Seeds of Algebraic Thinking comes from the Knowledge in Pieces (KiP) perspective of learning. KiP is a systems approach to learning that stems from the constructivist idea that people learn by building on prior knowledge. As people experience the world, they acquire small, sub-conceptual knowledge elements. When people engage in a particular…

A frequent concern about constructivist instruction is that it works well, mainly for students with higher domain knowledge. We present findings from a set of two quasi-experimental pretest-intervention-posttest studies investigating the relationship between prior math achievement and learning in the context of a specific type of constructivist…

Physics teachers need diagnostic competence to effectively identify students' misconceptions, but studies have yet to sufficiently investigate how this competence can be advanced through various instructional strategies. In this study, we compared a problem-solving strategy against an example-based strategy in advancing diagnostic competence in…

Background: In example-based learning, examples are often combined with generative activities, such as comparative self-explanations of example cases. Comparisons induce heavy demands on working memory, especially in complex domains. Hence, only stronger learners may benefit from comparative self-explanations. While static text-based examples can…

In 2 experiments, I studied the effects of an Equivalence Based Instruction (EBI) math intervention on the emergence of untaught selection responses and abstraction to production responses. In Experiment I, using a multiple baseline design, I implemented the EBI intervention among a group of 17 first grade participants with varying levels of math…

This study examined the effects of shared representational guidance (collaborative textural representative tool vs. collaborative graphical representative tool; TR vs. GR) and prior knowledge (low vs. high; LPK vs. HPK) on different levels of individuals' understanding of specific-domain knowledge after collaborative problem solving (CPS). A total…

Research on productive failure suggests that attempting to solve a problem prior to instruction facilitates conceptual understanding compared to receiving instruction prior to problem solving. The assumptions are that during the problem-solving phase, students activate their prior knowledge, become aware of their knowledge gaps, and discover deep…

Prior short-term studies have shown the effectiveness of worked examples and self-explanation prompts in improving elementary students' mathematics knowledge. However, year-long classroom-based interventions have been tested only with older students. The purpose of the current study was to explore the effectiveness of a year-long classroom…

The aim of this study is to provide data to better understand the processes of early mathematical modelling. According to this, an early mathematical modelling activity carried out by 21 Spanish schoolchildren aged 5-6 years is analysed, using the validated tool "Rubric for the Evaluation of Mathematical Modelling Processes" (REMMP). The…

The COVID-19 pandemic has affected student learning; according to the National Assessment of Educational Progress (2022), scores for students in the United States have shown a decline in mathematics. . One way educators can strengthen students' current mathematical knowledge and bridge to new content is through Number Talks. Number Talks are brief…

Comparison is an important mechanism for learning in general, and comparing two worked examples has garnered support over the last 15 years as an effective tool for learning algebra in mainstream classrooms. This study was aimed at improving our understanding of how Modified for Language Support-Worked Example Pairs (MLS-WEPs) contribute to…

In this article, the author describes strategies to support students in conjecturing in ways that promote their agency. Conjecturing involves justifying (i.e., a practice that promotes epistemic agency and engages students in mathematical reasoning to defend their claims; Bieda & Staples, 2020) by which a conjecture becomes proven, modified,…

At a research university near the east coast, a college algebra class has been restructured into two large lectures a week, an active recitation size laboratory once a week, and an extra day devoted to active group work called Supplemental Practice (SP). SP was added as an extra day of class where the SP leader has students work in groups on a…

This study examined the extent to which college algebra instructors employ pedagogical practices previously found to assist students master difficult STEM content material and address their own previous math deficits. Faculty classroom behaviors were assessed with a modified version of the Generalized Observation and Reflection Platform (GORP).…

Nowadays, mathematics teachers in K-12 strive to promote their students' mathematical knowledge and computational thinking (CT) skills. There is an increasing need for effective CT-embedded mathematics learning material and a better understanding of students' views toward them. In this work, we present the results of a research study, which…