so.....reasoning is only allowed to be called reasoning if it can be spoken or written? hmmmmmmm......that's really a disservice, i think, for people who are not articulate. and, glamador, im not sure that "old taught" people ever encountered real math. as an "old taught person, arithmetic was the norm and math was the exception. and i have a BS and MS in mathematics.
@lipsynchorswim The answers are different in Base 10 and in Base 8, because the starting numbers represent values that are fundamentally different. You can indeed verify that 147 (8) = 103 (10).
@YouGot4Shotted he is correct on what he is saying. the new math is rather for dummies. we are from a different day of teaching and teachers; a totaly different format. In the third grade the teacher will give the child the problem of 4+4= well we all now the answer is 8. if the child should be taught about reasoning and he answer with 8 it will be marked wrong because he did not use reasoning to get the answer.
As someone who has interacted on a regular basis with family members and friends who are "bad at math" I can state with 100% certainty that so-called "new math" is better for bringing up children with an understanding that better allows them to comprehend the concepts needed for more complicated mathematics than addition and subtraction.
It may seem like this is a stupid round-about way of doing things, but I see so many old-taught people with *ZERO* understanding of real math.
@Arexsos Yes, they're both equal, but because they are in different bases, the numbers are different. Just like how 100 in binary is equal to 4 in Base 10.
@lipsynchORswim He took the same digits in the base 8 example as he did in the base 10 example. Thus, the two numbers in the first example (in base 10) are NOT THE SAME AS the two numbers in the second example (in base 8). IF you convert the numbers beforehand, do the operations, and then convert the answer, THEN you will find the two answers are equivalent.
So no, the numbers are not equal. 169 in base 8 is 251. (Note: 2 * 8^2 + 5* 8^1 + 1* 8^0 = 128 + 40 + 1 = 169)
Take the example: 100 base 8 - 0 base 8 = 100 base 8; 100 base 10 - 0 base 10 = 100 base 10. 100 base 10 != 100 base 8. They are different numbers before hand, so you will get different numbers afterwards. If the problem was to give the answer in base 8, you could either change the operands to base 8 _before_ or the sum _after_, getting the same answer because you are still using the same numbers.
I've not bothered to read all the comments but to everyone who says the answer is wrong or the equation is wrong you're both right... You have to realise he takes the base 8 answers and converts them to base 10 in order to get the answer. He's using the irrational for of mathematical logic to dictate to you thru song that the new maths is incorrect. In other words he is trolling the maths community and for that I applaud him.
@VelaVenti92 That's not the point. Everybody should use the method he/she prefers (the're the same anyway). I'm claiming that the new method is ineffective to learn to pupils. I know this: I'm one of these pupils. Although I understood, many classmates didn't. IMO because the algorithm and the proof that it works are mixed in an over-complicated way. Separate the two (first learn to do it, then to understand it), and things should improve rapidly. I've tried this, it worked for my pupils.
Hey, I grew up with this song in the 1960s. No, I'm not a mathematician. I loved your video, it was pretty accurate as to what is going on. This was a great video.
this is one song that I showed my math teacher. She then proceeded to teach the class how to do calculations in base 8 (We ran out of material for the year)
He didn't, he's just lipsyncing with it. The guy who made this song is Tom Lehrer. He was mainly active in the 50's though I think, so he doesn't write songs anymore. Still, look him up, all his stuff is like this :D.
@jules7lego7dude7 This is a lip sync (Hence the up-loader's name is "lipsynchORswim"). The song is by Tom Lehrer. If you want more information on Tom Lehrer or want to hear more of his songs, I would suggest you go to the page @6funswede (Who is an absolute nut about Tom Lehrer).
That's not actually what "New Math" refers to. My understanding with the whole "new math" thing is that it was a movement to get kids to understand set theory and the laws of arithmetic before actually being taught how to do arithmetic. The idea was that kids would figure things out from the ground up and in doing so understand it better.
Personally I think trying to teach rigid exact mathematical concepts using the vague English language makes things far more difficult. It's much easier to understand math when you have an exact algorithm to study which conveys things much better than trying to figure out what someone means when they talk about abstract concepts I can't relate to and explicitly avoid using metaphor.
Can anyone explain "old math" to me? He says "3 from 2 is 9". Without doing it in "new math" I have no idea how you are supposed to know that. Just by looking at the equation I know I can't take 2 from 3 so I make it 12 - 3 = 9. Since I did that, I have to make 4 into 3 etc. But without doing that, I don't know how you get 9 without just knowing it and not doing any actual math. Anyone that remembers/understands "old math" want to clear this up for me?
2-3=9? 3-7=6? What? They used to really fuck up math. Now we actually understand what the hell we're doing, and we can understand other math systems that use different base numbers, like the Mayans. Progress kicks ass.
@YouDuck2345 Hmm... no, it's 147. They explain why in the video above, and my computer's calculator can confirm it's the right answer... Also, I really don't see how you'd get a decimal number by substracting two non-''decimalised'' numbers, no matter which base.
@ChevaliersEmeraude See, I tried to come up with an explanation for a quote I'd heard in a video game -- that 2+2=10 in base four. I noted that two plus two is, according to amateur math, four. It matched the base, so I assumed that any time a number matched its base, it must equal ten. Thus, I crunched the numbers, divided them by eight, and then multiplied the result by ten to make the number match my theory.
Obviously, my theory didn't work, but hey, it was worth trying.
@YouDuck2345 Well, you do have the base of the concept right, thought. Base 4, for instance, would be counted as 0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33, 100, 101, 102, 103, 110, 111, etc. See, 100, here, is the same as 16 (or 4-square) base 10. So yeah, you had the bases of the system right. But you can't just divide and multiply to change the base 4 to base 10, Truth be told, unless you're used to a system, it can be hard to switch any large number from any base to base 10.
A bit is a bit. I intended to show that we were using a 1,000 octal bit - that is, a physical matrix of ferrite cores that could be polarized in 2 different ways that was constructed to respond to octal arithmetic internally - memory. This was a cubic structure about 5 inches on a side, weighing about 2 pounds, made mostly of iron! We numbered each bit within the core with its octal address. Enough said.
I want to respond to the post by Cras17, re: Lehrer's joke that "...the idea is to understand what you're doing, rather than to get the right answer." I strongly disagree with Cras17's comment that "...this should be the focus of learning math." Mr. Lehrer's joke is a logical fallacy. I don't understand how it is possible to get the right answer if you DON'T know what you're doing. The "base" thing itself is too "Rube Goldbergish." Why go through that?
I hate how they are saying you need to know what you are doing. No, you just need to know if what you are doing will get you the right answer. It's math, all that matters is the answer. All you really need to know is what formula, if you will, to use in any situation.
After watching this two times, I realized what was happening in the beginning problem. It is funny because I am now in eighth grade and I did my subtraction problems the same way until second grade when they said my way was wrong and that I would get it wrong often. Well, I kept on doing it my own way until I got brainwashed into doing it their way..Honestly, the old way was faster and more efficient than the new way.
@abebou 12 is interesting in that we actually have natural English words for powers of 12 (up to 12^2 at least) which we don't have for the other bases (except 10 of course).
Consider "eight gross eleven dozen and ten". Using "A" for ten and "B" for eleven this is 8BA (base 12) = 1294 (base 10).
@SaphiraPorter the proper name is "exponents". Besides, base 8 isn't limited to substracion. You can do exponents the same way as base 10, binary, trinary, hexadecimal, and any other numerical system. for example 5^2 in base 8 would be 31.
@ciacho00000000 yea... you knew what i meant lol but i'm not thatgood at base 8 yet. you sound like a math genius, i feel dumb...but i know what base 8 math is... my math teacher doesn't XDXD hey your like me... i answer to 3-5 pestions or comment per vid lol i try not to.. but i do! lol
This video has inspired me and I am currently trying to figure out a way to make a song that contains similar stylistic elements to explain the proof converting ax^2+bx+c=0 to x=[-b-+(b^2-4ac)^(1/2)]/2a. Wish me luck on that.
Meanwhile, this easily one of my favorite videos on Youtube. I'm not easily swayed to bestow such honors because I know that they never mean much, but in my mind this has the makings of a classic. I hope to show all my friends in Calculus this video.
Ironically, there are human languages documented that use base 8. The hypothesis is that these languages were spoken by people who counted the spaces between their fingers, not their fingers.
@Telumel There are also areas of the world in which people use base 12 out of habit, because they counted using the sections of each finger on one hand, using the thumb as a pointer. You'd start at the base of your little finger and move up that, then go to the base of your ring finger, et cetera. I've taken to using that myself because if I need to count on my hands (and I don't often) it lets you go further on just one hand and requires fewer changes than using each finger as a binary digit.
Who would ever do anything in a base 8 system? That is totally counter-intuitive to our entire number system.
He did have a good point in there though. The idea is to understand what you are doing, not to get the right answer. He said it sarcastically but that actually is and should be the focus of learning math.
@TheShadowsLotus Base 8 works in a similar fashion to base 10 (which most of us are used to). Just like base 10 has a 1's place a 10's place and a 100's place, base 8 has a 1's place an 8's place and a 64's place. The reason why is because in each base, the places are to certain powers (0, 1, 2). So since something to the 0'th power is 1, something to the 1st power is that number, something to the 2nd power would be squared. Therefore, 10^2=100 and 8^2=64 That's where the 64 came from.
@lipsynchORswim "Base 8 works in a similar fashion to base 10 (which most of us are used to)" speaking of familiar number systems, I hear some people in central american used base 20.
@lipsynchORswim "Base 8 works in a similar fashion to base 10 (which most of us are used to)" speaking of familiar number systems, I hear some people in central america used base 20.
@TheShadowsLotus In base 10 you go 10,100,1000 because 10x10 = 100. 10x100=1000 etc. In base 8 ( again totally ridiculous and useless as a system) would be 8,64,512 because 8x8 is 64. 8x64 =512. It doesn't makes much sense because our numbers are not set up like that. There are 10 twenties, 10 thirties, 10 fourties for a reason. Anyway, it was still a funny video.
@TheShadowsLotus Another way to think about it in base8. There's the 1's place which goes up to 8, before rolling over. so the number 10 would represent 1. 8 or 8. 20 would represent the number 16 or 2 '8.s The 3rd slot would only roll over once it has 8 "(8s) so 100 would be 64. that would only roll over once you have 8 of those. So each place is essentially multiplying by 8 to move one over. Just like in base 10 you multiply by 10.
@TheShadowsLotus We have a hundreds place because ten tens is 100; in base eight (which is useful in some computing languages and other mathematical alleyways), you'd have a place for eight eights, and eight eights is 64. (He explained this in the song, you know.)
@cras17 Ironically enough, most computer systems use a variation of base 8 called hexidecimal (which is base 16, fyi). Different numbering systems are used when there is a need to store a lot of information in small numbers (or very large numbers as in base 2 (aka binary)).
@lipsynchORswim Ok, yes computers do it hexidecimal. That doesn't mean I'm happy about it :P and it still doesn't make it an intuitive or efficient way to count or to do math.
@cras17 You've got it the wrong way round, chap. It only seems intuitive and efficient because you're used to base-10. Bear in mind that if you were brought up on base 8, base 10 would seem just as difficult as base 8 does to you now. And that in base 8 it still goes 10,100,1000... and it's just that 10(base-8) != 10(base-10), and so on. !+ means "is not equal to", by the way. By extension, in base 8, the digit values for base 10 go 12,144,1750 - just as ridiculous as base-8 values in b10.
@VinuVonVin My point was that in our numbering system a base 8 system is not intuitive. It isn't because we are just used to base 10, it is because the way we talk about numbers is inherently rooted in the idea of 10.
We have 10 unique digits that repeat every 10 numbers, then every 100 numbers, then every 1000 numbers etc.
The only way a base 8 numbering system would be as intuitive as base 10 is if the numbers 9 and 0 did not exist and there was a switch every 8 numbers.I say we stick with10
@cras17 You're conflating intuitiveness with what-people-are-used-to. No number base is actually more intuitive than any other. Base-10 is in no way intuitive, but it /does/ mesh nicely with the language we've developed - which, incidentally, was at one point based on base-20. You can see remnants of that in French, which is a strong ancestor of modern English. For what it's worth, I like the base the Sumerians used - 60. Quite a lot of fractions are more convenient, for a start.
@VinuVonVin Well I guess I would say that a numbering system that meshes well with the language would be intuitive to speakers of that language. We could debate all day on the definition of what is intuitive but I would rather not :P
As for base 60, fractions may be easier in that system (never took a look myself) but I think knowledge of place value, and decimals which our friend the metric system is so heavily based upon comes easier with base 10 and is ultimately more useful for mathematics.
@cras17 Ah, well, there we must agree to disagree. It's true that base 60 is unwieldy, but I /do/ think base-12 is easier to deal with than base 10 once you get used to it. And place value is, of course, not base-specific in terms of whether it works or not. But you knew that. WRT the metric system, though - we use that because we use base ten, rather than the other way around! Again, I'm sure you know that.
@lipsynchORswim hex is actually a way of binary conversion, all computers at their very roots run in base 2, as you mentioned, hex just takes all those lines of 1 and 0s and turns them into something slightly more manageable
If a child is getting the right answer 100% of the time, does it matter in the real world if they are building a bridge whether or not they understand 'why'? It's -math-, not philosophy. If a child understands the 'theory' but is right only 70% of the time, they are useless for all practical real world applications.
I was a champion mental math student (won competitions) as a child -and- knew what I was doing, but as I grew older my schools did their best to 'train it out of me'
While my schools could not understand 'why' I was so far ahead of my peers in multiplication, division, math, etc (because I did math drills, could 'guess and check' fast, elimintate wrong answers quickly, memorize formulas, etc) - they thought it horrible that I never showed my work. The schools loved 'mental math' - making us spend 4 pages showing 'two different ways, with graphs' showing how we arrived to the conclusion of a given word problem. (It didn't matter if we got it right.)
@PersephonesFear Although I've never been particularly fast at math (largely due to inefficiencies in my thought processes) I was good at it and ran into the same issue. Our version of the "new math" also had us showing 2 different ways of looking at a word problem. I would figure it out and then have fun deciding which way I would illustrate it this time. I do, however, think that more formula work early on would have helped me, as it would have removed some of my aforementioned inefficiencies.
@cras17 most of the low level computing languages do arithmetic in base two (binary), base eight (octal), and base sixteen (hexadecimal)... if you were going to do some coding in ForTran back in the 70's for example, you'd need to know your octal. Octal is a useful way to express 8-bit values. Anyway, you can do whatever base you want. You simply select your digits and columns on that base. So base 5 has 5 digits (0,1,2,3,4) and columns in powers of 5 (1's, 5's, 25's, 125's) etc.
@PlasteredDragon "Octal is a useful way to express 8-bit values"
...No, it isn't. One octal digit equates to three bits. Three doesn't go into eight. Hex is a much better way of expressing 8-bit values, since one hexadecimal digit equates to four binary digits, so you can represent any 8-bit value with two hex digits. 8 bits equate to either two or three octal digits, so octal clearly /isn't/ a veryy good way. It's good for 9-bit values, though, and some machines used 9 bits to the byte
@skvakagud I live in Canada. I use the metric system. I think it is the greatest thing to happen to measurement ever. All SI units should become standard everywhere and always. Imperial influence does bleed in from the US but support for metric is growing there slowly. They just need to suck it up, get off the fence and make a full conversion.
@cras17 ah sorry from the comment i respondet on, i got the impression you were from the US, my bad. :D But yeah it really is silly concidering they've been "trying" to convert to the metric system, for more than 100 years now. But i think it has to do, with the way more and more workspaces, are working with milimeters, and therefor want cm as well, rather than inches. + They use tonn more and more, and pounds, ft, yards & such. Are just hill-billy meassure systems if u ask me :p
@cras17 true, the general public of the United States does rely on the English system of measurement, however a lot of the scientific community has switched to metric, mainly to provide uniformity with the scientific community outside the US, but also because it's sometimes easier to teach something in metric than it is in English units.
@lipsynchORswim Somewhat amusing that people refer to the imperial system of units as "English units" when we've converted to metric for everything except (for some reason) distances on roads, where we still use miles. Nobody seems to know quite why.
@cras17 Well, it's important to understand process because that's the fundamental portion of mathematics, but you cant' just say, "oh, well, you can't multiply 5 by 8 correctly but you understand multiplication, so it's all good" because if you send that kid into a job, they're totally boned, because their boss isn't going to care that they understand multiplication, they're going to care that the nuclear reactor is on fire because they multiplied wrong. (Future math teacher here)
@snoddymcsnodson True, but it's very annoying to lose 9 marks on a maths paper because you accidentally timed something by 11 instead of 1.1 half way through. Or if you read the question wrong. It makes tests more valid as they look at whether you can do maths rather than whether you read the question well enough.
I actually had to use Base 8 when studying early computer structures and programming. In fact, we manually input data in the ferrite core of our Univac Digital Trainer, which had 1,000 octal bits, or 512 decimal bits, of RAM that we had to address individually to input to the various registers. Our output device was a teletype writer, and we input the data by pushing binary coded decimal groups of lighted switches to change the states of the octal bits. Not user friendly...
@paulrank: Last time I checked, a bit is a bit, there is no such thing as a "decimal" bit or an "octal" bit, since a bit is a single quanta of data in a binary system. So a bit can either be on or off, true or false, 1 or 0. So not sure where you are getting decimal/octal bit from. Even if you meant byte, that still doesn't make sense.
@cras17: You must be joking right? Base-8 has a huge number of applications, not least to pretty much the entire field of computer science. Its incredibly important for a number of reasons, which I won't go into detail here. Just know that base-8 is used way more than you think. Google "octal" if you are interested, and take a look at the wikipedia entry on it.
@cras17 Sure, it is important to understand what you are doing, and it's OK in my oppinion to make a mistake every now and then, I mean, we're only human, but that doesn't mean the focus of the lesson is not to get the right answer. If you understand the concept, yet get the wrong answers, how much have you really accomplished?
@cras17 There shouldn't be a difference between "knowing what you're doing" and "getting the right answer" in learning something if the teacher is good.
@cras17 well it should be the goal of everything not only math but do you know math if you cant get the right answer cuz like the wrong answer might lead to another math problem or so I've heard :D
@cras17 The idea was to train america's kids to eventually get to the level of the soviet's engineers.
In regards to how it should be taught--different people learn in different ways, personally I use my own methods to get to the answers; being in college math this is fine. Some students thrive when they're taught this method though, it's all a matter of what works best.
@cras17 The focus of math education should always be about understanding the structure of problems, I agree. However, if you want pupils to perform arithmetics quickly and correctly, the new methods are completely useless. What's worse, even at the point of making pupils understand what they're doing, the new methods completely and utterly fail. Conclusion: back to the old methods. Whenever pupils have mastered those, it is not very hard to make them see why the old methods are correct.
odd.....i just wrote the last post, and it showed up under someone else's name.....mine is sailinglady and i am obviously female!
steve19541000 1 week ago
so.....reasoning is only allowed to be called reasoning if it can be spoken or written? hmmmmmmm......that's really a disservice, i think, for people who are not articulate. and, glamador, im not sure that "old taught" people ever encountered real math. as an "old taught person, arithmetic was the norm and math was the exception. and i have a BS and MS in mathematics.
steve19541000 1 week ago
[SYNTAX ERROR]
SHTrough 2 months ago
@lipsynchorswim The answers are different in Base 10 and in Base 8, because the starting numbers represent values that are fundamentally different. You can indeed verify that 147 (8) = 103 (10).
ui0ekim52 3 months ago 4
@ui0ekim52 147 (8) = 169 (10).
kodo690690 2 months ago
This has been flagged as spam show
@kodo690690 i think you converted it wrong.... 147 (8) = 103 (10)
MrBigEnchilada 1 month ago
"Is that clear?"
Yes, yes it is!
wolfmyth007 3 months ago
Oh I laught my head of!
29197yyy 3 months ago
Heresy! More witches.
showjumper57 3 months ago
great dubbing and deadpanning :)
MatthewCBridges 3 months ago
That's it, this is going to happen at my school talent show. AND I WILL WIN, DAMMIT.
LABBChaCha 3 months ago
What's a pirate minus the ship?
YouGot4Shotted 4 months ago 28
@YouGot4Shotted An alcoholic bum.
lipsynchORswim 4 months ago 24
@lipsynchORswim Just a creative homeless guy.
NumbFlynn 3 months ago
@YouGot4Shotted A pirate minus the ship equals a creative homeless guy!!! I guess we both have odd music choices.
wertacus 4 months ago in playlist wertacus's favorites
@YouGot4Shotted just a creative homeless guy
TylerrJayy 3 months ago
@YouGot4Shotted a creative homeless guy :)
LegiterallyAwesome 3 months ago
@YouGot4Shotted he is correct on what he is saying. the new math is rather for dummies. we are from a different day of teaching and teachers; a totaly different format. In the third grade the teacher will give the child the problem of 4+4= well we all now the answer is 8. if the child should be taught about reasoning and he answer with 8 it will be marked wrong because he did not use reasoning to get the answer.
sjsingleton68 2 months ago
As someone who has interacted on a regular basis with family members and friends who are "bad at math" I can state with 100% certainty that so-called "new math" is better for bringing up children with an understanding that better allows them to comprehend the concepts needed for more complicated mathematics than addition and subtraction.
It may seem like this is a stupid round-about way of doing things, but I see so many old-taught people with *ZERO* understanding of real math.
Glamador 3 weeks ago
But would the answer in Base 10 be equal to the one in Base 8?
Arexsos 4 months ago
@Arexsos Yes, they're both equal, but because they are in different bases, the numbers are different. Just like how 100 in binary is equal to 4 in Base 10.
lipsynchORswim 4 months ago 3
@lipsynchORswim He took the same digits in the base 8 example as he did in the base 10 example. Thus, the two numbers in the first example (in base 10) are NOT THE SAME AS the two numbers in the second example (in base 8). IF you convert the numbers beforehand, do the operations, and then convert the answer, THEN you will find the two answers are equivalent.
So no, the numbers are not equal. 169 in base 8 is 251. (Note: 2 * 8^2 + 5* 8^1 + 1* 8^0 = 128 + 40 + 1 = 169)
IgnantUTubeUser 4 months ago
@lipsynchORswim Oh jeez, I need a refresher course... For a moment there I thought 4 in Base 2 was 10...
XaleManix 4 months ago in playlist More videos from lipsynchORswim
@lipsynchORswim
Incorrect....
Take the example: 100 base 8 - 0 base 8 = 100 base 8; 100 base 10 - 0 base 10 = 100 base 10. 100 base 10 != 100 base 8. They are different numbers before hand, so you will get different numbers afterwards. If the problem was to give the answer in base 8, you could either change the operands to base 8 _before_ or the sum _after_, getting the same answer because you are still using the same numbers.
minime1235able 3 months ago
@minime1235able try that one more time, with all bases.
TheKhive 3 months ago
@TheKhive I am sorry, I am not sure I understand there :P What part would you like me to do again?
minime1235able 3 months ago
@Xyzyxx The joke is that he got the wrong answer but the right concept.
leafvillagerox 5 months ago
huh?
lionsruleu 5 months ago
Comment removed
Xyzyxx 5 months ago
I don't get the joke... 13 minus 7 is 6, not 5.
Xyzyxx 5 months ago
@Xyzyxx it becomes 69. . . .
ferdianeary92 5 months ago
Bye Bye Brain -.-
Mose1604 6 months ago
screw the logic, it's catchy and funny... that's all, goodnight...
TheWolfJeff 6 months ago
@TheWolfJeff the answer is correct in base 8
NPKing123 6 months ago
@NPKing123
I said "screw the logic" and "goodnight" I wasn't expecting a reply...
TheWolfJeff 6 months ago
I've not bothered to read all the comments but to everyone who says the answer is wrong or the equation is wrong you're both right... You have to realise he takes the base 8 answers and converts them to base 10 in order to get the answer. He's using the irrational for of mathematical logic to dictate to you thru song that the new maths is incorrect. In other words he is trolling the maths community and for that I applaud him.
Maharaghi 6 months ago
This has been flagged as spam show
This video was just added to a music playlist at JustPlay.fm
justplayfm 7 months ago
I like the song but I would like to dislike the person.
Excaliburproxy 7 months ago
@VelaVenti92 That's not the point. Everybody should use the method he/she prefers (the're the same anyway). I'm claiming that the new method is ineffective to learn to pupils. I know this: I'm one of these pupils. Although I understood, many classmates didn't. IMO because the algorithm and the proof that it works are mixed in an over-complicated way. Separate the two (first learn to do it, then to understand it), and things should improve rapidly. I've tried this, it worked for my pupils.
MathTine 7 months ago
Ohhhhhh I want to hear this problem done in base 12!
SleepyMongoose 7 months ago
Haha, you sort of look like Mike from Friends :')
CaitlinJF 7 months ago
How exactly was math taught before? did you memorize tables? 3-7=6? Where does that come from?
13Penguins 7 months ago
Hey, I grew up with this song in the 1960s. No, I'm not a mathematician. I loved your video, it was pretty accurate as to what is going on. This was a great video.
benreaves 8 months ago
69
RJtheStAr1 8 months ago 27
this is one song that I showed my math teacher. She then proceeded to teach the class how to do calculations in base 8 (We ran out of material for the year)
NPKing123 8 months ago
I'm dyscalculic.
My numerically retarded brain is burning...
EmmaAndHerKind 8 months ago
@EmmaAndHerKind me too whaaa
still love the song though
Tamashi88 8 months ago
Brilliant
blokeinthekitchen 8 months ago
" '64? Where did 64 come into this?' I hear you cry."
eviethebunny 9 months ago
i remember this video from my calc class lmao
babx13 9 months ago
Your awesome dude how did you come up with the song?! il love the part when you say "new hoo hoo math!" lol!
jules7lego7dude7 9 months ago
@jules7lego7dude7
He didn't, he's just lipsyncing with it. The guy who made this song is Tom Lehrer. He was mainly active in the 50's though I think, so he doesn't write songs anymore. Still, look him up, all his stuff is like this :D.
GrimEncounter 9 months ago
@jules7lego7dude7 This is a lip sync (Hence the up-loader's name is "lipsynchORswim"). The song is by Tom Lehrer. If you want more information on Tom Lehrer or want to hear more of his songs, I would suggest you go to the page @6funswede (Who is an absolute nut about Tom Lehrer).
mabarry3 7 months ago
nice glasses bro
flowiepanda 10 months ago
oh, my head. ow! LOL! Thanks y'all, this is awesome. Looking forward to looking at the rest of your videos!
Pemigewasset 10 months ago
headache @_@
beesweet0224 10 months ago
I remember being taught "new math" back in the 60s. I always thought it was nothing more than a new way to torture school children.
MegaNeilFan 10 months ago
LOL @ "ed biz"
pchapman905 10 months ago
@Menarkh Oh! So 'new math' is just writing out the steps rather then doing it in your head and writing it down. That makes sence, thanks!
sagbag 10 months ago
@sagbag
That's not actually what "New Math" refers to. My understanding with the whole "new math" thing is that it was a movement to get kids to understand set theory and the laws of arithmetic before actually being taught how to do arithmetic. The idea was that kids would figure things out from the ground up and in doing so understand it better.
Ziiiv 10 months ago
@Ziiiv
Personally I think trying to teach rigid exact mathematical concepts using the vague English language makes things far more difficult. It's much easier to understand math when you have an exact algorithm to study which conveys things much better than trying to figure out what someone means when they talk about abstract concepts I can't relate to and explicitly avoid using metaphor.
Ziiiv 10 months ago
Tom Lehrer, i have a math problem for you. 0/0
Halfpintbiwin 11 months ago
Can anyone explain "old math" to me? He says "3 from 2 is 9". Without doing it in "new math" I have no idea how you are supposed to know that. Just by looking at the equation I know I can't take 2 from 3 so I make it 12 - 3 = 9. Since I did that, I have to make 4 into 3 etc. But without doing that, I don't know how you get 9 without just knowing it and not doing any actual math. Anyone that remembers/understands "old math" want to clear this up for me?
sagbag 11 months ago
education is not a business. if it is, school would be HARD
jecho024 11 months ago
press 7 and 8 really fast :)
IPNdoctor20 11 months ago
2-3=9? 3-7=6? What? They used to really fuck up math. Now we actually understand what the hell we're doing, and we can understand other math systems that use different base numbers, like the Mayans. Progress kicks ass.
TacticusPrime 11 months ago
Well my math teacher has showed this movie before. I want to share it with my family.
924Gz 11 months ago
Wait. "Base 8 is like base 10... if you're missing two fingers" I thought the thumbs didn't count as fingers...?
KXMelodyKX 1 year ago 18
@KXMelodyKX
Yeah, but the people who made up our base ten counting system didn't have different words for fingers and thumbs.
carultch 7 months ago
@KXMelodyKX but what is base 8 and 10?
xboxliveShadow144450 5 months ago
uh-oh, hes wearing dark glasses, must be blind, right?
kozmosnotgay 1 year ago
he's mouthing the song.
mr13579100 1 year ago
@mr13579100 Duh. Not the username
thedoctorsfeziscool 11 months ago
@thedoctorsfeziscool *Note
thedoctorsfeziscool 11 months ago
is he blind?
no thumbs!
Mageneated 1 year ago
is he blind?
Mageneated 1 year ago
I remember my dad playing this and other Tom Lehrer songs when I was a kid. Awesome job!
Nicronon 1 year ago
1:51 69 xD anybody get the joke?
TinklyButtons 1 year ago
6 people are missing 2 fingers...
thetrueminigaara 1 year ago 2
Judging by what I know, and I could be VERY wrong since I don't know this concept yet, wouldn't 342-173 in base 8 be 211.25?
YouDuck2345 1 year ago
@YouDuck2345 Hmm... no, it's 147. They explain why in the video above, and my computer's calculator can confirm it's the right answer... Also, I really don't see how you'd get a decimal number by substracting two non-''decimalised'' numbers, no matter which base.
ChevaliersEmeraude 1 year ago
@ChevaliersEmeraude See, I tried to come up with an explanation for a quote I'd heard in a video game -- that 2+2=10 in base four. I noted that two plus two is, according to amateur math, four. It matched the base, so I assumed that any time a number matched its base, it must equal ten. Thus, I crunched the numbers, divided them by eight, and then multiplied the result by ten to make the number match my theory.
Obviously, my theory didn't work, but hey, it was worth trying.
YouDuck2345 1 year ago
@YouDuck2345 Well, you do have the base of the concept right, thought. Base 4, for instance, would be counted as 0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33, 100, 101, 102, 103, 110, 111, etc. See, 100, here, is the same as 16 (or 4-square) base 10. So yeah, you had the bases of the system right. But you can't just divide and multiply to change the base 4 to base 10, Truth be told, unless you're used to a system, it can be hard to switch any large number from any base to base 10.
ChevaliersEmeraude 1 year ago
This is brilliant lip-syncing. Great job! Also, great song!
Avyncentia 1 year ago
Gaaah you're such a cutie I can't stop watching this xD
I think I've memorized the song by now lol
My best friend got me into Tom Lehrer lol
peachesNpears95 1 year ago
A bit is a bit. I intended to show that we were using a 1,000 octal bit - that is, a physical matrix of ferrite cores that could be polarized in 2 different ways that was constructed to respond to octal arithmetic internally - memory. This was a cubic structure about 5 inches on a side, weighing about 2 pounds, made mostly of iron! We numbered each bit within the core with its octal address. Enough said.
paulrank 1 year ago
I'm horribly confused... What's wrong with this style of subtraction?
LGuardinal 1 year ago
isnt base eight called octal and really only used in hexidecimal computer systems?
smrterthenu 1 year ago
You ask a silly question you get a silly answer LOL
FerchOMulder 1 year ago
I want to respond to the post by Cras17, re: Lehrer's joke that "...the idea is to understand what you're doing, rather than to get the right answer." I strongly disagree with Cras17's comment that "...this should be the focus of learning math." Mr. Lehrer's joke is a logical fallacy. I don't understand how it is possible to get the right answer if you DON'T know what you're doing. The "base" thing itself is too "Rube Goldbergish." Why go through that?
Thundermist04167 1 year ago
Whats wrong with just using a calculator?
ms18ins 1 year ago
gr8 video, havent heard tom lehrer for along time.
seshold 1 year ago
win.
peachesNpears95 1 year ago
I hate how they are saying you need to know what you are doing. No, you just need to know if what you are doing will get you the right answer. It's math, all that matters is the answer. All you really need to know is what formula, if you will, to use in any situation.
15Smartie15 1 year ago
@15Smartie15 That is what you need to get through school. True mathematics requires a good imagintaion.
ciacho00000000 1 year ago
@ciacho00000000 Well said. In math education do we just want student to 'get through it' or do we want them to actually learn how math works.
cras17 1 year ago
After watching this two times, I realized what was happening in the beginning problem. It is funny because I am now in eighth grade and I did my subtraction problems the same way until second grade when they said my way was wrong and that I would get it wrong often. Well, I kept on doing it my own way until I got brainwashed into doing it their way..Honestly, the old way was faster and more efficient than the new way.
ShadowDetiny 1 year ago
Hmm.. Computer videos are more primitive than blackboards. True, true.
That guy is creeping me out with him mouthing everything.
mabarry3 1 year ago 2
I like base 12. Which is what our measurement of time is based on, btw.
abebou 1 year ago
@abebou and which was created by the Sumarians. It is a very useful system, considering 12 is divisible by 2, 3, and 4
ciacho00000000 1 year ago
@abebou 12 is interesting in that we actually have natural English words for powers of 12 (up to 12^2 at least) which we don't have for the other bases (except 10 of course).
Consider "eight gross eleven dozen and ten". Using "A" for ten and "B" for eleven this is 8BA (base 12) = 1294 (base 10).
softy8088 1 year ago
OMG!!! THANK YOU!!! I FINALLY UNDERSTAND BASE 8 MATH!!!!
SaphiraPorter 1 year ago
after watching this video i have decided to do all of my math hw for a week in base 8
LondonsBurning5 1 year ago 14
@LondonsBurning5 I SHOULD DO THAT!!! lol.. but i am doing powers....so umm uhhhhhhhhhh SHOOT!
SaphiraPorter 1 year ago
@SaphiraPorter Actually, the proper name is exponents.
ciacho00000000 1 year ago
@SaphiraPorter the proper name is "exponents". Besides, base 8 isn't limited to substracion. You can do exponents the same way as base 10, binary, trinary, hexadecimal, and any other numerical system. for example 5^2 in base 8 would be 31.
ciacho00000000 1 year ago
@ciacho00000000 yea... you knew what i meant lol but i'm not thatgood at base 8 yet. you sound like a math genius, i feel dumb...but i know what base 8 math is... my math teacher doesn't XDXD hey your like me... i answer to 3-5 pestions or comment per vid lol i try not to.. but i do! lol
SaphiraPorter 1 year ago
@LondonsBurning5 next time try binary. It's much more fun. Trinary isn't bad either
ciacho00000000 1 year ago
@LondonsBurning5
Yeah, good luck that!
924Gz 11 months ago
I love this video. :) Always makes me smile.
Xiakoda 1 year ago
This video has inspired me and I am currently trying to figure out a way to make a song that contains similar stylistic elements to explain the proof converting ax^2+bx+c=0 to x=[-b-+(b^2-4ac)^(1/2)]/2a. Wish me luck on that.
Meanwhile, this easily one of my favorite videos on Youtube. I'm not easily swayed to bestow such honors because I know that they never mean much, but in my mind this has the makings of a classic. I hope to show all my friends in Calculus this video.
Monosmith 1 year ago 5
@Monosmith look for the quadratic formula song
geraldinglin 11 months ago
@cras17: I agree
Krinek87 1 year ago
haha my teacher made us watch this today in maths class! we all laughed. nice job matey!
sprouseluver1234 1 year ago
Ironically, there are human languages documented that use base 8. The hypothesis is that these languages were spoken by people who counted the spaces between their fingers, not their fingers.
Telumel 1 year ago
@Telumel There are also areas of the world in which people use base 12 out of habit, because they counted using the sections of each finger on one hand, using the thumb as a pointer. You'd start at the base of your little finger and move up that, then go to the base of your ring finger, et cetera. I've taken to using that myself because if I need to count on my hands (and I don't often) it lets you go further on just one hand and requires fewer changes than using each finger as a binary digit.
VinuVonVin 1 year ago
Wait a second, I feel like previous versions of this had less laughter in them... or am I just going crazy?
Kebert442 1 year ago 4
@Kebert442 You're going crazy.
lipsynchORswim 1 year ago 24
Who would ever do anything in a base 8 system? That is totally counter-intuitive to our entire number system.
He did have a good point in there though. The idea is to understand what you are doing, not to get the right answer. He said it sarcastically but that actually is and should be the focus of learning math.
cras17 1 year ago 15
@cras17 i never learned the base 8 system so i'm lost and where did he get the 64?
TheShadowsLotus 1 year ago 11
@TheShadowsLotus Base 8 works in a similar fashion to base 10 (which most of us are used to). Just like base 10 has a 1's place a 10's place and a 100's place, base 8 has a 1's place an 8's place and a 64's place. The reason why is because in each base, the places are to certain powers (0, 1, 2). So since something to the 0'th power is 1, something to the 1st power is that number, something to the 2nd power would be squared. Therefore, 10^2=100 and 8^2=64 That's where the 64 came from.
lipsynchORswim 1 year ago 21
@lipsynchORswim thanx ^.^
TheShadowsLotus 1 year ago
@lipsynchORswim "Base 8 works in a similar fashion to base 10 (which most of us are used to)" speaking of familiar number systems, I hear some people in central american used base 20.
CrabTastingMan 1 year ago
This has been flagged as spam show
@lipsynchORswim "Base 8 works in a similar fashion to base 10 (which most of us are used to)" speaking of familiar number systems, I hear some people in central america used base 20.
CrabTastingMan 1 year ago
@TheShadowsLotus In base 10 you go 10,100,1000 because 10x10 = 100. 10x100=1000 etc. In base 8 ( again totally ridiculous and useless as a system) would be 8,64,512 because 8x8 is 64. 8x64 =512. It doesn't makes much sense because our numbers are not set up like that. There are 10 twenties, 10 thirties, 10 fourties for a reason. Anyway, it was still a funny video.
cras17 1 year ago
@TheShadowsLotus Another way to think about it in base8. There's the 1's place which goes up to 8, before rolling over. so the number 10 would represent 1. 8 or 8. 20 would represent the number 16 or 2 '8.s The 3rd slot would only roll over once it has 8 "(8s) so 100 would be 64. that would only roll over once you have 8 of those. So each place is essentially multiplying by 8 to move one over. Just like in base 10 you multiply by 10.
jilow 1 year ago
@TheShadowsLotus 8x8 is 64
medmondson2466 1 year ago
@medmondson2466 the proper way to say it is 8^2=64
ciacho00000000 1 year ago
@TheShadowsLotus listen to the song again...
ciacho00000000 1 year ago
@TheShadowsLotus Well, 64 is 8 squared, don't you see?
commentreviewman 1 year ago
@TheShadowsLotus
Well... 8^2 is 64, you ask a silly question you get a silly answer.
XMarkDemon 1 year ago
@TheShadowsLotus We have a hundreds place because ten tens is 100; in base eight (which is useful in some computing languages and other mathematical alleyways), you'd have a place for eight eights, and eight eights is 64. (He explained this in the song, you know.)
Wolfrover 1 year ago
@cras17 Ironically enough, most computer systems use a variation of base 8 called hexidecimal (which is base 16, fyi). Different numbering systems are used when there is a need to store a lot of information in small numbers (or very large numbers as in base 2 (aka binary)).
lipsynchORswim 1 year ago 11
@lipsynchORswim Ok, yes computers do it hexidecimal. That doesn't mean I'm happy about it :P and it still doesn't make it an intuitive or efficient way to count or to do math.
cras17 1 year ago
@cras17 You've got it the wrong way round, chap. It only seems intuitive and efficient because you're used to base-10. Bear in mind that if you were brought up on base 8, base 10 would seem just as difficult as base 8 does to you now. And that in base 8 it still goes 10,100,1000... and it's just that 10(base-8) != 10(base-10), and so on. !+ means "is not equal to", by the way. By extension, in base 8, the digit values for base 10 go 12,144,1750 - just as ridiculous as base-8 values in b10.
VinuVonVin 1 year ago
@VinuVonVin My point was that in our numbering system a base 8 system is not intuitive. It isn't because we are just used to base 10, it is because the way we talk about numbers is inherently rooted in the idea of 10.
We have 10 unique digits that repeat every 10 numbers, then every 100 numbers, then every 1000 numbers etc.
The only way a base 8 numbering system would be as intuitive as base 10 is if the numbers 9 and 0 did not exist and there was a switch every 8 numbers.I say we stick with10
cras17 1 year ago
@cras17 You're conflating intuitiveness with what-people-are-used-to. No number base is actually more intuitive than any other. Base-10 is in no way intuitive, but it /does/ mesh nicely with the language we've developed - which, incidentally, was at one point based on base-20. You can see remnants of that in French, which is a strong ancestor of modern English. For what it's worth, I like the base the Sumerians used - 60. Quite a lot of fractions are more convenient, for a start.
VinuVonVin 1 year ago
@VinuVonVin Well I guess I would say that a numbering system that meshes well with the language would be intuitive to speakers of that language. We could debate all day on the definition of what is intuitive but I would rather not :P
As for base 60, fractions may be easier in that system (never took a look myself) but I think knowledge of place value, and decimals which our friend the metric system is so heavily based upon comes easier with base 10 and is ultimately more useful for mathematics.
cras17 1 year ago
@cras17 Ah, well, there we must agree to disagree. It's true that base 60 is unwieldy, but I /do/ think base-12 is easier to deal with than base 10 once you get used to it. And place value is, of course, not base-specific in terms of whether it works or not. But you knew that. WRT the metric system, though - we use that because we use base ten, rather than the other way around! Again, I'm sure you know that.
VinuVonVin 1 year ago
@lipsynchORswim Older systems used Octal quite a bit.
aymric 1 year ago
@lipsynchORswim Many many many computers use base-8 (Octal) directly.
pleappleappleap 1 year ago
@lipsynchORswim hex is actually a way of binary conversion, all computers at their very roots run in base 2, as you mentioned, hex just takes all those lines of 1 and 0s and turns them into something slightly more manageable
1metalnation 1 year ago
@cras17 The earliest computers [S-100 Bus] I worked with - and programmed assembly language a little - used base 8.
d455ave 1 year ago
@cras17
If a child is getting the right answer 100% of the time, does it matter in the real world if they are building a bridge whether or not they understand 'why'? It's -math-, not philosophy. If a child understands the 'theory' but is right only 70% of the time, they are useless for all practical real world applications.
I was a champion mental math student (won competitions) as a child -and- knew what I was doing, but as I grew older my schools did their best to 'train it out of me'
PersephonesFear 1 year ago
@cras17
While my schools could not understand 'why' I was so far ahead of my peers in multiplication, division, math, etc (because I did math drills, could 'guess and check' fast, elimintate wrong answers quickly, memorize formulas, etc) - they thought it horrible that I never showed my work. The schools loved 'mental math' - making us spend 4 pages showing 'two different ways, with graphs' showing how we arrived to the conclusion of a given word problem. (It didn't matter if we got it right.)
PersephonesFear 1 year ago
@PersephonesFear Although I've never been particularly fast at math (largely due to inefficiencies in my thought processes) I was good at it and ran into the same issue. Our version of the "new math" also had us showing 2 different ways of looking at a word problem. I would figure it out and then have fun deciding which way I would illustrate it this time. I do, however, think that more formula work early on would have helped me, as it would have removed some of my aforementioned inefficiencies.
jokester251 1 year ago
@cras17 most of the low level computing languages do arithmetic in base two (binary), base eight (octal), and base sixteen (hexadecimal)... if you were going to do some coding in ForTran back in the 70's for example, you'd need to know your octal. Octal is a useful way to express 8-bit values. Anyway, you can do whatever base you want. You simply select your digits and columns on that base. So base 5 has 5 digits (0,1,2,3,4) and columns in powers of 5 (1's, 5's, 25's, 125's) etc.
PlasteredDragon 1 year ago
@PlasteredDragon "Octal is a useful way to express 8-bit values"
...No, it isn't. One octal digit equates to three bits. Three doesn't go into eight. Hex is a much better way of expressing 8-bit values, since one hexadecimal digit equates to four binary digits, so you can represent any 8-bit value with two hex digits. 8 bits equate to either two or three octal digits, so octal clearly /isn't/ a veryy good way. It's good for 9-bit values, though, and some machines used 9 bits to the byte
VinuVonVin 1 year ago
@cras17 Lots of ancient civilizations used base 8 (counting the spaces between fingers instead of the fingers themselves).
Alephantiasis 1 year ago
@cras17 well, concidering how the metric system is based, compared to what you guys use :) Then talk about counter intuitive :p ..
skvakagud 1 year ago
@skvakagud I live in Canada. I use the metric system. I think it is the greatest thing to happen to measurement ever. All SI units should become standard everywhere and always. Imperial influence does bleed in from the US but support for metric is growing there slowly. They just need to suck it up, get off the fence and make a full conversion.
cras17 1 year ago
@cras17 ah sorry from the comment i respondet on, i got the impression you were from the US, my bad. :D But yeah it really is silly concidering they've been "trying" to convert to the metric system, for more than 100 years now. But i think it has to do, with the way more and more workspaces, are working with milimeters, and therefor want cm as well, rather than inches. + They use tonn more and more, and pounds, ft, yards & such. Are just hill-billy meassure systems if u ask me :p
skvakagud 1 year ago
@skvakagud Imperial - A hillbilly measuring system. Couldn't have put it better myself.
cras17 1 year ago
@cras17 You're as full-of-shit as he makes it obvious you are.
pleappleappleap 1 year ago
@pleappleappleap change can be difficult, but change for the better is worth it. The US needs to grow a pair and leave the inch behind.
cras17 1 year ago
@cras17 true, the general public of the United States does rely on the English system of measurement, however a lot of the scientific community has switched to metric, mainly to provide uniformity with the scientific community outside the US, but also because it's sometimes easier to teach something in metric than it is in English units.
lipsynchORswim 1 year ago
@lipsynchORswim Somewhat amusing that people refer to the imperial system of units as "English units" when we've converted to metric for everything except (for some reason) distances on roads, where we still use miles. Nobody seems to know quite why.
VinuVonVin 1 year ago
@cras17 Well, it's important to understand process because that's the fundamental portion of mathematics, but you cant' just say, "oh, well, you can't multiply 5 by 8 correctly but you understand multiplication, so it's all good" because if you send that kid into a job, they're totally boned, because their boss isn't going to care that they understand multiplication, they're going to care that the nuclear reactor is on fire because they multiplied wrong. (Future math teacher here)
snoddymcsnodson 1 year ago
@snoddymcsnodson True, but it's very annoying to lose 9 marks on a maths paper because you accidentally timed something by 11 instead of 1.1 half way through. Or if you read the question wrong. It makes tests more valid as they look at whether you can do maths rather than whether you read the question well enough.
Asperger1993 1 year ago
I actually had to use Base 8 when studying early computer structures and programming. In fact, we manually input data in the ferrite core of our Univac Digital Trainer, which had 1,000 octal bits, or 512 decimal bits, of RAM that we had to address individually to input to the various registers. Our output device was a teletype writer, and we input the data by pushing binary coded decimal groups of lighted switches to change the states of the octal bits. Not user friendly...
paulrank 1 year ago
@paulrank: Last time I checked, a bit is a bit, there is no such thing as a "decimal" bit or an "octal" bit, since a bit is a single quanta of data in a binary system. So a bit can either be on or off, true or false, 1 or 0. So not sure where you are getting decimal/octal bit from. Even if you meant byte, that still doesn't make sense.
sutasman 1 year ago
@cras17: You must be joking right? Base-8 has a huge number of applications, not least to pretty much the entire field of computer science. Its incredibly important for a number of reasons, which I won't go into detail here. Just know that base-8 is used way more than you think. Google "octal" if you are interested, and take a look at the wikipedia entry on it.
sutasman 1 year ago
@cras17 Sure, it is important to understand what you are doing, and it's OK in my oppinion to make a mistake every now and then, I mean, we're only human, but that doesn't mean the focus of the lesson is not to get the right answer. If you understand the concept, yet get the wrong answers, how much have you really accomplished?
portalguy15837 1 year ago
@cras17 There shouldn't be a difference between "knowing what you're doing" and "getting the right answer" in learning something if the teacher is good.
njpadinha 1 year ago
@cras17 well it should be the goal of everything not only math but do you know math if you cant get the right answer cuz like the wrong answer might lead to another math problem or so I've heard :D
jokemas7er 1 year ago
@cras17 The idea was to train america's kids to eventually get to the level of the soviet's engineers.
In regards to how it should be taught--different people learn in different ways, personally I use my own methods to get to the answers; being in college math this is fine. Some students thrive when they're taught this method though, it's all a matter of what works best.
drumstyx6464 1 year ago
@cras17 The point is not to work in Base 8, it's to work in an arbitrary base. Base 8 is just an easy base to work with.
NegiJew 1 year ago
@cras17 The focus of math education should always be about understanding the structure of problems, I agree. However, if you want pupils to perform arithmetics quickly and correctly, the new methods are completely useless. What's worse, even at the point of making pupils understand what they're doing, the new methods completely and utterly fail. Conclusion: back to the old methods. Whenever pupils have mastered those, it is not very hard to make them see why the old methods are correct.
MathTine 1 year ago