Reunitising hundredths: prototypic and nonprototypic representations

This paper reports on a study in which 29 Year 6 students (selected from the top 30% of 176 Year 6 students) were individually interviewed to explore their ability to reunitise hundredths as tenths (Behr, Harel, Post & Lesh, 1992) when represented by prototypic (PRO) and nonprototypic (NPRO) models. The results showed that 55.2% of the students were able to unitise both models and that reunitising was more successful with the PRO model. The interviews revealed that many of these students had incomplete, fragmented or non-existent structural knowledge of the reunitising process and often relied on syntactic clues to complete the tasks. The implication for teaching is that instruction should not be limited to PRO representations of the part/whole notion of fraction and that the basic structures (equal parts, link between name and number of equal parts) of the part/whole notion needs to be revisited often.

Place value, multiplicativity, reunitizing and effective classroom teaching of decimal numeration

Student understanding of decimal number is poor (e.g., Baturo, 1998; Behr, Harel, Post & Lesh, 1992). This paper reports on a study which set out to determine the cognitive complexities inherent in decimal-number numeration and what teaching experiences need to be provided in order to facilitate an understanding of decimal-number numeration. The study gave rise to a theoretical model which incorporated three levels of knowledge. Interview tasks were developed from the model to probe 45 students’ understanding of these levels, and intervention episodes undertaken to help students construct the baseline knowledge of position and order (Level 1 knowledge) and an understanding of multiplicative structure (Level 3 knowledge). This paper describes the two interventions and reports on the results which suggest that helping students construct appropriate mental models is an efficient and effective teaching strategy.

Telomere length in circulating leukocytes is associated with lung function and disease

Several clinical studies suggest the involvement of premature ageing processes in chronic obstructive pulmonary disease (COPD). Using an epidemiological approach, we studied whether accelerated ageing indicated by telomere length, a marker of biological age, is associated with COPD and asthma, and whether intrinsic age-related processes contribute to the interindividual variability of lung function. Our meta-analysis of 14 studies included 934 COPD cases with 15 846 controls defined according to the Global Lungs Initiative (GLI) criteria (or 1189 COPD cases according to the Global Initiative for Chronic Obstructive Lung Disease (GOLD) criteria), 2834 asthma cases with 28 195 controls, and spirometric parameters (forced expiratory volume in 1 s (FEV1), forced vital capacity (FVC) and FEV1/FVC) of 12 595 individuals. Associations with telomere length were tested by linear regression, adjusting for age, sex and smoking status. We observed negative associations between telomere length and asthma (β= −0.0452, p=0.024) as well as COPD (β= −0.0982, p=0.001), with associations being stronger and more significant when using GLI criteria than those of GOLD. In both diseases, effects were stronger in females than males. The investigation of spirometric indices showed positive associations between telomere length and FEV1 (p=1.07×10−7), FVC (p=2.07×10−5), and FEV1/FVC (p=5.27×10−3). The effect was somewhat weaker in apparently healthy subjects than in COPD or asthma patients. Our results provide indirect evidence for the hypothesis that cellular senescence may contribute to the pathogenesis of COPD and asthma, and that lung function may reflect biological ageing primarily due to intrinsic processes, which are likely to be aggravated in lung diseases.