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Promoting deep understanding of fraction concepts continues to be a challenge for mathematics education. Research has demonstrated that students whose concept of fractions is limited to part-whole have difficulty with advanced fraction concepts. We conducted teaching experiments to study how students can develop a measurement concept of fractions and how task sequences can be developed to promote the necessary abstractions. Building particularly on the work of Steffe and colleagues and aspects of the Elkonin-Davydov curriculum, we focused on fostering student reinvention of a measurement concept of fractions. As a study of the Learning Through Activity research program, our goal was to promote particular activity on the part of the students through which they could abstract the necessary concepts. Charalambous and Pitta-Pantazi (2005) articulated a conclusion of many mathematics education researchers, "Fractions are among the most complex mathematical concepts that children encounter in their years in primary education" (p. 233). In this article, we report on a teaching experiment focused on promoting an important concept of fraction, fraction-as-measure.

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... However, there is relatively little research that has examined elementary students' understanding of fractions from a measurement approach (Davydov & Tsvetkovich, 1991), which examined the ratio between units and quantities. In addition, there is much less research that has investigated elementary students' understanding of fractions as measure within a technological environment (Simon, Placa, Avitzur, & Kara, 2018). ...

... Both whole numbers (the case of length A) and fractions (the case of length B) have the same structure of generating numbers as a result of the measurement. The incorporation of a measurement approach to fraction learning has been investigated sporadically in developing instructional materials (Dougherty, 2008), reflecting current approaches to introduce fractions (Kang & Ko, 2003), identifying learning trajectory through teaching experiments (Simon et al., 2018), implementing to classroom settings (Schmittau & Morris, 2004), and comparing international textbooks (Alajmi, 2012). ...

... Although hands-on manipulatives (e.g., Cuisenaire rods) can be accessible in the classroom and familiar with children to explore numerical concept development, such dynamic features can be only implemented in a digital environment. The Dynamic Ruler tasks were stemmed from prior research on the teaching and learning of fractions from a measurement approach (Dougherty, 2008;Simon et al., 2018) and have been refined after the pilot studies. Although Simon and his colleagues (2018) employed the JavaBars tool to explore children's fraction as measure concept, I devised a new tool for two reasons. ...

This study investigates students' conceptions of fractions from a measurement approach while providing a technological environment designed to support students' understanding of the relationships between quantities and adjustable units. 13 third-graders participated in this study and they were involved in a series of measurement tasks through task-based interviews. The tasks were devised to investigate the relationship between units and quantity through manipulations. Screencasting videos were collected including verbal explanations and manipulations. Drawing upon the theory of semiotic mediation, students' constructed concepts during interviews were coded as mathematical words and visual mediators to identify conceptual profiles using a fine-grained analysis. Two students changed their strategies to solve the tasks were selected as a representative case of the two profiles: from guessing to recursive partitioning; from using random units to making a relation to the given unit. Dragging mathematical objects plays a critical role to mediate and formulate fraction understandings such as unitizing and partitioning. In addition, static and dynamic representations influence the development of unit concepts in measurement situations. The findings will contribute to the field' s understanding of how students come to understand the concept of fraction as measure and the role of technology, which result in a theory-driven, empirically-tested set of tasks that can be used to introduce fractions as an alternative way.

... Transitioning from whole number concepts to rational numbers traditionally poses a considerable challenge to the mathematically developing mind and may become a stumbling block in the way of maturation in number conceptualisation (Durkin & Ritle-Johnson, 2014;Simon, Placa, Avitzur, & Kara, 2018). ...

... Experience with whole numbers makes the transition to rational numbers somewhat abstract (Simon et al., 2018). Bruce, Bennett and Flynn (2014) explain that fractions are multiple digits (numerator and denominator) that represent one quantity, making different interpretations of fractions possible. ...

... Conceptual understanding in the context of this article refers not to isolated facts, but to the ability of learners to make meaningful connections between fractional elements such as the numerator and the denominator. For example, in their studies, Deringöl (2019) and Simon et al. (2018) found that learners faced challenges in terms of viewing the numerator and the denominator as representing a whole together. Learners tend to see a fraction as a pair of numbers representing quantities with no relationship implied between those quantities (Stafylidou & Vosniadou, 2004). ...

... For example, in a study by Castro-Rodríguez et al. (2016), 98% of the participating prospective teachers (N = 358) used the area model to demonstrate the meaning they provide to fractions, where almost all these responses involved illustrative figures of circles and rectangles. This tendency to use the part-whole interpretation could be partly explained when considering that textbooks typically introduce the concept of fractions with circle or rectangle part-whole representations (Pantziara & Philippou, 2012;Simon, Placa, Avitzur, & Kara, 2018). However, while the part-whole conceptualisation is fundamental to the understanding of fractions, overreliance on this model may be a learning obstacle when being required to work with other models, and can inhibit the conceptualisation of improper fractions when learners find it impossible to have more parts than the whole (Simon et al., 2018;Stafylidou & Vosniadou, 2004). ...

... This tendency to use the part-whole interpretation could be partly explained when considering that textbooks typically introduce the concept of fractions with circle or rectangle part-whole representations (Pantziara & Philippou, 2012;Simon, Placa, Avitzur, & Kara, 2018). However, while the part-whole conceptualisation is fundamental to the understanding of fractions, overreliance on this model may be a learning obstacle when being required to work with other models, and can inhibit the conceptualisation of improper fractions when learners find it impossible to have more parts than the whole (Simon et al., 2018;Stafylidou & Vosniadou, 2004). ...

... With respect to fractions, since the area model is the most frequently used model in the introductory teaching of fractions (Simon et al., 2018), we argue that it serves as the most prevalentor evoked concept image (Tall & Vinner, 1981)associated with the concept of fractions. However, prevalent examples of a concept image may become an obstacle for learning, particularly when a more abstract, or simply a different, image is introduced. ...

We explore prospective teachers’ attempts to explain atypical fraction representations, particularly when non-integers appear in the numerator or denominator. As practical implication, we advocate the use of the set model to interpret atypical representations and expand learners’ understanding of fractions.

... However, instead of being of a geometric type, the content underlying our study is of an arithmetic nature, specifically on the rational number. We chose this mathematical object because it is important for the curriculum (Real Decreto 126/2014), and to know whether the teaching of this object to our preservice teachers when in elementary and high school might have limited them when making explanations from its different interpretations, as previous studies have already detected (Clarke et al., 2008;Escolano & Gairín, 2005;Freudenthal, 1983;Gairín, 2001;Martínez-Juste et al., 2017;Olive & Vomvoridi, 2006;Simon et al., 2018;Shield & Dole, 2013). ...

... Of all the previous interpretations, the most frequent in Spanish textbooks is the interpretation of the part-whole (Gairín & Muñoz, 2005;Olive & Vomvoridi, 2006;Simon et al., 2018), despite the disadvantages this poses for students. Indeed, Freudenthal (1983) analyses, both phenomenologically and mathematically, the limitations posed by the exclusive adoption of partwhole interpretation in teaching, which implies, among other disadvantages, a mechanical learning of algorithms and difficulties in understanding the improper fraction, since, in part-whole interpretation, the quantity of magnitude is both the total and the unit. ...

... The most common error is what we call "confusion of the part with the whole" (items a and c), possibly caused by one of the disadvantages of traditional teaching, generally limited to part-whole interpretation, in which the student does not need to recognise the unit because it coincides with the total quantity of magnitude (Gairín, 2001;Gairín & Muñoz, 2005). Simon et al. (2018) also highlight this and other difficulties in some advanced concepts of fractions when teaching is limited exclusively to part-whole interpretation, and how they can be mitigated via sequences of tasks in interpretation as a measure, as Olive and Vomvoridi (2006) also assure. On the other hand, the error "incorrect application of the operator" (item c) could be influenced by the false conception that multiplication increases the quantity, as stated by Clarke et al. (2008). ...

... The theoretical model is presented as shown in Figure 1 below. Behr, et.al (1983) Source: (Charalambous & Pitta-Pantazi, 2005) Part of the whole is the interpretation of fraction that starts learning about fraction (Simon, Placa, Avitzur, & Kara, 2018). Some experts justify that this happens because part of the whole is considered the basis for developing an understanding of the other four subcontracts (Charalambous & Pitta-Pantazi, 2005). ...

... Fraction is the distance from point 0 to a certain point obtained by partitioning a unit into several subunits (Chapin & Johnson, 2006;Lamon, 2012). In this interpretation, # $ and % $ have meanings that are independent of the whole (Simon et al., 2018). Interpretation of fractions as measures provides an alternative way of introducing fractions. ...

... Interpretation of fractions as measures provides an alternative way of introducing fractions. This has been developed in the Japan text series, which focuses on fraction learning not with model areas or discrete models, but with measurements of fluid length or volume (Simon et al., 2018). ...

... For example, in a study by Castro-Rodríguez et al. (2016), 98% of the participating prospective teachers (N = 358) used the area model to demonstrate the meaning they provide to fractions, where almost all these responses involved illustrative figures of circles and rectangles. This tendency to use the part-whole interpretation could be partly explained when considering that textbooks typically introduce the concept of fractions with circle or rectangle part-whole representations (Pantziara & Philippou, 2012;Simon, Placa, Avitzur, & Kara, 2018). However, while the part-whole conceptualisation is fundamental to the understanding of fractions, overreliance on this model may be a learning obstacle when being required to work with other models, and can inhibit the conceptualisation of improper fractions when learners find it impossible to have more parts than the whole (Simon et al., 2018;Stafylidou & Vosniadou, 2004). ...

... This tendency to use the part-whole interpretation could be partly explained when considering that textbooks typically introduce the concept of fractions with circle or rectangle part-whole representations (Pantziara & Philippou, 2012;Simon, Placa, Avitzur, & Kara, 2018). However, while the part-whole conceptualisation is fundamental to the understanding of fractions, overreliance on this model may be a learning obstacle when being required to work with other models, and can inhibit the conceptualisation of improper fractions when learners find it impossible to have more parts than the whole (Simon et al., 2018;Stafylidou & Vosniadou, 2004). ...

... With respect to fractions, since the area model is the most frequently used model in the introductory teaching of fractions (Simon et al., 2018), we argue that it serves as the most prevalentor evoked concept image (Tall & Vinner, 1981)associated with the concept of fractions. However, prevalent examples of a concept image may become an obstacle for learning, particularly when a more abstract, or simply a different, image is introduced. ...

We explore prospective elementary-school teachers’ attempts to provide signs and symbols with mathematical meaning in a case involving non-integers as numerator or denominator. The data comprises 33 responses to a task inquiring about the existence of numbers between 1/6 and 1/7, in which the participants were asked to compose a hypothetical classroom dialogue addressing this issue, and provide explanations on the mathematics involved. The analysis utilised the theoretical frameworks of “semiotic representation” and “concept image”. The findings indicate a strong reliance on prevalent fraction images and difficulties in assigning meaning to “6½” when it appears in the numerator or denominator of a fraction representation. These difficulties are examined via three approaches demonstrated in the scripts, relating to the symbolic representation of fractions, visual representation of fractions, and permissible operations on fractions. We suggest teacher-education courses may put further emphasis on the set model of fractions and atypical semiotic representations.

... In the case of fractions, this cross-national analysis is even more important as the introduction of fractions at earlier grades have implications for developing a more robust understanding of fractions at the later grades (Simon et al., 2018). For example, as Simon et al. (2018) have argued, introducing fractions as "part of a whole" is limiting and may hinder students in developing more "powerful conceptions of fractions" (p. ...

... In the case of fractions, this cross-national analysis is even more important as the introduction of fractions at earlier grades have implications for developing a more robust understanding of fractions at the later grades (Simon et al., 2018). For example, as Simon et al. (2018) have argued, introducing fractions as "part of a whole" is limiting and may hinder students in developing more "powerful conceptions of fractions" (p. 123). ...

... Among these representations, the part-whole construct is the most common conceptualization encountered in mathematics textbooks when fractions are first introduced. As highlighted by Simon et al. (2018), "fractions are typically introduced with circle or rectangle representations in which a fraction 1/n is one of n identical parts of the given figure" (p. 123). ...

This exploratory study focuses on analyzing three mathematics textbooks in Germany, Singapore and South Korea to reveal similarities and differences in their introductions of fraction concepts. Findings reveal that all three countries' textbooks introduce fraction concepts predominantly by using pictorial representations such as area models, but the introductions of multiple fraction constructs vary. The Singaporean and South Korean textbooks predominantly used a part-whole construct to introduce fractional concepts while the German textbook introduced various constructs sequentially in the first pages using several scenarios from different real-life situations. The findings were represented using visual representations, which we called textbook signatures. The textbook signatures provided configurations of the textbook features across the three countries. At the end of paper, we share insights and limitations about the use of textbook signatures in the research on textbook analysis.

... Las fracciones, como objeto matemático, se han destacado por ser un tema de investigación y enseñanza privilegiado en la didáctica de la matemática. Una comprensión profunda de los conceptos de fracción sigue siendo un desafío para la educación matemática (Simon, Placa, Avitzur & Kara, 2018). Las fracciones pueden verse como un mega-concepto que contiene una serie de significados e interpretaciones, que se admiten y se adquieren según el contexto en que se emplean (Malet, 2010). ...

... Las fracciones pueden verse como un mega-concepto que contiene una serie de significados e interpretaciones, que se admiten y se adquieren según el contexto en que se emplean (Malet, 2010). Por lo tanto, el concepto de fracción no se puede reducir únicamente a la exploración de sistemas concretos y tampoco se puede asociar exclusivamente a la relación parte-todo (Simon et al., 2018). El estudio de la fracción debe extenderse a otras magnitudes, de tal manera que aporte nuevos significados (Gairín y Escolano, 2015). ...

... En la investigación desarrollada por Simon et al. (2018) señalan cuatro tipos de limitaciones sobre la comprensión de las fracciones en edad escolar identificadas en diversos estudios. La primera de ellas está relacionada con la "ausencia de la fracción como cantidad", es decir, los estudiantes a menudo tienen la concepción de la fracción como un arreglo en la que m/n (m < n) representa un arreglo en el que un todo está dispuesto en "n" partes idénticas y "m" de esas partes están designadas. ...

Marín-Grisales, J. P. y Osorio-Cárdenas, A. M. (2020). Caracterización de los tratamientos en la solución de situaciones sobre la fracción como medida. Revista Latinoamericana de Estudios Educativos, 16 (2), 184-208. Resumen Este artículo presenta los resultados obtenidos en la investigación: caracterización de tratamientos que emplean docentes de básica primaria en la solución de situaciones relacionadas con la fracción como medida. El objetivo de esta investigación fue describir procedimientos, registros de representación y transformaciones de tratamiento en situaciones de la fracción como medida. La investigación se realizó bajo un enfoque cualitativo de tipo descriptivo a partir de un estudio de caso, empleando el cuestionario escrito con preguntas abiertas y la entrevista no estrucuturada. Para el análisis de los datos se empleó la codificación axial, abierta y selectiva de las respuestas de los docentes, obtenidas en los cuestionarios y las entrevistas. Se evidencia en los resultados del estudio el rol que cumple el tratamiento en la asignación, conservación y modificación de sentidos a través del uso de modificaciones intrínsecas, mereológicas y posicionales enmarcadas en los procesos de visualización de los docentes. Palabras clave: aprendizaje de la fracción, fracción como medida, representaciones semióticas de objetos matemáticos, docente en servicio, visualización.

... Clarke, Roche, and Mitchell (2011) indicate that the part-whole model is insufficient to provide a good foundation for fraction understanding, as does Kerslake (1986), whose research with 12-14 year-olds suggests that the part-whole models used alone prompts learners to think of fractions as shaded parts over a total number of pieces in the whole unit. Simon, Placa, Avitzur, and Kara (2018) identify four conceptual limitations commonly encountered when learning fractions: seeing a fraction as a visual arrangement, rather than a quantity; seeing a fraction as two unrelated numbers; seeing a fraction as constructed from parts of a single whole; and having limited understanding of the reference unit. A fundamental implication of the third limitation is that partitioning a single whole generates a limited number of parts from which the fraction quantity may be constructed, leading to the learner being only able to generate proper fractions and finding it difficult to make sense of fractions greater than 1. ...

... One possibility for such iteration is to construct fraction quantities from multiple wholes, which is the approach taken in our research. In the case of working with collections of multiple wholes, the fourth limitation identified by Simon et al. (2018) implies that learners may find it difficult to distinguish the reference unit (the whole) from the totality of objects that can be seen in the collection, an observation supported by Larson (1987) when using the number line as a model for learning fractions. ...

... We use the term 'unit' to denote a quantity generated by an even subdivision of the whole, which may be used as the measurement unit to represent rational quantities as fractions. Thus, a fraction may be interpreted as a measure and the process of generating fraction representations for quantities regarded as a process of measurement (Simon et al., 2018). Finally, the term 'fractioning' is used as a general term to describe the process of subdividing a whole (evenly or unevenly). ...

Learning rational number concepts is acknowledged as an important task but many learners find it difficult to make sense of them. This paper reports on a case study of the learning in a short (nine-lesson) learning programme for Grade 8 learners in a Namibian school, which sought to use visual models (circle area, bar area and number line) to deepen learners’ understanding of fractions as a means to represent rational quantities. The initial benchmark test indicated a number of ways of working with fraction representations, many of which were inappropriate to the rational quantity presented. Although most learners were able to use a fraction to appropriately describe a part–whole area diagram with a single whole, few were able to appropriately label a similar diagram with multiple wholes, or a quantity greater than 1 on the number line. In the learning programme learners worked with visual models that incorporated multiple reference wholes, to explicitly identify the reference whole, to quantify the size of appropriately subdivided units using unit fraction names and to use these units in a measurement process to quantify the quantities these models indicated. A final test showed a sound conceptualisation of the use of fractions to represent rational quantities less than and greater than 1, in such models.

... Transitioning from whole number concepts to rational numbers traditionally poses a considerable challenge to the mathematically developing mind and may become a stumbling block in the way of maturation in number conceptualisation (Durkin & Ritle-Johnson, 2014;Simon, Placa, Avitzur, & Kara, 2018). ...

... Experience with whole numbers makes the transition to rational numbers somewhat abstract (Simon et al., 2018). Bruce, Bennett and Flynn (2014) explain that fractions are multiple digits (numerator and denominator) that represent one quantity, making different interpretations of fractions possible. ...

... Conceptual understanding in the context of this article refers not to isolated facts, but to the ability of learners to make meaningful connections between fractional elements such as the numerator and the denominator. For example, in their studies, Deringöl (2019) and Simon et al. (2018) found that learners faced challenges in terms of viewing the numerator and the denominator as representing a whole together. Learners tend to see a fraction as a pair of numbers representing quantities with no relationship implied between those quantities (Stafylidou & Vosniadou, 2004). ...

... Transitioning from whole number concepts to rational numbers traditionally poses a considerable challenge to the mathematically developing mind and may become a stumbling block in the way of maturation in number conceptualisation (Durkin & Ritle-Johnson, 2014;Simon, Placa, Avitzur, & Kara, 2018). ...

... Experience with whole numbers makes the transition to rational numbers somewhat abstract (Simon et al., 2018). Bruce, Bennett and Flynn (2014) explain that fractions are multiple digits (numerator and denominator) that represent one quantity, making different interpretations of fractions possible. ...

... Conceptual understanding in the context of this article refers not to isolated facts, but to the ability of learners to make meaningful connections between fractional elements such as the numerator and the denominator. For example, in their studies, Deringöl (2019) and Simon et al. (2018) found that learners faced challenges in terms of viewing the numerator and the denominator as representing a whole together. Learners tend to see a fraction as a pair of numbers representing quantities with no relationship implied between those quantities (Stafylidou & Vosniadou, 2004). ...

Our research with Grade 9 learners at a school in Soweto was conducted to explore learners’ understanding of fundamental fraction concepts used in applications required at that level of schooling. The study was based on the theory of constructivism in a bid to understand whether learners’ transition from whole numbers to rational numbers enabled them to deal with the more complex concept of fractions. A qualitative case study approach was followed. A test was administered to 40 learners. Based on their written responses, eight learners were purposefully selected for an interview. The findings revealed that learners’ definitions of fraction were neither complete nor precise. Particularly pertinent were challenges related to the concept of equivalent fractions that include fraction elements, namely the numerator and denominator in the phase of rational number. These gaps in understanding may have originated in the early stages of schooling when learners first conceptualised fractions during the late concrete learning phase. For this reason, we suggest a developmental intervention using physical manipulatives to promote understanding of fractions before inductively guiding learners to construct algorithms and transition to the more abstract applications of fractions required in Grade 9.

... Researchers have explored alternative instructional approaches to develop a fundamental understanding of fractions as representing single quantities (Clarke & Roche, 2009;Empson, 2003). One alternative is based on the concepts of measurement units and unit relations (Davydov & Tsvetkovich, 1991;Simon et al., 2018), drawing not only on children's experience and the meaning of fractions but also on the historical development of fractions through measurement situations (Davydov & Tsvetkovich, 1991). This measurement interpretation emphasizes the relationship between units of measure and the magnitude of quantities (Vysotskaya et al., 2021); that is, a measure of a quantity is the ratio between the size of the quantity and the size of a unit (Brousseau et al., 2014). ...

... Drawing upon a measurement approach to fractions (Davydov & Tsvetkovich, 1991), researchers developed a set of learning activities through teaching experiments (Simon et al., 2018) and classroom studies (Schmittau & Morris, 2004). The activities introduced the concept of numbers in the context of measurement by using continuous quantities (e.g., water, paper strips). ...

... The activities introduced the concept of numbers in the context of measurement by using continuous quantities (e.g., water, paper strips). For example, Simon et al. (2018) designed a series of tasks in which students were asked to measure a beam with multiple given length units. This teaching experiment provided evidence of children developing flexible fraction conceptions through measurement tasks without encountering the difficulties that often accompany typical fraction instruction. ...

In this study, we examine students’ mathematical reasoning within a technological environment designed to support understanding of relationships between quantities with adjustable measuring units. In particular, we provide a cross-sectional snapshot of how 30 elementary students (Grades 3–5) engaged in a series of fraction-as-measurement tasks using a “Dynamic Ruler” that could continuously dilate unit sizes. Screencast recordings were collected from a task-based clinical interview and analyzed to investigate children’s mathematical actions and mathematical ideas. Students’ reasoning patterns were characterized using four distinct types (low attending, holistic estimating, determining, and commeasuring) based on their solution strategies. Findings suggest that the Dynamic Ruler tool can elicit rich conceptions of fractions and even prompt novel approaches such as commeasurement. We conclude by drawing insights into how elementary students might use dynamic technology meaningfully.

... al., 2020). However, some researchers criticize it(Simon et al., 2018;Watanabe, 2006;Wilkins & Norton, 2018), and in fact, the curriculum is more focused and only limited to part-whole subconstruct(Rahmawati et al., 2020).Simon et al. (2018) revealed that fractions as measures are more effective than part-whole and provides several benefits. However, studies also explain that students encounter obstacles using number lines(Charalambous & Pitta-Pantazi, 2007;Izsák et al., 2008). ...

... al., 2020). However, some researchers criticize it(Simon et al., 2018;Watanabe, 2006;Wilkins & Norton, 2018), and in fact, the curriculum is more focused and only limited to part-whole subconstruct(Rahmawati et al., 2020).Simon et al. (2018) revealed that fractions as measures are more effective than part-whole and provides several benefits. However, studies also explain that students encounter obstacles using number lines(Charalambous & Pitta-Pantazi, 2007;Izsák et al., 2008). ...

Learning emphasizing fractions as a part-whole concept causes several limitations in developing fraction knowledge and inhibits proportional reasoning. We use fractions as quotients as the first context introduced in our learning trajectory. We report the teaching experiment results using the improved learning trajectory on thirty 4th grade students in Jakarta's public schools. The findings of this study indicate that the fractions as quotients used as the first stage in the learning trajectory can lay a solid foundation for the concept of partitioning in a variety of strategies and the concept of fractional parts. Besides, the developed learning trajectory has provided opportunities for students to learn about fractional mental operations, which are interrelated and serve as the basis for the development of proportional reasoning.

... ). Furthermore, we note that the most predominant conceptualization of fractions by learners adheres to the part-whole area model involving equal parts in a circle, which is most likely due to how learners typically encounter fractions in school for the first time (Pantziara & Philippou, 2012;Simon et al., 2018). Accordingly, as formal mathematical definitions are not typically presented in elementary school, we suggest that this prevalent model serves as an in-action definition (Ouvrier-Buffet, 2011) for fractions, that is, as a tool that "enables students to be operational without explicit definition" (p. ...

... The predominant reliance of learners on the part-whole model of fractions has been repeatedly noted in prior research, while explaining this is likely due to how students typically encounter fractions in school for the first time (Simon et al., 2018). Accordingly, in scripts such as the one above, it seems that the option of considering 1 6.5 as a fraction was rejected based on not being able to convert the semiotic representation of 1 6.5 in the symbolic register to a corresponding representation in the visual register, which adheres to the part-whole model involving equal pieces in a circle. ...

We explore the responses of 26 prospective elementary-school teachers to the claim “1/6.5 is not a fraction” asserted by a hypothetical classroom student. The data comprise scripted dialogues that depict how the participants envisioned a classroom discussion of this claim to evolve, as well as an accompanying commentary where they described their personal understanding of the notion of a fraction. The analysis is presented from the perspective of productive ambiguity, where different types of ambiguity highlight the prospective teachers’ mathematical interpretations and pedagogical choices. In particular, we focus on the ambiguity inherent in the aforementioned unconventional representation and how the teachers reconciled it by invoking various models and interpretations of a fraction. We conclude with a description of how the perspective of productive ambiguity can enrich teacher education and classroom discourse.

... There is no doubt that experiencing the use of fractions along with their decimal and percent homologues in diverse situations will deepen students' understanding of rational numbers as they progress through their learning of mathematics. For example, introducing rational numbers in measurement contexts rather than with slices of pizza has been shown to be more effective in developing students' understanding of fractions (Simon, Placa, Avitzur, & Kara, 2018). However, yet again, using measurement as a context for introducing rational numbers does not serve to define the numbers. ...

... There is no doubt that experiencing the use of fractions along with their decimal and percent homologues in diverse situations will deepen students' understanding of rational numbers as they progress through their learning of mathematics. For example, introducing rational numbers in measurement contexts rather than with slices of pizza has been shown to be more effective in developing students' understanding of fractions (Simon, Placa, Avitzur, & Kara, 2018). However, yet again, using measurement as a context for introducing rational numbers does not serve to define the numbers. ...

The aim of this article is to provide a unified framework for the ratio concept which in turn provides an alternative foundation for understanding rational numbers. In order to achieve this, a new perspective on counting is presented. Creating a conceptual hub that houses all the real numbers helps connect as well as distinguish their diverse natures and purposes. In particular, given the extent of the difficulties learners have with rational numbers, the classification proposed here may support teachers in demystifying the nature of fractions, decimal numbers, and percents by situating them in the larger scheme of sizing amounts. Indeed, the proposed co-counting structure may provide a simple yet comprehensive foundation for proportional reasoning.

... 2013). Menurut Stafylidou dan Vosniadou (2004), murid melihat pecahan sebagai sepasang nombor yang mewakili kuantiti tertentu tanpa memahami bahawa wujudnya hubungan antara keduadua kuantiti tersebut. Miskonsepsi terhadap konsep pecahan yang abstrak ini telah menyebabkan murid tidak mempunyai pemahaman dan pengetahuan konseptual pecahan yang baik (Simon et. al., 2018;Zakiah et. al., 2013). Penggunaan Minecraft dilihat berpotensi dalam mengukuhkan pemahaman dan pengetahuan konseptual pecahan disebabkan Minecraft membolehkan aktiviti manipulasi secara hands-on dilakukan ke atas blok 3D dalam dunia maya Minecraft terutama semasa melakukan penyelesaian masalah pecahan. Blok 3D dalam dunia Minecraft dapat ...

Keupayaan untuk memahami dan menguasai konsep pecahan akan menyediakan asas yang kukuh untuk memahami konsep matematik yang lebih kompleks serta membantu dalam melakukan penyelesaian masalah yang melibatkan pecahan dalam kehidupan seharian. Namun demikian, dapatan pentaksiran antarabangsa Trends in Mathematics and Science Studies (TIMSS) dan Programme for International Student Assessment (PISA) menunjukkan bahawa pencapaian pecahan di kalangan murid di Malaysia adalah di tahap kurang memuaskan iaitu berada di bawah skor purata antarabangsa. Dengan mengambil kira perkembangan inovasi dan kecanggihan permainan digital masa kini, tujuan kajian ini adalah untuk mengkaji kesan pembelajaran berasaskan permainan digital menggunakan Minecraft terhadap peningkatan pencapaian murid dalam pecahan. Kuasi-eksperimen dengan reka bentuk ujian pra dan ujian pasca kumpulan rawatan dan kumpulan kawalan yang melibatkan 65 orang murid tahun lima dalam dua buah kelas sedia ada telah dijalankan. Melalui pensampelan kluster, sebuah kelas terpilih sebagai kumpulan rawatan manakala kelas yang satu lagi terpilih sebagai kumpulan kawalan. Kumpulan rawatan terdiri daripada 31orang murid yang didedahkan dengan kaedah pembelajaran berasaskan permainan digital menggunakan Minecraft manakala kumpulan kawalan terdiri daripada 34 orang murid yang didedahkan dengan kaedah konvensional. Data dianalisis menggunakan ujian-t dua kumpulan sampel tak bersandar bagi membandingkan min skor pencapaian ujian pasca bagi pecahan antara kumpulan rawatan dengan kumpulan kawalan. Terdapat perbezaan signifikan dalam min skor pencapaian antara kumpulan rawatan (m = 51.096, sp = 17.242) dengan kumpulan kawalan (m = 35.235, sp = 18.171). Hasil kajian ini akan lebih menggalakkan pelaksanaan pembelajaran berasaskan permainan digital menggunakan Minecraft dalam pengajaran matematik bagi meningkatkan pencapaian murid dalam pecahan.

... Korea has been documented to differ substantially from the US in students' mathematics achievements (OECD 2014;Mullis et al. 2012), fraction naming system (Miura et al. 1999), and pedagogical arrangement and emphasis on the interchangeability of rational numbers (Lee et al. 2016). Vast differences have also been observed between Russia and the US; Russian math curriculum places dramatically greater emphasis on abstraction and measure (Simon et al. 2018;Tyumeneva et al. 2018). Despite these dramatic differences, Korean and Russian college students exhibited a similar pattern of semantic alignment with their US counterparts (Lee et al. 2016;Tyumeneva et al. 2018). ...

Rational numbers can be represented in multiple formats (e.g., fractions, decimals, and percentages), and a rational number notation can be used to express different concepts in different contexts. The present study investigated the distribution of the multiple concepts expressed by these different rational number notations in real-world contexts as well as the semantic alignment between entity type (discrete vs. continuous) and rational number format (decimal vs. fraction) in the Chinese context. Textbook analysis and two paper-and-pencil experiments yielded the following four major findings: (1) Decimals were more likely used to represent numerical magnitudes, while fractions were more likely used to represent relations between two numerical magnitudes. (2) Decimals were more often used to represent continuous entities while fractions were preferred to represent discrete entities. (3) The strength of the association between different formats of rational numbers and their preferred conceptual meanings seemed more pronounced than the semantic alignment between number type and entity type. (4) Percentages were used in a way more similar to fractions than decimals in terms of the concepts they express. These findings indicate that different formats of rational numbers differ dramatically in their use in real-world contexts both in terms of the conceptual meanings they express and the entities they model. Educational implications of this study are discussed.

... On the other hand, difficulties with learning proportionality-based concepts might be connected to the way in which students are taught. Researchers suggest various explanations: whole number dominance (Behr, Wachsmuth, Post, & Lesh, 1984) or natural number bias (Ni & Zhou, 2005), part-whole relationships as a starting context (Simon, Placa, Avitzur, & Kara, 2018), or equipartition as a didactical obstacle (Cortina, Visnovska, & Zuniga, 2014). Researchers conclude that the main problem in learning such concepts is the differentiation of the attributes that form each ratio, and their integration into an appropriate structure (e.g., Siegler & Chen, 2008;Smith, Snir, & Grosslight, 1992). ...

We are developing an approach to teaching important proportionality-based concepts to first grade students in a way that supports students’ future progress in the domain. We consider the proportionality between magnitudes as a basic relationship behind multiple cases, usually described mathematically as ratio or rate. The core of our strategy is the modelling of a situation of proportionality and its transformations by creating a compound unit. The key action is coordinated measurement (co-measurement): students work in pairs, and each student is in charge of changing one of two magnitudes, while preserving proportionality. Based on our successful experiments in Grades 2–6, we incorporate shared responsibility work organization (“joint actions”) and the idea of compound unit into Davydov’s mathematics curriculum for the first grade. We built a new module based on Davydov’s idea about the role of rule-mediated counting by sets. In this paper, we present the results of our study, showing that first graders can learn the idea of compound unit and work with two magnitudes of different kinds while preserving proportionality.

... Fractions play an important role in the teaching and learning of mathematics (Siegler et al. 2013). However, Simon et al. (2018) have indicated that promoting a deep understanding of the concept of fractions continues to be a challenge for both teachers and learners. Moreover, learners who have limited foundational knowledge concerning solving problems involving concepts of fractions have challenges with the understanding of advanced concepts within fractions (Loc, Tong & Chau 2017). ...

Background: Within the ambit of the Fourth Industrial Revolution (4IR), the use of technology-based tools within teaching and learning is advancing rapidly at education institutions globally, including the teaching and learning of mathematics. Learners and teachers have challenges with teaching and learning fractions in mathematics. A learner’s understanding of fractions is fundamental for the understanding of key concepts in other mathematics sections.
Aim: This qualitative, interpretive study examined the perceptions of Grade 5 learners about the use of technology-based tools, more specifically videos and PowerPoint presentations when learning fractions in mathematics.
Setting: This study was located at one primary school in KwaZulu-Natal, South Africa.
Methods: The study was framed within the ambit of social constructivism, and data were generated via task-based worksheets, interactive technology-based lessons and focus group interviews.
Results: Based on the results of this study, it was evident that the participants valued the use of the technology-based tools during the teaching and learning of fractions. Based on an interpretive analysis of the data generated, two major themes emerged. Participants indicated that using videos and PowerPoint presentations inspired an appealing and fun way of learning fractions and inspired an encouraging atmosphere for learning fractions. These results may be of value to teachers, teacher educators, researchers, curriculum developers and learners of mathematics.
Conclusion: The concluding comments of this article mention research implications and recommendations for further research within this area. These recommendations are significant as there is a need for educational institutions globally to embrace the 4IR within teaching and learning.

... Öğretmenlerin derslerde kesirlerin diğer kullanım anlamlarını da öğrencilerine öğretmesi bu açıdan faydalı olabilir. Bununla birlikte parça-bütün anlamını kullanma eğilimi, ders kitaplarının kesir kavramını tipik olarak daire veya dikdörtgen parça-bütün temsilleriyle tanıtmasından da kaynaklanıyor olabilir (Pantziara ve Philippou, 2012;Simon, Placa, Avitzur ve Kara, 2018). Bu durumun sebeplerinin daha fazla araştırılması gerekmektedir. ...

This study aimed to examine fourth-grade students' mental structures related to the concept of fractions. Twelve fourth-grade students participated in the study, which was conducted in the case study model. The data were collected through triangulation to ensure that the mental structures of the students were examined in depth. Data collection tools are mind maps, fraction concept image test, and fraction modeling test. The data were analyzed using content analysis and the rubric developed in the study. The analysis results of mind maps showed that students' mental structures related to the concept of fractions were gathered mostly under the themes of fraction types, fraction parts, and fraction meanings. The other themes of mind maps are number operations, mathematical notation, mathematics, and modeled objects. With the analysis of the fraction concept image tests, it was revealed that the students had part-whole, quotient, and ratio fraction concept images. However, it was determined that they never used fractions in terms of operators and measures. In addition, it was found that the students used the area model in modeling all of the fraction problems, but they never used the number line and cluster model. This result showed that students associate fractions more with the area model in their minds. It was also seen that the students' fraction modeling skills are at a medium level. The results are discussed in the light of the related studies.

... However, instead of focusing on whether mathematics education should start with fractions or algebra, we would like to conclude that a task design allowing interplay between the two mathematical areas would be fruitful. Such a conclusion is supported by previous studies (e.g., Davydov, 2008;Izsák & Beckman, 2019;Schmittau, 2011;Simon et al., 2018;Venenciano & Heck, 2016). What we can offer, compared with previous research, is further description and more detailed analysis of the arguments, allowing us to highlight not only the types and aspects of algebraic and fractional thinking that are in focus but also patterns evident within the mathematical reasoning. ...

This study examines the collective mathematical reasoning when students and teachers in grades 3, 4, and 5 explore fractions derived from length comparisons, in a task inspired by the El´konin and Davydov curriculum. The analysis showed that the mathematical reasoning was mainly anchored in mathematical properties related to fractional or algebraic thinking. Further analysis showed that these arguments were characterised by interplay between fractional and algebraic thinking except in the conclusion stage. In the conclusion and the evaluative arguments, these two types of thinking appeared to be intertwined. Another result is the discovery of a new type of argument, identifying arguments, which deals with the first step in task solving. Here, the different types of arguments, including the identifying arguments, were not initiated only by the teachers but also by the students. This in a multilingual classroom with a large proportion of students newly arrived. Compared to earlier research, this study offers a more detailed analysis of algebraic and fractional thinking including possible patterns within the collective mathematical reasoning. An implication of this is that algebraic and fractional thinking appear to be more intertwined than previous suggested.

... Öğretmenlerin derslerde kesirlerin diğer kullanım anlamlarını da öğrencilerine öğretmesi bu açıdan faydalı olabilir. Bununla birlikte parça-bütün anlamını kullanma eğilimi, ders kitaplarının kesir kavramını tipik olarak daire veya dikdörtgen parça-bütün temsilleriyle tanıtmasından da kaynaklanıyor olabilir (Pantziara ve Philippou, 2012;Simon, Placa, Avitzur ve Kara, 2018). Bu durumun sebeplerinin daha fazla araştırılması gerekmektedir. ...

... Diverse research projects have highlighted the difficulties students have at learning fractions (Lamon, 2020;Simon et al., 2018). These difficulties usually result from a lack of conceptual comprehension, in which fractions are seen by many students as senseless symbols (Fazio & Siegler, 2011). ...

In this work we will analyze the relation between registers of representation and the construction of the fraction concept. Ninety-six students from first year of compulsory secondary education participated in the study, and performed equal share tasks in the context of Egyptian fractions (unit fractions with different denominators). The aim was to determine if, with these types of tasks, students could improve their learning of the different meanings of the fraction concept. Our results indicate that there seems to be a relationship between the meaning used and the representation chosen. Similarly, we found that—with these tasks—students significantly increased the number of registers of representation they used. Students who used distinct representations had to coordinate several registers, which might be interpreted as proof of the development of conceptual understanding.

Our microgenetic research methodology was developed for two purposes: 1) to develop greater understanding of conceptual learning and instructional design that promotes conceptual learning and 2) to develop empirically-based hypothetical learning trajectories for particular mathematical concepts. The challenge in developing the methodology was to afford analysis of the learning process (e.g., the transition), not just identification of conceptual steps through which learners progress. The methodology, developed by the Learning Through Activity research program , was based on constructivist teaching experiment methodology. Modifications were made to data collection for the purposes outlined and a multi-level retrospective analysis was developed .

Promoting an understanding of multiplication of fractions has proved difficult for mathematics educators. I discuss a research study aimed at developing a concept of multiplication that supports both multiplication of whole numbers and multiplication of fractions. The study demonstrates how domain-specific theories grounded in two major psychological theories contribute to the development of an empirically based approach to developing this concept. Specifically, the researchers used Learning Through Activity, grounded in constructivism, and aspects of the Elkonin-Davydov Curriculum, grounded in Russian activity theory (sociocultural theory).

The Measurement Approach to Rational Number (MARN) Project, a project of the ongoing Learning Through Activity (LTA) research program, produced eleven hypothetical learning trajectories (HLTs) for promoting fraction concepts. Four of these HLTs are the subject of research reports. In this article, we present the other seven HLTs We judged that the data and analyses of these seven would not separately make sufficient contributions to merit individual research reports. However, presenting these seven HLTs together was intended to meet the following goals:
1. To give a broad set of examples of HLTs developed based on the LTA theoretical framework.
2. To complete a set of HLTs that provide a comprehensive example of HLTs built on prior HLTs.
3. To make available for future research and development the full set of HLTs generated by the MARN Project.
LTA researchers have focused on how learners abstract a concept through their mathematical activity and how the abstractions can be promoted. The MARN Project continued this inquiry with rigorous single-subject teaching experiments.

We report on a teaching experiment intended to foster a concept of multiplication that would
both subsume students’ multiple-groups concept of whole number multiplication and provide a
conceptual basis for understanding multiplication of fractions. The teaching experiment, which
used a rigorous single-subject methodology, began with an attempt to build on students’ multiple-
groups concept by promoting generalizing assimilation. This was not totally successful and
led to a redesign aimed at promoting reflective abstraction. Analysis of this latter phase led to
several significant conclusions, which in turn led to a revised hypothetical learning trajectory.
The revised trajectory aims to foster a concept of multiplication as a change in units.

This commentary raises and discusses questions based on some of the agreements, disagreements, and themes found in the four chapters on fractions. It considers (1) the importance of tasks that are based in perception and readily available activity in light of an emphasis on problem solving in mathematics education, and the role that theories about thinking and learning play in designing such tasks; (2) some potential connections among various theories about thinking and learning as they relate to fractions; (3) the natural number bias and how ideas about natural numbers could serve as a foundation for fractions; and (4) the roles that magnitude, measurement, and linear representations of number play for fractions.

In this chapter, I propose a stance on learning fractions as multiplicative relations through reorganizing knowledge of whole numbers as a viable alternative to the Natural Number Bias (NNB) stance. Such an alternative, rooted in the constructivist theory of knowing and learning, provides a way forward in thinking about and carrying out teaching-learning of fractions, while eschewing a deficit view that seems to underlie the ongoing impasse in this area. I begin with a brief presentation of key aspects of NNB. Then, I discuss key components of the alternative framework, called reflection on activity-effect relationship, which articulates the cognitive process of reorganizing one’s anticipations as two types of reflection that give rise to two stages in constructing fractions as numbers. Capitalizing on this framework, I then delineate cognitive progressions of nine fractional schemes, the first five drawing on operations of iterating units and the last four on recursive partitioning operations. To illustrate the benefits of the alternative, conceptually driven stance, I link it to findings from a recent brain study, which includes significant gains for adult participants and provides a glance (fMRI) into circuitry recruited to process whole number and fraction comparisons.

The number and its basic operations can be conceptualised within a general system of relations. Children need to construct a system of numbers within which they can add, subtract, multiply and divide any rational number. Products and quotients can be defined in terms of general relational schemes. In this study, we examine whether elementary school children can construct a system of numbers such that fraction multiplication and division are based on the construction of general relational schemes. Groups of students are not homogeneous and children progress at different rates. For reliable assessment teachers need methods to examine developmental and individual differences in cognitive representations of mathematical concepts and operations. A logistic regression curve offers a visualisation of the learning process as a function of average marks. The analysis of fraction multiplication and division items shows an improvement on correct response probability, especially for students with a higher average mark.

Learning progressions represent the relationship between concepts within a domain and how students develop increasingly sophisticated thinking therein. Typical evidence sources used to validate theorized learning progressions are also used to validate the use and interpretation of assessments, such as student cognitive interviews and psychometric analyses of item responses on assessments (Alonzo, 2018; Duschl et al., 2011). However, evidence from student responses to assessment items may reflect an interaction with the assessment itself more so than students’ domain-specific knowledge and understanding (Lai et al., 2017; Penuel et al., 2014). In this manuscript, we propose that educators’ perspectives may serve as an independent source of evidence that can be integrated with traditional evidence sources (e.g., cognitive interviews with students, psychometric data) to overcome this shortcoming. This manuscript describes two studies that used surveys to draw on educator knowledge of students to identify upper and lower bounds of a learning progression (MMaRS study) and to understand the order of intermediary phases of learning (ESTAR study). For both studies, participants included mathematics educators who were classroom teachers or curriculum and assessment developers in relevant grades. Survey results yielded meaningful information to support or modify the hypothesized learning progressions for the respective studies, supporting the proposition that educators’ perspectives can meaningfully complement commonly used evidence to validate the structure of learning progressions. Advantages and limitations to this approach are described.

Development of the cardinality principle, an understanding that the last number-word recited in counting a collection of items specifies the number of items in that collection, is a critical milestone in developing a concept of number. Researchers in early number development have endeavored to theorize its development. Here we critique two widely respected hypotheses that explain cardinality-principle development as building on young learners’ ability to subitize small numbers. These hypotheses consider subitizing to be the basis of cardinality-principle development. We argue that there is a qualitative and significant difference between subitizing and the cardinality principle and that the explanation provided are insufficient to account for a change of this magnitude. We then propose a conjecture intended to better explain this change. The conjecture describes counting as the medium for a series of reflective abstractions leading to the cardinality principle.

The purpose of this study was to investigate the extent students compute fractions with correct understanding of the concept of fractions in Ethiopia. Participants of this survey were 1159 grade 7 and 8 students selected from 16 primary schools in four regions. Items of fractions from the MUST (Ethiopian and JICA Project) achievement tests were adapted and used for data gathering. Mann-Whitney statistical test, wrong answer analysis of individual item and descriptive statistic were used to analyse the data. The findings were that there was no statistical significant gender difference of answering fractional question items. Secondly, most students understood Part-Whole fractions wrongly. Further, the percentage of students who correctly calculated operations of fraction without correct understanding Part-Whole fraction was higher than the percentage of students who correctly calculated operations of fractions with correct understanding of Part-Whole Relationship. The overall results show that students' knowledge of fractions is limited by procedural knowledge; particularly their mathematical knowledge of fractions is limited by formulas. The implication was that students calculated operations of fractions by memorizing the formula without making sense as a replication of teacher characteristic "operationalism" drills and practices in classroom. The emergent of typical wrong fractional operations attempted by students are the consequence of quickly using formulas without referring the meaning. Therefore, teachers should see beyond student's response (the right and the wrong answers) and use the typical wrong fractional operations as a tool of teaching to improve students' learning. It is therefore recommended that teacher development programs and trainings focus these areas in the voyage of teaching.

Projektet, som tar sin utgångspunkt i de utmaningar som lärare i dag står inför när det gäller att planera, genomföra och utvärdera en matematikundervisning som lägger en grund för utvecklingen av de yngsta elevernas taluppfattning, problemlösningsförmåga och matematiska tänkande, avser att utveckla ett professionsstöd i relation till positionssystemet. Lärarnas utmaningar accentueras av att skolan präglas av heterogenitet avseende såväl elevernas språkliga och kulturella bakgrund (Adler 2019; Bunar 2010; Hansson 2012; Norén 2015) som deras matematiska erfarenheter (Roos 2019; Skolforskningsinstitutet 2018). Denna bild stärks exempelvis i den rapport som Dahlberg från intresseorganisationen Sveriges Ingenjörer (2021) skrivit och som visar att skolan med sin matematikundervisning ännu inte förmår överbrygga eller kompensera för elevernas bakgrund uttryckt i föräldrarnas utbildningskapital. De konstaterar att undervisningen redan i de tidigaste skolåren måste utvecklas för att tillgodose den variation som finns bland eleverna ur ett mångfaldsperspektiv samt kompensera elevgrupper som kommer från mindre studievana miljöer. Trots de satsningar som gjorts-t.ex. utökad tid för matematikundervisningen och Matematiklyftet-finns det alltjämt ett behov att utveckla den svenska matematikundervisningen så att den i högre utsträckning kan erbjuda vad Lindvall m.fl. (2021) talar om som en innehållsligt rik undervisning. Lindvall m.fl. konstaterar att Matematiklyftet ännu inte lett till synbara resultat gällande elevernas prestationer. Detta samtidigt som Österholm m.fl. (2016) visar att lärarna har upplevt tillfredsställelse med att delta i det kollektiva arbetet som utgjorde kärnan i Matematiklyftet. Lyftet med sina moduler och uppgifter upplevdes dock som splittrat. Ett skäl till att det inte syns några egentliga resultat på elevernas prestationer kan vara att Matematiklyftet riktade in sig på att förändra lärarnas undervisning i generella termer istället för att fokusera ett specifikt kunskapsinnehåll och systematiskt prövande av konkreta uppgifter och lektionsdesigns (Lindvall m.fl. 2021). Att satsningar som Matematiklyftet inte ser ut att ge resultat på elevernas prestationer stärks också av den forskning som visar att det finns flera svårigheter kopplade till utformningen av professionsprogram. En svårighet ligger i att de konceptuella och lärandeteoretiska ideer som finns inbyggda i de flesta program, behöver göras möjliga för lärare att förstå utan att förenklas (Brørup Dyssegaard m.fl. 2017; Ryve & Hemmi 2019). Lindvall m.fl. (2021) argumenterar därmed för att finns ett behov av program som innefattar konkreta modeller och exempel för nya undervisningshandlingar riktade mot elevernas lärande av ett specifikt kunskapsinnehåll (jfr även Boesen m.fl. 2014) istället för att försöka förändra undervisningen som helhet. Ett angeläget område undervisning gällande taluppfattning, problemlösning och matematiskt tänkande är elevernas fördjupade förståelse av positionssystemet.
Finansierat av Skolforskningsinstitutet 2022-2024

Whereas proficiency in performing the canonic multiplication-of-fractions algorithm is common, understanding of the algorithm is much less so. We conducted a teaching experiment with a fifth-grade student, based on an initial hypothetical learning trajectory (HLT), to promote reinvention of the multiplication-of-fractions algorithm. The instructional intervention built on two concepts, recursive partitioning and distributive partitioning. As a study of the Learning Through Activity research program, our goal was to promote particular activity on the part of the student through which she could abstract the necessary concepts. The results of the teaching experiment were analyzed and, based on conclusions from the research, a revised HLT was generated. Recursive partitioning and distributive partitioning proved to be a strong foundation for construction of the algorithm.

Mathematical understanding continues to be one of the major goals of mathematics education. However, what is meant by “mathematical understanding” is underspecified. How can we operationalize the idea of mathematical understanding in research? In this article, I propose particular specifications of the terms mathematical concept and mathematical conception so that they may serve as useful constructs for mathematics education research. I discuss the theoretical basis of the constructs, and I examine the usefulness of these constructs in research and instruction, challenges involved in their use, and ideas derived from our experience using them in research projects. Finally, I provide several examples of articulated mathematical concepts.

We present a synthesis of findings from constructivist teaching experiments
regarding six schemes children construct for reasoning multiplicatively
and tasks to promote them. We provide a task-generating platform
game, depictions of each scheme, and supporting tasks. Tasks must
be distinguished from children’s thinking, and learning situations must
be organized to (a) build on children’s available schemes, (b) promote
the next scheme in the sequence, and (c) link to intended mathematical
concepts.

Twelve students across Grades 2 to 6 were interviewed individually using a range of tasks, where the mathematical focus was a conceptual understanding of fractions. Careful listening established that despite giving a correct answer and appearing to have conceptual understanding, further probing sometimes revealed that the child had only a faulty procedural understanding. Similarly, success on one task did not guarantee success on a different but related task. Conversely, a task involving a continuous quantity enabled a child to move between discrete and continuous interpretations of fractional parts. This study supported the claimed advantages of one-to-one interviews over pen and paper tests, but also highlighted the importance of careful listening and the need for multiple tasks in the one mathematical domain in eliciting understanding. The structured interview provides an opportunity to listen closely to children's explanations of their mathematical understandings. In the present study, a child successfully completed a reversibility task, finding the whole when given a fractional part, by stating that given the three frogs in the picture in front of her, and seeing that they were three quarters of the frogs in the pond (the others being underwater), then there were four frogs in the pond altogether. Her understanding, however, was much more complex than her straightforward explanation conveyed. She added, as an afterthought while answering a different question, "oh with the frogs right, if they're the same weight, length, height, everything, and then I draw a little frog, it's not a quarter." This observation demonstrated that not only was her part whole knowledge of fractions stable but also that she had made connections to the measurement domain, in order to allude to the fact that rational numbers could involve more than a discrete whole number count. While three quarters could mean three out of four discrete items, in this case, frogs, three quarters could also mean a part of a whole that had continuous attributes like weight and length and height. This comment, an afterthought, was made possible by the one-to- one setting of a fraction interview, and the opportunity to respond to a range of tasks addressing similar content. Research has indicated that fractions as rational numbers are more than just shorthand for a part whole relationship based on equal parts. Rational numbers encompass the subconstructs: decimals, equivalent fractions, ratio numbers, multiplicative operators, quotients, and measures or points on a number line (Kieren, 1976). With minor variations

Although mathematics educators seem to agree on the importance of teaching mathematics for understanding, what they mean by understanding varies greatly. In this article, I elaborate and exemplify the construct of key developmental understanding to emphasize a particular aspect of teaching for understanding and to offer a construct that could be used to frame the identification of conceptual learning goals in mathematics. The key developmental understanding construct is based on extant empirical and theoretical work. The construct can be used in the context of research and curriculum development. Using a classroom example involving fractions, I illustrate how focusing on key developmental understandings leads to particular, potentially useful types of pedagogical thinking and directions for inquiry.

Constructivist theory has been prominent in recent research on mathematics learning and has provided a basis for recent mathematics education reform efforts. Although constructivism has the potential to inform changes in mathematics teaching, it offers no particular vision of how mathematics should be taught; models of teaching based on constructivism are needed. Data are presented from a whole-class, constructivist teaching experiment in which problems of teaching practice required the teacher/researcher to explore the pedagogical implications of his theoretical (constructivist) perspectives. The analysis of the data led to the development of a model of teacher decision making with respect to mathematical tasks. Central to this model is the creative tension between the teacher's goals with regard to student learning and his responsibility to be sensitive and responsive to the mathematical thinking of the students.

We discuss an emerging program of research on a particular aspect of mathematics learning, students’ learning through their own mathematical activity as they engage in particular mathematical tasks. Prior research in mathematics education has characterized learning trajectories of students by specifying a series of conceptual steps through which students pass in the context of particular instructional approaches or learning environments. Generally missing from the literature is research that examines the process by which students progress from one of these conceptual steps to a subsequent one. We provide a conceptualization of a program of research designed to elucidate students’ learning processes and describe an emerging methodology for this work. We present data and analysis from an initial teaching experiment that illustrates the methodology and demonstrates the learning that can be fostered using the approach, the data that can be generated, and the analyses that can be done. The approach involves the use of a carefully designed sequence of mathematical tasks intended to promote particular activity that is expected to result in a new concept. Through analysis of students’ activity in the context of the task sequence, accounts of students’ learning processes are developed. Ultimately a large set of such accounts would allow for a cross-account analysis aimed at articulating mechanisms of learning.

Our microgenetic research methodology was developed for two purposes: 1) to develop greater understanding of conceptual learning and instructional design that promotes conceptual learning and 2) to develop empirically-based hypothetical learning trajectories for particular mathematical concepts. The challenge in developing the methodology was to afford analysis of the learning process (e.g., the transition), not just identification of conceptual steps through which learners progress. The methodology, developed by the Learning Through Activity research program , was based on constructivist teaching experiment methodology. Modifications were made to data collection for the purposes outlined and a multi-level retrospective analysis was developed .

Promoting deep understanding of equivalent-fractions has proved problematic. Using a one-on-one teaching experiment, we investigated the development of an increasingly sophisticated, sequentially organized set of abstractions for equivalent fractions. The article describes the initial hypothetical learning trajectory (HLT) which built on the concept of recursive partitioning (anticipation of the results of taking a unit fraction of a unit fraction), analysis of the empirical study, conclusions, and the resulting revised HLT (based on the conclusions). Whereas recursive partitioning proved to provide a strong conceptual foundation, the analysis revealed a need for more effective ways of promoting reversibility of concepts. The revised HLT reflects an approach to promoting reversibility derived from the empirical and theoretical work of the researchers.

We discuss the theoretical framework of the Learning Through Activity research program. The framework includes an elaboration of the construct of mathematical concept, an elaboration of Piaget's reflective abstraction for the purpose of mathematics pedagogy, further development of a distinction between two stages of conceptual learning, and a typology of different reverse concepts. The framework also involves instructional design principles built on those constructs, including steps for the design of task sequences, development of guided reinvention, and ways of promoting reversibility of concepts. This article represents both a synthesis of prior work and additions to it.

We report on a teaching experiment intended to foster a concept of multiplication that would
both subsume students’ multiple-groups concept of whole number multiplication and provide a
conceptual basis for understanding multiplication of fractions. The teaching experiment, which
used a rigorous single-subject methodology, began with an attempt to build on students’ multiple-
groups concept by promoting generalizing assimilation. This was not totally successful and
led to a redesign aimed at promoting reflective abstraction. Analysis of this latter phase led to
several significant conclusions, which in turn led to a revised hypothetical learning trajectory.
The revised trajectory aims to foster a concept of multiplication as a change in units.

Whereas proficiency in performing the canonic multiplication-of-fractions algorithm is common, understanding of the algorithm is much less so. We conducted a teaching experiment with a fifth-grade student, based on an initial hypothetical learning trajectory (HLT), to promote reinvention of the multiplication-of-fractions algorithm. The instructional intervention built on two concepts, recursive partitioning and distributive partitioning. As a study of the Learning Through Activity research program, our goal was to promote particular activity on the part of the student through which she could abstract the necessary concepts. The results of the teaching experiment were analyzed and, based on conclusions from the research, a revised HLT was generated. Recursive partitioning and distributive partitioning proved to be a strong foundation for construction of the algorithm.

This book is a product of love and respect. If that sounds rather odd I initially apologise, but let me explain why I use those words. The original manuscript was of course Freudenthal’s, but his colleagues have carried the project through to its conclusion with love for the man, and his ideas, and with a respect developed over years of communal effort. Their invitation to me to write this Preface e- bles me to pay my respects to the great man, although I am probably incurring his wrath for writing a Preface for his book without his permission! I just hope he understands the feelings of all colleagues engaged in this particular project. Hans Freudenthal died on October 13th, 1990 when this book project was well in hand. In fact he wrote to me in April 1988, saying “I am thinking about a new book. I have got the sub-title (China Lectures) though I still lack a title”. I was astonished. He had retired in 1975, but of course he kept working. Then in 1985 we had been helping him celebrate his 80th birthday, and although I said in an Editorial Statement in Educational Studies in Mathematics (ESM) at the time “we look forward to him enjoying many more years of non-retirement” I did not expect to see another lengthy manuscript.

Tzur and Simon (2004) postulated 2 stages of development in learning a mathematical concept: participatory and anticipatory. In this article, we discuss the affordances for research of this stage distinction related to data analysis, task design, and assessment as demonstrated in a 2-year teaching experiment. We describe our modifications to and further explicate and exemplify the theoretical underpinnings of these stage constructs. We introduce a representation scheme and use it to trace the development of a concept from initial activity, through the participatory stage, and to the anticipatory stage.

In this constructivist teaching experiment with 2 fourth graders I studied the coemergence of teaching and children's construction of a specific conception that supports the generation of improper fractions. The children's posing and solving tasks in a computer microworld promoted a modification in their fraction schemes. They advanced from thinking about a unit fraction as a part of a whole to thinking about it as standing in a multiplicative relationship with a reference whole (the iterative fraction scheme). In this article I report an intertwined analysis of the children's construction of this multiplicative relationship and an examination of the teacher's adaptation of learning situations (tasks) and teacher-learner interactions to fit within the constraints of the children's mathematical activity.

Six fifth-grade students came to instruction with informal knowledge related to partitioning. This knowledge initially focused on partitioning "units of measure one" into a specific number of parts. Students were able to build on their informal knowledge to reconceptualize and partition units to solve problems involving multiplication of fractions in ways that were meaningful to them. Students built their knowledge by developing mental processes related to focusing on fractional amounts and to partitioning units in different ways. Students also frequently returned to their initial focus on the number of parts and to ideas embedded in equal-sharing situations.

Children's Fractional Knowledge elegantly tracks the construction of knowledge, both by children learning new methods of reasoning and by the researchers studying their methods. The book challenges the widely held belief that children's whole number knowledge is a distraction from their learning of fractions by positing that their fractional learning involves reorganizing not simply using or building upon their whole number knowledge. This hypothesis is explained in detail using examples of actual grade-schoolers approaching problems in fractions including the schemes they construct to relate parts to a whole, to produce a fraction as a multiple of a unit part, to transform a fraction into a commensurate fraction, or to combine two fractions multiplicatively or additively. These case studies provide a singular journey into children's mathematics experience, which often varies greatly from that of adults. Moreover, the authors' descriptive terms reflect children's quantitative operations, as opposed to adult mathematical phrases rooted in concepts that do not reflect-and which in the classroom may even suppress-youngsters' learning experiences. Highlights of the coverage: Toward a formulation of a mathematics of living instead of beingOperations that produce numerical counting schemes Case studies: children's part-whole, partitive, iterative, and other fraction schemes Using the generalized number sequence to produce fraction schemes Redefining school mathematics This fresh perspective is of immediate importance to researchers in mathematics education. With the up-close lens onto mathematical development found in Children's Fractional Knowledge, readers can work toward creating more effective methods for improving young learners' quantitative reasoning skills. © Springer Science+Business Media, LLC 2010 All rights reserved.

This work presents one of the original and fundamental experiments of Didactique, a research program whose underlying tenet is that Mathematics Education research should be solidly based on scientific observation. Here the observations are of a series of adventures that were astonishing for both the students and the teachers: the reinvention of fractions and of decimal numbers in a sequence of lessons and situations that permitted the students to construct the concepts for themselves. The book leads the reader through the highlights of the sequence's structure and some of the reasoning behind the lesson choices. It then presents explanations of some of the principal concepts of the Theory of Situations. In the process, it offers the reader the opportunity to join a lively set of fifth graders as they experience a particularly attractive set of lessons and master a topic that baffles many of their contemporaries. © 2014 Springer Science+Business Media B.V. All rights are reserved.

Fractions are among the most complex mathematical concepts that children encounter in their years in primary education. One of the main factor contributing to this complexity is that fractions comprise a multifaceted notion encompassing five interrelated subconstructs (part-whole, ratio, operator, quotient and measure). During the early 1980s a theoretical model linking the five interpretations of fractions to the operations of fractions and problem solving was proposed. Since then no systematic attempt has been undertaken to provide empirical validity to this model. The present paper aimed to address this need, by analyzing data of 646 fifth and sixth graders’ performance on fractions using structural equation modeling. To a great extent, the analysis provided support to the assumptions of the model. Based on the findings, implications for teaching fractions and further research are drawn.

Textbooks play an important part in the design of instruction. This study analyzed the presentation of fractions in textbooks
designed for the elementary grades in Kuwait, Japan, and the USA. The analysis focused on the physical characteristics of
the books, the structure of the lessons, and the nature of the mathematical problems presented. Findings showed USA and Kuwaiti
mathematics textbooks are larger than Japanese textbooks; this larger size is consistent with a great deal of repetition.
The Japanese texts do not address fractions until the third grade; they use linear models and connect fractions with measurement.
In the USA and Kuwait, fractions are introduced in the first grade. The Harcourt text uses concrete material to help students
learn fraction procedures. For the Kuwaiti series, the lessons depend on using some pictorial representation of the area model
to illustrate fraction ideas. All these textbooks focus on standard algorithms as the main computational methods.

The basic hypothesis of the teaching experiment, The Child’s Construction of the Rational Numbers of Arithmetic (Steffe & Olive, 1990) was that children’s fractional schemes can emerge as accommodations in their numerical counting schemes. This hypothesis is referred to as the reorganization hypothesis because when a new scheme is established by using another scheme in a novel way, the new scheme can be regarded as a reorganization of the prior scheme. In that case where children’s fractional schemes do emerge as accommodations in their numerical counting schemes, I regard the fractional schemes as superseding their earlier numerical counting schemes. If one scheme supersedes another, that does not mean the earlier scheme is replaced by the superseding scheme. Rather, it means that the superseding scheme solves the problems the earlier scheme solved but solves them better, and it solves new problems the earlier scheme didn’t solve. It is in this sense that we hypothesized children’s fractional schemes can supersede their numerical counting schemes and it is the sense in which we regarded numerical schemes as constructive mechanisms in the production of fractional schemes (Kieren, 1980).

Discussed are the background and implementation of a method of teaching fractions to primary children referred to as "The Visual Concept of Fractions." The advantages and criticisms of the use of this method are stressed. (CW)

This research examined the presentation of fractions in textbooks used by fifth and sixth graders in Singapore, Taiwan, and the United States. The specific textbooks examined were My Pals Are Here! Maths (MPHM) in Singapore; Kung Hsung (KH) in Taiwan; and Mathematics in Context (MiC) in the USA. Results show the problems posed in MiC put more emphasis on real-life situations than KH textbooks in Taiwan and MPHM in Singapore. Designing materials that provide opportunities to connect mathematics content with applications in real life is consistent with recommendations from professional organizations. The activities in KH and MPHM tended to emphasize procedures, while the activities of MiC focused more on conceptual understanding and less on the development of procedures. An examination of the mathematics textbooks revealed that MPHM introduced and developed fractions the earliest among the three countries investigated and the content taught in MPHM was about one grade earlier than when the same content was experienced by students in KH and MiC.

An experiment is reported that investigated the development of students' understanding of the numerical value of fractions. A total of 200 students ranging in age from 10 to 16 years were tested using a questionnaire that required them to decide on the smallest/biggest frac-tion, to order a set of given fractions and to justify their responses. Students' responses were grouped in categories that revealed three main explanatory frameworks within which frac-tions seem to be interpreted. The first explanatory framework, emerging directly from the initial theory of natural numbers, is that fraction consists of two independent numbers. The second considers fractions as parts of a whole. Only in the third explanatory framework, students were able to understand the relation between numerator and denominator and to consider that fractions can be smaller, equal or even bigger than the unit.

In this study, selected Chinese, Japanese and US mathematics textbooks were examined in terms of their ways of conceptualizing
and organizing content for the teaching and learning of fraction division. Three Chinese mathematics textbook series, three
Japanese textbook series, and four US textbook series were selected and examined to locate the content instruction of fraction
division. Textbook organization of fraction division and other content topics were described. Further analyses were then conducted
to specify how the content topic of fraction division was conceptualized and introduced. Specific attention was also given
to the textbooks’ uses of content constructs including examples, representations, and exercise problems in order to show their
approaches for the teaching and learning of fraction division. The results provide a glimpse of the metaphors of mathematics
teaching and learning that have been employed in Chinese, Japanese, and US textbooks. In particular, the results from the
textbook analyses demonstrate how conceptual underpinnings were developed while targeting procedures and operations. Implications
of the study are then discussed.

This paper critically examines the discrepancies among the pre-requisite fractional concepts assumed by a curricular unit on operations with fractions, the teacher's assumptions about those concepts and a particular student's understanding of fractions. The paper focuses on the case of one student (Tim) in the teacher's 6th grade class who was interviewed by one of the authors once a week during the teaching of the unit. The teaching materials and the teacher's instruction were based on the assumption that students understood the concept of a unit fraction as being one of several equal parts of a given whole. The teacher neither emphasized the need for equal parts nor the part-to-whole relation. The teacher's reasonable assumptions about her students’ understanding of fractions were severely challenged by the cognitive constructs that Tim exhibited during his first two interviews. When she viewed tapes of the class instruction and the interviews with Tim she realized Tim lacked essential constructs to make sense of her instruction. She subsequently made adjustments in her instruction, making effective use of more appropriate representations based on tasks from the unit that we modified and used with Tim in our interviews. These adjustments helped Tim to construct partitioning operations and an appropriate unit fractional scheme. This study illustrates the importance of coming to understand a student's mathematical activity in terms of possible conceptual schemes and modifying instructional strategies to build on those schemes. The coordinated design of the research study facilitated these instructional modifications.

Rational number, ratio, and proportion

- M J Behr
- G Harel
- T Post
- R Lesh

Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.). Handbook of research on mathematics teaching and
learning (pp. 296-333). New York: Macmillan.

Units of quantity: A conceptual basis common to additive and multiplicative structures

- M Behr
- G Harel
- T Post
- R Lesh

Behr, M., Harel, G., Post, T., & Lesh, R. (1994). Units of quantity: A conceptual basis common to additive and multiplicative structures. In G. Harel, & J. Confrey (Eds.).
The development of multiplicative reasoning in the learning of mathematics (pp. 121-176). Albany, NY: SUNY Press.

JavaBars [Computer software

- B Biddlecomb
- J Olive

Biddlecomb, B., & Olive, J. (2000). JavaBars [Computer software]. Retrieved from http://math.coe.uga.edu/olive/welcome.html.

Developing fractions knowledge

- A J Hackenberg
- A Norton
- R J Wright

Hackenberg, A. J., Norton, A., & Wright, R. J. (2016). Developing fractions knowledge. London: SAGE Publications Ltd.

The Singapore model method for learning mathematics

- K T Hong
- Y S Mei
- J Lim

Hong, K. T., Mei, Y. S., & Lim, J. (2009). The Singapore model method for learning mathematics. Singapore: Ministry of Education.

More about fractions

- Khan Academy

Khan Academy (2017). More about fractions. Retrieved from http://www.khanacademy.org/math/arithmetic/fractions/understanding_fractions/v/introduction-tofractions.

A teaching experiment: Introducing fourth graders to fractions from the viewpoint of measuring quantities using Davydov's mathematics curriculum

- A K Morris

Morris, A. K. (2000). A teaching experiment: Introducing fourth graders to fractions from the viewpoint of measuring quantities using Davydov's mathematics
curriculum. Focus on Learning Problems in Mathematics, 22(2), 33-84.

Rational number, ratio, and proportion

- Behr

Units of quantity: A conceptual basis common to additive and multiplicative structures

- Behr

Interaction and children's fraction learning (Doctoral dissertation, University of Georgia, 1995)

- Tzur