Why did Tom Lehrer swap fame for obscurity?
theguardian.com
theguardian.com
> This winter I'm just going to do a math course. I'm doing a three-unit, as opposed to five-unit course on infinity, which I've never done before. I'm planning to study like crazy. It's for non-math majors. I'm trying to bring in the fact that infinity is when things get complicated. In calculus, algebra, probability, geometry, everything, so I'm trying to learn things like how perspective drawing uses infinity. So that'll take me three months. They won't appreciate it, but I will. I'll have fun with it. I've been teaching a course for non-mathematicians for years, and a lot of the stuff has already been covered there.
1997 math lecture performance (13m), including "That's Mathematics" for kids, https://archive.org/details/lehrer/lehrer_high.wmv
GEO: I was surprised to learn that you enlisted in the Army back in 1955.
TOM LEHRER: That's one way of putting it, but probably not the appropriate verb. The point is that they were drafting people up to the age of 35. So I dodged the draft for as long as anybody was shooting at anybody. And then when I realized that I would have to go -- there was really no way out of it except getting an essential full time job, which I didn't really want to do -- I waited until everything was calm and then surrendered to the draft board. I wouldn't call it "enlist". "Enlist" means that you have to spend another year. I allowed myself to be drafted. I was 27 at the time and there were a lot of graduate students who were like me who had gotten deferred as graduate students and now had to pay up. So it was a kind of an odd group there, a lot of educated people in my "outfit", I believe is the word. And we had a lot of fun. So I did that for two years in Washington DC and had a great time -- especially since there was no war -- though vice president Nixon was trying to get us into one in Indo-China even then. So there was that little threat. And there was Suez and a few other little things that looked a little tricky. But it didn't look like there was going to be a real war. So it seemed to be safe to go in. And I'm sure that a lot of my cohort felt the same way.
GEO: And what did you do?
TOM LEHRER: It was NSA. I think I'm allowed to say that now. I asked around before I surrender to be sure that I would not be in special services or something playing volleyball with the troops in Korea. I wanted to make sure that I got a nice cushy job. We were called "The Chair Borned". And I found out that they were hiring mathematicians. So I arranged to be hired.
GEO: Do you find that your training as a mathematician influenced your song writing. Writing a song seems to me to be like creating a puzzle.
TOM LEHRER: Not Mathematics itself, but the kind of mind that likes mathematic. Stephen Sondheim has that kind of mind. He was a mathematics major in college, too. Having that kind of a mind, you look for organization, and rhyming, and pattern, and prosody -- all those things that are fun to do in a song, rather than -- which is what a lot of comedy songs are -- just couplets. Working all that out, if not "mathematical", is at least "logical".
GEO: As a mathematician did you ever make any brilliant discoveries?
TOM LEHRER: Oh,nonono. I have no desire to extend the frontier of human knowledge; retract them, if anything. I like to teach it and I like to think about it, but that's about it.
What is this latter? Is it https://en.wikipedia.org/wiki/Cubic_equation#Cardano's_formu... ?
Yes, that's exactly right—no solution of the general quintic in terms of radicals. This study introduced the term "soluble", and the group-theoretic version of the insolubility of the general quintic boils down to the fact that S_5 is not soluble, whereas S_2, S_3, and S_4 all are.
One of the episodes is called “Sets and Infinity”. Lots of my friends watched it and I got a lot of positive feedback, they subscribed to the channel! It takes one hour, not three months. Check it out.
https://youtu.be/xzaqERDWt9Q?si=cy1x9iVdem9KjdMM
If you watch it — please tell me what you thought! Anything good/bad etc.
https://en.wikipedia.org/wiki/Artie_Shaw
Gerry Rafferty (probably more commercially successful than Lehrer, at least) also didn't like the business or performing aspects of music.
Bill Murray said (paraphrasing), "I always say to someone who wants to be rich and famous, 'try being rich first, and see if that doesn't get you 90% of what you want.' Being famous is a 24-hour-a-day job."
That's why, people whom are smart and/or lazy won't touch it.
I was a science major and I said to myself “Adam, I don’t care what Tom Lehrer says, there’s no way you you’re not taking a math class from Tom Lehrer.”
He was bluffing. I stayed and loved every minute of it.
The kind of quiet fame that Lehrer managed, known to a smallish segment of the population, for being really good at something, and then going off and living a normal life sounds pretty great to me.
As much as I love the maths based songs, my favourites are still Oedipus Rex and The Vatican Rag.
Tom Lehrer and Flanders & Swann were the musical background to my childhood. If you enjoy Lehrer you might enjoy F&L too.
While they don't have the science/maths background that makes Lehrer an obvious win for the HN crowd, F&Ls songs were razor sharp satires of the time. One or two have not aged so well, but most are great, although knowing a bit of British history helps. Like Lehrer they wrote songs about the insanity of war, nuclear weapons, and prejudice (A Song of Patriotic Prejudice is an awkward listen because of the terms used, but a great representation of English exceptionalism in the post colonial era).
https://www.youtube.com/watch?v=sLkGSV9WDMA
A 2009 follow-up to their original song from 1993, the sun song, in which they erroneously claimed the sun is a mass of incandescent gas, to weigh the follow-up song is a correction.
(presumably Brian May would have had his phases of astral matter correct in the first set of lyrics?)
True, but they did one on thermodynamics https://www.youtube.com/watch?v=VnbiVw_1FNs
"I always want to say to people who want to be rich and famous: 'try being rich first'. See if that doesn't cover most of it. There's not much downside to being rich, other than paying taxes and having your relatives ask you for money. But when you become famous, you end up with a 24-hour job."
-- Bill Murray
>You had to admire these folk singers,” he says on the live LP. “It takes courage to get up in a coffee house or a student auditorium and come out in favour of the things everyone else is against, like peace and justice and brotherhood, and so on.”
It's pretty much my reaction to every protest or social concern story in the media.
I am also amused that I learned "new math" in elementary school, but I think it's actually different from the "new math" in his song of that name. I didn't hate it, honestly. "Now the book wants you to do it in base 8." This is actually relevant to my day to day work. (Though in base 8, I have to say that I only ever use bitmasks. chmod 755 foobar)
It was a fool's game. Just like every other "innovation" in mathematics education since, up to and including Common Core (one of which you probably encountered). At the elementary school level, the only way to increase math proficiency is drilling. Drill the basic math facts and standard algorithms until the kid knows them by heart and can do problems as easily as breathing. Only then will they be ready for the higher level, conceptual stuff.
The Soviets were ahead in math and science because they drilled their kids harder. Any kid who didn't want to drill was a traitor to the working class. (Were I a right-wing conspiracy theorist, I'd say New Math and its successors were Soviet psyops designed to sabotage math education in order to weaken the west. Instead I think it's more likely a psyop by the bourgeoisie to make us more compliant and exploitable slaves through mass innumeracy.)
No surprise, then, that today, when Americans really want their kids to learn math, they use the curriculum from a country where they cane you for minor infractions. They use Singapore Math. Math is hard, and hard things can only be mastered through discipline.
(I had a geometry teacher who had been excited because his daughter wanted him to sign something saying she'd be allowed to take "Sets Education". Imagine, finally sets being introduced at the Jr High level! ... and then he realised he'd misheard: there was an "x" at the end of the first word)
Lagniappe: https://www.smbc-comics.com/comic/why-i-couldn39t-be-a-math-...
* or undergraduate discrete maths, of the sort you'd want for any halfway decent CS programme. I'm glad I got a cheap'n'cheerful intro to lattices in 5th grade.
So you had this really ugly failure of teachers not really necessarily being prepared to teach the material combined with the rush to roll out the curriculum across the board instead of expanding it year by year so there wouldn't be a sudden change in expected education.
To this day, that enthusiasm for those cases, for understanding both how each side of these cases is both convinced they’re in the right and how often the cases pivot on very narrow aspects of the law has carried over for me decades later. Those lessons and the insights have shaped not only my passing interests in law and politics, but how I approach day to day conflict and debate.
If they had been unenthusiastic and dry, like so many other teachers I’d had, theirs would have been just another boring US history class with a jerk of a “difficult” teacher.
It's a bad thing if you never learn how to actually get the right answer. Which unfortunately seemed to be a common consequence of the New Math as a teaching method.
I would argue that if your "understanding" doesn't actually enable you to get the right answer, you don't really understand.
> you will never do a long multiplication in your life since a 3$ chip does it in a tenth of a microsecond
And how do you know the chip's answers are accurate?
Or what if you want to design the chip?
Or what if there's an EMP and all of the chips are fried?
More generally, if you're satisfied with just some conceptual-level "understanding" of anything that doesn't actually enable you to tell right answers from wrong, you are setting yourself up to be manipulated, misled, conned, etc. Critical thinking is a valuable life skill, and it requires you to be able to tell right from wrong answers.
There are certainly some things you can only learn effectively by doing them a lot. There are also other things that you've sort of got to learn by rote memorization (e.g. times tables, various formulas). I'm not aware of anything you can only learn effectively by the threat of physical violence.
For what it's worth, I was taught logic and set theory well before I learned things like long division. Somehow, it made me like math more, and I had no problems with long division either. Maybe it might have helped more if I'd been beaten though- not sure.
It was fun for a budding math geek. However I kept failing miserably at the tests of memorization of the multiplication table. I knew how to add, so I didn't see the point of memorizing something that I could simply derive by repeated addition.
edit: I just googled it and apparently New Math was during the 1950s-1970s. This confuses me even further since reading an analog clock seems even more important in those times?
(non-American here, I may be lacking context that makes it obvious)
The only skill required is remembering that the shorter hand is hours and the longer hand is minutes.
https://images.thdstatic.com/productImages/f220d887-1a05-418...
Hour 6 is also minute 30. There’s some arithmetic for you! Six times five, if you like, or six times ten divided by two if you prefer.
Many don’t have numbers at all, so you’ll need to build a good intuition for fractions and converting those to hours and minutes if you want to read them fast. Most of us do it so automatically we don’t notice, but some of that’s plausibly fraction multiplication.
Have you actually looked into it? I was skeptical too, but then I saw they were trying to teach kids the way I do math. For example, what's 4001 - 3989. The old way would be to borrow and carry three times. But change question to 1+4000-3990+1 and the answer is perfectly simple. Kids I went to school with would literally write out 43-39 with 13-9 and 3-3. Maybe they're just dumb, but you don't have to be a whiz to use these techniques if someone shows you them.
I hope they're still doing some times table memorization.
We also practiced this stuff regularly, and had mental maths quizzes at least once a week. And (at least when we weren't learning a specific technique), it didn't matter how we did the calculations, so if you felt more comfortable with the traditional methods, you could do that, you were usually just slower. (I was one of those slower people very often!)
The point isn't to go "here's how you do maths, it's always like this". It's about (a) helping you do arithmetic more quickly, and (b) helping you understand why numbers behave the way they do, in terms of bases and pairs and factors and other things like that.
I did not expect this short-sighted way to look at maths education here in this website.
For example, the curriculum indicates a teacher must teach 4 strategies for multiplication. Totally reasonable. But then the test will have questions like "Perform this multiplication using strategy foo" which seems like putting the cart before the horse. Isn't the whole point of teaching multiple strategies is so that at least one sticks?
[edit]
And no, they aren't doing times (nor even addition) memorization in school. We did them at home and the benefits were absurd.
I’ve had three kids go through the early grades with it. I was on the fence at first. It turned out to be awful. The kids hate it. We hate it. There’s weird unhelpful bullshit vocabulary everywhere (“let’s use ‘number sentence’ in kindergarten before we’ve taught kids what a regular sentence is, that’ll surely help!”). Solving the same problem five ways which is infuriating to a kid who “gets it” already and has been very harmful to their opinion of school in general. Their deeper mathematical understanding doesn’t seem to be any more advanced than mine was in elementary school, which was supposed to be the point, and we’re having to supplement the “bad” stuff like multiplication tables so they’re not lacking the very most important math skills in every day life and needed to make actual progress on wrapping one’s head around even simple stuff like, say, algebra involving fractions.
Terrible, way worse than even my more-pessimistic guesses would have been.
That was not how the Soviet schools worked. There was no corporal punishment, and struggling students were given help.
Instead, it was quite competitive. There was a system of academic competitions ("Olympiads") that went all the way up from schools to the international level. Students were encouraged to compete.
At the same time, school sports competitions were pretty much absent.
Punchline 1: (little Johnny sent to the principal after making a comment about physical attributes of the teacher instead of Pushkin's next line, says to the visiting teacher evaluators) "Next time, sirs, when you don't know, don't prompt!"
Punchline 2: (a former maths prof turned shipbuilder, trying to coast, read literature, and earn some extra cash in "maths for the proletariat" evening school, is stuck at the brownboard having calculated a negative circumference of a circle = -2πr; his classmate helps) "Psst, colleague, swap order of integration!"
Lagniappe: https://www.youtube.com/watch?v=VogwWGEO1-c&t=10574s
* I distinctly remember being taught the "right" way to raise hands in 4th grade, and in particular placing one's elbow on the desk and putting a forearm perpendicular to it was "wrong". Imagine my surprise upon discovering, many decades later, that they for sure didn't want us somehow (despite the iron curtain — as if we'd had had a clue anyway) copying soviet classroom protocol.
New math sounds like it would have been idea for me.
I recommend reading "A Mathematician's Lament" https://maa.org/sites/default/files/pdf/devlin/LockhartsLame...
[edit] my suspicion is because it’s both entirely contrary to my experience, and always seems to come from people who like math so much that they majored in it and started to really love it right around the time the folks I’m thinking of above—a far larger set—have decided not just that they dislike math, but that they can’t do it and also that hardly matters because it seems to be pointless.
You can't play a Chopin sonata, at least not smoothly and beautifully, without having played lots of boring scales and finger exercises. And you can't get to the fun, creative, beautiful bits of mathematics without having drilled in the fundamentals. Not unless you're Gauss or somebody.
> fundamental knowledge ingrained into the fiber of his being
I agree! That knowledge being: mathematical reasoning, logic, capacity for abstraction, rigour... and NOT doing stupid mindless work over and over again until you can recite multiplication tables by heart. Sorry, that's accounting, not mathematics.
https://gist.github.com/seanjensengrey/f971c20d05d4d0efc0781...
https://www.youtube.com/watch?v=27PSHASlGUU
He nails the Boston Brahmin accent so perfectly lol.
And of course the elements song is a total banger. https://www.youtube.com/watch?v=U2cfju6GTNs
That's an odd one to mention; he has nothing to do with its musical qualities. He didn't compose it.
Antimony, arsenic, aluminum, selenium…
Of course, I only know the lyrics due to the fact that I am the very model of a modern major-general…
But regardless, how much do you think you'd enjoy listening to the words absent the music?
Tom Lehrer has plenty of clever wordplay going on, of various types in different songs. When You Are Old And Gray has a fun example of wording for the sake of wording.
But I don't think wording is a strength of The Elements.
The "right" way that he presents at the beginning of the song is a way that I've never encountered anyone doing. It's actually fairly interesting:
> Consider the following subtraction problem, which I will put up here: 342 minus 173. Now, remember how we used to do that…Three from two is nine, carry the one, and if you’re under 35 or went to a private school you say seven from three is six but if you’re over 35 and went to a public school you say eight from four is six, and carry the one, and we have 169.
"Three from 2 is 9, carry the one"; it seems to be a completely algorithmic way of doing the calculation, where you end up with the right answer at the end, but it's completely detached from what's actually happening. Tom Lehrer - a math teacher, it should be noted - was mocking the idea of teaching people what was actually happening with subtraction. We see a similar thing today, where people mock the idea of teaching math concepts with Common Core because they think people should just use an algorithm to get the answer, even if they don't understand what's happening.
Here's a good animated version of the song that shows the different methods. You'll notice that the "silly" New Math method is the way that makes sense to Americans today, and the "simple" method preferred by Lehrer is very confusing for anyone who's not used to it:
https://www.youtube.com/watch?v=UIKGV2cTgqA
Here's more of the intro which makes it clear what's going on. My explanations in brackets:
> Some of you who have small children may have perhaps been put in the embarrassing position of being unable to do your child’s arithmetic homework because of the current revolution in mathematics teaching known as the “New Math”.
> So as a public service here tonight, I thought I would offer a brief lesson in the New Math. Tonight, we’re gonna cover subtraction.
[ New Math is confusing, and he's going to go over how confusing it is in a humorous way.]
> Consider the following subtraction problem, which I will put up here: 342 minus 173. Now, remember how we used to do that…
> Three from two is nine, carry the one, and if you’re under 35 or went to a private school you say seven from three is six but if you’re over 35 and went to a public school you say eight from four is six, and carry the one, and we have 169.
[This is how it was done before the confusing New Math, the "right" way to do it. "Three from two is nine, carry the one."]
> But in the new approach, as you know, the important thing is to understand what you’re doing rather than to get the right answer. Here’s how they do it now…
[The New Math approach is silly because it's so focused on trying to teach kids concepts that it leaves them unable to do basic arithmetic.]
After this comes the first verse, which is designed to make the idea of "borrowing" sound so overly complex that few people would be able to understand it.
Just focus on the ideas, maths is beautiful, pen and paper accounting is not.
> "Three from 2 is 9, carry the one"; it seems to be a completely algorithmic way of doing the calculation, where you end up with the right answer at the end, but it's completely detached from what's actually happening.
I can't understand what you're trying to say. The only difference in the patter is whether you carry the one before or after you subtract 3 from 2. "Both" approaches have you do the same thing in the same way. What's the contrast?
Look at this video if you haven't already:
https://www.youtube.com/watch?v=UIKGV2cTgqA
The first way he does it seems INCREDIBLY confusing to people who learned arithmetic in the last 30 or so years (the sentence "8 from 4 is 6" is nonsensical to most such people), but the "modern" way he's mocking is perfectly understandable.
I think the song is very funny and charming, but I do think this is a rare case where Lehrer is just wrong, the "borrowing" style is a much better and clearer way to explain subtraction. It is just as fast, and it gives you a much better intuition about what's actually happening, instead of just learning the steps by rote.
Anyone who understands "7 from 3 is 6" must necessarily also understand "8 from 4 is 6". Nobody learns how to subtract from numbers that have digits of 3 without simultaneously learning how to subtract from numbers that have digits of 4.
> I do think this is a rare case where Lehrer is just wrong, the "borrowing" style is a much better and clearer way to explain subtraction. It is just as fast, and it gives you a much better intuition about what's actually happening, instead of just learning the steps by rote.
But I already pointed out that the steps aren't different. They're the same style. Whether you use the term "carry" or "borrow" makes no difference to anyone.
Obviously, it does. Like, that's what the song is about. You're not just disagreeing with the other commenters, you're disagreeing with the concept of the song itself.
It's a different way of doing it, even if the underlying principle is the same. This stuff matters a lot in pedagogy, even if there's no difference in the underlying mathematics. I could say this: "you subtract two decimal numbers a_n a_n-1 ... a_1 and b_n b_n-1 ... b1 by successively calculating c_n = a_n - b_n - K_n-1 if a_n >= b_n, where K_n is the carry from the nth digit, or c_n = a_n - b_n + 10 - K_n-1 if a_n < b_n, and you set K_n to 1" or whatever. That's the same method, but it's a TERRIBLE way to teach a child how to do subtraction.
There's no difference in the steps being performed. If you go through each approach, you'll notice that you do the same things in the same order, with the possible exception that carries might appear before or after the actual subtraction with which they are associated. All of your intermediate calculations are the same. Everything you write down is the same in both cases. Someone presented with your worked solution would have no way to determine whether you had "old math" or "new math" in mind as you worked it. Someone who watched you solve the problem would also have no way to determine that, because there is literally no difference in the method.
The "new math" part of the problem isn't that you do the base-10 subtraction differently. It's that you're expected to be able to do the same subtraction in base 8 too.
That's the second verse. The entire first verse is mocking the "New Math" idea of borrowing, showing New Math subtraction in base 10.
Beyond the intriguing assumption that an adult man might purchase this book, a manual of basic arithmetic, for the purpose of self-improvement, it's pretty much indistinguishable from what we have today. This is the treatment of subtraction:
> If any figure [digit] in the subtrahend is a number greater than the one above it in the minuend, it cannot be subtracted directly and the following method is used. A single unit (1) is "borrowed" from the next figure to the left in the minuend and written (or imagined to be written) before the figure which is too small. The figure of the subtrahend is then subtracted from the number so formed and the remainder figure written down in the usual way.
> The minuend figure from which the 1 was borrowed is now considered as a new figure, 1 less than the original, and its corresponding subtrahend figure subtracted in the usual way. If the minuend figure is again too small, the process just described is repeated.
> As an illustration of the procedure just described, let it be required to subtract 26543 from 49825. The operation is written out as follows:
7₁
Minuend: 49/25 [the 8 is struck through; I don't know how to type this]
Subtrahend: 26543
-------
Remainder: 23282
(pp. 10-11)
What do you believe were the New Math revisions to this? There weren't any; what made it New Math was insisting that people be familiar with the theoretical background that the textbooks had always provided. The algorithm, and the explanation of it, were not changed in any way.
(Older textbooks do use the "8 from 4 is 6" model instead, where carries are done into the subtrahend instead of being taken from the minuend, and they have a different explanation. They still provide that explanation for those students who care to know, which is very few people.)
Borrowing and base 8 weren't created by New Math. Teaching them to students, at least according to the song, was part of New Math. Lehrer specifically says this. In the intro he gives the way it was taught ("Now, remember how we used to do that…Three from two is nine, carry the one"), then he says "But in the new approach, as you know, the important thing is to understand what you’re doing rather than to get the right answer. Here’s how they do it now…", then he immediately shows the borrowing approach, which is followed by the chorus "Hooray for New Math!" After showing "how they do it now", he then goes on to show the same problem in base 8 for the second verse (followed by a repetition of the chorus, and then the song ends).
I get that you don't think the approaches are different, or that they're tied to New Math. Lehrer and his audience did, which is the entire point of the song.
No, he doesn't.
So first, we can observe with our own eyes that borrowing and carrying are the same thing, with only the label being changed.
But we can also observe that what was taught to students, as reflected in their textbooks, is the same thing that was taught under the label New Math and the same thing that is still taught today. Go ahead and look at the textbook.
The part that is specific to the New Math is the conversion of the problem to base 8. If you want to stick closely to the lyrics of the song, you might notice that they specify that the base-8 subtraction is the only problem posed by the New Math textbook; the base-10 version is something that Tom Lehrer provides to the audience to aid their understanding of the base-8 version.
This isn't just the clear message of the song, it's also what you'll learn if you read retrospective or contemporaneous coverage of New Math. You can see discussion in precisely these terms on the rather perfunctory Wikipedia page.¹ But most importantly, you might notice that working in alternative bases is actually new, in that - unlike the working of the base-10 problem in the first verse of the song - it doesn't appear in textbooks written a hundred years before the New Math was developed.²
The joke in the first verse is just that it's hard to follow a rapid patter. One specific joke in that verse is the set of lines "And you know why four plus minus one plus ten is fourteen minus one, 'cause addition is commutative. Right." Again, there's nothing new about this material, it's just that the explanation is superfluous to the process and paced in a manner that makes it hard to follow.
¹ Admittedly, the page's view of what was salient in New Math is pretty likely to have been influenced by Tom Lehrer's song, but that's still a radically different and more plausible interpretation of the song than what you're pushing for.
² That far back, it's all carrying into the subtrahend, but the approach of "here's an example showing each step of the process in detail, accompanied by a theoretical discussion of why it works" is already present. To get carrying out of the minuend, you need to go to just decades before the development of New Math, as the patter notes.
> But I already pointed out that the steps aren't different.
They are, with borrowing you make the change to the tens place first, "getting" the extra ten ones, then explicitly add it to the ones place, then do the subtraction.
With the old way (Lehrer's preferred method), you don't even look at the tens place, and you do - something. I'm still not sure what they were actually doing with "3 from 2 is 9, carry the one." You could mentally change the 2 to a 12 and subtract three (which would be closer to borrowing, though the steps are out of order), but the fact he doesn't say 12 and says 3 from 2 makes me wonder if this wasn't the case. Because you could also take the tens complement of the number being subtracted and add it to the number you're subtracting from. Or simply memorize a subtraction table, the same way people memorize a multiplication table.
He mentions two ways people are taught to do the next step - the first is that after "carrying the one," you subtract it from the number in the 10's place. This basically creates a situation where subtraction is the same as addition - if you have extra with addition, you carry the one and add it. If you have an extra with subtraction, you carry the one and subtract it. The other way is carrying the one to the number you're taking away and adding it to it. So 4 - 7 in the tens place becomes 4 - 8 when you "carry the one."
[I'm using tens and ones place for clarity, it could also be the hundreds and tens place, the thousands and hundreds place, etc.]
So there are certainly differences. These might not seem like big differences to you, but they're big enough that Lehrer, and apparently others, felt that people couldn't understand it when one was used rather than the other. You see the same thing when Common Core approaches come up - it might be fundamentally the same thing, but the changes in the steps that you take can throw people off.
> you could also take the tens complement of the number being subtracted and add it to the number you're subtracting from
> Or simply memorize a subtraction table, the same way people memorize a multiplication table
Well, you can't take the ten's complement of the number being subtracted, because it's infinitely large. One obvious difference between subtracting 3 and adding ...99999999999999999999999999999997 is that it's possible to write "3".
You definitely can memorize a subtraction table, and that's the approach being taken in all cases you've mentioned so far. Including the new math approach; indexing your table entry under "12" and "3" is not a different approach from indexing the same entry under "2" and "3". As with "borrowing" versus "carrying", it is a purely cosmetic difference, where you have the same literal object with a slightly different name.
That's the reason the textbook wants you to do the same problem in a different numerical base; the author is making an attempt to force the student to solve the problem from first principles instead of relying on a memorized algorithm. This doesn't work unless the student cares about the material. But note that the author recognizes, as you seem not to, that regardless of how much theoretical background you provide for why the subtraction algorithm works, the student won't pay any attention to it unless they have to. And the algorithm itself hasn't changed - what's changed is the inclusion of the followup problem "same numbers, base 8".
Tom Lehrer implies that this approach to pedagogy is misguided; under the old system, students learned to produce correct solutions to subtraction problems and didn't know why their approach worked, whereas under the new system, we asked tricky questions that successfully revealed that the students didn't know why the approach they were being taught worked, and therefore couldn't apply it to problems of the kind that never come up. He is correct that this is pointless; we already knew that the students didn't know why the math worked.
> These might not seem like big differences to you, but they're big enough that Lehrer, and apparently others, felt that people couldn't understand it when one was used rather than the other.
As I just said, Lehrer knew that people couldn't understand it either way. The contrast is between "getting the right answer" and "understanding what you're doing"; there is no implication that people who learned the old approach understood what they were doing. But they got better marks than the new math students, because they weren't graded on whether they understood.
I am aware of one other contemporary record of societal struggles with "new math"; it came up a fair amount in Peanuts. The only example given was the problem "write the 'new math' sentence for 'three is less than five'", and the correct answer was "3 < 5".
The ten's compliment of 3 is 7.
> Including the new math approach; indexing your table entry under "12" and "3" is not a different approach from indexing the same entry under "2" and "3".
12 - 3 = 9 is quite different from 2 - 3 = 9 carry the one. The latter requires a separate explanation for what's actually happening.
> But they got better marks than the new math students, because they weren't graded on whether they understood.
People seem to do subtraction just fine with borrowing, and I've never heard anyone claim that the old method is superior outside of Lehrers song.
> As I just said, Lehrer knew that people couldn't understand it either way.
This is clearly false, though. Most people today understand borrowing just fine, while (at least according to Lehrer's song) people who studied the old approach had so little understanding of what was happening that they couldn't even grasp the concept of borrowing. If you look at what's actually being said, all of the stuff in the first verse that Lehrer is presenting as mindlessly complex for adults is completely intuitive to anyone with a decent grasp of modern elementary school math:
"You can't take three from two Two is less than three So you look at the four in the tens place Now that's really four tens So you make it three tens Regroup, and you change a ten to ten ones And you add 'em to the two and get twelve And you take away three, that's nine Is that clear?"
The sarcastic "is that clear?" is there to show how confusing this is. But it's actually quite clear for people with a modern education. The problem is 342 - 173. You don't do 2 - 3 ("You can't take three from two, Two is less than three"), so you borrow a ten from the 40, changing it to a 30 and the 2 to a 12 ("So you look at the four in the tens place, Now that's really four tens, So you make it three tens, Regroup, and you change a ten to ten ones, And you add 'em to the two and get twelve").
Not a good look for someone extolling the benefits of understanding the theory behind an algorithm. This is only true if you're working modulo 10.
I'm suddenly very curious what you think the ten's complement of 12 is.
Also try doing something like 2000-1111 in "new" method and you go on huge side quest to propagate the borrows and go back to the beginning. Compared to "old" method where you progress one digit at a time without backtracking.
How much of that is that you're familiar with this method (the same way Tom was familiar with the old method)?
But who knows, I wasn't taught the other system, maybe it's equally obvious. I do think it's indicative that the "borrowing" system is nowadays much more common (that's how I learned it in Sweden in the 90s), which probably indicates that it does have some pedagogic value. I don't think for a second either way is "more efficient" than the other: once you get the hang of the borrowing system, you do it very fast.
----
[What if the digit in the subtrahend is bigger than the one in the minuend?]
Imagine adding 10 to both numbers. Obviously, the difference between them will not be changed.
But adding 10 to the digit in the subtrahend is the same as adding 1 to the digit immediately to its left.
So, add 10 to the [current] digit in the minuend, add 1 to the [next] digit in the subtrahend, and then perform the subtraction.
Both "measure" and "subtract" can mean doing any number of things in mathematics, depending on what you're dealing with. Intuition for things in the physical world specifically shouldn't be a goal of teaching mathematics at school. It's a prerequisite. If a child already can't grasp the concept of taking away from something, throwing mathematics and numbers at them is not going to help one bit.
The text on his website suggests it won't be around for long. The archive is a git repo, you can help out by hosting it somewhere.
Back in the 80's, the MIT freshman chemistry class had A Thing where if you'd stand up in the lecture hall and sing The Elements, you'd get extra credit (I think it was an automatic A on the first quiz? something small.) I'd already placed out, but helped one of my housemates practice, which was fun (he did succeed, on the day.)
It put all the rest of our "talents" to shame!
Having known a couple of very famous people and seeing what that brought to them, I'd prefer obscurity. I don't know if that's related to his decision-making, of course.
Truer words have never been sung.
(The preambles are generally at least as funny as the songs, at least in my estimation.)
He seems to be personally, very happy.
That’s something that I find inspirational.
"Tom Lehrer Is Teaching Math and Doesn’t Want to Talk to You", brilliant.
He once said very clearly and seriously, "if you can live your life without an audience, do it."
I'm sure Tom experienced lots of the unfavorable aspects of the attention (and perhaps scrutiny) he garnered.
Personally I think his contributions would be welcome in any era, and either way “this is the year that was” is a certified banger
It's possible he thought he didn't have anything more to say. But I doubt that's the whole reason he stopped making songs and performing.
He lived through a time when the US defeated Nazi Germany, and then... hired prominent Nazis to work for them. This is what the Wernher von Braun song is about.
My take is, he thinks humanity as a whole doesn't deserve him — which may very well be the case.
> we do know that he believed satire changed nothing. He quoted approvingly Peter Cook’s sarcastic remark about the Berlin cabarets of the 1930s that did so much to prevent the rise of Hitler and the second world war.
compare "Our Long National Nightmare": https://legacy.npr.org/assets/news/2013/onion-bush.pdf
As I understand it the US was preoccupied with Japan until the USSR had Berlin within reach. US racial policy being an inspiration for european fascists, like Hitler, kind of makes the affiliation with nazis both in the US and West Germany (and NATO) easier to understand than the USSR keeping some nazis in important positions on their side.
So, what about it? Why don't you bring up the invasion of Sicily instead?
edit: found a copy: https://pbs.twimg.com/media/DeTFuNkUQAAHIwa?format=jpg
He posts a lot about Tom Lehrer!
https://twitter.com/search?q=from%3A%40ericrweinstein%20lehr...
Here's Tyler Cohen claiming Tom Lehrer would have been a part of the "Intellectual Dark Web" https://marginalrevolution.com/marginalrevolution/2018/12/to...
There's a recording of "Tomfoolery" performance on iTunes!
Tom Lehrer DAT Recordings - https://news.ycombinator.com/item?id=38778749 - Dec 2023 (2 comments)
That's Mathematics – Tom Lehrer Songs - https://news.ycombinator.com/item?id=38471908 - Nov 2023 (1 comment)
Tom Lehrer puts all music and lyrics in public domain - https://news.ycombinator.com/item?id=34038206 - Dec 2022 (1 comment)
Looking for Tom Lehrer, Comedy's Mysterious Genius - https://news.ycombinator.com/item?id=34034896 - Dec 2022 (1 comment)
Tom Lehrer has released all of his songs into the public domain - https://news.ycombinator.com/item?id=34024968 - Dec 2022 (130 comments)
Tom Lehrer – We Will All Go Together When We Go - https://news.ycombinator.com/item?id=30509279 - March 2022 (2 comments)
Tom Lehrer – So Long, Mom (A Song for World War III, 1967) - https://news.ycombinator.com/item?id=30496103 - Feb 2022 (1 comment)
Tom Lehrer on Kurt Weill's Broadway Music (1999) - https://news.ycombinator.com/item?id=27743713 - July 2021 (1 comment)
Tom Lehrer Puts His Music into the Public Domain - https://news.ycombinator.com/item?id=24882384 - Oct 2020 (1 comment)
Tom Lehrer releases song lyrics to public domain - https://news.ycombinator.com/item?id=24833683 - Oct 2020 (132 comments)
Tom Lehrer's Mathematical Songs (1951) - https://news.ycombinator.com/item?id=24279151 - Aug 2020 (44 comments)
Tom Lehrer’s memorable “Revue” session - https://news.ycombinator.com/item?id=18036813 - Sept 2018 (6 comments)
Tom Lehrer at 90: a life of scientific satire - https://news.ycombinator.com/item?id=16774608 - April 2018 (83 comments)
Looking for Tom Lehrer, Comedy's Mysterious Genius - https://news.ycombinator.com/item?id=10684409 - Dec 2015 (3 comments)
Tom Lehrer - https://news.ycombinator.com/item?id=10675682 - Dec 2015 (32 comments)
Tom Lehrer's last (math) class (2001) - https://news.ycombinator.com/item?id=1914399 - Nov 2010 (1 comment)
So, a few years ago, he put all his lyrics in the public domain, and then more recently, all his original melodies too.
Top man.
Still a fun read though.