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from the how-to-learn-mathematical-thinking dept.
Technology Review is running an unusual book review -- books about learning math, pure math, not applied math, https://www.technologyreview.com/2023/04/24/1071371/book-reviews-math-education/
The author admits to being adrift,
As a graduate student in physics, I have seen the work that goes into conducting delicate experiments, but the daily grind of mathematical discovery is a ritual altogether foreign to me. And this feeling is only reinforced by popular books on math, which often take the tone of a pastor dispensing sermons to the faithful.
An initial attempt led to a MasterClass by a "living legend of contemporary math", but the master is seated in a white armchair with no blackboards, pens or paper and does not enlighten.
A side story covers a writer for the New Yorker who plans a year to go back and learn the high school algebra/geometry/calculus that escaped him, but mostly fails. For backup he has a niece who is a math professor...but after months without getting it, he complains. Her answer?
"For a moment, think of it as a monastic discipline. You have to take on faith what I tell you." Where his niece and others see patterns and order, he perceives only "incoherence, obfuscation, and chaos"; he feels like a monk who sees lesser angels than everybody around him.
I won't spoil the end, but the author does make some progress with books by mathematician and concert pianist Eugenia Cheng, starting with "Cakes, Custard and Category Theory", where each chapter starts with an analogy to baking.
Unfortunately, for the SN audience, the article does not include any car analogies...
(Score: 5, Interesting) by JoeMerchant on Sunday June 11 2023, @02:35PM (25 children)
>think of it as a monastic discipline. You have to take on faith what I tell you
My high school geometry teacher put it this way:
"You can't use that for a proof because I haven't taught that yet. It doesn't matter if it's true or not, we do this the same exact way every time."
Then at some point she proceeded to skip ahead and start using theorems that she hadn't taught, because that's what she wanted to do.
One lesson of many in: grades indicate how well you follow instructions, repeat what your are told and toe the line. Thinking is not only optional, but actually discouraged.
Then, if you have reached the pinnacle, regurgitated all the litanies correctly, you are permitted to attempt to show something new that builds on top of all that exists. And, like the einstein tile guy, be prepared for the world to piss in your cornflakes when you do show them something new. I wonder if he had the non-flipping einstein in his back pocket and just released the flipper first to see what would happen?
🌻🌻🌻🌻✌️ [google.com]
(Score: 2) by krishnoid on Sunday June 11 2023, @03:24PM (1 child)
I hope someone called her out on that and meted out a proportionate punishment [youtu.be], possibly with follow-ups if she continues in her intransigence. But seriously, having to answer for that inconsistency (at least asking after class is over) would probably reveal something about educational methods for a classroom of multiple skill levels, or maybe just about rigor in delivering the material and method -- considering "we do this the exact same way every time".
(Score: 2) by JoeMerchant on Sunday June 11 2023, @05:02PM
She was teaching high school 10th graders, she looked to be in her late 50s / early 60s, shaped like a tomato with a tight belt on. In 1982 her eyeglasses were high fashion from 1958. She was clearly there for the paycheck, and a little power tripping, nothing more.
🌻🌻🌻🌻✌️ [google.com]
(Score: 5, Insightful) by VLM on Sunday June 11 2023, @03:26PM (2 children)
School is seen as vocational training for the workplace.
(Score: 2) by JoeMerchant on Sunday June 11 2023, @07:14PM
My workplace actively encourages us to innovate, both products and our internal processes.
The real test is continuing to do what is required to collect the paychecks as the decades roll by and you watch 90% of the innovations sit on the shelf while the most active 5% of process innovations swing back and forth in one reorganization after another.
It's nice to be non-redundant. I keep trying to cross train my skills to others but the more I do that the more essential I seem to become.
🌻🌻🌻🌻✌️ [google.com]
(Score: 0) by Anonymous Coward on Monday June 12 2023, @12:33AM
Sounds correct.
Shut up. Sit at a desk. Accept you are being graded. When the angry man at the front yells at you, bow your head. "Because I say so" is all the motivation you need.
(Score: 2) by krishnoid on Sunday June 11 2023, @03:52PM
They won't if you show them something just new enough [youtu.be].
LOL, passive-aggression in modern mathematics. Nice.
(Score: 5, Interesting) by Anonymous Coward on Sunday June 11 2023, @04:03PM (6 children)
We certainly had different high school math experiences. My geometry teacher was friendly and let several of us better students sit in the back. She posted the next day's assignment at the beginning of class--and the four of us in the back would take the first bit of class to get that done (so we rarely had homework to do at home). Then we quietly passed notes back and forth while the others in the class took in her lecture. She saw exactly what was going on and let us goof off (quietly) because she also knew that we understood the material.
Many years later I read her obit. Turns out she was a PhD math major at a top university and had done some original math research--but was locked out of an academic career by the glass ceiling for women back then. She never discussed her misfortune, although she had every reason to be bitter.
Now, AP Calculus (senior year high school) was another matter all around. It was taught by the football coach who was (by his own admission) barely qualified. He more or less said, "Here's the book, go to it." At that point I really needed some help and didn't get it. The result is that, while I can intellectually make the jump from the slope of a small interval to an infinitesimal (differential, for one example), I never really believed it. Without that jump, the motivation to understand the rest of it wasn't there.
(Score: 5, Interesting) by HiThere on Sunday June 11 2023, @04:51PM (4 children)
It's not really clear that you SHOULD believe in continuity, infinitesimals, infinities, etc. They follow from a certain set of assumptions, but I, personally, don't believe them. I believe that the universe is discontinuous at around the Planck length. This has certain problems in it's own right, but they are different problems than the ones that continuity entails. (And I think Cantor's red and blue number lines is just goofy. You can't assign colors to numbers, only to a more complex structure. What he really did was create two instances of the same [infinite] set, but then his reasoning isn't about the set, but rather about instances of the set. Either that, or he was creating a struct {int value; color bool;} and reasoning about that rather than about integers.)
Javascript is what you use to allow unknown third parties to run software you have no idea about on your computer.
(Score: 1) by shrewdsheep on Monday June 12 2023, @05:02PM (3 children)
Mathematics makes no claims about the universe. The deep point about calculus is about limits. Limits mean that you assign a single number to an infinite sequence of numbers. As a matter of fact most numbers are defined as sets of infinite sequences (equivalence classes). Mathematics is a game that allows you to manipulate sequences of symbols - no more, no less.
(Score: 1, Interesting) by Anonymous Coward on Monday June 12 2023, @08:02PM
> Mathematics is a game that allows you to manipulate sequences of symbols - no more, no less.
Interesting comment, thanks. If true, then my difficulty with math is common with my lack of interest in most games. I just don't get why most people like games.
(Score: 3, Interesting) by HiThere on Monday June 12 2023, @09:11PM (1 child)
Yes. And that's why I said "they follow from a certain set of assumptions". I'm not denying that they *do* follow from those assumptions, just that the assumptions properly describe the universe. (They do come pretty close, though.)
Javascript is what you use to allow unknown third parties to run software you have no idea about on your computer.
(Score: 4, Insightful) by hendrikboom on Tuesday June 13 2023, @01:48AM
And that's the difference between math and physics.
And that's why physicists still use math.
(Score: 3, Insightful) by JoeMerchant on Sunday June 11 2023, @05:06PM
The teacher makes all the difference. If you can teach yourself, with enthusiasm, that's a teacher you never lose (at least not until senility sets in...)
🌻🌻🌻🌻✌️ [google.com]
(Score: 3, Interesting) by mhajicek on Sunday June 11 2023, @04:49PM (4 children)
I had a very similar experience in high school geometry. Also, for me, I need to know an application in order to be motivated and really understand something. When I asked the teacher how this might be used in the professional world, she said "Trust me, you'll use it."
Then I went to tech school for machining, and I had applications for everything I learned. Vector math was a breeze, because I knew it was needed for polar coordinate programming on a live-tool lathe, for example.
The spacelike surfaces of time foliations can have a cusp at the surface of discontinuity. - P. Hajicek
(Score: 3, Interesting) by JoeMerchant on Sunday June 11 2023, @07:21PM (3 children)
I don't know that I have ever used high school geometry type of proofs. I know that I have never used path integrals around a singularity despite studying them for two semesters. I sort of forced diffeqs into service a couple of times but could easily have ignored them.
Integrals: yes, definitely, particularly discrete summations. Trigonometry: a lot. Discrete mathematics proofs by induction and similar... Not really.
🌻🌻🌻🌻✌️ [google.com]
(Score: 1, Funny) by Anonymous Coward on Sunday June 11 2023, @10:18PM
Q: Why did the mathematician name his dog Cauchy?
A: Because it would leave a residue at every pole.
(Score: 2) by aafcac on Sunday June 11 2023, @11:47PM
They are a complete and utter waste of effort at that point. It was mostly a matter of being able to refer to various proofs in the book and those mostly didn't have standardized names, so if you weren't working out of the same book, you probably wouldn't be able to understand the proofs. It's wonderful that there's been more focus on word problems, as those at least attempt to deal with a real problem people have. It's rare to be handed an equation to be solved, usually, you have to write and solve your own problem. The word problems in math tend to be rather artificial, but there's not much way around that if you're expecting to not require methods that haven't been taught.
(Score: 0) by Anonymous Coward on Monday June 12 2023, @12:39AM
Well it's all fucking irrelevant now we have rockstar CEO scientists that sell their visions to funding agencies and oversee a sweatshop of grad students on visas under threat of failing to meet the standards of a committee full of CEO mentors.
(Score: 4, Interesting) by RamiK on Sunday June 11 2023, @05:18PM (4 children)
A professor in uni forced me to retake a test over using (yet to be taught) Laplace transforms to solve a test in ordinary (surely it wasn't complex ones as that would have been unavoidable? can't recall...) differential equations. So, in the redo, I developed the transforms I needed from the axioms that were on the material and they were forced to accept the answer.
I don't know how your teacher would have responded to that but I'm guessing they wouldn't have objected too fiercely...
compiling...
(Score: 3, Interesting) by JoeMerchant on Sunday June 11 2023, @05:42PM (2 children)
I took every (Calculus II and up) math class offered in my uni math department, then I took one more from the physics department that taught three subjects: Green's functions (a way to solve some differential equations) something about closed path integrals around singularities, and something else about differential equations. The final exam counted as 60% of the final grade, and was pre-announced to be: select 5 of 11 questions. I never did learn how to use Green's functions, but probability told me that the likelihood of 6/11 questions being on Green's functions was slim. The prof also handed us sample previous exams to study, which I did.
Exam day arrives and 5/11 questions explicitly stated "using Green's functions..." and one of the remaining questions was a "solve this differential equation" which could have used Green's functions. However, I recognized the equation as being a simple transform of a question from a previous exam, so I proved my "solution by inspection/recognition" and flew through the other four questions. 5/5 perfect answers, and yet I still received a B...
Seems that the prof saw through my shortcut and, having failed to demonstrate knowledge of Green's functions 100% earns a B.
🌻🌻🌻🌻✌️ [google.com]
(Score: 2) by RamiK on Sunday June 11 2023, @05:52PM (1 child)
I think you sorta vindicated your high-school teacher there... :D
compiling...
(Score: 3, Funny) by JoeMerchant on Sunday June 11 2023, @06:34PM
By the time I was taking Mathematics for Physics 517, I had learned how little the difference between an A grade and a B matters - to me, to my future, etc. I certainly enjoyed waterskiing on Tuesdays and Thursdays much more than mastering Green's functions could ever have done for me.
🌻🌻🌻🌻✌️ [google.com]
(Score: 2) by bzipitidoo on Monday June 12 2023, @03:14PM
I found the approach of deriving things took too long for tests. The timed format forces the student to memorize a lot more and consequently understand less, in order to be fast enough to finish the tests.
Math is one of my favorite subjects, because it is objective. The right answer is the right answer. Teachers who hate smart students have much less room for subjective bull excuses to justify giving a good student a bad grade. They can still do some crap ("you didn't show your work!") but as your anecdote shows, that's harder.
(Score: 1) by khallow on Monday June 12 2023, @01:48AM
All the einstein tile guy had to do was deliver what they promised. It wasn't a high bar. And note that they actually did some weeks later.
(Score: 1, Touché) by khallow on Monday June 12 2023, @04:23AM