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…

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…

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…

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…

This study explored the relationships between task beliefs about the empty number line (ENL), mathematical ability, gender, and voluntary ENL use in multi-digit subtraction and addition. One hundred twenty-three Dutch third-grade students and nine teachers from six schools participated in this study. The multilevel path analysis showed that task…

In this paper, we report how 5-year-olds' arithmetic skills developed through participation in an 8-month-long intervention. The intervention program aimed to enhance the children's ways of experiencing numbers' part-part-whole relations as a basis for arithmetic skills and was built on principles from the variation theory of learning. The report…

The question of teaching a standard written algorithm for the four arithmetical operations in the first years of elementary school is increasingly raised as a consequence of the possibility of using calculators, tablets, and computers. This article first presents arguments both against and for maintaining this teaching in the first school years.…

Although students' invented strategies typically prove to be meaningful and effective in improving the students' mathematical understanding, much remains unexplored in the current literature. This study examined, through a teaching-scenario task, the nature of 80 preservice teachers' reasoning and responses to students' informal and formal…

At the start of mathematics education children are often presented with addition and subtraction problems in the form of pictures. They are asked to solve the problems by filling in corresponding number sentences. One type of problem concerns the representation of an increase or a decrease in a depicted amount. A decrease is, however, more…

Children who experience difficulties with learning mathematics should be taught by teachers who focus on the child's best way of learning. Analyses of the mathematical difficulties are necessary for fine-tuning mathematics education to the needs of these children. For this reason, an instrument for Observing and Analyzing children's Mathematical…

Subtractions of the type M - S = ? can be solved by various strategies, including subtraction by addition. In this study, we investigated children's use of subtraction by addition by means of reaction time analyses. We presented 106 third to sixth graders with 32 large non-tie single-digit problems in both subtraction (12 - 9 = .) and addition…

This paper reports an intervention phase of a design study aimed to assist second-grade Filipino children in solving addition word problems in English, a language they primarily encounter only in school. With Filipino as the medium of instruction, an out-of-school pedagogical intervention providing linguistic and representational scaffolds was…

Subtraction can be understood by two basic models--taking away (ta) and determining the difference (dd)--and by its inverse relation to addition. Epistemological analyses and empirical examples show that the two models are not relevant only in single-digit arithmetic. As curricula should be developed in a longitudinal perspective on mathematics…

The basis of this intervention study is a distinction between numerical calculus and relational calculus. The former refers to numerical calculations and the latter to the analysis of the quantitative relations in mathematical problems. The inverse relation between addition and subtraction is relevant to both kinds of calculus, but so far research…

In this study, we examined special education students' use of indirect addition (subtraction by adding on) for solving two-digit subtraction problems. Fifty-six students (8- to 12-year-olds), with a mathematical level of end grade 2, participated in the study. They were given a computer-based test on subtraction with different types of problems.…