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An algebraic model uses a set of algebraic equations to describe a situation. Constructing such models is a fundamental skill, but many students still lack the skill, even after taking several algebra courses in high school and college. For such students, we developed instruction that taught students to decompose the to-be-modelled situation into schema applications, where a schema represents a simple relationship such as distance-rate-time or part-whole. However, when a model consists of multiple schema applications, it needs some connection among them, usually representedby letting the same variable appear in the slots of two or more schemas. Students in our studies seemed to have more trouble identifying connections among schema applications than identifying the schema applications themselves. We developed several tutoring systems and evaluated them in university classes. One of them, a step-based tutoring system called OMRaaT (One Mathematical Relationship at a Time), was both reliably superior (p = 0.02, d = 0.67) to baseline and markedly superior (p < 0.001, d = 0.84) to an answer-based tutoring system using only commercially available software (MATLAB Grader).
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This content will become publicly available on November 16, 2025
The Semantics and Expression of Apprehensional Modality
ABSTRACT Apprehensionalityrefers to the semantic domain of negative prospective epistemic modality and the formal expressions used to encode it. In this article, we break down theapprehensional situationinto five prototypical components: (I) future reference, (II) possibility, (III) negative evaluation, (IV) avertive intent, and (V) a preferred action. In doing so, we propose a framework for comparing synchronic and diachronic aspects of the semantics and expression of apprehensional modality across languages. We discuss several recent formal accounts of three apprehensional morphemes—Francez's account of Biblical Hebrewpen, Dąbkowski and AnderBois's of A'ingae‐sa'ne, and Phillips's of Australian Kriolbambai—relate their formalisms to the apprehensional situation schema, and evaluate their predictions. We summarise previous findings on the grammaticalisation pathways towards and among apprehensional morphemes. We find that apprehensionals grammaticalise from a wide range of sources, including components I–IV of the apprehensional schema. Among the apprehensional functions themselves, both subordination and insubordination are commonly encountered.
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- Award ID(s):
- 1911348
- PAR ID:
- 10558473
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Language and Linguistics Compass
- Volume:
- 19
- Issue:
- 1
- ISSN:
- 1749-818X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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