Republished from the old blog, to be used as a draft for a paper I am currently writing.

In Britain, everyone is already familiar with the new look of a 20 pounds note. Perhaps, the origin of words

prominently displayed on the note is less known. Of course, they are from the famous book The Wealth of Nations by Adam Smith, whose portrait is on the banknote:

Smith’s analysis starts, in Book I, Chapter I: Of The Division of Labour, with the following description of pin manufacturing:

One man draws out the wire; another straights it; a third cuts it; a fourth points it; a fifth grinds it at the top for receiving the head; to make the head requires two or three distinct operations; to put it on is a peculiar business; to whiten the pins is another; it is even a trade by itself to put them into the paper; and the important business of making a pin is, in this manner, divided into about eighteen distinct operations.

and comes to the conclusion that separation of the pin production process into 18 operations

increases the productivity by factor of 240.

The history of Western civilisation is the history of ever deepening division of labour. And we reached a unique point in history when 95% of people have no vaguest idea about the working of 95% of technology in their immediate use. The schism is profound: mathematics built-in in a mobile phone or MP3 player is beyond understanding by most graduates from mathematics departments in British universities.

In the ever deepening division of intellectual labour, mathematics is a 21st century equivalent of sharpening a pin. Of course, the same is true about physics, chemistry, biology — although biology is perhaps not sharpening the pin but attaching a head, which, as Adam Smith remarks, in itself consists of two or three operations.

We have to admit that 95% of population do not need any mathematics beyond use of a calculator.

But what are the implications for mathematical education? First of all, collapse of the traditional pyramid of maths education. This is how it looked in the mid 20th century, with pupils / students / grauate students at every level of education being selected from the much larger pool of students at the previous level:

And this is what we should expect in the future:

Mathematical education had always been selective — but now it lost the critical mass of the selection pool.

Perhaps mathematics is in need of a rebranding exercise. If only a small percentage of people need to know mathematics, but know well, why not try to create an up-market brand of maths learning? Why not rebrand mathematics as what it actually is — a tool of personal development and a spiritually enhancing activity?

Selective mathematical education will not work unless we know:

- What are mathematical abilities?
- What is the nature of mathematical intuition?
- What do children actually do when they learn mathematics?
- What do mathematicians actually do when they do mathematics?

The era of extensive mathematical education of majority is over — we have to develop a model of intensive mathematical education of minority. Unfortunately, the mathematics education community is totally unprepared to this tectonic shift.

I’ve commented before; I’m a secondary teacher (high school) of math for students with mild disabilities in the US. The last five years in the US have seen federal law mandating improvement in using standards-based instruction, where the standards are meaningful, rigorous, and assessed to death.

My state has just overhauled the math standards for grades K-12, and it’s about to hit my grade level next year (9th grade – the first year of high school). What’s interesting to me is that the INTENT of all the new standards and overhaul is to do the opposite of what you’ve predicted for the future. “All students will have mastered algebra by the end of 8th grade.” “All students will have mastered Algebra 3, Statistics, and Geometry in order to graduate.” And they mean all. Or at least they intend all.

I have learned some new frameworks for planning for next year that I hope will indeed result in better instruction from me and mastery for my students, but if there’s any group that’s expected to struggle, it’s my group.

By:

Adrienneon July 3, 2008at 5:39 pm