STUDENTS\u27 DEVELOPMENT IN PROOF: A LONGITUDINAL STUDY

Despite importance of teaching proof in any undergraduate mathematics program, many students have difficulties with proof (Dreyfus, 1999; Harel & Sowder, 2003; Selden & Selden, 2003; Weber, 2004). In this qualitative case study, nine undergraduate students were each interviewed once every two weeks over the course of an academic year. During each interview, the students were asked to complete, evaluate or discuss mathematical proofs. The results of these interviews were then analyzed using two different frameworks. The first focused on proof type, which refers to what kind of proof is created and how it came about. The second framework addressed identifying each student\u27s proof scheme, which constitutes ascertaining and persuading for that person (Harel & Sowder, 1998). Using these structures as a guide, the question I sought to answer is: What, if any, identifiable paths do students go through while learning to prove? Unfortunately, the data from this study failed to demonstrate any identifiable path that was common to all participants. In fact, only a single student made clear progress as judged by the criteria laid out at the beginning of this study. Specifically, the way she attempted proofs changed which was reflected in a greater tendency to use a particular proof type as time passed: semantic. Of the other students, six entered the study with a fairly mature view of proof that remained unchanged and thus had little progress to make relative to the frameworks used in the study. These students were also generally successful with the proofs they attempted and were more likely to use semantic proofs. The remaining two students were generally less successful and used semantic proofs rarely. This seems to imply that as students become more comfortable with proof, they become inclined toward the semantic proof type and this coincides with becoming more successful with proof in general

The history of mathematics as a pedagogical tool: Teaching the integral of the secant via Mercator’s projection

This article explores the use of the history of mathematics as a pedagogical tool for the teaching and learning of mathematics. In particular, we draw on the mathematically pedigreed but misunderstood development of the Mercator projection and its connection to the integral of the secant function. We discuss the merits and the possible pitfalls of this approach based on a teaching module with undergraduate students. The appendices contain activities that can be implemented as an enrichment activity in a Calculus course

Exploring gender and forest, tree and agroforestry value chains: Evidence and lessons from a systematic review

This systematic review of literature on gender and value chains of forest, tree and agroforestry (FTA) products examined gender differences and inequalities in FTA value chains, factors that influence these differences, and interventions to foster greater gender equity. There is limited information available on gender in FTA value chains, and a strong bias in the literature towards African countries. Gender differences in participation mainly owe to social-cultural factors, including gendered access rights, and to the physical nature of value chain activities. Cultural norms and overlapping customary and formal regulatory arrangements often position men in more favorable positions than women in FTA value chains. Interventions in FTA value chains largely focus on enhancing women’s participation and benefits, but rarely consider the relationships between men and women. Raising awareness of gender biases, relations and potential trade-offs among those involved in value chains and those supporting inclusive value chain development should accompany technological innovations, and should occur across multiple stages of the value chain

Feshbach spectroscopy and scattering properties of ultracold Li+Na mixtures

We have observed 26 interspecies Feshbach resonances at fields up to 2050 G in ultracold $^6$Li+$^{23}$Na mixtures for different spin-state combinations. Applying the asymptotic bound-state model to assign the resonances, we have found that most resonances have d-wave character. This analysis serves as guidance for a coupled-channel calculation, which uses modified interaction potentials to describe the positions of the Feshbach resonances well within the experimental uncertainty and to calculate their widths. The scattering length derived from the improved interaction potentials is experimentally confirmed and deviates from previously reported values in sign and magnitude. We give prospects for $^7$Li+$^{23}$Na and predict broad Feshbach resonances suitable for tuning.Comment: 8 pages, 4 figures, version as published in PR

Char-forming behavior of nanofibrillated cellulose treated with \u3ci\u3eglycidyl phenyl\u3c/i\u3e POSS

Cellulose-reinforced composites have received much attention due to their structural reinforcing, light weight, biodegradable, non-toxic, low cost and recyclable characteristics. However, the tendency for cellulose to aggregate and its poor dispersion in many polymers, such as polystyrene, continues to be one of the most challenging roadblocks to large scale production and use of cellulose-polymer composites. In this study, nanofibrillated cellulose (NFC) is modified using GlycidylPhenyl-POSS (a polyhedral oligomeric silsesquioxane). The product yield, morphology, and crystallinity are characterized using a variety of spectroscopy and microscopy techniques. Thermal analyses are performed using thermal gravimetric analysis and pyrolysis combustion flow calorimetry

Three dimensional ink-jet printing of biomaterials using ionic liquids and co-solvents

1-Ethyl-3-methylimidazolium acetate ([C2C1Im][OAc]) and 1-butyl-3-methylimidazolium acetate ([C4C1Im][OAc]) have been used as solvents for the dissolution and ink-jet printing of cellulose from 1.0 to 4.8 wt%, mixed with the co-solvents 1-butanol and DMSO. 1-Butanol and DMSO were used as rheological modifiers to ensure consistent printing, with DMSO in the range of 41–47 wt% producing samples within the printable range of a DIMATIX print-head used (printability parameter < 10) at 55 °C, whilst maintaining cellulose solubility. Regeneration of cellulose from printed samples using water was demonstrated, with the resulting structural changes to the cellulose sample assessed by scanning electron microscopy (SEM) and white light interferometry (WLI). These results indicate the potential of biorenewable materials to be used in the 3D additive manufacture process to generate single-component and composite materials