The aim of this study was to examine how pre-service teachers performed in tasks related to three specific aspects of curricular noticing. The participants completed a two-part written task in which they solved three pattern generalization problems and sequenced them for teaching purposes. Inductive content analysis was used to analyze the…

This study investigated how 155 pre-service teachers solved three pattern generalization problems in a two-part written test and sequenced them for teaching purposes to demonstrate their curricular noticing. Participants' solutions were analyzed using inductive content analysis, which showed that only 8.4% of PSTs produced correct answers to all…

In this paper, we explore how students' algebraic noticing's and explanations changed across a two-year period with the introduction of designed instructional material. The data in this report is drawn from n=53 Year 7-8 students' responses to a free-response assessment task across two different years. Analysis focused on how students noticed and…

Generalizing patterns is an important feature of algebraic reasoning that is accessible to students across grade-levels because it connects their numerical reasoning to algebraic reasoning. In this article, the authors describe how teachers can use the game Crack the Code to introduce generalizing to their students or can extend students'…

Feedback on student answers and even during intermediate steps in their solutions to open-ended questions is an important element in math education. Such feedback can help students correct their errors and ultimately lead to improved learning outcomes. Most existing approaches for automated student solution analysis and feedback require manually…

Due to their abstract nature, representation of mathematical concepts through different registers favors their understanding. In the case of ''sequences and regularities'', it becomes propitious the exploration of different registers of representation in the institution of topics, such as term, order, formation law, and generating expression.…

A few case studies have suggested students' struggles with the "temporal order" of epsilon and delta in the formal limit definition. This study problematizes this hypothesis by exploring students' claims in different contexts and uncovering productive resources from students to make sense of the critical relationship between epsilon and…

Linear pattern is the primary material in learning number patterns in junior high schools, but there are still many students who fail to generalize the linear pattern. The students' failure in generalizing the pattern occurred when the students ended to view the problems globally without breaking them into the constructors' components such as the…

The authors who are mathematics teacher educators, have found that classifying, composing, and transforming shapes (in particular, rotations and reflections) are areas of difficulty for adults as well as for children. However, these are also some of the most important geometric ideas. They are fundamental topics in the K-8 Geometry and Measurement…

In this article we advance characterizations of and supports for elementary students' progress in generalizing and representing functional relationships as part of a comprehensive approach to early algebra. Our learning progressions approach to early algebra research involves the coordination of a curricular framework and progression, an…

The National Council of Teachers of Mathematics [NCTM] noted that middle and high school students are expected to be able to both explain inequalities by using mathematical symbols and understand meanings by interpreting the solutions of inequalities. Unfortunately, research has revealed that not only do middle and high school students hold…

Traditional mathematics education focuses on teaching rote procedures to solve problems, though these procedures are not usually motivated by goals. As a result, students have trouble flexibly using procedures and generalizing their knowledge to solve novel problems that differ from the problems they practice during instruction. In the following…

Six fifth-grade students with learning and/or behavioral/emotional disabilities received peer tutoring on multiplication and division facts using 3 drill and practice conditions varying in the ratio of unknown to known facts (30:70, 50:50, and 70:30). Five of the students mastered facts at a faster rate in the 70% unknown/30% known condition. (DB)