Editorial: Single-blind or double-blind review processes?

Educational Studies in Mathematics - The investigation at scale of the tensions that teachers need to manage when deciding to follow recommendations for practice has been hampered by the problem of...     

Connecting characterizations of equivalence of expressions: design research in Grade 5 by bridging graphical and symbolic representations

Educational Studies in Mathematics - One typical challenge in algebra education is that many students justify the equivalence of expressions only by referring to transformation rules that they...     

Taking away and determining the difference—a longitudinal perspective on two models of subtraction and the inverse relation to addition

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 learning processes, the article highlights some exemplary steps in which the inverse relation is discussed in light of the two models, namely mental subtraction, the standard algorithms for subtraction, negative numbers and manipulations for solving algebraic equations. For each step, the article presents educational considerations for fostering a flexible use of the two models as well as of the inverse relation between subtraction and addition. In each section, a mathe-didactical analysis is conducted, empirical results from literature as well as from our own case studies are presented and consequences for teaching are sketched.     

How to develop mathematics-for-teaching and for understanding: the case of meanings of the equal sign

What kind of mathematical knowledge do prospective teachers need for teaching and for understanding student thinking? And how can its construction be enhanced? This article contributes to the ongoing discussion on mathematics-for-teaching by investigating the case of understanding students’ perspectives on equations and equalities and on meanings of the equal sign. It is shown that diagnostic competence comprises didactically sensitive mathematical knowledge, especially about different meanings of mathematical objects. The theoretical claims are substantiated by a report on a teacher education course, which draws on the analysis of student thinking as an opportunity to construct didactically sensitive mathematical knowledge for teaching for pre-service middle-school mathematics teachers.     

Students’ individual schematization pathways - empirical reconstructions for the case of part-of-part determination for fractions

According to the design principle of progressive schematization, learning trajectories towards procedural rules can be organized as independent discoveries when the learning arrangement invites the students first to develop models for mathematical concepts and model-based informal strategies; then to explore the strategies and to discover pattern for progressively developing procedural rules. This article contributes to the theoretical and empirical foundation of the design principle of progressive schematization by empirically investigating students’ individual schematization pathways on the micro-level for the specific case of part-of-part determination of fractions. In design experiments series in laboratory settings, nine pairs of sixth graders explored the part-of-part determination and progressively schematized their graphical strategies before discovering the procedural rule. The qualitative in-depth analysis of 760 min of video shows that progressive schematization is a multi-facetted process that cannot be described by internalization of graphical procedures alone. Instead, the compaction of concepts- and theorems-in-action is crucial, especially for the goal of justifiable procedural rules.     

Writing reviews: perspectives from the editors of Educational Studies in Mathematics

Educational Studies in Mathematics -     

Brauchen mehrsprachige Jugendliche eine andere fach- und sprachintegrierte Förderung als einsprachige?

Which kinds of content- and language-integrated interventions can better support mathematical learning, depending on students’ language background, as measured by communication on the discourse level or additional lexical training on the word level? A control trial compared interventions with two different materials (discursive versus lexical-discursive, each 5 × 90?min.) with respect to the dependent variable of conceptual and procedural knowledge for fractions. The effects are investigated differentially for four language groups: monolinguals versus bilinguals, each with higher versus lower German language proficiency (n = 343). For both interventions, the ANOVA shows an increase of mathematical knowledge for the experimental group, which is significantly higher than for the control group, but no significant difference between the interventions. The intervention with lexical-discursive materials led to a slightly higher increase of knowledge in the post test, but the discursive intervention was superior in the follow up test. Monolingual and multilingual students had similar patterns of growth without differential pattern. However, the proficient monolingual students tend to profit more from the interventions, especially from the lexical-discursive intervention.