Some thoughts about self-education

[symbolipoint]

Astronuc, #30
You were and are(maybe) a very unusual person. I cannot say that your frustrating teachers were right or wrong, but seemingly misunderstandings were happening from both first and second parties.

I really hope that, at least the music teacher gave your confiscated book back to you at the end of that class meeting.

 

[OmCheeto]

... My class teacher told me that I was too far ahead of the rest of the class, and that I need to keep pace.
...

Do they teach this to teachers in teacher school?
"If you have a student who is too far advanced, do your utmost to hold them back."

hmmm..... This is kind of making me want to start a new thread: "What are your 'triggers'?"

per google AI; "A trigger is a stimulus that sets off a reaction, most commonly referring to something (a sound, sight, memory) that causes intense emotional distress, often linked to past trauma..."

 

[berkeman]

Astronuc, #30
You were and are(maybe) a very unusual person.

Unusual in the best ways.

I feel very fortunate that my small highschool had an advanced track in math. Without that I would have been far behind when I went to undergrad uni.

 

[Astronuc]

I really hope that, at least the music teacher gave your confiscated book back to you at the end of that class meeting.

The music teacher gave my math workbook to my class teacher who then had a talk with me after class. I promised not to do math during music lessons. She recognized my ability at math and encouraged me. She also asked me to read books of fiction, which was a requirement for our English track, but I don't remember if I did. I explained that fiction didn't appeal to me, but that I rather read books on science, technolgy or history and geography.

At the end of the school year, I was put in a special academically able (AA) summer program in which we did higher level math, like matrix algebra. I did the AA program after 4th, 5th and 6th grade. During my secondary school program, 7th through 11th grade, I did a 6 week program at a local university, where we did three classes per day, usually 1.5 hours each, which were college level courses in various subjects, e.g., history, math, physics, computer programming. Between 11th and 12th grade, I attended a summer program funded by the National Science Foundation at the Colorado School of Mines, an 8-week program in electrical and nuclear engineering. One of the fellow students went on to Princeton and UC Berkeley, did some graduate work in cosmology, did experiments in cosmic microwave background (CMB), and eventually became head of the Physics Department at Caltech, among other achievements.

 

[Hornbein]

I'm self taught in piano but that's different. I learned about neutron stars by reading research papers -- much more interesting than black holes -- but never attempted to even skim the math. That's what peer review is for, yes?

My only mathematical talent was multidimensional geometry. I also have a natural ability at the game of golf -- applied 3D geometry. I never did much with it. why bother?

Some acquaintances from my grade school math class are now heads of science departments. I was in a blues band with the head of LIGO. He played the harmonica.

 

[jedishrfu]

@Astronuc, your experience in school is not uncommon. I had a friend who completed all of high school math by eighth grade. The school had no intermediate programs like today's IB or AP courses, so they had to fund him to attend a local college so that by 12th grade, he had completed four years of undergraduate math.

In my case, I was nowhere near his accomplishments. I do recall the grade school structured classroom. In first grade, you get grouped into one of three reading classes and remain in that group as you meander through grade school. The teacher would schedule the reading groups: advanced, medium, and slow. Other courses like math, science and geography were structured around the three reading groups.

As one group read, the others did math or science. The other subjects were arranged in the same way, so that my group, group three, was always the last to learn. One time, I saw the kids doing long division, which looked really interesting, and I wanted to learn it, but I was told I had to wait until we got to that lesson.

To this day, I've never forgotten about that rebuke and how sad I felt about it. Once you were assigned a reading group, there was no means of advancing beyond the other students. We were the rearguard of learning, and we always felt left behind.

When sixth grade came around, my teacher encouraged me in science. My first report was on atomic energy ie basically structuring my project after an article in the Parade magazine, a popular Sunday paper insert. It featured striking images of a future powered by atomic energy.

In the closing days of sixth grade, my teacher met with each student and their parents. In my case, he told my mom I was college material, reading at an 8th-grade level. He gave me the remaining books we had to read for the elementary school curriculum and said, "Take these home and come back and tell me you read them." Huh?

He placed me on equal footing with the fast readers for middle school, breaking the spell of the slow-reading group. I am forever grateful for his encouragement. It changed my outlook on learning.

Basically, school is about grouping kids into smaller groups so that the teacher can juggle teaching all the required material for that grade. However, for slow readers, we might learn it in the next higher grade because we were the last to learn it.

Also, I think it had to do with resource management, where each class had a limited set of reading books, say about 12 for each book we needed to read. The fast group would get first crack, finishing in two weeks; then the middle group would get those books and maybe take a month; and finally the slow group would get those books, 2 months later, while behind the other students in all subjects.

This was in the late 1950s and early 1960s. As an aside, my mom wanted to improve my reading, so she had me read The Bobbsey Twins, a book series she had read in grade school.

In response, I started getting Scholastic Book Club books on Rocks and Minerals, Codes and Ciphers, Strangely Enough, and others to escape the curse of the Bobbsey Twins.

I had wanted to read Tom Swift, but our library didn't have them, or I didn't know where they were. The next best thing was the Childhood of Famous Americans series: Albert Einstein, Abraham Lincoln, Thomas Edison, and other interesting people.

and so it goes...

 

[jtbell]

I had wanted to read Tom Swift

Tom Swift Sr. or Tom Swift Jr.?

When I was in elementary school, my parents bought me some of the Tom Swift Jr. series which was more or less current at the time. Then when I got to junior high (what we called middle school in those days) I found some of the older series in the school library.

 

[jedishrfu]

I don't remember which it was. I was in 3rd and 4th grade and titles like that didn't register.

——
@BillTre

It was Tom Swift Jr. I saw a reference to his electric rifle. Some covers had the The New Adventures of Tom Swift Jr
But the book title said Tom Swift and His Electric Rifle.

 

[Astronuc]

I also had my math workbook confiscated by a music teacher during music class, because I was supposed to be learning music, not doing math. I liked to do math while listening to music. My class teacher told me that I was too far ahead of the rest of the class, and that I need to keep pace.

I should indicate that it was a music lesson in 4th grade class. As I recall, the lesson was part of a music appreciation in which we listened to 20 pieces of classical music over several weeks, and about mid-year there was a test in which we had to identify each piece. I scored 100%.

I thought listening to classical music was a perfect opportunity to do math.

 

[symbolipoint]

I should indicate that it was a music lesson in 4th grade class. As I recall, the lesson was part of a music appreciation in which we listened to 20 pieces of classical music over several weeks, and about mid-year there was a test in which we had to identify each piece. I scored 100%.

I thought listening to classical music was a perfect opportunity to do math.

That part in bold is a reason why I said a few days ago that you are unusual. As for myself, I find no desire to listen to any music while I study anything (or almost and most frequently no desire); but I do drink coffee.

That other part where you re-quoted what yourself wrote, reminds me of a course in which I studied ahead, because I learned the material several years previously, and I used the course syllabus to do the studies and the assignments before they were reached on the class schedule. This seemed to be a little annoying for the teacher, but I did score fairly well and earned A in the course.

 

[mathwonk]

As a high school student long ago in the US south our school offered no calculus, very inadequate algebra and geometry, and the local college ridiculed a request to enroll in a class there as a high school senior. There were no supplementary programs available, to my knowledge. So I amused myself by reading books in the college library stacks, some recommended in the one good book encountered in a “special” high school class: Principles of Mathematics, by Allendoerfer and Oakley. Unexpectedly admitted to Harvard, (via “geographical distribution”?) I placed into an honors calculus class, without the background possessed by my classmates from the northeast, based entirely on the self-acquired knowledge of Cantor’s theory of uncountable infinite sets. The interviewer summed up his sneering assessment: “at least you have some interest in mathematics”!
So self - learning apparently does demonstrate motivation, which some teachers value.

 

[jedishrfu]

My favorite anecdote was while in middle school in Spanish class, I finished up some homework slipped it away.

I decided to draw from memory the Lost in Space robot, B9 and got out a blank sheet of paper. About five minutes before the end of class, the teacher surreptitiously walked by and scooped up the paper.

At the end of class, I sheepishly went to see her and asked if I could have my robot drawing back. She opened the paper saw the robot and handed it back, mildly shocked that it wasn't homework.

I learned a valuable lesson that day: never be too greedy in doing your homework and always finish up with a cool drawing.

 

[symbolipoint]

jedishrfu
I was studying & doing assignments ahead in part because of fear of being short on time. I had a part time or more job at that time, and you can also guess, I was motivated to learn (in part "relearn" too).

 

[mathwonk]

I think vela hit the nail on the head. Becoming educated, i.e. learning to understand something, is for me a long, difficult process requiring significant effort from the learner. A friend who teaches reading emphasizes to her students the need to “engage their thinking mind”.

One receives information from a source such as a book or lecture, then must turn it over and over in ones own mind and grapple with it at length. During this process it is very helpful, maybe essential, to have someone else, intelligent and/or knowledgable, to bounce ones thoughts off of, to help correct errors of comprehension.

In my experience it is also wonderfully enlightening at some point, maybe late in the process, to consult the original works of the genius who created the theory. I spent my whole professional career studying and using the Riemann - Roch theorem, (a formula for the vector dimension of the space of meromorphic functions on a closed connected surface, and having only given poles). I read, and listened to, many versions at every level of sophistication and abstractness, by all the modern experts. Still I never felt I actually understood it in simple, clear terms until late in my career I was asked to review a translation of Riemann’s own works, and read his version. It seemed suddenly that the scales fell from my eyes. I had similar experiences reading Euclid, Archimedes, Euler, Poincare', Gauss, Maxwell, and (in connection with a recent PF thread) Galois.
On the other hand, being almost entirely self taught in physics, even reading Einstein, Feynman, Born, Taylor/Wheeler, deBroglie, and Pauli did little to dent my lack of understanding.

 

[jedishrfu]

There's a book on learning strategies called Make It Stick by Peter C. Brown, Henry L. Roediger III, and Mark A. McDaniel.

It distills decades of cognitive science into practical lessons about how people actually learn.

Many of its recommendations contradict what feels intuitive.

• Replace rereading with self-testing
• Space study sessions
• Mix problem types
• Explain ideas aloud or in writing
• Seek feedback
• Embrace difficulty as a signal of learning

https://www.amazon.com/dp/067472901...make it s&tag=pfamazon01-20&tag=pfamazon01-20

 

[martinbn]

Still I never felt I actually understood it in simple, clear terms until late in my career I was asked to review a translation of Riemann’s own works, and read his version. It seemed suddenly that the scales fell from my eyes.

I am curious about this. Can you easily give a reference? (Don't spend too much time on it if you don't have it readily, I am just curious, it is not important)

 

[mathwonk]

@martinbn:

Riemann's original discussion is reproduced on pages 105-108 of:
the dover reprint of Riemann’s works in the original German: especially sections 4,5 of part I of his paper on abelian functions. In the following link, it occurs on pages 19-22, or more fully on 15-22 of the pdf file, with background beginning on page 1:
https://www.emis.de/classics/Riemann/AbelFn.pdf

[remark: Riemann is notoriously hard to read. The scales fell from my eyes eventually, after spending at least one day per page reading him, but it was worth it. It is helpful to read also as many preliminary pages as possible.]

For Riemann’s exposition in English, see pages 98-99 of the English translation of his works published by Kendrick Press. The whole book seems to be available here:
https://dokumen.pub/bernhard-riemann-collected-papers-0974042722-0974042730.html
or for purchase here:
https://kendrick.press/Bernhard-Riemann-Collected-Papers.htm

By the way, Roch's paper completing the theorem is much clearer and easier to read than Riemann, even though in German. (AI gives this):

• Original Source: G. Roch, "Ueber die Anzahl der willkührlichen Constanten in theorethischen Functionen," J. Reine Angew. Math. 64 (1865). (pages 372-376)

https://gdz.sub.uni-goettingen.de/id/PPN243919689_0064?tify={"pages":[376],"pan":{"x":0.461,"y":0.659},"view":"info","zoom":0.382}https://gdz.sub.uni-goettingen.de/id/PPN243919689_0064?tify={"pages":[376],"pan":{"x":0.461,"y":0.659},"view":"info","zoom":0.382}

One reason Roch's paper is easier is probably that it contains none of the deep existence proofs, but "just" a residue calculation, essentially using Green's theorem.

The classical argument of Riemann and Roch is given, following closely the original ones, in Griffiths and Harris: Principles of Algebraic Geometry, pp.244-245. They also work out some details omitted by Roch, on pages 240-243.

I sketch my own insight in this answer to a question on mathoverflow:
https://mathoverflow.net/questions/...ch-for-compact-riemann-surfaces/253187#253187

In the next post I summarize what I learned from reading Riemann.

 

[mathwonk]

Here is what I learned from reading Riemann.
Recall the Riemann- Roch theorem is a formula for the vector dimension of the space L(D) of meromorphic functions on a compact connected complex manifold M of complex dimension one (i.e. a "Riemann surface"), having poles only at most at a given set D of points.

Even if you do not read much German, just perusing Riemann's writeup shows the right way to look at this. Namely the first thing to study is the differential one -forms on M having given poles. This question has a simpler answer. Riemann lays it out in clear fashion: the one-forms are divided into three classes:

[Recall the fundamental residue theorem, that for any meromorphic differential, the sum of the residues over all poles must be zero.]

erster Gattung (first kind): these have no poles at all, i.e. they are holomorphic everywhere. Riemann proved the dimension of this vector space equals the topological genus g of M.

zweiter Gattung (second kind): these have a single pole at one given point, with pole of order two there (it cannot have order one, since the residue summed over all poles must be zero). Riemann showed the dimension of the space of differentials with at worst this singularity equals g+1, i.e. non holomorphic ones exist, and the difference of any two is (after scaling) a differential of first kind. Linear combinations of these, including those with higher order poles, are now also called differentials of second kind, i.e. meromorphic differentials with no residues at any pole.

dritter Gattung (third kind): for Riemann, these have two poles, at two given points, of order one, with equal and opposite residues at them. The dimension of the space of differentials with at worst these singularities, is also g+1, so they also all equal the sum of a holomorphic differential plus a scalar multiple of one non -holomorphic one. [These are not needed for the proof of the Riemann Roch theorem, but the point is that it seems then all meromorphic differentials are linear combinations of these three kinds.]

[In more modern times, Hermann Weyl recommends making the 3 divisions into successively larger subspaces: 1st kind: holomorphic differentials; 2nd kind: meromorphic differentials with zero residues at every pole; 3rd kind: arbitrary meromorphic differentials.]

Recall we can pass back and forth between meromorphic functions and meromorphic differentials, by taking d of a function, and by integrating a form.

Recall also, an "exact" differential df, i.e. obtained by taking the differential of a meromorphic function, has not only the sum of its residues equal to zero, but each individual residue itself is zero, since the differential of z^(-n) is a multiple of z^(-n-1). Thus exact differentials are a subspace of differentials of second kind.

The other characteristic of exact differentials is they have zero "periods". I.e. when integrating a meromorphic form to get a function, the function is globally single valued on M if and only if its integral around every pole (its residue at the pole) is zero, and also its integral around every non trivial loop on M is zero. Riemann did (i.e. invented) the topology to show that a surface of genus g has a "homology basis" of exactly 2g distinct loops, such that the integral over any loop is a linear combination of integrals over these. The integrals over these loops are called periods.

Since forms of the second kind are by definition those with zero residues, as a corollary, Riemann observed that one can calculate a space of exact differentials on M as a subspace of those differentials of the second kind that also have zero periods.

Thus given a set D of (say distinct) points of M, if we define W(D) as the space of forms of second kind with poles only at points of D and of order 2, and if we define a linear "period map" P: W(D)-->C^2g, by integrating each form over all 2g loops in a homology basis of M, then the space of differentials of functions in L(D), is the kernel of the period map P. Moreover, by Riemann's analysis above of forms of second kind, dimW(D) = g+#(D), where #(D) = the number of points in the set D.

Since the map, taking the differential of a function, from L(D) to W(D), has kernel only the one dimensional subspace of constants, consequently, we have from the rank -nullity theorem of linear algebra, that dim.ker(P) ≥ dimW(D) - 2g = #(D) - g. Thus dimL(D) = 1+ dimker(P) ≥ #(D) + 1-g. This is called "weak Riemann Roch" or Riemann's inequality.

To make the formula more precise, one needs to calculate the exact rank of the period map P. As might be guessed, this calculation of the period integrals is best done by computing residues. This was carried out by Riemann's student Roch. The result is that the period map's failure to be surjective, i.e. the dimension. of the "cokernel" of the period map, has dimension equal to that of the subspace of those holomorphic differentials, i.e. those differentials of first kind, which vanish identically on the points of the set D. This error term gives us the precise rank of P, as 2g-dimK(-D), and hence also the precise dimension of its kernel, as
dimker(P) = (#(D)+g) - (2g-dimK(-D)) = #(D) - g + dimK(-D).

Hence if K(-D) = space of those holomorphic differentials which vanish identically on D, then dimL(D) = 1-g + #(D) + dimK(-D), the full classical Riemann-Roch theorem.

Furthermore, if #(D) > 2g-2, then K(-D) = {0}, and hence in that case, i.e. if #(D) > 2g-2, then dimL(D) = 1-g + #(D), so Riemann's inequality becomes an equality for large D.

Riemann proves his inequality, and much more, in a few sentences in section 5 of part I of Abelian Functions. E.g. he shows that on a Riemann surface of genus g, there always exists a meromorphic function with no more than 1 + p/2 poles, a result sometimes attributed to Brill and Noether, writing much later. E.g. a surface of genus 2 always has a function with only two poles, but a surface of genus 3 need not have such a function. I.e., every Riemann surface of genus g has a meromorphic function with at most d poles if and only if g ≤ 2(d-1).

The hard part of Riemann's approach to these results technically is the proof of existence of the relevant differential forms. For this Riemann appealed to the "Dirichlet principle" which was not rigorously justified for some years after. Modern discussions of this principle appear in The Concept of a Riemann Surface, by Hermann Weyl; Topics in Complex Analysis, vol.1, by Carl L. Siegel; and Introduction to Riemann Surfaces, by George Springer. Griffiths and Harris appeal to the Kodaira vanishing theorem, which they discuss, but some people have criticized their treatment of these foundations. So there is some hard analysis underlying all these beautiful results. (I myself have read Siegel, and once plowed through Kodaira's proof of his vanishing theorem as an energetic young man. Nowadays there are other approaches to KVT, such as the "cyclic covering" proof by Kolla'r.)

In the case however of Riemann surfaces obtained by desingularizing plane curves, the relevant one- forms can simply be written down in terms of the coordinates of the plane and the equation of the curve, at least for the forms of first kind, which Riemann does. By the trick of "Brill-Noether reciprocity", explained also in Griffiths and Harris, one can get away with just those. Riemann states that those of second kind are also easy, but easy for him is not as easy for me, and he did not bother to do it in his paper.
I hope this helps to understand what Riemann himself says much more briefly.