There is a difference between mathematical thinking and simple arithmetic or performing calculations. Mathematical thinking is being able to abstract and organize significant ideas (data) from apparently "messy" reality into cogent and coherent statements. Mathematical thinking allows one to construct not only original equations and sequencing of numeral manipulations but also to perceive any of type of patterns and to make technical/ethical judgements outside of our visceral reactions using tenable and defensible reasoning behind arguments.

Rote-learning tends to be rather mechanical and prescriptive: "have faith, 1/2 is greater than 1/4"; "that 2 + 2 = 4", etc. It does not engage the student in the reasoning aspects, the organizing principles behind these arithmetical statements, and everything is particular - ie, the discussion centers around inductive thinking (reasoning from particular facts) rather than deductive thinking (reasoning from cause to its consequences).

To be sure, mastery of both types of reasoning are required for learning to take hold. But these can only happen in covert and overt discussion of philosophy behind the deeper connections between arithematical and geometrical facts that impel such statements.

Tobias Dantzig, in his book called, Number: the language of science, wrote:

*The genesis of the natural number, or rather of the cardinal numbers, can be traced to our matching faculty, which permits us to establish correspondence between collections*

*. The notion of*equal-greater-less

*precedes the number concept. We learn*to compare

*before we learn*to evaluate.

*Arithmetic does not begin with numbers; it begins with*criteria.

*Having learned to apply these criteria of equal-greater-less, man's next step was to devise models for each*type

*of plurality...*

*The principle of correspondence generates the integer and through the integer dominates all arithmetic.*(pp. 216-217) -emphases by Dantzig

In crude and simplicist terms, I've learned to use the concept of the number (or real) line to gain some understanding and insights into the thinking behind magnitudes of fractions, the nature of irrational numbers and/or radicands and how these concepts may be, and are, extended to serve multifarious purposes in scientific inquiry. When I first don't understand arithmetical reasons or manipulations, I've learned to seek out their geometrical causes and demonstrations if they exist. And, they usually exist. Somewhere.

The active seeking of them helps me to learn much more (even about other things) and my appreciation of mathematic elegance and beauty constantly grows. I may not create but I can see certain abstract things more clearly and appreciate them more deeply.

The unfortunate confusion caused by certain English terms that have vague and various meanings in laity also become easier to avoid for me in my thinking of them

*in Inuktitut*(or, Jaypeetese) as I learn more about mathematics and its philosophical scaffolds. For instance, the seemingly "natural" dichotomies between 'real' and 'imaginary' numbers, 'rational' and 'irrational' numbers have very specific meanings in maths, and their diametric oppositions are more apparent than real as suggested in the English terms.

It was the great Gauss who suggested that we use terms like 'direct unity'; 'indirect unity' and 'lateral unity' for the symbols 1, -1, and √-1, respectively, rather than the oft times confusing: positive 1, negative 1 and the 'imaginary square root' of negative 1. I personally think that Gauss' suggested term for the imaginary number line as 'lateral' is very intuitive, being as it is often represented as the north-south line laterally perpendicular to the x-axis (the y-axis is the up-down line, not treated as lateral to the x-axis).

Geometric demonstrations, in conjunction with deductive reasoning, behind mathematical facts tends to dispel the daunting mysteries of mathematics. Going about it this way also makes the historical developments of the ideas almost unavoidable; the seemingly perfected mathematical structures are human creations after all and, as such, the discourse on the historical developments of their logical foundations is often more interesting and instructive and inspirational than passive prescription and recitations of multiplication tables. In historical-based discourse, one also learns how to negotiate one's way through the many different interpretations inherent in all human endeavours. this is critical thinking par excellence!

Jay

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