This study lies within the field of early-age algebraic thinking and focuses on describing the functional thinking exhibited by six sixth-graders (11- to 12-year-olds) enrolled in a curricular enhancement program. To accomplish the goals of this research, the structures the students established and the representations they used to express the…

Recent research has highlighted the role of functional relationships in introducing elementary school students to algebraic thinking. This functional approach is here considered to study essential components of algebraic thinking such as generalization and its representation, as well as the strategies used by students and their connection with…

This study investigated how three fifth-grade students' understanding of fraction and decimal magnitude evolved over the course of a five-week teaching experiment. Students participated in teaching and learning sessions focused on developing concepts of fraction and decimal magnitude. The following questions guided this study: (1) How do fifth…

Given the relevance of graphs of functions, we consider their inclusion in primary education from the functional approach to early algebra. The purpose of this article is to shed some light on the students' production and reading of graphs when they solved generalization problems from a functional thinking approach. We aim to explore how 3rd and…

We conducted a descriptive exploratory study in which we analyzed 313 sixth to eighth grade students' answers to a word problem, accompanied by diagrams, involving generalization in an algebraic functional context. In this research, we jointly addressed two objectives: (a) to determine the strategies deployed by students to generalize and (b) to…

This study investigates sixth-grade Turkish students' pattern-generalization approaches among arithmetical generalization, algebraic generalization, and naïve induction. A qualitative case study design was employed. The data was collected from four sixth-grade students through the Pattern Questionnaire (PQ) and individual interviews based on the…

This study aims to understand individual differences between children in metacognitive monitoring and control processes and the developmental trajectories of metacognition over one year. Three indicators of procedural metacognition were used: monitoring accuracy (discrimination of confidence judgments between correct and incorrect test responses),…

We describe 24 third (8-9 years old) and 24 fifth (10-11 years old) graders' generalization working with the same problem involving a function. Generalizing and representing functional relationships are considered key elements in a functional approach to early algebra. Focusing on functional relationships can provide insights into how students…

Functional thinking is an established route into algebra. However, the learning mechanisms that support the transition from arithmetic to functional thinking remain unclear. In the current study we explored children's pre-instructional intuitive reactions to functional thinking content, relying on a conceptual change perspective and using mixed…

Incorporating student choice is an effective intervention to improve task engagement that may also promote skill and behavioral generalization. In this study, we evaluated the effectiveness of a reading fluency intervention that incorporated student choice of topic and instructional passages using a delayed multiple baseline across student design.…

Differential Item Function (DIF) analysis is commonly employed to examine potential bias produced by a test item. Since its introduction DIF analyses have focused on potential bias related to broad categories of oppression, including gender, racial stratification, economic class, and ableness. More recently, efforts to examine the effects of…

Large scale international assessments depend on invariance of measurement across countries. An important consideration when observing cross-national differential item functioning (DIF) is whether the DIF actually reflects a source of bias, or might instead be a methodological artifact reflecting item response theory (IRT) model misspecification.…

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'…

Recent research provides evidence that an intersectional approach to defining reference and focal groups results in a higher percentage of comparisons flagged for potential DIF. The study presented here examined the generalizability of this pattern across methods for examining DIF. While the level of DIF detection differed among the four methods…

Verbal praise is frequently used as motivation by teachers in class and truly effective verbal praise plays an important role in maintaining classroom order and stimulating positive student behavior. In this study, students from three classes of X Primary School of Jiangxi Province were interviewed and 328 verbal praise expressions collected. The…