In junior high school, had a very basic understanding of algebra (1+?=3 - what is ?). Anything above that is more complicated than I am able to understand. I could never have taken an academic course in high school because I would have failed math. So, I took a commercial course, became a legal secretary, and did quite well for myself over the past 58 years.
I took the required courses in Algebra and Geometry and ended up an internal auditor. Fortunately auditing doesn't require higher math. Add subtract divide multiply was I all I ever needed to figure out what was wrong with something and how to fix it! But I did take an Academic Course and took the obligatory classes. Did you not take Geometry L? Thank you for your reply and Happy Saturday still. Sunday toon! :)
Oh. "Commercial"? I'm not familiar with that. There was ACADEMIC for those who planned to go on to college but I don't remember what the other courses were called. I mean it was a very long time ago. I know there were home economics classes where you learned to sew and bake. And there was shop for guys who planned on working on cars to earn a living. I guess there was also business for those who were going to work in corporations at whatever level. No matter what ckasses you took the usual first job for a female was as a "secretary". Later on they got hi falutin and called them "administrative assistants" but it was the same thing. What's in a name? Whatever you want! Thank you for your reply L! :)
There were academic courses, if you planned on going to college, commercial courses, if you planned to go into business - yes, generally as a secretary or in a steno pool, and there were general courses for those who had passing, but low grades - if you were planning to be a file clerk or messenger.
Administrative assistants generally had more experience, worked for a mid-level or higher executive in the company and had more responsibilities than just taking steno and typing. I started out as a low-level secretary in a law firm which made me a "legal secretary" because I learned to assist my associates with many things they did, such as drafting documents for their review, making appointments for depositions (I was working for negligence attorneys) and the like.
Which also likewise helped you too by making your job a lot more interesting. In fact I got into the accounting end by "helping out" in Accounting when I had time. I got drawn to it so later on I took bookkeeping at Night School but the person I learned the most from ON THE JOB was a darling CPA who worked for the firm that audited our books at the end of every year. She was young enough to be my daughter and she showed up for a few years at audit time and she was VERY GOOD at teaching me how to do the working papers correctly. Her name was MIKI and she was about 5 feet tall..slender sweet never once talked down to me. She could see that I really wanted to do a good job so she gave me handy hints along the way and also went out of her way to explain WHY. I think her last name was Okamura or something. Because of her I learned enough to become an internal auditor in several companies. NOT A CPA but that wasn't required. The CPA always was available by phone to ask if U wasn't sure which way to go on something. And it was the CPA who audited my work at the end of every year which I really looked forward too so I had the advantage of learning on the job from the best. Miki worked for Price Waterhouse and I will forever be indebted to her for her patience and for educating me! Thank you for your reply L. :).
This post was edited by RosieG at January 17, 2021 7:50 AM MST
It's always nice when someone is willing to mentor you to be better at what you do rather than try and keep you down. That signifies her confidence in her own position. I learned on the job, too. After a couple of years, I was doing what is now a paralegal job. I could have looked for a job as a paralegal, but by that time, my salary had increased significantly and paralegals were paid less so I stuck with the secretarial end.
Arithmetic and Mathematics were always my weakest subjects.
Arithmetic in Infants' School introduced simple money sums, but Primary School included Compound Arithmetic. That was calculating the cost of given quantities of commodities like coal, at given cost per bulk weight; but for my generation still all pre-decimal currency and pre-metric measures, so £.s.d. and Tons, cwt, qrs.
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When it came to Maths though, two of the many topics that took years to sink in were Logarithms and Differentiation. They did so eventually because I met them again, but now in real-life, respectively at work (not as times-sums tools - we had calculators for that) and in my hobbies.
I could use logs for arithmetical calculations at school but without understanding them, and it was only in the last 20-30 years when I encountered them in the guise of the ratio called the deciBel, that I twigged their principle.
Calculus was a mystery at school. Decades later I was in a geology-society tutorial on analysing river gradients to spot possible geological boundaries hidden under the valley floor sediment, farming etc. The tutor displayed a simple formula to determine the gradient of each short sample length, from memory something of [H / (L1 - L2) ] construction; H being altitude drop. Something made me write it in calculus notation, ( H / dL ), and I suddenly realised this gradient measuring was a basic Differentiation. Before any hydrologist reading this comments, I am not sure now if the full numerator is (H1-H2), hence (dH / dL). Integration still eludes me though.
THE one maths topic that really stays in the dark is Matrices, which I encountered first in a refresher school-level course about 20 years ago. These were boxes of simple sums done in a certain order; but utterly abstract, lacking all obvious meaning or link to any other maths or physical things. A scientist who tried to help me enthusiastically said Matrices do have a use... as in her work, with a computer programme handling simultaneous equations of typically 100 unknowns...
Most of what you wrote I do not comprehend my friend. It's like some ancient language. I became an internal auditor through the work years but that is simple math or arithmetic. At least insofar as my job entailed. Add subtract multiply divide. I expect there may be auditors who require higher math for their work though. I took the obligatory algebra and geometry in high school because I was on an academic track. Languages too. French for 4 years...that is two years in high school and then two years in junior college. I think it's so neat that at some point what seemed a mystery to you years before became clearer later on in your real world experiences. I guess no knowledge is ever lost. Never know when it will come in handy. Thank you for your reply Durdle and Happy Saturday almost Sunday. How's the U.K. lockdown going viruswise? Still doing okay under the circumstances?
This post was edited by RosieG at January 19, 2021 2:34 AM MST
That was good, having someone who could really help you.
I don't know how complicated auditing becomes - I would imagine percentages and interest calculations would be important too - but I think economics also uses those pesky Matrices I found so baffling.
On thing I did discover about those. was that though their principles were centuries old by then, one of their 19C developers was Prof. Charles Dodgson, author of Alice in Wonderland under the pseudonym of Lewis Carroll. It thought that rather appropriate, that researchers into such an abstract and puzzling mathematical tool as matrices, should also write literary fantasies!
One thing I have noticed, and you seem to hint at it there, was that American schools appear to treat the different topics of mathematics as totally separate curriculum subjects. In the UK, although each topic - algebra, geometry, trigonometry, etc. - is taught more or less on its own, the whole field and the culminating examinations combine them. So we are not taught Algebra and Geometry and Trig. etc. courses; but are taught a single curriculum subject called "Mathematics" whose syllabus includes those. Maths is maths though, (or "math" in your country!) so I imagine the overall contents and levels are similar; just a different approach.
I found at work a coffee-mug decorated to commemorate the 18C English mathematician George Green. Around it is printed an equation that I assume is quoted from his work. I can identify it as integral calculus and guess it refers to three-dimensional geometry, but beyond that it is totally beyond anything I could manage to learn. Since Element says he enjoyed Calculus I'll see if I can add it below, as a picture, for him to have a bit more fun!
I struggled with French, too. I was allegedly highly intelligent when young but was always a slow learner unable to grasp most things to more than very modest levels. It seemed needless to me, too. Despite the French coast being only 70 miles South from the English Channel coast close to my home, holidays abroad were still quite exotic to most Britons in the 1960s; and I never thought I would ever visit the country. Also the language was taught as if just a school-leaving examination subject, not something millions of people live their lives by. Barely five years after I left school, my new-found hobby of caving took me on the first of quite a number of holidays with friends to France.
You are right though - no knowledge is lost. It might suffer from neglect, so we forget parts of it, but it can often return with new use.
' The lock-down? Grim! Most people are understandable weary of it and of its results, but we grin and bear it as best we can. Some try to bend or break the rules and end up being fined for it, but the majority of Britons do obey them. I am lucky in many ways. The ones who are really suffering are those living with families in cramped accommodation, struggling to get by on very limited means with their employment (often low-paid anyway) temporarily stopped, but without being certain of the business surviving. Although the vaccinations schemes are now under way the virus and its hyperactive offspring is still touring the country, and it will be months yet before any sort of "normal" life can return.
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Here we are, to tickle Element's fancy... One of Green's equations. (Sorry the typesetting's a bit weird - it is hard to know how the symbols will appear on screen.)
That equation is obviously baffling to me. A language I could never learn. I know my limitations. It looks sanscrit or greek or hieroglyphics even. Something borne of ancient times. That each one of those characters symbolizes something? Who discovered or invented something that COMPLEX? Thinking upon that gives me a headache. Here's a question that seems nonsense but I don't think it is. Do you have to know where you want to go to get there? Do you have to know your destination and then figure out the best path there when it comes to math equations. Or do you STUMBLE upon it but doing WHAT IFS? I'm going to ask. Thank you for your thought and informative reply Durdle! Another day we survived. May it continue! :)
I know the field of mathematics it occupies (Calculus) but that's about all.
You are on the right track with the symbols; and yes, they all mean something in that equation.
σ, δ and π (Pi) are Greek letters, and they are used simply to extend the range of symbols available, and to separate them clearly from all the Arabic letters. The letter Pi is an exception in that it is only ever used to symbolise that magical value 3.1416... familiar to us from school when we first meet it in the dimensions of circles.
"Magical".... I read somewhere that in mathematical language the value of pi is called a "Transcendental and Irrational Number" - proper technical words, but rather romantic!
ʃ is actually English, the old form of the letter 'S', and you see it in historical documents. It is in fact, the letter S here, standing for "Sum", which is effectively what it is telling us to calculate: a special sort of "adding sum".
Well, the one who discovered the relationship that that equation describes is George Green, but he was developing on work by others previously, including Isaac Newton.
How they do it though, I have no idea! Those are good questions: I think the mathematicians must see things that looks as if they can be described by regular numerical rules; or they see problems that might yield to mathematics. There might be a lot of "what ifs", but the basic "what if" is that first idea - "IF there is a numerical pattern or solution to this, what methods can I use to investigate it?"
"Stumble upon it..." I daresay that happens sometimes. There are plenty of accidental discoveries in science, and anecdotes of people mentally chewing over a problem in their sleep, so I expect there have been plenty of accidental revelations in maths.
In practical science and engineering, the pattern often emerges in the results of many careful measurements. If a graph of those gives a smooth, straight line or elegant curve, there is a probably a numerical pattern in there, and it will be revealed by careful combinations of standard mathematical techniques. Statistics is often used to remove the wobbliness natural to measuring real things, and show the real relationship properly.
It might be something really simple - the principle of the pendulum is credited to Galileo Galilei, who is said to have used his own wrist-pulse to time the gentle swinging of a large chandelier, showing him as he'd guessed that the time of swing stays constant even though the swinging is gradually reducing. I do not know if he discovered the formula for it, but simply realising that constancy hints at a numerical relationship between a pendulum's length and time, waiting for experiments to reveal it. Sure enough there is, it is fairly simple - and oh look, it includes our old Greek friend Pi.
(All Greek? I once had a fortnight holiday on Crete, and by the end could just about string together the Greek for Please and Thank you, from the phrase-book. Near our holiday flats was a large building with what looked like a lot of Physics algebra on it. Pi and all, I think. When I pieced the letters together, from the alphabet in the book, the first syllable seemed familiar from scientific publications. Phylle.. or Phyto... I forget which but knew it meant plant, as in gardens! Biologists use Greek words in certain contexts, e.g. "Chlorophyll" ("Green [of] plants"). The building was a garden-centre!)
Crete? Small world. When the Armenian genocide was in progress my dad and his parents fled to the island of Crete where they lived for a few years before they emigrated to America!
Are you aware of "the universal mind" which houses the entirety of all the knowledge that ever existed? Some "genius" types allegedly have been able to tap into that repository and bring us the astounding astonishing "discoveries" they have. It is said that anyone can tap into it if you know how. What d'ya think of that? That is mystical and magical and really enticing don't you think? Thank you for your extremely thoughtful and informative reply Durdle. Apologies for the delay but I am always behind. Mea culpa! :(
Multiple integral calculus and differential equations. I enjoyed it, but I only used it a few times during grad school. I remember very little, but I can still do algebra, trig and geometry. Math was fun.
There are people who really GROOVE on math E and it's my honor to know one. You. Danica McKellar is a former child star and still acts occasionally. I believe she is a math whiz and travels around trying to get girls to become interested in Math. I am very impressed by her and by what she is trying to do. We KNOW that there were women involved in the space program who were math whizzes. Did you see the movie HIDDEN NUMBERS? I loved that movie. Women like that are superstars to me. I mean I EXPECT some men to be math geniuses. But women too? What a revelation that is. Higher math has always been a mystery to me. To know that there are brilliant women who are math geniuses too? It really make me proud even though I will never be part of that crowd. And your saying "math was fun" is exactly what Danica is trying to get across to other girls. Thank you for your reply! Happy Tuesday to thee and thine! :)
I'm with E99. Integral calculus was not only fun but useful, particularly when pricing the more grandiose designs of architects. I enjoyed math. Helps that I was rather good at it, and being the son of a former schoolmarm (who drilled me daily in arithmetic until I was proficient) also helped. I balked at tensor calculus and multidimensional arrays. There comes a point in math where the symbols relate only to each other and have no application to real space-time events. I couldn't see the point, and lost interest.
This post was edited by Slartibartfast at January 19, 2021 11:43 AM MST
SpunkySenior 