Fourth graders’ adaptive strategy use in solving multidigit subtraction problems

Using the choice/no-choice methodology we investigated Dutch fourth graders’ (N = 124) adaptive use of the indirect addition strategy to solve subtraction problems. Children solved multidigit subtraction problems in one choice condition, in which they were free to choose between direct subtraction and indirect addition, and in two no-choice conditions, in which they had to use either direct subtraction or indirect addition. Furthermore, children were randomly assigned to mental computation, written computation, or free choice between mental and written computation. One third of the children adaptively switched their strategy according to the number characteristics of the problems, whereas the remaining children consistently used the same strategy. The likelihood to adaptively switch strategies decreased when written computation was allowed or required, compared to mandatory mental computation. On average, children were adaptive to their own speed differences but not to the accuracy differences between the strategies.     

Dutch sixth graders’ use of shortcut strategies in solving multidigit arithmetic problems

Strategy flexibility, adaptivity, and the use of clever shortcut strategies are of major importance in current primary school mathematics education worldwide. However, empirical results show that primary school students use such shortcut strategies rather infrequently. The aims of the present study were to analyze the extent to which Dutch sixth graders (12-year-olds) use shortcut strategies in solving multidigit addition, subtraction, multiplication, and division problems, to what extent student factors and task instructions affected this frequency of shortcut strategy use, and to what extent the strategies differed in performance. A sample of 648 sixth graders from 23 Dutch primary schools completed a paper-and-pencil task of 12 multidigit arithmetic problems, designed to elicit specific shortcut strategies such as compensation. Based on the students’ written work, strategies were classified into whether a shortcut strategy was used or not. Results showed that the frequency of shortcut strategies ranged between 6 and 21% across problem types, and that boys and high mathematics achievers were more inclined to use shortcut strategies. An explicit instruction to look for a shortcut strategy increased the frequency of these strategies in the addition and multiplication problems, but not in the subtraction and division problems. Finally, the use of shortcut strategies did not yield higher performance than using standard strategies. All in all, spontaneous as well as stimulated use of shortcut strategies by Dutch sixth graders was not very common.     

优化小学数学课堂设计,提升课程质量

小学阶段的数学知识主要包括应用题和加减乘除,对于小学生而言这些知识是比较枯燥的。因此为了提升课程质量,教师在教学中必须要不断优化小学数学课堂设计。基于此,本文主要介绍了优化小学数学课堂教学的重要意义,而且提出了优化小学数学课堂设计的有效策略,希望可以给有需要的人提供参考意见。     

Solution strategies and adaptivity in multidigit division in a choice/no-choice experiment: Student and instructional factors

Adaptive expertise in choosing when to apply which solution strategy is a central element of current day mathematics, but may not be attainable for all students in all mathematics domains. In the domain of multidigit division, the adaptivity of choices between mental and written strategies appears to be problematic. These solution strategies were investigated with a sample of 162 sixth graders in a choice/no-choice experiment. Children chose freely when to apply which strategy in the choice condition, but not in the no-choice conditions for mental and written calculation, so strategy performance could be assessed unbiasedly. Mental strategies were found to be less accurate but faster than written ones, and lower ability students made counter-adaptive choices between the two strategies. No teacher effects on strategy use were found. Implications for research on individual differences in adaptivity and the feasibility of adaptive expertise for lower ability students are discussed.     

Fostering the acquisition of subtraction strategies with interleaved practice: An intervention study with German third graders

Interleaved practice is a promising approach to foster the adaptive use of subtraction strategies. By intermixing strategies, comparison processes are evoked, which prompt more task-based strategy use. However, the effectiveness of interleaved practice in primary school mathematics has not been investigated yet. In the current study, 236 German third graders were randomly assigned to either an interleaved or a blocked condition. Both groups were instructed in using number-based strategies and the standard written algorithm for solving subtraction problems over 14 lessons. The students in the interleaved condition were prompted to compare strategies (between-comparison), while the students in the blocked condition compared the adaptivity of one strategy for different tasks (within-comparison). Our findings show that the students in the interleaved condition solved subtraction tasks with greater adaptiveness and accuracy. The effect on correctness was mediated by greater adaptive strategy use in the interleaved condition.     

Are first graders' arithmetic skills related to the quality of mathematics textbooks? A study on students’ use of arithmetic principles

A major goal of the first years of schooling is students' development from using counting strategies to using calculation strategies – or even recall – to solve addition or subtraction problems. It requires the perception and usage of arithmetic principles. Although the continued use of counting strategies is problematic for further learning progress, they are frequently used by many students beyond Grade 1. While learning resources possibly play a role in students' development in this field, our knowledge about their impact is limited. In this study we investigated the presented learning opportunities of four different textbooks regarding arithmetic principles in Grade 1, and subsequently the relation to first graders' strategy use by analyzing a one-year dataset of 1614 students from 86 classes. The analyses show differences in the textbooks' quality concerning arithmetic principles as well as a considerable connection to students' strategy use at the end of Grade 1.     

Counting strategies from 5 years to adulthood: Adaptation to structural features

The aim of this study was twofold. First, it evaluated the developmental changes on frequency of use, efficiency, and choice of counting strategies from childhood to adulthood. Second, it determined the adaptation of the counting strategies to the structural features of the task. Participants in seven age groups ranging from 5-year-old to adulthood were asked to count dots in arrays varying on size, arrangement, density and size of the subgroups. The nature of the strategies used and their efficiency, i.e. speed, accuracy and rate of manual pointing, were recorded using the overt behavior technique. The developmental pattern of results for the four main strategies, counting by ones, by ns, addition and multiplication, was in line with Siegler’s overlapping waves model. The structural features affected the use of the strategies since 13 years, except for the multiplication strategy. Finally, intra- and inter-individual variability in strategy showed a monotonous increase with age. Implications for understanding the development of counting skill are discussed.     

From Informal Strategies to Structured Procedures: mind the gap!

This paper explores written calculation methods for division used by pupils in England (n = 276) and the Netherlands (n = 259) at two points in the same school year. Informal strategies are analysed and progression identified towards more structured procedures that result from different teaching approaches. Comparison of the methods used by year 5 (Group 6) pupils in the two countries shows greater success in the Dutch approach, which is based on careful progression from informal strategies to more structured and efficient procedures. This success is particularly not able for the girls in the sample. For the English pupils, whose written solutions largely involved the traditional algorithm, the discontinuity between the formal computation procedure and informal solution strategies presents difficulties.This revised version was published online in September 2005 with corrections to the Cover Date.     

Subtraction by addition in children with mathematical learning disabilities

In the last decades, strategy variability and flexibility have become major aims in mathematics education. For children with mathematical learning disabilities (MLD) it is unclear whether the same goals can and should be set. Some researchers and policy makers advise to teach MLD children only one solution strategy, others advocate stimulating the flexible use of various strategies, as for typically developing children. To contribute to this debate, we compared the use of the subtraction by addition strategy to mentally solve two-digit subtractions in children with and without MLD. We used non-verbal research methods to infer strategy use patterns, and found that both groups of children switch between the traditionally taught direct subtraction strategy and subtraction by addition, based on the relative size of the subtrahend. These findings challenge typical special education classroom practices, which only focus on the routine mastery of the direct subtraction strategy.     

Sex differences in mental strategies for single-digit addition in the first years of school

Abstract

Strategy use in single-digit addition is an indicator of young children’s numeracy comprehension. We investigated Danish primary students’ use of strategies in single-digit addition with interview-based assessment of how they solved 36 specific single-digit addition problems, categorised as either ‘error’, ‘counting’, ‘direct retrieval’ or ‘derived facts’. The proportional use of each strategy was analysed as multi-level functions of school age and sex. In a first study (260 interviews, 147 students) we found decreasing use of counting and increasing use of direct retrieval and derived facts through years 1–4, girls using counting substantially more and the other two strategies substantially less than boys, equal to more than 2 years’ development. Similar results appeared in a subsequent study (155 interviews, 83 students), suggesting that the pattern is pervasive in Danish primary schools. Finally, we ask whether sex differences in strategy use is generally under-reported since many studies do not explicitly address them.     

Addition and Subtraction of Three-digit Numbers: German Elementary Children's Success, Methods and Strategies

The present paper describes the success, the methods (mental, informal written, standard algorithm) and the strategies of informal written arithmetic to be observed when 300 elementary students worked on six addition and six subtraction problems with three-digit numbers. These twelve problems were administered repeatedly by means of a class test: in February (grade 3; nine-year-olds) before the standard algorithms were introduced, in June after they had been dealt with and in October at the beginning of grade 4. This revised version was published online in July 2006 with corrections to the Cover Date.     

Using mathematics strategies in early childhood education as a basis for culturally responsive teaching in India

The objective of this small study was to elicit responses from early childhood teachers in India on mathematics learning strategies and to measure the extent of finger counting technique adopted by the teachers in teaching young children. Specifically, the research focused on the effective ways of teaching mathematics to children in India, and examined teachers’ approach to number counting. In India, children were taught by their parents or by their teachers to use fingers to count. The qualitative study conducted by the researcher further enriched the topic with first‐hand comments by the teachers. Although the finger counting method was not the only process that teachers would adopt, it was embedded in the culture and taken into consideration while infusing mathematics skills. The teachers confirmed adopting the Indian method of finger counting in their teaching strategy; some specified that the method helped children to undertake addition and subtraction of carrying and borrowing, as counting by objects could not be available all the time. Although the study is limited by its small sample to the unique mathematics learning experience in India, it provides readers with a glimpse of culturally responsive teaching methods and an alternative mathematics teaching strategy.     

7-8岁数学学习困难与正常儿童加法策略比较研究

选取二、三年级数学学习正常和学习困难的儿童各 30名 ,共 12 0名被试。采用实验法、观察法和口语报告法相结合的方式 ,考察了两类儿童在加法任务中 ,策略选择和执行的差异及特点。研究表明 :小学低年级儿童的策略选择具有多样性、适应性和简约性的特点。从策略选择上看 ,出声、竖式、分解、对位和提取策略是小学 2— 3年级两类儿童的主选策略 ;数学学习困难儿童较多使用手指、数数、放弃和猜测等策略 ;数学学习正常儿童则较多使用提取、分解、凑数、换位和乘法策略。从策略执行上看 ,小学低年级数学学习困难儿童比正常儿童策略执行的正确率低 ,反应时长 ,有效性差     

Instructional approaches to foster third graders’ adaptive use of strategies: an experimental study on the effects of two learning environments on multi-digit addition and subtraction

The adaptive use of strategies, that is selecting a strategy which allows an efficient solution for a given problem, can be considered as an important individual ability relevant in various domains. Based on models of subjects’ skills of adaptive use of strategies, two idealized instructional approaches are suggested to foster students in their strategy development. The explicit approach aims at reducing cognitive load by demonstrating and practicing strategies combined with an explicit identification of criteria for strategy efficiency by contrasting problem solutions. The implicit approach capitalizes on the generation effect and stimulates students to generate their own strategies and efficiency criteria based on the analysis of task characteristics and the comparison of problem solutions. In a 1-week experimental study (16 lessons) with 73 third-graders, we examined the effectiveness of these instructional approaches in the domain of multi-digit addition and subtraction. Results from post- and two follow-up tests after 3 and 8?months did not yield different effects of the two approaches on students’ skills in adaptive use of strategies. A comparison of strategies used by the students showed that the students of the explicit approach more frequently applied complex strategies whereas the students from the implicit approach showed a more sustainable use of self-generated strategies. Hence, for the adaptive use of those strategies students are able to generate, the implicit approach turned out to be more effective than the explicit approach. However, this generation effect does not hold for strategies which are too complex to be generated by students.     

Self‐regulation strategies improve self‐discipline in adolescents: benefits of mental contrasting and implementation intentions

Adolescents struggle with setting and striving for goals that require sustained self‐discipline. Research on adults indicates that goal commitment is enhanced by mental contrasting (MC), a strategy involving the cognitive elaboration of a desired future with relevant obstacles of present reality. Implementation intentions (II), which identify the action one will take when a goal‐relevant opportunity arises, represent a strategy shown to increase goal attainment when commitment is high. This study tests the effect of mental contrasting combined with implementation intentions (MCII) on successful goal implementation in adolescents. Sixty‐six 2nd‐year high school students preparing to take a high‐stakes exam in the fall of their third year were randomly assigned to complete either a 30‐minute written mental contrasting with implementation intentions intervention or a placebo control writing exercise. Students in the intervention condition completed more than 60% more practice questions than did students in the control condition. These findings point to the utility of directly teaching to adolescents mental contrasting with implementation intentions as a self‐regulatory strategy of successful goal pursuit.     

A catalog of models for multiplication and division of whole numbers

Summary The above catalog contains fifteen headings, each of which indicates a collection of families of models for multiplication and division of whole numbers. The catalog refers to somewhat more than sixteen families of models which are easily distinguished one from the other.Not included in the catalog thus far developed are several interpretations of multiplication and division that are also of interest. Among these are models based on the equivalency of denominations of money and various units of measurement. Other interpretations which are of historical interest are those of McLellan and Dewey [15] and Thorndike [24]. The relation between models of operations on whole numbers and models of operations defined on larger universal sets is also of interest. One aspect of this area of interest is the process of constructing models of multiplication and division of whole numbers from such models by altering the rules of the model or delimiting its universal set. For example, one can begin with one of Diénès' models of multiplication of integers [8, pp. 57–58] and make approapriate adjustments and result in a model of multiplication of whole numbers. Other interpretations developed by Diénès are of interest because they involve concretizations of whole numbers which are operators as opposed to states [8, pp. 12, 30; 9, p, 36].These are a great many strategies available for the use of models in teaching the operations on whole numbers. In one such strategy, an educator can define either multiplication or division on some basis (most likely in terms of a model) and then the other can be defined as its inverse.Another strategy is to define each operation in terms of a different model. For example, one might define multiplication in terms of the repeated addition model and division in terms of the repeated subtraction model.Still another type of procedure involves a multiple embodiment strategy in which several interpretations are taught as representing each operation.The choice of a particular strategy would depend upon a great many factors. Some of the factors would be the type of culture and students for which the program is written, the psychological assumptions adopted by the writer, and the writer's knowledge of the domain of models for the operations as well as their relation to the abstract mathematical domain which they represent. This article has contributed to a basis for intelligent decisions in this area by presenting a characterization of the domain of models for multiplication and division of whole numbers and their relation to the abstract operations.     

Teaching Kindergarten Children to Solve Word Problems

A kindergarten teacher's practice was investigated in order to understand her knowledge of her children's mathematical thinking, the ways in which she acquired that knowledge, and the uses she made of that knowledge in making instructional decisions. The focus of the investigation was the teacher's knowledge of her children's thinking about numbers, including counting and addition, subtraction, multiplication, and division. The teacher had attended Cognitively Guided Instruction workshops at which she had the opportunity to learn about research on children's mathematical thinking. She gathered information on her own children's thinking by posing word problems, listening to children as the described their strategies for solving the problems, and talking to other adults about her children. She used that information to select problems to pose in subsequent lessons.     

论珠心算在小学数学计算教学中的作用

计算能力是人类终身必备的基本能力,培养学生的计算能力应该是学校数学教学的主要任务之一。珠心算与笔算有着紧密的联系和内在规律。珠心算是在脑中运用象数思维的模型进行快速计算的方法,而笔算是借助阿拉伯数码来计算的模型。珠心算是从左向右运算的,与人的思维方式相应,符合人脑反映的顺逆关系,又与读数、写数、看数顺序一致,因而优于从低位向高位计算的笔算。在小学低年级数学计算教学中,利用珠心算能大大加快学生的计算速度,比笔算的运算过程直观、形象、快捷。