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Learning to calculate with natural numbers by structuring seems promising but how this can be taught in a sustainable manner remains an open question. An eight-month-long intervention based on the idea of using a structural approach to addition and subtraction, and particularly bridging through ten, was implemented in four Swedish first-grade classes. One goal was that by the end of first grade, students would be able to solve tasks such as subtracting 8 from 15 by using part-whole number relations. In this paper, we report on learning outcomes from task-based interviews with intervention and control groups before, immediately after, and one year after the intervention, in order to investigate long-term effects and whether students used a structural approach when solving tasks in a higher number range in the second grade. In comparison to controls, students in the intervention group showed higher increases in their learning outcomes. Moreover, the intervention group used a structural approach to a larger extent when solving tasks in higher number ranges, whereas students in the control group more commonly used single-unit counting to solve such tasks. These findings have implications both for teaching and for research on students' development of arithmetic skills. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we report the findings of a study conducted with 6 Grade 8 students in the United States. The students posed stories for open number sentences involving addition and subtraction of integers. We analysed the stories posed by the students to build models that describe the conceptual structures behind these posed stories – the conceptual models for integer addition and subtraction. These four conceptual models for thinking about and using integer addition and subtraction include Bookkeeping, Counterbalance, Relativity, and Translation, and are generated from the students' posed stories. We also provide profiles of conceptual model use for two of the 6 students that describe how the students posed stories to accommodate conceptual model use, such as posing unconventional or unrealistic stories or changing the structure of the number sentences. The conceptual models and descriptions of how the students used them provide perspective into student thinking about integers and contexts, highlighting the mathematics of the students, and calling for a re-examination of contexts used in school mathematics. [ABSTRACT FROM AUTHOR]

An instructional activity for primary-grade students to boost their relational thinking as they reason about subtraction is presented.

This study examined the impact of response repetition as an error-correction strategy in teaching subtraction facts to three students with learning difficulties. Written response repetition (WRR) and oral response repetition (ORR) were compared using an alternating treatments design nested in a multiple baseline design across participants. Introduction of the error-correction procedures resulted in increases in subtraction facts mastered and correct responding relative to baseline for all participants. All participants mastered more facts in the WRR condition, with varying levels of differentiation. Two of three participants demonstrated higher levels of correct responding in the WRR condition. Results are discussed in terms of consistency with other studies of response repetition and implications for school-based personnel developing and delivering intensive academic interventions. [ABSTRACT FROM AUTHOR]

National Council of Teachers of Mathematics (NCTM) recommends that students be able to communicate mathematics, using correct and appropriate language, by eighth grade [8]. Mathematics teachers at all levels agree that they have the responsibility to teach their students content-specific writing, but many feel that they don't have the tools to do this work. This article offers a foundation and methodologies for different writing assignments that can be used in mathematics classes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to showcase a handful of mathematical challenges found in classical literature and to offer possible ways of integrating classical literature in mathematics lessons. We analyze works from a range of authors such as Jules Verne, Anton Chekhov, and others. We also propose ideas for further tasks. Most of the problems can be restated in terms of simple mathematical equations, and they can often be solved without a computer. Nevertheless, we use the computer program Mathcad to solve the problems and to illustrate the solutions to enhance the reader's mathematical experience. [ABSTRACT FROM AUTHOR]

The article describes an activity program for third grade students which offers lessons on the standard algorithm for subtraction.

Math proficiency is related to math calculation fluency. Explicit timing provides repeated practice for math fluency. It is enhanced through goal setting, graphic feedback, and rewards. Self-selected goals have potential to increase performance for math fluency. This study compared the effect of goal lines, and researcher goals versus self-selected goals. The authors compared three groups: (a) researcher-selected goals only, (b) researcher-selected goals with goal lines, and (c) self-selected goals with goal lines. First-grade students completed subtraction probes and colored in a bar graph with their performance. When participants met or exceeded their goals, participants received rewards twice per week. Results indicated that the researcher-selected goals with goal lines procedures with explicit timing had the greatest effect on subtraction fluency for first-grade students. The researcher-selected goals with goal lines and self-selected goals with goal lines groups outperformed the researcher-selected goals-only group. [ABSTRACT FROM PUBLISHER]

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The article offers information on the authors' findings and recommendations regarding a word problem, one of the twelve expected of kindergarten, first-grade, and second-grade students to solve, which had not been studied more closely. Discussed is the both-addends-unknown problem which is part of the 12 addition and subtraction problems contained in the Common Core State Standards for Mathematics. Included are recommendations in selecting and implementing both-addends-unknown problems.

The article looks at possible interpretations of a standard described in the Common Core State Standards for Mathematics (CCSSM) about subtraction of rational numbers. It discusses the use of the keep-change-change approach in the middle school education, along with its disadvantage for students. The opportunities provided by writing algorithm for students and the range and complexity of student thinking related to the computation of rational numbers are explored.

Based on Common Core Standards (2010), mathematics interventions should emphasize conceptual understanding of numbers and operations as well as fluency. For students at risk for failure, the concrete-representational-abstract (CRA) sequence and the Strategic Instruction Model (SIM) have been shown effective in teaching computation with an emphasis on conceptual understanding. No studies have investigated the effects of CRA and SIM for teaching multiplication with regrouping. Therefore, the purpose of this study was to replicate and extend the literature, teaching subtraction and multiplication with regrouping using CRA and SIM. Three students receiving tier three mathematics interventions participated. A multiple-probe across behaviors design was used to show a functional relation. All of the students demonstrated increases in fluency across all regrouping tasks. [ABSTRACT FROM AUTHOR]

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. [Copyright &y& Elsevier]

Achieving fluency with basic subtraction and add-with-8 or -9 combinations is difficult for primary grade children. A 9-month training experiment entailed evaluating the efficacy of software designed to promote such fluency via guided learning of reasoning strategies. Seventy-five eligible first graders were randomly assigned to one of three conditions: guided subtraction (e.g., If 5 + 3 = 8, then 8 – 3 is 5), guided use-a-10 (e.g., If 10 + 7 = 17, then 9 + 7 is 16), or an unguided-practice condition for 30-minute sessions twice a week for 12 weeks. An ANCOVA revealed that at the delayed posttest, the guided-subtraction group outperformed both comparison groups on unpracticed subtraction combinations. Analyses of gains in slow but appropriate use of reasoning and decreases in inefficient strategy use indicated that both types of guided training promoted the learning of a targeted reasoning strategy. [ABSTRACT FROM AUTHOR]

The article presents information on the study of subtraction in education. The authors look at the focus of conjecturing through reasoning and proving in fourth and fifth grade education in the U.S. The article also discusses activity programs for students in mathematics education that focuses on mathematical exploration.

The article discusses results of investigations to examine the strategies used for solving addition and subtraction operations. Topics discussed include seven adolescents diagnosed with deep deafness due to peer interaction, improvement of mathematical performance of the population, and improvement of teacher's professional development. Other topics discussed include solving strategies.

Copyright of Teoría de la Educación. Revista Interuniversitaria is the property of Ediciones Universidad de Salamanca and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Mathematics continues to be one of the most challenging aspects of the school curriculum for students with learning disabilities. The provision of pedagogically sound foundational computation and word problem instruction in the elementary grades is critical for students to progress through the mathematics curriculum successfully. A multiple-probe-across-participants design was used to investigate the effects of explicit instruction using the concrete-representational-abstract sequence with integrated cognitive strategies to teach five fourth- and fifth-grade students how to subtract when regrouping is required. All five participants benefited from the scripted instructional lessons. Practical implications and suggestions for future research in this area are provided. [ABSTRACT FROM AUTHOR]

An adapted alternating treatments design was used to compare the effects of class-wide applications of Taped Problems (TP) and Cover, Copy, and Compare (CCC) procedures designed to enhance subtraction fact fluency in an intact third-grade classroom. During the TP procedure, a tape provided an auditory prompt (i.e., the problem), followed by a 2-second delay and then the correct answer. Students attempted to supply the correct answer before the answer was provided by the tape. During the CCC procedure, students would cover a math-fact family triangle, attempt to write one of two possible problems and answer, write the reciprocal fact and answer, and evaluate their responses. Results show that TP caused greater increases in subtraction fact fluency than CCC or a control condition. Discussion focuses on preventing student math deficits, issues regarding alternating treatments designs, and future directions for research. © 2012 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

The research study this article is based on aims to implement research knowledge to teaching, that is, the concept of critical aspects and dimensions of variation used in the variation theory. To do this, the researchers worked with willing teachers to explore how to make mathematics teaching more effective. This paper illustrates how teachers make use of a learning theory, the variation theory, as well as their own professional expertise and collaboration to help students improve their mathematical understanding of subtraction as well as their learning of it. The students' tests, examinations of students' mathematical work, the teachers' lessons plan and reports of the instructions for lessons form the data base for the article. The analysis indicates that one of the critical aspects in the process of implementation of the variation theory in the teachers' practice was to identify the critical aspects in students' learning. Another critical aspect in the implementation of the variation theory was to open up dimensions of variation in the identified critical aspects of the students. By giving teachers the possibility to develop the ability to identify critical aspects in students' learning, dimensions of variation are opened up in these aspects, and by applying this knowledge in the daily teaching, they have the possibility to improve students' learning. The findings suggest that developing an understanding of the students' critical aspects can be a productive basis in helping teachers make fundamental changes in their instructions and improve students' learning. [ABSTRACT FROM AUTHOR]

Abstract: In considering mathematics problem solving as a model-eliciting activity (), it is important to know what students are modeling for the problems: situations or solutions. This study investigated Grade 3 students’ mathematization process by examining how they modeled different types of multi-digit subtraction situation problems. Students’ modeling processes differed from one problem type to another due to their prior experiences and the complexity of the problems. This study showed that students make their own distinctions between solution and situation models in their mathematization process. Mathematics curricula and teaching should consider these distinctions to carefully facilitate different model development of and support student understanding of a content topic. [Copyright &y& Elsevier]

While addition and subtraction is a key mathematical skill for young children, a typical activity for them in classrooms involves doing repetitive arithmetic calculation exercises. In this study, we explore a collaborative way for students to learn these skills in a technology-enabled way with wireless computers. Two classes, comprising a total of 52 students in Grade 4 (ages 10 or 11) participated in the study. They used the Group Scribbles software to run an adapted version of the "Cross Number Puzzle" that was designed with the "feedback" mechanism to assist students' problem solving. In one class, students played the game individually and in the other class, students played the game collaboratively. The low-ability students in the collaborative class were found to have made the most significant progress in arithmetic skills through playing this game. Three dominant interactive collaboration patterns, one contributing to productive interactions and two to less productive interactions, were also identified in the students' collaboration. [ABSTRACT FROM AUTHOR]

The article offers tips for mathematics teachers in helping their students master subtraction with regrouping. It mentions that computational fluency in math and decoding skills in reading are both necessary for comprehension and high-order thinking. It suggests that students must understand that, although an addition problem combines parts to come up with the total amount, a problem on subtraction starts with the total amount and takes part of the total away.

The article discusses the number line approach to negative numbers. It mentions that the knowledge on subtraction involving numbers either side of zero can be carried by a conception of the subject combined with a visual image of the number line. It adds that the number line model for operations on positive numbers is extended to account for negative numbers and to deal with the difficulty of the students to the operations and negative or positive signs.

Although basic math skill deficits are commonly encountered across elementary and secondary school students, few empirically validated, group-administered interventions are available for educators attempting to prevent or remedy such problems. This study compared the effectiveness of two theoretically distinct interventions for improving the subtraction-fact fluency of a general education class of second-grade students. An alternating treatments design was used to compare a behavioral intervention. Cover, Copy, and Compare (CCC). to an intervention from a constructivist- oriented resource. Facts That Last (FTL). Results demonstrated that CCC led to increases in math-fact fluency, whereas the classwide response to FTL activities did not differ from the control condition. Two months postintervention, maintenance data revealed that the fluency increases associated with CCC were sustained. Discussion focuses on practical implications of the study as well as possible theoretical explanations for the documented results. [ABSTRACT FROM AUTHOR]

This study compared the fluency and error rates produced when using the Cover, Copy, and Compare (CCC) and a modified CCC procedure (MCCC) called Copy, Cover, and Compare to complete subtraction math problems. Two second-grade classrooms consisting of 47 total students participated in the study. The following items were administered to participants: (a) a timed pretest, (b) a timed CCC worksheet, (c) a timed MCCC worksheet, and (d) a timed posttest. Then the participants were asked which procedure they liked best. Results revealed significantly higher digits correct per minute (i.e., fluency scores) on the posttest when compared with the pretest scores. Likewise, students' fluency scores were significantly higher under the CCC condition when compared to the MCCC condition. There were no significant differences in accuracy from pretest to posttest nor between the CCC and MCCC conditions. Discussion focuses on implications for modifying instructional strategies, measuring student progress, implications for practice, and directions for future research. [ABSTRACT FROM AUTHOR]

Solving arithmetic and algebraic word problems is a key component of the Singapore elementary mathematics curriculum. One heuristic taught, the model method, involves drawing a diagram to represent key information in the problem. We describe the model method and a three-phase theoretical framework supporting its use. We conducted 2 studies to examine teachers' perceptions and children's application of the model method. The subjects were 14 primary teachers from 4 schools and 151 Primary 5 children. The model method affords higher ability children without access to letter- symbolic algebra a means to represent and solve algebraic word problems. Partly correct solutions suggest that representation is not an all-or-nothing process in which model drawing is either completely correct or completely incorrect. Instead, an incorrect solution could be the consequence of misrepresentation of a single piece of information. Our findings offer avenues of support in word problem solving to children of average ability. [ABSTRACT FROM AUTHOR]

Although preservice elementary school teachers (PSTs) lack the understanding of multidigit whole numbers necessary to teach in ways that empower students mathematically, little is known about their conceptions of multidigit whole numbers. The extensive research on children's understanding of multidigit whole numbers is used to explicate PSTs' conceptions of these numbers. A grounded theoretical approach leads to the development of a framework for PSTs' conceptions of multidigit whole numbers, and that framework facilitates description of their conceptions and their difficulties in the context of the standard algorithms. The framework also enables discussion of the PSTs' performance in other contexts. [ABSTRACT FROM AUTHOR]

The article focuses on the use of alternative subtraction and multiplication algorithm in solving mathematical problems. The author notes that she encourages her students to use multiple approaches in solving problems. The use of equal-addition algorithm in solving a subtraction problem is discussed. The introduction of Mathias' method for solving double-digit multiplication problems is also tackled. The author states that the learning exercise she introduced to the class allows students to reflect on their understanding of important mathematical concepts they had previously learned.

Presents an article about teaching addition and subtraction in children. Problems associated with mathematics education; Ways to improve computational fluency; Strategies in teaching addition and subtraction.

The algorithms used for subtraction vary. In the United States, three different algorithms were used almost equally until the 1940s: the equal additions, the decomposition, and the Austrian method. After William Brownell modified the decomposition algorithm, the use of other algorithms was practically eliminated in mathematics textbooks. Today, some texts introduce the equal additions algorithm, but the decomposition is still the predominant algorithm. This article reviews the historical use of each of the three algorithms in the United States, discusses the modification to the decomposition algorithm, and gives a brief overview of the current status of subtraction algorithms. [ABSTRACT FROM AUTHOR]

Presents experiments on whether chidren who had demonstrated difficulty in learning basic subtraction facts by rote could learn them more rapidly through the use of a simple rule. Generalize the use of the rule to new number facts; Counter-balanced treatment design; Multiple baseline design.

Presents a study to design and field-test instruction intended to help students construct knowledge about addition and subtraction story problems and determine if this knowledge would transfer to actually solving problems. Two related hypotheses; Support of the effectiveness of instruction based on a structured approach to authoring arithmetic word problems.

Presents an alternative solution method to substraction of mixed numbers with regrouping devised by Krystal Demmons. Process of solving a problem using the method; Sample problems using the method. INSET: Diversity, equity, and standards: An urban agenda in Mathem....

The article compares the efficacy of the additive and the borrowing method of subtraction in teaching subtraction to elementary school students. According to Joseph S. Taylor, District Superintendent of New York City, the desirability of use of the additive method among students is subject to discussion. A survey, conducted in the sixth, seventh and eighth grade students of San Francisco Normal School, California, revealed that students using the additive method are more accurate, but are slower than students using the borrowing method.