PROSPECTIVE TEACHERS UNDERSTANDING FRACTION DIVISION USING RECTANGLE REPRESENTATION

Fraction division is one of the most difficult subjects in elementary school. Not only elementary students but many prospective teachers don’t understand the fraction division concept yet—most of them using a keep-change-flip algorithm to solve fraction division problems. A study using rectangle representation was conducted by us to prospective teachers. This study aims to see whether this rectangle representation will make prospective teachers understand or not. To do so, we made a mixed-method study with 80 prospective teachers as participants. The results show that 53,75% of prospective teachers use the keep-change-flip algorithm without understanding the concept of fraction division, and just 15% of prospective teachers understand fraction division. We assume that most prospective teachers still can’t imagine how fraction division works in a real-life context. They remember what they used to do to finish the fraction division problem that their teacher has introduced in primary school. Based on the results, we conclude that the study with rectangle representation still needs an improvement, whether the teacher’s explanation or the rectangle media.

PARTITIVE FRACTION DIVISION: REVEALING AND PROMOTING PRIMARY STUDENTS’ UNDERSTANDING

Students show deficient understanding on fraction division and supporting that understanding remains a challenge for mathematics educators. This article aims to describe primary students’ understanding of partitive fraction division (PFD) and explore ways to support their understanding through the use of sequenced fractions and context-related graphical representations. In a design-research study, forty-four primary students were involved in three cycles of teaching experiments. Students’ works, transcript of recorded classroom discussion, and field notes were retrospectively analyzed to examine the hypothetical learning trajectories. There are three main findings drawn from the teaching experiments. Firstly, context of the tasks, the context-related graphical representations, and the sequence of fractions used do support students’ understanding of PFD. Secondly, the understanding of non-unit rate problems did not support the students’ understanding of unit rate problems. Lastly, the students were incapable of determining symbolic representations from unit rate problems and linking the problems to fraction division problems. The last two results imply to rethink unit rate as part of a partitive division with fractions. Drawing upon the findings, four alternative ways are offered to support students’ understanding of PFD, i.e., the lesson could be starting from partitive whole number division to develop the notion of fair-sharing, strengthening the concept of unit in fraction and partitioning, choosing specific contexts with more relation to the graphical representations, and sequencing the fractions used, from a simple to advanced form.

Mathematical Modeling of Diesel Fuel Hydrotreating

The principles of mathematical modeling of Hydrotreating diesel fuel in the representation of raw materials in the form of a set of narrow fractions in which the total content of various organosulphuric components is considered as a pseudocomponent are considered. As a result of the analysis of schemes of reactor blocks of Hydrotreating plants the most perspective two-reactor systems characterized by separate desulfurization of streams are revealed. It is shown that the preliminary fractionation of Hydrotreating raw materials into light and heavy fractions with the choice of the optimal boundary of fraction division can minimize the loading of the catalyst into the reactor unit. An algorithm for solving this problem, including experimental and computational fragments, is presented.