When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of…

This article illustrates and differentiates the unique role cognitive interviews and think-aloud interviews play in developing and validating assessments. Specifically, we describe the use of (a) cognitive interviews to gather empirical evidence to support claims about the intended construct being measured and (b) think-aloud interviews to gather…

This study was carried out to examine proportional reasoning problem solving processes of seventh grade students. This study was conducted with the explanatory sequential mixed method design. In this respect, firstly, quantitative data from 56 students were collected and analyzed. Then, qualitative data of the study was collected from six students…

Previous studies have explored students' understanding of the relationship between definite integrals and areas under curves, but not their metacognitive experiences and skills while solving such problems. This paper explores students' mathematical performance, metacognitive experiences and metacognitive skills when solving integral-area tasks by…

The purpose of this study was to gain insight into 30, first year calculus students' understanding of the relationship between the concept of vertex of a quadratic function and the concept of the derivative. APOS (action-process-object-schema) theory was applied as a guiding framework of analysis on student written work, think-aloud and follow up…

In classroom learning, students need mathematical cognitive flexibility to be able to solve mathematical problems with the various ideas they express. To solve the problems, they must be able to grasp the problem, see it from various points of view, and should not be rigid thinking with one solving method. In fact, the students still lack the…

This study discusses the construction of deductive warrant derived from inductive warrant in mathematical argumentations expressed by pre-service teacher. In completing a mathematics task, a problem solver needs argumentation to determine, reveal, and support a reasonable solution. A mathematical argumentation can be analyzed by Toulmin scheme…

Comparing datasets, that is, sets of numbers in context, is a critical skill in higher order cognition. Although much is known about how people compare single numbers, little is known about how number sets are represented and compared. We investigated how subjects compared datasets that varied in their statistical properties, including ratio of…

The study explores the development of 11-year-old students' informal inference about random bunny hops through student talk and use of computer simulation tools. Our aim in this paper is to draw on dialogic theory to explain how students make shifts in perspective, from intuition-based reasoning to more powerful, formal ways of using probabilistic…

The ability to reason with formal logic is a foundational skill for computer scientists and computer engineers that scaffolds the abilities to design, debug, and optimize. By interviewing students about their understanding of propositional logic and their ability to translate from English specifications to Boolean expressions, we characterized…

The Institute of Education Sciences (IES) publishes practice guides in education to bring the best available evidence and expertise to bear on current challenges in education. Authors of practice guides combine their expertise with the findings of rigorous research, when available, to develop specific recommendations for addressing these…

How does the type of learning material impact what is learned? The current research investigates the nature of students' learning of math concepts when using manipulatives (Uttal, Scudder, & DeLoache, 1997). We examined how the type of manipulative (concrete, abstract, none) and problem-solving prompt (metacognitive or problem-focused) affect…

This study explores the mathematical problem-solving process of trainee teachers in a teachers' training institute. This research adopts a constructivist perspective that views learning mathematics as a process of constructing meaningful representation and knowledge as being constructed by the individual. The research methodology employs the case…

This descriptive study was aimed at looking into how Primary 5 pupils solve pre-algebra problems concerning patterns and unknown quantities. Specifically, objectives of this study were to describe Primary 5 pupils' solution strategies, modes of representations and justifications in: (a) discovering, describing and using numerical and geometrical…

Investigates the ways children used their schemes of fractions to interpret fraction situations. Records clinical interviews with nine children (grades three through five). Reports that children do not establish relationships among different schemes when giving meaning to a given situation. (Author/YP)