The number line is a versatile tool. When used in estimation, it can serve as a visual representation of number. This study evaluated the relationship between sorting numbers by magnitude and number line estimation performance. Fifty-eight participants in Grades 1, 2, and 3 estimated values on a 0-100 number line over four sessions. During two intervention sessions they sorted numbers into 5 categories either linearly or nonlinearly before they estimated the same target values. The linear group’s number line estimates had less error than the nonlinear group’s estimates at posttest. In particular, the participants who started with low numeracy scores in the linear group outperformed their counterparts in the nonlinear group on the number line estimation task. Computer-based number categorization supports computer-based number line estimation skills when numbers are categorized linearly. This finding extends the representational mapping hypothesis to computer-based scaffolds.

Starting from the Swedish curriculum’s goals of a school for all children this study’smain purpose is to contribute to the understanding of how mental number-lines arebeing described and related to by people with visualized number-forms. The study aimsto answer the two questions:- How are the mental number-lines described by people with visualized numberformsand in what way do these descriptions differ from a conventional numberline?- What relations can be found between the visualized number-line and the choiceof counting-strategies on number-tasks?The study used a qualitative method in which three participants with visual mentalnumber-lines described their preferred counting-strategies and relation to their mentalnumber-lines in three semi-structured interviews. The data was then analyzed byconnecting counting-strategies to the described mental number-lines and by comparingthe described mental number-lines to conventional, linear, number-lines. Two of theparticipants in the study displayed a distinct connection between counting-strategies andnumber-line-visualization. This was shown by the way they adapted their countingstrategiesto fit their personal visualization of the mental number-line. The resulthighlights the importance of understanding how students visualize their number-lines.For further studies this study suggests further investigations regarding how mentalnumber-line-visualizations hinders students from achieving important steps in theirunderstanding of mathematics.

Absolute value has often been taught procedurally. Many students struggle with solving absolute value equations and inequalities because they do not have an understanding of the underlying concepts. This study was designed to determine to what extent solving absolute value equations and inequalities by using the concept of distance on a number line is an effective method. The claim is that if students use the distance concept on a number line, they will develop the necessary conceptual understanding in addition to just a procedural knowledge that will lead to the success with and flexibility in the use of strategies for more challenging problems. The following questions were addressed in this study: How and to what extent can using a number line develop a conceptual understanding of absolute value equations and inequalities? What solution strategies do students tend to use to solve absolute value equations and inequalities? Does the strategy depend on the complexity of the problem? To what extent do they exhibit flexibility in their use of strategies? What extensions are students able to make? What misconceptions do they have? In this study, lessons and assessments were implemented based on the “best practice” of using multiple representations with a focus on conceptual understanding of absolute value. The lessons were consistent with current content standards. Students completed a pre and post assessment, and some students were selected to participate in a 15 minute interview based on their responses from their assessments. The results were analyzed qualitatively and show that students struggled with remembering the procedure for solving absolute value equations and inequalities. The results also show that students were more successful when using the distance concept on a number line.

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The present thesis deals with the mental representation of numbers in space. Generally it is assumed that numbers are mentally represented on a mental number line along which they ordered in a continuous and analogical manner. Dehaene, Bossini and Giraux (1993) found that the mental number line is spatially oriented from left­-to­-right. Using a parity­-judgment task they observed faster left-hand responses for smaller numbers and faster right-hand responses for larger numbers. This effect has been labelled as Spatial Numerical Association of Response Codes (SNARC) effect. The first study of the present thesis deals with the question whether the spatial orientation of the mental number line derives from the writing system participants are adapted to. According to a strong ontogenetic interpretation the SNARC effect should only obtain for effectors closely related to the comprehension and production of written language (hands and eyes). We asked participants to indicate the parity status of digits by pressing a pedal with their left or right foot. In contrast to the strong ontogenetic view we observed a pedal SNARC effect which did not differ from the manual SNARC effect. In the second study we evaluated whether the SNARC effect reflects an association of numbers and extracorporal space or an association of numbers and hands. To do so we varied the spatial arrangement of the response buttons (vertical vs. horizontal) and the instruction (hand­related vs. button­-related). For vertically arranged buttons and a button­related instruction we found a button-­related SNARC effect. In contrast, for a hand-­related instruction we obtained a hand­-related SNARC effect. For horizontally arranged buttons and a hand­related instruction, however, we found a button­related SNARC effect. The results of the first to studies were interpreted in terms of weak ontogenetic view. In the third study we aimed to examine the functional locus of the SNARC effect. We used the psychological refractory period paradigm. In the first experiment participants first indicated the pitch of a tone and then the parity status of a digit (locus­-of-­slack paradigma). In a second experiment the order of stimulus presentation and thus tasks changed (effect­-propagation paradigm). The results led us conclude that the SNARC effect arises while the response is centrally selected. In our fourth study we test for an association of numbers and time. We asked participants to compare two serially presented digits. Participants were faster to compare ascending digit pairs (e.g., 2-­3) than descending pairs (e.g., 3-­2). The pattern of our results was interpreted in terms of forward­associations (“1­-2-­3”) as formed by our ubiquitous cognitive routines to count of objects or events. / Die vorliegende Arbeit beschäftigt sich mit der räumlichen Repräsentation von Zahlen. Generell wird angenommen, dass Zahlen in einer kontinuierlichen und analogen Art und Weise auf einem mentalen Zahlenstrahl repräsentiert werden. Dehaene, Bossini und Giraux (1993) zeigten, dass der mentale Zahlenstrahl eine räumliche Orientierung von links­-nach­-rechts aufweist. In einer Paritätsaufgabe fanden sie schnellere Links-hand­ Antworten auf kleine Zahlen und schnellere Rechts-hand Antworten auf große Zahlen. Dieser Effekt wurde Spatial Numerical Association of Response Codes (SNARC) Effekt genannt. In der ersten Studie der vorliegenden Arbeit ging es um den Einfluss der Schriftrichtung auf den SNARC Effekt. Eine strenge ontogenetische Sichtweise sagt vorher, dass der SNARC Effekt nur mit Effektoren, die unmittelbar in die Produktion und das Verstehen von Schriftsprache involviert sind, auftreten sollte (Hände und Augen). Um dies zu überprüfen, forderten wir Versuchspersonen auf, die Parität dargestellter Ziffern durch Tastendruck mit ihrem rechten oder linken Fuß anzuzeigen. Entgegen der strengen ontogenetischen Hypothese fanden wir den SNARC Effekt auch für Fußantworten, welcher sich in seiner Charakteristik nicht von dem manuellen SNARC Effekt unterschied. In der zweiten Studie gingen wir der Frage nach, ob dem SNARC Effekt eine Assoziation des nicht-­körperbezogenen Raumes und Zahlen oder der Hände und Zahlen zugrunde liegt. Um dies zu untersuchen, variierten wir die räumliche Orientierung der Tasten zueinander (vertikal vs. horizontal) als auch die Instruktionen (hand-­bezogen vs. knopf­-bezogen). Bei einer vertikalen Knopfanordnung und einer knopf-­bezogenen Instruktion fanden wir einen knopf­bezogenen SNARC Effekt. Bei einer hand-­bezogenen Instruktion fanden wir einen hand-­bezogenen SNARC Effekt. Mit horizontal angeordneten Knöpfen gab es unabhängig von der Instruktion einen knopf-­bezogenen SNARC Effekt. Die Ergebnisse dieser beiden ersten Studien wurden im Sinne einer schwachen ontogenetischen Sichtweise interpretiert. In der dritten Studie befassten wir uns mit dem funktionalen Ursprung des SNARC Effekts. Hierfür nutzten wir das Psychological Refractory Period (PRP) Paradigma. In einem ersten Experiment hörten Versuchspersonen zuerst einen Ton nach welchem eine Ziffer visuell präsentiert wurde (locus-­of-­slack Paradigma). In einem zweiten Experiment wurde die Reihenfolge der Stimuluspräsentation/Aufgaben umgedreht (effect­-propagation Paradigma). Unsere Ergebnisse lassen vermuten, dass der SNARC Effekt während der zentralen Antwortselektion generiert wird. In unserer vierten Studie überprüften wir, ob Zahlen auch mit Zeit assoziiert werden. Wir forderten Versuchspersonen auf zwei seriell dargebotene Zahlen miteinander zu vergleichen. Versuchspersonen waren schneller zeitlich aufsteigende Zahlen (z.B. erst 2 dann 3) als zeitlich abfolgenden Zahlen (z.B. erst 3 dann 2) miteinander zu vergleichen. Unsere Ergebnisse wurden im Sinne unseres vorwärtsgerichteten Mechanismus des Zählens („1-­2-­3“) interpretiert.

The purpose of this report is to describe the course, Analysis of the Real Line, taught at The University of Texas at Austin. Course materials are presented using the inquiry based learning method. Students work a series of warm up problems before being presented rigorous problems in calculus, including topics on integration, exponential functions, and real number line analysis. Additionally, students consider aspects of these problems that could be incorporated into a high school curriculum. Typical problems in several major areas are summarized along with warm up problems that introduce or extend the topics. / text

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Syftet med denna studie är att ta reda på hur lärare i årskurserna 1–6 beskriver begreppen tallinje och taluppfattning, samt hur de använder tallinjen i matematikundervisning för att utveckla elevers taluppfattning. Studien är en kvalitativ studie som baseras på semistrukturerade intervjuer. Vi valde att intervjua sju informanter med samma yrke. Studiens resultat visar att samtliga lärare beskriver att taluppfattning handlar om att förstå talraden, värdet av talet, vad som är större och mindre, förståelse för storleken med mera. Vidare i resultat visas det att tallinjen ses som ett bra verktyg i undervisningen som kopplas bland annat till linjalen. Tallinjen medför att eleverna utvecklar en god taluppfattning, de får upptäcka mer av det konkreta till exempel genom aktiviteter i klassrummet. Studiens resultat visar att samtliga lärare beskriver att elever utan de grundläggande kunskaperna kommer möta på svårigheter i ämnet matematik. Studiens slutsats blir därmed att elever måste ha en god taluppfattning utifrån de olika grunderna för att sedan kunna operera med de fyra räknesätten tillsammans med verktyget tallinjen. Det innebär att tallinjen bidrar till att eleverna får en större talförståelse. / The purpose of this study is to find out how teachers in grades 1–6 describe the concepts number line and speech perception, and how they use the number line in mathematics teaching to develop student speech perception. The study is based on a qualitative study with semi structured interviews. We chose to interview seven informants with the same professions. The study's results show that all teachers describe that speech perception is about understanding the number line, the value of the speech, what is bigger and smaller, understanding the size and so on. Further in the results, it is shown that the number line is shown as a good tool in teaching which is linked among other things such as the ruler. The number line means that the students develop a good speech perception, they can discover more of the concrete for example through activities in the classroom. The results also show that all teachers mention that students without the basic knowledge for a good speech perception will encounter difficulties in the subject of mathematics. The study's conclusion becomes that students must have a good speech perception on the basis of the different bases in order to then be able to operate with the four methods of calculation together with the tool number line. This means that the number line helps students gain a greater understanding of speech.

Raketmatte is an App designed to enhance children’s numerical magnitude. This study describes the agile process behind the scene from the making of the App. The focus of the app is the transition from a logarithmic to a linear view on the mental number line. Theoretically the app is based on the integrated theory of numerical development, and the app aims to train the users mental number line. The study also describes how elements from gamification has been incoroprated in the final product.

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