Analysis in univalent type theory

Some constructive real analysis is developed in univalent type theory (UTT). We develop various types of real numbers, and prove several equivalences between those types. We then study computation with real numbers. It is well known how to compute with real numbers in intensional formalizations of mathematics, where equality of real numbers is specified by an imposed equivalence relation on representations such as Cauchy sequences. However, because in UTT equality of real numbers is captured directly by identity types, we are prevented from making any nontrivial discrete observations of arbitrary real numbers. For instance, there is no function which converts real numbers to decimal expansions, as this would not be continuous. To avoid breaking extensionality, we thus restrict our attention to real numbers that have been equipped with a simple structure called a \emph{locator}. In order to compute, we modify existing constructions in analysis to work with locators, including Riemann integrals, intermediate value theorems and differential equations. Hence many of the proofs involving locators look familiar, showing that the use of locators is not a conceptual burden. We discuss the possibility of implementing the work in proof assistants and present a Haskell prototype

Comparing and Contrasting Knowledge on Mules and Hinnies as a Tool to Comprehend Their Behavior and Improve Their Welfare.

Mules and hinnies are the hybrids between donkeys (Equus asinus) and horses (Equus caballus). For centuries, mankind has used them for agrarian purposes, the military, or recreation. Contrasting literature with behavioral observations, we seek a better behavioral understanding andthus comprehensive solutions for their welfare enhancement. Over the past 6 years, we have assessed physical and behavioral welfare in over 900 mules by surveying owners from Egypt, Peru, Portugal, Spain, Mexico, and the U.S. These mules participated in shows, brick kiln work, cart-pulling, packing, tourism, and cattle herding. Observations are discussed alongside facts from the literature. Unfortunately, their behavior has been misunderstood by many, and harsh treatment and equipment has been used to control them. Few studies have attempted to define or use learning theory to understand how and why mules and hinnies behave as they do. Hence, understanding their health considerations, natural behavior, and training theory is crucial for those who work with them.Solutions to welfare improvement partially lie in an individual's ability to handle mules and hinnies from birth, and to proceed slowly through training. Conclusively, this review sets forth a clearer understanding of these hybrids' behaviors and promotes positive handling, allowing their access to more routine healthcare and ultimately, a higher welfare standard

The P scales: level descriptors P1 to P8

"These are the level descriptors for pupils working below level 1 of the national curriculum... To support teachers in making judgements about pupils’ attainment below level 1 of the national curriculum" - Back cover. This booklet is part of the DVD and print booklet package 'Using the P scales: assessing, moderating and reporting pupil attainment at levels P1 to P8' (QCA/09/4060

Exploring behaviour in the online environment: student perceptions of information literacy

The aim of this paper is to show how information literacy can be conceptualised as a key learning process related to discipline and academic maturity, rather than as a generic skill. Results of a small-scale study including questionnaires and observation of student behaviour are reported and analysed in relation to Bruces 'seven faces of information literacy' framework. The findings illustrate that information literacy is a highly situated practice that remains undeveloped through mandatory schooling. Some methodological issues are considered in relation to researching information literacy, including the limits of the Bruce model as a framework for analysis. We also show how decontextualised courses can foreground and privilege certain behaviours that are beneficial but that developing higher-level information literate attitudes is likely to be an iterative and contextualised process

'Sounds of Intent' : Mapping musical behaviour and development in children and young people with complex needs

This article reports on the first year of an Esmae Fairbairn Foundation-funded research project into the design and evaluation of an original 'framework' for mapping the behaviour and development in, and through, music for children with complex needs, specifically those with profound and multiple learning difficulties (PMLD). An initial four-month design and pilot phase critiqued and evaluated a framework that was grounded in video-based iterative analyses of individual case studies that had been collected during the previous two years. The piloting phase was followed by a sustained period of classroom-based music lesson observation in five special schools over a period of seven months. A total of 630 observations were made using the framework for 68 participants whose ages ranged from 4 years 7 months to 19 years 1 month. Subsequent analyses support the general design features of the observational framework and provide new evidence of PMLD musical behaviour and development