustrate them further. It is proposed that the different uses stem from different pedagogical principles. The implications of the English use are discussed in relation to the presentation of mental calculation strategies in an algorithmic way.

ICT‐based dynamic assessment to reveal special education students’ potential in mathematics

This paper reports on a research project on information and communication technology (ICT)‐based dynamic assessment. The project aims to reveal the mathematical potential of students in special education. The focus is on a topic that is generally recognised as rather difficult for weak students: subtraction up to 100 with crossing the ten. The students involved in the project were 8–12 years old. Their mathematical level was one to four years behind the level of their peer group in regular schools. In special education, teachers often use standardised written tests to assess their students’ mathematical understanding and computational skills. These tests do not allow students to use auxiliary resources. In the research project, an assessment instrument was developed and used in which items taken from a standardised written test were placed in an ICT environment. The digitised items were enriched with an optional auxiliary tool that students could use for solving the problems. Currently, two sub‐studies, investigating two different auxiliary tools for the same set of subtraction problems, have been completed. In the first study (n = 37), the tool provided students with digital manipulatives. In the second study (n = 43), an empty number line was included in the ICT environment. In both studies the students’ scores in the ICT version of the test items were compared with the standardised written version of the test items. The results of both sub‐studies show that the students could benefit from using an auxiliary tool. Consequently, the ICT‐based dynamic assessment could help to reveal weak students’ potential in subtraction. In addition, the studies disclosed that in the manipulatives study there was a clear preference for direct subtraction procedures (taking away), whereas in the number line study indirect addition or subtraction procedures (e.g. adding on) were frequently used as well.

Fostering the development of mental number representations and arithmetic competencies in the first school year

Event Abstract Back to Event Fostering the development of mental number representations and arithmetic competencies in the first school year Andreas Obersteiner1*, Stefan Ufer1 and Kristina Reiss1 1 Technische Universität München, Heinz Nixdorf-Stiftungslehrstuhl für Didaktik der Mathematik, School of Education, Germany The development of arithmetic competencies is a main goal of early mathematics instruction. These competencies comprise exact mental calculation as well as approximate calculation and number estimation [1]. Various manipulative materials are used in primary schools in order to support students’ development of mental number representations. These materials can represent either exact numerosities (e.g., a pegboard with two rows of ten pegs each) or analogue quantities (e.g., empty number line). While theoretical considerations support their use, the empirical foundation of effects on students’ learning is scarce. This is the starting point for our study. We investigate students’ mathematical achievement after a training phase aimed at fostering either exact or approximate mental number representations. It is of particular interest whether the two trainings have different effects on a) the development of the respective mental number representations and b) general arithmetic achievement. Furthermore, it is explored if c) general cognitive abilities, e.g., spatial abilities, are related to these learning processes. According to cognitive psychological and neuropsychological theories, which suggest that exact and approximate number processing rely on distinct mental representations, different learning effects can be expected. For implementing the training sessions, modified versions of the adaptive computer game “The Number Race”[2] are used. One version requires exact and the other one approximate number processing. The present study can be regarded as an example how neuropsychological findings and cognitive theories are used in educational contexts to support students’ learning at school. We will present the theoretical background of the study as well as first results of piloting studies.

Learning from “Didactikids”: An impetus for revisiting the empty number line

This article discusses students’ perceptions as a source for understanding education. It addresses what didactically experienced children, called “didactikids”, taught us about the empty number line as a didactical model for teaching whole number calculations. The article mainly reports on a student consultancy study carried out in the Netherlands. The findings are similar to what was revealed in an Australian study. Both studies explain what can go wrong when the number line is applied rigidly and wrongly implemented. In this article I reflect upon the empty number line (a blank number line with no numbers or markers) as a didactical model for teaching calculations with whole numbers in the early grades of primary school. Children whom I call “didactikids” prompted this reflection. These are didactical experts who have an inclination to understand teaching, perhaps having developed this talent because a parent or relative works in education. Every child, though, has the potential to become a didactikid when given the opportunity to reflect upon education. As will be shown in this paper, teachers, teacher educators, and researchers and developers of mathematics education can learn a great deal from didactikids. Based on what such didactical experts brought to the fore about their experience with the empty number line, I will discuss some critical issues to do with the use of the number line and what can go wrong when this didactical model is not implemented in the way it was originally conceptualised. In other words, this article has two foci: the issue of learning from children, and the empty number line. In current thinking about mathematics education, children’s contributions to the development of mathematics teaching are highly valued. In the Netherlands, it was Freudenthal who laid the foundation for this appreciation with inspiring observations of his children and grandchildren. His goal was to build knowledge about children’s cognitive

The Empty Number Line: A Useful Tool or Just Another Procedure?

A grade 4 mathematics homework question instructed students, “Use the empty number line to solve the following,” causing nineyear- old Emily to respond, “No one should tell me what strategy to use. I should be allowed to make up my own mind!” Emily found it easier to solve some two-digit addition and subtraction computations by splitting them into tens and ones rather than using a strategy that could be recorded on a number line.

Assessing pupil knowledge of the sequential structure of numbers

Research on children’s mental strategies for multidigit addition and subtraction identifies two categories of strategy. Collections-based strategies involve partitioning numbers into tens and ones, and can be modeled with base-ten materials. Sequence-based strategies involve keeping one number whole, and using the sequential structure of numbers. They can be modeled as jumps on an empty number line. Studies have found sequence-based strategies to be more successful, and to correlate with more robust arithmetic knowledge, particularly among low-attaining pupils. Studies also suggest that sequence-based strategies and sequential structure are not explicitly developed in many primary mathematics classrooms. This report draws on results from a three-year project which has the goal of developing pedagogical tools for intervention in the number learning of low-attaining 3rd- and 4th-graders (8- to 10-year-olds). These tools include assessment tasks to inform intervention. The report focuses on four groups of assessment tasks that collectively enable documenting in detail, pupils’ knowledge of the sequential structure of numbers. Tasks and pupils’ responses are described in detail. Some examples follow. When asked to count back from 52, pupils said, ‘52, 51, 40, 49, 48, and so on’. When asked to count back by tens from 336, pupils had difficulty continuing after 326. Thus teen numbers in the hundreds (316) presented particular difficulties. Pupils had difficulty saying the number that is ten less than 306. Pupils had difficulty with locating the numbers 50, 25, 62 and 98 on a number line on which zero and 100 were marked. The report provides insight into assessing knowledge of sequential structure and argues that this is important basic number knowledge.

Enhancing primary mathematics teaching

Part one: Subject content Progression from mental to written strategies The empty number line Shape and space/geometry Part two: Teaching approaches/pedagogy Early years/foundation Thinking and talking Whole class teaching/lesson structure Problem solving word problems Problem solving AT1 and investigative approaches Part three: Assessment Assessment for learning Mistakes, misunderstandings and misconceptions Part four: Intervention Intervention strategies Gifted and talented Part five: Research and related issues Calculators ICT Place value Brain based research The Leverhulme Project Index.

Introducing the empty number line

The empty number line (ENL) is an essential element of Dutch mathematics education, already internationally acknowledged as a successful model for developing mental strategies. The concept is a deceptively simple one but is based upon a complex structure of previous research findings. Several recent and influential UK publications have already misused the concept. This article is based upon the original work of Beishuisen from Leiden University and sets out a very simplified version of how the ENL is derived and built up for use in mental mathematics.

Which Mental Strategies in the Early Number Curriculum? A Comparison of British Ideas and Dutch Views

AbstractMental arithmetic has recently gained a higher profile in primary mathematics. The question then arises whether mental strategies should be left to spontaneous development or should be taught in some didactic order? In reaction to a School Curriculum and Assessment Authority (SCAA) discussion paper and other publications a comparison is made between British ideas and Dutch views. In the Netherlands the Realistic Mathematics Education theory has, since the 1980s, inspired textbook design and teaching practice, but also discussions on mental mathematics. Better balancing between ‘mental recall and mental strategies’ led to an improved National Programme (1990) and a second generation of revised textbooks. A great difference is the radical choice for postponing (formal) place value and written algorithms and building first on pupils’ (informal) mental strategies. This principle of ‘progressive mathematisation’ is developed further with the empty number line as a new model for mental strategies up to 100. Some British experts have called for a similar curriculum change. Although current suggestions for everyday mental activities and for mental methods in relation to written procedures mark a beginning, they are analysed as not going far enough. The assumption is made that British pupils also develop informal mental strategies, which deserve more substantial attention.