Students need to create mental models for different types of addition and subtraction situations in order to develop a broad and viable understanding of these operations. Although most students succeed when changing or combining sets, situations that demand a quantitative comparison of sets seem considerably more difficult in the first school year. Textbooks represent the most important learning resource for elementary school mathematics teachers. However, research on their impact on students’ knowledge is limited. Hence, we examined textbooks’ role in students’ ability to model quantitative comparisons by analyzing the learning opportunities presented by four German textbooks for Grade 1 and by conducting a secondary analysis of a dataset based on 1513 students from 84 classes that used one of these textbooks. The results revealed differences in the textbooks’ topic-specific instructional quality as well as a significant relation between this quality and student achievement in quantitative comparisons.

Children and adults use various strategies to solve arithmetic problems such as 701–698. Many studies have investigated students’ strategy use at a specific time point, but our knowledge about the longitudinal changes in students’ strategy use in elementary school is limited. Using a multilevel latent transition analysis, we examined individual differences in strategy use in multi-digit addition and subtraction problems of 1,947 students measured at the end of Grade 3 and the end of Grade 4. The analysis identified five different strategy use profiles, with the three largest profiles (mostly written algorithms, mostly mental computations, and a combination of written algorithms and mental computations) characterizing the strategy use of the large majority of students. For strategy use change over time, the analysis identified heterogeneous learning paths from Grade 3 to Grade 4. Nearly half of the students remained in the same strategy profile, but the other half moved to another profile. Students’ prior arithmetic knowledge and gender affect both strategy use in Grade 3 and strategy change between Grade 3 and 4, with amplifying effects. The classroom characteristics classroom intellectual composition and mathematics textbook had an effect on students’ strategy use in Grade 3, but not on strategy change. In conclusion, the current study shows how students differ not only in their arithmetic strategy use at one specific point in time but also in their learning paths, and that both student and classroom factors have impact.

Supporting students with difficulties in learning mathematics is a challenge for teachers and educational administrators. Formative assessment is considered to play a successful role in supporting at-risk students as well as students without difficulties in mathematics. There is a need for intervention programs, including formative assessment techniques, that (a) are easy to implement in the regular classroom without requiring radical changes in teachers’ individual teaching style, and (b) are effective in supporting at-risk students at the earliest stage possible in their school careers. This article analyzes an effectiveness trial of a formative assessment program developed to meet these goals and conducted in the first two years of elementary school. The examination of the longitudinal dataset from Grades 1–3 (N = 2,330) revealed an effect after the implementation, which was maintained at nearly the same effect size one year after completion of the program. The findings imply that formative assessment can foster the arithmetic achievement of students at risk as well as that of the entire class without changing the curriculum or teachers’ individual teaching style.