Research suggests that troubleshooting activities that require students to reflect on teacher-crafted erroneous examples; i.e., erroneous solutions to problems that correspond to widespread naïve ideas, are beneficial to learning. One possible explanation to these beneficial effects is that troubleshooting activities encourage students to test the…

Graded Troubleshooting (GTS) is a powerful routine that teachers can use easily to engender students' metacognitive thinking and boost their understanding of mathematics concepts and procedures. This article describes a new GTS activity designed to prompt students to efficiently exploit worked examples when asked to diagnose erroneous examples…

Developing both conceptual and procedural knowledge is important for students' mathematics competence. This study examined whether U.S. Grade 9 general education mathematics teachers' self-reported use of concept-focused instruction (CFI) and procedure-focused instruction (PFI) were associated differently with ninth graders' algebra achievement…

This study addresses a significant gap in educational research by employing statistical methods to measure the sequencing of different components and content topics in textbooks. It is well-documented that the structure and sequencing features of textbooks play a crucial role in enhancing students' learning. However, existing literature has…

This paper reports on the development of complex unfamiliar mathematics questions using conventional mathematics questions. The autoethnographic research data included retrospective reflections of two educators and review of literature. Data analysis resulted in a three-phase development process: identifying conventional questions and subject…

Sixty (35 girls and 25 boys) 9th-grade students' conceptual understanding of the number line was qualitatively assessed through verbal explanations and visual representations. The assessment included an open-ended question focused on students' number line descriptions and the explanations coalesced around six features: sequential ordering (i.e.,…

Misconceptions occur as a result of learners' wrong beliefs and experiences, and because subsequent learning is built on these misconceptions, they cause new concepts to be learned incorrectly. Studies show that students have problems in making sense of many mathematical concepts in mathematics teaching processes and this situation can be…

This study examined "backward transfer," which we define as how students' ways of reasoning about previously encountered concepts are modified when learning about new concepts. We examined the backward transfer produced when students learned about quadratic functions. We were specifically interested in how backward transfer may vary for…

Making sense of algebraic expressions in context is multifaceted and extends beyond representing real-world situations using algebra. As part of Project MathTalk, the authors created videos that feature a pair of Grade 9 algebra 1 students coming to understand algebraic expressions in context. The students found three aspects of making sense of…

This large-scale study characterizes the algebraic learning of a cohort of nearly 800 French students during the last year of middle school (9th grade, 14-15 years old) and examines their evolution in relation to the composition of the classes to which the students belong. Based on the Anthropological Theory of the Didactic (ATD), the study is…

This study aims (1) to develop an instructional model to enhance mathematical literacy for secondary students and (2) to examine the effects of using this instructional model. The sample group consisted of students from Lampabpla Wittayakarn School, under the jurisdiction of the Surin Secondary Educational Service Area Office in Thailand by using…

Recent research syntheses have identified several potentially high-leverage teaching strategies for improving low-attaining secondary school students' learning of mathematics. These strategies include the structured use of representations and manipulatives and an emphasis on derived facts and estimation. This paper reports on 70 semi-structured…

This research investigates errors and misconceptions among learners in algebraic education by utilizing Koch's error analysis method alongside the Structure of the Observed Learning Outcome (SOLO) taxonomy. The primary aim of the investigation is to discern the kinds of errors and cognitive stages demonstrated by Grade 9 students when engaged in…

The bring-your-own-device policy (BYOD) in schools has recently attracted considerable research interest. BYOD allows students to learn subjects like mathematics using personal mobile devices. Accordingly, BYOD can increase students' desire to learn mathematics in school. In the study we report on here, we assessed 9th grade students' (n = 500)…

A controversial, equity-focused mathematics reform in the San Francisco Unified School District (SFUSD) featured delaying Algebra I until ninth grade for all students. This study examines student-level longitudinal data on mathematics course-taking across successive cohorts of SFUSD students who spanned the reform's implementation. We observe…