In this paper, we elaborate the
Algebra readiness and algebraic structure as foundational ideas for algebraic learning
Algebra readiness and algebraic structure have been core ideas in our curriculum development work to help students develop conceptual understandings of algebra. This research brief uses Kaput’s (2008) definition of algebra to describe algebra readiness and algebraic structure, as they relate to thinking about algebra and thinking with algebra.
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- Award ID(s):
- 2021414
- PAR ID:
- 10547969
- Editor(s):
- Walker, WS; Bryan; LA; Guzey; SS; Suazo-Flores, E
- Publisher / Repository:
- Proceedings of the Seventh Annual Indiana STEM Education Conference
- Date Published:
- Subject(s) / Keyword(s):
- Algebraic Thinking
- Format(s):
- Medium: X
- Location:
- West Lafayette, IN
- Sponsoring Org:
- National Science Foundation
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Abstract seeds of algebraic thinking perspective, drawing upon Knowledge in Pieces as a heuristic epistemological framework. We argue that students’ pre-instructional experiences in early childhood lay the foundation for algebraic thinking and are a largely untapped resource in developing students’ algebraic thinking in the classroom. We theorize that seeds of algebraic thinking are cognitive resources abstracted over many interactions with the world in children’s pre-instructional experience. Further, we provide examples to demonstrate how the same seeds of algebraic thinking present in early childhood can be invoked in reasoning across contexts, grade levels, and different levels of formality of algebraic instruction. The examples demonstrate how the seeds perspective differs from other accounts of the relationship between children’s early activity and their engagement in algebraic reasoning processes. We anticipate this new theoretical direction for characterizing the nature and development of algebraic thinking will lay the foundation for a robust agenda that sheds light on the development of algebraic thinking and informs algebra instruction, particularly how teachers notice and respond to children’s developing algebraic thinking. -
Algebraic statistics uses tools from algebra (especially from multilinear algebra, commutative algebra, and computational algebra), geometry, and combinatorics to provide insight into knotty problems in mathematical statistics. In this review, we illustrate this on three problems related to networks: network models for relational data, causal structure discovery, and phylogenetics. For each problem, we give an overview of recent results in algebraic statistics, with emphasis on the statistical achievements made possible by these tools and their practical relevance for applications to other scientific disciplines.more » « less
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W. S. Walker, III ; null ; null ; null (Ed.)Thinking with Algebra (TWA) is a project to develop algebra curriculum for college classes. The overall goal is to develop curricular materials that will help prepare students conceptually for college algebra. The project is described and the importance of algebraic structure as a theoretical consideration is explained.more » « less
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Is intelligence realized by connectionist or classicist? While connectionist approaches have achieved superhuman performance, there has been growing evidence that such task-specific superiority is particularly fragile in systematic generalization. This observation lies in the central debate between connectionist and classicist, wherein the latter continually advocates an algebraic treatment in cognitive architectures. In this work, we follow the classicist’s call and propose a hybrid approach to improve systematic generalization in reasoning. Specifically, we showcase a prototype with algebraic representation for the abstract spatial-temporal reasoning task of Raven’s Progressive Matrices (RPM) and present the ALgebra-Aware Neuro-Semi-Symbolic (ALANS) learner. The ALANS learner is motivated by abstract algebra and the representation theory. It consists of a neural visual perception frontend and an algebraic abstract reasoning backend: the frontend summarizes the visual information from object-based representation, while the backend transforms it into an algebraic structure and induces the hidden operator on the fly. The induced operator is later executed to predict the answer’s representation, and the choice most similar to the prediction is selected as the solution. Extensive experiments show that by incorporating an algebraic treatment, the ALANS learner outperforms various pure connectionist models in domains requiring systematic generalization. We further show the generative nature of the learned algebraic representation; it can be decoded by isomorphism to generate an answer.more » « less
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Thinking With Algebra (TWA) is a curriculum development project for college students needing introductory/intermediate algebra. Unique guiding principles of the project are a focus on algebraic structure, mixed review, equity, and a classroom approach emphasizing small-group work and whole- class discussion (Feikes et al., 2022). Results are derived from a survey given to one class of fifteen students taking an intermediate algebra summer course. The paper shares both Likert and open-ended responses. Data demonstrates that the students believed that the TWA curriculum was beneficial.more » « less
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