Looking Back in Mathematical Modeling: Classroom Observations and Instructional Strategies

Theorists often characterize modeling as a cyclic problem-solving process. One builds the model, assesses its validity with regard to the underlying problem situation, and revises accordingly. The process halts when, in the opinion of the modeler, the model generates a valid solution to the underlying problem. Recent research suggests that students, like experts, employ cyclic modeling processes. Extensive observations of university and high school students’ modeling efforts, however, suggest the use of linear rather than cyclic modeling strategies. That is, novice modelers often fail to look back or revise their initial models. This paper offers empirical evidence on behalf of the linear modeling theory and identifies five factors that promote the use of linear modeling strategies: students’ conceptions of models and the modeling process, the perceived objectives of the modeling activity, constraints on time and resources, statistical misconceptions, and an overall lack of interest. The paper concludes with several promising instructional strategies (strategies that address students’ difficulties and promote reflective modeling behavior), as well as suggestions for future research.     

Comparison between biokinetics of inhaled plutonium nitrate and gadolinium oxide in humans and animals

Due to the paucity of human data after inhalation of different chemical forms of radionuclides, the implications for human exposure are often based on animal studies. This paper describes biokinetic studies of plutonium nitrate and gadolinium oxide in human volunteers and rats. The results, together with information from other studies with radionuclides, suggests that animal studies can be used with advantage for assessing the biokinetic behavior in humans, and for providing guidance on the assessment of intake and optimal monitoring regimens.     

A comparison of double-focusing sector field ICP-MS, ICP-OES and octopole collision cell ICP-MS for the high-accuracy determination of calcium in human serum

Human serum is routinely measured for total calcium content in clinical studies. A definitive high-accuracy and low-uncertainty method is required for reference measurements to underpin medical diagnoses. This study presents a novel octopole collision cell ICP-MS, high-accuracy, methodology and comparison of that technique with double-focusing sector field ICP-MS and an ICP-OES method. Double-matched isotope dilution mass spectrometry (IDMS) was employed for ICP-MS techniques and an exact matching bracketing technique using scandium as an internal standard was used for ICP-OES analysis. Medium resolution mode was utilised for double-focusing sector field ICP-MS analysis to resolve the dominant interferences on the ⁴⁴Ca/⁴²Ca isotope pair. Hydrogen reaction gas was employed to chemically resolve a number of polyatomic interferences predominantly through charge transfer reactions in the octopole collision cell. Comparison data presented for NIST CRM 909b human serum analysis from all three techniques demonstrates highest accuracy (99.6%) and lowest uncertainty (1.1%) for octopole collision cell ICP-MS. Data from ICP-OES using a non-IDMS technique produces comparably accurate data and low-uncertainties. The much higher total expanded uncertainties for double-focusing sector field ICP-MS compared with octopole collision cell data are explained by lower precision on the measurement of the ⁴⁴Ca/⁴²Ca isotope ratio. Data for octopole collision cell ICP-MS submitted for an international blind trial comparison (CCQM K-14) demonstrated excellent agreement with the mean of all participants with a low expanded uncertainty.     

Preparation of polymer colloids by chemical reactions in aerosols. I. Poly(p-tertiarybutylstyrene)

A procedure was developed by which polymer colloids can be prepared by polymerizing aerosol droplets of monomers with an initiator in the vapor state. This work describes the formation of poly(p-tertiarybutylstyrene) particles by exposing the corresponding monomer droplets to trifluoromethanesulfonic acid. It was found that the monomer-to-initiator mass ratio is the critical factor in determining the uniformity and morphology of the particles. Under ideal conditions spheres of narrow size distribution are obtained. Otherwise porous particles or soft particles with whiskers may develop. The aerosol was prepared in a falling film generator, but other procedures for producing monomer droplets may be used.     

The structure and crystallization of thin water films on Pt(111)

When water is adsorbed on Pt(111) above 135 K several different ice structures crystallize, depending on the thickness of the ice layer. At low coverage water forms extended islands of ice with a (square root(37) x square root(37))R25(o) unit cell, which compresses as the monolayer saturates to form a (square root(39) x square root(39))R16(o) structure. The square root(39) low-energy electron diffraction (LEED) pattern becomes more intense as the second layer grows, remaining bright for films up of 10-15 layers and then fading and disappearing for films more than ca. 40 layers thick. The ice multilayer consists of an ordered square root(39) wetting layer, on which ice grows as a crystalline film which progressively loses its registry to the wetting layer. Ice films more than ca. 50 layers thick develop a hexagonal LEED pattern, the entire film and wetting layer reorienting to form an incommensurate bulk ice. These changes are reflected in the vibrational spectra which show changes in line shape and intensity associated with the different ice structures. Thin amorphous solid water films crystallize to form the same phases observed during growth, implying that these structures are thermodynamically stable and not kinetic phases formed during growth. The change from a square root(39) registry to incommensurate bulk ice at ca. 50 layers is associated with a change in crystallization kinetics from nucleation at the Pt(111) interface in thin films to nucleation of incommensurate bulk ice in amorphous solid water films more than 50 layers thick.     

The scope of catalytic enantioselective tandem carbonyl ylide formation-intramolecular [3 + 2] cycloadditions

Catalytic enantioselective tandem carbonyl ylide formation-intramolecular 1,3-dipolar cycloaddition reactions of 2-diazo-3,6-diketoesters show promising scope in terms of asymmetric induction as the tethered alkene/alkyne dipolarophile component is varied. Cycloadditions were found to occur in moderate to very good yields, with a difference in ee exhibited by the electronically different 2-diazo-3,6-diketoesters 1, 25 and 33, 34. Values for ee of up to 90% for alkene dipolarophiles and up to 86% for alkyne dipolarophiles were obtained.     

Vitamin B12 as an allosteric cofactor; dual fluorescence, hysteresis, oscillations and the selection of corrin over porphyrin

Studies of the emission spectra of four Co(III) cobinamides (diaquo-, aquohydroxo-, dihydroxo- and dicyano-) show (1) that the excited states corresponding to the alphabeta and epsilon absorption bands behave like the S(1) and S(2) levels in the non-alternant hydrocarbon azulene (with emission from S(2)> S(1) in violation of Kasha's rule) and (2) that the excited states include a TICT (twisted intramolecular charge transfer) mechanism, as in the simpler cyanines, but where the TICT state gives rise to dual fluorescence instead of cis-trans isomerisation. Combined with the previously reported dual fluorescence from the S(1) level in synthetic metal corrinoids and in the naturally-occurring metal-free corrin, this provides evidence that the existence of an additional (metastable) ground state with a significantly different vibronic splitting and nuclear configuration is an intrinsic property of the basic corrin ligand (irrespective of the nature of the side-chains and the metal ion or even the absence of a metal) which distinguishes it from porphyrin. The occurrence of hysteresis (and its associated oscillations) in redox reactions of the cobinamides involving both the Co(III/II) and Co(II/I) couples indicates that the corrin ligand also has an intrinsic ability to exist in different conformations or "allosteric" forms with differing redox potential, which further distinguishes it from the porphyrin ligand. Possible links between the existence of an additional metastable ground state and of allosteric changes and the likely reasons for the selection of corrin over a porphyrin for the vitamin B(12)-dependent enzymes are discussed.     

Deprotonation-electrophile trapping of terminal epoxides

Organolithium-induced deprotonation of terminal epoxides in the presence of appropriate diamine ligands allows trapping with a range of electrophiles, yielding functionalised di- and tri-substituted epoxides in good yields and with control of stereochemistry at the epoxide.