Why Don't We Live Forever?

The older one gets, the more one's life expectancy exceeds the population's given expectancy (at birth). Yet longevity is finite. This apparent paradox is analysed probabilistically with reference to empirical demographic data.     

A Closer Look at a Relatively Neglected Mean

The self‐weighted mean (WS) – in which each value is weighted by itself – is presented in several contexts and illustrated. It is incorporated in a set of more familiar means. Intuitions concerning WS are explored.     

Numbers defy the law of large numbers

As the number of independent tosses of a fair coin grows, the rates of heads and tails tend to equality. This is misinterpreted by many students as being true also for the absolute numbers of the two outcomes, which, conversely, depart unboundedly from each other in the process. Eradicating that misconception, as by coin‐tossing experiments, should be incorporated early on into learning the law of large numbers.     

A potential for learning probability in young children

Sixty-one children, from 4 to 11 years old, were presented with two sets, each containing blue and yellow elements. Each time, one colour was pointed out as the payoff colour (POC). The child had to choose the set from which he or she would draw at random a POC element in order to be rewarded. The sets were of varying sizes with different proportions of the two colours. The problem was to select the higher of the two probabilities. Three kinds of materials were used: Pairs of urns with blue and yellow beads, pairs of roulettes divided into blue and yellow sectors, and pairs of spinning tops, likewise divided into two colours.Roughly around the age of six, children started to select the greater of the two probabilities systematically. The dominant error was selecting the set with the greater number of POC elements. Verbal concepts of probability and chance were explored and some egocentric thought processes were described. The study indicates that probability concepts could be introduced into school teaching even in the first grades. The deterministic orientation in the instruction for young ages should be attenuated, permitting concepts of uncertainty right from the beginning.     

Consensus is Unfair

A coefficient of unfairness in the allocation of goods to people can be extended to measuring consensus among judges. The notion of relative variability underlies the formation of these measures.     

THINK: A Game of Choice and Chance

A game of chance in which players can maximise their expected gain by making appropriate decisions is described. Developing an optimal strategy involves computing probabilities and conditional expected values.     

Direct Instruction in Written Expression: The Effects on English Speakers and English Language Learners with Disabilities

Students with disabilities often struggle with writing tasks. In order to improve the written expression performance of high school students with deficits in written expression, a Direct Instruction writing program was implemented. The participants were six high school students in programs for individuals with learning disabilities. Three of the six students were served in programs for students who are English Language Learners. Using a multiple‐probe across‐participants design, the effect of the writing program was examined. The intervention was implemented over a 5‐week period with maintenance checks conducted 2 and 4 weeks after the termination of instruction. Results were variable, but there appeared to be a positive trend in student writing performance as measured by correct word sequence, length of text, and the TOWL‐3. Implications for practice and future directions are also provided.