Learning about and understanding statistical concepts

In Australia, school attendance is compulsory up to Year 10 (~16 years old) with Mathematics as a core subject for the whole period. The Australian National Curriculum for Mathematics was endorsed and released in 2010. It sets out what is taught as common content across the country at each year level from pre-school (~5 years old) to Year 10 (~16 years old). A separate document specifies the advanced mathematics curriculum for the final two years of school. This means that the minimum level of mathematics the next generation of Australian adults will have been exposed to is, in theory, that of the common Year 10 curriculum. Having been taught a concept is, of course no guarantee of understanding it. Before the release of the Australian National Curriculum each State and Territory had its own broadly similar curriculum. Using the current curriculum as a guide, Table 2-1 below shows a selection of when various mathematical and statistical concepts are taught

(Australian Curriculum Assessment and Reporting Authority (ACARA) 2010).

Table 2-1: Mathematical and statistical concepts taught in Australian Schools

(compiled from the Australian National Curriculum) School Year

and approx. age

Selection of the Mathematical and statistical concepts taught

Foundation and Year 1

~6-7 years old

By the end of the Foundation year, students make connections between number names, numerals and quantities up to 10. They compare objects using mass, length and capacity.

By the end of Year 1, students describe number sequences resulting from skip counting by 2s, 5s and 10s. They identify representations of one half. Students count to and from 100 and locate numbers on a number line. They carry out simple additions and subtractions using counting strategies. They partition numbers using place value. Students classify outcomes of simple familiar events (will happen, won’t happen, might happen). They collect data by asking questions and draw simple data displays.

Year 6 ~12 years old

By the end of Year 6, students recognise the properties of prime, composite, square and triangular numbers. They describe the use of integers in everyday contexts. They solve problems involving all four operations with whole numbers. Students connect fractions, decimals and percentages as different representations of the same number. They solve problems involving the addition and subtraction of related fractions. Students make connections between the powers of 10 and the multiplication and division of decimals.

Students list outcomes of chance experiments with equally likely outcomes and assign probabilities between 0 and 1.

Students compare observed and expected frequencies. They interpret and compare a variety of data displays including those displays for two categorical variables. They evaluate secondary data displayed in the

School Year and approx. age

Selection of the Mathematical and statistical concepts taught

media. Year 7

~13 years old By the end of Year 7, students solve problems involving the comparison, addition and subtraction of integers. They make the connections between whole numbers and index notation and the relationship between perfect squares and square roots. They solve problems involving percentages and all four operations with fractions and decimals. Students represent numbers using variables. They connect the laws and properties for numbers to algebra. They interpret simple linear representations and model authentic information. Students use fractions, decimals and percentages, and their equivalences. They express one quantity as a fraction or percentage of another. Students solve simple linear equations and evaluate algebraic expressions after numerical substitution.

Students identify issues involving the collection of continuous data. They describe the relationship between the median and mean in data displays. Students determine the sample space for simple experiments with equally likely outcomes and assign probabilities to those

outcomes. They calculate mean, mode, median and range for data sets. They construct stem-and-leaf plots and dot-plots.

Year 10 (standard)

~16 years old By the end of Year 10, students solve problems involving linear equations and inequalities. They make the connections between algebraic and graphical representations of relations. Students expand binomial expressions and factorise monic quadratic expressions. They find unknown values after substitution into formulas. They perform the four operations with simple algebraic fractions. Students solve simple quadratic equations and pairs of simultaneous equations. They compare data sets by referring to the shapes of the various data displays. They describe bivariate data where the independent variable is time. Students describe statistical relationships between two

continuous variables. They evaluate statistical reports. Students list outcomes for multi-step chance experiments and assign probabilities for these experiments. They calculate quartiles and inter-quartile ranges.

As can be seen from Table 2-1, there is a strong focus in mathematics education on links to concrete objects and the everyday world. The early work on probability focuses on the concepts of ‘likely’ and ‘unlikely’, with formal, numerical odds being taught later. The concepts become distinctly more abstract in Year 7 with the introduction of formal statistical language. The idea of independent trials is not taught until Year 8 (~14 years old).

A major report into literacy and numeracy skills in Australia and their impact on labour market outcomes was published by the Productivity Commission in May 2014 (Shomos and Forbes 2014). It gives a different perspective to the national curriculum

on what statistical knowledge might be expected in the Australian population as it attempts to describe what is known rather than what has been taught. Using data from the Australian Bureau of Statistics’ (ABS) Programme for the International Assessment of Adult Competencies (PIAAC) it uses six bands for describing a person’s level of numeracy. A score of Level One and below means that someone has only the most basic of numeracy skills: counting, basic arithmetic simple percentages and simple graphs. From Level Two up to Level Five the amount of skills in statistics steadily increases from very simple (similar to mid-primary school in the National

Curriculum) to very complex including the ability to critique, evaluate choices of models and representations of data. In 2011-12 almost 22% of Australians aged between 15 and 74 years had a numeracy level of one or below, meaning that the had essentially no understanding of statistics. A further 32.5% had a numeracy level of two, meaning that their understanding of statistics is limited to interpretation of relatively simple data and statistics in texts tables and graphs (Shomos and Forbes 2014). This leaves under half the adult population with the skills for the

interpretation and basic analysis of data and statistics texts, tables and graphs. (Shomos and Forbes 2014)

Research into the phenomena of anchoring and adjustment suggests that when dealing with a range of probabilities people tend to choose a few points in the range to which they attach a verbal meaning, adjust the actual point they are given so that it matches one of the anchors and then act on the result (Lichtenstein and Slovic 1971; Tversky and Kahnemann 1974). For example, a range of probabilities from 0% to 100% in 5% increments might be reduced to three anchor points 0% = not going to happen, 50% = don’t know, and 100% = will happen. Any amount in that range is then adjusted to the anchor point that seems nearest which may not be the closest numerically. The end points, with their definite outcomes are stronger

‘attractors’ that the indeterminate middle. This has strong implications for the use of waiting times as the clinical guidelines for different priorities of surgery will act as anchors.