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Models of Problem Solving: A Study of Kindergarten Children's Problem-Solving Processes
Abstract
Seventy kindergarten children who had spent the year solving a variety of basic word problems were individually interviewed as they solved addition, subtraction, multiplication, division, multistep, and nonroutine word problems. Thirty-two children used a valid strategy for all nine problems and 44 correctly answered seven or more problems. Only 5 children were not able to answer any problems correctly. The results suggest that children can solve a wide range of problems, including problems involving multiplication and division situations, much earlier than generally has been presumed. With only a few exceptions, children's strategies could be characterized as representing or modeling the action or relationships described in the problems. The conception of problem solving as modeling could provide a unifying framework for thinking about problem solving in the primary grades. Modeling offers a parsimonious and coherent way of thinking about children's mathematical problem solving that is relatively straightforward and is accessible to teachers and students alike.
... A sociopolitical perspective creates rehumanizing mathematical learning experiences that allow MTEs to work toward dismantling and disrupting oppressive structures (Louie et al., 2021) while attending to issues of power, status, agency, and positive identity development (Gutiérrez, 2018;Louie et al., 2021;Morales & DiNapoli, 2018). In this conceptual article, we use a sociopolitical perspective informed by critical theories (i.e., Critical Theory and Critical Race Theory) to consider the bounds of the use of Cognitively Guided Instruction (CGI: Carpenter et al., 1989Carpenter et al., , 1993Fennema et al., 1996) as a widely used researchbased mathematical instructional framework. CGI is one of the most influential frameworks currently in use in mathematics teacher education and professional development (Battista et al., 2009;Celedón-Pattichis et al., 2018). ...
... Three decades ago, CGI investigations presented "one of the first models for integrating research on learning with research on teaching," and has since "played a vital role in the national reform movement" (Jacobs, 2004, p. 134). Initial studies focused on meticulously analyzing children's intuitive ideas about number and arithmetic operations, and revealing the complexity of children's thinking and the varied ways that children respond to different types of mathematics problems (Carpenter, 1985;Carpenter et al., 1993Carpenter et al., , 1998. Also, concurrently investigated alongside teachers, was the implementation and use of children's mathematical thinking for classroom instruction, with a research-based professional development program named CGI (Carpenter et al., 1989;Fennema et al., 1996). ...
... When we talk about children's solution strategies, we are careful not to present them as hierarchical and instead, explicitly challenge hierarchical interpretations in several ways. First, we give special attention to the power and beauty of direct modeling strategies, a powerful research finding (Carpenter et al., 1993), to counter common perceptions of modeling as low-level or basic. Concrete models support students to "move beyond stating the mathematical relationships that exist and justify why they exist . . . ...
Elementary mathematics teacher education often draws on research-based frameworks that center children as mathematical thinkers, grounding teaching in children’s mathematical strategies and ideas and as a means to attend to equity in mathematics teaching and learning. In this conceptual article, a group of critical mathematics teacher educators of color reflect on the boundaries of Cognitively Guided Instruction (CGI) as a research-based mathematical instructional framework advancing equity through a sociopolitical perspective of mathematics instruction connected to race, power, and identity. We specifically discuss CGI along the dominant and critical approaches to equity outlined by Gutiérrez’s (2007, 2009) framework. We present strategies used to extend our work with CGI and call for the field to continue critical conversations of examining mathematical instructional frameworks as we center equity and criticality.
... Having examined what is required to understand multiplication and division concepts, we turn our attention to international studies in which children's reasoning was examined. It has been reported that young children can solve multiplication and division problems by combining direct modelling with counting and grouping skills, and with thinking based on addition and subtraction (Anghileri, 1989;Becker, 1993;Bicknell et al., 2016;Carpenter et al., 1993;Kouba, 1989;Mulligan, 1992;Mulligan & Mitchelmore, 1997). The aforementioned studies were conducted prior to any formal instruction in multiplication and division and often in a one-to-one interview setting. ...
... The aforementioned studies were conducted prior to any formal instruction in multiplication and division and often in a one-to-one interview setting. Problems were either presented in a physical context (e.g., Anghileri, 1989) or on cards, and children could use materials to find solutions or construct a physical representation (e.g., Bicknell et al., 2016;Carpenter et al., 1993;Kouba, 1989;Mulligan & Mitchelmore, 1997). In some studies, it was reported that children used intuitive counting-based strategies to find a solution (Becker, 1993;Carpenter et al., 1993;Nunes & Bryant, 1996;Sophian & Madrid, 2003). ...
... Problems were either presented in a physical context (e.g., Anghileri, 1989) or on cards, and children could use materials to find solutions or construct a physical representation (e.g., Bicknell et al., 2016;Carpenter et al., 1993;Kouba, 1989;Mulligan & Mitchelmore, 1997). In some studies, it was reported that children used intuitive counting-based strategies to find a solution (Becker, 1993;Carpenter et al., 1993;Nunes & Bryant, 1996;Sophian & Madrid, 2003). These strategies included direct modelling (using physical objects, fingers, or drawings), counting by ones, one-to-many correspondence or many-to-one count strategy, and rhythmic counting. ...
Children’s multiplicative thinking as the recognition of equal group structures and the enumeration of the composite units was the subject of this research. In this paper, we provide an overview of the Multiplication and Division Investigations project. The results were obtained from a small sample of Australian children ( n = 21) in their first year of school (mean age 5 years 6 months) who participated in a teaching experiment of five lessons taught by their classroom teacher. The tasks introduced children to the “equal groups” aspect of multiplication. A theoretical framework of constructivist learning, together with research literature underpinning early multiplicative thinking, tasks, and children’s thinking, was used to design the research. Our findings indicate that young children could imagine equal group structures and, in doing so, recognise and enumerate composite units. As the children came to these tasks without any prior formal instruction, it seemed that they had intuitive understandings of equal group structures based on their life experiences. We argue that the implications for teaching include creating learning provocations that elicit children’s early ideas of multiplication, visualisation, and abstraction. The research has also shown the importance of observing children, listening to their explanations of their thinking, and using insights provided by their drawings.
... Smith and Smith (2006) described the four basic interconnected concepts that underlie multiplication as: quantity, multiplicative problem situations, equal groups, and units, and also suggested that these can develop from experiences using counting and grouping strategies when solving contextualized problems in the early grades. Likewise, Carpenter, Ansell, Franke, Fennema and Weisbeck (1993) demonstrated that children in their first year of school in the United States could solve simple multiplication and division word problems. Most children in their study modelled the problem using freely available counters, though a small number of children solved them without modelling. ...
... 421). International studies that reported the ways in which young children in the early years of primary school, and prior to formal instruction in multiplication and division, solved such problems, often did so through task-based one-on-one interviews (Anghileri, 1989;Bicknell, Young-Loveridge, & Nguyen, 2016;Carpenter et al., 1993;Clark & Kamii, 1996;Kouba, 1989). The studies used prompts such as bare number facts (e.g., 3 × 4), or closed word problems, and materials to find solutions or model the mathematical construct e.g., equal groups, arrays. ...
Children’s multiplicative thinking as the visualization of equal group structures and the enumeration the composite units was the subject of this study. The results were obtained from a small sample of Australian children (n = 18) in their first year of school (mean age 5 years 6 months) who participated in a lesson taught by their classroom teacher. The 12 Little Ducks problem stimulated children to visualize and to draw different ways of making equal groups. Fifteen children (83 %) could identify and create equal groups; eight of these children (44 %) could also quantify the number of groups they formed. These findings show that some young children understand early multiplicative ideas and can visualize equal group situations and communicate about these through their drawings and talk. The study emphasises the value of encouraging mathematical visualization from an early age; using open thought-provoking problems to reveal children’s thinking; and promoting drawing as a form of mathematical communication.
... In this study, while developing the mathematical problem types test, the stages specified by Turgut Baykul (2015), which also includes the steps given above, were applied. • Determining the purpose of the test • Determining the behaviors to be measured by the test • Writing the items • Reviewing the written items • Preparing the trial form to be pre-applied • Pre-application • Evaluating the results of the pre-application and making item analysis • Item selection • Revealing the final version of the test and making statistics In this study, based on the classification of Carpenter et al. (1993), an application was carried out by taking into account the researches of Van de Walle, Karp andBay-Williams (2014) andMarshall (1991; as cited in Kasap & Ergenekon, 2017). Since the difference between "sharing" and "grouping division" within the classification made by Carpenter et al. (1993) is not clear, the questions in this class are classified as "split division" by Van de Walle, Karp and Bay-Williams (2014). ...
... • Determining the purpose of the test • Determining the behaviors to be measured by the test • Writing the items • Reviewing the written items • Preparing the trial form to be pre-applied • Pre-application • Evaluating the results of the pre-application and making item analysis • Item selection • Revealing the final version of the test and making statistics In this study, based on the classification of Carpenter et al. (1993), an application was carried out by taking into account the researches of Van de Walle, Karp andBay-Williams (2014) andMarshall (1991; as cited in Kasap & Ergenekon, 2017). Since the difference between "sharing" and "grouping division" within the classification made by Carpenter et al. (1993) is not clear, the questions in this class are classified as "split division" by Van de Walle, Karp and Bay-Williams (2014). Van de Walle, Karp and Bay-Williams (2014) have been preferred as the only classification name, as the "split division" classification includes both types of questions included in the classification of "sharing" and "grouping division". ...
Çalışmanın amacı ortaokul öğrencilerinin matematiksel problem türlerinde ortaya koydukları performanslarını belirlemek için bir ölçek geliştirmektir. Literatür incelendiğinde ölçek geliştirme çalışmalarının çoğunda ortak olarak belirtilen aşamalar; madde havuzu oluşturma, uzman görüşü, deneme uygulaması ve geçerlik- güvenirlik aşamalarıdır. Bu çalışmada içerisinde belirtilen aşamaların da yer aldığı Turgut & Baykul (2015) tarafından ifade edilen aşamalar kullanılmıştır. “Matematiksel Problem Türleri Testi” öğrencilerin literatürde geçen matematiksel problem türlerinden çözebildikleri veya çözmekte zorlandıkları problem türlerini belirlemeyi amaçlamaktadır. Test, alt boyutlarını da oluşturan, literatürden yola çıkarak belirlenen 8 matematiksel problem türünü içeren toplam 30 çoktan seçmeli maddeden oluşmaktadır. Belirlenen problem türleri ve aynı zamanda testin alt boyutları, “ayırma”, “birleştirme”, “çarpma”, “karşılaştırma”, “parçalamalı bölme”, “kalanlı bölme”, “çok adımlı”, “rutin olmayan” problemlerdir. Sonuç olarak “Matematiksel Problem Türleri Testi”nin geçerli ve güvenilir olduğu ortaya konmuştur.
... This study shows that even in problems with difficult ratios, almost half of the children can already use one-to-many correspondences, an important step in the development of proportional reasoning. Carpenter et al. (1993) used easier ratios (e.g., 4:1, 3:1, and 2:1) in a variety of basic word problems and obtained higher success rates than Kouba (1989), approximately three-fourths of the children used appropriate strategies. Becker (1993) assessed preschooler's use of counting in many-to-one situations (e.g., "How many cookies do you need to give every doll two cookies?"). ...
... Becker (1993) assessed preschooler's use of counting in many-to-one situations (e.g., "How many cookies do you need to give every doll two cookies?"). He used the same ratios as Carpenter et al. (1993) and obtained a similar success rate, which led him to the conclusion that children's appropriate use of counting in many-to-one situations develops during the period from 4 to 5.5 years old. Nunes and Bryant (2010) also pointed out that many children already use the schema of one-to-many correspondence even before being taught multiplication and division in school. ...
The present study cross-sectionally investigated proportional reasoning abilities in 5-to 9-year-old children (n = 185) before they received instruction in proportional reasoning. This study addressed two important aspects of the development of proportional reasoning that remain unclear in the current literature: (1) the age range in which it develops and (2) the influence of the nature of the quantities (discrete or continuous) on children's performance. Three proportional reasoning tasks (i.e., one with two discrete quantities, one with a discrete and a continuous quantity, and one with two continuous quantities) were used. A two-step cluster analysis was conducted on the groups of children based on qualitative differences in understanding. Six different early stages of proportional reasoning were revealed, showing differences in understanding depending on the nature of the quantities involved and which quantity was unknown. The development of proportional reasoning starts at a very early age but it is not yet fully mastered at the age of 9.
... [24]). The ability to solve mathematical problems develops in later preschool years [24,31], however, studies have shown that already four to five-year-old children are able to solve mathematical problem-solving tasks including problems that require arithmetic skills [30,32] as well as geometrics and combinatorics [30]. ...
... Squire and Bryant (2002b) mention that teachers' introduction of division problems usually involves sharing problems. Research evidence shows that teachers are prone to using more sharing actions instead of grouping actions (Carpenter et al. 1993;Fischbein et al. 1985). In South Africa, Roberts (2012) posits in a theoretical paper that when teachers introduce the idea of division, the idea of division is normally linked to a sharing model. ...
Background: Internationally, the teaching of division has noted that the use of sharing situations with sharing actions (one-by-one distribution) is the predominant division model at the beginning of schooling. In South Africa, research suggests a sharing situation with sharing actions is also preferred in the early grades.Aim: This paper aims to look at the predominant approaches to the use of division actions that teachers offer in teaching division tasks.Setting: The study is set in three government schools in Gauteng, South Africa.Methods: In this qualitative study, the teachers were observed through video recording, and then the video recording was transcribed, and semiotics was used to make sense of their teaching.Results: The findings of this article suggest that grouping actions and group-based approaches to teaching division tasks are more prevalent than sharing through one-by-one distribution actions, even when sharing situations are used.Conclusion: This study concludes that grouping actions and group-based approaches are part of how teachers solve sharing situations.Contribution: This study concludes that in a South African context, identifying the grouping actions and group-based approaches linked to sharing situations is a more efficient way of solving sharing situations and will assist teachers in explaining division tasks more coherently.
... According to Carpenter et al. (1993), if given the opportunity, children as young as kindergarten can invent strategies to solve various problems. Hence, the instruction should build upon students' existing knowledge on informal and intuitive strategies (Carpenter & Fennema, 1992). ...
This study examined 91 preservice mathematics teachers’ strategy repertoire that they referred to when solving one direct and one inverse proportion missing-value word problems. When encouraged to provide multiple solutions, the preservice teachers exhibited the ability to solve the two problems using more than one strategy. However, they used a significantly greater number of strategies for solving the direct than for the inverse proportion problem. The most frequently used strategies for the two problems were the cross-multiplication and across-multiplication, respectively, and many of the preservice teachers also used these two strategies as their first strategy. On the other hand, the number of strategies applied by the preservice teachers did not differ significantly according to their first choice of strategies. A key finding of this study was that the preservice teachers possessing less common strategies in their repertoire had a significantly larger strategy repertoire than those who had more common strategies.
... This late introduction is remarkable considering the increasing evidence that points towards the early emergence of proportional reasoning abilities. Many studies have shown that the understanding of one-to-many correspondence and multiplicative relations-two important precursors of proportional reasoning (Nunes & Bryant, 2010;-already emerges in the last year of kindergarten and the first years of elementary school, while the operation of multiplication is typically only introduced in second grade, and proportional reasoning even a few years later (e.g., Becker, 1993;Carpenter et al., 1993;Frydman & Bryant, 1988;Nunes & Bryant, 2010). Although there are not many studies that investigated the early development of proportional reasoning abilities, those that did so indicate that it already starts between the ages of 5 and 9 and that these young children can already reason proportionally in certain situations (Boyer & Levine, 2012;He et al., 2018;Hurst & Cordes, 2018;Resnick & Singer, 1993;Van Dooren et al., 2018;. ...
Not only children but also adolescents and adults encounter great difficulties in learning to reason proportionally. Despite these difficulties, research increasingly shows that proportional reasoning emerges early, before it is being instructed in school. There have however been very few attempts to stimulate this early emerging ability. The aim of the present study was to stimulate proportional reasoning in second graders. We developed an intervention program focusing on quantitative reasoning and promoting different strategies to solve proportional missing-value problems. The effectiveness of the program was evaluated in a pretest-intervention-posttest study with a control group (n = 139). Results showed a large effect of the intervention program on children’s proportional reasoning abilities in fair-sharing situations and a small transfer effect to word problem solving. There was also a moderate effect on the proportional vocabulary that was explicitly taught in the intervention program, but no transfer effect to proportional vocabulary not explicitly taught.
... Seriam essas propostas com etapas tão específicas aplicáveis à Educação Infantil? Carpenter et al (1993) realizaram um estudo no qual constataram que as crianças da Educação Infantil podem resolver uma gama de problemas incluindo problemas que envolvem as operações matemáticas básicas e, que em geral, a maioria das crianças utilizam estratégias nas quais representam ou modelam uma ação ou as relações descritas nos problemas propostos. Syaodih et al (2018) apud Dyah e Setiawati (2019) explicam que a habilidade de resolução de problemas no início da infância é essencial porque quando as crianças resolvem problemas, elas podem criar a capacidade de pensar logicamente, criticamente e sistematicamente. ...
Este trabalho aborda uma pesquisa qualitativa e apresenta uma proposta de intervenção em sala de aula da Educação Infantil, em uma turma de crianças pequenas da faixa etária de 4 anos a 5 anos e 11 meses. A proposta integra uma pesquisa de Mestrado em andamento no Programa de Pós-Graduação em Ensino de Ciências e Matemática da Universidade Federal de Alagoas. Esta proposta é composta por uma atividade que consiste na aplicação de um problema verbalizado oralmente pelo professor, cuja resolução é acompanhada de material concreto, com a finalidade de contribuir para o desenvolvimento do sentido de número, primando pelo desenvolvimento dos processos mentais, sendo um caminho que leve ao desenvolvimento das habilidades de resolução de problemas na Educação Infantil. Os resultados da pesquisa demonstraram que a aplicação da resolução de problemas verbalizados oralmente é possível, sendo essencial que o professor planeje a atividade valorizando a ludicidade e a interação entre as crianças, favorecendo o desenvolvendo da linguagem e das manifestações espontâneas das ideias matemáticas ligadas aos processos mentais que desencadeiam a construção do sentido de número.
... Previous studies have demonstrated that young children (4-5-year-olds) can model division problems using concrete materials before having any formal instruction (e.g., Carpenter et al., 1993), and that children's early understanding of division is underpinned by their experiences of sharing and allocating portions (Correa et al., 1998;Squire & Bryant, 2002). However, Correa et al. argue that an understanding of sharing is not the same as having an understanding of division, as division requires as understanding of the inverse relation between the divisor and the quotient. ...
The purpose of this study is to configure the landscape of empirical mathematics educational research on problem-solving in teacher education, and thereby disentangle how mathematical problem-solving is understood and used. The method consists of a configurative review of empirical mathematics education research on problem-solving in teacher education. A two-dimensional model is presented to illustrate how different aspects of problem-solving in teacher education are connected to and complement each other. Using the model, the configuration results in the proposition of four major categories of research on problem-solving in teacher education. The result indicates an almost equal distribution of research which views problem-solving as an aim for mathematics education versus research which views problem-solving as a means for learning mathematics. However, within the former, roughly three quarters of the articles focus on content knowledge, and only a quarter on pedagogical content knowledge. Implications for teacher education and future research are discussed.
... For example, Gelman (2006) and Zur and Gelman (2004) found that children aged 2½ to 4 years could demonstrate some arithmetic reasoning, and a grasp of the cardinality of numbers. Children as young as 3 years of age show themselves to be competent logical problem-solvers (Andrews & Trafton, 2002;Carpenter et al., 1993) and even 2-year-olds are able to grasp concepts such as one-one correspondence (Tirosh et al., 2020). Ginsburg et al. (2008) found children aged 3 to 5 years old to be capable of grasping and exploring a range of mathematical concepts, including abstract ones as well as ones relating to objects. ...
Mathematics in early years settings is often restricted to learning to count and identifying simple shapes. This is partly due to the narrow scope of many early years curricula and insufficient teacher training for exploring deeper mathematical concepts. We note that geometry is an area particularly neglected. In an innovative year-long project, a group of university-based mathematicians and early years teachers collaborated on a child-led exploration of ‘patterns in nature.’ The early years teachers ran the project within the setting, meeting regularly with the mathematicians to discuss potential areas of interest, and to highlight the children’s mathematical thinking. We found that, with the appropriate environment and guidance, the children naturally displayed deep levels of geometrical thinking and found enjoyment and satisfaction in the exploration of mathematical ideas. We define what we mean by the term ‘deep geometrical thinking’ and demonstrate this by looking at three excerpts through the lens of the van Hiele levels of geometric thought, finding that the children are capable of exhibiting thought at level 3 (abstraction), more advanced than previously thought of children of this age. Using a second taxonomy we also assess the range of skills across which they are demonstrating such geometric thought.
... This article explains, among other aspects, how a teacher proposes seven problems to her 5-year-old students and the development of the sessions where they solve them. The problems and the resolution strategies used by the children were classified based on a table prepared by the authors of the reading article and based on the typology of Carpenter et al. (1993). Table 2 summarized task A. ...
This work aims to identify the criteria to design activities based on problem-solving tasks that emerge when future early childhood education teachers jointly plan their activities and reflect on them. The participants were 76 students from the Didactics of mathematics subject that was carried out in the 2nd year of the Early Childhood Education Degree of a Catalan public university. This is qualitative research in which the phases of the thematic analysis have been adapted: familiarizing with the data; systematically applying the categories to identify the student criteria emerged; triangulating the analysis with experts; reviewing and discussing the results. The Didactic Suitability Criteria (DSC), from the Ontosemiotic approach (OSA) framework to design tasks and their indicators, were used to categorise and analyse the tasks performed by future teachers. As a result, it was identified that when the future teachers adopt consensually design their activities, they are implicitly based on the Didactic Suitability Criteria (DSC). Still, not all their indicators emerge since their reflection is spontaneous and is not guided by an explicit guideline that serves them to show their didactic analysis in detail. The study concludes that it would be convenient to offer future teachers a tool, such as DSC, to have explicit criteria to guide the designs of their mathematical tasks. In this sense, a future line of research opens, much needed, to adapt the DSC to the singularities of this educational stage.
... It seems clear that this ability may be related with the capability of understanding the semantic structure of the mathematical problems statements and increases with age. Children could be able to solve real world problems at early age [14], but the acquisition of academic language comes later, after they dominate everyday language [15]. The relation between problem solving and linguistic comprehension is explored in [16]. ...
The objective is to study the evolution of different characteristics of a population through time. These response variables may be related for each experimental unit, and in addition, the observations for each response may as well be correlated with time, producing a complex correlation structure. The number of responses that can be observed is usually limited for budget, resources, or time reasons, and thus the selection of the most informative time points when data must be taken is quite convenient. This will be performed by using the optimal design of experiments techniques. Some analytical results will be shown, and the results will be applied to obtain the most convenient points when tests about two variables related with the capability of the resolution of mathematical problems in primary school students should be performed.
... He put 3 guppies in each jar. How many jars did Tad put guppies in?" children demonstrated many-to-one counting by counting out 15 guppies into groups of 3, and then counting the number of groups (Carpenter et al., 1993). However, these strategies usually require external support and small set sizes. ...
Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to perform symbolic and non-symbolic approximate division. Subjects were presented with non-symbolic (dot array) or symbolic (Arabic numeral) dividends ranging from 32 to 185, and non-symbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children (Experiment 1 N = 89, Experiment 2 N = 42) and adults (Experiment 3 N = 87) were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide non-symbolically mediated the relation between Approximate Number System (ANS) acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction.
... Several studies have shown that the development of this kind of one-step or simple multiplicative reasoning involving the schema of one-to-many correspondence starts in early childhood, before the start of formal instruction in multiplicative reasoning or proportional reasoning (e.g., Becker, 1993;Carpenter et al., 1993;Frydman & Bryant, 1988;Kouba, 1989;Nunes & Bryant, 2010). Others reported evidence of the emergence of actual proportional reasoning abilities in early childhood as well (Boyer & Levine, 2012;He et al., 2018;Hurst & Cordes, 2018;Resnick & Singer, 1993;Vanluydt et al., 2020). ...
The present study longitudinally investigated proportional reasoning abilities in early elementary school before the start of its instruction. Three aims were put forward: (a) distinguishing the different developmental states in young children’s understanding of missing-value proportional situations, (b) investigating how children transition through these states, and (c) exploring possible predictors that explain individual differences in young children’s development of proportional reasoning abilities. We longitudinally investigated 5- to 8-year-olds’ (n = 315) proportional reasoning abilities in a fair-sharing missing-value proportional reasoning task. First, results showed that the development of proportional reasoning already starts at a very early age and is still ongoing when children are in their third year of elementary school. Second, latent class analysis revealed five different early states of proportional reasoning. The understanding of one-to-many correspondence was identified as an essential stepping-stone toward success in more difficult proportional reasoning problems with many-to-many correspondences. Third, exploratory analyses revealed that the large individual differences in children’s development of proportional reasoning abilities were associated with socioeconomic status, language, spatial abilities, and numerical abilities. Theoretical, methodological, and educational implications are discussed.
... If the hypotheses drawing from the analysis of action schemas are correct, one should continue to observe similar relative difficulty of problem types across different variations of tasks. Mathematical problems can be presented with concrete materials (e.g., Carpenter et al., 1993 ;Squire & Bryant, 2002a ) and pictures (e.g., Squire & Bryant, 2002b ) and so on. Previous studies show that different ways of representations can influence the ease with which children solve the problems ( Ching & Nunes, 2017b ;Gilmore & Papadatou-Pastou, 2009 ;Sophian et al, 1997 ). ...
This study examined the longitudinal development of young children's informal understanding of division through fair sharing. Two hundred and twelve children were followed from the age of 4.5–5.5 over three waves of assessment. Division understanding was measured with different types of problems (same- versus different-divisor trials) presented with different situations (partitive vs quotitve). Several key findings emerged. First, different-divisor problems were significantly more difficult than same-divisor problems at all age levels, which suggests that tasks that demand children's understanding of the inverse relation between divisor and quotient are more difficult than tasks that demand the understanding of equivalence principle. Second, children found it easier to understand the inverse relation between divisor and quotient in the context of partitive than quotitive situation. Third, with age, there was a sharper improvement in solving different-divisor problems than same-divisor problems, whereas the progress was more prominent for different-divisor problems presented in partitive situations, compared with other problems. Finally, working memory made the most consistent and unique contribution to solving all kinds of division problems at all age levels, compared with other cognitive factors. Theoretical and practical implications in early childhood mathematics education are discussed.
... Carpenter et al. [26] also corroborated that minority students performed at a lower level in a study of accountancy, and this effect was partially attributed to "lower performance expectations". Cole and Ahmadi [27] investigated whether or not the religious preference of Islam (being a Muslim) had a significant influence on the academic achievement of students. ...
Academic success in undergraduate programs is indicative of potential achievements for graduates in their professional careers. The reasons for an outstanding performance are complex and influenced by several principles and factors. An example of this complexity is that success factors might change depending on the culture of students. The relationship of 32 factors with the reported academic performance (RAP) was investigated by using a survey distributed over four key universities in Saudi Arabia. A total of 3565 Saudi undergraduate students completed the survey. The examined factors included those related to upbringing, K-12 education, and structured and unstructured activities. Statistical results validate that many factors had a significant relationship with the RAP. Among those factors, paternal’s education level and work field, type of intermediate and high schools, and the attendance of prayers in mosques were significantly associated with the reported performance. This study provides important insights into the potential root causes of success so that they can be targeted by educators and policy makers in the effort to enhance education outcomes.
... The proposed approach allows modeling physics to be integrated into a typical introductory college mechanics course. A third study developed models of problem-solving to study children's problem-solving process [17]. According to the study, the conception of modeling the problem-solving process could provide a unifying framework for thinking about problem-solving in children. ...
Problem-solving focuses on defining and analyzing problems, then finding viable solutions through an iterative process that requires brainstorming and understanding of what is known and what is unknown in the problem space. With rapid changes of economic landscape in the United States, new types of jobs emerge when new industries are created. Employers report that problem-solving is the most important skill they are looking for in job applicants. However, there are major concerns about the lack of problem-solving skills in engineering students. This lack of problem-solving skills calls for an approach to measure and enhance these skills. In this research, we propose to understand and improve problem-solving skills in engineering education by integrating eye-tracking sensing with virtual reality (VR) manufacturing. First, we simulate a manufacturing system in a VR game environment that we call a VR learning factory. The VR learning factory is built in the Unity game engine with the HTC Vive VR system for navigation and motion tracking. The headset is custom-fitted with Tobii eye-tracking technology, allowing the system to identify the coordinates and objects that a user is looking at, at any given time during the simulation. In the environment, engineering students can see through the headset a virtual manufacturing environment composed of a series of workstations and are able to interact with workpieces in the virtual environment. For example, a student can pick up virtual plastic bricks and assemble them together using the wireless controller in hand. Second, engineering students are asked to design and assemble car toys that satisfy predefined customer requirements while minimizing the total cost of production. Third, data-driven models are developed to analyze eye-movement patterns of engineering students. For instance, problem-solving skills are measured by the extent to which the eye-movement patterns of engineering students are similar to the pattern of a subject matter expert (SME), an ideal person who sets the expert criterion for the car toy assembly process. Benchmark experiments are conducted with a comprehensive measure of performance metrics such as cycle time, the number of station switches, weight, price, and quality of car toys. Experimental results show that eye-tracking modeling is efficient and effective to measure problem-solving skills of engineering students. The proposed VR learning factory was integrated into undergraduate manufacturing courses to enhance student learning and problem-solving skills.
... It is clear that a lower language comprehension reduces the ability to correctly solve mathematical problems. At that development stage, children are able to solve real world problems [6], but academic language is acquired slowly, long after the children develop the domain of practical everyday language [7]. ...
For primary school students, the difficulty in solving mathematical problems is highly related to language capacity. A correct solution can only be achieved after being able to deal with different abstract concepts through several stages: comprehension, processing, symbolic representation and relation of the concepts with the right mathematical operations. A model linking the solution of the mathematical problems (PS) with the mental representation (MR) of the problem statement, while taking into account the level of the students (which has influence in the linguistic abilities), is presented in this study. Different statistical tools such as the Analysis of Covariance (ANCOVA), ROC curves and logistic regression models have been applied. The relation between both variables has been proved, showing that the influence of MR in PS is similar in the different age groups, with linking models varying just in the constant term depending on the grade level. In addition, a cutoff in the mental representation test is provided in order to predict the student’s ability in problem resolution.
... Math problem solving includes combining and analyzing skills (Cawley & Miller, 1986) and consists of one and/or more steps (Fuchs et al., 2004). It requires necessary calculation operations to be used in the solution process (Carpenter et al., 1993) and rarely contains irrelevant or distracting information (Passolunghi et al., 2005). The components of metacognition play a crucial role in math ...
The purpose of this study was to examine the effects
of cognitive strategies and metacognitive functions of
students with learning disabilities (LD), students with lowachieving
(LA), and students with average-achieving (AA)
over their math problem-solving performance. The study
sample consisted of 150 students with 50 students from
each group. Study data were collected through Think-
Aloud Protocols, Metacognitive Experiences Questionnaire,
Math Problem Solving Assessment-Short Form, and 10 math
problems. Study findings revealed that the significant
predictors of math problem-solving performance were
metacognitive strategies and experiences regarding
students with LD, metacognitive strategies and knowledge
considering students with LA, and metacognitive strategies
in students with AA. A statistically significant relationship
was found between problem-solving performance of
students with LD and their metacognitive strategies and
metacognitive experiences. Problem-solving performance
and metacognitive strategies of students with LA were
found to be close to a high level, and their metacognitive
knowledge had a moderate relationship. It was also
observed to be moderately related to problem-solving
performance and metacognitive strategies in students
with AA. The findings were discussed within the relevant
literature scope, and suggestions were made for teachers in
terms of implementation and researchers for further studies
... Antes de recibir instrucción formal sobre las operaciones aritméticas, los estudiantes pueden resolver una variedad de problemas mediante estrategias basadas en el modelo de la situación. Estas estrategias se ponen de manifiesto cuando, para encontrar la solución, los estudiantes hacen alguna simulación de la acción descrita en el problema junto con algún procedimiento de conteo (Carpenter et al., 1993;Mulligan y Mitchelmore, 1997;Brissiaud y Sander, 2010). Por ejemplo, los estudiantes encuentran la solución a un problema de dividir mediante una acción de reparto. ...
Los estudiantes diagnosticados con Trastornos del Espectro Autista (TEA) suelen desarrollar un interés especial sobre áreas que no son habituales en cuanto a su intensidad o temática. Este trabajo profundiza en comprender la influencia que tienen las áreas de interés especial en los procesos de resolución de problemas aritméticos verbales de multiplicación y división, mediante un estudio de caso único con un estudiante de 11 años diagnosticado con TEA y discapacidad intelectual. Se utiliza un cuestionario formado por 15 problemas enunciados en tres tipos de contextos: de interés especial, familiar y no familiar. Siguiendo una metodología cualitativa, se clasifican las estrategias informales y el éxito en la obtención de la solución. Los resultados muestran que las áreas de interés especial han supuesto una mayor implicación del estudiante, pero no han logrado una mejora efectiva respecto de los contextos familiares, ya que en ambos contextos el estudiante ha resuelto los problemas de multiplicación y de división-agrupamiento pero no ha logrado resolver los problemas de división-reparto. En el contexto no familiar no ha resuelto ningún problema. Estos hallazgos contribuyen a completar la literatura existente sobre la utilidad educativa de las áreas de interés especial en estudiantes diagnosticados con TEA.
... Math problem solving forms a large part of the general and special education curriculum (Parmar & Cawley, 1997;Rivera, 1997). It is a process that involves problem-solving, combining and analysing skills (Cawley & Miller, 1986), consists of one and/or more steps (Fuchs, Fuchs & Prentice, 2004), requires the necessary calculation processes to be used in the solution process (Carpenter et al., 1993), and rarely contains irrelevant or distracting information (Passolunghi, Marzocchi & Fiorillo, 2005). This process involves the implementation of knowledge, skills, and strategies (Fuchs et al., 2004). ...
Focusing on students with mild disabilities, this study aimed to examine the effect of STAR problem solving strategy on their a) solving change problems involving one-step addition and subtraction, b) maintaining their acquisition of solving change problems involving one-step addition and subtraction after 1, 3, and 5 weeks, c) generalizing their performance in solving problems to the classroom environment. Three students with mild mental disabilities participated in the study. A multiple probe across participants design was used in the study. The number of problems that students solved correctly was determined by scoring the data. The data are shown graphically and analysed visually. Findings emphasized the effectiveness of STAR strategy for students with mild mental disabilities when solving change problems that involve a one-step addition and subtraction, indicating that those who acquired this strategy could demonstrate the same problem solving performance 1, 3, and 5 weeks after the intervention. Also, students were observed to generalize their strategy performance to the classroom environment. The findings of the research were discussed within the framework of the relevant literature and theoretical views, and suggestions were made for teachers in terms of interventions and for researchers considering further studies.
... Antes de recibir instrucción formal sobre resolución de problemas, los estudiantes pueden resolver una variedad de problemas mediante estrategias informales basadas en el conteo (Carpenter, Ansell, Franke, Fennema y Weisbeck, 1993). Pero para ser eficientes en la resolución de problemas deben desarrollar otras estrategias relacionadas con el reconocimiento y ejecución de las operaciones aritméticas que resuelven cada problema. ...
En este artículo se examina la efectividad de un enfoque de Instrucción Basada en Esquemas para mejorar el rendimiento matemático de resolución de problemas verbales aditivos por un estudiante con trastorno del espectro autista. Se llevó a cabo una intervención adaptada a las necesidades educativas del alumno. Este mostró una mejoría en la resolución de problemas verbales aditivos de cambio con incógnita en las tres cantidades (inicial, cambio y final) en dos aspectos: la identificación de la operación aritmética y la obtención del resultado numérico correcto. El estudiante abandonó sus primeras estrategias informales, que fueron sustituidas por hechos numéricos aditivos. Además, generalizó las habilidades adquiridas a problemas verbales de dos operaciones. Los logros se mantuvieron cuatro semanas después de finalizar la instrucción.
... We are also motivated by the works (and datasets) of [4], [20], and [21] who developed systems handling all four basic types of operations and we gained insights into the diversity of multiplication and division problem types to enrich our classification framework from their work. Our feature engineering, particularly for the division and multiplication categories, is inspired by the work of [4], [19], [20], and [22]. Our proposed word problem taxonomy contains four arithmetic word problem categories (level-1) with 12 subcategories (level-2) (see Table I). ...
Solving Mathematical (Math) Word Problems (MWP) automatically is a challenging research problem, which has gained momentum in the recent years in natural language processing (NLP), machine learning (ML), education (learning) technology, etc. Applications of solving varieties of MWPs can increase the efficacy of teaching-learning systems such as--E-learning Systems, Intelligent Tutoring Systems (ITS), etc., to help improve learning (or teaching) to solve word problems by providing interactive computer support for peer Math tutoring. Our work is specifically intended to benefit any teaching-learning systems such as ITS on teaching and learning arithmetic word problem solving by adding an interactive intelligent word problem solver as a major system component, as well as assessing an individual's learning performance. This paper presents Arithmetic Mathematical Word Problems Solver (AMWPS), an educational software application for solving arithmetic word problems involving single equations with single operations. Our work is based on a combination of a machine learning-based (classification) approach and a rule-based approach. We start with classification of arithmetic word problems into 4 categories Change, Compare, Combine, and Division-Multiplication) along with their sub-categories, followed by classification of operations (+, -, *, and /) related to different sub-categories. Our system processes an input arithmetic word problem, predicts the category and sub-category, predicts the operation, identifies and retrieves the relevant quantities within the problem with respect to answer generation, formulates and evaluates the mathematical expression to generate the final answer. AMWPS outperforms similar systems on the standard AddSub and SingleOp datasets and produces new state-of-the-art results (94.22% accuracy).
... Problem solving is one of the basic skills of mathematics. Despite various definitions related to the mathematical problem solving process, it generally refers to a process that includes combining and analysing skills (Cawley & Miller, 1986), consists of one and/or more steps (Fuchs et al., 2004), requires the distinguishing of necessary calculation operations to be used in the solving process (Carpenter et al., 1993), and rarely contains irrelevant or distracting information (Passolunghi et al., 2005). As with all academic skills, math problem solving skills require using cognitive strategies and operations (Montague, 1992;Rosenzweig et al., 2011;Sweeney, 2010). ...
Giriş: Öğrencilerin matematik problemi çözmede kullandıkları bilişsel ve üstbilişsel stratejilerin belirlenmesi problem çözme öğretiminde yapılacak düzenlemeler açısından önemlidir. Bu araştırmanın amacı altıncı sınıfa devam eden öğrenme güçlüğü olan öğrenciler ile düşük ve ortalama başarılı öğrencilerin matematik problemi çözerken kullandıkları bilişsel ve üstbilişsel stratejileri karşılaştırma ve belirtilen stratejiler arasındaki farklılığın incelenmesidir.Yöntem: Araştırmaya, kaynaştırma ortamında bulunan ve altıncı sınıfa devam eden 50 öğrenme güçlüğü, 50 düşük başarılı ve 50 ortalama başarılı olmak üzere toplam 150 öğrenci katılmıştır. Öğrenme güçlüğü olan öğrenciler ile düşük ve ortalama başarılı öğrencilerin kullandıkları bilişsel ve üstbilişsel stratejileri belirlemek amacıyla Sesli Düşünme Protokolleri kullanılmıştır. Araştırma sonucunda elde edilen veriler, ‘R Programlama Dili’ kullanılarak analiz edilmiştir.Bulgular: Araştırma sonuçlarına göre öğrenme güçlüğü olan öğrenciler farklı zorluk düzeyinde matematik problemleri çözerken düşük ve ortalama başarılı olan akranlarından daha az bilişsel ve üstbilişsel strateji kullandıkları sonucuna ulaşılmıştır.Tartışma: Araştırmanın sonuçları ilgili alanyazın ve teorik görüşler çerçevesinde tartışılmış, öğretmenlere uygulamaya ve alanda çalışan araştırmacılara da ileride yapılacak araştırmalara yönelik önerilerde bulunulmuştur.
... En general, son muchos los autores que previenen de una introducción prematura de los símbolos. Carpenter (1985) y Carpenter et al. (1993) señalan que además de ser innecesaria puede ser contraproducente, ya que la asociación de los símbolos "+" y "-" a situaciones aditivas desde el principio puede ocasionar una pérdida de flexibilidad y confianza al resolver problemas. Si bien no es necesario haber alcanzado el nivel máximo de dominio del recitado de las palabras numéricas (nivel cadena numerable bidireccional en la terminología de Fuson, Richards y Briars, 1982) para resolver situaciones aditivas, el recitado y el conteo son anteriores. ...
Presentamos a los lectores una serie de actualidad pero que, desafortunadamente, solo se encuentra en inglés. Se trata de Numberblocks, una producción británica pensada para niños y niñas de 4 a 7 años en la que los números son los protagonistas, dicho esto último en el sentido más literal posible. Son episodios muy breves, de unos cinco minutos de duración, comienzan con la presentación de los números, empezando por el uno, y llegan a tratar situaciones multiplicativas. Como veremos, cada número tiene su propia personalidad, la cual se relaciona con sus propiedades matemáticas. Uso del lenguaje muy cuidado desde el punto de vista matemático, descomposiciones de los números y un simbolismo adicional en segundo plano para espectadores en el límite superior del rango de edad, hacen de esta serie un recurso a tener en cuenta para los más pequeños. No en vano, ha contado con la colaboración del NCETM del Reino Unido, en cuyo sitio web hay materiales adicionales. Describimos en este artículo las características principales de la serie, contenidos que aborda y… qué piensan los niños.
... LT instruction. Instructors created opportunities for children to represent the objects, actions, and relationships that define the 12 types of arithmetic story problems (Carpenter, Ansell, Franke, Fennema, & Weisbeck, 1993) within the LTs model (Clements & Sarama, 2014;Sarama & Clements, 2009). The intention was to support children's progression through the arithmetic LT, with the goal of reaching three levels above each child's pretest LT level. ...
Although basing instruction on a learning trajectory (LT) is often recommended, there is little evidence regarding a premise of a LT approach—that to be maximally meaningful, engaging, and effective, instruction is best presented 1 LT level beyond a child’s present level of thinking. We evaluated this hypothesis using an empirically validated LT for early arithmetic with 291 kindergartners from four schools in a Mountain West state. Students randomly assigned to the LT condition received one-on-one instruction 1 level above their present level of thinking. Students in the counterfactual condition received 1-on-1 target-level instruction that involved solving story problems three levels above their initial level of thinking (a skip or teach-to-target approach). At posttest, children in the LT condition exhibited significantly greater learning, including target knowledge, than children in the teach-to-target condition, particularly those with low entry knowledge of arithmetic. Child gender and dosage were not significant moderators of the effects.
... He put 3 guppies in each jar. How many jars did Tad put guppies in?" children demonstrated a strategy of using many-to-one counting by counting out 15 guppies into groups of 3, and then counting the number of groups (Carpenter, Ansell, Franke, Fennema, & Weisbeck, 1993). However, these strategies usually require external support and may only work for small set sizes. ...
Symbolic mathematics allows humans to represent and describe the logic of the world around us. Although we typically think about math symbolically, humans across the lifespan and a wide variety of animal species spontaneously exhibit numerical competence without reference to formal mathematics. This intuitive ability to approximately compare, estimate, and manipulate large non-symbolic numerical quantities without language or symbols is called the Approximate Number System. The four chapters of this dissertation explore whether non-symbolic, approximate calculation can function as a bridge between our Approximate Number System and symbolic mathematics for children at the beginning of formal math education and university undergraduates. Chapter 1 explores how non-symbolic and symbolic ratio reasoning relates to general math skill and Approximate Number System acuity in elementary school children. Chapter 2 examines whether children and adults can perform a non-symbolic, approximate division computation, and how this ability relates to non-symbolic and symbolic mathematical skill. Chapter 3 tests the robustness and mechanism of a non-symbolic, approximate addition and subtraction training paradigm designed to improve arithmetic fluency in university undergraduates. Chapter 4 investigates whether the negative relation between math anxiety and symbolic math performance extends to approximate, non-symbolic calculation. Together, Chapters 1 and 2 provide evidence that non-symbolic calculation ability functions as a mechanism of the relation between Approximate Number System acuity and symbolic math. Chapters 3 and 4 identify populations of students for whom practice with non-symbolic calculation may or may not be beneficial. In sum, this dissertation describes how non-symbolic, approximate calculation allows students harness their intuitive sense of number in a mathematical context.
... Although there are many definitions on the concept of problem solving, it is generally defined as a process that usually includes problem solving, combining, and analyzing skills (Cawley & Miller, 1986). Additionally, problem solving includes one and/ or more than one step (Fuchs et al., 2004), and requires the differentiation of the calculations to be used in the solution process (Carpenter et al., 1993). The concept of problem solving may also contain information that is rarely unrelated or distracting (Passolunghi, Marzocchi, & Fiorillo, 2005). ...
Being a cognitive strategy instruction model called 'Solve It!' involves cognitive and metacognitive elements. The model was developed by
Montague (1992) as one of the process-based teaching strategies. The purpose of 'Solve It!' strategy is to teach the following seven cognitive
strategy steps: read, paraphrase, visualize, hypothesize, predict, calculate, and check. Each cognitive strategy step has the following three
metacognitive steps: ask, say, and check. 'Solve It!' strategy has been used to teach students with special needs on how to solve word problems.
This study aimed to evaluate the studies using 'Solve It!' strategies. Therefore, this study reviewed studies by examining electronic databases,
journal indexes, and references part of relevant studies. A total of 48 studies were found. These studies were reviewed in terms of inclusion and
exclusion criteria, and 12 of them were used for descriptive analysis. The findings of the study revealed that 'Solve It!' was effective in teaching
mathematical problem-solving skills for students with special needs. The findings were discussed in line with relevant literature, and some
suggestions for future research, and practitioners were presented at the end of the paper.
... 380). Carpenter, Ansell, Franke, Fennema, and Weisbeck (1993) found that, by the end of kindergarten, children in their study could solve a variety of problems by modeling the action or relations described in the problems. They concluded that children as young as kindergarten can invent direct model strategies to solve a variety of problems if they are given the opportunity to do so. ...
In recent years, considerable attention has been given to the knowledge teachers ought to hold for teaching mathematics. Teachers need to hold knowledge of mathematical problem solving for themselves as problem solvers and to help students to become better problem solvers. Thus, a teacher’s knowledge of and for teaching problem solving must be broader than general ability in problem solving. In this article a category-based perspective is used to discuss the types of knowledge that should be included in mathematical problem-solving knowledge for teaching. In particular, what do teachers need to know to teach for problem-solving proficiency? This question is addressed based on a review of the research literature on problem solving in mathematics education. The article discusses the perspective of problem-solving proficiency that framed the review and the findings regarding six categories of knowledge that teachers ought to hold to support students’ development of problem-solving proficiency. It concludes that mathematics problem-solving knowledge for teaching is a complex network of interdependent knowledge. Understanding this interdependence is important to help teachers to hold mathematical problem-solving knowledge for teaching so that it is usable in a meaningful and effective way in supporting problem-solving proficiency in their teaching. The perspective of mathematical problem-solving knowledge for teaching presented in this article can be built on to provide a framework of key knowledge mathematics teachers ought to hold to inform practice-based investigation of it and the design and investigation of learning experiences to help teachers to understand and develop the mathematics knowledge they need to teach for problem-solving proficiency.
... Brosvic, Dihoff, Epstein, & Cook, 2006). Arguably, these methods of instruction reduce student learning and create barriers that deny them access to mathematical practices (e.g., Carpenter, Ansell, Franke, Fennema, & Weisbeck, 1993) and rich discourse (e.g. Moschkovich, 1999Moschkovich, , 2015 that can promote mathematical understandings (e.g. ...
I present study findings that depict three Latinx emergent bilinguals' mathematical agency with identified learning disabilities in 12 discussions centered on the framework of children's mathematical thinking. I utilized qualitative analysis methods to illustrate how they exhibited agency as the power to express their mathematical thinking and make sense of story problems. I argue that when children are given opportunities to voice their thinking in any language and choose how to solve word problems, they are capable of understanding the content and are able to justify their thinking. Implications are offered for the math instruction of emergent bilingual children identified with a learning disability.
... When faced with a remainder, children will often ignore it or incorporate it back into their whole number answer (e.g., Guerrero & Rivera, 2001;Lautert & Spinillo, 2004. Fortunately, there is evidence demonstrating the benefit of intervention training to help children properly incorporate the remainder (Carpenter, Ansell, Franke, Fennema, & Weisbeck, 1993;Guerrero & Rivera, 2001;Lautert, Spinillo, & Correa, 2012;Spinillo & Lautert, 2006). More than simply recognizing the remainder for what it is, however, these situations allow for exploring the notion of fractions by subdividing the remaining pieces to then be shared. ...
This chapter reviews the literature examining children’s early understanding of proportional reasoning and division, and how these early conceptions contribute to children’s later understanding of fractions. Although the earliest literature on proportional reasoning suggested that only adolescents have a true understanding of proportional reasoning, more recent research suggests that very young children, if asked properly, do demonstrate a basic or intuitive understanding of proportion. Similarly, young children also have an intuitive understanding of division concepts, stemming from the concept of sharing, in particular, through a sharing action schema. Children’s understanding of both proportion and division concepts does vary depending on whether the stimuli used are continuous or discrete, with continuous stimuli being favorable for proportional reasoning and discrete stimuli being favorable for division. Children also demonstrate an intuitive understanding of how the changes to the numerator and denominator affect the proportion, division, or fraction, and they also demonstrate an intuitive understanding of half. Taken as a whole, the research on the early learning of proportion and division suggests that the factors that help children with these tasks can also help them with their understanding of fractions, and we offer some recommendations about how to do so.
Background
The sophistication of young children's arithmetic problem-solving strategies can be influenced through experience and instructional intervention. One potential pathway is through encountering story problems where the location of the unknown quantity varies.
Aims
The goal of the present study is to characterize how arithmetic problem-solving strategy sophistication can evolve through opportunities to solve story problems.
Sample
We used microgenetic principles to guide the coding of arithmetic problem-solving behavior (8843 attempts) across three timescales (time within-session, attempt to solve, and between sessions) for nine story problem structures (N = 40, 19 girls). Data come from a teaching experiment conducted in a Mountain West US state in Spring 2018.
Methods
We employed a Bayesian hierarchical ordinal regression with a nine-level response variable. The model contained fixed effects for session, attempt, story problem structure; a smooth time within session effect; and random effects for student, instructor, and equation.
Results
Our analysis indicates which transitions from less to more sophisticated strategies are better supported by additional attempts to solve the same problem vs. additional instructional sessions. Strategy sophistication also varied by the location of the unknown quantity (result unknown, find difference, start unknown), but not operation (join, separate, part-whole).
Conclusions
If confirmed by other studies, including experiments, what teachers offer children in terms of learning opportunities (more attempts within the same problem or more problems across work sessions) should vary based on the transition they are making.
Програмске садржаје предшколске математике, карактерише апстрактност, која условљава њихово формирање мисаоним поступцима, односно развијањем појмова. У предшколској математици посебно су значајни проблемски задаци који су најсложенији међу текстуалним задацима и чије рјешавање представља врхунац математичког образовања. Проблемским задацима дјеца стичу математичка знања. Према Каменову (2010), на успјех у рјешавању проблема утичу, не само умне способности дјетета, него и бројне друге његове особине, посебно се истиче интелектуална иницијативност. Ријеч метода потиче од грчке ријечи methodos и означава начин или поступак рада на остваривању постављених циљева и задатака у сваком организованом процесу и дјелатности. Моделовање прати дидактички круг који почиње упознавањем дјетета са стварним проблемом из живота и преко својих фаза води ка развоју математичког мишљења и рјешењу проблема (Прибишев Белеслин, Милинковић, Шиндић, 2017). Циљ овог рада јесте да се укаже на значај методе моделовања при рјешавању проблемских задатака на предшколском узрасту, како да дјеца лакше и успјешније рјешавају проблемске задатке.
The transition from arithmetic to algebra in middle school can be critical for students having persistent difficulties in mathematics. The leading perception in our project is that holistic relational thinking is an effective problem-solving tool in both arithmetic and algebra. We worked in collaboration with two teachers of special classes to implement elements of this approach with Secondary I students. We implemented several problem-solving-related activities with students. We collected data through individual interviews with students before and after the intervention. Our data shows the positive effects of the intervention in students changing their strategies as well as overall success in solving word problems requiring relational analysis. Le passage de l’arithmétique à l’algèbre à l’école intermédiaire peut s’avérer déterminant pour les élèves qui éprouvent des difficultés persistantes en mathématiques. Le principe qui oriente notre projet suppose que la pensée relationnelle holistique est un outil de résolution de problème efficace à la fois en arithmétique et en algèbre. Nous avons travaillé en collaboration avec deux enseignants de classes d’éducation spécialisée de première année de l’enseignement secondaire afin de mettre en œuvre des éléments de l’approche relationnelle holistique. Durant cette intervention, nous avons présenté aux élèves plusieurs activités liées à la résolution de problèmes puis nous avons recueilli des données par le biais d’entrevues individuelles avant et après l’intervention. Nos résultats indiquent que l’intervention a eu, dans l’ensemble des effets positifs sur la façon dont les élèves composent avec les problèmes écrits. Nous avons relevé des changements dans les stratégies adoptées par les élèves ainsi que dans leur taux global de réussite pour résoudre les problèmes écrits qui requièrent de l’analyse relationnelle.
Purpose
This paper aims to study the strategies used by ten students diagnosed with autism when solving multiplication and division problems because these operations are rarely studied in students with this condition.
Design/methodology/approach
This study conducted an exploratory study with ten students diagnosed with autism to explore and describe the strategies used in solving equal group problems. The authors also describe in detail the case of a student whom the authors deem to be representative because of the reasoning the student employed.
Findings
The informal strategies that they used are described, as well as the difficulties observed in the various problems, depending on the operation required to solve them. The strategies used include direct modeling with counting and others that relied on incorrect additive relationships, with strategies based on multiplication and division operations being scarce. Difficulties were observed in several problems, with measurement division being particularly challenging for the study participants.
Practical implications
The detailed description of the strategies used by the students revealed the meanings that they associate with the operations they are executing and brought to light potential difficulties, which can help teachers plan their instruction.
Originality/value
This research supplements other studies focusing on mathematical problem-solving with autistic students.
This dissertation is an investigation into the nature of teachers’ formative assessment responses to students as they learn addition and subtraction. Teachers’ background experiences, including classroom experience and professional learning opportunities, were considered as factors which could play a role in accounting for that variation, both when teachers responded to individual students’ thinking and when they determined goals for group discussion based on students’ thinking. In particular, this study investigates whether the responses from teachers who had been trained in a learning trajectory for early addition and subtraction reflected a quality that had the potential to extend student learning opportunities. Data for the study came in the form of practicing elementary teachers’ responses to a multimedia scenario-based survey. In a series of classroom scenarios, participant teachers were shown instances of students solving problems of early addition and subtraction. Those teachers were asked to describe those instances of student thinking, indicate how they would respond to the student, and what learning goal they would set forth for the student. After seeing two individual students’ solutions, the teachers were also asked to choose a problem and set an instructional goal for a discussion of the problem with a group of students that included the two just observed. Twenty-two teachers teaching at the time in elementary schools in a Midwestern state participated; some of those teachers had previously participated in professional development related to a learning trajectory for early addition and subtraction. The results of the study indicate that teachers’ classroom and professional learning experiences were associated with higher rates of teachers interpreting student thinking. In addition to this, those teachers who taught in an early elementary classroom and had training in a learning trajectory were more likely to describe responses to student thinking that showed a potential to extend learning opportunities. Some differences were found among the instructional goals set for the group discussion of addition and subtraction word problems: Some early elementary teachers were open to students’ use of multiple methods, and a small number of early elementary teachers who had been trained in the learning trajectory discussed those multiple methods by connecting them in discussion in ways that attended to the mathematical sophistication of those methods. The findings suggest that when supporting or studying teachers’ formative assessment practices, a content-specific lens may be useful for informing and analyzing those practices. In addition, the findings may provide insight into teachers’ mathematical knowledge for teaching and the measures used to determine quality of teaching responses.
Giriş: Öğrenme güçlüğü olan öğrencilere yönelik matematik problemi çözme müdahaleleri içeren çalışmaların incelenmesi, bu öğrencilere destek sağlayacak uygulamaların belirlenmesi için önemli görülmektedir. Ayrıca müdahale çalışmalarının yöntemsel olarak belirli kalite standartları çerçevesinde değerlendirilmesi, uygulamaların yinelenebilirliği ve sonuçların güvenirliği hakkında bilgi sağlamaktadır. Bu araştırmada, a) son 20 yılda, ortaokul düzeyinde bulunan öğrenme güçlüğü olan öğrencilere yönelik uygulanan matematik problemi çözme müdahalelerinin derlenmesi, b) bu araştırmaların özelliklerinin betimsel olarak listelenmesi ve c) bu araştırmaların kalite göstergeleri açısından incelenmesi amaçlanmaktadır.Yöntem: Elektronik veri tabanları, dergi indeksleri ve araştırma referansları temel alınarak kapsamlı bir tarama gerçekleştirilmiştir. Katılımcı makalelerin bu araştırmaya dahil edilmesine ve dışlanmasına ilişkin bazı temel seçim ölçütleri doğrultusunda toplam 9 makale; betimsel analiz ve kalite göstergeleri bağlamında incelenmiştir. Araştırmaların; betimsel analizi yapılarak özellikleri ortaya konulmuş, tek denekli deneysel desenlerin kullanıldığı araştırmalar için belirlenen kalite göstergeleri doğrultusunda incelenerek kalite düzeyleri belirlenmiştir.Bulgular: Bu araştırmada incelenen çalışmaların görsel grafiklerinin ve yazılı bulgularının incelenmesi sonucunda, çeşitli problem çözme öğretim müdahalelerinin (doğrudan öğretim, somut-yarı somut-soyut stratejisi, şema temelli yaklaşım, öz düzenleme stratejisi ve ipucu kartları, STAR stratejisi, açık anlatım, LAP stratejisi, görsel stratejiler, Solve It! ve SOLVE stratejisi) problem çözme becerilerinin kazanımında etkili olduğu belirlenmiştir. Çalışmalar kalite göstergeleri açısından genel olarak değerlendirildiğinde, temel kalite göstergelerinin başlama düzeyi, deneysel kontrol/iç geçerlik ve dış geçerlik bileşenlerinde tüm çalışmalar belirlenen ölçütleri karşılamaktadır. Diğer bileşenler için %44 ile %78 arasında değişen orandaki çalışmanın, belirlenen ölçütleri karşıladığı görülmüştür. Sadece bir çalışmanın kalite göstergelerinin tümünü karşıladığı görülmüştür.Tartışma: Araştırma bulguları, ilgili alanyazın ve teorik görüşler çerçevesinde tartışılmıştır. Öğretmenlere, uygulamaya ve alanda çalışan araştırmacılara ileride yapılacak araştırmalara yönelik önerilerde bulunulmuştur. Bu doğrultuda, öğrenme güçlüğü olan öğrenciler ile çalışan uzmanların, problem çözme aşamaları, şematik düzenleyiciler, bilişsel ve üstbilişsel stratejiler ile sesli düşünme teknikleri gibi öğrencilerin başarılı bir şekilde problem çözümünü kolaylaştıracak stratejileri içeren müdahale programları oluşturması gerektiği belirtilmiştir.
Bu araştırmada çocukların okula hazırbulunuşluk düzeyleri ile problem çözme becerileri arasındaki ilişkinin incelenmesi amaçlanmıştır. Araştırmada yöntem olarak nicel araştırma yöntemlerinden ilişkisel deseni kullanılarak tasarlanmıştır. Çalışma grubunu 5 yaşında toplam 174 çocuk oluşturmaktadır. Araştırmada Demografik Bilgi Formu, Bracken Temel Kavram Ölçeği Gözden Geçirilmiş Formunun Okula Hazırbulunuşluk Alt Ölçeği ve Çocuklar için Problem Çözme Becerisi Ölçeği veri toplama aracı olarak kullanılmıştır. Normallik testi sonucunda normal dağılım gösteren veriler Basit Doğrusal Regresyon analizi kullanılarak analiz edilmiştir. Yapılan analizler sonucunda çocukların okula hazırbulunuşluk düzeylerinin çocukların problem çözme becerilerini açıklama gücünün yüksek olduğu, okula hazırbulunuşluk düzeyi arttıkça problem çözme becerilerinde de artışın olduğu bulunmuştur.
Counting is fundamental to early mathematics. Most studies of teaching counting focus on teachers observing children count. The present study compares mathematical ideas that 12 PK, transitional kindergarten (TK), and kindergarten teachers noticed from observing their own students count during a classroom session of Counting Collections with ideas that they noticed outside class time in the same students' representations of counting on paper. Inviting teacher noticing in representations (a) drew attention to distinct conceptions that children required to represent counting; (b) increased the number of mathematical ideas that participants perceived in students' thinking; and (c) helped participants perceive different levels in, and their own uncertainties about, students' understanding. This study suggests that teacher noticing in children's representations of counting can deepen teachers' understanding of students' mathematical thinking.
Young children’s problem posing and problem solving are rarely the focus of research. In stories told by teachers, in focus group interviews about photos that they took of children engaging in mathematics in their kindergarten, we also found that problem posing and problem solving were rarely discussed explicitly. However, an analysis using Bishop’s universal mathematical activities of Explaining and Playing enabled four different components that the teachers paid attention to, to be identified. These components were to do with the routine or non-routine nature of the problems; known or unknown problem-solving strategies; body actions or verbal explanations; playing by exploring different scenarios or following rules. Identifying what teachers noticed about children’s problem posing and problem solving provides insights into their professional knowledge.
In this chapter, we discuss characteristics of problem-solving situations in outdoor activities considered as mathematical. Although kindergarten children often participate in outdoor activities, these activities are rarely considered as being mathematical. We use theories about instrumental and pedagogical situations in kindergarten to identify the kinds of problem-solving that is occurring and how mathematics comes to play in these situations. We analyse one spontaneous episode in an outdoor setting to gain a better understanding of mathematical problem-solving in outdoor activities. Our findings indicate that although the children only used mathematics implicitly as a tool to solve a practical task in the outdoor environment, the kindergarten teacher was able to support their understanding of the mathematical ideas. This indicates that the teacher had a pedagogical purpose when contributing to the interaction. We argue that awareness of the features of mathematical problem-solving could support kindergarten teachers to be able to support and develop mathematical problem- solving in the outdoor environment.
The aim of this study was to investigate how pictorial representations with different semiotic characteristics affect additive word problem solving by kindergartners. The focus of the study is on three categories of additive problems (change problems, combine problems and equalize problems) and on representational pictures with different semiotic characteristics: (a) pictures in which the problem quantities are represented in pictorial form, that is, as groups of illustrated objects (PP pictures), (b) pictures in which the quantities are represented partly in pictorial form and in symbolic form (PS pictures), and (c) pictures in which the quantities are represented in symbolic form (SS pictures). Data were collected from 63 kindergartners using a paper-and-pencil test. Results showed that the semiotic characteristics of representational pictures had a strong and significant effect on performance. Children’s performance was higher in the problems with PP pictures but declined in the problems with PS and SS pictures. However, the differences in children’s performance across the problems with different representational format varied between the problem categories and their mathematical structures. The semiotic characteristics of representational pictures had an important role in the establishment of close relations between children’s solutions in problems in different categories. Detailed analysis of children’s answers to the problems revealed a number of picture-related difficulties. Findings are discussed and directions for future research are drawn considering the methodological limitations of the study.
This research aims to find: 1) the relation between the motivation and the achievement in learning Mathematics. 2) The relation between the perception to Mathematics and the achievement in learning Mathematics. 3) The relation between the parents’ education level and the achievement in learning Mathematics. 4) The role of motivation and perception in learning Mathematics and parents’ education level in the achievement in learning Mathematics.The hypotheses in this research are: 1) There is a positif relation between the motivation and the achievement in learning Mathematics. 2) There is a positif relation between the perception to Mathematics and the achievement in learning Mathematics. 3) There is a positif relation between the father’s education level and the achievement in learning Mathematics. 4) There is a positif relation between the mother’s education level and the achievement in learning Mathematics 5) The motivation in learning Mathematics, the perception to Mathematics, and the parents’ education level have a role in the achievement in learning Mathematics. The research’s subject is 100 students class X and XI of SMAN 2 Yogyakarta. The data were collected by using scale, documentation, and identity. The scale used was math motivation scale and math perception scale. The data analysis were done by using multiple linear regression and product moment correlation.Based on the correlation analysis done was known that there was positive and significant relation between motivation and achievement in learning Mathematics and between the perception of Mathematics and the achievement in learning Mathematics, but there is no positive and significant relation between father and mother’s education level and the achievement in learning Mathematics. Based on the result of multiple linear regressions analysis, it was known that motivation and perception in learning Mathematics as well as parents’ education level have a role in the achievement in learning Mathematics. It is based on the significant result of F testing, i.e. α = 0.05. The determination of coefficient was 0.162 meant that the percentage of the effect of motivation and perception in learning Mathematics and parents’ education level to the achievement in learning Mathematics was 16.2%, while the rest of 83.8% was influenced by other variables which were not included in this research.Key words: Motivation in learning Mathematics, perception to Mathematics, parents’ education level, the achievement in learning Mathematics.
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