Search

An important approach for developing children's algebraic thinking involves introducing them to generalized arithmetic at the time they are learning arithmetic. Our aim in this study was to investigate children's attention to and expression of generality with the subtraction-compensation property, as evidence of a type of algebraic thinking known as relational thinking. The tasks involved subtraction modelled as difference and comparing the heights of towers of blocks. In an exploratory qualitative study, 22 middle primary (9-11-year-old) students from two schools participated in individual videoed interviews. The tasks were designed using theoretical perspectives on embodied visualization and concreteness fading to provide multiple opportunities for the students to make sense of subtraction as difference and to advance their relational thinking. Twelve out of 22 students evidenced conceptual understanding of the comparison model of subtraction (subtraction as difference) and expression of the compensation property of equality. Four of these students repeatedly evidenced relational thinking for true/false tasks and open equivalence tasks. A proposed framework for levels of attention to/expression of generality with the subtraction-compensation property is shared and suggestions for further research are presented.

Fraction arithmetic facility is fundamental to learning more advanced math topics. However, attaining the ability to add and subtract fractions is hard for many students. The present longitudinal study examined students' growth on simple addition and subtraction word problems between fourth and sixth grades (N = 536). Latent class growth analyses revealed three empirically distinct growth trajectories: (a) "consistently accurate" students who displayed high accuracy across time; (b) "high-growth" students who demonstrated low initial accuracy but made subsequent growth; and (c) "low-growth" students who demonstrated consistently poor accuracy across time. Age and whole number calculation fluency predicted membership in the consistently accurate and high-growth classes relative to membership in the low-growth class. Language and nonverbal reasoning skills were identified as predictors of student membership in the consistently accurate versus low-growth class. Low-growth students displayed more whole number bias and calculation errors compared with students who demonstrated growth. Findings reveal that a concerning subset of students make little to no progress in simple fraction arithmetic computation, despite several years of fractions instruction in school.

Adults shift their attention to the right or to the left along a spatial continuum when solving additions and subtractions, respectively. Studies suggest that these shifts not only support the exact computation of the results but also anticipatively narrow down the range of plausible answers when processing the operands. However, little is known on when and how these attentional shifts arise in childhood during the acquisition of arithmetic. Here, an eye-tracker with high spatio-temporal resolution was used to measure spontaneous eye movements, used as a proxy for attentional shifts, while children of 2nd (8 y-o; N = 50) and 4th (10 y-o; N = 48) Grade solved simple additions (e.g., 4 + 3) and subtractions (e.g., 3 - 2). Gaze patterns revealed horizontal and vertical attentional shifts in both groups. Critically, horizontal eye movements were observed in 4th Graders as soon as the first operand and the operator were presented and thus before the beginning of the exact computation. In 2nd Graders, attentional shifts were only observed after the presentation of the second operand just before the response was made. This demonstrates that spatial attention is recruited when children solve arithmetic problems, even in the early stages of learning mathematics. The time course of these attentional shifts suggests that with practice in arithmetic children start to use spatial attention to anticipatively guide the search for the answer and facilitate the implementation of solving procedures.

As students have struggled to use the "chip model" (i.e., red and yellow chips representing positive and negative numbers) to model integer addition and subtraction and have found it confusing, the authors developed a series of activities based on adding and removing opposite objects to and from a boat to better help students in this area of mathematics. In this article, they describe a sequence of four Floats and Anchors activities that they use to introduce integer addition and subtraction. This "Construct It!" task typically takes between 50 to 75 minutes of class time and requires no prior knowledge of integers. This task sets students up to successfully use both the chip model and a vertical number line as tools for reasoning about integers.

The decomposition of numbers when solving subtraction tasks is regarded as more powerful than counting-based strategies. Still, many students fail to solve subtraction tasks despite using decomposition. To shed light upon this issue, we take a variation theoretical perspective (Marton, 2015) seeing learning as a function of discerning critical aspects and their relations of the object of learning. In this paper, we focus on what number relations students see in a three-digit subtraction task, and how they see them. We analyzed interview data from 55 second-grade students who used decomposition strategies to solve 204-193=. The variation theory of learning was used to analyze what number relations the students experienced and how they experienced them, aiming to explain why they made errors even though they used presumably powerful strategies in their problem-solving. The findings show that students who simultaneously experienced within-number relations and between-number relations when solving the task succeeded in solving it, whereas those who did not do this failed. These findings have importance for understanding what students need to discern in order to be able to solve subtraction tasks in a proficient way.

Researchers determined virtual manipulatives, on their own or as part of an instructional sequence, to be an evidence-based practice for students with intellectual and developmental disabilities (IDD; Long et al., 2022). Yet, the majority of research looking at virtual manipulative-based instructional sequences involves the phases learned in a specific order rather than an integrated fashion, which researchers suggested may be more effective and efficient (Root et al., 2021; Strickland & Maccini, 2012). This single-case design study explored use of the virtual-representational-abstract integrated (VRI) instructional sequence to teach three middle school students with IDD to solve computational subtraction with regrouping problems. Researchers found a functional relation between the intervention package of the VRA-I instructional sequence and corrective feedback for incorrect probe answers and student accuracy. Researchers also found mixed results for each student with regards to generalization to solving word problems involving subtraction with regrouping.

This article illustrates how teachers can use number lines to support students with or at risk for learning disabilities (LD) in mathematics. Number lines can be strategically used to help students understand relations among numbers, approach number combinations (i.e., basic facts), as well as represent and solve addition and subtraction problems. The authors draw from the existing research base to describe practices for incorporating number line representations to (a) support students' conceptual understanding of number and (b) teach addition and subtraction computational strategies. Specific examples are provided for teachers to integrate number line representations with other mathematical models to support the development of whole number understanding for students with LD in mathematics.

This study, which used a multiple baseline across participants' design, examines the effectiveness of a modified schema-based instructional approach to improve the mathematical word problem solving performance of three students with mild intellectual disability, two of whom had an autism spectrum disorder. Following the intervention, the three students improved their performance when solving addition and subtraction change word problems; however, their performance was inconsistent on change word problems. The effects of the instruction were generalized to two-step addition and subtraction word problems for the three participants. Moreover, the results were generalized to an untrained setting and were maintained 8 weeks after the instruction. The implications of these findings for teaching problem-solving skills to students with intellectual disability are discussed.

This article presents a case study of two Grade 5 boys' argumentation concerning addition and subtraction of negative numbers while using an interactive tablet-based application simulating positive and negative tiles. We examine the properties of integers they conjectured, and the kinds of evidence and arguments they used to support their conjectures. The proof-based teaching theory used to develop the tasks, and the features of the virtual manipulative environment, are described. The results show that the tasks, in combination with the virtual manipulative environment, allowed the boys to perform calculations that they had not been able to perform previously, that in one instance one boy used a deductive argument to explain a conjecture they had made, and that a known weakness of the counterbalance model of integers that was used was not a problem in this case.

This study is the first to examine the associations between the occurrence, frequency, and adaptivity of children's subtraction by addition strategy use (SBA; e.g., 712 - 346 = ?; 346 + 54 = 400, 400 + 300 = 700, 700 + 12 = 712, and 54 + 300 + 12 = 366) and their underlying conceptual knowledge. Specifically, we focused on two rarely studied components of conceptual knowledge: children's knowledge of the addition/subtraction complement principle (i.e., if a + b = c, then c - b = a and c - a = b) and their knowledge of different conceptual subtraction models (i.e., understanding that subtraction can be conceived not only as "taking away" but also as "determining the difference"). SBA occurrence was examined using a variability on demand task, in which children had to use multiple strategies to solve a subtraction. SBA frequency and strategy adaptivity were investigated with a task in which children could freely choose between SBA and direct subtraction (e.g., 712 - 346 = ?; 712 - 300 = 412, 412 - 40 = 372, and 372 - 6 = 366) to solve 15 subtractions. We measured both children's knowledge of the addition/subtraction complement principle, and whether they understood subtraction also as "determining the difference." SBA occurrence and frequency were not related to conceptual knowledge. However, strategy adaptivity was related to children's knowledge of the addition/subtraction complement principle. Our findings highlight the importance of attention to conceptual knowledge when teaching multi-digit subtraction and expand the literature about the relation between procedural and conceptual knowledge.

Mathematical writing is one way for primary students to communicate their mathematical thinking. Research in the field of writing has shown that to become an effective teacher of writing, preservice teachers must have experience engaging in the kinds of writing given to their students. The study reported in this paper explored how 27 preservice teachers enrolled in an elementary mathematics methods course explained their reasons for selecting strategies to solve equations that required making either addition or subtraction calculations. Element scoring was used to analyse the preservice teachers' reasons for selecting the strategy and the composition of their mathematical writing. Results indicated that the majority of preservice teachers included a mathematical reason for selecting their strategy, however; the reasons varied in their connection to the mathematics discipline. This study contributes to the existing literature on mathematical writing by providing insight into how preservice teachers explain their mathematical reasons for selecting a computation strategy; thereby, demonstrating the need for increased opportunities for preservice teachers to write.

This qualitative case study investigates how, and to what extent, primary school teachers notice students' mathematical thinking in the context of whole number subtraction. A task involving a student's invented strategy was used to collect data. Three noticing questions connected to the task were asked to 45 teachers. Their written answers were analyzed to reveal teachers' noticing skills based on attending, interpreting, and deciding how to respond. The study results revealed that most participants provided limited evidence of attending and interpreting skills, and some responses showed a lack of evidence. More specifically, they could not identify the relationship between digits in ones and tens places. Further, their interpretations did not directly focus on students' solutions and consisted of general statements and misconceptions.

The concept of number sense first appeared theoretically in the report published by NCTM in 1989. Although it is an important concept that includes the meaning of numbers and the relationship between numbers, it is very difficult to make a clear definition of it. This could also be true for the concept of integer, which we cannot place in our world of meaning in daily life or which we have difficulty in finding an equivalent for. For integers that have an abstract world of meaning, a misconception may occur between the integer sign and the sign used for operation. In this respect, this study aimed to examine the effect of subtraction, which can create pre-learning, on the concept of integers. Moreover, it was requested to demonstrate that the operational knowledge used by students in subtraction can be utilized in teaching integers. A total of 17 students studying in the 5th grade of a secondary school in Turkey participated in the study. As the research design, the action research method, one of qualitative research methods, was used, and the data were collected with the trilogy technique. The data obtained were analyzed using descriptive analysis and content analysis. The findings revealed that the teaching technique of subtraction opened the door for students to form and make sense of the concept of minus. In this context, it is recommended to provide mathematical skills in a cumulative manner.

Background/purpose: Investigation into the misconceptions of preschool students in mathematics and their differences between the ages of 4-5 and 5-6 years old helps form appropriate developmental mathematics teaching programs. However, although several studies have been conducted examining preschoolers' previous knowledge and misconceptions about mathematics, no corresponding research has been found in Greece. This study aims to investigate preschoolers' misconceptions about numbers and operations and to reveal differences between preschoolers aged 4-5 and 5-6 years old. Materials/methods: Data were collected through semi-structured interviews and analyzed according to content analysis methodology. Results: The study's results showed that 5-6-year-old preschoolers perform better than those aged 4-5 years old. Most misconceptions of the latter group appeared to be related to reverse counting, identifying arithmetic symbols and their matching quantities, adding and removing numbers without using auxiliary objects and multiplication. On the contrary, some preschoolers aged 5-6 years old needed help adding or subtracting two-digit numbers without the use of auxiliary objects. There were also a few cases where cardinality, division, and multiplication were observed. Conclusion: The misconceptions identified in the two groups of students regarding numbers and operations and their distinct needs that emerged through the research will allow teachers to offer differentiated instruction and personalize teaching support.

The notion that mental arithmetic is associated with shifts of spatial attention along a spatially organised mental number representation has received empirical support from three lines of research. First, participants tend to overestimate results of addition and underestimate those of subtraction problems in both exact and approximate formats. This has been termed the operational momentum (OM) effect. Second, participants are faster in detecting right-sided targets presented in the course of addition problems and left-sided targets in subtraction problems (attentional bias). Third, participants are biased toward choosing right-sided response alternatives to indicate the results of addition problems and left-sided response alternatives for subtraction problems (Spatial Association Of Responses [SOAR] effect). These effects potentially have their origin in operation-specific shifts of attention along a spatially organised mental number representation: rightward for addition and leftward for subtraction. Using a lateralised target detection task during the calculation phase of non-symbolic additions and subtractions, the current study measured the attentional focus, the OM and SOAR effects. In two experiments, we replicated the OM and SOAR effects but did not observe operation-specific biases in the lateralised target-detection task. We describe two new characteristics of the OM effect: First, a time-resolved, block-wise analysis of both experiments revealed sequential dependency effects in that the OM effect builds up over the course of the experiment, driven by the increasing underestimation of subtraction over time. Second, the OM effect was enhanced after arithmetic operation repetition compared to trials where arithmetic operation switched from one trial to the next. These results call into question the operation-specific attentional biases as the sole generator of the observed effects and point to the involvement of additional, potentially decisional processes that operate across trials.

Years and Centuries are the measurement units used to quantify a longer time duration, while subtraction is the operation required to determine the duration based on two given time points. However, subtraction of time is a difficult skill to be mastered by many elementary students. To identify the root cause of the student's failure in performing subtraction involving the unit of time, we developed and validated the three cognitive models related to this skill by conducting a descriptive study which involved 119 Grade Five students from three Malaysian elementary schools. The cognitive diagnostic assessment developed based on the three cognitive models was used to elicit the participants' responses. Then, Attribute Hierarchy Method and Classical Test Theory were employed to analyse the data. The findings indicated that the hierarchical structures of all cognitive models are supported by the student's responses. The three student-based cognitive models were also highly consistent with the corresponding expert-based cognitive models. The cognitive models developed could guide diagnostic assessment development and diagnostic inference making.

The aim of this research is to determine the mental processing strategies used by 3rd grade primary school students in addition and subtraction using the empty number lines. The model of the research is a case study, one of the qualitative research methods. The participants consist of 143 students; 75 girls and 68 boys, attending the third grade (8-years-old). Content analysis method was used in data analysis. Considering the results of the research, the mental strategies used by primary school 3rd-grade students were gathered in four categories: counting, balancing, first process step and partitioning. It was concluded that the students mostly used the mental processing strategy of using "multiples of 10", which is in the first process category, and this strategy was followed by "counting all" and "partitioning the second number", the least used mental processing strategy by the students was "using equivalent numbers".

The objective of this study was to evaluate and characterize the argumentation patterns used by seventh-grade students in the context of algebraic addition and subtraction problems. A qualitative case study was conducted using the Claim-Evidence-Reasoning (CER) model (McNeill & Krajcik, 2008) and argument quality framework. To describe the arguments put forth by the participants, a sample of ten students with high mathematics proficiency and ten students with low mathematics proficiency were selected from the target population of junior high schools in a large region of West Java, Indonesia. Data collection was carried out using written argumentation frame for algebraic operations (AFAO). The results revealed that, although none of the students employed the CH+EH+RH pattern (which represents the highest quality of arguments), students with high mathematics proficiency levels exhibited a greater prevalence of these argumentation patterns compared to those with low mathematics proficiency levels. The findings have implications as a valuable resource for teachers in monitoring the advancement of their students and preventing or alleviating diverse difficulties or inaccuracies that they may face. Based on the findings, a specific design for a classroom teaching activity is proposed.

This study investigated whether mathematics education based on a multi-tiered response to intervention (RTI) model can support students' arithmetic competence in primary schools in Sweden. The intent was to identify and support students at risk of failure. In this study, 113 students participated in the intervention, and 30 students participated in the control group. Both groups were followed from Grade 1 to the end of Grade 2 and compared. During the first semester in Grade 1, all students were taught basic addition and subtraction with explicit instructions in Tier 1. Those who did not respond to Tier 1 after one semester were provided support within Tier 2 during the second semester. The same was repeated in grade 2 and the students that did not respond to Tier 2 were supported within Tier 3. At the end of Grade 2, students in the intervention group performed significantly higher on the basic arithmetic competence in the number range 1-9 than the control group. No significant difference was found in a test measuring basic arithmetic competence in the number range 10-19. This study shows that using multi-tiered RTI might be sufficient to identify and support students at risk in early arithmetic competence.

The relation to operands (RO) principles describe the relation between operands and answers in arithmetic problems (e.g., the sum is always larger than its positive addends). Despite being a fundamental property of arithmetic, its empirical relation with arithmetic/algebraic problem solving has seldom been investigated. The current longitudinal study aimed to address this issue. Two-hundred-and-two Chinese fifth graders (57% boys) were assessed on their RO understanding. Their arithmetic/algebraic problem solving were assessed multiple times over 2 years. Results from latent growth curve modeling showed that RO understanding predicted the growth in arithmetic/algebraic problem solving when other known predictors of arithmetic/algebraic problem solving were controlled. The findings highlight the role of RO understanding in children's mathematical development. Interventions should be developed to enhance children's RO understanding.

Background: In several countries, children's math skills have been declining at an alarming rate in recent years and decades, and one of the explanations for this alarming situation is that children have difficulties in establishing the relations between arithmetical operations. Aim: In order to address this question, our goal was to determine the predictive power of previously taught operations on newly taught ones above general cognitive skills and basic numerical skills. Samples: More than one hundred children in each school level from Grades 2 to 5 from various socio-cultural environments (N = 435, 229 girls) were tested. Methods: Children were assessed on their abilities to solve the four basic arithmetic operations. They were also tested on their general cognitive abilities, including working memory, executive functions (i.e., inhibition and flexibility), visual attention and language. Finally, their basic numerical skills were measured through a matching task between symbolic and nonsymbolic numerosity representations. Additions and subtractions were presented to children from Grade 2, multiplications from Grade 3 and divisions from Grade 4. Results and Conclusions: We show that addition predicts subtraction and multiplication performance in all grades. Moreover, multiplication predicts division performance in both Grades 4 and 5. Finally, addition predicts division in Grade 4 but not in Grade 5 and subtraction and division are not related whatever the school grade. These results are examined considering the existing literature, and their implications in terms of instruction are discussed.

One of the most fundamental subjects of primary school mathematics, the concept of fraction plays a key role in students' educational background. Students experience difficulties and misconceptions regarding this concept. Therefore, it is of great importance to understand this subject well and to use different teaching methods and techniques. Various methods can be used in learning mathematics, such as invention method, computer-assisted teaching, problem solving, creative drama and discussion. Argumentation method is a scientific discussion of a claim on a subject by providing data, warrant and backing, and by revealing the rebuttal and the invalidity of the claim. Students gain more skills by interacting with teachers or collaborating with their peers, and that they possess skills that have not been obtained before or that are too difficult to achieve on their own. Therefore, the argumentation method was chosen for its effect on the development of primary school students' fraction calculation strategies. Having a case study design, this study was conducted with 28 primary school fourth-grade students and 10 open-ended questions prepared by the researchers were used during interviews. The results suggested that primary school students were able to develop fraction calculation strategies through the argumentation method. In general, inventing strategies, using representations, number sense, addition and subtraction, using unit fractions and proportional reasoning were used as fraction calculation strategies. It recommended to use the argumentation method in other critical subjects of mathematics as well.

The effects of whole number computation interventions among school students with learning disabilities in Grades K to 5 were examined using a multilevel meta-analysis. Applying a correlated and hierarchical effect model of robust variance estimation, we examined the intervention effects among 15 peer-reviewed articles and dissertations (two group-design and 13 single-case design studies) published between 1989 and 2021. Whole number computation interventions demonstrated a statistically significant and large effect on whole number computation outcomes ([g-bar]= 3.74). The type of mathematical operations, type of whole number computation measures, and the number of instructional components did not significantly affect the magnitude of the effect size estimate. However, the results showed slightly larger average effect sizes for the addition problem, the accuracy measure, and the additional number of instructional components by one. The limitations and implications for the practice of the meta-analysis are discussed, and future research directions are proposed.

This study examined the influence of addition and subtraction practice embedded within a word-problem intervention for Grade 3 students experiencing word-problem difficulty. Students (N = 448) were randomly assigned to one of two variants of a word-problem intervention or a business-as-usual comparison condition. Within the word-problem interventions, students participated in targeted addition and subtraction fact and computation practice for approximately 3 minutes of each 30-minute session across 47 intervention sessions, but also continually practiced addition and subtraction through word-problem solving. At posttest, students who participated in the intervention demonstrated improved single-digit addition and subtraction performance as well as improved double-digit addition and subtraction performance. At follow-up in Grade 4, persistence of effects was only identified for double-digit addition. Considering addition and subtraction are foundational to many other mathematics skills, addition and subtraction practice serves as an important component of mathematics intervention.

Integer addition and subtraction entail significant conceptual shifts for many students, and teachers often offer rules intended to help mitigate this challenge. When these rules are not connected to sense making, a misremembered rule can function as a barrier to student learning. In this article, the authors highlight the work of an expert teacher who leveraged context as a resource for student sense making about integer subtraction. The authors highlight her use of zero pairs as a content-specific instructional strategy for teaching integer subtraction. Additionally, the authors illustrate how this teacher supported students through the two non-content-specific instructional practices of supporting connection and engagement between student context and the mathematics learning environment and attending to mathematical language. These practices have been related to more equitable learning outcomes among students from historically marginalized groups.

In mathematics, the expectation to set up and solve word problems emerges as early as kindergarten; however, many students who experience mathematics difficulty (MD) and dual-language learners often present with specific challenges in this area. To investigate why these populations experience word-problem difficulty, we examined the word, problem solving and oral explanations of third-grade dual-language learners (DLLs; n = 40) and non-DLLs (n = 40), all of whom were identified as experiencing MD. Students solved five additive word problems and provided oral explanations of their work, which were transcribed and coded for the number of words in each explanation, type of mathematics vocabulary terms used, inclusion of correct numbers in explanations, and descriptions of addition or subtraction. We identified no significant differences in word-problem scores between DLLs and non-DLLs with MD. For both DLLs and non-DLLs, students who answered problems correctly used more words in each explanation and used more mathematics vocabulary terms within their explanations. For incorrectly answered problems, the most common mistake for both DLLs and non-DLLs involved using the incorrect operation to solve the problem.

The objectives of this research were to construct and verify a mathematics diagnostic assessment on addition and subtraction basis for grade 4 students in Buriram Primary Educational Service Area Office 3. The research samples consisted of 440 Prathomsuksa 4 students selected by adapting the multi-stage random sampling method. The research instruments employed in this research comprised of two types of examinations constructed according to strand 1: number and algebra in Mathematics 1.1. The tests were to examine test and diagnostic test. The examining test was a subjective test, consisted of 80 items which divided into addition part for 40 items and subtraction part for 40 items. The diagnostic test was a multiple-choice with 4 choices tests, consisted of 48 items which divided into addition part for 24 items and subtraction part for 24 items. The diagnostic test. The results showed that the content validity, determined by the expert, had the IOC score at 0.60-1.00. The addition part in the 1st test had a difficulty score of 0.21-0.79, the discrimination score at 0.23-0.89, and the reliability score at 0.89. The subtraction part in the 2nd test had the difficulty score at 0.31-0.72, the discrimination score at 0.32-0.72 and the reliability scores at 0.32-0.81.

As national and international assessments continue to show students struggle with rational concepts, which are seen as a major stumbling block for future career in STEM fields, we used the certainty of response index (CRI) as a method of analysis to investigate a group of 40 United Arab Emirates (UAE) seventh grade students' understanding of basic fraction concepts. The types of errors and misconceptions displayed by students show that over 60% of all participants assumed that adding fractions procedurally is the same as adding whole numbers with many not being able to distinguish a numerator from a denominator. The results lead us to believe that the manner in which students were taught did not ensure the necessary transitional shift from learning whole numbers to learning rational numbers. Applying the theories of Dienes (1960) and Bruner (1966), we propose a method of teaching addition and subtraction of fractions that will allow students to build their own understanding of the rules using the graphing calculator as the medium for concept formation.

Solving arithmetic word problems has been a challenge for primary students due to difficulties in understanding the problem structure and relating the quantities in the problem to each other. This paper reports on an action research study to enhance students' subtraction word problem-solving skills. The authors observed that their students had difficulties in representing the situations in word problems and solving the problems correctly. They designed, implemented, and analyzed an intervention to scaffold their students' subtraction word problem-solving skills. As part of the intervention, a digital subtraction game was developed and used with second and third-grade students. The game involves three different representations: a discrete visual model, a bar model, and a number sentence. The students played the game and solved additional problems to strengthen their skills for representing and solving subtraction word problems. Twenty-four 2nd grade students in China and two 3rd grade students in Turkey participated in the study. Data sources included a pre-test, a post-test, student worksheets, and teachers' filed notes. Data analysis showed an increase in students' subtraction word problem-solving performance. They also effectively used a variety of representations to represent problem situations. The design and implementation processes of the intervention are discussed in the paper. We share suggestions for future implementation.

Several working memory processes have been hypothesized to influence different arithmetic operations. Working memory has been compartmentalized into a number of different sub-processes, such as phonological memory and visuospatial memory that are believed to have unique contributions to the performance of two distinct arithmetic operations: multiplication and subtraction. A previous dual task experiment produced these effects, but subsequent experiments have yielded inconsistent results. Because the reasons for these inconsistencies are not immediately apparent, the current study systematically reviewed these subsequent attempts and attempted to replicate this effect in a within-subjects dual task experiment using tasks developed from prior work across a number of different subsamples. In contrast to the original finding, we observed no differential impact of specific working memory secondary tasks by arithmetic operation in any of our analyses. However, our analyses do not entirely rule out the possibility of differential effects of working memory tasks. Our findings suggest that the working memory facet by arithmetic operation interactions observed in previous work may be idiosyncratic in nature and difficult to predict a priori in subsequent experiments.

This paper reports on the design, implementation, and evaluation processes of an activity that focused on the comparison meaning of the subtraction of integers and included the number line model as the primary representation. Finding the answer and determining the sign of the answer in the subtraction questions were targeted as the learning objectives. Four different situations that can be encountered in the subtraction questions were determined considering the magnitude and sign of minuend and subtrahend. Through the designed activity, it is expected that the students work on example questions first and then make generalizations by reflecting on the questions they solve. The activity was implemented in an elementary school with the participation of 27 seventh grade students. The analysis of the implementation process revealed that the activity was effective in supporting the students' performance of finding the difference and the sign of the difference when subtracting integers.

Central elements of adaptive expertise in arithmetic problem solving are flexibility, using multiple strategies, and adaptivity, selecting the optimal strategy. Research shows that the strategies children actually use do not fully reflect the strategies they know: there is hidden potential. In the current study a sample of 147 third graders from the Netherlands completed a comprehensive assessment of adaptive expertise in the domain of multidigit subtraction, designed to measure, first, the strategies students know and use to solve subtraction problems (potential and practical flexibility). Second, it measured to what extent students know which strategy is optimal and to what extent they use the optimal strategy (potential and practical adaptivity). Findings for flexibility showed that most students consistently used the same strategy across all problems: practical flexibility was low. When prompted, students knew more strategies than they used spontaneously, suggesting hidden potential in flexibility. Findings for adaptivity showed that students hardly ever spontaneously used the task-specific strategy that is efficient for specific problems since it has the fewest and easiest steps. However, almost half of the students could select this strategy from a set of given strategies at least once. Furthermore, an innovative, personalized version of the choice/no-choice method showed that the task-specific strategy was usually not the optimal strategy (fastest strategy leading to a correct answer) for individual students. Finally, students used the strategy with which they performed best more often than the other strategies, but there is hidden potential for the adaptive use of task-specific strategies.

This study examined adults' frequent, efficient and adaptive use of direct subtraction (DS) and subtraction by addition (SBA) in mental multi-digit subtraction with the choice/no-choice method. Participants were offered subtractions in one choice condition (choice between DS and SBA) and two no-choice conditions (mandatory use of either DS or SBA). SBA was used as frequently as DS in the choice condition. DS was most accurate on subtractions with a large difference (e.g., 502 - 18), while SBA was fastest on subtractions with a small difference (e.g., 903 - 886). In general, participants were adaptive for task characteristics and their personal speed characteristics. We further analyzed task-based adaptivity on an individual level via a Latent Class Analysis. Results showed that two-thirds of the participants were adaptive to task characteristics, and that these adaptive participants were the most proficient in accuracy and speed in the choice condition. We further examined whether executive functions (updating, inhibition, shifting) were related to individual differences in strategy efficiency and task-based adaptivity. In line with our hypothesis, updating was related to strategy efficiency, such that participants with higher updating skills were more accurate. In contrast to our expectations, inhibition and shifting were not related to task-based strategy adaptivity. This study highlights adults' efficient and adaptive use of arithmetic strategies, and its association with their proficiency and executive functions.

In this paper it is investigated if the mental computation strategies in the research literature are enough to satisfactorily categorize the mental computation strategy use by preservice elementary teachers (PETs) in subtraction on two-digit natural numbers. The PETs' use of mental computation strategies is measured operationally with a written questionnaire. The paper indicates that the strategies used by PETs are generally seen contained in the research literature on elementary school pupils' mental computation strategy use, but that there are additional strategies used by PETs, which are presented and discussed in the paper.

In mathematics, the expectation to set up and solve word problems emerges as early as kindergarten; however, many students who experience mathematics difficulty (MD) and dual-language learners often present with specific challenges in this area. To investigate why these populations experience word-problem difficulty, we examined the word, problem solving and oral explanations of third-grade dual-language learners (DLLs; n = 40) and non-DLLs (n = 40), all of whom were identified as experiencing MD. Students solved five additive word problems and provided oral explanations of their work, which were transcribed and coded for the number of words in each explanation, type of mathematics vocabulary terms used, inclusion of correct numbers in explanations, and descriptions of addition or subtraction. We identified no significant differences in word-problem scores between DLLs and non-DLLs with MD. For both DLLs and non-DLLs, students who answered problems correctly used more words in each explanation and used more mathematics vocabulary terms within their explanations. For incorrectly answered problems, the most common mistake for both DLLs and non-DLLs involved using the incorrect operation to solve the problem. [This article was published in "Learning Disability Quarterly" (EJ1323542).]

Skills and understanding of operations with negative numbers, which are typically taught in middle school, are crucial aspects of numerical competence necessary for all subsequent mathematics. To more swiftly and coherently develop the field's understanding of how to foster this critical competence, we need shared measures that allow us to compare results across studies with diverse populations and theoretical perspectives. Yet, to date no validated instrument exists to assess all four primary operations (addition, subtraction, multiplication and division). Thus, we conducted a Rasch analysis of the Integer Test of Primary Operations (ITPO) with 187 middle school students to provide a valid and reliable assessment with good person and item fit. The implications of this study are numerous for multiple stakeholders including scholars, test and textbook developers, as well as teachers. First, we validated three forms of the ITPO to foster future longitudinal studies of how integer arithmetic knowledge ismaintained or decays as well as how such knowledge might be related to success in STEM disciplines. Second, our analysis provides trustworthy insights about relative difficulty of integer problem structures because regardless of test form similar problem structures loaded together. For instance, sums of additive inverses were the easiest structure, whereas division by -1 was more difficult than multiplying or dividing by any other integer. We discuss each of these and other findings that have practical implications for learning and teaching integers. Third, for broader mathematics assessments in which minimal items can be included to measure integer knowledge, this study informs which items would serve the intended assessment purpose. Finally, we provide the three forms as an appendix in printable formats to ensure these validated tests are practical to implement for teachers as well as scholars.

This study replicates and extends prior research on multisensory mathematics instruction (Grünke, Urton, & Karnes, 2018) by integrating a touch points strategy, performance feedback, reward system, and a reinforcing card game into an instructional package. A multiple baseline design across participants was used to evaluate the effects of the touch points package on the subtraction skills of four German female first year students at-risk for learning disabilities. During intervention, the students were administered eight to eleven treatment sessions to learn how to subtract a one-digit subtrahend from a two-digit minuend up to 18 crossing over the tens barrier. Results indicated that all students made substantial increases in their subtraction performance during intervention. Student performance improved from 0 to 2 out of 10 math problems solved correctly during baseline to between 8 and 10 problems correct by the end of the intervention. Effect sizes observed across the four participants indicated the effectiveness of the intervention ranged from high to very high.

The purpose of this study is to electrophysiologically assess the effect of an individualized education program supported with musical activities on the success of primary school students with computational difficulties. A mixed-methods design consisting of qualitative and quantitative methods was employed as the research model. The research group consists of four students with mild special learning difficulties and learning difficulties in mathematics, among the students attending the third and fourth grades at a primary school in Kütahya city center. By examining the primary school mathematics curriculum, 12 mental processing gains were identified for addition and subtraction, one of the third-grade mental processing gains. An assessment form, training module, and math songs were prepared in line with these acquisitions. An assessment form was applied to the participants as a pre-test and post-test to determine the effect of the training module and math songs on the students' success. Additionally, electroencephalogram (EEG) of the participants were recorded before and after the training module, and math songs were applied for 12 weeks. During the EEG recordings, 10 questions were asked to the participants that would enable them to make mental operations. The power densities of the EEG data were calculated using the Welch method in the MATLAB. To analyze the qualitative data of the research, descriptive analysis technique was used. As a result of the study, when the effects of the evaluation form of the participants' mental processing skills were examined, it was seen that the training module and math songs positively affected the mental processing skills of the students. In addition, it has been shown that the prepared training module and math songs increase the success of the participants, supported by electrophysiological evaluations.

A converging body of evidence from neuroimaging, behavioral, and neuropsychology studies suggests that different arithmetic operations rely on distinct neuro-cognitive processes: while addition and subtraction may rely more on visuospatial reasoning, multiplication would depend more on verbal abilities. In this paper, we tested this hypothesis in a longitudinal study measuring language and visuospatial skills in 358 preschoolers, and testing their mental calculation skills at the beginning of middle school. Language skills at 5.5 years significantly predicted multiplication, but not addition nor subtraction scores at 11.5 years. Conversely, early visuospatial skills predicted addition and subtraction, but not multiplication scores. These results provide strong support for the existence of a double dissociation in mental arithmetic operations, and demonstrate the existence of long-lasting links between language/visuospatial skills and specific calculation abilities.

In numerical cognition research, the operational momentum (OM) phenomenon (tendency to overestimate the results of addition and/or binding addition to the right side and underestimating subtraction and/or binding it to the left side) can help illuminate the most basic representations and processes of mental arithmetic and their development. This study is the first to demonstrate OM in symbolic arithmetic in preschoolers. It was modeled on Haman and Lipowska's (2021) non-symbolic arithmetic task, using Arabic numerals instead of visual sets. Seventy-seven children (4-7 years old) who know Arabic numerals and counting principles (CP), but without prior school math education, solved addition and subtraction problems presented as videos with one as the second operand. In principle, such problems may be difficult when involving a non-symbolic approximate number processing system, whereas in symbolic format they can be solved based solely on the successor/predecessor functions and knowledge of numerical orders, without reference to representation of numerical magnitudes. Nevertheless, participants made systematic errors, in particular, overestimating results of addition in line with the typical OM tendency. Moreover, subtraction and addition induced longer response times when primed with left- and right-directed movement, respectively, which corresponds to the reversed spatial form of OM. These results largely replicate those of non-symbolic task and show that children at early stages of mastering symbolic arithmetic may rely on numerical magnitude processing and spatial-numerical associations rather than newly-mastered CP and the concept of an exact number.

The virtual-representational-abstract (VRA) instructional sequence presents as a viable mathematics intervention for students with disabilities, including students with autism spectrum disorder (ASD). Yet, to date, all research has been conducted face-to-face with students. However, virtual manipulatives provide an opportunity for educators to provide effective interventions virtually, both during emergency situations as well as to support students who have limited access to services. The purpose of this study was to examine the effects of the intervention package of the VRA instructional sequence with explicit instruction and the system of least prompts delivered virtually, on the accuracy and independence of four individuals with ASD solving double-digit subtraction with regrouping problems. Researchers found a functional relation between the intervention package and student accuracy.

The aim of the study was to determine the concepts that primary school fourth-grade students reveal with their mind maps related to four operations (addition, subtraction, multiplication, division) and to compare the mind map and achievement test scores. In the study, a mixed method was used in which quantitative and qualitative data were collected and presented together. The study was carried out with a total of 14 students. There were eight girls and six boys, who studying in the fourth-grade of a primary school in Mentese district of Mugla, in the spring semester of the 2019-2020 academic year. In the research, mind maps created for four operations and success test were used as data collection tools. The data collection process was carried out simultaneously. In the analysis of the data, qualitative data were transformed into the quantitative analysis, and quantitative analyzes were made. Concepts in mind maps created for four operations were subjected to qualitative analysis and photos were added as evidence. As a result of the research, it was determined that the majority of the students adopted the concepts of addition, subtraction, multiplication, and division. Besides, it was revealed that there was a high-level positive relationship between the scores of the students obtained from the achievement test and the scores from the mind maps they created. The evaluation made using the mind map for primary school fourth-grade students is more advantageous than the achievement test in terms of determining the conceptual understanding.

Developing textbooks of optimal quality is crucial for enriching the students' learning and understanding. This study examined fraction addition and subtraction problems in the U.S. and South Korean mathematics textbooks according to the types of denominators. In particular, we investigated Everyday Mathematics (EM) and South Korean mathematics (KM) textbooks revised in 2015 and developed an analytic framework encompassing horizontal and vertical dimensions to examine the learning opportunities presented to students by the textbooks. We assessed their topic sequence and frequency of fraction addition and subtraction contents with regard to the former and examined their contextual features, cognitive demands, and mathematical activities with regard to the latter. We observed that EM provided inadequate learning opportunities for fraction subtraction problems, representation problems, and high-order cognitive abilities. However, KM provided more even learning opportunities for fraction addition and subtraction problems, various contextual features, and high and low thinking skills. Moreover, we found that EM emphasized understanding and resolving activities, whereas KM underscored exploring and explaining activities. The findings of this study suggested updating fraction addition and subtraction contents in the U.S., South Korea, and other countries by considering horizontal and vertical dimensions.

We describe how mathematical argumentation supports curiosity and exploration by sharing a first-grade lesson in which students explored decomposition with subtraction. We also reflect on the conditions that supported the inclusion of mathematical argumentation.

Students with learning disabilities in mathematics often struggle with the underlying concepts of multidigit addition and subtraction. To help students build a conceptual understanding of these computations, teachers can utilize evidence-based practices such as the concrete-semi-concrete-abstract framework and the use of multiple visual representations. This column presents five key strategies that incorporate evidence-based practices and teach whole number operations that rely on place value understanding.

Researchers established virtual manipulatives as an evidence-based practice for students with intellectual and developmental disabilities (IDD). However, much of the existing research for this population targets middle school students with IDD. This study examined a virtual number line paired with teacher modeling and the system of least prompts to support high school students with IDD in acquiring, maintaining, and generalizing addition and subtraction problems. Through a single case research study with three high school students with IDD, the researchers found a functional relation between the intervention package and student accuracy. Students also became more independent but struggled to maintain their skills when prompting was not provided. Further, the students were unable to generalize the skills of addition and subtraction to real-life scenarios targeting time and money. These findings and their implications for the field of IDD are examined.

The National Council of Teachers of Mathematics (NCTM) promotes creating resources that build procedural fluency from conceptual understanding through intentionally sequenced tasks that draw on students' prior knowledge and move from simple, concrete representations to more complex and abstract representations (Boston et al., 2017). Liljedahl (2020) and Wiemen et al. (2021) provide frameworks for designing such tasks, called Thin Slicing and Problem Strings, offering teachers an accessible entry point to create tasks that build from students' existing mathematical understandings. To illustrate how to integrate these design frameworks into instruction, the authors demonstrate a task sequence for adding and subtracting like terms--grounded in the concepts of equivalence and algebra as generalized arithmetic--that helps students see connections between concepts and procedures in algebra.