Principled by the Embodied, Situated, and Distributed Cognition paradigm, the study investigated the impact of using a research-based curriculum that employs multiple modalities on the performance of grade 5 students on 3 subscales: concept of unit, fraction equivalence, and fraction comparison. The sample included five schools randomly selected from a population of 14 schools in Lebanon. Eighteen 5th grade classrooms were randomly assigned to experimental (using multimodal curriculum) and control (using a monomodal curriculum) groups. Three data sources were used to collect quantitative and qualitative data: tests, interviews, and classroom observations. Quantitative data were analyzed using two methods: reliability and MANOVA. Results of the quantitative data show that students taught using the multimodal curriculum outperformed their counterparts who were instructed using a monomodal curriculum on the three aforementioned subscales (at an alpha level =. 001). Additionally, fine-grained analysis using the semiotic bundle model revealed different semiotic systems across experimental and control groups. The study findings support the multimodal approach to teaching fractions as it facilitates students’ conceptual understanding.

Key words: embodied, situated, and distributed learning, fractional knowledge, multimodality, solution schemes, theorems-in-action.

Three iterative, after school design experiments with small groups of middle school students were conducted to investigate how students represent fractional relationships between two unknowns and whether they construct reciprocal reasoning with unknowns. Of the 22 students who participated, 9 had constructed fractions as multiples of unit fractions. Seven of these 9 students constructed reciprocal reasoning with unknowns. They did so by constructing a multiplicative relationship between a unit fraction of one unknown and the other unknown. The other 2 students did not make this construction. Instead, they represented relationships between unknowns with whole number multiplication and division, and by adding unknowns and fractional parts of unknowns. The students who constructed reciprocal reasoning demonstrated a link between fractions as measures and fractions as operators, one benefit of working on rational number knowledge and algebraic reasoning together. Implications for teaching include recommendations for supporting students to construct reciprocal reasoning with unknowns.

In this study, we challenge the deficit perspective on mathematical knowing and learning for children labeled as LD, focusing on their struggles not as a within student attribute, but rather as within teacher-learner interactions. We present two cases of fifth-grade students labeled LD as they interacted with a researcher-teacher during two constructivist-oriented teaching experiments designed to foster a concept of unit fraction. Data analysis revealed three main types of interactions, and how they changed over time, which seemed to support the students’ learning: Assess, Cause and Effect Reflection, and Comparison/Prediction Reflection. We thus argue for an intervention in interaction that occurs in the instructional process for students with LD, which should replace attempts to “fix” ‘deficiencies’ that we claim to contribute to disabling such students.

Key words: constructivism, student-teacher interaction, mathematics, intervention, learning difference

This commentary addresses the role of theoretical frameworks in building models of students’ mathematics. Specifically, it compares ways that the Learning Through Activity framework (LTA) and scheme theory explain and predict students’ mathematical activity. Both frameworks rely on Piagetian constructs – especially reflective abstraction – to build explanatory models for teaching and learning. LTA attempts to provide the teacher-researcher with a greater degree of determination in student learning trajectories, but then the teacher-researcher must address constraints in the students’ available ways of operating. These issues are exemplified in the case of teaching students about multiplying fractions. Additional theoretical issues arise in explaining logical necessity in students’ ways of operating and the role of reflective abstraction in organizing new ways of operating.

Discusses criteria for evaluating clinical research in mathematics education. It is suggested that any set of criteria must be consistent with the underlying epistemology from which that research is carried out. The implications of environmentalist vs constructionist world views of mathematics education for establishing criteria are examined, and a set of criteria applicable to constuctivist mathematical-education research is presented.