For the longest time I thought mathematics was arithmetic and mindless algorithms (BORING!), and after I learned in first grade how to add and subtract I completely lost interest in it (or, more precisely, what I thought was "it"). I know I'm not alone in this indictment on the (public) education system. There is even a word for lack of interest in maths and the sciences in general: Meh.

When I came across what is called "pebble notation"—a method of arranging numbers (pebbles) into geometric shapes invented (or, adopted) by the ancient Greeks—I realized how much is robbed of us starting from day 1 by inadvertently unimaginative elementary school teachers.

For eg, square numbers are not just N

For eg, square numbers are not just N

^{2}, nor is the definition even really:
"...a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it can be written as 3 × 3" (http://en.wikipedia.org/wiki/Square_number)

A square number can literally be arranged into a real square! For eg, nine dots (or pebbles) can be arranged like so:

For other numbers there are other shapes in which they can be arranged: squares, triangles, rectangles, and simple lines (or, numbers that cannot be arranged into perfect rectangles—ie, prime numbers!).

There is much more to this seemingly simple pebble notation, much much more. In fact, this is the very beginning of "number theory"—what Gauss called "queen of mathematics".

In Mathematical Mysteries: the beauty and magic of numbers, Calvin C Clawson writes of one of the greatest of number theorists from England:

...[Godfrey Harold]

But it is what Clawson says in the continuation of the passage above that I'm interested in:

Given this state of affairs in teaching mathematics it is hardly surprising that our children suffer the drudgery of what is passed for mathematics as we ourselves suffered before them. Yet, this laying down of "foundations of mathematics" is absolutely essential to what I call "mathematical literacy".

It gets even worse. It is not just this basic subject that suffers thusly. The language arts (at least, in the aboriginal experience) is likewise an utterly alienating experience where the mechanics of reading and writing (becoming a copyist of mindless word lists seems to be the goal here) is heavily emphasized, full stop. This, in lieu of regarding and treating the field of language arts as a means to enhance the humanity of our students.

In my efforts to learn in the style of "the liberal arts", I've had a happy accident in discovering an appreciation for good writing. Under this broad rubric of "good writing" I would include not only literary classics, great historical documents such as the "Declaration of Independence" but also movies, tv and radio.

It is the ideas that get me. These ideas, in turn, further enhance my appreciation of what and how the subject matter is expressed/articulated. It is a snowball effect. Some experiences are so exquisite it is very much like a spiritual euphoria.

I know of someone who is likewise affected by well-crafted linguistic expression, and, whom I would consider rightfully belongs to a master class of French letters. My attraction to and admiration for this person has transcended the physical, into the deeply spiritual, if I may unabashedly betray my true affections.

Literacy, thus defined above (ie, as mathematical literacy, as linguistic literacy, as any kind of literacy that inspires self-improvement), is less a skill than a state of being, and it truly goes beyond the act of reading into a realm of becoming. Northrop Frye said that up to the secondary grades one takes a subject, and at the college level the subject takes you. I have always found this to be a truism.

Literacy as such may not save our lives (that is not its point after all), it may not promote us to the status of a rock star (who but the very banal would regard art as such anyhow), but it truly has the capacity to raise us above ourselves and provides us space to imagine the possible.

Jay

There is much more to this seemingly simple pebble notation, much much more. In fact, this is the very beginning of "number theory"—what Gauss called "queen of mathematics".

In Mathematical Mysteries: the beauty and magic of numbers, Calvin C Clawson writes of one of the greatest of number theorists from England:

...[Godfrey Harold]

*Hardy believed a mathematical idea is good because it is beautiful, beautiful because it is serious, and serious because it is connected*[in a deep way]*to many other mathematical ideas...Hardy's claim that beauty is central to the enjoyment of mathematics is fervently believed by the majority of all who are enthralled by mathematics. In this he seems to have captured the essence of our love for this subject matter. Jerry King, in*The Art of Mathematics,*points out, "Mathematicians know beauty when they see it for that is what motivates them to do mathematics in the first place.*(Clawson, Mathematical Mysteries, Perseus Books, 1996, p. 213)But it is what Clawson says in the continuation of the passage above that I'm interested in:

*Hardy's idea that pure (good) mathematics should be devoid of meaningful applications has been adopted by many mathematicians at our universities. Unfortunately, this idea has caused some mathematicians to become elitists, casting disdain on all other branches of knowledge. This, in turn, has tended to alienate mathematicians from the rest of the academic community. Most elementary and secondary teachers we send out of our universities are not professional mathematicians, and they feel this alienation between themselves and what they see as snobbish old men barricaded in the ivory towers of academia.These same teachers, who feel alienated from higher mathematics, are asked by us to teach our children the foundations of mathematics. Do you imagine they embrace the task with enthusiasm?*(ibid, pp. 213-214)Given this state of affairs in teaching mathematics it is hardly surprising that our children suffer the drudgery of what is passed for mathematics as we ourselves suffered before them. Yet, this laying down of "foundations of mathematics" is absolutely essential to what I call "mathematical literacy".

It gets even worse. It is not just this basic subject that suffers thusly. The language arts (at least, in the aboriginal experience) is likewise an utterly alienating experience where the mechanics of reading and writing (becoming a copyist of mindless word lists seems to be the goal here) is heavily emphasized, full stop. This, in lieu of regarding and treating the field of language arts as a means to enhance the humanity of our students.

In my efforts to learn in the style of "the liberal arts", I've had a happy accident in discovering an appreciation for good writing. Under this broad rubric of "good writing" I would include not only literary classics, great historical documents such as the "Declaration of Independence" but also movies, tv and radio.

It is the ideas that get me. These ideas, in turn, further enhance my appreciation of what and how the subject matter is expressed/articulated. It is a snowball effect. Some experiences are so exquisite it is very much like a spiritual euphoria.

I know of someone who is likewise affected by well-crafted linguistic expression, and, whom I would consider rightfully belongs to a master class of French letters. My attraction to and admiration for this person has transcended the physical, into the deeply spiritual, if I may unabashedly betray my true affections.

Literacy, thus defined above (ie, as mathematical literacy, as linguistic literacy, as any kind of literacy that inspires self-improvement), is less a skill than a state of being, and it truly goes beyond the act of reading into a realm of becoming. Northrop Frye said that up to the secondary grades one takes a subject, and at the college level the subject takes you. I have always found this to be a truism.

Literacy as such may not save our lives (that is not its point after all), it may not promote us to the status of a rock star (who but the very banal would regard art as such anyhow), but it truly has the capacity to raise us above ourselves and provides us space to imagine the possible.

Jay

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