In this paper we consider how mathematics instruction that values, attends to, and builds on students' mathematical ideas is realized through discourse. We describe interactions that build on students' thinking and in which students help to determine the direction of mathematics lessons as responsive. Using a framework we developed to characterize…

This paper contributes to the discourse in stochastic education of how young students deal with learning settings that allow a data-based approach to probability. By using the micro-structure of arguments by Toulmin (1958), it explores which arguments students use and which role they play in the learning process. The data stems from design…

The ultimate goal for many choral directors is to develop independent musicians within the ensemble. The ability to sing a series of pitches and rhythms at first sight is widely understood to be a fundamental building block of independent musicianship. Yet sight-reading is not simply a holistic skill. There are separate components of…

Currently, language competences in mathematics lessons gain more attention in Germany. The paper reports an interdisciplinary study of linguistics and mathematics education on reasoning. A model to rate the competences in arithmetic reasoning at primary level will be presented for discussion: mathematical reasoning is coded separately from its…

Results of the last thirty years in mathematics education have shown the importance of an operational concept development. One of the geometrical concepts that has been researched for years already, however, not with the particular focus on its systematic teaching in school mathematics, is the concept of angle. In this paper we focus on children's…

In this study we explored to what extent middle school teachers were able to appropriately identify proportional situations when presented with various mathematical structures and if there were relationships between attributes of the teachers and their ability to identify proportional situations. Interestingly, there were no strong relationships…

In this proposal, I introduce a method for modeling the dynamics of a sixth-grade student's accommodation of his fractions scheme to include a disembedding operation (Steffe & Olive, 2010). I will describe a three-part approach consisting of a constructivist teaching experiment, retrospective analysis, and stochastic modeling of the student's…

This article develops three different types of student explanations and studies how teachers respond to these. The data come from five classrooms at upper grade 5-7 (ages from eleven to thirteen) where all mathematics teaching for one week was filmed. These films were transcribed and student explanations identified. Through a close inspection of…

This study compares Singaporean Grade 6 students' performance and strategy preference on two graphic-rich mathematics tasks, presented via pencil-and-paper and iPad modes. There were statistically significant differences between students' performances on the two tasks, one in favour of the paper mode and the other in favour of the iPad. Students…

Both additive and proportional reasoning are types of quantitative analogical (QA) reasoning. We investigated the development and nature of primary school children's QA reasoning by offering two missing-value word problems to 3rd to 6th graders. In one problem, ratios between given numbers were integer, in the other ratios were non-integer. These…

This paper demonstrates the explanatory power of Kieren's framework for rational number knowing (1988, 1992, 1993, 1995), renamed here the four-three-four model, by describing the different approaches of Grade 6 students to a quotient context task (sharing three or seven custard tarts between five people) using Kieren's terminology of…

Working collaboratively with the researchers, a small team of teachers developed and taught a lesson based on the Teaching for Abstraction model (White & Mitchelmore, 2010). This paper reports how one teacher learned about the model and implemented it in her lesson. It was found that she had assimilated several key features of the model, such…

We detail a learning progressions approach to early algebra research and how existing work around learning progressions and trajectories in mathematics and science education has informed our development of a four-component theoretical framework consisting of: a curricular progression of learning goals across big algebraic ideas; an instructional…

This paper describes Singapore and Australian Grade 6 students' (n=1,187) performance on a symmetry task in a recently developed Mathematics Processing Instrument (MPI). The MPI comprised tasks sourced from Australia and Singapore's national assessments, NAPLAN and PSLE. Only half of the cohort solved the item successfully. It is possible that…

We introduce a cognitive framework for measuring reading comprehension that includes the use of novel summary-writing tasks. We derive NLP features from the holistic rubric used to score the summaries written by students for such tasks and use them to design a preliminary, automated scoring system. Our results show that the automated approach…