Probability and stats in GCSE Maths

Written by David Spiegelhalter on . Posted in Features

It's an important time for the teaching of probability and statistics at GCSE level. The Department of Education has just carried out a consultation on the subject content and assessment objectives for mathematics GCSE which features major changes for probability and statistics - the detailed proposals are given at the bottom of this article.

Compared to the current curriculum, the new proposals will split probability and statistics. In probability, the proposal is to emphasise multiple representations. This is in addition to including more attention to conditional probabilities, plus risk and expectation. In statistics, histograms with unequal intervals and the ‘data cycle’ will be dropped (although mentions of ‘limitations of sampling’ will be retained). Calculating line of best fit will be included though.

Perhaps the most controversial element is the non-inclusion of the ‘data-cycle’ (or 'statistics cycle'), of problem analysis, data collection, data presentation, data analysis. There has been a long argument within the statistics community of whether this belongs in GCSE mathematics. The 2004 Smith Inquiry into post-14 maths education ‘Making Mathematics Count’ recommended:

'The Inquiry recommends that there be a radical re-look at this issue and that much of the teaching and learning of Statistics and Data Handling would be better removed from the mathematics timetable and integrated with the teaching and learning of other disciplines (eg biology or geography). The time restored to the mathematics timetable should be used for acquiring greater mastery of core mathematical concepts and operations.'

Indeed, the proposed science GCSE subject content and assessment objectives, now includes:

• Apply the cycle of collecting, presenting and analysing data, including:

Present observations and data using appropriate methods.

Carry out and represent mathematical and statistical analysis.

Represent random distributions of results and estimations of uncertainty.

Interpret observations and data, including identifying patterns and trends, make inferences and draw conclusions.

Present reasoned explanations including of data in relation to hypotheses.

Evaluate data.

Use an appropriate number of significant figures in calculations

• Communicate the scientific rationale for investigations, methods used, findings and reasoned conclusions through written and electronic reports and presentations.

However the Royal Statistical Society's recently-commissioned Porkess Report said:

• Recommendation 5: School and college mathematics departments should ensure they have the expertise to be the authorities on statistics within their institutions. Mathematics departments should be centres of excellence for statistics, providing guidance on correct usage and good practice.

• Recommendation 6: Under present conditions, statistics is best placed in the mathematics curriculum.

Essentially the view is that if this vital element were not in mathematics, it will either not be taught or taught badly. This is tricky. My personal view is that the ‘data cycle’ is absolutely vital, but that it is better placed within understanding of the ‘scientific method’ than within core mathematics. I feel that GCSE mathematics should provide the tools for analysis that can be used in empirical investigations, but techniques for carrying out those experiments should not be part of the assessment criteria.

Obviously there is opportunity for cross-subject activity, say with geography or science, featuring experimental design, data-collection, analysis, presentation and interpretation of real-world numerical evidence: it is inevitably tempting to look to a different type of qualification that took a broader cross-disciplinary perspective, but we appear stuck with the rigid subject demarcations of GCSEs.

At A-level the link between probability and formal statistical inference can be revealed in all its glory. If a post-16, non-A-level maths qualification is developed, then this could also include real-world investigation into the appropriate interpretation of numerical evidence.

 

The detailed proposals are as follows:

Probability:

• Record and describe the frequency of outcomes of probability experiments using tables and frequency trees.

• Apply ideas of randomness, fairness and equally likely events to calculate expected outcomes of multiple future experiments.

• Relate relative expected frequencies to theoretical probability, using appropriate language and the 0-1 scale.

• Apply the property that the probabilities of an exhaustive set of mutually exclusive outcomes sum to one.

• Enumerate sets and combinations of sets systematically, using tables, grids, tree diagrams and Venn diagrams.

• Construct theoretical possibility spaces for single and combined events with equally likely and mutually exclusive outcomes and use these to calculate theoretical probabilities.

• Calculate the probability of independent and dependent combined events, including tree diagrams and other representations and know the underlying assumptions.

• Calculate and interpret conditional probabilities through representation using two-way tables, tree diagrams, Venn diagrams and by using the formula.

• Understand that empirical samples tend towards theoretical probability distributions, with increasing sample size and with lack of bias.

• Interpret risk through assigning values to outcomes. (e.g. games, insurance)

• Calculate the expected outcome of a decision and relate to long-run average outcomes.

 

Statistics:

• Apply statistics to describe a population or a large data set, inferring properties of populations or distributions from a sample, whilst knowing the limitations of sampling.

• Construct and interpret appropriate charts and diagrams, including bar charts, pie charts and pictograms for categorical data, and vertical line charts for ungrouped discrete numerical data.

• Construct and interpret diagrams for grouped discrete data and continuous data, i.e. histograms with equal class intervals and cumulative frequency graphs.

• Interpret, analyse and compare univariate empirical distributions through:

Appropriate graphical representation involving discrete, continuous and grouped data.

Appropriate measures of central tendency, spread and cumulative frequency. (median, mean, range, quartiles and inter-quartile range, mode and modal class)

• Describe relationships in bivariate data: sketch trend lines through scatter plots; calculate lines of best fit; make predictions; interpolate and extrapolate trends.

 

This piece first appeared on the ‘Understanding Uncertainty’ blog that David publishes along with other contributors.

The views expressed in the Opinion section of StatsLife are solely those of the original authors and other contributors. These views and opinions do not necessarily represent those of The Royal Statistical Society.

 

David Spiegelhalter National curriculum GCSE

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