I denna studie undersöks vad som sker i intensivundervisning där en SUM-elev, elev med särskilda utbildningsbehov i matematik, erbjuds multisensorisk undervisning. Vilken utveckling som sker vad gäller elevens taluppfattning, samt olika uttryck för lärande i undervisningsprocessen är också något som studeras. Studien har en kvalitativ ansats och har genomförts som en fallstudie med en SUM-elev i årskurs ett. Fallstudien innehåller en intervention med matematiktester vilka genomförts före och efter en intensivundervisning i grundläggande taluppfattning. Deltagande observationer av undervisningssekvensen har genomförts, vilka också har filmats. Till analysen används Bruners teori om representationer tillsammans med Vygotskijs teori om lärande ur ett sociokulturellt perspektiv. Representationerna i Bruners teori beskrivs som den enaktiva som är handlingsbaserad, den ikoniska som är bildbaserad och den symboliska som är språkbaserad. Resultatet visar i detta fall att en intensivundervisning med multisensorisk approach påverkar elevens taluppfattning positivt och elevens lösningsfrekvens mellan för- och eftertest tredubblats. En utveckling av bland annat matematiska förmågor var synlig. Eleven utvecklade lärande i de olika representationerna där inledningsvis den enaktiva representationen användes och succesivt även den ikoniska- och symboliska representationen. Resultatet visar en positiv påverkan på elevens självförtroende i matematiska situationer, främst gällande symbolhantering. / This study examines what happens in the situation of intense, multi-sensory instruction for a SEN-student in the subject of mathematics, a student with special education needs in mathematics. What kind of development is happening in terms of the student's number sense, as well as various forms of learning in the teaching process is also something that is studied. The study has a qualitative approach and was implemented as a case study with a SEN-student in the subject of mathematics in year one. The case study includes an intervention with math tests which were taken before and after an intensive teaching of basic number sense. Participant observation of the teaching sequence has been implemented, which also have been filmed. For the analysis Bruner's theory of representations along with Vygotsky's theory of learning from a sociocultural perspective is used. The representations in Bruner's theory is described as the enactive that is action-based, the iconic that is image based and the symbolic which is language-based.  The result in this case shows that an intensive instruction with a multi-sensory approach affects the student's number sense in a positive way. The pupils solution frequency between pre- and posttest tripled. The Pupil developed learning in the different representations where initially the enactive representation was used and gradually even the iconic- and symbolic representation. A development including mathematical competences were visible. The results show a positive impact on the pupil self-confidence in mathematical situations, primarily in situations for symbol processing.

This study was conducted to analyse aspects of mental computation within primary school mathematics curricula and to formulate recommendations to inform future revisions to the Number strand of mathematics syllabuses for primary schools. The analyses were undertaken from past, contemporary, and futures perspectives. Although this study had syllabus development in Queensland as a prime focus, its findings and recommendations have an international applicability. Little has been documented in relation to the nature and role of mental computation in mathematics curricula in Australia (McIntosh, Bana, & Farrell, 1995,p. 2), despite an international resurgence of interest by mathematics educators. This resurgence has arisen from a recognition that computing mentally remains a viable computational alternative in a technological age, and that the development of mental procedures contributes to the formation of powerful mathematical thinking strategies (R. E. Reys, 1992, p. 63). The emphasis needs to be placed upon the mental processes involved, and it is this which distinguishes mental computation from mental arithmetic, as defined in this study. Traditionally, the latter has been concerned with speed and accuracy rather than with the mental strategies used to arrive at the correct answers. In Australia, the place of mental computation in mathematics curricula is only beginning to be seriously considered. Little attention has been given to teaching, as opposed to testing, mental computation. Additionally, such attention has predominantly been confined to those calculations needed to be performed mentally to enable the efficient use of the conventional written algorithms. Teachers are inclined to associate mental computation with isolated facts, most commonly the basic ones, rather than with the interrelationships between numbers and the methods used to calculate. To enhance the use of mental computation and to achieve an improvement in performance levels, children need to be encouraged to value all methods of computation, and to place a priority on mental procedures. This requires that teachers be encouraged to change the way in which they view mental computation. An outcome of this study is to provide the background and recommendations for this to occur. The mathematics education literature of relevance to mental computation was analysed, and its nature and function, together with the approaches to teaching, under each of the Queensland mathematics syllabuses from 1860 to 1997 were documented. Three distinct time-periods were analysed: 1860-1965, 1966-1987, and post-1987. The first of these was characterised by syllabuses which included specific references to calculating mentally. To provide insights into the current status of mental computation in Queensland primary schools, a survey of a representative sample of teachers and administrators was undertaken. The statements in the postal, self-completion opinionnaire were based on data from the literature review. This study, therefore, has significance for Queensland educational history, curriculum development, and pedagogy. The review of mental computation research indicated that the development of flexible mental strategies is influenced by the order in which mental and written techniques are introduced. Therefore, the traditional written-mental sequence needs to be reevaluated. As a contribution to this reevaluation, this study presents a mental-written sequence for introducing each of the four operations. However, findings from the survey of Queensland school personnel revealed that a majority disagreed with the proposition that an emphasis on written algorithms should be delayed to allow increased attention on mental computation. Hence, for this sequence to be successfully introduced, much professional debate and experimentation needs to occur to demonstrate its efficacy to teachers. Of significance to the development of efficient mental techniques is the way in which mental computation is taught. R. E. Reys, B. J. Reys, Nohda, and Emori (1995, p. 305) have suggested that there are two broad approaches to teaching mental computation,,Ya behaviourist approach and a constructivist approach. The former views mental computation as a basic skill and is considered an essential prerequisite to written computation, with proficiency gained through direct teaching. In contrast, the constructivist approach contends that mental computation is a process of higher-order thinking in which the act of generating and applying mental strategies is significant for an individual's mathematical development. Nonetheless, this study has concluded that there may be a place for the direct teaching of selected mental strategies. To support syllabus development, a sequence of mental strategies appropriate for focussed teaching for each of the four operations has been delineated. The implications for teachers with respect to these recommendations are discussed. Their implementation has the potential to severely threaten many teachersf sense of efficacy. To support the changed approach to developing competence with mental computation, aspects requiring further theoretical and empirical investigation are also outlined.

In his keynote address to the National Council of Teachers of Mathematics research pre-session, Sloane (2006b) challenged mathematics education researchers to ‘quantify qualitative insights’. This quasi-experimental study used blended methods to investigate the development of two-digit addition and subtraction strategies. Concurrent classroom teaching experiments were conducted in two intact first grade classrooms (n = 41) in a mid-Atlantic American public school. From a pragmatic emergent perspective, design research (Gravemeijer & Cobb, 2006) was used to develop local instructional theory. An amplified theoretical framework for early base-ten strategies is explicated. Multilevel modelling for repeated measures was used to evaluate the differences in strategy usage between classes across occasions and the association of particular pedagogical practices with the emergence of incrementing and decrementing by ten (N10) or decomposition (1010) strategies (Beishuizen, Felix, & Beishuizen, 1990).The two matched classes were not different in terms of gender, poverty, race, pre-assessment performance, and special education services. After the first unit of instruction with differentiated pedagogical tools, the collection class was significantly (p = .001) more likely to use 1010 than the linear class. No difference was demonstrated during the post-assessment. Students in both classes were more likely to use N10 during the last structured interview than in the first (p < .0001). Furthermore, there was no difference between the two classes in using any advanced strategy; however, students in both classes were more likely to use an advanced strategy at the conclusion of the study than they were initially (p = .033). The order of emergence of 1010 and N10 was not associated with the ability to develop both strategies, but there was an association (p < .001) between use of an advanced strategy and success on a district-mandated written assessment of two-digit addition and subtraction.Two original instructional sequences of contextually-based investigations are presented. Protocols transcribed from videotaped lessons and dynamic assessment interviews are presented to illuminate specific constructs detected and to illustrate the pedagogical techniques. An amplified framework for early place value constructs is proposed. Recommendations for future studies, curricular changes, and the need of early intervention are discussed.

In his keynote address to the National Council of Teachers of Mathematics research pre-session, Sloane (2006b) challenged mathematics education researchers to ‘quantify qualitative insights’. This quasi-experimental study used blended methods to investigate the development of two-digit addition and subtraction strategies. Concurrent classroom teaching experiments were conducted in two intact first grade classrooms (n = 41) in a mid-Atlantic American public school. From a pragmatic emergent perspective, design research (Gravemeijer & Cobb, 2006) was used to develop local instructional theory. An amplified theoretical framework for early base-ten strategies is explicated. Multilevel modelling for repeated measures was used to evaluate the differences in strategy usage between classes across occasions and the association of particular pedagogical practices with the emergence of incrementing and decrementing by ten (N10) or decomposition (1010) strategies (Beishuizen, Felix, & Beishuizen, 1990).The two matched classes were not different in terms of gender, poverty, race, pre-assessment performance, and special education services. After the first unit of instruction with differentiated pedagogical tools, the collection class was significantly (p = .001) more likely to use 1010 than the linear class. No difference was demonstrated during the post-assessment. Students in both classes were more likely to use N10 during the last structured interview than in the first (p < .0001). Furthermore, there was no difference between the two classes in using any advanced strategy; however, students in both classes were more likely to use an advanced strategy at the conclusion of the study than they were initially (p = .033). The order of emergence of 1010 and N10 was not associated with the ability to develop both strategies, but there was an association (p < .001) between use of an advanced strategy and success on a district-mandated written assessment of two-digit addition and subtraction.Two original instructional sequences of contextually-based investigations are presented. Protocols transcribed from videotaped lessons and dynamic assessment interviews are presented to illuminate specific constructs detected and to illustrate the pedagogical techniques. An amplified framework for early place value constructs is proposed. Recommendations for future studies, curricular changes, and the need of early intervention are discussed.

Kan intensivträning med digitala verktyg påverka elevers taluppfattning och motivation? - En interventionsstudie på högstadiet. Can intensive practice with digital tools affect students number sense and motivation? – An intervention study in Secondary School.   Utgångspunkten för denna studie är de negativa konsekvenser matematiksvårigheter kan få för den enskilda eleven, som lämnar grundskolan utan godkänt betyg i matematik. Viktiga faktorer för att lyckas är god taluppfattning och motivation för att jobba med matematik.   Syftet med studien är att undersöka intensivträningens effekter på taluppfattning och motivation för att arbeta med matematik, genom att använda det webbaserade träningsprogrammet Mattelek Flex. Studien är gjord som fallstudie, där intervention, intervjuer och observationer ingått. Fem SUM-elever, som vid studiens början gick i åk 8, deltog. För att undersöka interventionens effekter på taluppfattningen gjordes en diagnos före, direkt efter samt efter sommarlovet. Diagnosen som användes var test 6 i Förstå och använda tal (McIntosh, 2008). Interventionens effekter på motivationen synliggjordes genom intervjuer och observationer, som analyserats utifrån några påverkansfaktorer som Jenner(2004) lyfter fram.   Resultatet visar att arbetet med Mattelek Flex stärker elevernas taluppfattning samtidigt som datorns effekter gör att motivationen ökar. / Can intensive practice with digital tools affect students number sense and motivation?  – an intervention study in Secondary School.   The starting point in this study is the negative consequences difficulties in mathematics may have on the individual student, who leaves primary school without passing there math grades. Important factors to succeed are good number sense and motivation to work with mathematics.   The purpose of this examine is to notice what effect intensive training has on number sense and the motivation to work with mathematics, by using the web based training program Mattelek Flex. The study was made as a case study, concluded interventions, interviews and observations. Five SEM-students, as in the beginning of the study were in the eighth grade, participated. To examine the interventions effects on the number sense, a test was made before, directly after and after a summer breake. The used test was test 6 in “Förstå och använda tal (McIntosh, 2008). The interventions effects on the motivation were made visible through interviews and observations. They were analysed based on some influencing factors by Jenner(2004).   The result shows that, working with Mattelek Flex strengthens the students number sense and the effects of the computer increases the motivation.

Applikationer har blivit en tillgång för att utveckla elevernas kunskaper inom taluppfattning. Men innehåller applikationer didaktiska eller pedagogiska styrkor som andra läromedel inte innehåller? Vilka matematiska kunskaper är möjliga att utveckla genom att spela en applikation inom området? Med dessa frågeställningar tog denna studie avstamp som syftar till att studera applikationers layout och innehåll inom taluppfattning. Studien är en kvalitativ innehållsanalys där 39 applikationer har analyserats genom en analysmodell. Alla applikationerna används av elever i förskoleklass eller grundskolans tidigare år 1–3 någonstans bland landets grundskolor inom ramen för matematikundervisningen. Studien visar att applikationerna är utformade med en layout som har en stimulerande och intresseväckande effekt hos eleverna. Applikationernas möjlighet till individuell direkt återkoppling kan dessutom bidra till att elevernas motivation ökar. Vilket matematiskt innehåll som applikationerna erbjuder varierar. De områden inom taluppfattning som minst antal applikationer behandlar är mönster och rimlighetsbedömning. Däremot erbjuder delar av applikationerna möjligheten att utveckla elevernas aritmetiska kunskaper. / Applications have become a resource for developing students' knowledge in number sense. But do applications have didactical or educational strengths that other teaching materials don´t? What mathematical skills are possible to develop by playing an application in the field? With these questions, this study took off with the aim to study applications' layout and content within number sense. The study was conducted thru a qualitative content analysis, where 39 applications were analyzed by a framework. All applications are used by students in pre-school or primary school's somewhere among the country's elementary schools within the framework of mathematics education. The study shows that the applications are designed with a layout, that has a stimulating and interesting effect on the students. What mathematical skills these applications contains varies. Patterns and reasonability are the two most uncommon components of number sense apparent in the applications. However a large part of the applications develop the students' arithmetic skills.

Forskning visar att taluppfattning är grundläggande för elevers förståelse och utveckling av matematik. Läraren har en central roll i att utveckla elevers taluppfattning. Denna kvalitativa studie undersöker hur nio lärare arbetar med och för taluppfattning hos elever i förskoleklass och årskurs 1. Studien syftar till att komplettera det aktuella forskningsläget med lärares kunskaper och erfarenheter och utgår från frågeställningen: Hur uttrycker dessa lärare att de arbetar med och för taluppfattning i förskoleklass och årskurs 1? Kvalitativa semistrukturerade intervjuer med lärarna genomfördes och analyserades genom tematisk analys varpå resultatet ytterligare analyserades genom det teoretiska ramverket FONS och dess åtta komponenter av taluppfattning. Sedan diskuterades resultatanalysen i relation till det aktuella forskningsläget. Resultatet visar att lärares arbete med och för taluppfattning till stor del överensstämmer med både tidigare forskning och det teoretiska ramverket FONS. Lärarnas arbete sammanfattas inom följande teman: planering, grunden, begrepp, instruktioner, samlärande, matteboken, metoder, färdighetsträning och talsystemet. Det resultat som särskiljer sig från tidigare forskning och det teoretiska ramverket är lärarens avsikt och förmåga att arbeta med automatisering av taluppfattning, användandet av matteböcker för utveckling av taluppfattning, samt förmågan att anpassa undervisningen utefter elevgruppen. / Research shows that foundational number sense is vital to children’s understanding and development of mathematics. Teachers have an essential role in the development of children’s foundational number sense. This qualitative study investigates how nine teachers in Sweden work to develop foundational number sense with six- and seven-year-olds (corresponding grade 0 and grade 1 in Sweden). The research question: How do these teachers express their work to develop foundational number sense for six- and seven-year-olds? is answered by qualitative semi structured interviews with the teachers. The teacher interviews were analyzed using thematic analysis, establishing the following themes: Planning, The foundation, Concept, Instructions, Co-learning, The math book, Methods, Skill training and Number system. These themes were then analyzed by the theoretical framework FONS and its eight components of foundational number sense, establishing several corresponding results. The results were discussed in relation to previous research within the field and concluded that the teachers' work to develop foundational number sense is mainly in line with both previous research and the theoretical framework FONS. Results that stand out are the teachers' abilities to adjust their teachings to their particular group of students/children, their work with automatization of number sense and the use of math books.

Det råder en brist på en entydig definition av begreppet taluppfattning som är grunden för all matematik. Genom att veta vilka vanligt förekommande problem som finns inom ett ämne kan lärare förebygga problem och främja förståelse. Den här systematiska litteraturstudien syftar därmed till att sammanställa hur begreppet taluppfattning definieras inom forskning, samt undersöka vilka återkommande problem som enligt forskning finns inom taluppfattning i årskurs F-3. Genom en systematisk litteratursökning har relevant litteratur valts ut och granskats med hjälp av en innehållsanalys. Resultatet visar att det finns många skilda definitioner av taluppfattning som i denna studie sammanställs till 13 olika aspekter. Definitionerna skiljer sig beroende på vilket teoretiskt perspektiv utvald forskning använder sig av, samt vilken ålder utvalda forskare utgår ifrån i definition av begreppet. Resultatet visar även brist på forskning kring återkommande problem inom taluppfattning. Trots spekulationer är det inte möjligt att redogöra vilka vanliga problem som finns inom taluppfattning med anledning av brist på tidigare forskning.

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There is a worldwide development, better to say a non-development: We teach paper & pencil skills in primary schools almost like we did 30 or 50 or 100 years ago. Till today the primary school teachers spend up to more than 100 hours in the class room to teach and to train old fashioned algorithms though in daily life situations and for business purposes everybody uses a calculator. Why do we waste so much time of our children to teach them things which later on they will not need? We see an emotional dichotomy. Despite the research results from many research projects in many countries there still is the fear that the use of calculators in primary grades will harm mental arithmetic and estimation skills. To explain and to overcome that fear we will reflect the nature of number sense and of paper&pencil skills more carefully. We realize that the development of number sense is an intuitive and unconscious mental process while the ability to get an exact calculation result is trained logically and consciously. To overcome the above dichotomy we must solve the hidden dichotomy number sense versus precise calculation result. We need a new balance. Different types of examples will be given how we can further the development of number sense in a technology dominated curriculum.