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Margherita Piroi – Educational Studies in Mathematics, 2025
This study aims at elaborating a well-established theoretical framework that distinguishes three modes of thinking in linear algebra: the analytic-arithmetic, the synthetic-geometric, and the analytic-structural mode. It describes and analyzes the bundle of signs produced by an engineering student during an interview, where she was asked to recall…
Descriptors: Undergraduate Students, Engineering Education, Case Studies, Algebra
Charles Hohensee; Laura Willoughby; Sara Gartland – Mathematical Thinking and Learning: An International Journal, 2024
Backward transfer is defined as the influence that new learning has on individuals' prior ways of reasoning. In this article, we report on an exploratory study that examined the influences that quadratic functions instruction in real classrooms had on students' prior ways of reasoning about linear functions. Two algebra classes and their teachers…
Descriptors: Prior Learning, Abstract Reasoning, Mathematical Concepts, Algebra
Sara Ebner; Mary K. MacDonald; Paulina Grekov; Kathleen B. Aspiranti – Learning Disabilities Research & Practice, 2025
The concrete-representational-abstract (CRA) approach is an instructional framework for teaching math wherein students move from using concrete materials to solve problems to using visual representations of the materials, and finally abstract concepts. This study provides a literature synthesis and meta-analysis of the effectiveness of the CRA…
Descriptors: Meta Analysis, Mathematics Instruction, Teaching Methods, Abstract Reasoning
Fangli Xia; Mitchell J. Nathan; Kelsey E. Schenck; Michael I. Swart – Cognitive Science, 2025
Task-relevant actions can facilitate mathematical thinking, even for complex topics, such as mathematical proof. We investigated whether such cognitive benefits also occur for action predictions. The action-cognition transduction (ACT) model posits a reciprocal relationship between movements and reasoning. Movements--imagined as well as real ones…
Descriptors: Undergraduate Students, Geometry, Mathematical Concepts, Mathematics Instruction
Shipman, Barbara A.; Stephenson, Elizabeth R. – PRIMUS, 2022
Point-set topology is among the most abstract branches of mathematics in that it lacks tangible notions of distance, length, magnitude, order, and size. There is no shape, no geometry, no algebra, and no direction. Everything we are used to visualizing is gone. In the teaching and learning of mathematics, this can present a conundrum. Yet, this…
Descriptors: Mathematics Instruction, College Mathematics, Undergraduate Students, Topology
Mark A. Creager – Australian Mathematics Education Journal, 2023
Mark Creager noticed that how we teach students to reason mathematically may be counter-productive to our teaching goals. Sometimes a linear approach, focusing on sub-processes leading to a proof works well. But not always. Students should be made aware that reasoning is not always a straight forward process, but one filled with false starts and…
Descriptors: Secondary School Students, Mathematical Concepts, Mathematics Instruction, Logical Thinking
Mara Cotic; Daniel Doz; Matija Jenko; Amalija Žakelj – International Electronic Journal of Mathematics Education, 2024
The evolution of mathematics coincided with advancements in its teaching. The 19th and 20th centuries marked a pedagogical revolution in mathematics education. This paper argues that Bruner's (1966) model, Gagné's (1985) taxonomy, innovative teaching methods emphasizing research and problem-solving, and the inclusion of data analysis topics have…
Descriptors: Mathematics Education, Mathematics Instruction, Educational History, Mathematics Achievement
Simon, Martin A. – Mathematical Thinking and Learning: An International Journal, 2020
The goal of our research program is to explicate the learning of mathematical concepts in ways that are useful for instructional design and to develop design principles based on those explications. I review one type of concept and our elaboration of reflective abstraction, coordination of actions (COA) that accounts for its construction. I then…
Descriptors: Instructional Design, Mathematics Instruction, Mathematical Concepts, Abstract Reasoning
Luke T. Reinke; Michelle L. Stephan; Jerold R. Griggs – Mathematics Teacher: Learning and Teaching PK-12, 2024
Many teachers use problems set in real or imaginary contexts to make mathematics engaging, but these problems can also be used to anchor conceptual understanding. By constructing an understanding of mathematical ideas through solving problems in contexts that make sense to students, they have a better chance of actually understanding those…
Descriptors: Middle School Mathematics, Middle School Students, Middle School Teachers, Mathematical Concepts
Khatin-Zadeh, Omid; Farsani, Danyal; Yazdani-Fazlabadi, Babak – Cogent Education, 2022
Since formal mathematics is discussed in terms of abstract symbols, many students face difficulties to acquire a clear understanding of mathematical concepts and ideas. Transforming abstract or dis-embodied representations of mathematical concepts and ideas into embodied representations is a strategy to make mathematics more tangible and…
Descriptors: Mathematics Instruction, Mathematical Concepts, Concept Formation, Problem Solving
Darío González – North American Chapter of the International Group for the Psychology of Mathematics Education, 2023
This paper introduces two theoretical constructs, open-loop covariation and closed-loop covariation, that combine covariational reasoning and causality to characterize the way that three preservice mathematics teachers conceptualize a feedback loop relationship in a mathematical task related to climate change. The study's results suggest that the…
Descriptors: Preservice Teachers, Cognitive Processes, Abstract Reasoning, Thinking Skills
Donovan, Andrea Marquardt; Fyfe, Emily R. – Educational Psychology, 2022
Children often learn abstract mathematics concepts with concrete manipulatives. The current study compared different ways of using specific manipulatives -- base-ten blocks -- to support children's place value knowledge. Children (N = 112, M age = 6.88 years) engaged in place value learning activities in one of four randomly assigned conditions in…
Descriptors: Children, Mathematical Concepts, Manipulative Materials, Mathematics Activities
Copur-Gencturk, Yasemin; Baek, Clare; Doleck, Tenzin – International Journal of Science and Mathematics Education, 2023
Teachers' mathematical knowledge has important consequences for the quality of the learning environment they create for their students to learn mathematics. Yet relatively little is known about how teachers reason proportionally, despite the fact that proportional reasoning is foundational for several mathematics concepts and that ratios and…
Descriptors: Mathematics Skills, Pedagogical Content Knowledge, Mathematics Instruction, Mathematical Concepts
Elias, Dafna; Dreyfus, Tommy – Teaching Mathematics and Its Applications, 2022
We investigated how two didactical tools assist high school students in constructing knowledge about convergence and limits. The first tool is manual plotting of the terms of selected sequences, and the second, a technological applet. Student pairs worked in an interview setting on an activity designed for the purpose of this research. The…
Descriptors: High School Students, Mathematical Concepts, Mathematics Instruction, Abstract Reasoning
Hinton, Vanessa; Flores, Margaret – Rural Special Education Quarterly, 2022
Mathematics is crucial to the educational and vocational success of students. The concrete-representational-abstract (CRA) approach is a method to teach students mathematical concepts. The CRA involves instruction with manipulatives, representations, and numbers only in different lessons (i.e., concrete lessons include manipulatives but not…
Descriptors: Mathematics Instruction, Addition, Mathematical Concepts, Teaching Methods