Now the University View video is working again. Oddly, at Ken's site it says it has been removed when clicking on the link. I hope the mystery will enhance its appeal.
I like the new video a lot since it deals with the devastaing effects of the fuzzy math plague on higher ed and marshals anecdote AND data. A winning combination. I only wish the good prof would appear to speak more spontaneously and naturally. Planned and rehearsed spontaneity, a winning oxymoron! There is something to be said for being a trained broadcaster as in the first video.
Now we need a video on the high school view. I am thinking of providing the middle grades view. Just kidding, but I see the devastating effects of fuzzy math victims daily. More later. Gotta rush for another day trying to bring real math to the pupils.
The link to the other video has been fixed. They revised the video and had taken it down. It is now up again on a new link, which is in the post I did on this yesterday.
The Gram Schmidt Identity video provides a necessary, but complicated connection that is missing in the first 2 Where's the Math? videos. It responds to those fuzzies that believe it doesn't matter how you learn to multiply and divide. These other alternative methods work just fine.
People have argued for years that it does matter because kids are unprepared to do algebra, geometry, etc. But the arguments as I've seen them, have been conclusory and general, even at MathematicallyCorrect (although someone there tried to show the need for long division in higher level math). I like his point that the identity he was proving was mid-level math, and long division was essential. It really gets to the heart of why fuzzy math is bad. I agree, the video has limited appeal. But it appeals to me.
The lattice method seems like it's very unfriendly to children learning place value. For instance, writing 26 horizontally and 31 vertically, places the 6 of the 26 next to the 3 of the 31, and then the calculations all slide sideways and down. Is that any improvement over the vertical layout of the standard algorithm?
Vertical alignment is so useful! My boys were doing decimal sums in the Singapore 4B Intensive Practice book. Some problems were written horizontally, e.g. 31.12 + 7.089. My boys were trying to do the separate sums of single digits in their heads -- scanning between the two numbers and starting from the rightmost digit of each number -- and needless to say, they took a lot of casualties. They got a lot of problems wrong.
Vertical alignment is so helpful when kids move from operations on whole numbers (bigger numbers are longer) to operations on numbers with a decimal fraction component (longer numbers are not necessarily bigger).
I had my boys go back and rewrite the sums in the margin, but aligned vertically so they could be registered horizontally by decimal point.
I think the maker of the video is confusing the Gram Schmidt procedure with ... something. It really doesn't make sense for that identity to have "Gram Schmidt" attached to it, historically or even mathematically.
In any case, using long division to introduce geometric series is not very natural at all. In fact, a more standard approach can be found on wikipedia under "geometric series."
Besides, students with no understanding of the workings of "long division" of numbers isn't going to understand "long division" involving polynomials.
I think the argument in the video ends up shooting itself in the foot.
(The first part, timing how long the different methods took for a sample problem, was good, especially since he was whipping through all three tries.)
As Anonymous says, it is unusual to rely on the standard long division algorithm to get at geometric series, although people do use that for dividing polynomials (note: 1/(1-x) is not a polynomial). The most natural way is to say, gee, for small x, 1/(1-x) is close to 1. How close? Let's examine 1/(1-x) - 1 and write 1 as (1-x)/(1-x) (common denominator! you need to know fractions to do algebra!)Do the semi-obvious. Then every time you see a factor of 1/(1-x), repeat! It never ceases to amuse and impress. What you need is basically hands-on experience manipulating fractions and things that look like fractions, not a standard algorithm.
He shoots himself in the foot by making easily refuted claims that the long division is need for geometric series. It isn't needed for geometric series, therefore ... it's must not be needed for anything. (Ow! my foot!) The last part was facetious. I'm a big believer in fluency in standard algorithms, but let's be real.
Also, it is easy to oversell the need for long division in dividing polynomials. The only times I have ever had to do such a thing in the course of solving some other problem, I did it "by hand". I think if you asked a random math graduate student to divide some polynomials they would do it "by hand". The analog of doing it by hand for multiplying 26x31 is exactly the first method the video showed with so much derision (apply distribution a few times and add 'em up). It's good to be able to do that sort of thing. It's just not good if that's the only way you can multiply big numbers.
Not very effective all by itself. I liked the other video, but it appears to have disappeared. What happened to it?
ReplyDeletewhat??
ReplyDeleteWhich video disappeared???
The one linked to earlier today. When I go to YouTube, this is the message I get:
ReplyDelete"This video has been removed by the user."
Now the University View video is working again. Oddly, at Ken's site it says it has been removed when clicking on the link. I hope the mystery will enhance its appeal.
ReplyDeleteI like the new video a lot since it deals with the devastaing effects of the fuzzy math plague on higher ed and marshals anecdote AND data. A winning combination. I only wish the good prof would appear to speak more spontaneously and naturally. Planned and rehearsed spontaneity, a winning oxymoron! There is something to be said for being a trained broadcaster as in the first video.
Now we need a video on the high school view. I am thinking of providing the middle grades view. Just kidding, but I see the devastating effects of fuzzy math victims daily. More later. Gotta rush for another day trying to bring real math to the pupils.
The link to the other video has been fixed. They revised the video and had taken it down. It is now up again on a new link, which is in the post I did on this yesterday.
ReplyDeleteThe Gram Schmidt Identity video provides a necessary, but complicated connection that is missing in the first 2 Where's the Math? videos. It responds to those fuzzies that believe it doesn't matter how you learn to multiply and divide. These other alternative methods work just fine.
ReplyDeletePeople have argued for years that it does matter because kids are unprepared to do algebra, geometry, etc. But the arguments as I've seen them, have been conclusory and general, even at MathematicallyCorrect (although someone there tried to show the need for long division in higher level math). I like his point that the identity he was proving was mid-level math, and long division was essential. It really gets to the heart of why fuzzy math is bad. I agree, the video has limited appeal. But it appeals to me.
There is something to be said for being a trained broadcaster as in the first video.
ReplyDeletelol!
This is an excellent demonstration of the importance of expertise.
Our middle school wants to go to a pure interdisciplinary approach, which is exactly what one does not want.
Being smart is one thing.
Having expertise within a field or discipline (or a profession or practice, like teaching) is another.
TEN YEAR RULE
I see the devastating effects of fuzzy math victims daily. More later.
ReplyDeleteYou should definitely fill us in when you have time.
The Gram Schmidt Identity video provides a necessary, but complicated connection that is missing in the first 2 Where's the Math? videos.
ReplyDeleteThis video is extremely important, I think - and very helpful for me.
I've known, simply because I've encountered articles about it, that long division is used somehow in the teaching of much more advanced concepts.
I do now understand exactly why one wants a student to know long division in order to teach the division of one polynomial by another.
But this steps it up to the college level - AND gives me language and a visual I can use.
I think it's incredibly important.
The amateur hour effect is great, too.
Watching this video a parent is going to think "Oh sh**"
The lattice method seems like it's very unfriendly to children learning place value. For instance, writing 26 horizontally and 31 vertically, places the 6 of the 26 next to the 3 of the 31, and then the calculations all slide sideways and down. Is that any improvement over the vertical layout of the standard algorithm?
ReplyDeleteVertical alignment is so useful! My boys were doing decimal sums in the Singapore 4B Intensive Practice book. Some problems were written horizontally, e.g. 31.12 + 7.089. My boys were trying to do the separate sums of single digits in their heads -- scanning between the two numbers and starting from the rightmost digit of each number -- and needless to say, they took a lot of casualties. They got a lot of problems wrong.
Vertical alignment is so helpful when kids move from operations on whole numbers (bigger numbers are longer) to operations on numbers with a decimal fraction component (longer numbers are not necessarily bigger).
I had my boys go back and rewrite the sums in the margin, but aligned vertically so they could be registered horizontally by decimal point.
they took a lot of casualties
ReplyDeleteI love it!
What the hell is this Gram Schmidt identity? Never heard about it...
ReplyDeleteI think the maker of the video is confusing the Gram Schmidt procedure with ... something. It really doesn't make sense for that identity to have "Gram Schmidt" attached to it, historically or even mathematically.
ReplyDeleteIn any case, using long division to introduce geometric series is not very natural at all. In fact, a more standard approach can be found on wikipedia under "geometric series."
Besides, students with no understanding of the workings of "long division" of numbers isn't going to understand "long division" involving polynomials.
I think the argument in the video ends up shooting itself in the foot.
ReplyDelete(The first part, timing how long the different methods took for a sample problem, was good, especially since he was whipping through all three tries.)
As Anonymous says, it is unusual to rely on the standard long division algorithm to get at geometric series, although people do use that for dividing polynomials (note: 1/(1-x) is not a polynomial). The most natural way is to say, gee, for small x, 1/(1-x) is close to 1. How close? Let's examine 1/(1-x) - 1 and write 1 as (1-x)/(1-x) (common denominator! you need to know fractions to do algebra!)Do the semi-obvious. Then every time you see a factor of 1/(1-x), repeat! It never ceases to amuse and impress. What you need is basically hands-on experience manipulating fractions and things that look like fractions, not a standard algorithm.
He shoots himself in the foot by making easily refuted claims that the long division is need for geometric series. It isn't needed for geometric series, therefore ... it's must not be needed for anything. (Ow! my foot!) The last part was facetious. I'm a big believer in fluency in standard algorithms, but let's be real.
Also, it is easy to oversell the need for long division in dividing polynomials. The only times I have ever had to do such a thing in the course of solving some other problem, I did it "by hand". I think if you asked a random math graduate student to divide some polynomials they would do it "by hand". The analog of doing it by hand for multiplying 26x31 is exactly the first method the video showed with so much derision (apply distribution a few times and add 'em up). It's good to be able to do that sort of thing. It's just not good if that's the only way you can multiply big numbers.