The Students–Professors Problem—The Reversal Error and Beyond

This paper first introduces and reviews the existing research on the well-known “students–professors (S/P) problem”, which was first formulated in 1979. Next, it presents an empirical study of Danish upper secondary students’ answers to two mathematical modeling versions of the S/P-problem; a mathematization version (296 students), and a de-mathematization version (658 students). Besides reproducing several previously reported findings, e.g., the so-called reversal error, the study identifies new error types not previously reported in the literature. The mathematical modeling perspective adopted, along with a mixed-methods design, give rise to new potential explanations of the reversal error as well as explanations of the new error types. Our study shows that interpreting the linguistic formulation of the S/P-problem statement is not only related to language but is intrinsically of a mathematical – and cognitive – nature as well. Altogether, there is still more to be said about the S/P-problem forty years after its emergence. The impact sheet to this article can be accessed at 10.6084/m9.figshare.16610104.

A Stakeholder Analysis of the Development of a Large-Scale Professional Development Program

The development of a large-scale professional development project for Swedish mathematics teachers is retrospectively examined. By referring to documentation produced by stakeholders in the development process, the stakeholder’s design recommendations and underlying assumptions on teacher development are described. Seeing the development as a co-determination process explains how research-based principles appearing early in the process gradually change to become something different in the end, without the reasons for this shift ever being explicitly discussed in stakeholders’ documentations. It is discussed whether the distributed way of constructing the program might cause difficulties in sticking to an explicit theory of change. The impact sheet to this article can be accessed at 10.6084/m9.figshare.16610113.

Emerging Organizations When Implementing a Collaborative Professional Development Program

In this article, we explore an emerging organization that unfolds during the implementation of a collaborative and practice-oriented professional development program (PD) called Action Learning. In Action Learning, local mathematics supervisors facilitate meetings where mathematics teachers collaboratively discuss and develop interventions in their own teaching. Thereafter, teachers carry out their interventions and are observed by the team, who afterwards provide feedback in an evaluation meeting, thereby taking on a central role in the PD program. Drawing on qualitative interviews of teachers, local supervisors, and school managers and observations of meetings in the PD program, we investigate what roles emerge for local supervisors, and how their contributions are framed by colleagues and school managers. This identifies three simultaneously present logics among the stakeholders, positioning the supervisors in roles as project leaders, academic beacons, and equal coaches, confronting each of them with different and mutually exclusive expectations. The impact sheet to this article can be accessed at 10.6084/m9.figshare.16610119.

Two Replication Studies of the Relationships between Mathematical Knowledge for Teaching, Mathematical Quality of Instruction, and Student Achievement

In this paper, we share two conceptual replications of Hill et al.’s (2012c) study linking Mathematical Knowledge for Teaching (MKT), Mathematical Quality of Instruction (MQI), and student assessment scores. In study 1, we share data from 4th and 5th grade teachers in an urban school district. In study 2, we share data from middle school teachers in a school district with a relatively high proportion of emergent bilingual students. By varying contexts, we found that Hill et al.’s (2012c) suggested use of the MKT cutoff points was not warranted in our differing settings. Further, we found some significant relationships between MKT, MQI, and student assessments; however, we were not able to reproduce these consistently with our data. We suggest that the relationship between teaching practice and MKT may be quite sensitive to contextual factors including grade level, demographics, school effects, and assessments. We recommend that policymakers and researchers take caution when using such instruments to evaluate program initiatives and identify teachers for remediation or leadership positions. The impact sheet to this article can be accessed at 10.6084/m9.figshare.16610080.

Implementation Studies in Mathematics Education: What Theoretical Resources?

This article is devoted to the theoretical resources likely to support implementation studies. After the introductory section, we focus on the potential offered by resources internal to the field of mathematics education, especially those offered by design-based research, by the systemic and ecological perspective underlying the anthropological theory of the didactic, and by the documentational approach to didactics regarding the teachers’ relationship with educational resources. We then examine particular examples of implementation studies, paying specific attention to the way they combine internal and external resources. For that purpose we consider on the one hand, European projects aiming at the large-scale implementation of inquiry-based learning and teaching in STEM education, especially the PRIMAS project, and on the other hand, the contributions to the thematic working group devoted to implementation and replication studies at CERME 10 and CERME 11. We conclude with more general reflections.

Upper Elementary School Children’s Adaptive Use of Subtraction by Addition: A Choice/No-Choice Replication Study Involving Two Choice Conditions

The current study examined upper elementary school children’s frequent, efficient and adaptive use of direct subtraction (DS) and subtraction by addition (SBA) when mentally solving multi-digit subtractions, replicating and expanding previous research by Torbeyns et al. (2018). First, children were offered subtractions in two choice conditions in which they had to indicate whether they would use DS or SBA to solve each item. In the choice-compute condition they were allowed to make actual calculations, in the choice-decide condition they were not allowed to do so. Thereafter, two no-choice conditions were offered (mandatory use of either DS or SBA). Results showed that children made frequent, efficient and adaptive use of the untrained SBA strategy. Furthermore, children were most adaptive for task and subject characteristics in the choice-compute condition. The current study confirms previous findings on SBA, and questions current classroom practices that focus heavily on the development of DS.

An Empirically Grounded System of Supports for Improving the Quality of Mathematics Teaching on a Large Scale

Research on the teaching and learning of mathematics has made significant progress in recent years. However, this work has had only limited impact on classroom instruction in many countries. We report on an eight-year project in which we partnered with several large urban school districts in the US that were attempting to support mathematics teachers’ development of ambitious, inquiry-oriented instructional practices. These partnerships provided contexts in which we could iteratively test and revise conjectures about instructional improvement strategies intended to support teachers’ and others’ learning. The product of this work is a theory of action for instructional improvement at scale that spans from the classroom to school system instructional leadership. We present project findings as they relate to key elements of the theory including: teachers’ knowledge, perspective and practices; instructional materials and student assessments; participatory supports for teachers’ learning; and additional supports for currently struggling students.

In Pursuit of a Replication Culture in Mathematics Education

As we celebrate the launch of a new mathematics education journal focused (at least in part) on replication studies, I add to a conversation initiated by others (e.g., Aguilar, 2020) around what it would mean for our field to be more amendable to replication. This paper begins by examining several replication studies in mathematics education as a way to reflect upon types of replication studies and the importance of replication in our field. I then problematize the idea of a conceptual replication, and I explore the boundary between a replication and a follow-up study. Finally, I consider what it might take for the field of mathematics education to become a more replication-friendly culture by introducing a distinction between idea-initiated research and results-initiated research.