A Mathematician's Lament:

TL;DR

>>But don’t we need third graders to be able to do arithmetic?
>Why? You want to train them to calculate 427 plus 389? ... most adults don’t fully understand decimal place-value arithmetic
>>Then what should we do with young children in math class?
>Play games! ...
>>But surely there is some body of mathematical facts of which an educated person should be cognizant.
>Yes, the most important of which is that mathematics is an art form done by human beings for pleasure!
What a colossal jackass.

>Alright, yes, it would be nice if people knew a few basic things about numbers and shapes, for
instance. But this will never come from rote memorization, drills, lectures, and exercises. You learn things by doing them
What does he think drills and exercises are?

This is some real "common core" shit. Of course, he's fantasizing about a great mathematician teaching a classroom full of little geniuses with unlimited time for "enrichment", rather than a second-rate babysitter teaching the usual crowd of barely-trainable normies who need to get their multiplication tables out of the way so they can get back to learning how to spell.

I would like to see more of the equivalent of creative writing in math, where the student is not only fed word problems, but encouraged to invent their own scenarios where math would be useful and then do the math for them.

Why the negativity? I think his point is fair. We teach mechanical plug and chug from an early age which builds little to no proper mathenatical intuition at the long run. We end with people who can't really solve problems and believe math skims down into a bunch of formulas. I do believe there is beauty in solving mathematical problems and it's far more interesting than repeating whatever is put forward in the class. That was my upbringing, because I had a wonderful tutor who helped really understand the underlying principles behind many formulas. I can tell you I was failing math until he realized my issue was that blurry concepts made it difficult for me to remember them and after that I had to fo half as much work (even less) than my peers. And no, it was no bullshit handwavy common core bullshit, but actual principles he new I was going to encounter in more advanced math problems. This has also worked for me and my students from all ages.

Current year schools don't teach. They babysit you inattentively for 13 years while the redpilled kids focus on something else like mastering piano or sit on Khan Academy all day and go on to buy better teaching material at the bookstore. The only government schools that whipped your ass when it came to STEM were those existing under the iron curtain.

I absolutely recommend his marvelous book on ODE and another one on mathematical principles in physics or something like that.

>math should be taught proof based!
>explanations should be terse, cold, and left to the reader to discover on their own!

do you fucking retarded faggots actually believe this shit?
98.999... + -12(1+2+3+4+5...)% of people do not need anything past basic algebra in their lives, and people who need to take calculus or above are already gifted/interested in math

Most people that pass those courses are just dedicated to their studies and are in no way. And if you want only learn basic shit so you can survive you shouldn't even be in middle school. Education has no clear ulterior purpose but it's supposed to guide you on the different human activities out there with emphasis on shit that requires an academic environment which is a lot of them. No one uses anything all the time, but promoting learning for the sake of it can help many people reach things that sometimes look as impossible pipedreams. And yea, having an emphasis on math for math itself can be a way to get interesting problems into the math curriculum. The only reason it's hard is because crybaby parents feel that if they can't help their children with their homework, it must be too difficult.

And no, this is in no way what common core tries to do. This is teaching from an early age what math actually is instead of 1000 different methods to """intuitively""" understand math.

>Do you know any math books that fall somewhat within Lockhart's ideals?

Yes, Ian Stewart's Concepts of Modern Mathematics does this exquisitely.

"Modern Mathematics" was the British version of "New Math", the project failed there too (largely because high school teachers don't know real math and fail at teaching it) but this book is more of a historical record of the sort of material that was taught and the way it was intended to be taught.

The book gives a brief introduction to tons of different areas of mathematics including abstract algebra, analysis, topology (and algebraic topology in a separate chapter), etc.. and it does all this by focusing on explaining concepts clearly without doing anything by rote methods.

That said, it is important to distinguish between a number of similar but different beliefs about how mathematics should be taught.

V. I. Arnold for example doesn't believe that mathematics should be anything more than a tool of science. He believes that mathematics should only be taught in the context of real world applications and has written a number of books on this including one derived from a class on Algebra (up through some basic Galois Theory) that he gave to children.

I don't agree with V. I. Arnold's perspective at all but I can't deny that there exist people who find his approach valuable.

There has recently been some increase in interest into this area and while there doesn't seem to be any organized group there does seem to exist a loose network of mathematicians, philosophers, and educators who are attacking the problem on different fronts (aesthetics/functional beauty, processing fluency, etc...). I'm really passionate about this and wish there were a more organized effort going on.

This. Also notice this, these faggots are professional complainers, always complaining about how maths ought to be taught, yet, look at his books, is there any textbook worthy of notice? Would anyone pick up Godel, Escher Bach as a substitute for an AI textbook or a CS automata book? No. You do not learn jack SHIT with the other methods, they're great AFTER you've done the boring heavy lifting (or before if you never intended to learn much anyway).

>...maths is taught at school so that students would learn math. This is false.

I think it may be sufficient to support the claim that
>math ought to be taught at school so that students learn math.

I'm this guy so you can see how I feel about V. I. Arnold.

Multiplication tables, drills, and exercises do not teach math.

Math is about understanding concepts, not about working out computations.

>I would like to see more of the equivalent of creative writing in math
There is some writing on the subject, particularly in the context of proof writing.
>where the student is not only fed word problems, but encouraged to invent their own scenarios where math would be useful and then do the math for them.
There is also writing on this subject but it has very little to do with math.

I agree to some extent in the sense that I believe most math books out there attempt to do this. However I disagree in the sense that I don't believe many math books actually achieve this. Instead what you get are books that are too wordy and informal to be clear and too terse and formal to be interesting.

>...of 1000 different methods to """intuitively""" understand computations.
fixed
If Common Core focused on math instead of computations it would be this.

Another great book to check out is Ralph Abraham's
Dynamics: The Geometry Of Behavior.
The book uses a novel style that focuses primarily on images in a sequential explanatory style to give an intuition for Dynamics concepts (particularly differential equations, though Ralph has other literature on Chaos Theory and other areas of Dynamics).

>I read the entire thing.
Actually you only read a short 25 page essay which was later expanded into an entire book.

Also, he is a teacher.
maa.org/external_archive/devlin/devlin_03_08.html

GEB is a shit-tier pop-sci book and you are a retard for thinking it has anything to do with this.

>GEB is a shit-tier pop-sci book and you are a retard for thinking it has anything to do with this.

Nice undergraduate superiority complex. Are you enjoying being mediocre student nº1000000?

>superiority complex
Have you even fucking read that book? It's not a textbook. It tries to make a vague "argument" about self-referential loops via alternating chapters of exposition and chapters of "allegorical" dialogue between a tortoise and a hare.

I doubt a retard like yourself managed to read a 700 page book when you haven't even managed to read my post. I haven't claimed it was a textbook, what I'm saying is that the books that showcase the beauty of math are never textbooks. I'm saying that to actually become a good mathematician you HAVE to do the heavy lifting boring work.

He's pretty spot on about what's wrong with maths education.
Fixing the problem, though, would involve maths teachers actually understanding maths, which is quite rare.

Please, we teach maths in school so every student gets at least some abstract thinking into his brain. Art and creative writing is also taught in school because we want them to be able to understand that these fields have some basic principles and that you can analyze them, not to turn them into artists.

And for the average student, the best way to do is is boring repetition of exercises. Sure, the students who are actually interested are wasted on this, but schools have the task to produce adults capable of basic understanding of society around them and the ability to chose a direction to continue learning, so naturally they aim for a middleground between teaching retards and teaching interested students.

>GEB
>showcasing beauty of math
lmao

I have read it. I even have a copy on my bookshelf. It has nothing to do with what Lockhart is talking about and no one doing research in the area would even consider GEB in that context.

As I said in another post, there is a small but growing body of research on the topic. I actually recently attended a conference where one of the talks attempted to give a formal description of "mathematical beauty" from the perspective of analytic philosophy (particularly aesthetics). If anyone has a superiority complex about stuff they have no familiarity with it is clearly you.

>And for the average student, the best way to do is is boring repetition of exercises.
This is precisely the claim being contested, user.

No the claim is that they can't see the beauty of maths and never learn real maths by doing that. That's true, but that's is not what they are supposed to do.

>no one doing research in the area would even consider GEB in that context.
First of all, GEB is commonly associated with mathematical beauty and it is a beautiful book, your contrarian ideas are best kept to yourself. Second, why do the researchers matter? They already think the field is beautiful, Lockhart is talking about the general population, nothing better than a pop-sci book example, it is the kind of mathematics that people enjoy reading about.

Do you see how you completely misunderstand everything and are so fast to dismiss things? This is what I mean by superiority complex.

>I actually recently attended a conference where one of the talks attempted to give a formal description of "mathematical beauty" from the perspective of analytic philosophy (particularly aesthetics)

Huh, cool? Really irrelevant and just a way to pat yourself on the back.

Basically what I'm getting out of this is that:
>What you call 'math' has nothing to do with what anyone formally trained in math calls 'math'.
>What you call 'mathematical beauty' has nothing to do with what anyone formally trained in math calls 'mathematical beauty'.
>You only care about pop-science texts meant to give the general public a nice story about not-math but not actually meant to educate anyone in any sense of the term.
>You do not care about people who are actually interested in math being provided with better resources nor about research being invested in figuring out better guidelines for creating said resources.

You are correct in that Lockhart is talking about the general population but he is talking about them in an educational setting. Lockhart is a former researcher and PhD holder who now works as a school teacher that subversively teaches 'real math' to students. He is talking about education reform. To reiterate: Lockhart is absolutely NOT talking about pop-sci books and certainly not pop-sci books like GEB. Even truly beautiful pop-sci math books about actual math like 'Art of the Infinite' or 'Concepts of Modern Mathematics' are not really what Lockhart is talking about.

The claim is that they should learn and understand some basic principles of mathematics. Moreover such basic principles are poorly taught by repetition of exercises. By teaching in other ways students not only obtain a better understanding but they are exposed to the beauty of mathematics and can begin to understand the bigger picture (i.e. real math).

Looks interesting to me. I like when basic assumptions are questioned and/or proved like that.

You're very dumb. And I have basically decided that people who get hung up on the beauty of math are just tards who can't actually do math so they stick to telling everyone about how beautiful they think it is.

Now I'll proceed to destroy all of your brainlet misinterpretations and not reply any further.

>What you call 'mathematical beauty' has nothing to do with what anyone formally trained in math calls 'mathematical beauty'.
So mathematicians find mathematics beautiful, how surprising! You're very stupid.

>What you call 'math' has nothing to do with what anyone formally trained in math calls 'math'.
I am formally trained in mathematics. And again if the issue is with people who aren't trained in formal mathematics why are you talking about mathematicians as if they were the ones we need to make see mathematical beauty?

>You only care about pop-science texts meant to give the general public a nice story about not-math but not actually meant to educate anyone in any sense of the term.
GEB is a great book, It's quite obvious that you only dislike it because it's popular. And if we're to discover what the general audience finds beautiful wouldn't the popularity of a book be a good indicator? Simply put, the general audience don't like your books. We already know this because they don't buy them.

That aside, you and this lockhart fool are delusional. People do not like "real" math, they don't like math at all. Maybe if you weren't such a socially inept fucktard you'd know this. Who hasn't heard a student say "Why do we need to know this?" when they get a sneak peak into the more theoretical math? If anything, students care about the applications of math, very few would enjoy pure math.

>Lockhart is a former researcher and PhD holder who now works as a school teacher that subversively teaches 'real math' to students.
Don't care.

I have no more patience to entertain your delusion of grandeur because MaThEMaTiCz iS mY FeeeWeeeWeess!!!

>So mathematicians find mathematics beautiful, how surprising! You're very stupid.
There are beautiful proofs/constructions/definitions just as there are ugly counterparts to those.

>I am formally trained in mathematics.
>I think GEB is mathematically beautiful.
Choose one, retard.

>popular = beautiful, disregard the fact that GEB is a meme

>theoretical math
Kek, applied math/sciencefag detected. If I would have known that from the start I would have never bothered talking to you as you clearly don't understand real math either. Go masturbate to GEB some more, retard.

>all this arrogance because you """like""" pure math

Don't be mad because I destroyed every single stupid point you tried to make. Now you obviously resort to bullshit answers like "X is a meme". Do your smart colleagues know you're this much of a fraud?

>being this buttblasted because someone pointed out that GEB is shit tier and has nothing to do with math.

Yep, should've known you were just a fraud memer. I'm glad you decided to be honest now and not pretend to be smart.

>user posted a picture on an image board
>that invalidates their argument
top zoz

The only fraud here is the retard trying to pass of GEB as mathematically beautiful while claiming to be mathematically trained.

Then, what is "real" math?

Remember, guys.

>Multiplication tables, drills, and exercises do not teach math.
Except they obviously do, you fucking monkey.

>Math is about understanding concepts, not about working out computations.
You don't get to redefine "math" to whatever you like. Arithmetic, including basic manual computation skills, is an area of math, and the most useful part to master, which is why we teach it to everyone.

What you're doing here is taking something everyone gets taught, and dismissing the importance of teaching it because everyone knows it. It seems you somehow simply can't imagine the difficulties people would face if nobody sat them down and drilled them in the nuts and bolts of addition, subtraction, multiplication, division, fractions, decimals, and percentages.

Make no mistake, teaching children practical arithmetic is not something you can skip without serious consequences in their lives.

There's really no excuse for the fact that math is okay to hate in the US. People need to learn the vital skills like basic algebra and arithmetic but they should also be introduced to math properly.

I hated math for most of my childhood because it just seemed like pointless bullshit, the teachers didn't care about teaching it and the students never learned to care about learning it. If countries that are way poorer than us such as Russia can foster an appreciation for math in their high schoolers then why can't we? No excuse really other than our plutocratic government

only geometric proofs apparently

You can teach arithmetic without rote techniques.

More importantly, the school system is meant to give students the tools they'll need to survive in the world they're going to grow up into. It doesn't make much sense to teach kids to do precise computations quickly using shitty algorithms when everyone is already walking around with a computer in their pocket.

There was a time when kids were taught all sorts of ridiculous logarithmic tables and shit because they made certain computations faster. Nowadays everyone will unanimously call you a retard for suggesting we teach it to future generations.

Mathematics boils down to communicating abstract concepts. Definitions, theorems, and proofs are some but not all of the commonly used techniques for achieving this.

Arithmetic and computations are not really math in this sense since they don't really communicate anything meaningful.

>You can teach arithmetic without rote techniques.
Just about the only rote learning in arithmetic is memorizing the multiplication table, and you can't be even halfway decent at head math without learning that.

>It doesn't make much sense to teach kids to do precise computations quickly using shitty algorithms when everyone is already walking around with a computer in their pocket.
If you can't do basic arithmetic without hesitation while your hands are occupied with something else, you're basically crippled. Teaching kids to do unlimited-precision arithmetic on paper prepares them to develop as much head math as they need, as well as making them familiar with the concept of algorithms and training them the discipline to follow a procedure exactly when necessary.

>There was a time when kids were taught all sorts of ridiculous logarithmic tables and shit because they made certain computations faster.
Ever notice how aerospace progress dropped off sharply as the first people trained to use calculators rather than slide rules for engineering math entered the workforce? How technological progress in general shifted from new inventions to incremental refinement of existing technologies?

Depending on black boxes to do your calculations for you is terrible for your sense of intuition and ability to rapidly estimate values while exploring possibilities. We've gained very little by setting aside advanced manual computation methods and aids with open workings, and lost a lot.

>Depending on black boxes to do your calculations for you is terrible for your sense of intuition and ability to rapidly estimate values while exploring possibilities. We've gained very little by setting aside advanced manual computation methods and aids with open workings, and lost a lot.

Learning computation-as-math instead of math-as-math including computation-as-technique-subset essentially treats maths as a black box, out of which - magically - computation arises. Learning computation-as-math, then, must be terrible for the sense of intuition and ability to innovate.

Would not engineering be much easier if all engineers understood math well, rather than some portion of engineers having roughly wiggled through a faulty math curriculum without deeply learning anything?

Why does this guy want dummies to mud up mathematics?

>We need to turn math into finger painting so the dummies can see how beautiful it is!!!

Actually, he's arguing the opposite - that math has already been turned into finger painting.

>You can teach arithmetic without rote techniques.

HAHAHAHA YOU STUPID DUMBFUCK

>Hmm... 426 + 793. Let me analyze these two numbers...
>Ah! 426 is one more than 425. Now what do I do with it?
>It's obvious! I'll take that 1 I removed off of 426 and add it on to 793
>Now I have 425 + 794. Now I'm getting somewhere! Well I know 425 is a multiple of 25... But what can I do with that?
>Hmmm... Perhaps nothing.
>Should I have even removed the 1 from 426? Let me get back to my original form
>426 + 793. There we go. That's better. But it seems I've gotten nowhere.
>Perhaps I can draw some squares.
>Draws squares
>There we go! 426 squares and 793 other squares. Let me count them
>427
>428
>429
>
>
>

two hours later

>1219! There we go. First problem done. I'm so glad none of my teachers taught me to do any rote memorization!!!

We're talking about a middle school classroom. The point is not to make mathematicians; it's to learn middle school math.

More like
>426 + 793
>6 + 20 + 400 + 3 + 90 + 700
>6 + 3 + 20 + 90 + 400 + 700
>9 + 110 + 1100
>119 + 1100
>1219

Which isn't a rote technique, it is just logic. If you are going to do arithmetic in base 10 you might aswell make the best of it.

You've missed the point and are creating some weird non-existent dichotomy between rote memorization and not being taught anything ever.

Or
>let's analyze 426 + 793
>Before showing you how to solve it lets decompose it into 400 + 20 + 6 +700 + 90 + 30
>Ignore 0s and add like we know the like terms 400 +700 = 1100 etc
>After some examples and generalizations, now let's show you the most efficient way.

Now you have a look at why the efficient way works and after some practice you get the grip of it. That's quite doable.

how the fuck do you make a box out of triangular sides??? What the fuck am I even reading

Take the diagonal you brainlet.

>he's fantasizing about a great mathematician teaching a classroom full of little geniuses with unlimited time for "enrichment", rather than a second-rate babysitter teaching the usual crowd of barely-trainable normies who need to get their multiplication tables out of the way so they can get back to learning how to spell.
this.

>and you can't be even halfway decent at head math without learning that.
That's where you're wrong. You can memorize a few simple multiplications and decompose complicated problems into simple ones in your head. That is what common core teaches.

>If you can't do basic arithmeti
>Teaching kids to do unlimited-precision arithmetic on paper
lel

>Ever notice how...
Are you for real implying that technological progress is slower now than it was then. You are a legit fucking moron.

>Depending on black boxes to do your calculations for you is terrible for your sense of intuition and ability to rapidly estimate values while exploring possibilities. We've gained very little by setting aside advanced manual computation methods and aids with open workings, and lost a lot.
Of course, but depending on black-box algorithms is just as bad if not worse because it gives people the impression that they understand something when they really don't. In truth the best way to teach that sort of 'basic intuition' is by introducing students to the notion of a field.

>fantasizing
Lockhart is a teacher (formally a mathematics researcher/professor with a PhD). He argues that teaching this way is better because it is the way he teaches.

Don't worry, you'd understand had you been taught maths properly. As you are, you'll have to make do with revising the triangle-box conversion algorithm you should have memorised years ago.

>You can memorize a few simple multiplications and decompose complicated problems into simple ones in your head. That is what common core teaches.
Common core is garbage which wastes time teaching inferior, idiosyncratic methods instead of giving people a solid foundation they can customize.

The point of the arithmetic changes in common core is to invent a lot of easier things to test other than actually performing arithmetic, so you have excuses to give kids passing grades when they haven't actually learned the subject, while also being able to produce tests that conventionally-educated kids will fail despite having much better understanding and skills.

>Are you for real implying that technological progress is slower now than it was then.
In many areas, it certainly has been. Again: look at aerospace. By the early 1970s, we got the Boeing 747 and 737, C-130, B-52, SR-71, X-15, F-4, F-14, Atlas/Centaur, Saturn V, Apollo spacecraft, Titan, Proton, and Soyuz. Each in their own way unsurpassed to this day.

>depending on black-box algorithms
It's not a "black-box" when you see every part of how to do it and it's explained to you why you do it that way, you bloody monkey.

You clowns are talking about how conventional arithmetic training is just "rote learning", but it's not. You don't just give kids multiplication tables and tell them to memorize them, you show them how to construct them. They memorize them for speed, not because they'll be helpless to multiply if they forget an entry in the table.