Neville Davies

Teaching, Learning and Assessing Statistical Problem Solving.

In this paper we report the results from a major UK government-funded project, started in 2005, to review statistics and handling data within the school mathematics curriculum for students up to age 16. As a result of a survey of teachers we developed new teaching materials that explicitly use a problem-solving approach for the teaching and learning of statistics through real contexts. We also report the development of a corresponding assessment regime and how this works in the classroom. Controversially, in September 2006 the UK government announced that coursework was to be dropped for mathematics exams sat by 16-year-olds. A consequence of this decision is that areas of the curriculum previously only assessed via this method will no longer be assessed. These include the stages of design, collection of data, analysis and reporting which are essential components of a statistical investigation. The mechanism outlined here could provide some new and useful ways of coupling new teaching methods with learning and doing assessment — in short, they could go some way towards making up for the educational loss of not doing coursework. Also, our findings have implications for teaching, learning and assessing statistics for students of the subject at all ages.

The Role of Technology in Teaching and Learning Statistics

In this chapter the merits, or otherwise, of using technology in teaching and learning statistics are considered. The many affordances that technological advances offer to teachers of statistics and the issues that hinder their widespread use in classrooms are summarised. When statisticians do statistics they get involved with far deeper concepts and carry out activities that require a wider range of cognitive skills compared with just applying techniques. It seems that pedagogic developments have not kept pace with those in software design, in that the opportunity to use computers to engage students in the full statistical enquiry cycle is not being exploited. The authors believe that beginning teachers must be exposed to such opportunities if they are to appreciate the key role that technology could have in facilitating the development of students’ understanding of statistics.

A new Bayesian approach to quantile autoregressive time series model estimation and forecasting

This paper proposes a Bayesian approach to quantile autoregressive (QAR) time series model estimation and forecasting. We establish that the joint posterior distribution of the model parameters and future values is well defined. The associated Markov chain Monte Carlo algorithm for parameter estimation and forecasting converges to the posterior distribution quickly. We also present a combining forecasts technique to produce more accurate out-of-sample forecasts by using a weighted sequence of fitted QAR models. A moving window method to check the quality of the estimated conditional quantiles is developed. We verify our methodology using simulation studies and then apply it to currency exchange rate data. The results obtained show that an unequally weighted combining method performs better than other forecasting methodology.

Computer-based learning in statistics: a problem solving approach

Summary. Software which allows interactive textual and graphical material to be created relatively easily has now existed for several years. This has promoted considerable interest in computerbased learning. This paper describes an approach which uses these tools to create problem-based material for use in laboratory sessions in association with courses in statistics. Illustrations are given, and the presentation of one particular problem, based on the design and analysis of a simple clinical trial, is developed in detail. Important issues of design, construction and evaluation are also discussed. It is argued that, although it is expensive to produce, material of this type can provide real benefits as a teaching resource.

An International Resource for Learning and Teaching

Summary This article compares the national curriculum data-handling specifications of the UK, South Africa, Australia and New Zealand and shows how data from the CensusAtSchool project can be used to enhance the data-handling capabilities of pupils in those countries. These data can also provide enhanced opportunities for the integration of ICT into core curriculum activities. Some ideas to enable teachers of statistics to create classroom teaching material with an international flavour are also provided.

Monitoring the parameter changes in general ARIMA time series models

We propose methods for monitoring the residuals of a fitted ARIMA or an autoregressive fractionally integrated moving average (ARFIMA) model in order to detect changes of the parameters in that model. We extend the procedures of Box & Ramirez (1992) and Ramirez (1992) and allow the differencing parameter, d to be fractional or integer. Test statistics are approximated by Wiener processes. We carry out simulations and also apply our method to several real time series. The results show that our method is effective for monitoring all parameters in ARFIMA models.

An International Project for the Development of Data Handling Skills of Teachers and Pupils

In this paper we provide an overview of the international CensusAtSchool project, designed, written and implemented first in the UK from October 2000 - April 2001 for pupils aged 7 -16 in primary and secondary schools. It has been adapted for similar aged school children in South Africa and Australia and was implemented in those countries in July and October 2001, respectively. We present our motivation, aims and objectives for carrying out such a project and show some results of analysis from the returns we have received from all three countries. Key outputs from the project include: worksheets that are suitable for enhancing data handling skills of pupils; a training course that wraps information and communications technology with data handling skills that is suitable to enhance the professional development of teachers; a raised awareness amongst pupils and teachers of the need to properly collect, present and analyze primary data; a contribution to improving the statistical numeracy and thinking skills of both teachers and pupils.

Testing for randomness in stream ciphers using the binary derivative

The binary derivative has been used to measure the randomness of a binary string formed by a pseudorandom number generator for use in cipher systems. In this paper we develop statistical properties of the binary derivative and show that certain types of randomness testing in binary derivatives are equivalent to well-established tests for randomness in the original string. A uniform method of testing randomness in binary strings is described based on using the binary derivative. We show that the new tests are faster and more powerful than several of the well-established tests for randomness.

A Simple Bootstrap Method for Time Series

In this article we present a simple bootstrap method for time series. The proposed method is model-free, and hence it enables us to avoid certain situations where the bootstrap samples may contain impossible values due to resampling from the residuals. The method is easy to implement and can be applied to stationary and nonstationary time series. The simulation results and the application to real time series data show that the method works very well.

Web-Based Project and Key Skills Work

Summary Pupils in England and Wales are increasingly being asked to undertake investigative-type work, be it the new compulsory projects in data handling for GCSE Mathematics (age 14–16) (see Browne 2002) or the Key Skills topic application of number. This article shows how teachers can generate realistic project scenarios using real data and produce indicative model solutions from the same data. The projects range from simple presentational problems for data, through hypothesis testing to complex modelling scenarios.