This study analyzed the relative importance of different cognitive abilities for solving complex mathematical word problems (CWPs)-a demanding task of high relevance for diverse fields and contexts. We investigated the effects of spatial, verbal, numerical, and general reasoning abilities as well as gender on CWP performance among N = 1282 firstyear university engineering students. Generalized linear mixed models unveiled significant unique effects of spatial ability, β = 0.284, verbal ability, β = 0.342, numerical ability, β = 0.164, general reasoning, β = 0.248, and an overall gender effect in favor of male students, β = 0.285. Analyses revealed negligible to small gender effects in verbal and general reasoning ability. Despite a gender effect in spatial ability, d = 0.48, and numerical ability, d = 0.30-both in favor of male students-further analyses showed that effects of all measured cognitive abilities on CWP solving were comparable for both women and men. Our results underpin that CWP solving requires a broad facet of cognitive abilities besides mere mathematical competencies. Since gender differences in CWP solving were not fully explained by differences in the four measured cognitive abilities, gender-specific attitudes, beliefs, and emotions could be considered possible affective moderators of CWP performance.

Diagnostic competences are an essential facet of teacher competence. Many studies have investigated the quality of teachers' judgments of students' competences. However, little is known about the processes that lead to these judgments and about the ways to promote these processes in the early phase of teacher training. The aim of the research project on which we report in this paper was to develop a simulated computer-based environment that allows assessing and promoting the diagnostic processes of prospective teachers. In the simulated environment, 'virtual third-graders' solve mathematical problems. Participants are asked to diagnose the students' competence levels according to a theoretical model, which has been empirically validated. Participants can repeatedly select mathematical problems of varying difficulty levels, assign them to a virtual student, and then receive the student's written solution. In this paper, we present the conceptualization of the simulated environment. We also report on the results of a pilot study with 91 prospective primary school mathematics teachers to analyze whether the environment allows an assessment of individual differences in diagnostic processes. The majority of participants rated the environment as authentic and as one in which they could become immersed. Overall, participants were fairly accurate in their diagnoses concerning the student's competence level. However, log data and participants' written notes indicated that there was large variability in their diagnostic processes. Participants varied greatly in the number of mathematical problems they assigned to a student during their diagnostic process, and in how strongly the difficulty of these problems deviated from the student's true competence level. Overall, the data suggest that the simulated environment has the potential to assess diagnostic processes in a valid way. We discuss open questions and issues for further development.

No abstract

Analyzing students’ documents (e.g., their homework) can serve as a basis for diagnosing students’ learning status and thus also for adaptive teaching. When making diagnostic judgments about students’ learning status in mathematics, teachers may benefit from using theoretical models of mathematical competence because such models illustrate what tasks students should have mastered on each level of competence. Based on students’ documents and a model of mathematical competence at the primary level, we developed a simulated learning environment for (1) analyzing and (2) supporting pre-service teachers’ diagnostic processes and results. When working in the simulated environment, pre-service elementary teachers are asked to assess virtual third graders’ learning status by diagnosing their mathematical competence levels as well as their misconceptions (e.g., misconception regarding multiplication) based on the competence model. To do so, pre-service teachers analyze students’ solutions to mathematical problems that they can select from a set of problems varying in content and difficulty. First data analyses suggest that the environment can capture differences in pre-service teachers’ diagnostic processes. A better understanding of these processes can serve as a basis for further developing the learning environment.