This Khan Academy classic is worth projecting in a
classroom. I know I would be eager to share it, if back in high school
teaching, like in the good old days
I'm not suggesting I'd turn my whole curriculum over to Sal, just that I'm up for sampling choice pieces by authors I admire, Vi Hart another one (good at math video as an art form). This retrospective by Khan, combined with a polemic at the end, seems a great high point to cite.
The math classroom of the future, in one of its many forms, casts the teacher in the role of VJ (video DJ) in a lot of ways.
Most clips might be of lesser length than Khan's average, although some teachers might invest in a full series of something, with their own time at the podium, as emcee and expert, to apply local spin, relate it to actual students more directly (personalize, make it real, take questions).
It's up to the teachers and lesson planners to glue the samplings into coherent pathways that come together to produce whole concepts. The teacher may improvise a good portion of the presentation, yet there's a known terrain, the school's agreed-upon curriculum, what it's known for.
The fact remains that "citing video" is a lot like a VJ's job and editing video is the "anime" equivalent of doing "manga" or "stills" (picture Jupyter Notebooks with still graphics courtesy of matplotlib).
In the graphical arts, manga equals comic books, anime equals cartoons.
In mathematics, one needs both.
Sal covers a lot of ground in the above video, looking back, reviewing trig functions, radians, the whole idea of pi.
Then comes his impassioned defense of tau (as 2 pi). He uses tau's appearance on the stage as an excuse to quickly review the whole board once again, having set us up the first time with a critique of Euler's Formula, arguably the most beautiful in math, but having a flaw, a blemish.
I don't see any either/or here really. Use tau in place of pi when it's prettier. Why not?
However at this juncture it's worth mentioning another low intensity tug-o-war in which tau is involved. Versus Phi. There's been some debate about whether tau stands for .618... i.e. the reciprocal of 1.618...
Some authors use tau for the latter, however the convention I'm most familiar with assigns Greek letter phi to 1.618... and tau to its reciprocal (0.618...). In English or other romanized ("ascii-fied") texts, we may use phi in place of the Greek letter. Then comes the low intensity debate whether to pronounce it "fee" or "fie". A contingent says it both ways. Maybe one should say "fee" on odd days of the week?
When Sal comes back around again, reviewing the board, the trig functions, the unit circle, he shows that Euler's formula may be made even more beautiful. By that time some of us are ready to vote for Sal's argument, especially with the caveat that pi is not hereby banned. These constants do not compete so much as reinforce one another.
As for phi versus tau, I think we've pretty much settled on tau being the reciprocal of the golden mean, which golden mean is itself > 1, i.e. is 1.618... or (1 + rt2(5))/2 -- note that I sometimes replace sqrt() with rt2() as I don't want to push the mental model to favor "squares" too prejudicially, given the triangle-friendly balance of this curriculum (Martian Math inspired)).
Especially if tau is going of to do yeoman's service as 2 pi, replacing 2 pi r with tau r in many a Jupyter Notebook, all the more reason, then, to not force it to do double duty as the golden ratio as well.
That'd only add to the post Babel confusion, not that I expect any posting by me to serve as an edict. The post Babel confusion (not everyone on the same page) is more what I'm drawing attention to, as another lens for viewing math.
Tune in the debates for a change. Listen to arguments.
Kirby
Additional reading / links
[originally a post to mathfuture, a Google Group]
I'm not suggesting I'd turn my whole curriculum over to Sal, just that I'm up for sampling choice pieces by authors I admire, Vi Hart another one (good at math video as an art form). This retrospective by Khan, combined with a polemic at the end, seems a great high point to cite.
The math classroom of the future, in one of its many forms, casts the teacher in the role of VJ (video DJ) in a lot of ways.
Most clips might be of lesser length than Khan's average, although some teachers might invest in a full series of something, with their own time at the podium, as emcee and expert, to apply local spin, relate it to actual students more directly (personalize, make it real, take questions).
It's up to the teachers and lesson planners to glue the samplings into coherent pathways that come together to produce whole concepts. The teacher may improvise a good portion of the presentation, yet there's a known terrain, the school's agreed-upon curriculum, what it's known for.
The fact remains that "citing video" is a lot like a VJ's job and editing video is the "anime" equivalent of doing "manga" or "stills" (picture Jupyter Notebooks with still graphics courtesy of matplotlib).
In the graphical arts, manga equals comic books, anime equals cartoons.
In mathematics, one needs both.
Sal covers a lot of ground in the above video, looking back, reviewing trig functions, radians, the whole idea of pi.
Then comes his impassioned defense of tau (as 2 pi). He uses tau's appearance on the stage as an excuse to quickly review the whole board once again, having set us up the first time with a critique of Euler's Formula, arguably the most beautiful in math, but having a flaw, a blemish.
I don't see any either/or here really. Use tau in place of pi when it's prettier. Why not?
However at this juncture it's worth mentioning another low intensity tug-o-war in which tau is involved. Versus Phi. There's been some debate about whether tau stands for .618... i.e. the reciprocal of 1.618...
Some authors use tau for the latter, however the convention I'm most familiar with assigns Greek letter phi to 1.618... and tau to its reciprocal (0.618...). In English or other romanized ("ascii-fied") texts, we may use phi in place of the Greek letter. Then comes the low intensity debate whether to pronounce it "fee" or "fie". A contingent says it both ways. Maybe one should say "fee" on odd days of the week?
Quoting from Wikipedia:
What I suggest coming
from a liberal arts background is we remind students of mathematics
that debate is a feature (not a bug) of their discipline and circle
these simple examples, in preparation for investigating bigger debates
in lessons to come.
Since the 20th century, the golden ratio has been represented by the Greek letter φ (phi, after Phidias, a sculptor who is said to have employed it) or less commonly by τ (tau, the first letter of the ancient Greek root τομή—meaning cut).
When Sal comes back around again, reviewing the board, the trig functions, the unit circle, he shows that Euler's formula may be made even more beautiful. By that time some of us are ready to vote for Sal's argument, especially with the caveat that pi is not hereby banned. These constants do not compete so much as reinforce one another.
As for phi versus tau, I think we've pretty much settled on tau being the reciprocal of the golden mean, which golden mean is itself > 1, i.e. is 1.618... or (1 + rt2(5))/2 -- note that I sometimes replace sqrt() with rt2() as I don't want to push the mental model to favor "squares" too prejudicially, given the triangle-friendly balance of this curriculum (Martian Math inspired)).
Especially if tau is going of to do yeoman's service as 2 pi, replacing 2 pi r with tau r in many a Jupyter Notebook, all the more reason, then, to not force it to do double duty as the golden ratio as well.
That'd only add to the post Babel confusion, not that I expect any posting by me to serve as an edict. The post Babel confusion (not everyone on the same page) is more what I'm drawing attention to, as another lens for viewing math.
Tune in the debates for a change. Listen to arguments.
Kirby
Additional reading / links
[originally a post to mathfuture, a Google Group]
