How a book written in 1910 could teach you calculus better than several books of today

I sub some calculus & pre-calc classes in high school.
This might provide a way to help in a simpler way than how I can explain it now.

"Even Isaac Newton, said to be the discoverer of gravity, knew there were problems with the theory. He claims to have invented the idea early in his life, but he knew that no mathematician of his day would approve his theory, so he invented a whole new branch of mathematics, called fluxions, just to "prove" his theory. This became calculus, a deeply flawed branch having to do with so-called "infinitesimals" which have never been observed.

"Then when Einstein invented a new theory of gravity, he, too, used an obscure bit of mathematics called tensors. It seems that every time there is a theory of gravity, it is mixed up with fringe mathematics. Newton, by the way,was far from a secular scientist, and the bulk of his writings is actually on theology and Christianity. His dabbling in gravity, alchemy, and calculus was a mere sideline, perhaps an aberration best left forgotten in describing his career and faith in a Creator."

https://ncse.ngo/gravity-its-only-theory

The best explanation I've encountered for why there are two high tides per day is part of a 1966 short story by Larry Niven, titled Neutron Star, which can be read here. The explanation is on page 3.


The Moon has the same effect on the Earth, trying to pull it apart, but in this case the ocean's response is most visible because it's liquid and can flow. Whether the Moon is overhead or below our feet, it tries to pull the earth apart. Therefore there are two tide cycles per day.

There is an annotated version that explains many of the jokes:
https://skepticalteacher.wordpress.com/tag/ellery-schempp/

"The existence of tides is often taken as a proof of gravity, but this is logically flawed. Because if the moon’s “gravity” were responsible for a bulge underneath it, then how can anyone explain a high tide on the opposite side of the earth at the same time? Anyone can observe that there are two-not just one-high tides every day. It is far more likely that tides were given us by an Intelligent Creator long ago and they have been with us ever since. In any case, two high tides falsifies gravity.

"The gravitational force(1) depends upon the mass of each object and upon the distance between the objects. The closer they are to each other, the more strongly they attract. In the case of the moon and the earth, the moon pulls the ocean water on the nearer side of the earth more strongly than it pulls the rest of the earth, so the water forms a bulge. On the far side of the earth, the water experiences the least pull because it is farthest from the moon. The rest of the earth is pulled away from the far-side water, leaving a bulge of water opposite to the moon, though this bulge is not quite as large as the first one.(6) The moon holds these bulges (more or less) in place while the earth rotates beneath them. The net effect is the tides we experience at the seashore. By the way, there are also small land tides--these are about a centimeter."

A very complete explanation of the tides is presented here.(7)
http://www.coops.nos.noaa.gov/restles3.html
-----------------------
footnotes:
(5) Observant readers might ask, if the earth-moon center of mass
system is constant in its path around the sun (ignoring perturbations
from Jupiter, et al.), then doesn’t the earth wiggle a little in its
path? The answer is yes, the ripple in the earth’s path is similar to
that from above, but even smaller. The center of mass in the
earth-moon system is about 1000 miles under our feet (when the moon is overhead), or about ¾ of the way from earth’s center.

(6) High and low tides often have a time delay from when the moon is
overhead. For example, it takes 2-4 hours to fill/empty San Francisco
Bay through the Golden Gate. Cf. high tide on the Pacific side of the
peninsula vs. San Jose.
---------------------------------
The author of this parody on creationists is Dr Ellery Schempp--who originated the landmark Supreme Court case that stopped Bible-reading and prayer in public schools (1963). See Wiki.

At all!




For ref, I did a whole five or six weeks of the 9th grade. 25 years later I went to college and came out with two degrees, but I still suck at math and the like, dyslexia gets me often. I can understand it, just don't ask me to do it because I rarely come out with the same answer twice for complex operations.

Because that’s exactly, almost word for word, how it was explained to me.

I didn’t realize anyone thought it was difficult until I entered the working world.

Perish the thought! Instead textbooks cost hundreds and are good for a year or a semester.

esp college, such a waste of resources.

All the students in our local high schools who don't have their own laptops, or don't want to bring them to school, get Chromebooks. My wife's sister has gone paperless in her high school classroom. She's not the only teacher doing that. Chromebooks cost less than some textbooks now.

My children had quite a few college classes without traditional textbooks.

Culturally here in the U.S.A. we are always loathe to set up government agencies that are in direct competition with private industry, even when we should. Health care is the prime example. The result is we have the most expensive health care in the world and it's nowhere near the best, not even for wealthy people with "platinum level" insurance. Many of our textbooks fail in a similar way, or worse, are written so as not to offend the anti-intellectual rubes, racists, and religious cultists of shit-hole U.S.A.. History, biology, and health texts are especially crippled by this phenomena.

The primary focus of any K-12 history textbook should be the struggle for civil rights. The primary focus of any K-12 biology textbook should be evolution and the natural environment. The primary focus of any health text for middle and high school students should be frank and realistic education about sex and relationships.

My own middle and high school textbooks back in the anti-communist war days of the twentieth century were utter crap. I was fortunate to have a few teachers who worked around those limitations, but mostly I hated everything about high school and quit for college at sixteen. Thanks largely to my parents my curiosity and love of learning hadn't been extinguished.

About 2/3rds of the schools where I sub are almost all, to all, chromebook/iPad.
A few schools have them, but they still do handouts which, as a sub, I like. I can have my own copy, go around checking work, and help out more.
Oddly, I see way less of it in science classes than in math. I'd say 75% of those schools still have science textbooks.

I just paid $70 for a "workbook" that sends you all over the internet to do much of the "research" for the class. The class was six weeks, and had very scant information on the topic. However, you need the workbook in order to take the tests. Same with chemistry. $100 for an online workbook that you need to do the homework.

Some classes still need the texts.

Which is why god invented z library . . .

I saved it on my iPhone… The whole text that is…

Many years ago, my daughter and I wereout to dinner with a guy I knew and his daughter, both kids were about 11years old. This fellow was an attorney and made some reference to the impossibility of explaining calculus to anyone, and I countered that I could explain it to the two kids. So I took a napkin, drew axes, and a curved line above the x-axis, and demonstrated how you could find the area under the curved line by using successive rectangular iterations until you hit an infinite number and then that would give you the answer. He was stunned that the kids got it right away and I was wondering if there was a book that would explain it so plainly. Apparently there is and I thank you for posting it here…

One might say that the entire manifest Universe is the sum of infinite little bits.
Indeed, the search for 'what-is-it?' continuously finds smaller and smaller 'little bits'; it is a quantum mystery.
Odd that in the search for some 'unified field theory', one 'little bit' (un-measurable and formless) is essentially ignored.
The consciousness of the searcher, the consciousness of the knower of little bits, the wielder of the math that reveals that the outcome is not accurately measured when consciousness is ignored.
Non-negatable 'consciousness' is the integral unified-field by which the measure of all little bits is known.

means a little bit of u …

🤔



✌🏻

that will be a relief for many students who can't follow much of the stuff.

The major hurdle is knowing what sorts of problems can be solved with calculus or any other sort of math.

Here in the U.S.A. we tend to teach math in isolation, no thanks to the idiotic reading, 'riting, and 'rithmetic crowd. None of these subjects can be taught well in isolation. It's almost like we want children to hate math, or at least think they are no good at it.

A child ought to have had some exposure to calculus and what it's used for by the time they finish middle school. In most of the world calculus is first introduced with basic physics, just as Isaac Newton and Gottfried Wilhelm Leibniz first applied it.

One nice thing I can say about our new age of social media is that it encourages some level of literacy. Every child knows that reading and writing are useful skills. Too bad there's no similar incentives for numeracy beyond knowing how much money you've got.

I'd completed college algebra and took a statistics class, and was enjoying the math so much that I decided to go for calculus. Loved it! Understood it completely.

I asked several of the math teachers why, after failing miserably the first time in high school, I was doing so well thirty years later. To a person they said, "Oh, Poindexter. People don't realize that it's a developmental thing. A lot of students just aren't ready for it when they're 17, but give it a year or two and they will be." Clearly that was the case with me.

And found it easy. (Okay, didn't quite get related rates for a while, but when I came back to them I thought, "Oh, where was the problem?"

Mostly because my college algebra teacher was great and made sure we under functions and limits and how to "see" most common kinds of equations.

Had to help some students last fall with their calculus and explained that the capital Greek S (sigma) "sum" symbol is good for a countable number of elements and usually uses a Greek "d" (delta), the long S is a sum as the size of the elements approaches zero and are "infinitesimals". I teach high school science.

Years ago there was a Japanese program in which they taught calculus to 6th graders. I find that students suck at calculus when they ignore the math and think of it as (just) a series of tricks and techniques.

When you take classes in your 30s 40s 50s... you have lived some life, you appreciate education, and you care about learning and how it can change your life. This is not a critique of young people because I sure didn't give a fuck when I was 17 about half the classes I took and as a high school teacher I so love them. But I do think the difference in perspective is huge.

I was taking math classes because I was planning to take degree that required math through college algebra.

At my local junior college I tested into algebra 2. I enrolled, and the instructor assigned homework problems that had the answers in the texts, not the ones without the answers. Thank whatever deity is appropriate. I went to class, took good notes, and then at home. trying to work the problems I was totally lost. I'd look at the answers and then could work backwards to figure out how to get at the answer. Next problem, same thing.

The instructor kept on telling the class that if any of us weren't really prepared for algebra 2, we still had time to backtrack. In the second week we had our first exam. At the end of that period, I went up to her and professed my doubts about being in that class. She asked, "How do you think you did on the test?" I told her I was pretty sure I got a B, and she said, that's good enough. And she was right. I wound up with an A in the class, to my intense surprise and delight.

She was about 4 years older than me at that point, and I quickly connected to her. In high school, my math program was UICSM, which stood for the University of Illinois Committee on School Mathematics. I have never, in all these many years since, run into anyone who has any clue about it. Here's the thing. It was a "new math" program. This was 1962. We didn't have regular textbooks, but paperback ones that we wrote in. We proved EVERYTHING. Trust me on this. It's why I remember a lot of specifics from that high school math, and why I could test into 2nd year algebra some 30 years later. All during the algebra 2, and the next semester college algebra, both with the same wonderful teacher, I'd be sitting in class, we'd be covering something or another, and I'd remember what I'd learned decades earlier.

One interesting thing was that in my high school class we'd learned that something was true "if and only if" something else was true. At some point I asked my college teacher about this, and she told me that that was the kind of language normally used in rather advanced classes.

I am very glad I took UICSM math all those years ago. I actually remember various specifics, but more to the point I learned a system of thinking about things that has stayed with me.

Oh, and every so often, just to keep in shape, I re-work the Fahrenheit to Celsius formula.

We need to do something to get more people into STEM careers. College algebra and then Calculus are the first exposure many students get to problem solving in real world situations. The world and industry are full of story problems and we need more engineers to solve them.

We’re way behind in producing scientists, mathematicians, and engineers. If this isn’t addressed we’re not going to be able to compete in the economies of the future.

sortof use this sort of teaching style. When he explained multivariable calculus, rather than have 2 dimensional plots of an "area", you go 3-dimensional by just "rotating a (2-axis) curve around a 3rd axis" and get the "volume", and I was like

Have just downloaded this volume!

...instead of using needlessly-updated, ever-changing (but for no good reason) textbooks as a profit center.

Calculus was developed by Newton and Leibnitz independently in the late 17th century, and it was put on a rigorous basis by Riemann and others in the mid 19th century. That is: mathematicians have been writing about calculus for a very, very long time, and the odds are that somebody in that span of time has written a clearer and better explanation of it than people writing about it today. (New is not necessarily better.)

That I said, I would give a loud shout out to Howard Anton's _Calculus with Analytic Geometry_. Very well written, covers the ground more thoroughly than any other text I've seen. Forty years on, it's right above my desk for reference.

J.