Using addition to solve large subtractions in the number domain up to 20

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 < D), and the numerical distance between subtrahend and difference. When the subtrahend was larger than the difference and S and D were far from each other (e.g., 12 − 9 = .), problems were solved faster in the addition than in the subtraction format; when the subtrahend was smaller than the difference and S and D were far from each other (e.g., 12 − 3 = .), problems were solved faster in the subtraction than in the addition format. However, when the subtrahend and the difference were close to each other (e.g., 13 − 7 = .), there were no significant reaction time differences between both formats. These results suggest that adults do not rely exclusively and routinely on addition to solve large single-digit subtractions, but select either addition-based or subtraction-based strategies depending on the relative size of the subtrahend.     

Use of indirect addition in adults' mental subtraction in the number domain up to 1,000

This study examined adults' use of indirect addition and direct subtraction strategies on multi-digit subtractions in the number domain up to 1,000. Seventy students who differed in their level of arithmetic ability solved multi-digit subtractions in one choice and two no-choice conditions. Against the background of recent findings in elementary subtraction, we manipulated the size of the subtrahend compared to the difference and only selected items with large distances between these two integers. Results revealed that adults frequently and efficiently apply indirect addition on multi-digit subtractions, yet adults with higher arithmetic ability performed more efficiently than those with lower arithmetic ability. In both groups, indirect addition was more efficient than direct subtraction both on subtractions with a subtrahend much larger than the difference (e.g., 713 - 695) and on subtractions with a subtrahend much smaller than the difference (e.g., 613 - 67). Unexpectedly, only adults with lower arithmetic ability fitted their strategy choices to their individual strategy performance skills. Results are interpreted in terms of mathematical and cognitive perspectives on strategy efficiency and adaptiveness.     

Inverse reference in subtraction performance: An analysis from arithmetic word problems

Studies of elementary calculation have shown that adults solve basic subtraction problems faster with problems presented in addition format (e.g., 6?+?_?=?13) than in standard subtraction format (e.g., 13 – 6?=?_). Therefore, it is considered that adults solve subtraction problems by reference to the inverse operation (e.g., for 13 – 6?=?7, “I know that 13 is 6?+?7”) because presenting the subtraction problem in addition format does not require the mental rearrangement of the problem elements into the addition format. In two experiments, we examine whether adults' use of addition to solve subtractions is modulated by the arrangement of minuend and subtrahend, regardless of format. To this end, we used arithmetic word problems since single-digit problems in subtraction format would not allow the subtrahend to appear before the minuend. In Experiment 1, subtractions were presented by arranging minuend and subtrahend according to previous research. In Experiment 2, operands were reversed. The overall results showed that participants benefited from word problems where the subtrahend appears before the minuend, including subtractions in standard subtraction format. These findings add to a growing body of literature that emphasizes the role of inverse reference in adults' performance on subtractions.     

Strategies in subtraction problem solving in children

The aim of this study was to investigate the strategies used by third graders in solving the 81 elementary subtractions that are the inverses of the one-digit additions with addends from 1 to 9 recently studied by Barrouillet and Lépine. Although the pattern of relationship between individual differences in working memory, on the one hand, and strategy choices and response times, on the other, was the same in both operations, subtraction and addition differed in two important ways. First, the strategy of direct retrieval was less frequent in subtraction than in addition and was even less frequent in subtraction solving than the recourse to the corresponding additive fact. Second, contrary to addition, the retrieval of subtractive answers is confined to some peculiar problems involving 1 as the subtrahend or the remainder. The implications of these findings for developmental theories of mental arithmetic are discussed.     

Inverse reference in subtraction performance: an analysis from arithmetic word problems

Studies of elementary calculation have shown that adults solve basic subtraction problems faster with problems presented in addition format (e.g., 6 ± =?13) than in standard subtraction format (e.g., 13 - 6?=?). Therefore, it is considered that adults solve subtraction problems by reference to the inverse operation (e.g., for 13 - 6?=?7, "I know that 13 is 6?+?7") because presenting the subtraction problem in addition format does not require the mental rearrangement of the problem elements into the addition format. In two experiments, we examine whether adults' use of addition to solve subtractions is modulated by the arrangement of minuend and subtrahend, regardless of format. To this end, we used arithmetic word problems since single-digit problems in subtraction format would not allow the subtrahend to appear before the minuend. In Experiment 1, subtractions were presented by arranging minuend and subtrahend according to previous research. In Experiment 2, operands were reversed. The overall results showed that participants benefited from word problems where the subtrahend appears before the minuend, including subtractions in standard subtraction format. These findings add to a growing body of literature that emphasizes the role of inverse reference in adults' performance on subtractions.     

Children's understanding of the relation between addition and subtraction: inversion, identity, and decomposition.

In order to understand addition and subtraction fully, children have to know about the relation between these two operations. We looked at this knowledge in two studies. In one we asked whether 5- and 6-year-old children understand that addition and subtraction cancel each other out and whether this understanding is based on the identity of the addend and subtrahend or on their quantity. We showed that children at this age use the inversion principle even when the addend and subtrahend are the same in quantity but involve different material. In our second study we showed that 6- to 8-year-old children also use the inversion in combination with decomposition to solve a + b - (b + 1) problems. In both studies, factor analyses suggested that the children were using different strategies in the control problems, which require computation, than in the inversion problems, which do not. We conclude that young children understand the relations between addition and subtraction and that this understanding may not be based on their computational skills.     

Patterns of problem‐solving in children's literacy and arithmetic

Patterns of problem‐solving among 5‐to‐7 year‐olds' were examined on a range of literacy (reading and spelling) and arithmetic‐based (addition and subtraction) problem‐solving tasks using verbal self‐reports to monitor strategy choice. The results showed higher levels of variability in the children's strategy choice across Years 1 and 2 on the arithmetic (addition and subtraction) than literacy‐based tasks (reading and spelling). However, across all four tasks, the children showed a tendency to move from less sophisticated procedural‐based strategies, which included phonological strategies for reading and spelling and counting‐all and finger modelling for addition and subtraction, to more efficient retrieval methods from Years 1 to 2. Distinct patterns in children's problem‐solving skill were identified on the literacy and arithmetic tasks using two separate cluster analyses. There was a strong association between these two profiles showing that those children with more advanced problem‐solving skills on the arithmetic tasks also showed more advanced profiles on the literacy tasks. The results highlight how different‐aged children show flexibility in their use of problem‐solving strategies across literacy and arithmetical contexts and reinforce the importance of studying variations in children's problem‐solving skill across different educational contexts.     

Self‐regulated learning of basic arithmetic skills: A longitudinal study

Background. Several studies have examined young primary school children's use of strategies when solving simple addition and subtraction problems. Most of these studies have investigated students’ strategy use as if they were isolated processes. To date, we have little knowledge about how math strategies in young students are related to other important aspects in self‐regulated learning. Aim. The main purpose of this study was to examine relations between young primary school children's basic mathematical skills and their use of math strategies, their metacognitive competence and motivational beliefs, and to investigate how students with basic mathematics skills at various levels differ in respect to the different self‐regulation components. Sample. The participants were comprised of 27 Year 2 students, all from the same class. Method. The data were collected in three stages (autumn Year 2, spring Year 2, and autumn Year 3). The children's arithmetic skills were measured by age relevant tests, while strategy use, metacognitive competence, and motivational beliefs were assessed through individual interviews. The participants were divided into three performance groups; very good students, good students, and not‐so‐good students. Results. Analyses revealed that young primary school children at different levels of basic mathematics skill may differ in several important aspects of self‐regulated learning. Analyses revealed that a good performance in addition and subtraction was related not only to the children's use of advanced mathematics strategies, but also to domain‐specific metacognitive competence, ability attribution for success, effort attribution for failure, and high perceived self‐efficacy when using specific strategies. Conclusions. The results indicate that instructional efforts to facilitate self‐regulated learning of basic arithmetic skills should address cognitive, metacognitive, and motivational aspects of self‐regulation. This is particularly important for low‐performing students.     

小学生表征数学应用题策略的实验研究

通过一个2(成功与否)×2(提示与否)×2(题型)的混合实验设计,对小学五年级学生解决和差应用题的表征策略进行了研究.结果表明:(1)与比较应用题的表征相类似,小学生对和差应用题的表征也存在着直译策略和问题模型策略;(2)不成功组解题者在表征和差应用题时倾向于运用直译策略,而成功组的解题者更倾向于运用问题模型策略,这导致了成功者与不成功者在列式上的差异,特别是在不一致题型上表现得更明显;(3)在读题前给以“请注意理解这道题的意思”这样简单的提示,对不成功的解题者对和差问题的正确表征并不能起到作用;(4)成功的和差应用题解题者和不成功的解题者在列式正确性的自我评价上存在显著差异.     

The association between children's numerical magnitude processing and mental multi-digit subtraction

Children apply various strategies to mentally solve multi-digit subtraction problems and the efficient use of some of them may depend more or less on numerical magnitude processing. For example, the indirect addition strategy (solving 72–67 as “how much do I have to add up to 67 to get 72?”), which is particularly efficient when the two given numbers are close to each other, requires to determine the proximity of these two numbers, a process that may depend on numerical magnitude processing. In the present study, children completed a numerical magnitude comparison task and a number line estimation task, both in a symbolic and nonsymbolic format, to measure their numerical magnitude processing. We administered a multi-digit subtraction task, in which half of the items were specifically designed to elicit indirect addition. Partial correlational analyses, controlling for intellectual ability and motor speed, revealed significant associations between numerical magnitude processing and mental multi-digit subtraction. Additional analyses indicated that numerical magnitude processing was particularly important for those items for which the use of indirect addition is expected to be most efficient. Although this association was observed for both symbolic and nonsymbolic tasks, the strongest associations were found for the symbolic format, and they seemed to be more prominent on numerical magnitude comparison than on number line estimation.     

Subtraction by addition

University students’ self-reports indicate that they often solve basic subtraction problems (13?6=?) by reference to the corresponding addition problem (6+7=13; therefore, 13?6=7). In this case, solution latency should be faster with subtraction problems presented in addition format (6+_=13) than in standard subtraction format (13+6=_). In Experiment 1, the addition format resembled the standard layout for addition with the sum on the right (6+_=13), whereas in Experiment 2, the addition format resembled subtraction with the minuend on the left (13=6+_). Both experiments demonstrated a latency advantage for large problems (minuend > 10) in the addition format as compared with the subtraction format (13+6=_), although the effect was larger in Experiment 1 (254 msec) than in Experiment 2 (125 msec). Small subtractions (minuend ≤ 10) in Experiment 1 were solved equally quickly in the subtraction or addition format, but in Experiment 2, performance on small problems was faster in the standard format (5?3=_) than in the addition format (5=3+_). The results indicate that educated adults often use addition reference to solve large simple subtraction problems, but that they rely on direct memory retrieval for small subtractions.     

小学生图形推理策略发展特点的研究

本研究用我们修订的瑞文推理为材料,对145名儿童图形推理策略进行研究。结果发现：小学生在解决图形推理问题时使用六种策略,它们分别是分析策略、不完全分析策略、知觉分析策略、知觉匹配策略、格式塔策略和自主想象策略;不同年龄儿童在解决不同类型题目时的策略使用表现出不同的特点,儿童在解决数量规则题中,知觉分析策略在整个小学儿童阶段占主导地位,而在解决加减规则题中,分析策略占主导地位,随年龄增长而呈上升趋势;小学二年级开始出现图形推理能力发展的飞跃,二年级儿童开始能够同时观察到两种规则,五六年级儿童更能够不受题目形式的影响．而是从本质上把握逻辑规则。     

Flexibility in strategy use: Adaptation of numerosity judgement strategies to task characteristics

In previous investigations we documented that people use several strategies to determine numerosities of blocks that are presented in a square grid. One of these strategies is the clever subtraction strategy, wherein the number of empty squares in the grid is subtracted from the total number of squares in the grid. In the present study we investigated participants' flexibility in strategy use when varying the size of the grids. Results are described in terms of the theoretical framework of Lemaire and Siegler (1995) regarding strategic change, and show that this contextual variable affected the frequency, execution time, and accuracy of subjects' use of the subtraction strategy. The usefulness of this framework for analysing the nature of the adaptation to contextual variations is discussed. From a methodological point of view, this study documents the potential of Beem's (1993, 1999) segmented linear regression models for assessing subjects' strategy use in cognitive tasks.     

Explaining inappropriate strategy selection in a simple reasoning task

People's strategy selections appear to reflect attempts to maximize performance by selecting the most effective option for a particular task or format. Theories that account for such behaviour will be named rational models of strategy selection. However, it is possible to find instances where people are apparently biased towards using less effective strategies, and such behaviour appears to go against these models. Two experiments are reported in which participants were instructed to use first one, and then the other of two possible strategies for solving a compass point directions task (the instructed phase), and were subsequently permitted to use any strategy (the free‐choice phase). A substantial minority of participants selected the less effective spatial strategy during the free‐choice phase. Overall, it was found that people who rely on the spatial strategy when given a free‐choice tend to be those who: (1) have not been given particular incentive to perform as well as possible; (2) have difficulty executing the better alternative, cancellation and (3) are particularly prone to making errors at the spatial strategy. Hence, although evidence was found in support of rational models of strategy selection, it is also suggested that these must additionally take account of the motivational and conceptual difficulties that people may have with a task.     

The Five Factor Mindfulness Questionnaire in Norway

The aim of this study was to adapt the Five Factor Mindfulness Questionnaire (FFMQ) for use in Norway. Three studies involving three different samples of university students (mean age 22 years, total N = 792) were conducted. Confirmatory factor analyses showed that a five factor structure provided an acceptable fit to the data. All five factors loaded significantly on the overall mindfulness factor. As expected, correlations between the FFMQ total scores and subscales were positive and significant, ranging from 0.45 to 0.65. Correlations between FFMQ total/subscales and Mindful Attention Awareness Scale (MAAS) were significant and negative (since low scores on the MAAS indicate high mindfulness), ranging from r = ?0.17 to ?0.69. The Norwegian FFMQ total score was inversely correlated with all indicators of psychological health: neuroticism (r = ?0.61), ruminative tendencies (r = ?0.41), self‐related negative thinking (r = ?0.40), emotion regulation difficulties (r = ?0.66) and depression (r = ?0.46 to r = ?0.65). In contrast to the other FFMQ subscales, the FFMQ Observe subscale did not have a positive relation to psychological health in our mostly non‐meditating sample. However, being able to non‐judgmentally observe one's inner life and environment is a part of the mindfulness construct that might emerge more clearly with more mindfulness training. We conclude that the Norwegian FFMQ has acceptable psychometric properties and can be recommended for use in Norway, especially in studies seeking to differentiate between different aspects of mindfulness and how these may change over time.     

小学生图形推理策略个体差异

选取一至六年级儿童145名为对象,根据项目中图形关系的规则,把瑞文测验的项目分成6类,选择出14道题目为实验材料,探讨小学生图形推理策略的个体差异.结果发现,数学能力不同的小学生在解决简单图形推理问题时,基本上使用知觉算法策略,没有表现出差异.但在较难图形推理问题时差异显著,中高数学水平儿童主要使用分析策略和知觉分析策略;而低数学水平儿童基本上使用知觉匹配策略;除了格式塔类型的题目以外,推理水平高的儿童在解决这五类题目时主要使用分析策略和知觉分析策略,而推理水平低的儿童主要使用知觉匹配策略;推理水平高的儿童在解决较为简单的图形推理问题时的策略使用很集中,随着题目难度的加大,策略变得越来越分散,而推理水平低的儿童则无论题目难易如何变化,他们的策略主要使用知觉匹配策略,而且表现出随机性特点.     

Doing Despite Disliking: Self‐regulatory Strategies in Everyday Aversive Activities

We investigated the self‐regulatory strategies people spontaneously use in their everyday lives to regulate their persistence during aversive activities. In pilot studies (pooled N = 794), we identified self‐regulatory strategies from self‐reports and generated hypotheses about individual differences in trait self‐control predicting their use. Next, deploying ambulatory assessment (N = 264, 1940 reports of aversive/challenging activities), we investigated predictors of the strategies' self‐reported use and effectiveness (trait self‐control and demand types). The popularity of strategies varied across demands. In addition, people higher in trait self‐control were more likely to focus on the positive consequences of a given activity, set goals, and use emotion regulation. Focusing on positive consequences, focusing on negative consequences (of not performing the activity), thinking of the near finish, and emotion regulation increased perceived self‐regulatory success across demands, whereas distracting oneself from the aversive activity decreased it. None of these strategies, however, accounted for the beneficial effects of trait self‐control on perceived self‐regulatory success. Hence, trait self‐control and strategy use appear to represent separate routes to good self‐regulation. By considering trait‐ and process‐approaches these findings promote a more comprehensive understanding of self‐regulatory success and failure during people's daily attempts to regulate their persistence. © 2018 European Association of Personality Psychology     

Girls’ Spatial Skills and Arithmetic Strategies in First Grade as Predictors of Fifth-Grade Analytical Math Reasoning

This study investigated longitudinal pathways leading from early spatial skills in first-grade girls to their fifth-grade analytical math reasoning abilities (N = 138). First-grade assessments included spatial skills, verbal skills, addition/subtraction skills, and frequency of choice of a decomposition or retrieval strategy on the addition/subtraction problems. In fifth grade, girls were given an arithmetic fluency test, a mental rotation spatial task, and a numeric and algebra math reasoning test. Using structural equation modeling, the estimated path model accounted for 87% of the variance in math reasoning. First-grade spatial skills had a direct pathway to fifth-grade math reasoning as well as an indirect pathway through first-grade decomposition strategy use. The total effect of first-grade spatial skills was significantly higher in predicting fifth-grade math reasoning than all other predictors. First-grade decomposition strategy use had the second strongest total effect, while retrieval strategy use did not predict fifth-grade math reasoning. It was first-grade spatial skills (not fifth-grade) that directly predicted fifth-grade math reasoning. Consequently, the results support the importance of early spatial skills in predicting later math. As expected, decomposition strategy use in first grade was linked to fifth-grade math reasoning indirectly through first-grade arithmetic accuracy and fifth-grade arithmetic fluency. However, frequency of first-grade decomposition use also showed a direct pathway to fifth-grade arithmetic reasoning, again stressing the importance of these early cognitive processes on later math reasoning.     

The value of proactive goal setting and choice in 3‐ to 7‐year‐olds' use of working memory gating strategies in a naturalistic task

Rule‐guided behavior depends on the ability to strategically update and act on content held in working memory. Proactive and reactive control strategies were contrasted across two experiments using an adapted input/output gating paradigm (Neuron, 81, 2014 and 930). Behavioral accuracies of 3‐, 5‐, and 7‐year‐olds were higher when a contextual cue appeared at the beginning of the task (input gating) rather than at the end (output gating). This finding supports prior work in older children, suggesting that children are better when input gating but rely on the more effortful output gating strategy for goal‐oriented action selection (Cognition, 155, 2016 and 8). A manipulation was added to investigate whether children's use of working memory strategies becomes more flexible when task goals are specified internally rather than externally provided by the experimenter. A shift toward more proactive control was observed when children chose the task goal among two alternatives. Scan path analyses of saccadic eye movement indicated that giving children agency and choice over the task goal resulted in less use of a reactive strategy than when the goal was determined by the experimenter.     

Moving attention along the mental number line in preschool age: Study of the operational momentum in 3‐ to 5‐year‐old children's non‐symbolic arithmetic

People tend to underestimate subtraction and overestimate addition outcomes and to associate subtraction with the left side and addition with the right side. These two phenomena are collectively labeled 'operational momentum' (OM) and thought to have their origins in the same mechanism of 'moving attention along the mental number line'. OM in arithmetic has never been tested in children at the preschool age, which is critical for numerical development. In this study, 3–5 years old were tested with non‐symbolic addition and subtraction tasks. Their level of understanding of counting principles (CP) was assessed using the give‐a‐number task. When the second operand's cardinality was 5 or 6 (Experiment 1), the child's reaction time was shorter in addition/subtraction tasks after cuing attention appropriately to the right/left. Adding/subtracting one element (Experiment 2) revealed a more complex developmental pattern. Before acquiring CP, the children showed generalized overestimation bias. Underestimation in addition and overestimation in subtraction emerged only after mastering CP. No clear spatial‐directional OM pattern was found, however, the response time to rightward/leftward cues in addition/subtraction again depended on stage of mastering CP. Although the results support the hypothesis about engagement of spatial attention in early numerical processing, they point to at least partial independence of the spatial‐directional and magnitude OM. This undermines the canonical version of the number line‐based hypothesis. Mapping numerical magnitudes to space may be a complex process that undergoes reorganization during the period of acquisition of symbolic representations of numbers. Some hypotheses concerning the role of spatial‐numerical associations in numerical development are proposed.