Thesis advisor: Elida V. Laski / Use of base-10 decomposition strategy for addition in first grade is related to mathematics advantage in the later years (Geary et al., 2013), yet we know little about the strategy’s prevalence among first graders or factors contributing to its use. The present study sought to bridge this gap by testing 87 first graders in the greater Boston area. The results confirmed previous findings that showed that in the last 10 years first graders in the US have increased in frequency of base-10 decomposition. Children who had better knowledge of basic number facts used it more frequently, particularly on problems with smaller addends. Further, the study tested whether an instructional intervention would be effective in increasing reliance on base-10 decomposition. 61 of the original participants were selected to take part in an experimental intervention that taught them to execute the strategy while relying on external representations – sometimes known as manipulatives. Informed by two lines of research, the present study tested the hypothesis that the efficacy of the intervention may depend on whether one or multiple external representations are used for instruction. The results showed a dramatic increase in first graders’ mental base-10 decomposition use as a result of the intervention. Children grew in their use of the strategy at the same rates across genders, levels of basic arithmetic fluency, and working memory. Overall, the results showed that relying on multiple representations during instruction appears more beneficial to strategy use on mental arithmetic, but this benefit may be conditional on how well the children have mastered and abstracted the strategy. Implications to classroom interventions aimed to increase the use of advanced arithmetic strategies are discussed. / Thesis (PhD) — Boston College, 2016. / Submitted to: Boston College. Lynch School of Education. / Discipline: Counseling, Developmental and Educational Psychology.

As part of their mathematical knowledge for teaching (Ball et al., 2008), teachers must have a well-connected understanding of the subject matter they teach and must know this content in deeper and different ways than other adults. This is essential for quality teaching and learning, as teachers’ knowledge and understanding impact the nature and effectiveness of instruction (e.g., Hill et al., 2005). Since decimal concepts are part of elementary curriculum (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010), and can be difficult for children and adults (e.g., Jacobson et al., 2020; Kastberg & Morton, 2014; Steinle & Stacey, 1998), mathematics teacher educators must consider how we can strengthen support for prospective teachers of elementary school (PTs), to deepen their mathematical knowledge for teaching decimals. This is of particular importance as existing research provides few rich characterizations of PTs’ decimal understanding and is limited in explorations into connections and mechanisms that may improve that understanding. In this dissertation, I attend to the research question, following engagement with rich conceptually focused decimal instruction, how may PTs’ conceptualizations of decimal place value and magnitude, and factors which have influenced this understanding, be characterized? I address some gaps in current literature by considering how robust decimal understanding for PTs may be connected to and grounded in their broader knowledge of the base-ten place value system, and to the decimal models which they use. Following an introduction to the problem in Chapter One, and a review of relevant literature in Chapter Two, Chapter Three reports on a study in which I examined how PTs characterized the base-ten place value system, distinguishing between responses crafted by PTs who had demonstrated different levels of decimal understanding. This allowed me to identify elements of base-ten place value understanding which likely supported PTs’ thinking about decimal place value and magnitude. In the study in Chapter Four, I explored the nature of PTs’ decimal understanding and its relationships with decimal square or number line models that they used, finding that certain model features facilitated PTs’ ability to think about decimal place value and magnitude in ways that are more likely to be productive and appropriate for teaching. These two empirical studies are both qualitative content analyses (Hsieh & Shannon, 2005) carried out in the context of the Elementary Mathematics Project (Chapin et al., 2021). Though implications for teachers and teacher educators are incorporated in Chapters Three and Four, Chapter Five is a practitioner article in which I focus more directly on these implications, making recommendations about important model features and areas of emphasis for decimal instruction. Chapter Six looks across the dissertation, discussing overarching themes and directions for future research. Results of this research may be used to support mathematics teacher educators in carrying out effective decimal instruction with their PT students, since better understanding of PTs’ thinking can help mathematics teacher educators to make informed curricular and pedagogical decisions to foster PT development. This is of high importance, since as PTs increase and enrich their decimal understanding, their students’ opportunities to learn will also expand. / 2027-04-30T00:00:00Z

This is an action research study which focuses on a didactical model founded on base ten decomposition as an algorithm for performing division on naturals. Base ten decomposition is used to enhance the algebraic structure of division on naturals in an attempt to cross the cognitive divide that currently exists between arithmetic long division on naturals and algebraic long division on polynomials. The didactical model that is proposed and implemented comprises three different phases and was implemented over five one hour lessons. Learners’ work and responses which were monitored over a fiveday period is discussed in this report. The structure of the arithmetic long division on naturals formed the conceptual basis from which shorter methods of algebraic long division on polynomials were introduced. These methods were discussed in class and reported on in this study.

Understanding base-ten numbers is one of the most important mathematics topics taught in the primary school, and yet also one of the most difficult to teach and to learn. Research shows that many children have inaccurate or faulty number conceptions, and use rote-learned procedures with little regard for quantities represented by mathematical symbols. Base-ten blocks are widely used to teach place-value concepts, but children often do not perceive the links between numbers, symbols, and models. Software has also been suggested as a means of improving children's development of these links but there is little research on its efficacy. Sixteen Queensland Year 3 students worked cooperatively with the researcher for 10 daily sessions, in 4 groups of 4 students of either high or low mathematical achievement level, on tasks introducing the hundreds place. Two groups used physical base-ten blocks and two used place-value software incorporating electronic base-ten blocks. Individual interviews assessed participants' place-value understanding before and after teaching sessions. Data sources were videotapes of interviews and teaching sessions, field notes, workbooks, and software audit trails, analysed using a grounded theory method. There was little difference evident in learning by students using either physical or electronic blocks. Many errors related to the "face-value" construct, counting and handling errors, and a lack of knowledge of base-ten rules were evident. Several students trusted the counting of blocks to reveal number relationships. The study failed to confirm several reported schemes describing children's conceptual structures for multidigit numbers. Many participants demonstrated a preference for grouping or counting approaches, but not stable mental models characterising their thinking about numbers generally. The independent-place construct is proposed to explain evidence in both the study and the literature that shows students making single-dimensional associations between a place, a set of number words, and a digit, rather than taking account of groups of 10. Feedback received in the two conditions differed greatly. Electronic feedback was more positive and accurate than feedback from blocks, and reduced the need for human-based feedback. Primary teachers are urged to monitor students' use of base-ten blocks closely, and to challenge faulty number conceptions by asking appropriate questions.

Den här licentiatuppsatsen handlar om hur en undervisning kan främja elevers förståelse av tiobassystemet genom att de bjuds in till en undervisning där de kan delta i ett kollektivt utforskande av bassystemet, som en övergripande nivå av tiobassystemet. Syftet är att exemplifiera och diskutera hur en undervisning, som främjar elevernas möjligheter att utveckla ett tänkande om tiobassystemet på en generell eller övergripande nivå, kan utformas. Vidare är syftet att studera vad i en undervisning, designad enligt principerna för lärandeverksamhet, som kan förstås som förebyggande i specialpedagogisk mening. Studien bygger på resultatet av två artiklar med följande frågeställningar: 1) Vilka aspekter i uppgifternas utformning och genomförande främjar elevers möjligheter att pröva relationen mellan olika bastal och övergång till successivt större respektive mindre talenheter? 2) Vad i elevernas resonemang och arbete med speciellt designade uppgifter kan ses som tecken på teoretiskt tänkande om bassystemet? Learning study har använts som forskningsramverk och genererat data för uppsatsens två artiklar. Learning studyn genomfördes i årskurs 4, tillsammans med tre matematiklärare. Design och analys är inspirerad av El’konins och Davydovs matematiska program och några principer från lärandeverksamhet. Analysen i de bägge artiklarna är genomförd med stöd av teoretiska principer för lärandeverksamhet. Datamaterialet består av videoinspelningar från tre lektioner (totalt cirka 420 minuter), transkriptioner och datasammanställningar av uppgifter, transkriptioner och anteckningar från utvärderingarna i det iterativa arbetet med lektionerna. Resultatet består av tre aspekter som behöver synliggöras genom uppgifternas utformning och genomförande för att eleverna ska kunna arbeta teoretiskt med bassystemets struktur: (1) Bastalet, (2) Tal som mätetal, och (3) Talenheternas representationer. Eleverna behöver urskilja aspekterna för att kunna identifiera att det fattas en lämplig talenhet och för att kunna pröva och reflektera över relationen mellan bastalet och övergången till successivt större respektive mindre talenheter i olika baser. Resultatet består också av ett antal exempel på tecken på teoretiskt tänkande inom tre identifierade kategorier: 1) basens funktion för det värde som siffrorna anger i talet, 2) positionsväxling, och (3 entalet som ett av en kvantitet. I analysen har Davydovs definition av teoretiskt tänkande, som något som kan komma till uttryck i form av teoretisk reflektion, analys och planering samt reproduktion av grundläggande principer för ett specifikt ämnesinnehåll, använts. Resultatet ger ett bidrag till den matematikdidaktiska forskningen och till den specialpedagogiska forskningen med inriktning mot matematik genom beskrivningar av de tre aspekterna och av tecken på teoretiskt tänkande. Vidare kan beskrivningarna av uppgifternas utformning och genomförande användas i undervisning och i fortsatta studier. I diskussionen behandlas hur en undervisning kan utformas som främjar elevernas möjligheter att utveckla ett teoretiskt tänkande om bassystemet. Diskussionen behandlar också vad i en undervisning designad enligt principerna för lärandeverksamhet som kan förstås som förebyggande i specialpedagogisk mening genom att skapa möjligheter för elever att redan tidigt i grundskolan kollektivt utforska och förstå hela positionssystemet som en struktur. Slutligen diskuteras implikationer för specialundervisning och studiens bidrag. / The subject of this licentiate thesis is about how a teaching can promote students' understanding of the base-ten system by inviting them to a teaching where they can collectively explore the base system, as an overall level of the base-ten system. The aim is to exemplify and discuss how teaching, which affords students opportunities to develop thinking about the base-ten system on a general or overall level, can be designed. Furthermore, the aim is to study teaching designed according to the principles of learning activity in terms of how it can be preventive from a special pedagogical perspective. The study is based on the results of two articles with the following questions: 1) Which aspects in the design and implementation of the tasks afford students opportunities to explore the relationship between different base numbers and the transition to successively larger and smaller number units? 2) What in the students' reasoning and work with specially designed tasks can be seen as signs of theoretical thinking about the base system? Learning study has been used as a research framework and generated data for the two articles. The learning study was carried out in year 4, together with three mathematics teachers. Design and analysis are inspired by El’konin’s and Davydov’s mathematical program and some principles from learning activity. The theoretical principles for learning activity were used as tools for analysis. Data consists of video recordings from three lessons (in total approximately 420 minutes), transcriptions and data compilations of tasks, transcriptions, and notes from the evaluations in the iterative work with the lessons. The results consist of three aspects that need to be made visible through the design and implementation of the tasks so that the students can work theoretically with the structure of the base system: (1) The base number, (2) Number as a measurement number, and (3) The representations of the number units. The students need to distinguish the aspects in order to discern that a bigger or smaller number unit is missing and to test and reflect on the relationship between the base number and the transition to successively larger and smaller number units, constructed in different bases. The result also includes several examples of signs of emerging theoretical thinking within three identified categories: 1) the bases function to the digits´ value in the number, 2) position switching, and (3) the unit as one of a quantity. In the analysis, Davydov's definition of theoretical thinking, as something that can be expressed in the form of theoretical reflection, analysis, and planning as well as the reproduction of fundamental principles for specific subject content, has been used. The results contribute to mathematics didactic research and to special educational research with a focus on mathematics through descriptions of the three aspects and signs of theoretical thinking. Further the descriptions of how tasks can be designed and carried out be used in teaching and continued research. The discussion deals with how teaching can be designed that afford student opportunities to develop theoretical thinking about the base system. The discussion also deals with what in a teaching designed according to the principles of learning activity can be understood as preventive in a special pedagogical perspective, by creating opportunities for students to collectively explore and understand the entire position system as a structure already early in elementary school. Finally, implications for special education and the contribution of the study are discussed. / <p>Sammanläggningen i Pdf innehåller båda artiklarna. </p>