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Mental strategies used by school students in solving problems involving operations with whole numbers have been documented for some time. There is, however, little information about how students solve fraction, decimal, and percent -or part-whole -number problems mentally. This is despite the fact that there is an increased emphasis on facility wit...
Contexts in source publication
Context 1
... the sample of students is relatively small, a wide variety of strategies was exhibited for the problems answered. Table 4 summarizes the strategies observed for part-whole numbers and presents examples of each strategy. Many strategies that are employed in solving part-whole number problems appear to overlap with those documented for whole numbers. ...
Context 2
... were employed singularly, for example using a memorized rule, or put together as a combination of strategies in a sequential fashion, for example, changing from decimals to fractions, and then bridging to a whole. Twelve problems, including decimals, fractions, and percents, have been selected to demonstrate the different Instrumental or Conceptual processes students display when working with part-whole numbers and to further illustrate the mental part-whole number strategies documented in Table 4. A summary of frequencies of Instrumental and Conceptual responses is presented at the end of this section in Table 5. ...
Context 3
... avenue of future research could be associated with the classification of responses based on structural models from cognitive psychology (e.g., Biggs & Collis, 1982;Case, 1985). The single and multiple use of elemental strategies identified in Table 4 appears to be a potentially rewarding starting point. ...
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Citations
... In multiplication, one or both numbers can be split using the distributive property (e.g., 12×16=10×16+2×16=160+32=192), but in division we only can split the dividend (e.g., 168÷14; 140÷14=10; 28÷14=2; 10+2=12). In varying strategy, compensation strategies can be used (e.g., distributive property can be used in multiplication, such as 120 × 19; 120 × 20=2400; 120 × 1=120; 2400 − 120=2280, or in division, 475 ÷ 25=19; 500 ÷ 25=20; 25÷25=1; 20−1=19) as well as doubles and halves (e.g., 12×20=24×10) or inverse operations where, for example, to calculate 320÷80, the student says 4 because 4×8 is equal to 32, so 4×80 is 320 (Caney & Watson, 2003;Hartnett, 2007;Thompson, 2009). The column-based strategy involves a vertical representation of the decomposition strategy in multiplication and repeated subtraction strategy in division, and "may act as a fruitful stepping stone, or even alternative, to the digit-based algorithms" (Hickendorff et al., 2019, p. 557). ...
This paper presents a "Day Number" routine involving the four arithmetic operations with multi-digit numbers. Students are challenged to use numbers and operations, according to their knowledge, to create a train of calculations in which the answer to one calculation is used to start the next one. The result of the last calculation needs to be the number of the day where the lesson takes place. Using a qualitative approach, we undertook an exploratory study that aims to identify the arithmetic knowledge that 3 rd grade students activate when they are free to use any numbers and operations to calculate mentally in the "Day Number" routine. We focused our analysis on the numbers, operations and mental calculation strategies used by the students. Data were collected through direct observation, video recording, field notes and students' own productions. The analysis of students' work shows that most of them combined the four arithmetic operations envisioning to construct long trains. In all the four arithmetic operations, students used decomposition strategies; in addition/subtraction they also used sequential strategies, and in multiplication/division their options involved varying strategies.
... Estratégias baseadas em relações numéricas refletem o pensamento relacional dos alunos (Empson, Levi & Carpenter, 2010) ao contemplarem a mudança de representação (Caney & Watson, 2003), entre números racionais (fraçãoàdecimal; decimalàfração; fraçãoàpercentagem; percentagemà fração; percentagemàdecimal e decimalàpercentagem) ou de um racional para um número natural (decimalànúmero natural referente a ); a relação parte-todo ou parte-parte; a equivalência entre expressões; a relações entre operações inversas, etc. O pensamento relacional é um aspeto importante do cálculo mental pois refere-se à capacidade para usar propriedades fundamentais das operações e a noção de igualdade, para analisar e resolver problemas tendo em conta o seu contexto (Empson et al., 2010). Baseia-se em relações numéricas e o seu desenvolvimento serve de suporte à transição da aritmética para a álgebra (Carpenter, Franke & Levi, 2003). ...
... Para a análise das estratégias, consideramos três categorias: (i) factos numéricos; (ii) regras memorizadas; e (iii) relações numéricas. Estas categorias (e suas subcategorias) foram construídas com base em estudos anteriores (e.g., Caney & Watson, 2003) e na análise dos dados recolhidos. O nome dado à estratégia do aluno foi escolhido em função do elemento mais forte presente na sua estratégia (por exemplo, se faz um uso forte de relações numéricas, nomeadamente da relação parte-todo é considerada uma estratégia de categoria "relações numéricas" e subcategoria "relação parte-todo"). ...
... No cálculo mental com números racionais os alunos optam por recorrer a estratégias que envolvem relações numéricas, embora estratégias baseadas na aplicação de factos numéricos (tabuadas) e regras memorizadas (aplicação mental do algoritmo para a adição de numerais decimais) surjam igualmente, Nas questões analisadas, as relações numéricas centram-se na relação parte-todo para operar com frações e percentagens e na mudança de representação (Caney & Watson, 2003) para operar com frações e numerais decimais. ...
Resumo: O indivíduo constrói representações mentais do mundo que o rodeia, às quais recorre para compreender a realidade e fazer inferências. Estas representações mentais refletem os conhecimentos matemáticos e do mundo real dos alunos e tanto são fundamentais para a realização de cálculo mental como para estabelecer relações entre números e operações. O objetivo deste estudo é analisar as estratégias de cálculo mental dos alunos e as representações mentais que lhes estão subjacentes em questões de cálculo mental com números racionais nas representações fracionária, decimal e percentagem. O estudo seguiu uma metodologia de design research, tendo sido realizados dois ciclos de experimentação com a participação de duas professoras e 39 alunos do 6.º ano. As estratégias de cálculo mental dos alunos centradas em relações numéricas, aplicação de factos ou regras memorizadas parecem ter subjacentes representações representativas (modelos e imagens) e descritivas (representações proposicionais), embora as representações descritivas se associem mais a relações numéricas.
... That is that, a student or an adult can find mechanically the correct answer for an operation without understanding the meaning of the numbers or the operation. Caney and Watson (2003) conducted a study with 24 students, from grades 3-10, in order to record the strategies they used in performing mental calculations with fractions, decimals and percents. Despite the fact that the sample of students was relatively small, many mental strategies were exhibited in their effort to answer the problems. ...
... c. mental picture. For example, in substraction ¾ -½, a student divides an imagined picture of rectangle into 4 parts (Caney and Watson, 2003) and d. converting a fraction or percentage to a decimal. ...
Problem solving and number sense are two of the core subjects on which strong emphasis is given in contemporary mathematics curricula of compulsory education. In this study, we examined fifth and sixth grade Greek students’ number sense concerning the mental calculation with rational numbers and specifically fractions and percents. We attempt to analyze the behaviors of fifth and sixth graders in mental calculations with fractions and percent examining the performance of students, categorizing the mental strategies used by them.
Despite of the educational importance of these two mathematical areas – problem solving and number sense in mental calculations with rational number – there are not studies which examine directly the relation of students' performance in these two areas.
This study has shown that the majority of students use rule-based strategies in operations with fractions and percents. Another result is that the students’ strategy choice (number sense or rule based) relates to their performance in problem solving.
Keywords: number sense, rational numbers, problem solving.
... Diversos autores (e.g., Bell, 1993;Callingham & Watson, 2004;Caney & Watson, 2003;Galen, Feijs, Figueiredo, Gravemeijer, Herpen & Keijzer, 2008;Lamon, 2006) apontam aspetos relativos à aprendizagem dos números racionais que importa ter em conta no design de tarefas de cálculo mental. Um desses aspetos, como refere Bell (1993), é o contexto em que os números racionais são apresentados e abordados com os alunos. ...
... Outro aspeto prende-se com o uso de diversas representações dos números racionais (decimal, fração, percentagem) com o intuito de ajudar os alunos a relacionarem diferentes representações (Caney & Watson, 2003) e a estabelecerem relações entre representações e imagens mentais de determinados conceitos matemáticos (Swan, 2008). ...
... Os números de referência (e.g., O conhecimento acerca das estratégias dos alunos e níveis de cálculo mental com números racionais (Callingham & Watson, 2004;Caney & Watson, 2003) apoia o professor na construção de tarefas que possam desenvolver determinado tipo de estratégias, bem como na compreensão das estratégias e dos conhecimentos matemáticos usados pelos alunos. Quando estes calculam mentalmente, por vezes, as suas estratégias dão origem a soluções incorretas, fruto de erros que cometem. ...
O cálculo mental contribui para o desenvolvimento, nos alunos, de diversas
capacidades importantes para a aprendizagem da Matemática. O objetivo deste estudo,
que tem por base uma experiência de ensino, é identificar aspetos nas estratégias dos
alunos que possam apoiar a definição de princípios orientadores para o design de tarefas
de cálculo mental com números racionais positivos para alunos do 6.º ano. As tarefas
apresentadas foram criadas e aperfeiçoadas no quadro de um projeto baseado nos
princípios do design research. A análise das estratégias dos alunos permitiu formular
quatro princípios orientadores do design de tarefas de cálculo mental, que realçam a
importância do uso de contextos, de diversas representações dos números racionais, do
nível cognitivo das tarefas e de conhecimentos sobre estratégias e erros dos alunos.
... The student who only uses instrumental mental computation strategies, just involving known facts and memorized rules (Caney & Watson, 2003), does not show relational thinking. To compute 3/4-1/2 the student may apply the rule of subtraction of fractions with different denominators with no understanding of the quantities involved. ...
... To compute 3/4-1/2 the student may apply the rule of subtraction of fractions with different denominators with no understanding of the quantities involved. In contrast, the student who uses knowledge about numbers, their relationships and operations provides evidence of conceptual strategies (Caney & Watson, 2003) and builds an important conceptual foundation for learning algebra (Empson et al., 2010). For example, to do this computation, a student may decompose 3/4 in 1/2+1/4 and, by subtracting the halves, see that the result is 1/4, showing some understanding about numbers. ...
... In data analysis the dialogues (audio and video recorded) that show students' mental computation strategies in collective discussions, were transcribed to identify how these strategies evolved during the teaching experiment. The students' mental computation strategies were categorized as instrumental or conceptual (Caney & Watson, 2003), and students' use of relational thinking (Empson et al., 2010) was analyzed. ...
Carrying out mental computation with positive rational numbers represented as fractions may contribute to students' development of rational number sense. We analyze the development of grade 6 students' mental computation strategies with fractions through a teaching experiment based on mental computation tasks with rational numbers involving the four operations and the discussion of strategies. In the beginning of the study, the students used mainly instrumental strategies and, along the teaching experiment, they used more and more conceptual strategies.
... This paper reports on a small part of a larger project whose aims included working with teachers to develop modules for enhancing students' strategy use in performing mental computation tasks, as well as collecting data over three years to document the progression in mental computation skill levels over the Grades 3 to 10. Interviews were also conducted with some students (Caney & Watson, 2003), which helped to provide structure for correct strategies for problems involving rational numbers. ...
The analysis of errors while completing mental computation tasks reported in this paper represents the first stage of analysis of 5535 test responses from students in Grades 3 to 10 over a period of three years to various subsets of 374 items. Following a previous analysis that suggested the items represented eight increasing levels of difficulty covering nine sub-domains of basic number skills, this report focuses on responses to items at Level 6. Of particular interest are the performances across the grades, the types of errors diagnosed, and the relationship of errors from different types of operations. In analysing errors a developmental approach is adopted, suggesting that more than the "right-wrong" nature of responses is involved. Some errors appear to demonstrate a "partial number sense" that could be used to help construct more complete understanding. Suggestions for future research and classroom practice are made.
