Tuesday, March 26, 2013

why stop at four?

The Ontario Social Studies Curriculum says:

By the end of Grade 2, students will:
– recognize and use pictorial symbols
(e.g., for homes, roads), colour (e.g., blue
line/river), legends, and cardinal directions
(i.e., N, S, E,W) on maps of Canada and
other countries;

I was helping out with some Grade 2 homework the other day, and when it came time to mark the cardinal directions I reached for an old trigonometry text book to show a much more detailed compass rose. Young students can figure out how many points you have each time you sub-divide the compass (4, 8, 16, 32, ...), and the naming conventions for the points also make sense to kids (what's between North and East? North East! What's between North and North East? North North East!).

Thursday, March 21, 2013

calculated thought experiment

I always feel that I come away with something new whenever I read Ludwig Wittgenstein's Remarks on the Foundations of Mathematics - likely because I understood so little on each previous read. In the book, one thing  he tries to get at is what we mean by words "mathematics" and "calculation," and in doing so he asks questions that are so basic that they call into question our implicit assumptions about what these words mean. One of these sets of questions ask about whether our mental state and attitude in any way influences whether or not we are actually "doing mathematics."

For example,
"Imagine the geometry of four-dimensional space done with a view to learning about the living conditions of spirits. Does that mean that it is not mathematics?... Could people be imagined, who in their ordinary lives only calculated up to 1000 and kept calculations with higher numbers for mathematical investigations about the world of spirits?" 
Does it matter what we think we are doing when we are doing math? As long as we are moving the symbols around correctly does it still count as mathematics?

Elsewhere he asks "What would happen, if we rather often had this: we do a calculation and find it correct; then we do it again and find it isn't right; we believe we overlooked something before - then we go over it again and our second calculation doesn't seem right, and so on. Now should I call this calculating, or not?"

Does calculation require a social convention - if one person performed something once, could it be considered an algorithm? "What about this consensus - doesn't it mean that one human being by himself could not calculate? Well, one human being could at any rate not calculate just once in his life."

Some of the most fascinating thought experiments that Wittgenstein proposed (way back in 1942-1944) were about (what we would now call) computers or "mobile devices":
"Does a calculating machine calculate? Imagine that a calculating machine had come into existence by accident; now someone accidentally presses its knobs (or an animal walks over it) and it calculates the product 25 x 20."
Has any calculation happened in this case? Later he suggests a scenario that now seems quite familiar:
"Imagine that calculating machines occurred in nature, but that people could not pierce their cases. And now suppose that these people use these appliances, say as we use calculation, though of that they know nothing. Thus e.g. they make predictions with the aid of calculating machines, but for them manipulating these queer objects is experimenting. These people lack the concepts which we have, but what takes their place?"
Very (unintentionally) prescient Ludwig! We are actually now living in a reality which closely resembles this thought experiment - and an environment that sounds like the classroom of the future as imagined by Computer Based Math. What will replace current concepts of number once our experience with calculation is mediated entirely by machines whose cases cannot be pierced? And will we even notice that they have been replaced once they are gone?

Sunday, March 3, 2013

syllabi old and new

If you can, please take a moment to look at the syllabus above. Here's a zoom in on one of the flows between third and fourth year:

I was prompted to dig out this pretty relic from my collection of old books after reading about the ongoing efforts to describe the ideal curricular progressions for Common Core State Standards for high school mathematics (via a post by +Raymond Johnson that linked to the Common Core Tools blog).

If you look carefully at the topics, and the emphasis on set theory and logical sequencing, you may be able to guess when this curriculum was in effect (maybe I'll provide the answer in a future post). What will the new CCSS progressions tell us about what we value in mathematics education?