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A Part-Part-Whole Curriculum for Teaching Number in the Kindergarten
Abstract
The set-subset relationship is an important mathematical construct that underlies many mathematical concepts developed by young children. Forty-two kindergarten children who received number concept instruction using a part-part-whole curriculum that stressed set-subset relationships developed a more mature concept of number, were more successful in solving addition and subtraction word problems, and developed greater understanding of place value in the base-10 numeration system than a comparable group of 44 kindergarten children who received standard instruction on number concepts.
... However, they conclude that some abilities related to number relations (subitizing and more-less comparison) seemed to be essential for the success. Others, take part-whole relations as the outset for developing arithmetic strategies, e.g., Fischer (1990), teaching the relation between small sets and subsets, and Cheng (2012), teaching decomposition strategies based in categorizing sets and subsets. Effective interventions that promote children's knowledge of number relations thus seem to build on preexisting knowledge that allows the children to apply arithmetic strategies in successful ways. ...
... Our intervention is an exception compared to many other intervention studies, in that the children have been encouraged to attend to numbers primarily as composite sets using finger patterns as an aid to visualize the part-whole relations. In other interventions described in the literature, children have most often counted to determine or compose sets (Fischer, 1990;Paliwal & Baroody, 2017). ...
... What our study shows is that the children in our study make a necessary change toward experiencing numbers as structure and we would suggest, through the systematic work affording the discernment of necessary aspects of numbers, particularly relating to a semi-decimal structure (five is a whole hand), rather than determining a quantity by counting single units (cf. Fischer, 1990). This pedagogy has provided the children opportunities to discern aspects of numbers and number relations that liberate how numbers can be experienced, rather than teaching them strategies as the primary goal. ...
We report here on an intervention implementing a structural approach to arithmetic problem-solving in relation to learning outcomes among preschoolers. Using the fundamental principles of the variation theory of learning for developing the intervention and as an analytical framework, we discuss teaching and learning in commensurable terms. The research question is how teaching grounded on a structural approach and designed based on principles of variation theory is reflected in children’s learning of numbers. To answer this, three analyses were conducted, addressing: i) how the children’s ways of experiencing numbers changed after participating in the intervention, ii) how the theoretical ideas were afforded in the intervention program, and iii) synthesizing how the affordance was associated with the children’s arithmetic learning. One group of eight children participating in the intervention program was chosen for thorough analysis. Progression was observed in how the children changed their ways of experiencing numbers during the intervention that allowed them to enact more advanced arithmetic strategies, which was associated with the structural approach in teaching. The results also show how analysis focusing on aspects discerned in learning and aspects afforded in teaching provides a way of describing arithmetic learning with significant implications for teaching practices.
... The terms whole and parts describe the set-subset relationship of a collection of items, and this relationship can be represented by numbers (Fischer, 1990). When children are to learn that within a collection (set) there are smaller groups of items (subsets), they need to understand the cardinality principle (in counting items, each number word refers both to the last counted item and to all the items together) (Fuson, 1982). ...
... Building on evidence both that understanding the part-whole relationship of numbers is fundamental for children's development and that this understanding is possible among preschool-aged children (Fischer, 1990;Jung et al., 2013), some early mathematics intervention programmes implement number activities that emphasise part-whole relations (e.g., Clements & Sarama, 2009;Dyson et al., 2011). Some interventions have included a narrower focus on a structural approach to part-whole relations. ...
... However, few studies have more explicitly examined teaching part-whole relations of numbers. In a preschool intervention programme, 4-to-5-year-olds who received instruction in part-whole relationships developed a better understanding of numbers and of solving additive relation word problems than the group of children following the more common "counting" programme (Fischer, 1990). In a 2013 study by Jung et al. in which activities focusing on part-whole relationships, number relations, and subitising were implemented, similar positive effects were found in the group that received instruction which emphasised number relationships and subitising, whereas in the control group no such effects were seen. ...
In this paper, differences in the implementation of a number activity called the snake game are studied. Nine Swedish preschool teachers worked in collaboration with a research team, enacting the same activity with their groups of 5-year-old children over a 3-month period. Variation theory forms the basis for the analysis of 67 videorecorded enactments. The results suggest that an activity such as the snake game can bring various aspects of numbers to the fore through differences in enactment. The activity became mathematically richer when the teacher compared children’s different finger patterns and used systematically varied examples of number relations. This study’s results contribute knowledge about characteristics of teaching that foregrounds numbers’ part-whole relations.
... Before the term number sense was coined, Payne and Rathmell (1975) proposed the terms 'whole' and 'parts' to explain relations of numbers. Part-whole relationship of numbers means that quantities are interpreted as being composed of other numbers where a quantity (a whole) can be partitioned into two or more parts (Fischer, 1990;Jung, 2011). Enabling pupils to interpret quantities and numbers in this way can be considered as a key conceptual achievement in the early years of their mathematics education. ...
... The part-whole concept facilitates interpretation of conceptual structures of different addition and subtraction problems. Young pupils can therefore deal with mathematics problems more flexibly (Fischer, 1990), demonstrating multiple and flexible representations of numbers. For example, as Figure 2 illustrates (Jung, 2011), number 9 can be represented in multiple ways as 1 + 8, 2 + 7, 3 + 3 + 3, and 4 + 5. Understanding the part-whole relationship can enhance the conception of place value (Fischer, 1990;Resnick, 1983), measurement, and fractions (Charlesworth, 2012). ...
... Young pupils can therefore deal with mathematics problems more flexibly (Fischer, 1990), demonstrating multiple and flexible representations of numbers. For example, as Figure 2 illustrates (Jung, 2011), number 9 can be represented in multiple ways as 1 + 8, 2 + 7, 3 + 3 + 3, and 4 + 5. Understanding the part-whole relationship can enhance the conception of place value (Fischer, 1990;Resnick, 1983), measurement, and fractions (Charlesworth, 2012). Pupils may hereby develop the ability to deconstruct quantities, keep track of the parts, put the parts ...
This review examines reasons for Malaysian pupils’ underperformance in solving mathematics problems that demand numerical estimation and mental computation. Their underdeveloped number sense appears to be the major reason. According to the Ministry of Education, the national mathematics curriculum is unlikely to account for this underachievement. By contrast, this article argues that the curriculum largely focusing on written computation skills, is likely to account for the underperformance, given that the development of number sense has yet to become the focus of attention. This review further suggests teaching and learning activities that may enable the Malaysian pupils, in year one of primary schools, to conceive part-whole relations of numbers and thus develop number sense. The earlier they understand the part-whole relationship, the
stronger they may develop number sense and thus the better can be their performance.
... Early Instruction of the Part-Whole Concept The integration of the described conceptual basic ideas into precise quantitative situations does not necessarily occur automatically and instruction should therefore not only begin with the entry to school, but rather before entering school (Sophian and McCorgray 1994). A study by Fischer (1990) with 4 to 6-year olds showed that early dealing with quantity relations strongly fosters the development of the concept of numbers. In Fischer's study, kindergarten children who were instructed on the basis of a "Part-Part-Whole Curriculum" were compared with a control group instructed on the basis of a "Count-Say-Write Curriculum". ...
... They were more successful in solving addition and subtraction word problems and developed greater understanding of place value in the base-10 numeration system than the reference group. According to Fischer (1990), the results indicate that early instruction that emphasizes set-subset relations is helpful for the development of basic number concepts and related skills. ...
... Gerlach and Fritz 2011;Krajewski et al. 2007). Several authors have already proved that explicit fostering instruction of contents of the part-whole concept is extremely important and effective (Steinke 2008;Young-Loveridge 2002;Fischer 1990). ...
For the development of mathematical competencies from preschool children and children at primary school age, the acquisition of a comprehensive part-whole concept is essential. The principle of understanding numbers as compositions of other numbers as well as being able to decompose numbers and interpret quantitative relations, enables a flexible understanding of operations. It is known that the part-whole concept is based on various schemas and is subject to a longer learning process. The aim of the study was to examine the non-numerical and numerical understanding of the part-whole concept more closely. To achieve this, the availability of the complex part-whole concept in children aged 4 to 8 years was examined by means of a sample of 181 children on the basis of word problems with non-numerical and numerical tasks on different part-whole contents (compensation, covariation, final amount, initial amount, finding and evaluating decompositions of amounts). A Rasch analysis resulted in a two-dimensional model. According to this, two content-related competency dimensions must be differentiated: A non-numerical dimension with the understanding of solving part-whole tasks in word problems without number relation, and a numerical dimension with the understanding of exact quantification with numbers. The two dimensions are considered as two different domains of knowledge, though they are highly correlated. They develop partly parallel, while at the beginning the non-numerical part-whole understanding develops a little earlier. Two levels of competencies could be identified within each dimension. The findings are discussed regarding the distribution of the children on the respective levels of competencies depending on their age.
... Precisamente desde ese enfoque de acompañamiento progresivo, y dado el propósito investigativo de favorecer la comprensión del valor posicional y dar cuenta del avance de los estudiantes, se consideraron en la estructuración del marco conceptual de este estudio, aquellos trabajos teóricos o empíricos, que asumen la comprensión del valor posicional y del SDN en términos del desarrollo gradual de habilidades, y de la construcción progresiva de unidades (simples o compuestas) para incrementar la flexibilidad en el conteo, en la realización de particiones, en la agrupación y en la comparación de cantidades; constructos que desde la investigación en educación matemática (Alba y Quintero;2016;Angulo y Herrera, 2009;Baturo, 2002;Bruno y Noda, 2014;Chandler y Kamii, 2009;Fischer, 1990;Fuson, Carroll y Drueck, 2000;Fuson et al, 1997;Kamii y Joseph, 1988;Salazar y Vivas, 2013;Zúñiga, 2015) han mostrado ser esenciales para analizar la enseñanza y el aprendizaje del valor posicional. ...
... Usualmente los niños realizan la partición señalando los objetos, agrupándolos separadamente o bien a través de la memoria visual. Cuando no se dispone de material concreto se requiere de un nivel de abstracción mayor para establecer la relación parte-parte-todo (Fischer, 1990), el cual está asociado al avance en la comprensión del concepto de valor posicional. ...
RESUMEN Se reporta un estudio en el cual se propone una estrategia de enseñanza para hacer frente y, eventualmente, superar las dificultades que presentan estudiantes de 2º de educación básica primaria (5-7 años) de dos colegios públicos de Bogotá (Colombia), en relación con el concepto de valor posicional. A partir de un análisis del contexto de aula de ambos grupos y de la revisión de literatura relacionada, se diseñó, implementó y evaluó una unidad didáctica bajo el enfoque de la Enseñanza para la Comprensión, con el propósito de que los estudiantes avanzaran en la comprensión del concepto de valor posicional. El estudio se realizó desde un enfoque cualitativo de alcance descriptivo, utilizando el diseño metodológico de investigación-acción. Los hallazgos evidencian que los estudiantes avanzaron en la comprensión del concepto de valor posicional, a través de los desempeños de comprensión propuestos en la unidad didáctica. Palabras clave: Educación matemática infantil, valor posicional, estrategia de enseñanza, Enseñanza para la Comprensión.
ABSTRACT
This paper reports a study that proposes an alternative teaching strategy to cope with and eventually overcome some difficulties presented by second graders (ages 5-7) at two public Elementary schools in Bogota (Colombia). Based on the analysis of the classroom contexts in both groups and on a literary review on the field, a teaching unit, founded on the Teaching for Understanding framework, was designed, implemented and evaluated. This strategy intended to enhance students' understanding of the concept of place value. This was a qualitative-descriptive study framed within the Action-Research design. The findings show that students made progress in understanding the concept of place value through the activities proposed in the teaching unit.
... During the preschool years children learn number words, recognize numerical symbols, and start using these symbols quantitatively. There is substantial research evidence that: (1) many young children know how to count (Arai, 1984;Baroody, 1992b; Baroody & White, 1983;Bertelli, Joanni, & Martlew, 1998;Clements, 1984a;Gallistel & Gelman, 1990;Newman & Berger, 1984;Wiegel, 1998); (2) many have some understanding of addition and subtraction (Baroody, 1987b(Baroody, , 1989Baroody & Wilkins, 1999;Campbell & Maertens, 1988;Fuson, 1992;Fuson & Kwon, 1992;Geary, 1994;Groen & Resnick, 1977;Hughes, 1986;Siegler, 1987); (3) they can compare things around them (e.g., state that 8 is more than 2) (Kotovsky & Gentner, 1996; R. C. Williams, 1980); (4) they use strategies in problem-solving (Carpenter et al., 1999;Geary, 1994;Hughes, 1986;Swanson, 1992); (5) many can match and sort shapes and sizes (Clements et al., 1999;Tyler, Allen, & Pasnak, 1983); and (6) they start acquiring some measurement concepts (e.g., they begin to understand the concept of half and a whole and its parts) (Boisvert, Standing, & Moller, 1999;Fischer, 1990;Hunting & Sharpley, 1988;Kimchi, 1993;Sophian, Garyantes, & Chang, 1997). ...
... Another curriculum was proposed by Fischer who trained 42 kindergartners on number concepts (Fischer, 1990). The children in the experimental group learned about cardinality by exploring parts into which sets of objects could be separated, counting sets and parts of objects, choosing numerals representing each picture set, and comparing parts among each other. ...
... For example, Fuson (1990) has suggested that place value instruction should be integrated with multi-digit addition or subtraction operations and postponed until at least second grade. Others have suggested that a complete understanding of place value requires a working knowledge of part-whole number relationships (Cobb and Wheatley 1988;Fischer 1990;Ross 1986). Consequently, foundational place value activities and instruction are rarely provided or studied in Pre-Kindergarten classrooms. ...
... In addition to helping students make connections among different representations of numbers (e.g., verbal, Arabic numeral, and pictorial), the MTP Math Number Chart is specifically designed to scaffold students in their understanding of the part-to-whole relationships within numbers (see Fig. 4), relationships that have been demonstrated to be foundational to place value understanding (Fischer 1990;Hunting 2003) as well as underlie the mathematical operations of addition and subtraction (Copley 2000). As part of its numeric and pictorial representations, the MTP Math Number Chart displays the component parts of each number in terms of groups of tens and ones or ''leftovers'' (e.g., 24 is 2 groups of 10 and 4 ''leftovers''). ...
Development of two-digit place value understanding in the elementary grades has been the subject of some study; however, research at the pre-kindergarten (Pre-K) level is limited. This two-part paper begins by providing an overview of two-digit place value instruction in Pre-K and describes the component parts of a research-based math curriculum, MyTeachingPartner Math (MTP Math). Part two presents the results of a video analysis of classroom interactions across four MTP Math place value activities facilitated by two high quality teachers. Particular attention is given to the primary conceptual hurdles faced by students, as well as the scaffolding strategies employed by teachers. Results indicate that students possess a conceptual understanding of the ones place prior to the tens place and initially struggle the concept of unitizing groups of ten. Considerations are discussed for improving the quality of teacher-child interactions in pre-kindergarten that can best support children’s thinking and learning.
... Third, five-frames present opportunities for children to establish connections between different numerical representations, a critical skill in one's mathematical development (NCTM 2008;NRC 2009). Fourth, five-frames allow children to explore combinations of numbers and observe part-towhole relationships, an important consideration during early mathematics (Copley 2000;Fischer 1990;Hunting 2003). Finally, five-frames are visually and conceptually similar to ten-frame representations, and therefore early exposure to five-frames will help to familiarize students with an instructional tool commonly used in later elementary mathematics. ...
... The rationale for this increased focus stems from a research base supporting both the need for number sense development (Jordan 2007) as well as the increasing realization that number sense is one of the most fundamentally important concepts to be developed in early mathematics (Baroody 2009;Jordan 2007;Kilpatrick et al. 2001;McGuire and Wiggins 2009;NCTM 2008;Van de Walle 2003). Research suggests that targeting number sense development in early mathematics prepares students to learn more complex mathematics concepts such as place value (Miura et al. 1993;Van de Walle 2003), part-to-whole number composition and decomposition (Fischer 1990;Hunting 2003), and basic arithmetic operations involving addition and subtraction (Levine et al. 1992;Wynn 1990). More generally, early exposure to key number sense concepts equips students with an understanding of core mathematical properties (NMAP 2008) and promotes numerical fluency (Baroody 2009). ...
Teachers in early childhood and elementary classrooms (grades K-5) have been using ten-frames as an instructional tool to support students’ mathematics skill development for many years. Use of the similar five-frame has been limited, however, despite its apparent potential as an instructional scaffold in the early elementary grades. Due to scant evidence of teacher use and a lack of systematic research we know little to nothing about both the developmental and pedagogical implications of using five frames and related instructional manipulatives in early childhood mathematics classrooms. In this paper, we provide an overview of five-frames and specifically demonstrate ways that five-frames, if used in conjunction with concrete manipulatives, can support pre-kindergarten (pre-K) children’s development of Gelman and Gallistel’s (1978) three basic counting principles: the stable-order principle, one-to-one correspondence, and cardinality. We conclude by discussing the developmental and instructional implications of using five-frames, as well as offer a set of teaching tips designed to help teachers maximize the potential advantages of integrating five-frames in the pre-K classroom.
... (p. 114) Empirical studies have shown that if children understand part-whole relationships, they are better able to understand the relationships among number, addition, and subtraction (Baroody, Ginsburg, & Waxman, 1983;Fischer, 1990); attain the most advanced level of addition and subtraction problem solving (Riley, Greeno, & Heller, 1983); and understand place value (Resnick, 1983a). ...
... Finally, since the instructional approaches discussed in this article are culture specific (e.g., use of an abacus), we are not suggesting that the Chinese model of teaching addition and subtraction is ideal and that other nations ought to immediately follow this model; however, both Chinese and U.S. researchers (e.g., Baroody et al., 1983;Fischer, 1990;Resnick, 1983a;Riley et al., 1983) agree that children's understanding of part-whole relationships is critical to their understanding of addition and subtraction. Thus, reformers of math instruction in the United States may want to develop ways of more systematically and thoroughly integrating these concepts in instruction on addition and subtraction. ...
Cross-cultural studies on mathematical cognition and education have suggested that curriculum and teaching have contributed to U.S. versus Asian differences in student performance; however, previous discussions of curriculum and teaching practices have been very general and have not focused on a detailed analysis of how mathematics concepts are taught and presented in textbooks. To address this limitation, this article focuses on the teaching of addition and subtraction to first-grade students in China. We discuss how curriculum topics are arranged, how concrete and abstract mathematical knowledge are reconciled in learning these concepts, and how teachers facilitate children's conceptual understanding of the logic behind addition and subtraction procedures. Toward this end, we analyze a research-based Chinese mathematics curriculum to highlight the sequencing of topics pertaining to addition and subtraction and the instructional practices used by teachers to help students develop an understanding of these concepts. The purpose of this article is not to suggest that other nations' curriculum and instructional practices ought to follow the Chinese model. Rather, we attempt to provide useful information about developing effective curriculum and instructional practices that could help young children make sense of the mathematics they learn in school. © 2005 Wiley Periodicals, Inc. Psychol Schs 42: 259–272, 2005.
... In a didactic study, Gardes (2020, 2020) mapped the early understanding of numbers. This study is based on various research results in math education (Fayol & Seron, 2005;Fischer, 1990;Fuson, 1988;Margolinas & Wozniak, 2012. Eleven types of tasks were identified to test the different areas of knowledge necessary for early understanding of numbers. ...
There is a paucity of research on Montessori Education's impact on learning, especially in France. In this article, we present a study comparing the development of early number learning in preschoolers through Montessori Education with “conventional” education in France. Using a cross-sectional design and random assignment of children, we evaluated 131 French preschoolers (aged 5-6) enrolled at the same public school following either Montessori Education or conventional education over three years. Students were evaluated with the Woodcock-Johnson III (WJ-III) Applied Problems sub-test and a test designed by researchers in math education. The Montessori curriculum was associated with outcomes that were comparable to the conventional curriculum on math.
... In einer Interventionsstudie von Fischer (1990) Cheng (2012) zur Wirksamkeit einer Förderung des Teile-Ganzes-Konzepts auf die arithmetischen Fähigkeiten. Die Teilnehmenden der Interventionsgruppe (N = 35) zeigten hier im Vergleich zur Kontrollgruppe (N = 33) nicht nur eine höhere Lösungsrate, sondern rechneten auch flexibler; sie wendeten häufiger Ableitungsstrategien und seltener zählende Strategien an. ...
Zusammenfassung
Die Entwicklung des Teile-Ganzes-Konzepts ist von grundlegender Bedeutung für die Entwicklung des Zahlbegriffs und das Rechnenlernen und deshalb eine zentrale Aufgabe des arithmetischen Anfangsunterrichts. Das Teile-Ganzes-Konzept wird insbesondere durch das Zerlegen von Zahlen und das numerische Erfassen der Beziehung zwischen Teilen und Ganzem entwickelt. Vor diesem Hintergrund werden in einer qualitativen typenbildenden Inhaltsanalyse zwölf Schulbücher der ersten Jahrgangsstufe daraufhin untersucht, in welcher Weise sie Zahlzerlegungen behandeln. Hierfür wurden zunächst die Lerngelegenheiten in den Schulbüchern umfassend kodiert und ausführliche Einzelfallanalysen erstellt. Auf dieser Basis konnten durch Fallvergleiche und Fallkontrastierung vier Typen von Schulbüchern bezüglich der Behandlung von Zahlzerlegungen identifiziert werden, die ein jeweils unterschiedliches Potenzial im Hinblick auf das Rechnenlernen bieten.
... Studies of intervention programmes have shown that it is possible to teach young children to handle novel and more advanced arithmetic tasks based on knowledge of numbers' partwhole relation (Fischer, 1990). Some researchers advocate that a structural approach, which primarily directs attention towards relationships between elements (Venkat et al., 2019) and making use of part-whole relations rather than single unit counting strategies (such as "counting all" and "counting on"), should be emphasised already in the early years (Brissiaud, 1992;Brownell, 1935;Davydov, 1982;Neuman, 1987;Schmittau, 2004). ...
In this paper, we present a way of describing variation in young children’s learning of elementary arithmetic within the number range 1–10. Our aim is to reveal what is to be learnt and how it might be learnt by means of discerning particular aspects of numbers. The Variation theory of learning informs the analysis of 2184 observations of 4- to 7-year-olds solving arithmetic tasks, placing the focus on what constitutes the ways of experiencing numbers that were observed among these children. The aspects found to be necessary to discern in order to develop powerful arithmetic skills were as follows: modes of number representations, ordinality, cardinality, and part-whole relation (the latter has four subcategories: differentiating parts and whole, decomposing numbers, commutativity, and inverse relationship between addition and subtraction). In the paper, we discuss particularly how the discernment of the aspects opens up for more powerful ways of perceiving numbers. Our way of describing arithmetic skills, in terms of discerned aspects of numbers, makes it possible to explain why children cannot use certain strategies and how they learn to solve tasks they could not previously solve, which has significant implications for the teaching of elementary arithmetic.
... The protoquantitative part-whole schema is the foundation for later understanding of binary addition and subtraction and for several fundamental mathematical principles, such as the commutativity and associativity of addition and the complementarity of addition and subtraction. (S. 32) Bobis (1993) und Fischer (1990) (Gaidoschik, 2015, S. 60). ...
... Further studies point to a connection between visual structuring ability and part-whole understanding (Young-Loveridge 2002). Numerous studies have also shown that part-whole understanding is essential for the development of numeral concepts (Benz et al. 2015;Fritz et al. 2013;Baroody et al. 2006) and successful mathematical learning (Fischer 1990;Resnick 1983). ...
The ability to perceive structures in sets and to use them to determine cardinality is one important basis for arithmetical learning. This study is based on a theoretical model that distinguishes between the two processes of perception and determination. A total of 95 5-year-old children were interviewed individually to find out whether and how children of this age perceive structures in a visually presented set and whether and how they use these structures to determine the cardinality of the set. To gain insights into the invisible process of perception, eye-tracking was used. Known structures, such as the pattern of a dice-four, seem to play a role in these processes. With the help of an analyzing process consisting of three different types of data, final interpretations were generated that suggest that 5-year-old children can already perceive structures and use them to determine cardinalities. There also seem to be children who are already aware of a structure, but cannot use it to determine the cardinality. This leads to the conclusion that perception and use of structures are possible elements for early mathematical education at this age.
... Siegler (1987) assumed that children automatically associate subtraction with addition, whereas Canobi (2004) suggested that an understanding of addition concepts precede that of subtraction and the addition-subtraction relationship. Baroody (1999) indicated that the inverse relationship between addition and subtraction is difficult for children to understand, thereby adversely affecting children's fluency in simple math-related tasks (Bryant, Christie, & Rendu, 1999;Fischer, 1990). Fluency is important; ultimately, when viewed as a combination of speed and accuracy, it indicates a level of skill mastery that allows unoccupied cognitive resources to be applied toward new and more advanced mathematical tasks (Binder, 1996;Skinner, 1998). ...
This study was designed to investigate the cognitive level of development and mathematical fluency of first grade children. Piaget's conservation-of-liquid task was administered to 97 6- and 7-year-old children from two low-socioeconomic-level elementary schools. Using a counterbalanced design, conserving and non- conserving children completed separately timed lists of addition and subtraction problems. Two one-way analyses of covariance (ANCOVAs) were conducted to evaluate the fixed-factor of conservation level (con- serving and non-conserving) and the two dependent variables (addition and subtraction fluency); age was a covariate. After controlling for the effect of age, the results suggest that conserving children were significantly more fluent in completing addition and subtraction problems than non-conserving children. Overall, children in the same grade and of the same age were at different levels of cognitive development; these levels had an effect on both addition and subtraction fluencies.
... Siswa yang memiliki kepekaan bilangan yang lemah cenderung memiliki masalah dalam memahami konsep-konsep dasar perhitungan matematika (aritmatika), mengalami kesulitan dalam perhitungan, dan juga mengalami masalah dalam penyelesaian masalah matematika yang lebih kompleks (Devlin, 2017;Neergaard 2013;Fosnot & Dolk, 2001). Lebih lanjut, berbagai riset telah menunjukkan dengan meyakinkan tentang hubungan dan pengaruh kepekaan bilangan siswa terhadap penguasaan konsep atau keterampilan matematika lainnya, seperti kemampuan kalkulasi secara mental (Hope & Sherrill, 1987;Trafton, 1992), kemampuan estimasi dalam perhitungan (Bobis, 1991;Case & Sowder , 1990), kemampuan pemecahan masalah (Cobb et.al., 1991), menentukan nilai relatif dari suatu bilangan (Sowder , 1988), dan mengenal hubungan sebagian atas keseluruhan dari suatu bilangan serta konsep nilai tempat (Fischer , 1990;Ross,1989). ...
The current research intends to both identify students' strategies and clarify the level of students' number sense expression of primary students in solving problems relating to addition, subtraction, and multiplication of integers. This is a qualitative research. Five respondents are involved which are randomly taken from a primary school in Lombok Island. The data are collected through test. Meanwhile, data analysis utilize three indicators of number senses in order to answer two main research questions. The findings indicate that, firstly, the strategies used by the students to solve problems relating to addition, subtraction, and multiplication are homogeneous, routine and standardized which are algorithm or procedure oriented, such as column addition, column subtraction, and column multiplication, respectively. Secondly, according to the four level of number sense expressions ─such as 'not yet evident', emerging, expressing, and excelling─, the findings indicate that the students' level of number sense expression is categorized as 'not yet evident', which is the lowest level of the four level.
Abstrak
Tujuan dari penelitian ini adalah untuk mengungkap strategi dan tingkat ekspresi kepekaan bilangan siswa sekolah dasar dalam menyelesaikan masalah terkait dengan operasi bilangan bulat, yang meliputi operasi penjumlahan, pengurangan, dan perkalian. Kajian ini menggunakan pendekatan kualitatif. Responden dalam penelitian ini terdiri atas 5 orang siswa yang diambil secara acak dari sebuah Sekolah Dasar di Pulau Lombok. Data dikumpulkan melalui tes. Analisis data menggunakan tiga indikator kepekaan bilangan untuk menjawab dua pertanyaan penelitian. Hasil penelitian ini menunjukkan bahwa: Pertama, strategi yang digunakan siswa dalam menyelesaikan masalah terkait operasi penjumlahan, pengurangan, dan perkalian bilangan bulat adalah seragam dan terpaku pada penggunaan algoritma atau prosedur berhitung yang bersifat baku dan rutin, yaitu prosedur penjumlahan bersusun, pengurangan bersusun, dan perkalian bersusun. Kedua, berdasarkan empat tingkatan ekspresi kepekaan bilangan, ─yaitu belum nampak, nampak, terungkap, dan terampil─, temuan penelitian ini menunjukkan bahwa tingkat kepekaan bilangan siswa termasuk dalam katagori belum nampak, yaitu level terendah dari 4 tingkatan tersebut.
... Studies related to the acquisition of the part-part-whole (PPW) concept (Clarke, Clarke, Grüßing & Peter-Koop, 2008;Fischer, 1990;Irwin, 1996;Resnick, 1992;Sophian and McCorgray, 1994) have demonstrated that the concepts of part-partwhole, compensation and covariance, are already available in pre-school children, if problems do not require numerical precision. Children are only able to recognise the compensation principle in tasks like '5 + 2' and '4 + 3' numerically precisely, after some months of schooling (Baroody, Wilkins & Tiilikainen, 2003). ...
... The tablet game focused on speeded responses, but also the relationships between quantities were trained (e.g., 9e5 ¼ 8e4 ¼ 4). This knowledge about part-part-whole relations could have encouraged the use of transformation strategies (LeFevre, Sadesky, & Bisanz, 1996) in which the child uses regrouping tricks for calculating the answer (e.g., 9e4 ¼ 8e4þ1) as an efficient alternative to counting (e.g., Fischer, 1990;Hunting, 2003;Putnam, de Bettencourt & Leinhardt, 1990; see also,; Baroody, Bajwa, & Eiland, 2009). Adults mainly use a combination of calculation and retrieval (Campbell & Fugelsang, 2001;Imbo & Vandierendonck, 2008), which may be most advantageous for good mathematics: One cannot memorize all possible sums, and calculation methods are crucial when retrieval fails. ...
In the present study, we aimed to playfully improve arithmetic fluency skills with a tablet game training. Participants were 103 grade 1 children from regular primary schools. The tablet game was tested with a pretest-posttest control group design, and consisted of a racing game environment in which the player competed against a virtual opponent by rapidly solving addition and subtraction problems up to 20. During the 5-week intervention, one group (n = 52) practiced with the game while another group (n = 51) continued regular education without the game. Before, directly after, and three months after the intervention, we applied an arithmetic test to measure simple addition and subtraction skills in both symbolic (Arabic; 4) and non-symbolic (dots; ::) number notations. The intervention group increased significantly more on dot-subtraction efficiency than the control group, an effect which was prominent directly after the intervention. Since i) dot-subtraction is considered to rely more on calculation than the other arithmetic types that we measured and ii) the dot problem-answer representations were not practiced during the intervention, our results suggest that the tablet game promoted arithmetic fluency by benefitting calculation efficiency rather than retrieval efficiency or the switch from calculation to retrieval.
... This study provided initial evidence that nonstandard strategies not only helped students with their math computational skills in terms of an increase in the proportion of correct answers, but also served to simultaneously reduce the time taken to solve computational problems. Fischer (1990) used a pretest-posttest control group design to examine the effectiveness of curricula that concentrated on the cardinal component of number. Participants included 86 kindergarten students broken into one treatment group (n=42) and one control group (n=44). ...
This article summarizes the findings of research studies focusing on number sense instruction to improvemathematics competence of school going children. Twenty-three studies were located that met the inclusion criteria.Interventions gleaned from the review were categorized based on type of instruction (i.e., constructivist, explicit or acombination of the two). Treatment outcomes are discussed in relation to the various instructional approaches,student characteristics (e.g., grade, age), instructional features (e.g., materials, treatment length), assessment (formal,informal) and methodological features. Implications for classroom practice and future research directions areprovided.
... According to him, number sense means knowing what resources are offered by an environment, how to find these resources in activities and how to use them, and to understand and comprehend hidden patterns. Furthermore, researchers reported that children with good number sense can do mental computation (Trafton, 1992), computing estimation (Bobis, 1991), determining the size of the numbers (Sowder, 1988), recognizing part-whole relationships and the concept of digits (Fischer, 1990), and problem solving (Cobb et al., 1991). ...
The purpose of this research is to examine the number sense performance of the classroom teacher candidates taking the Mathematics Education I and II courses. Moreover, it investigates whether there is a change in the number sense performance of the teacher candidates following the Mathematics Education I and II courses. Embedded experimental design was used a mixed methods research design. Pretest-posttest weak experimental design was used in the quantitative part of the study, and a case study for the qualitative part. A total of 74 teacher candidates in the third year of the Faculty of Education, Department of Classroom Teaching at a state university participated in the study. As a data collection tool, the 17-question number sense test was used in both quantitative and qualitative part of the research. The quantitative data showed that there is a significant increase in the number sense performance of teacher candidates after the Mathematics Education I and II courses. The qualitative data indicated that prior to the Mathematics Education I and II courses, the teacher candidates considered mathematics as a course in mathematical operations. They generally tended to use routine rules and algorithms in the number sense test, could not fully conceptualize some mathematical rules and phrases they use, and tried to compute instead of using number sense skills. It was found that after the Mathematics Education I and II courses, there was a decrease in the number of teacher candidates computing, whereas an increase in the number that use number sense.
... Studies related to the acquisition of the part-part-whole (PPW) concept (Clarke, Clarke, Grüßing & Peter-Koop, 2008;Fischer, 1990;Irwin, 1996;Resnick, 1992;Sophian and McCorgray, 1994) have demonstrated that the concepts of part-partwhole, compensation and covariance, are already available in pre-school children, if problems do not require numerical precision. Children are only able to recognise the compensation principle in tasks like '5 + 2' and '4 + 3' numerically precisely, after some months of schooling (Baroody, Wilkins & Tiilikainen, 2003). ...
A common theme of models of conceptual growth is to establish the hierarchical structures of abilities that can be interpreted along developmental lines. Integrating the literature on the development of mathematical concepts and skills in children, a comprehensive 6 level model for describing, explaining and predicting the development of key numerical concepts and arithmetic skills from age 4 to 8, is proposed. Two studies will be presented. In the first study, 1095 preschool children completed a mathematics test (MARKO-D0) based on a 5-level model. The test fitted with a one-dimensional Rasch model. The extension of the model to a sixth level was verified in a new study: 312 first-graders took part in a mathematics test based on the six levels (MARKO-D1). In order to check whether the data of both samples adhered to the principle of unidimensionality, the data of MARKO-0 and MARKO-1 were used in a common analysis for comparative purposes. The applicability of these findings for a qualitative diagnostics and an adaptive training will be discussed.
... We seek to devise instructional settings in which students can first, develop context-bound knowledge of combinations and non-counting strategies-such as identifying five red and three green dots as eight dots-and then, reflect on their activity, and generalise toward more formal reasoning about numbers-such as partitioning eight into five and three to solve the written task 8-3 without counting. Building knowledge of combining and partitioning numbers supports the development of facile calculation (Bobis, 1996;Fischer, 1990;Young-Loveridge, 2002). For addition and subtraction in the range 1 to 20, informal non-counting strategies commonly develop around doubles combinations, combinations with 5 and 10, and tens-complements (9+1, 8+2, 7+3, 6+4, 5+5) (Gravemeijer et al., 2000). ...
Nate was one of 200 participants in a research project aimed at developing pedagogical tools for use with low-attaining 3rd- and 4th-graders. This involved an intervention program of approximately thirty 25- minute lessons over 10 weeks. Focusing on the topic of structuring numbers to 20, the paper describes Nate’s pre- and post-assessments, including major gains on tests of computational fluency. Relevant instructional procedures are described in detail and it is concluded that the procedures are viable for use in intervention.
... Bobis (1996) found that activities with five-year-olds that focused on the visual identification of groups of numbers rather than counting one-by-one, helped children to develop part-whole relationships, especially the decomposition of ten, a key understanding in developing addition facts. Fischer (1990) found that instruction that emphasised partwhole number relationships aided the development of basic number concepts and children's ability to solve addition and subtraction word problems and to deal with place value, even though these applications were not specifically the focus of instruction. ...
As part of the Victorian Early Numeracy Research Project, over 1400 Victorian children in the first (Preparatory) year of
school were assessed in mathematics by their classroom teachers. Using a task-based, one-to-one interview, administered during
the first and last month of the school year, a picture emerged of the mathematical knowledge and understanding that young
children bring to school, and the changes in this knowledge and understanding during the first year of school. A major feature
of this research was that high quality, robust information on young children’s mathematical understanding was collected for
so many children. An important finding was that much of what has traditionally formed the mathematics curriculum for the first
year of school was already understood clearly by many children on arrival at school. In this article, data on children’s understanding
are shared, and some implications for classroom practice are discussed.
Bu araştırma; okul öncesi eğitim kurumuna devam eden 60-72 aylık çocukların matematik yeteneklerinin cinsiyet, daha önce okul öncesi eğitim alma durumu ve aile değişkenleri açısından incelenmesi amacıyla yapılmıştır. Araştırma, Uşak ilinde Millî Eğitim Bakanlığına bağlı bağımsız bir anaokulunda 60-72 ay arasındaki 100 çocuk üzerinde yürütülmüştür. Araştırmada veri toplama aracı olarak “Genel Bilgi Formu ve Erken Matematik Yeteneği Testi (TEMA-3)” kullanılmıştır. Verilerin analizinde bağımsız t testi ve tek yönlü varyans analizi ANOVA kullanılmıştır. Yapılan araştırmanın bulgularında cinsiyet, anne-babanın yaşı ve ebeveynlerin çocukla oyun oynama sıklığı ile çocukların matematik yetenekleri arasında anlamlı bir ilişki bulunmamıştır. Buna karşın çocuğun daha önce eğitim kurumuna gitmesi, annenin çalışması, ailedeki çocuk sayısı, anne ve babanın eğitim durumu, ailenin gelir düzeyi ve ebeveynlerin çocuklarına kitap okuma sıklığı değişkenlerinin çocukların matematik yetenekleri açısından anlamlı farklılıklar ortaya çıkardığı belirlenmiştir.
Teachers’ decision-making and pedagogical reasoning often become visible in their lesson planning sessions. This study explores how the professional learning opportunities presented by teacher-facilitator interactions in planning sessions. By tracing shifts in teachers’ discourse in planning sessions, we examine the development of their pedagogical reasoning. The findings of this study illustrate that these interactions create four kinds of professional learning opportunities: (1) reflecting on students’ learning in past lessons; (2) attending to the details of the complexity of teaching; (3) negotiating and adjusting lesson goals and activities by making direct and informed decisions for teaching; and (4) anticipating specific features of instruction. This study also illustrates how refining and elaborating teaching complexity supports teachers in re-conceptualizing and expanding their teaching knowledge.
This paper looks at a key aspect of numeracy, quantification, the process for determining how many things are in a group. Things can be quantified by counting or by subitizing (knowing just by looking). Many mathematics educators see counting as the first step towards more advanced mathematical understanding. However, there is some evidence to suggest that, for some children, subitizing is well-established before counting. There seems to be reasonable agreement that children need to understand about the relationships between the parts and the whole (part—whole thinking). This paper looks at ways to support children's part—whole thinking in the early childhood years, and the use of egg cartons to create three-dimensional tens-frames and six-frames for this purpose. Collecting up treasures to put in the compartments of an egg carton capitalises on young children's ‘accumulation intent’, and helps them to appreciate the way that numbers are composed of other numbers.
Resumen: Se presentan algunas estrategias de enseñanza diseñadas e implementadas en el marco de un estudio que tuvo como propósito favorecer la comprensión del valor posicional. Se adopta un enfoque conceptual de la comprensión en términos del desarrollo gradual de habilidades, y de la construcción progresiva de unidades (simples o compuestas) para incrementar la flexibilidad en el conteo, en la realización de particiones, en la agrupación y en la comparación de cantidades. Los hallazgos evidencian que las estrategias diseñadas permitieron hacer frente a las dificultades que presentaban los estudiantes, quienes avanzaron en su comprensión.
Palabras clave: Educación matemática infantil, valor posicional, estrategia de enseñanza, Comprensión.
This chapter focuses on the connections that can be made by using multiplication and division problem-solving contexts in early childhood education and school settings. Prior to starting school, young children experience many opportunities to make groups using familiar objects, beginning with groups of two and then moving to larger groups such as five and ten. Typically, children begin by using units of one, as shown in counting one-by-one. However, children should experience “groups of” objects larger than one (composite units) early on in their schooling. Another key idea for children to understand is the concept of additive composition, the way that numbers are composed of other numbers (part–whole relationships). The connections are explored between mathematics learning in informal and formal settings; ordinality and cardinality; composing and decomposing quantities; operations and processes; and word problems and representations. To illustrate these connections, we draw on a two-year study undertaken with 84 culturally and linguistically diverse five- to eight-year-olds. During the study, children participated in a series of lessons where they solved multiplication and division problems involving naturally occurring groups of twos, fives, and tens using a variety of materials and multiple representations. Results for the 35 five-year-olds showed improvement in number knowledge, addition and subtraction, early place-value understanding, as well as multiplication and division.
True personal stories are used to introduce some of the research into pre-school children’s development of number knowledge and skills. A range of conversations and stimulating environments illustrate how parents, grandparents, peers, and early childhood professionals support the mastery of new number words and concepts as well as mathematical actions, in everyday contexts and play situations. The stories discuss the learning of real children developing knowledge and skills in the pre-school years. They tell about early quantity identification along with some young children’s growth of interest in and skills with cardinal and ordinal number and counting; learning about more and less, then very simple addition and subtraction; early recognition and naming of multiplication “arrays”; written numeral identification; and one child’s earliest abstract understanding of the idea of infinity. For each of these topics, some research on pre-school learning is outlined. The growth of children’s self-concepts as they handle mathematics and the situatedness of learning in varied and everyday, informal learning contexts are supplementary themes of this chapter.
Numbers parts-whole relationship is mainly represented by number bonds in the Malaysian year one mathematics textbooks. Seldom tasks are given to encourage students to create their own representations. The purpose of this study was to investigate how year one students represent and solve word problems based on numbers parts-whole relationship. On this account a cross sectional survey was conducted to collect data from a purposive sample of 42 year one students chosen based on enrollment and their mathematics performance in LINUS. Their numbers parts-whole representation was examined with the ‘Number Pairs Instrument’ and a short interview. Results indicated that though the students have been taught about using representations for finding number pairs, they faced difficulty in producing accurate and complete representations by themselves to solve the word problems. This study shows that year one students need guidance and practice for creating efficient representations to solve parts-whole word problems which require multiple answers.
Nuestra propuesta parte de la necesidad de aportar al niño herramientas que le faciliten la comprensión del entorno que le rodea, respetando su singularidad y las capacidades innatas que tiene. A lo largo de estos últimos años, en nuestras visitas a colegios, hemos encontrado casos en uno u otro lugar de niños y niñas que tras la escolarización perdían capacidades intuitivas de cálculo o dejaban de reconocer por ejemplo propiedades de formas geométricas que conocían por descubrimiento; por ello, queremos que se lea el presente manual desde la intencionalidad del respeto al crecimiento personal de cada niño, de cada niña, tanto de forma individual como colectiva.
La matemática no puede continuar siendo un espejismo que distancia la escuela y la realidad, hemos de vivenciar el aprendizaje a través de situaciones simbólicas de la realidad u observando el entorno del niño. La escuela debe despertar en los niños y niñas la necesidad de investigar, de conocer, de descubrir, de imaginar, de buscar, … verbos todos ellos que inviten a la acción, tanto dentro como fuera del aula, que el maestro muestre dando ejemplo con su propia práctica.
Although the early development of children’s number sense is a strong
predictor of their later mathematics achievements, it has been
overlooked in primary schools in Malaysia. Mainly attributable to
underdeveloped number sense of Malaysian primary and secondary
school children, their inability to handle simple mathematics tasks,
which require the understanding of basic mathematical concepts,
numerical estimation, and mental computation, is a cause for worry. To
enhance the perception of curriculum designers and mathematics
teachers about why and how number sense ought to be developed, this
article serves as a review of essential components, theoretical
framework, and test of number sense.
The development of mathematical competence -- both by humans as a species over millennia and by individuals over their lifetimes -- is a fascinating aspect of human cognition.
Mathematics education emphasizes on nurturing number sense, but researches on this have been scarce, and most of them has been confined to elementary level students. This thesis, therefore, tried to analyze how elementary students solve mathematics sense problems in order to give some insight into how to teach number sense. For this, this thesis categorized into two ways of using number sense and algorithm as problem solving, and analyzed students` responses using test sheets. Accordingly, middle school students showed higher score on the number sense test and higher rates of using number sense than elementary students. In addition, students showing higher achievement used both number sense and algorithm, but those of lower achievement were more likely to use only algorithm. Plus, among students showing higher achievement, middle school students used more number sense than elementary school students, but there was not meaningful difference among those showing lower achievement. Lastly, It was shown that there was difference in the rate using number sense according to the number sense components.
As part of the Victorian Early Numeracy Research Project in Australia, over 1400 children in the first (Preparatory) year of school in the state of Victoria were assessed in mathematics by their classroom teachers. Using a task-based, one-to-one interview administered during the first and last month of the school year, a picture emerged of the mathematical knowledge and understanding that young children bring to school and how this develops during the first year of school. The same interview was conducted after translation into German with around 850 kindergarten children (five-year-olds) in the north-western region of Germany by preservice teachers from the University of Oldenburg. In this paper, based on an extensive international literature review that guided the development of the interview protocol, the data on children’s mathematical understanding and its development during the preparatory grade (Australia) and the final year in Kindergarten (Germany) are shared and analysed comparatively. Finally, implications for classroom practice are discussed.
The role of dynamic assessment involving two instructional strategies, a specific and a general instructional strategy, on experts' and novices' error patterns in solving part-whole addition and subtraction word problems was examined. Experts in the specific strategy group (SSG) showed a decrease in errors from the pretest to transfer posttests and maintained the low frequency of errors on the delayed posttest. In addition, the two strategies differentially influenced the kinds of errors made by experts on the posttests following dynamic assessment. However, error patterns of novices in the two groups did not show a specific trend of decrease in the number and kind of errors.
The ability to count has traditionally been considered an important milestone in children's development of number sense. However, using counting (e.g., counting on, counting all) strategies to solve addition problems is not the best way for children to achieve their full mathematical potential and to prepare them to develop more complex and advanced computational skills. In this experimental study, we demonstrated that it was possible to teach children aged 5–6 to use decomposition strategy and thus reduced their reliance on counting to solve addition problems. The study further showed that children’ ability to adopt efficient strategies was related to their systematic knowledge of the part–part–whole relationship of the numbers 1–10.
This study compared number sense instruction in three first-grade traditional mathematics textbooks and one reform-based textbook (Everyday Mathematics (EM)). Textbooks were evaluated with regard to their adherence to principles of effective instruction (e.g., big ideas, conspicuous instruction). The results indicated that traditional textbooks included more opportunities for number relationship tasks than did EM; in contrast, EM emphasized more real-world connections than did traditional textbooks. However, EM did better than traditional textbooks in (a) promoting relational understanding and (b) in- tegrating spatial relationship tasks with other more complex skills. Whereas instruction was more di- rect and explicit and feedback was more common in traditional textbooks than it was in EM, there were differences among traditional textbooks with respect to these two criteria. Although EM excelled in scaffolding instruction by devoting more lessons to concrete and semiconcrete activities, traditional textbooks provided more opportunities for engaging in all three representations. However, EM em- phasized (a) a variety of models to develop number sense concepts, (b) a concrete, or semiconcrete, to symbolic representational sequence, and (c) hands-on activities using real-world objects to enhance learner engagement. Finally, even though traditional textbooks excelled over EM in providing more opportunities to practice number sense skills, this finding may be an artifact of the worksheet format employed in traditional textbooks. At the same time, adequate distribution of review in subsequent lessons was evident in EM and in only one of the traditional textbooks. Implications for practice in ac- cessing the general education curriculum for students with learning problems are discussed.
This article reviews research on the achievement outcomes of three types of approaches to improving elementary mathematics: mathematics curricula, computer-assisted instruction (CAI), and instructional process programs. Study inclusion requirements included use of a randomized or matched control group, a study duration of at least 12 weeks, and achievement measures not inherent to the experimental treatment. Eighty-seven studies met these criteria, of which 36 used random assignment to treatments. There was limited evidence supporting differential effects of various mathematics textbooks. Effects of CAI were moderate. The strongest positive effects were found for instructional process approaches such as forms of cooperative learning, classroom management and motivation programs, and supplemental tutoring programs. The review concludes that programs designed to change daily teaching practices appear to have more promise than those that deal primarily with curriculum or technology alone.
Shows how mathematics is integrated into everyday hands-on activities and problem-solving situations in one preschool. Provides examples of how themes emerge from children's interests and are incorporated in classroom learning centers. Asserts that meaningful teaching methods and pictorial demonstrations and manipulatives engage preschoolers in active thinking about and use of mathematics. (KB)
Simplicity in number naming in a language (e.g. ‘ten‐two’ in Chinese is simpler than the irregular ‘twelve’ in English) has been used to explain cross‐cultural disparities in children's computational competence. In contrast to previous research focusing only on whether children can provide the correct answers, in this study (N = 117 and 92) we examined Chinese pre‐schoolers' computational strategies in depth and individually so as to examine their understanding of the base‐10 system and place value. The results showed that despite the fact that many can give the correct answers, there is strong evidence that Chinese pre‐schoolers do not have adequate understanding of the base‐10 number system and place value, suggesting that the advantages of the simpler number‐naming system are limited.
This study examined the effectiveness of a number sense program on kindergarten students' number proficiency and responsiveness to treatment as a function of students' risk for mathematics difficulties. The program targeted development of relationships among numbers (e.g., spatial, more and less). A total of 101 kindergarten students (not at risk: 22 control and 36 experimental; at risk: 18 and 25) from five classrooms in a high-poverty elementary school participated in the study. Using a quasi-experimental design, classrooms were randomly assigned to either the intervention (number sense instruction, NSI) or control condition. Results indicated significant differences favoring the treatment students on all measures of number sense (e.g., spatial relationships, more and less relationships, benchmarks of five and ten, nonverbal calculations) at posttest and on a 3-week retention test. Furthermore, the effects were not mediated by at-risk status, suggesting that NSI may benefit a wide range of students. Implications in terms of preventing early mathematical learning difficulties are discussed.
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