Reconceptualizing Early Mathematics Learning : The Fundamental Role of Pattern and Structure

The Pattern and Structure Mathematics Awareness Program (PASMAP) was developed concurrently with the studies of AMPS and the development of the Pattern and Structure Assessment (PASA) interview. We summarize some early classroom-based teaching studies and describe the PASMAP that resulted. A large-scale two-year longitudinal study, Reconceptualizing Early Mathematics Learning (REML) resulted. We provide an overview of the REML study and discuss the consequences for our view of early mathematics learning. A purposive sample of four large primary schools, two in Sydney and two in Brisbane, representing 316 students from diverse socio-economic and cultural contexts, participated in an evaluation of the PASMAP intervention throughout the 2009 school year and a follow-up assessment in 2010. Two different mathematics programs were implemented: in each school, two Kindergarten teachers implemented the PASMAP and another two implemented their regular program. The study shows that both groups of students made substantial gains on the ‘I Can Do Maths’ standardized assessment and the PASA interview, but highly significant differences were found on the latter with PASMAP students outperforming the regular group on PASA scores. Qualitative analysis of students’ responses for structural development showed increased levels for the PASMAP students. Implications for pedagogy and curriculum are discussed.

Exploring the fundamental theorem of arithmetic in excel 2007

This paper discusses how fundamentals of number theory, such as unique prime factorization and greatest common divisor can be made accessible to secondary school students through spreadsheets. In addition, the three basic multiplicative functions of number theory are defined and illustrated through a spreadsheet environment. Primes are defined simply as those natural numbers with just two divisors. One focus of the paper is to show the ease with which spreadsheets can be used to introduce students to some basics of elementary number theory. Complete instructions are given to build a spreadsheet to enable the user to input a positive integer, either with a slider or manually, and see the prime decomposition. The spreadsheet environment allows students to observe patterns, gain structural insight, form and test conjectures, and solve problems in elementary number theory.

Exploration of the fundamental equilibrium behaviour of calcium exchange with weak acid cation resins

This study evaluated the complexity of calcium ion exchange with sodium exchanged weak acid cation resin (DOW MAC-3). Exchange equilibria recorded for a range of different solution normalities revealed profiles which were represented by conventional “L” or “H” type isotherms at low values of equilibrium concentration (Ce) of calcium ions, plus a superimposed region of increasing calcium uptake was observed at high Ce values. The loading of calcium ions was determined to be ca. 53.5 to 58.7 g/kg of resin when modelling only the sorption curve created at low Ce values,which exhibited a well-defined plateau. The calculated calcium ion loading capacity for DOWMAC-3 resin appeared to correlate with the manufacturer's recommendation. The phenomenon of super equivalent ion exchange (SEIX) was observed when the “driving force” for the exchange process was increased in excess of 2.25 mmol calcium ions per gram of resin in the starting solution. This latter event was explained in terms of displacement of sodium ions from sodium hydroxide solution which remained in the resin bead following the initial conversion of the as supplied “H+” exchanged resin sites to the “Na+” version required for softening studies. Evidence for hydrolysis of a small fraction of the sites on the sodium exchanged resin surface was noted. The importance of carefully choosing experimental parameters was discussed especially in relation to application of the Langmuir–Vageler expression. This latter model which compared the ratio of the initial calcium ion concentration in solution to resin mass, versus final equilibrium loading of the calcium ions on the resin; was discovered to be an excellent means of identifying the progress of the calcium–sodium ion exchange process. Moreover, the Langmuir–Vageler model facilitated standardization of various calcium–sodium ion exchange experiments which allowed systematic experimental design.

Some remarks on the symplectic pairing on the moduli space of representations of the fundamental group of surfaces

This work grew out of an attempt to understand a conjectural remark made by Professor Kyoji Saito to the author about a possible link between the Fox-calculus description of the symplectic structure on the moduli space of representations of the fundamental group of surfaces into a Lie group and pairs of mutually dual sets of generators of the fundamental group. In fact in his paper [3] , Prof. Kyoji Saito gives an explicit description of the system of dual generators of the fundamental group.

Vibrational Spectra and Assignments for the Fundamental Vibrations of Imidazolidine-2-thione and -2-selenone

Infrared spectra of imidazolidine-2-thione (N,N?-ethylenethiourea, ETU) and its N,N?-deuterated (ETU-d2) and S-methylthiouronium iodides have been recorded from 4000 to 30 cm?1. Normal coordinate analyses of ETU and ETU-d2 have been made for all the fundamental frequencies, employing a Urey-Bradley potential function supplemented with valence type constants for the out of plane modes of the planar skeleton. Raman frequencies of ETU from literature have been utilised. The results of the vibrational analyses are discussed in relation to the group frequencies in structurally related molecules and frequency shifts on deuteration and S-methylation. The normal coordinate treatment is also performed for the planar vibrations of imidazolidine-2-selenone (N,N?-ethyleneselenourea, ESU) to propose assignments for ESU and so also to support the assignments of ETU.

Flat connections, geometric invariants and the symplectic nature of the fundamental group of surfaces

In this paper we associate a new geometric invariant to the space of fiat connections on a G (= SU(2))-bundle on a compact Riemann surface M and relate it tcr the symplectic structure on the space Hom(pi(1)(M), G)/G consisting of representations of the fundamental group pi(1)(M) Of M into G module the conjugate action of G on representations.