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When solving subtraction problems such as 83-46, children use the direct subtraction (DS) strategy (e.g., 83 - 40 = 43, 43 - 6 = 37) or the subtraction by addition (SBA) strategy (e.g., 46 + 4 = 50, 50 + 30 = 80, 80 + 3 = 83, so the answer is 4 + 30 + 3 = 37). This study is the first to use the choice/no-choice method to examine DS and SBA use in third-graders (8- and 9-year-olds) with varying mathematical achievement levels. All children (N = 66) solved a series of small difference subtractions (e.g., 72 - 64) and large difference subtractions (e.g., 94 - 8) in one choice condition (choice between DS and SBA) and two no-choice conditions (obligatory use of either DS or SBA). Results showed that, although only the DS strategy was taught to these children, in the choice condition nearly half the children already made use of the SBA strategy and that SBA was used on one in five subtractions. Whereas DS was the fastest strategy on large difference items, interestingly, children achieved a similar level of accuracy with the SBA strategy compared with DS. Finally, 1 in 5 children made problem-based adaptive strategy choices, and children were generally adaptive to their individual strategy speed. This study clearly demonstrates the nascent use of SBA in third-graders and provides evidence for the plea to introduce SBA as an alternative to DS already in the lower grades of elementary school.

Many studies have investigated the association between children’s spontaneous attention to number and their early mathematical abilities. This work has presented number in a non-symbolic format by exclusively using numerosities as their stimuli. Therefore, little is known about children’s spontaneous attention for Arabic number symbols. We aimed to address this gap by exploring, for the first time, whether we can observe individual differences in young children’s tendency to spontaneously focus their attention on Arabic number symbols (SFONS) using a newly developed Picture task. We investigated SFONS and its association with spontaneous focusing on numerosity (SFON), numerical abilities, and teacher ratings of mathematical competence in the three years of Flemish kindergarten. One hundred and eleven kindergartners (2 years 4 months – 6 years 2 months of age) completed a battery of tasks to measure their SFONS and SFON, as well as a range of numerical abilities (i.e., Arabic numeral identification, verbal counting, and counting objects). Kindergarten teachers were asked to rate their children’s mathematical competence on a 4-point Likert scale. Findings revealed that children in the three kindergarten years under study largely differed in their tendency to spontaneously attend to Arabic number symbols in the pictures they had to describe. We also observed differences in the levels of SFONS and SFON between the three kindergarten years. Although SFONS and SFON were not strongly associated in the three kindergarten years, significant associations were found between children’s SFONS and their numerical abilities in kindergarten year 1 and 2, and teacher ratings of their mathematical competence in kindergarten year 3. Together, these results suggest that SFONS is a relevant component of children’s early mathematical development. ispartof: Early Childhood Research Quarterly vol:48 pages:111-121 status: published

We examined children's number-based and digit-based strategy use profiles on multi-digit subtraction problems in relation to task performance and individual (gender, grade, mathematical achievement level) and contextual (country) characteristics. 318 third- to sixth-graders solved a multi-digit subtraction strategy use task, where half of the subtraction problems had number characteristics that stimulated the number-based compensation strategy. First, latent class analyses revealed five strategy use profiles. Most children consistently relied on one strategy (either digit-based strategies or the number-based decomposition or sequential strategies) across problems. Only a minority of children demonstrated strategy variety and flexibility – more often high achievers and children from higher grades. Second, children's strategy instruction history was related to their profile of strategy use, but this relation was less straightforward than generally assumed. Finally, consistent digit-based strategy users were more accurate than consistent decomposition strategy users and varied number-based strategy users.

Despite the general agreement regarding the importance of stimulating young children’s early mathematical skills, mathematical learning opportunities in preschool are limited. Educational technology (ET) may be an effective tool to address this problem. Taking into account the weaknesses of previous reviews, we conducted a systematic and comprehensive review of the research literature on the effectiveness of ET in early mathematics education, critically analyzing the findings of studies adhering to a media comparison approach (comparing an ET condition to a non-ET condition or another ET condition) versus studies following a value-added approach (comparing at least two ET conditions which only differ with respect to one feature in the ET or in its implementation). Reviewing 54 studies, we systematically analyzed (1) the effectiveness of ET, (2) the features of the ET and ET implementation associated with ET effectiveness, and (3) child characteristics associated with ET effectiveness. Our analyses indicated that media comparison studies comparing an ET to a non-ET condition provide evidence for the effectiveness of ET for enhancing young children’s mathematical competencies. Second, value-added studies pointed to ET implementation features associated with ET effectiveness, namely teacher support during ET use and an individual grouping structure. Finally, both media comparison and value-added studies revealed that ET effectiveness is associated with children’s performance in the ET and with their prior knowledge. To further advance the field, value-added studies on features of the ET and its implementation are required, as well as studies focusing on important child characteristics that are associated with ET effectiveness. ispartof: International Journal of Child-Computer Interaction vol:27 pages:1-26 status: published

Rational numbers are amongst the most difficult topics in the elementary and secondary school curriculum and teaching them requires an appropriate knowledge base of teachers to properly deal with students' difficulties. We investigated prospective teachers' content knowledge (CK) and pedagogical content knowledge (PCK) on rational numbers, the relationship between CK and PCK, and differences in CK and PCK among prospective elementary teachers (trained as general classroom teachers) and lower secondary teachers (trained as subject-specific classroom teachers). The results revealed gaps in prospective teachers' CK and PCK, a positive correlation between CK and PCK, and a better CK but not PCK for secondary compared to elementary school teachers.

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. publisher: Elsevier articletitle: Subtraction by addition in children with mathematical learning disabilities journaltitle: Learning and Instruction articlelink: http://dx.doi.org/10.1016/j.learninstruc.2013.11.001 content_type: article copyright: Copyright © 2013 Elsevier Ltd. All rights reserved. ispartof: Learning and Instruction vol:30 pages:1-8 status: published

The present study investigates adults' use of addition to solve two-digit subtractions. Inspired by research on single-digit arithmetic, we first examined regression models in which different problem characteristics predicted participants' reaction times. Second, we compared performance on two-digit subtractions presented in 2 presentation formats, i.e., the standard subtraction format (81-37=.) and an addition format (37+.=81). Both methods lead to the conclusion that the participants switched between direct subtraction and subtraction by addition depending on the relative size of the subtrahend: If the subtrahend was smaller than the difference, direct subtraction was mainly used; if the subtrahend was larger than the difference, subtraction by addition was the dominant strategy. However, this performance pattern was only observed when the distance between the subtrahend and the difference was large; when the subtrahend and the difference were close to each other, there was no subtrahend-dependent selection of direct subtraction vs. subtraction by addition. These data indicate that theoretical models of people's strategy choices in subtraction should include the relative size of the subtrahend as an important factor in the strategy selection process.

This study examined the development of indirect addition as an alternative for solving multidigit subtractions, by means of two learning environments in traditionally schooled third-graders in Flanders (Belgium). Thirty-five third-graders, who did not demonstrate mastery of indirect addition, participated in an Explicit (n ¼ 20) or Implicit (n ¼15) learning environment that aimed to encourage the development of indirect addition. Over a period of six weeks, children participated in nine individual sessions: four practice sessions, three test sessions, one transfer session, and one retention session. The results revealed that throughout the study indirect addition was rarely used, even by the children in the explicit learning environment. However, when indirect addition was used, it was executed very efficiently. 2009 Elsevier Ltd. All rights reserved.

This study examined 25 university students' use of addition to solve large single-digit subtractions by contrasting performance in the standard subtraction format (12-9=.) and in the addition format (9+.=12). In particular, we investigated the effect of the relative size of the subtrahend on performance in both formats. We found a significant interaction between format, the magnitude of the subtrahend (S) compared to the difference (D) (S>D vs. S

This article discusses the characteristics of the indirect addition strategy (IA) in the domain of multi-digit subtraction. In two studies, adults' use of IA on three-digit subtractions with a small, medium, or large difference between the integers was analysed using the choice/no-choice method. Results from both studies indicate that adults spontaneously apply IA on three-digit subtractions, and use IA with the same frequency as direct subtraction (DS) strategies. Furthermore, IA resulted in faster responses – without any loss in accuracy – than DS. Finally, adults flexibly applied IA and DS strategies on the basis of task and individual strategy performance characteristics. Our results support theoretical models on adaptive strategy choices, and question instructional practices focusing on the routine mastery of only DS.

The number line estimation task captures central aspects of children's developing number sense, that is, their intuitions for numbers and their interrelations. Previous research used children's answer patterns and verbal reports as evidence of how they solve this task. In the present study we investigated to what extent eye movements recorded during task solution reflect children's use of the number line. By means of a cross-sectional design with 66 children from Grades 1, 2, and 3, we show that eye-tracking data (a) reflect grade-related increase in estimation competence, (b) are correlated with the accuracy of manual answers, (c) relate, in Grade 2, to children's addition competence, (d) are systematically distributed over the number line, and (e) replicate previous findings concerning children's use of counting strategies and orientation-point strategies. These findings demonstrate the validity and utility of eye-tracking data for investigating children's developing number sense and estimation competence.

This study investigated ability-related differences in strategy use and development in the domain of simple arithmetic, in terms of the model of strategic change, using the choice/no-choice method and the chronological-age/ability-level-match design. Twenty-six second-graders with strong mathematical abilities (MA), 25 second-graders with weak MA, and 20 third-graders matched on MA with the strong second-graders solved 36 additions and subtractions with the bridge over 10 in 4 conditions. In the choice condition, children could choose between retrieval, decomposition to 10, and counting on. In the three no-choice conditions, they had to solve all problems with retrieval, decomposition to 10, or counting on. Results revealed differences in strategy use between strong and weak second-graders, but not between strong second-graders and matched third-graders, which indicates that the strategy development of children with weaker MA is marked by a delay. The theoretical, methodological, and practical implications of the study are discussed in detail.