When abstract thinking is applied to numbers, whole new sets of numbers begin to emrege.  A creation story of numbers could be envisioned as follows:

• Numbers are directly related to objects or events that exist in the tangible world.  eg. 2 means 2 apples or 2 days.
• Numbers are abstracted from the object or event that they describe in the sense that a meaning from the number may be applied to the objects or events that the number describes.  Symbolic thinking is emergent here in that there may be a special meaning of twoness

A major emphasis in the mathematical curriculum as we move into the 21^st century is for an integration of different areas of mathematics. Despite any preconceived notions you may have about computer programming, it has continued to be one of my students’ most enjoyable and rewarding experiences

Analytic, abstract thinking is ideal for reporting the regular, the expected, the normal, the ordinary, the unsurprising, the mundane, the things we often take so much for granted that we are hardly conscious that we know them at all

While linear algebra has many real-life applications, it also has its elegant side, a side which is somewhat abstract. This aspect of the subject also has a real-life application: to develop clarity of thought and expression

The following is an edited and supplemented response I gave to a teacher who wrote me the question and comments that are in bold font.

> My question is this: I moved up to bilingual third grade this year, and I'm trying desperately to
> teach them to think logically

Abstraction is the process of reducing the information content of a concept, typically in order to retain only information which is relevant for a particular purpose. For example, abstracting a leather soccer ball to a ball retains only the information on general ball attributes and behaviour. Similarly, abstracting an emotional state to happiness reduces the amount of information conveyed about the emotional state.

This report describes your results on the Abstract Reasoning test. This test looks at your

ability to identify the patterns and relationships between shapes.

When reading this report, the following points should be considered:

Simon Sample

Reasoning Abilities

The Aptitude Tests assess the candidate's ability to think logically using words, numbers

and abstract concepts. They assess the ability to both understand and develop logical

arguments and solve complex logical problems.

VR1: Verbal Reasoning