Educational Studies in Mathematics

The main thesis of this paper is that the construction and coordination of abstract units is central to mathematical activity in both numerical and geometric settings. Data were gathered from students in grade three through six, with four students being observed over a three year period. A consistent parallel was found in the sophistication of the types of units constructed in a geometric setting (tiling the plane) with their numeric activity. The tiling activity of the students was analyzed for evidence of the construction and coordination of units. Some students constructed rather sophisticated abstract composite units to facilitate their tiling with a particular shape while others had difficulty making a covering. Students who constructed abstract composite units in tiling did so also in adding and subtracting whole numbers. Evidence of unitizing and coordinating the units constructed was associated with advances in mathematical thinking. Unitizing seems to be a fundamental mental operation in coming to act mathematically. As mathematics activities are planned, it is important to provide opportunities for students to construct abstract composite units in both geometric and numeric settings. Emphasis on prescribed procedures such as subtracting with two-digit numerals may inhibit this construction process.

Models-and-modeling perspective (MMP) is a problem-solving and learning perspective in mathematics education. Although modeling processes have been addressed widely in the international discussion on mathematical modeling, a homogeneous understanding has not been established yet. Hence, the field needs studies addressing the epistemological grounds of model development. Therefore, in this study, I aimed to scrutinize the latent aspects of modeling in MMP-based research. Based on the analysis of 143 chapter-sized documents, I aimed to articulate the characteristics of the modeling process and the models. The thematic analysis that was incorporated in document analysis revealed four latent aspects, namely, modeling (1) is a subjective and also inter- and intra-subjective process, (2) encompasses both structural and systematic properties, (3) produces models that are both implicit and explicit to a certain degree, and (4) culminates in an incomplete — but not inadequate — model. These aspects of model development led me to reconceptualize what a model conveys in MMP-based research and articulate the potential and limits of the models. This is particularly important for teachers and researchers in understanding what models indicate in relation to students’ ways of thinking, and such a systematic and analytical investigation can contribute to the scholarly conversation about modeling in the field of mathematics education.

Research shows that students, and sometimes teachers, have trouble with fractions, especially conceiving of fractions as numbers that extend the whole number system. This paper explores how fractions are addressed in undergraduate mathematics courses for prospective elementary teachers (PSTs). In particular, we explore how, and whether, the instructors of these courses address fractions as an extension of the whole number system and fractions as numbers in their classrooms. Using a framework consisting of four approaches to the development of fractions found in history, we analyze fraction lessons videotaped in six mathematics classes for PSTs. Historically, the first two approaches—part–whole and measurement—focus on fractions as parts of wholes rather than numbers, and the last two approaches—division and set theory—formalize fractions as numbers. Our results show that the instructors only implicitly addressed fraction-as-number and the extension of fractions from whole numbers, although most of them mentioned or emphasized these aspects of fractions during interviews.

Based on insights into the nature of vocational mathematical knowledge, we designed a computer tool with which students in laboratory schools at senior secondary vocational school level could develop a better proficiency in the proportional reasoning involved in dilution. We did so because we had identified computations of concentrations of chemical substances after dilution as a problematic area in the vocational education of laboratory technicians. Pre- and post-test results indeed show that 47 students aged 16–23 significantly improved their proportional reasoning in this domain with brief instruction time (50–90 min). Effect sizes were mostly large. The approach of using a visual tool that foregrounds mathematical aspects of laboratory work thus illustrates how vocational mathematical knowledge can be developed effectively and efficiently.

Problems teaching probability in Tonga (in the South Pacific) led to the question on how language and culture affect the understanding of probability and uncertainty. The research uses a discursive approach to identify the endorsed narratives which underlie Tongans’ reasoning in situations of uncertainty. I aim to justify the claim that the Tongan language and the Tongan way of life interact to make the concept of uncertainty very different from that found in western countries and the concept of probability almost redundant in Tongan day-to-day discourse. There are very few cross-cultural studies concerning the ways in which probability and uncertainty are understood in different cultural contexts, and this article aims to make a small contribution to filling this gap.

This paper describes the problem-solving behaviors of 12 mathematicians as they completed four mathematical tasks. The emergent problem-solving framework draws on the large body of research, as grounded by and modified in response to our close observations of these mathematicians. The resulting Multidimensional Problem-Solving Framework has four phases: orientation, planning, executing, and checking. Embedded in the framework are two cycles, each of which includes at least three of the four phases. The framework also characterizes various problem-solving attributes (resources, affect, heuristics, and monitoring) and describes their roles and significance during each of the problem-solving phases. The framework’s sub-cycle of conjecture, test, and evaluate (accept/reject) became evident to us as we observed the mathematicians and listened to their running verbal descriptions of how they were imagining a solution, playing out that solution in their minds, and evaluating the validity of the imagined approach. The effectiveness of the mathematicians in making intelligent decisions that led down productive paths appeared to stem from their ability to draw on a large reservoir of well-connected knowledge, heuristics, and facts, as well as their ability to manage their emotional responses. The mathematicians’ well-connected conceptual knowledge, in particular, appeared to be an essential attribute for effective decision making and execution throughout the problem-solving process.

Of five structural variables that in previous studies accounted for a significant amount of the observed variance in the error rate in verbal arithmetic problems, the length variable (number of words in the problem statement) apparently was more important in the upper grades than in the lower grades. In this study three forms of a verbal problem set in which the number of words in the problem statements were systematically varied were administered to classes of students in Grades 4–9. Using regression analysis, the investigator found that although the length variable did not measure a significant amount of variance between two of the forms, with the third it did.

The purpose of this study was to explore how university students construct the relationship between the concepts of differential and derivative with the integration of GeoGebra and the ACODESA method. The participants in this study were 33 pre-service mathematics teachers. An open-ended questionnaire, knowledge test, tasks, and participants’ dynamic constructions were used as data collection tools. The analysis of participants’ products was based on descriptive analysis and Toulmin’s model. As a result of the analysis, it was found that the participants constructed the relationship between the concepts of differential and derivative as well as the concepts of tangent and slope by using them within the geometric framework. Due to the integration of GeoGebra and the ACODESA method, the participants explored the relationships among true change, estimated change, and error by using the geometric interpretation of the concept of differential. It was found that with this method, they deduced that Δx and dx were two different symbols for the same variable.