Individuals draw conclusions about possibilities from assertions that make no explicit reference to them. The model theory postulates that assertions such as disjunctions refer to possibilities. Hence, a disjunction of the sort, "A or B or both," where "A" and "B" are sensible clauses, yields mental models of an…

Mathematics education often grapples with the challenge of teaching abstract mathematical concepts, particularly those existing in 3D space. Visualizing, manipulating, and comprehending these abstract objects can be a formidable task for learners. While 3D printing technology has found applications in various fields, its utilization in mathematics…

This article reviews two aspects of human learning: (1) people draw inferences that appear to rely on hierarchical conceptual representations; (2) some categories are much easier to learn than others given the same number of exemplars, and some categories remain difficult despite extensive training. Both of these results are difficult to reconcile…

Humans are sensitive to the properties of individual items, and exemplar models are useful for capturing this sensitivity. I am a proponent of an extension of exemplar-based architectures that I briefly describe. However, exemplar models are very shallow architectures in which it is necessary to stipulate a set of primitive elements that make up…

Instruction in many STEM domains heavily relies on visual representations, such as graphs, figures, and diagrams. However, students who lack representational competencies do not benefit from these visual representations. Therefore, students must learn not only content knowledge but also representational competencies. Further, as learning…

Research in undergraduate physics and in K-12 science education has demonstrated challenges and successes in facilitating student engagement with reasoning practices associated with professional physicists. Here we focus on one important dimension of physics reasoning, using evidence to revise models. While this topic has been explored at the…

Analogical reasoning has played a significant role in the development of modern mathematical concepts. Although some perspectives in mathematics education have argued against the use of analogies and analogical reasoning in instructional contexts, some attempts have been made to leverage the pedagogical power of analogies. I assert that with a…

Mathematics teacher education programs in the United States are charged with preparing prospective secondary teachers (PSTs) to teach reasoning and proving across grade levels and mathematical topics. Although most programs require a course on proof, PSTs often perceive it as disconnected from their future classroom practice. Our design research…

The syntactic behaviour of human beings cannot be explained by analogical generalization on the basis of concrete exemplars: analogies in surface form are insufficient to account for human grammatical knowledge, because they fail to hold in situations where they should, and fail to extend in situations where they need to. [For Ben Ambridge's…

Building scientific arguments is a central ability for all scientists regardless of their specific domain. In organic chemistry, building arguments is a necessary skill to estimate reaction processes in consideration of the reactivities of reaction centres or the chemical and physical properties. Moreover, building arguments for multiple reaction…

Ambridge argues that the existence of exemplar models for individual phenomena (words, inflection rules, etc.) suggests the feasibility of a unified, exemplars-everywhere model that eschews abstraction. The argument would be strengthened by a description of such a model. However, none is provided. I show that any attempt to do so would immediately…

The authors argue that Ambridge's radical exemplar account of language cannot clearly explain all syntactic priming evidence, such as inverse preference effects ("greater" priming for less frequent structures), and the contrast between short-lived lexical boost and long-lived abstract priming. Moreover, without recourse to a level of…

We present a framework designed to help categorize various sense making moves, allowing for greater specificity in describing and understanding student reasoning and also in the development of curriculum to support this reasoning. The framework disaggregates between the mechanisms of student reasoning (the cognitive "tool" that they are…

By using high-leverage models to connect student learning experiences to overarching concepts in mathematics, teachers can anchor learning in ways that allow students to make sense of content on the basis of their own prior experiences. A rectangular area model can be used as a tool for understanding problems that involve multiplicative reasoning.…

In this work, we discuss the importance of underlying theoretical assumptions in research, focusing on the conclusions reached when analyzing data from a misconceptions constructivist (stable, unitary) perspective in contrast to a fine-grained constructivist (resources, knowledge-in-pieces) perspective. Both frameworks are rooted in the idea that…