Teacher and students’ joint production of a reversible fraction conception

Within a constructivist perspective, I conducted a teaching experiment with two fourth graders to study how a teacher and students can jointly produce the reversible fraction conception. Ongoing and retrospective analysis of the data revealed the non-trivial process by which students can abstract multiplicative reasoning about fractions. The study articulates a conception in a developmental sequence of iteration-based fraction conceptions and the teacher’s role in fostering such a conception in students.     

Parametric excitation of waves on a free boundary of a horizontal fluid layer

We consider the dynamical stability of horizontal fluid layer, performing harmonic oscillations in vertical direction. The continued fractions approach allowed us to avoid the conventional restriction to the case of small viscosity and almost-resonant frequencies. Our numerical results cover a wide range of the parameters (viscosity, amplitude and frequency of the oscillation, and depth of the layer). To cite this article: V.I. Yudovich et al., C. R. Mecanique 332 (2004).     

A task that elicits reasoning: A dual analysis

This paper reports on the forms of reasoning elicited as fourth grade students in a suburban district and sixth grade students in an urban district worked on similar tasks involving reasoning with the use of Cuisenaire rods. Analysis of the two data sets shows similarities in the reasoning used by both groups of students on specific tasks, and the tendency of a particular task to elicit numerous forms of reasoning in both groups of students. Attributes of that task and ways that those attributes can be replicated in other domains may have implications in the teaching of early reasoning.     

Fractional commensurate, composition, and adding schemes: Learning trajectories of Jason and Laura: Grade 5

A case study of two 5th-Grade children, Jason and Laura, is presented who participated in the teaching experiment, Children’s Construction of the Rational Numbers of Arithmetic. The case study begins on the 29th of November of their 5th-Grade in school and ends on the 5th of April of the same school year. Two basic problems were of interest in the case study. The first was to provide an analysis of the concepts and operations that are involved in the construction of three fractional schemes: a commensurate fractional scheme, a fractional composition scheme, and a fractional adding scheme. The second was to provide an analysis of the contribution of interactive mathematical activity in the construction of these schemes. The phrase, “commensurate factional scheme” refers to the concepts and operations that are involved in transforming a given fraction into another fraction that are both measures of an identical quantity. Likewise, “fractional composition scheme” refers to the concepts and operations that are involved in finding how much, say, 1/3 of 1/4 of a quantity is of the whole quantity, and “fractional adding scheme” refers to the concepts and operations involved in finding how much, say, 1/3 of a quantity joined to 1/4 of a quantity is of the whole quantity. Critical protocols were abstracted from the teaching episodes with the two children that illustrate what is meant by the schemes, changes in the children’s concepts and operations, and the interactive mathematical activity that was involved. The body of the case study consists of an on-going analysis of the children’s interactive mathematical activity and changes in that activity. The last section of the case study consists of an analysis of the constitutive aspects of the children’s constructive activity, including the role of social interaction and nonverbal interactions of the children with each other and with the computer software we used in teaching the children.     

Using Productive Disposition to Differentiate Between Students' Level of Precision When Critiquing a Peer's Work

This study examined the productive disposition of pre‐algebra students who demonstrated similar knowledge of the focal content but varied in other academic behaviors expected in the Common Core State Standards for Mathematics (CCSSM). Specifically, the study considered students' attention to precision when critiquing a peer's work. The comprehensive definition of productive disposition used included task values (interest, utility), an ability belief (efficacy), three personal achievement goals, and negative emotions. As hypothesized, the 61 students who provided a more precise critique reported higher productive disposition (in particular, significantly higher mastery‐approach personal achievement goals and less frequent negative emotions) than the 79 students who provided a basic critique. These findings illustrate how productive disposition can inform assessments of mathematical competence within the CCSSM recently implemented across the United States.     

Children’s sense-making of division of fractions

The purpose of this paper is to share some results from a year-long teaching experiment in which fourth grade students were given the opportunity to understand fraction concepts prior to the introduction of algorithmic instruction. In particular, this paper focuses on the means by which children solved problems involving division of fractions. Children were given a task-based activity specifically designed to promote solutions that would be grounded in conceptual understanding. Three distinct solution methods, all related to counting, emerged. When the activity was replicated as part of regular classroom practice seven and a half years later, the same solution methods were observed.     

Synthesis of polynitroalcohols by nucleophilic addition of butadiene-styrene nitrooligomer to acetaldehyde

Two types of secondary polynitroalcohols (PNA) were prepared by A_N-reaction of butadiene-styrene nitrooligomer (BSNO) and acetaldehyde. The corresponding yields were 58-64% and 25-33%. The reaction was conducted in water-ethanol solution in the presence of alkaline base as catalyst at temperatures from 40 to 60 °C, for 3.5-4 h. Introduction of hydroxyethyl groups by A_N-reaction of BSNO to acetaldehyde increased the polarity and thermal stability of PNA as compared to BSNO. By using IR spectroscopy and liquid absorption chromatography on silica gel, PNA were found to be polyfunctional compounds, that contained structurally and functionally heterogeneous fractions. The quantitative functional composition of the first PNA type as well as their main fractions were determined by evaluating the relative content of nitro-, carbonyl and hydroxyl groups in the products. PNA are considered to be starting materials for the preparation of polynitrourethanes and salts of N-containing sulphonic acids.     

Obstacles and challenges in preservice teachers’ explorations with fractions: A view from a small-scale intervention study

In this study, we implemented one-on-one fractions instruction to eight preservice teachers. The intervention, which was based on the principle of Progressive Formalization (Freudenthal, 1983), was centered on problem solving and on progressively formalizing the participants’ intuitive knowledge of fractions. The objectives of the study were to examine the potential effects of the intervention and to uncover specific difficulties experienced by the preservice teachers during instruction. Results revealed improvement on one measure of conceptual knowledge, but not on a transfer task, which required the teachers to generate word problems for number sentences involving fractions. In addition, the qualitative analysis of the videotaped instructional sessions revealed a number of cognitive obstacles encountered by the participants as they attempted to construct meaningful solutions and represent those solutions symbolically. Based on the findings, specific suggestions for modifying the intervention are provided for mathematics teacher educators.