The pre-op examination included visual acuity recordings on Snellen's visual acuity charts/logMAR charts, uncorrected and corrected, however converted into logarithm of the minimum angle of resolution (logMAR) and

decimal notations.

For example, in Singapore

decimal notation is taught through money in Year 3 and expanded rapidly to three places in Year 4 (Ministry of Education, 2007).

The example is not, of course, decisive--giving

decimal notation a right-adjoining syntax, rather than left-adjoining, makes a compositional semantics readily available (7)--but it, and others like it, when combined (a) with the absence of a positive argument for the Modest Principle and (b) with the possibility, entertained by Szabo, that linguistic understanding is a matter of inducting from structurally similar cases (rather than deriving from constituent semantics), suffices to cast serious doubt on the Modest Principle, and hence on the role of the argument from understanding in motivating a principle of compositionality.

In modem terms, incommensurable quantities are those which cannot be expressed by a common fraction (such as 1/3) and which, if put in

decimal notation, are expressed by an infinite decimal.

Although students first learn about

decimal notation in primary schools, it is well known that secondary students in many countries, including Australia, do not have an adequate knowledge of the concepts involved.

A longitudinal study of students' developing understanding of

decimal notation has been conducted by testing over 3000 students in Grades 4 to 10 up to 7 times.

Add the results in each column from left to right, and record the answer in standard

decimal notation.

In modern terms, incommensurable quantities are those which cannot be expressed by a common fraction (such as 1/3) and which, if put in

decimal notation, are expressed by an infinite decimal.

Many of our students are not actively listening to our explanations, partly because they are not convinced that they need to re-think their understanding of

decimal notation.

I proceeded to explain to Trissy the structure of

decimal notation (Figure 2):

The results of question 1-2 seem to further highlight a deficiency understanding

decimal notation.

The use of fractional language helps to create a strong link between the fraction notation, the

decimal notation and the decimal place value.

This paper describes features of a group of misconceptions about

decimal notation that lead to students selecting as larger, decimals that look smaller.

The lessons flow naturally and easily from the most basic idea of sharing food with a friend to the development of the ability to write fractions, use fraction vocabulary, and understand equivalent fractions and

decimal notation.

Finally, make sure all these experiences of thousandths, millionths, and billionths are sensibly linked with neatly made and labelled place-value columns, and with calculator displays, both using

decimal notation.