This article describes a learning trajectory on expressions and equations based on the sixth-grade Common Core State Standards for Mathematics (NGA/CCSSO, 2010). A classroom teaching experiment using design research (Lamberg & Middleton, 2009) was conducted. The data was analysed using Corbin and Strauss' (2014) Constant Comparative method and…

Empirical arguments rely on examples without necessarily addressing all cases. Students should be skeptical of empirical evidence and should seek more secure arguments for generalizations, such as those that explain why a generalization is true for all cases. Generalizing on the basis of patterns in data is an important mathematical practice;…

Online learning from video modeling examples, in which a human model demonstrates and explains how to perform a learning task, is an effective instructional method that is increasingly used nowadays. However, model characteristics such as gender tend to differ across videos, and the model-observer similarity hypothesis suggests that such…

This study developed learning progression-based measures of science teachers' content knowledge (CK) and pedagogical content knowledge (PCK). The measures focus on an important topic in secondary science curriculum using scientific reasoning (i.e., tracing matter, tracing energy, and connecting scales) to explain plants gaining weight and…

Thus far, it is unclear how students can learn most effectively from their own errors. In this study, reflections on the rationale behind self-made errors are assumed to enhance knowledge acquisition. In a field experiment with pre/post/follow-up design, the authors practiced fractions with 174 seventh- and eighth-grade students who were randomly…

For self-regulated learning to be effective, students need to be able to accurately assess their own performance on a learning task and use this assessment for the selection of a new learning task. Evidence suggests, however, that students have difficulties with accurate self-assessment and task selection, which may explain the poor learning…

Learning algebra is difficult for many students in part because of an emphasis on the memorization of abstract rules. Algebraic reasoners across expertise levels often rely on perceptual-motor strategies to make these rules meaningful and memorable. However, in many cases, rules are provided as patterns to be memorized verbally, with little overt…

Analogies are ubiquitous during instruction in science classrooms, yet research about the effectiveness of using analogies has produced mixed results. An aspect seldom studied is a model of instruction when using analogies. The few existing models for instruction with analogies have not often been examined quantitatively. The Teaching With…

Learning science requires the understanding of concepts and formal relationships, processes that--in themselves--have been proved to be difficult for students as they seem to encounter substantial problems with most of the inquiry-learning processes in which they engage. Models in inquiry-based learning have been considered as powerful…

Analogous thinking has been commonly discussed as being an inherent and distinguishing characteristic of human cognition (e.g., Gentner, 2010; Goldstone, Day, & Son, 2010; Holyoak, in press; Rittle-Johnson & Star, in press). Gentner (2003) has argued that as part of the human cognitive toolbox, comparison accompanied by the relational language to…

McCarthy (1985) constructed the 4MAT teaching model, an eight step instrument developed in 1980, by synthesizing Dewey's experiential learning, Kolb's four learning styles, Jung's personality types, as well as Bogen's left mode and right mode of brain processing preferences. An important implication of this model is that learning retention is…