Temple Grandin on Algebra (she’s agin it)

[Roger Feeley]

She is a fascinating person. Here, she argues that higher math is required for too many vocations that simply don’t need it. I tend to disagree but I respect her point of view.

I think of math like push ups. No vocation requires push ups but they do require upper body strength. I see algebra as calisthenics for the mind.


https://www.theatlantic.com/ideas/archive/2022/10/against-algebra/671643/ (https://www.theatlantic.com/ideas/archive/2022/10/against-algebra/671643/)

This isn’t political, is it?

[Tom M King]

Absolutely. Exercise, and sometimes you surprise yourself when the need pops up. Problem solving doesn't come naturally to many, and math is all problem solving exercise.

I like and respect Temple Grandin.

[Edwin Santos]

That was an interesting article, thanks for sharing.
I must say I thought it was going in one particular direction with the discussion of STEM curriculum and how it has resulted in the elimination of non-tested areas like art, industrial arts, and the other electives many of us took in school.

But then she turned the attention specifically toward algebra. I get why algebra was uniquely problematic for someone with autism like Temple. But I don't think that's enough of a case to eliminate it because algebra hones problem solving thinking even if the problem at hand is quantitative but not not necessarily "algebra".
So I'm in total agreement with the first part of the article, not so with the second half. But I appreciated her perspective and argument nonetheless.

[Tom Bender]

I am a retired mechanical engineer. In college I spent at least half of my effort and energy trying to pass calculus courses and engineering classes that were heavily calculus based. A typical lecture started with a problem followed by the way to solve it using calculus. Since then I have had a full and rewarding career as an engineer, never having even met another engineer who had ever used calculus at work. It is just an academic tool. The practical uses it is put to in academia are easily resolved with simplifications, charts and now software. A colossal waste has been made of a lot of brainpower.

Yes there are a few applications that call for calculus and had any of them been in front of me I'd have found someone with a proclivity for the art to help with it, just as I would have sought help with windmill design or ship architecture. Nobody can do it all and calculus (advanced algebra) has low value.

So much other content would have been more valuable; languages, law, patent law, ethics, environmental studies, economics, negotiation, the list is long.

[Prashun Patel]

I don't think everyone needs to know it. But algebra is the fundamental concept that is key to computer programming - er "coding". The ability to think in terms of generalized formulas and variables is not necessary for everyone, but is arguably necessary for more people than ever before.

That being said, I'm all for a world where judgment is less and alternate paths are seen as valid and worthy of pride.

[Lee DeRaud]

In the mid-70s, an engineer/physicist coworker of mine saw the way the wind was blowing wrt to computers, and had his high-school freshman son sign up for typing class instead of shop class.
He ended up in a six-month battle with the local school board over it. "What, you want your kid to be a secretary?!?"
(Oddly enough, they would let boys substitute home-ec for shop, but not typing.)

Knowing now how much code I would sling over the course of my career, I certainly wish that had been an option when I started HS.

[Andrew More]

Software engineer here. Software is eating the world, and Algebra is one of the best ways to teach the almost arbitrary rules and abstract thinking required to do it and do it well. I would agree with Tom about the real world applications of calculus, but then I thought the same thing about most of my first two years in college. Add onto that chemistry, history, psychology, electric networks, magnetic fields and a few other courses that I have never had a practical use for. However there's something to be said to learning things that aren't necessarily directly practical, though I'm not as sure about requiring them.

[Prashun Patel]

Um... denegrate calculus if you want, but it is the basis for machine learning. It is used to minimize the error of the equations the machine attempts to fit to the real world data. This didn't hit me in the face until I tried to help my 11th grader with Calculus while also trying to take the Coursera Andrew Ng Machine Learning course. The fundamental concepts in calculus are rates of change and rates of rates of change.

I'm not saying everyone needs to know it - but like algebra, there are MORE people that need to know it and know it critically at this nascent point of machine learning. People relying on black boxes and only a few people knowing how the engine works got us into some hot water in 2008. Now that I think about it, those were "derivatives" too!

Yikes!

[Mark Rainey]

I am unable to read the article - you have to subscribe to the magazine. Algebra can be important in today's world. For many it is not important. The idea of a challenging high school course to exercise the mind has merit. The problem comes when a significant percentage of high school students simply cannot grasp the concepts despite varied educational strategies. There are some brilliant students that are befuddled by algebra. The student is demoralized and sometimes labeled causing placement in a "non-academic" path. This is unfortunate and can be destructive.

[Maurice Mcmurry]

Dads shop when I was a kid was in-between the rail depo and the slaughter house. The miserable cattle on the way to die were a constant source of discomfort. I enjoyed learning about Temple Grandin as well.

[Ron Citerone]

My take on the article is less about algebra, and more about how standized test scores rule the day at the expense of other types of learning. In that regard she is correct. As to her point that not teaching Algebra because some students can't grasp is, IMO, wrong.

[Thomas Wilson]

I joke that Robert Fulghum claimed he learned all he really needed to know he learned in kindergarten. I, on the other hand, had to wait until. I finished differential equations. Math comes easily to me. I read equations as easily as reading a newspaper. I made a good living solving math problems for people who thought they could not do it themselves. I use math in designing woodworking. I see math as a tool to do a job, almost any job. But let me tell you something I learned about learning math when I was quite young. In the local library, I noticed a group of volumes, Men of Mathematics, by E. T. Bell. I read a few bios whenever I went by. I was in there a lot. The bios are problematic as they totally ignored the women and failed to tell the full story of the men as I learned later. But the narrative of the lives of the men’s stories were a framework on which I could hang the knowledge of math itself. The brain is associative and the association does not need to be so much logical as compelling. Having a strong peg in my brain on which to hang a concept meant that I could remember it.

Our brain did not evolve to do math, it evolved to survive. Retaining collective knowledge through storytelling is a survival skill. Retaining a mental image, a map, of the surrounding environment for its resources, risks, and ways the navigate it is a survival skill. Having a mental shorthand for simplifying a problem and manipulating it to find a resolution is a survival skill. The field of mathematics is built on these skills. I would observe that those skills vary in strength greatly in individuals. Whether that is a difference in innate ability and in development of the skill, I do not know. I do know that finding the right way to explain a mathematical concept is key to opening that door and no one single way works for every learner.

I tsk-tsk-tsk at all the woodworking articles that begin with doing some task with NO MATH. Math is a tool. At the least, it makes a job easier. At greatest leverage, it is the only way a job can be done. Solution by infinite experiment is no solution.

[Ron Citerone]

I joke that Robert Fulghum claimed he learned all he really needed to know he learned in kindergarten. I, on the other hand, had to wait until. I finished differential equations. Math comes easily to me. I read equations as easily as reading a newspaper. I made a good living solving math problems for people who thought they could not do it themselves. I use math in designing woodworking. I see math as a tool to do a job, almost any job. But let me tell you something I learned about learning math when I was quite young. In the local library, I noticed a group of volumes, Men of Mathematics, by E. T. Bell. I read a few bios whenever I went by. I was in there a lot. The bios are problematic as they totally ignored the women and failed to tell the full story of the men as I learned later. But the narrative of the lives of the men’s stories were a framework on which I could hang the knowledge of math itself. The brain is associative and the association does not need to be so much logical as compelling. Having a strong peg in my brain on which to hang a concept meant that I could remember it.

Our brain did not evolve to do math, it evolved to survive. Retaining collective knowledge through storytelling is a survival skill. Retaining a mental image, a map, of the surrounding environment for its resources, risks, and ways the navigate it is a survival skill. Having a mental shorthand for simplifying a problem and manipulating it to find a resolution is a survival skill. The field of mathematics is built on these skills. I would observe that those skills vary in strength greatly in individuals. Whether that is a difference in innate ability and in development of the skill, I do not know. I do know that finding the right way to explain a mathematical concept is key to opening that door and no one single way works for every learner.

I tsk-tsk-tsk at all the woodworking articles that begin with doing some task with NO MATH. Math is a tool. At the least, it makes a job easier. At greatest leverage, it is the only way a job can be done. Solution by infinite experiment is no solution.

Awesome Post Thomas! Well written. An Engineer friend of mine made an interesting comment about Math to me. He said "Technical people see Math as a tool, Mathematicians see Math as a subject." That made me think.

I love "Solution by infinite experiment is no solution."

Cheers

[Tom M King]

Wonderful post Thomas. I'm going to copy and paste it to send to some of my Scientist friends if you give permission.

Temple has given talks at NASA. There are quite a number of very high functioning Autistic folks, and some of them are top Mathematicians. NASA saw that there would probably be benefit from hearing Temple's views. She stayed with some friends of ours back then while she was there.

[Mike Henderson]

I'm an electrical engineer and took a lot of higher mathematics in college. The EE classes used a lot of mathematics (calculus) because there was no way to teach what was going on without mathematics.

Once I got out of school and working in communications, I learned that communications is all mathematics - and complex mathematics. For example, take digital error correction techniques - there are a bunch of different techniques and they involve really deep mathematics. People are still developing improved techniques and the underlying mathematics takes some serious effort to understand. You sure wouldn't understand it if you hadn't taken a lot of advanced mathematics.

I don't know about other engineering fields but it seems to me that just about any engineering task must require complex mathematics.

Mike

P.S. I took typing in High School and I'm glad I did. I learned on old, manual typewriters where you had to carriage return by hand.

[If you want some exposure to the complexities of FEC, take a look here (https://www.sciencedirect.com/topics/engineering/forward-error-correction). This is just a summary of a few papers and books on the subject. And FEC is just a small part of communications.]

[Maurice Mcmurry]

I continue to have embarrassing "should have paid more attention in school" experiences. Especially Math, Algebra and Trigonometry. What are all of those mysterious extra buttons on the kids calculators? I have had the building Inspector tell me to "Show engineering" On things like ridge beams and cantilevers. The kids were a big help showing me how to make and solve the equations. So far my tailgate engineering produces things that are ridiculously over built. I wish I could stamp my designs... Should have paid more attention in school!

[glenn bradley]

He ended up in a six-month battle with the local school board over it. "What, you want your kid to be a secretary?!?"
(Oddly enough, they would let boys substitute home-ec for shop, but not typing.)

A testament to the shortsightedness of that age 50-ish years ago. Sadly many current community leaders still cling to these views even today. The decline of the arts in our public schools is inexcusable but a reality for a few generations now trying to deal with the world as adults. My heart goes out to young people often dismissed as ignorant simply because we didn't expose them to a broader range of experiences in public school.

[Thomas Wilson]

Wonderful post Thomas. I'm going to copy and paste it to send to some of my Scientist friends if you give permission.

Temple has given talks at NASA. There are quite a number of very high functioning Autistic folks, and some of them are top Mathematicians. NASA saw that there would probably be benefit from hearing Temple's views. She stayed with some friends of ours back then while she was there.
Please correct the typing before sending it. I would love to meet Temple Grandin and get a hug.

[Tom M King]

I never met her, but was just told about that.

[Kevin Jenness]

An interesting discussion by Richard Feynman of the relationship of math and physics https://www.youtube.com/watch?v=obCjODeoLVw

The mathematicians are only dealing with the structure of reasoning...they don't even need to know what they are talking about

If you want to discuss nature...it's necessary to find out the language she speaks in

[Edwin Santos]

A few thoughts on this interesting subject in no particular order:

The claim of what is and isn't applicable to the real world can only be made in hindsight. Usually at the time the student is a student, his/her future isn't completely known. So who's to say their path won't take them to a place where the curriculum won't be used in their real world?

Then there's the issue of not every piece of subject matter needs to have direct practical applicability. Sometimes, the idea is that the brain is a muscle and exercising it in ways that might seem eccentric is part of building the foundation for future problem solving in general.

But I agree if a student cannot grasp a given subject for whatever reason, the school should be able to offer an alternative. When I was in high school, there were some kids that could not succeed at algebra. I recall the alternative was a class they called Business Math which was mostly an expansion of arithmetic applied to the real world i.e. percentages, interest rate calculations, how discounts/markups are calculated, how loans are calculated.

If a curriculum truly needs to be restricted to daily working practice, I think we're then into the realm of trade schools. Not a bad alternative at all for many. And for people like Temple Grandin who have particular situations that pose challenges, I think it's a great thing that alternatives be available so talent is not obstructed by the barriers of convention.

This past week an interesting dust-up got a lot of press. The gist of it is a professor of chemistry at NYU was fired when 82 of his 350 students (dissatisfied with their grades) signed a petition of complaint that his med school class in organic chemistry was unduly hard. Plus they had sundry other complaints about his teaching style. There are good arguments on both sides of the issue.

One is that the 80 year old professor was way past his prime, he was in the wrong role, wasn't showing a true commitment to the success of his students, and demonstrated curt sarcasm and dismissiveness in lecture.
And organic chemistry isn't used by most physicians on a daily basis in the real world anyway.

The other argument is that he is a distinguished professor whose textbook on organic chemistry is now in its 5th edition.
Organic chemistry is a weed out class, designed to be hard because succeeding in medical school is supposed to be rigorous.
And what precedent does it set when 82 out of 350 students, ostensibly dissatisfied with their grades can get a professor fired?
What about the students who were succeeding in the class and didn't see fit to sign the petition of complaint? Does their success suggest the professor couldn't have been all that bad?

Student centered education is a sound concept but taken too far, it can get into participation trophy mentality where by design every student must walk away a winner - definitely not the way the real world works.

[Ron Citerone]

A few thoughts on this interesting subject in no particular order:

The claim of what is and isn't applicable to the real world can only be made in hindsight. Usually at the time the student is a student, his/her future isn't completely known. So who's to say their path won't take them to a place where the curriculum won't be used in their real world?

Then there's the issue of not every piece of subject matter needs to have direct practical applicability. Sometimes, the idea is that the brain is a muscle and exercising it in ways that might seem eccentric is part of building the foundation for future problem solving in general.

But I agree if a student cannot grasp a given subject for whatever reason, the school should be able to offer an alternative. When I was in high school, there were some kids that could not succeed at algebra. I recall the alternative was a class they called Business Math which was mostly an expansion of arithmetic applied to the real world i.e. percentages, interest rate calculations, how discounts/markups are calculated, how loans are calculated.

If a curriculum truly needs to be restricted to daily working practice, I think we're then into the realm of trade schools. Not a bad alternative at all for many. And for people like Temple Grandin who have particular situations that pose challenges, I think it's a great thing that alternatives be available so talent is not obstructed by the barriers of convention.

This past week an interesting dust-up got a lot of press. The gist of it is a professor of chemistry at NYU was fired when 82 of his 350 students (dissatisfied with their grades) signed a petition of complaint that his med school class in organic chemistry was unduly hard. Plus they had sundry other complaints about his teaching style. There are good arguments on both sides of the issue.

One is that the 80 year old professor was way past his prime, he was in the wrong role, wasn't showing a true commitment to the success of his students, and demonstrated curt sarcasm and dismissiveness in lecture.
And organic chemistry isn't used by most physicians on a daily basis in the real world anyway.

The other argument is that he is a distinguished professor whose textbook on organic chemistry is now in its 5th edition.
Organic chemistry is a weed out class, designed to be hard because succeeding in medical school is supposed to be rigorous.
And what precedent does it set when 82 out of 350 students, ostensibly dissatisfied with their grades can get a professor fired?
What about the students who were succeeding in the class and didn't see fit to sign the petition of complaint? Does their success suggest the professor couldn't have been all that bad?

Student centered education is a sound concept but taken too far, it can get into participation trophy mentality where by design every student must walk away a winner - definitely not the way the real world works.

Nice well thought out post Edwin. It clearly makes sense that a student forced to take a class they can't possibly pass is counter productive. My friend taught Algebra to some of the top kids in our district when I taught. For many, it was the first time they had a math class they couldn't ace without working hard. Many wanted out. He always talked to the student and their parent to stick it out and work hard before they bail out. Most of them ended up succeeding and doing fine. Every time a kid struggles or doesn't get an A doesn't mean he should get an easy way out. My district also offered an Algebra class that ran longer at a slightly slower pace for some kids who were struggling and many made it through that way.

[Dan Friedrichs]

While an poignant example of the need for accommodations, the conclusion that "higher math isn't needed in most fields so shouldn't be widely taught" is bizarre. She gives the example that, "I teach veterinarians, but I couldn’t get into veterinary school myself, because I couldn’t do the math". Presumably veterinarians need math for dosing or other such tasks. There are many educators in professional programs who couldn't get into those programs - human anatomy, even at medical schools, is often taught by anatomists (non-MDs); chemistry (certainly key to pharmacology) taught by PhD chemists (non-pharmacists); even in fine arts, there is coursework taught by those who likely lack the artistic talent of their students; etc. Inability to do math precludes certain professional careers.

Some replies to this thread from engineers point out that they don't make use of higher math in their careers. I'd argue this violates a basic tenant of engineering, which is to do work based on an understanding of first scientific principles. Certainly I have not hand-calculated a Fourier transform since engineering school, and I am grateful that tools exist to do this work for me with the press of a button - but merely pressing the button without understanding the underlying math is the definition of technician work. Understanding the calculations that take place when the button is pressed (and being able to replicate them myself) enables troubleshooting, deeper understanding, prevents overreliance on opaque tools, etc.

[Bill Dufour]

My high school district requires passing algebra to graduate. No auto shop or metal shop classes are left. I think there are two home ec teachers left in seven schools. Typing class is gone replaced by computer lit or some such. Half the schools still have woodshop.
Bill D

[Maurice Mcmurry]

My high school district requires passing algebra to graduate. No auto shop or metal shop classes are left. I think there are two home ec teachers left in seven schools. Typing class is gone replaced by computer lit or some such. Half the schools still have woodshop.
Bill D

I posted before that my last math class was Fundamentals of Algebra in the 8th grade. It is embarrassing. The the rural school let me graduate.

[Alan Lightstone]

I'm of the belief that learning statistics is far more useful in daily life than calculus.

I'm an MIT grad, perfect Math SAT scores, blah, blah, blah so clearly did my part in taking high-level university math courses, calculus, partial differential equations, etc... And was good at it many moons ago. Plus I had to get A's in them at MIT to get into medical school. And the music majors there I knew had a perfect math SAT scores, so I perfectly understand that this wasn't a representative group.

Have I used any of that math since college? Precisely zero. And I'm a cardiothoracic anesthesiologist, so not exactly a low tech specialty in medicine. I did realize then that knowing statistics could prove useful in evaluating studies, etc... so I also took statistics in college. Great move. Use that knowledge all the time. Especially during COVID-19 looking at medical studies, not newspaper articles. Not going to get into politics, but the difference between medical articles and what is reported in the lay press is staggering.

I really think that Algebra and statistics should be mandatory in high school. Trigonometry and calculus, no.

One man's opinion.

[Bill Dufour]

My uncle was a rocket scientist, chemist, for the Navy then Lockheed. He said he only used above algebra one time in his career. It was to run a round pipe through a round fuel tank at an angle. There is math to cut the hole shape. Any errors the welder filled.
He liked the job so much when the navy said he had to stay away for 6 weeks a year of vaction. He quiot and went to Lockheed. They told him at the interview he could come in any time, any day but only get paid for scheduled work days.
Bill D
Bill D

[Stan Calow]

I first heard of her on NPR years ago. I thought her books on animals were amazing. I'm not sure she's an expert on education. I think I understand her point. I agree more non-college paths should be available. But I dont think we as a society are collectively smart enough to identify and select children before they've been exposed to the hard subjects.

A teacher friend said once that teaching was all about pushing and tricking children to learn things against their will. Unfortunately, our society has made being good at hard things, like algebra and science, a negative (nerds and geeks). How many times have you heard adults say "Im not good at math", bragging, as if taking pride in ignorance. That trickles drown and gives kids (especially girls) permission to fail. In Ms Grandin's case, there's a reason. But that doesn't mean we need to avoid teaching the hard subjects. Or to start sorting the kids who can from those that can't, before they even try.

I too suffered through 15 credit-hours of calculus and differential equations. Most of which I only temporarily understood. But I'm glad I persevered. No, I never used any in my job, but because I had gone through it, I understood that calculus was behind most of the science and technology with which I worked. A high school math teacher told us that "if you don't learn this, you'll be dependent on people who do, and wont even know it."

[Mark Rainey]

A teacher friend said once that teaching was all about pushing and tricking children to learn things against their will.
Interesting, but dark commentary on education today. I can see why we have a teacher shortage. We learn best when we are intrinsically ( self ) motivated. Almost all on this forum are self motivated to build good things. That is what pushes us - ourselves. Extrinsic motivation is when you are forced to learn about something you have no interest it. Sometimes you must learn about things you have no interest in...but the quality of learning pales in comparison. Good teachers facilitate learning. Force feeding of knowledge rarely achieves long term results.

[Ken Fitzgerald]

I was pretty immature during HS but managed to take most of the higher-level science and math courses offered with the exception being calculus. As an adult, I wished I had taken calculus. Professionally, working in electronics in air traffic control maintenance and then in diagnostic imaging (MR, CT and x-ray) the level of math needed made me glad I'd taken algebra and trig. But the HS class I used most was the typing class I took in HS. I took it because I thought I would go to college immediately after HS and would need it for term papers. My lack of maturity, passing the draft physical and car payments for the '64 SS Impala convertible I bought, got in the way of my initially entering college. In the age of digital electronics, the typing class was a great boon due to the amount of time spent on a keyboard to install, calibrate, align and troubleshoot the machines.

On one family of CT scanners, there existed a chassis, used to in the backprojection portion of the reconstruction process, that the electronic circuits actually performed a specific polynomial equation. The diagnostic programs for the chassis which had a dozen or more rather large circuit boards actually ran known data values though the chassis, analyzed the resultant errors to diagnose the problem and provide you a suspected board, with a chip map for that board indicating which of the socketed chips need to be replaced.

[Prashun Patel]

I think teaching should be about showing kids that what they think is hard may just be within their grasp. They are worthy and capable of understanding any concept at some level. And that level should be free from judgment. Knowledge is not a sport where we need a winner.

Education is a right ... and an obligation. Economics, math, literary critique, history, other cultures - we can all find something that we "just don't get". But pushing ourselves - and *encouraging* others to learn everything is the way we hold power to account and have a hand in solving the world's problems. I see these things as civic duties. We may not like it, but we can only make informed decisions and critiques if we educate ourselves.

[Ken Fitzgerald]

I think teaching should be about showing kids that what they think is hard may just be within their grasp. They are worthy and capable of understanding any concept at some level. And that level should be free from judgment. Knowledge is not a sport where we need a winner.

Education is a right ... and an obligation. Economics, math, literary critique, history, other cultures - we can all find something that we "just don't get". But pushing ourselves - and *encouraging* others to learn everything is the way we hold power to account and have a hand in solving the world's problems. I see these things as civic duties. We may not like it, but we can only make informed decisions and critiques if we educate ourselves.

I agree wholeheartedly! Well stated Prashun!

[Edwin Santos]

A teacher friend said once that teaching was all about pushing and tricking children to learn things against their will.


Interesting, but dark commentary on education today. I can see why we have a teacher shortage. We learn best when we are intrinsically ( self ) motivated. Almost all on this forum are self motivated to build good things. That is what pushes us - ourselves. Extrinsic motivation is when you are forced to learn about something you have no interest it. Sometimes you must learn about things you have no interest in...but the quality of learning pales in comparison. Good teachers facilitate learning. Force feeding of knowledge rarely achieves long term results.

I didn't take away a dark interpretation from Stan's comment. It reminded me of the Karate Kid movie where Mr. Miyagi made Ralph Macchio wax on/wax off, paint fences and sand floors in circles but in reality he was really teaching him karate.

Sometimes the whole mountain can seem overwhelming, so "tricking" a student into taking it in pieces, or even indirectly can help them overcome the psychological barriers.

You're 100% right, self motivation is irreplaceable, but every student at some point or another struggles with the confidence that they can do something and the best teachers can help them overcome those hurdles. I shouldn't speak for Stan, so this was just my interpretation, and I admit I don't like the "against their will" part.
Good teachers are a treasure!

[Stan Calow]

Yes Edwin, it wasn't meant to be dark, but rather recognition that kids have to be persuaded to learn things. My two grandkids would rather watch SpongeBob, recess, play video games, t-ball, etc. Reading a book wouldn't happen if they weren't pushed. How many kids would take violin lessons by choice? In later years, young adults decide that school is a waste of time, if their parents didn't push them. So I think algebra, is something only a few students would choose on their own. Raising kids without exposure to those complicated things, makes it unlikely that they will choose to engage in them later. But against their will may sound dark, but what kid has the ability to see the big picture enough to make good choices?

Other teachers I've known say that their goal is to have one or two kids a year that they can help point to a brighter future. And to Ms Grandin's point, yes we need better tools at identifying and supporting all kids to be successful. I am pessimistic that we as a society are willing to pay the cost of that.

[Edward Weber]

I'll say, interesting but misguided.
Learning algebra is learning a method of thinking, problem-solving.
This is the basis for just about every activity I can think of.
Deciding what variables there are, which ones matter and what to do with them. This is how you begin to understand how to think, instead of reaching for your phone and googling. IMO, If you can problem-solve and think your way through issues yourself, it's certainly a benefit in just about everything.
Algebra is far more than if A=2 and B=3, solve for C
I certainly don't solve equations with pencil and paper every day, but I'm sure I use some element of algebra daily.

[Mike Henderson]

I wonder if we're heading to a future where most of the good jobs require a college degree, and mostly a technically oriented degree. I have friends who work in the trades as plumbers and electricians and they did pretty well. But now that they're in their late 50's, they all say the same thing. Their bodies are breaking down from the work. It was easy when they were young but it gets harder every year, and they wonder if they can continue to work until they can afford to retire.

Additionally, while they did okay financially, they don't have a big cushion for retirement.

Most people in my age group who got a technical degree made good money during their working years. Many had company retirement plans and a 401(K). Unless they went crazy spending during their younger years, they have a comfortable retirement.

The only non-degreed people I know who did well were ones that started a business and were successful in growing it. There just aren't many jobs like the old days - working at a manufacturing plant where you had job security and a retirement plan.

Mike

[Mel Fulks]

Well , late 50s is not far from 65, and a lot of them would have some of the same problems from working in an office. But, YES with a
GOOD office job you would be OK. Lot of women have retired early from moving around files in law offices.

[mark mcfarlane]

You're never going to reach all of the students in the US' public school system in any given topic. That said, algebra can help develop some critical thinking skills that are very useful in life. For example, I believe learning to solve 'word problems' helps people ingest other data in life and can help develop a better understanding of the rigorousness of a claim. It can help people move from an initial gut reaction to life events to more critical thinking about what is happening.

Me, I loved math as a youngster. Took a half dozen classes beyond Diffeq. Although I've never had to solve a differential equation by hand in my professional life, the 'systems think / classification' mentality that finding a solution requires helped develop my analytical skills. I did however need most of 'that math' to stay current in my field, scientific journal authors love equations.

[Mark Rainey]

Bill Gates was excused from math classes because he was fascinated with computers.

[Edward Weber]

The only non-degreed people I know who did well were ones that started a business and were successful in growing it. There just aren't many jobs like the old days - working at a manugfacturing plant where you had job security and a retirement plan.

Mike

Made in America is back, leaving US factories scrambling to find workershttps://www.cnn.com/2022/10/09/economy/manufacturing-jobs/index.html
"But even though he’s paying $20 to $30 an hour he can’t find the workers he needs. "

I think that sums up the problem, thinking $30 per hour is a fair wage. Grossing 60k just doesn't cut it anymore these days. It's actually below the median household income by about 10k.

There is still a great need for actual labor in many disciplines, skilled craftsmen/women. All the math and computer science still can't do what a skilled worker can, and we're losing them with no means of replacing, certainly not at what some are willing to pay. There are simply things that there is no app for.
A college degree shouldn't be the determining factor in wages.

[Mike Henderson]

Made in America is back, leaving US factories scrambling to find workers

https://www.cnn.com/2022/10/09/economy/manufacturing-jobs/index.html
"But even though he’s paying $20 to $30 an hour he can’t find the workers he needs. "

I think that sums up the problem, thinking $30 per hour is a fair wage. Grossing 60k just doesn't cut it anymore these days. It's actually below the median household income by about 10k.

There is still a great need for actual labor in many disciplines, skilled craftsmen/women. All the math and computer science still can't do what a skilled worker can, and we're losing them with no means of replacing, certainly not at what some are willing to pay. There are simply things that there is no app for.
A college degree shouldn't be the determining factor in wages.

The determining factor is usually supply and demand. The reason computer engineers get big money is that there's only so many of them (people who can really code).

The reason manufacturing jobs don't pay a whole lot is that there are many people ready to take those jobs.

Additionally, it's a value add issue. The computer person may work on a product, such as a smartphone, where the code that s/he produced is sold by the millions, and there's no incremental computer engineer costs for each one. The cost of the computer people is spread over all the products sold. The more sold, the lower the allocation per unit for the code development.

The manufacturing worker puts a product together that will be sold at the market price. If the labor cost of building that product is too high, the product can't be sold. And the worker is at risk of being replaced by a machine/robot.

While you may not need a college degree to write code, it's a common path to the job. The real determining factor is what value you bring to the market.

Mike

[Steve Demuth]

I'm of the belief that learning statistics is far more useful in daily life than calculus.


Absolutely. Except that to actually understand much about statistics, you need integral calculus.


I really think that Algebra and statistics should be mandatory in high school. Trigonometry and calculus, no.


I mostly agree, although I think the course you want in high school is not statistics per se, but rather something like "quantitative analysis" - tools for understanding what quantities mean, how they are properly compared, and then, what derived quantities about other numerical subjects (that is, basic descriptive statistics) mean, and how to interpret them. It's not important for everyone to understand the derivation and fundamentals of basic statistics, but people should understand what they mean and have some intuition for where they come from. Hardly anyone coming out of college, unless from a mathematics degree, has any sense of what Bayes' rule means when they think about risk, or make a decision. Everyone should have a feel for such things.

[Steve Demuth]

Hardly anyone who studies calculus in high school, or for one or two courses in college, uses it when they leave school. And yet, calculus is foundational to vast tracts of science, technology, and economics. If you want to be able to turn the crank of the scientific understanding machine, you need calculus. If you just need to apply what others have already figured out to a slightly different situation, you probably won't need to be able to do calculus, although you might benefit from having once understood how in more subtle ways. The challenge that I find, when folks suggest maybe we don't need it is that if you're 18 years old and trying to get an education that fits your skills and interests, and you want to go down a technical or scientifically-informed path, how do we know whether you'll end up in one the branches that can handily discard calculus, or whether you're one who wants to go further, and to continue to develop new science or technology in one of those numerous calculus enable fields?

I have never absolutely needed calculus to do my job in the 48 years since I left school, but I have used it over and over again anyway, to deepen my understanding of a subject, or understand something new, or just to save the time of figuring things out without it.

[Prashun Patel]

I definitely think we should all be learning statistics more in school. I think it will go down more easily if we call it "Risk".

As for calculus, Any kind of sum we do is really integration (calculus). If we understand that concept on an atomic level, we can do sums of continuous things like areas under curves and volumes of objects. It changes how you look at the world and formulas. Conversely, derivatives are just an indication of the speed at which you are going in a particular direction. If you know a relationship between two things, the derivative can help you figure out the optimal combination of those two things. There are shortcuts, but if you understand this, you can apply 'sums' and 'rates of change' in lots of different ways.

We still have some problems to solve that - on the surface - don't appear to be math problems. For example, human behavior and group behavior will probably eventually be modeled and understood mathematically. This is only possible if we give ourselves more tools. Just like in wood working, if I learn a new technique like steam bending or using a lathe, it opens up the problems that can be solved and the art that can be created.

I just heard a fascinating episode of Radio Lab where they found out that a disproportionate number of people with perfect pitch in music are native Mandarin speakers. It's a tonal language that tunes speakers' ears from a young age to hear subtleties in inflection. Our minds are amazingly adaptable at a young age. There is some research that algebra and some math is hard to learn before a certain age. But on balance, the young mind can absorb a ton. The fact that it's hard for the majority of kids to learn things is not a reflection in the difficulty of the subject but in the deficiency in our teaching institutions and methods.

The esoteric nature of these concepts IMHO comes from the branding of education and advanced degrees. You have to achieve admission or 8 years of study to be a master, so it by definition is inaccessible to the masses.

BS! These concepts should be made less esoteric and more accessible. Risk, sums, rates of change are highly intuitive concepts. We should be encouraged to approach the math through how the world works.

I think this is largely a problem with one-size-fits-all education. We use the Internet like we teach in schools: content creation and broadcast. But I've learned an entire craft of woodworking not through videos or classes . I've learned from you wonderful people here. I ask a question, and I get an answer. I find mentors. I find partners. I teach others and learn something. The INternet has the power to (ironically) make the world more human. It's a people search engine at it's heart. I found mentors in Australia, California, Oregon, and right here in Princeton. People who met me at my level and allowed me to learn. I wouldn't have had the courage to pick up a tool were it not for this. We need to make school and education the same way. This will only be possible if we do this without monetizing and gaming it. But I do believe there are surgeons and librarians and home makers and grandparents who are out there who would love the opportunity to give and teach and mentor for no more remuneration than a human connection. At least I would.

[roger wiegand]

I'm absolutely in the "more statistics" camp. The time I spent in classes like differential equations would have been much more profitably spent understanding statistics at a much deeper level. I can't say that I ever used calculus over 50 years as a working scientist (though a basic understanding, just like my basic understanding of the differences between Aristotle and Plato, was certainly not wasted); I used statistics every single day, and much more advanced statistics was critical to the work I did in the last decade of my career. It was hard to learn at age 60!

I found algebra to be a whole 'nother thing, and I still don't have a clue of why anyone other than a mathematician would take it. One of my housemates who went on to become one of the world's leading experts in algebraic geometry and K-theory (whatever they are) convinced me I had to take it. Fortunately it only took two classes to figure out that that was not how my brain worked, so I was out well before the drop deadline. He still sends me his papers and I still can't figure out even what he was trying to prove or why, much less follow the proofs.

[Kev Williams]

......"But even though he’s paying $20 to $30 an hour he can’t find the workers he needs. "

I think that sums up the problem, thinking $30 per hour is a fair wage. Grossing 60k just doesn't cut it anymore these days. It's actually below the median household income by about 10k.

Just learned a bit about what 'median' wages mean: It's the $$ point where 50% of earners make less than that, and 50% of earners make more than that. Just my opinion but that's a horrible way to figure 'average' income, considering you're mixing in absurdly wealthy people's incomes with the rest of everyone's just-getting-by incomes. And depending on which website I've hit, 'average' income last year varies from around $44k to $67k- and just the $23k difference isn't exactly chicken feed!

Best income data I found that seems actually based in "reality", is this table from Wallethacks.com, that shows the percentage of earners in this country that are making MORE than the listed $ amount:



1%: $250,000
5%: $140,000
10%: $100,000
20%: $65,000
30%: $45,000
40%: $35,000
50%: $25,000
60%: $20,000
70%: $10,000
80%: $5,000
90%: $0.01 — $4,999


This is pretty telling to me... if true that only 30% of US earners make more than $45k per year, then the $67k 'median' figure that seems the most popular number I've found is just a pipe dream to 70% of people in this country...

Some quick math: $45,000/52weeks/40hours=$21.63 per hour... a bump to $30 per hour gets you $62,400. Since roughly 75% of earners in this country make LESS than that, well... I'm wondering if some of those holding out for $30+ per hour jobs aren't maybe shooting themselves in the foot... ?

And just for giggles, no algebra was used (or abused) in this post ;)

[roger wiegand]

The $67K median is for "household" income, so the combination of 2+ jobs for most households. As your percentile numbers show the individual median for employed people is about $31K.

Median is the better indicator, and is what your percentiles reflect. Averages are going to be much higher, and distort the picture-- when Elon Musk and I are together in the same room we have an average net worth of $130 billion, a far cry from reality.

[Edward Weber]

Best income data I found that seems actually based in "reality", is this table from Wallethacks.com, that shows the percentage of earners in this country that are making MORE than the listed $ amount:



1%: $250,000
5%: $140,000
10%: $100,000
20%: $65,000
30%: $45,000
40%: $35,000
50%: $25,000
60%: $20,000
70%: $10,000
80%: $5,000
90%: $0.01 — $4,999


This is pretty telling to me... if true that only 30% of US earners make more than $45k per year, then the $67k 'median' figure that seems the most popular number I've found is just a pipe dream to 70% of people in this country...

Some quick math: $45,000/52weeks/40hours=$21.63 per hour... a bump to $30 per hour gets you $62,400. Since roughly 75% of earners in this country make LESS than that, well... I'm wondering if some of those holding out for $30+ per hour jobs aren't maybe shooting themselves in the foot... ?

And just for giggles, no algebra was used (or abused) in this post ;)

Shooting themselves in the foot or not, living on (the best case scenario) 60k is not easy in many parts of the country.
If he can't find those willing to take a job at a certain rate, then something is wrong or unknown about the situation.
When I got out of school (40 years ago) and a couple of years later, laborer and factory jobs (strong back weak mind) were getting $10 per hour, adjust for inflation and that would be $27 today.

[Edwin Santos]

I definitely think we should all be learning statistics more in school. I think it will go down more easily if we call it "Risk".

As for calculus, Any kind of sum we do is really integration (calculus). If we understand that concept on an atomic level, we can do sums of continuous things like areas under curves and volumes of objects. It changes how you look at the world and formulas. Conversely, derivatives are just an indication of the speed at which you are going in a particular direction. If you know a relationship between two things, the derivative can help you figure out the optimal combination of those two things. There are shortcuts, but if you understand this, you can apply 'sums' and 'rates of change' in lots of different ways.


I think this is largely a problem with one-size-fits-all education.

I really like some of these ideas.
But if I may, I would not call the class "Risk", I would call it Decision Making. Everyone's life is mapped by the decisions they make. Algebra (and statistics) can be at the root of decision making but it's not all that obvious. I'm talking about how to weigh out pros and cons, asymmetric risk/benefits. If a class applied Algebra or statistical math to real world situations i.e. whether to buy this house or that house, whether to spend your money on this versus that. Even a concept as simple as understanding all the variables and considering contingencies. Biases even.

Many of these ideas are captured in euphemisms like "a bird in the hand is worth two in the bush" or "the juice isn't worth the squeeze". But some understanding of the mathematical underpinning of these concepts is a life skill IMO.

Not that I'm envisioning people whipping out a notepad to assign formulas and variables to the particular problem at hand.
But once you have learned fundamentals of analysis, it gets to the point where you do it in your head intuitively when presented with new situations. Have you ever noticed there are some people who constantly stub their toe with poor decisions, and others that always seem to land on their feet? It's my belief that the latter category are not necessarily smarter, just better at evaluating decisions and therefore making them more rationally.

Of course this is not to say it's realistic to expect a 100% success rate with decision making, but with some basic tools I believe you can hedge the risks and improve your odds of success.
And it is a skill equally important for a welder, nurse, attorney, or engineer or any other field including a stay-at-home mom.

[Lee DeRaud]

When I got out of school (40 years ago) and a couple of years later, laborer and factory jobs (strong back weak mind) were getting $10 per hour, adjust for inflation and that would be $27 today.
Inflation can sneak up on you.
I had a major "How did THAT happen?!?" moment in 2001 when I saw a poster at work listing the recently-adjusted California minimum wage of $6.25/hour.
Thing is, in 1974, with a fresh MS degree (in math, so on-topic), my starting salary there was $250/week, AKA $6.25/hour. Felt like a lot more at the time, after a couple years of living on a TA stipend.

[Kev Williams]

I got out of school exactly 50 years ago- My senior year I worked as a cook for Denny's, worked swing and graveyard while going to school days, and I was paid $1.60 per hour. Lead cook wages, highest this side of the managers, was $2.25 an hour. I got married (first time) in mid '73, and found a "good" job at St. Regis Paper company, stacking bags off a conveyor belt all day every day, getting paid $3.18 per hour. Quite a bump from the $1.60 I WAS making. But there was a reason for the high wage: The Work absolutely sucked. Turnover was incredibly high, the high wage just kept people coming in, and wasn't hard to teach people how to stack paper bags on pallets. So that didn't last, and I went back to work at Denny's, then Sambo's, then some hotel restaurant, all at the same $1.60 per hour. And oh yeah, because it was "food service", even though I routinely worked 60 hour weeks, never any time-and-a-half, as food service employees are exempt (or were, I have no idea about these days). Last restaurant job was cooking for Sizzler for awhile, they paid salary that worked out to about $2 an hour. Next up was working at re-worked mobile home mfr who got a contract to build mobile 4-plex sleeping quarters for workers on the Alaska Pipeline. Way up to $2.25 an hour, and wasn't worth it. 200% employee turnover per month due to the 'crack the whip' pace of trying to stay within deadlines. Quit, and got a job at Budget Rent-a-Car in '75, and back to $1.60 an hour, married with a kid on the way. -- So yes, Lee, $6.25 an hour back then WAS good money, so it SHOULD have felt like it! ;)

These days, depending on the job and the machine used to do it, at times I'm able to generate over $6.25 per MINUTE. Nice when that happens, but it's not often, and always short lived...

And also these days, inflation isn't just sneaking up, it's coming at us full steam ahead in an armored tank. And I must say, I'm a bit appalled at some of the price hikes of 'famous' every-day stuff lately like Little Caesars $5 and $6 pizzas are now $6 and $7-- thats a 20% and 17% price jump, a wee bit more than the 8.5% that's supposed to be 'average' these days. And Dollar Tree is now "Dollar-and-a-Quarter-Tree", that's a full 25% jump! Seems a bit greedy to me...

oh, and on topic: I didn't need or use algebra OR calculus to figure out those percentages ;) (good thing, cuz all I know is basic arithmetic!)

[Lee DeRaud]

...So yes, Lee, $6.25 an hour back then WAS good money, so it SHOULD have felt like it! ;)
Oh, absolutely. I deliberately did not go through the litany of jobs I had before that...way too depressing. Even the TA gig was near-minimum wage, depending on how you counted the hours. (Though relatively easy work, I'll admit. :) )

The inflation in the 1970s was at least as bad as it is now, and lasted a lot longer (so far). I don't recall "shrinkflation" being an official term in those days, but oddities like the coffee machines at work only filling to the 3/4 mark certainly qualified. (They 'fixed' that one rather quickly by using plastic cups with an inner shell to make it less obvious. We called them "WIN cups", after Gerald Ford's "Whip Inflation Now" campaign.)

And don't get me started on Nixon's "Wage and Price Controls" program in '71-'72.

[Lee DeRaud]

oh, and on topic: I didn't need or use algebra OR calculus to figure out those percentages ;) (good thing, cuz all I know is basic arithmetic!)
Possibly not in this specific instance, although there is more algebra baked into "basic arithmetic" than is immediately obvious.

But this whole thread makes me think of how concepts get internalized and used unconsciously. Saying "I don't use algebra" feels a lot like saying "I don't diagram sentences", a concept from grade school English classes (not sure if they still teach it formally) that, at some level, is critical to reading comprehension. The concepts exist and are used, even if one doesn't consciously think, "Hey, I remember learning how to do this in XYZ class!".

[Tom M King]

Remembering those days, in the early '70's you could buy a new car for $3,000. I was working at an airport, and could rent a plane with fuel, since I worked at the Cessna dealer that had a flight school, for half price at $7.50 an hour. I don't even remember what I was making, but think take home was about 200 a week. I flew most days after work. Pretty soon, I tired of being an employee though, and haven't been one ever since. I remember gas went to a dollar a gallon, and everyone was freaking out.

[Patty Hann]

Possibly not in this specific instance, although there is more algebra baked into "basic arithmetic" than is immediately obvious.

But this whole thread makes me think of how concepts get internalized and used unconsciously. Saying "I don't use algebra" feels a lot like saying "I don't diagram sentences", a concept from grade school English classes (not sure if they still teach it formally) that, at some level, is critical to reading comprehension. The concepts exist and are used, even if one doesn't consciously think, "Hey, I remember learning how to do this in XYZ class!".

Diagramming sentences (5th grade, Sr. Mary Alicia, thank you very much) is not just critical to sentence comprehension.
It is necessary for writing well, too.
It also was invaluable when I took Latin at university. First semester was nothing but grammar.
I couldn't "get it" until it suddenly struck me (one of those light bulb moments) that if I applied the diagramming structure to Latin grammar everything fell into place.
It was an easy 4 years after that. :)

[John K Jordan]

... Algebra can be important in today's world. For many it is not important. ...

Years ago I often gave fun talks and demonstrations at various grade and middle school classes that included things like the simple aspects of electricity, sound, astronomy, and even scuba diving. I always tried to slip in a little pep talk about how important math (algebra, trig, matrices, and more) has been in my work, both in years of software development and later when I switched to sci/tech computer modeling and animation. Most math including calculus was fun, statistics not so much fun.

Now, even long retired, the desk at this computer has plenty of hand-scratched calculations on pieces of paper in the "pile" (yes, I'm a professionally certified card-carrying slob).

[Steve Demuth]

Diagramming sentences (5th grade, Sr. Mary Alicia, thank you very much) is not just critical to sentence comprehension.
It is necessary for writing well, too.

I have never diagrammed a sentence in my life, and what I know about grammar wouldn't fill a page. I can, nevertheless, read English rapidly and for comprehension, and compose the language quite competently as well. None of this is a brag - there are millions just like me in this regard, because, in reality, people learn language, including reading, just fine without formal understanding of grammar. I need a formal understanding of grammar to speak, write, and understand language about as much as I need Shannon's Theorem to operate my cell phone.

[Roger Feeley]

To be fair, Ron, she doesn’t object to algebra being taught. She objects to it being required for certain degrees and certifications. I tend to agree with her to a point. Like I wrote in my original post, I think of things like this as calisthenics for the mind. I made a good living as a computer programmer for thirty years and I don’t think I ever needed any sort of higher math. Unlike many here, I hit a hard wall when I took trigonometry and my math education stopped there. If I had needed trig or calculus as a programmer, I would have known it.

I got into programming through the back door. I hated teaching so I took some night classes at the local jr. College. Then I lucked into a job that couldn’t have been a better situation. My luck held for three decades. My point here is that I don’t have a CS degree and I did well. I did see more than a few people with CS degrees who were hopelessly incompetent. Do I reject the requirement of higher math for a degree in Computer Science? No. But I think the requirement that most programmers need the degree is unnecessary.

[Lee DeRaud]

I did see more than a few people with CS degrees who were hopelessly incompetent.
Depends a LOT on how CS is defined and taught at any given school. I know of a local CS department full of PhDs who (1) consider programming a "trade" and (2) refuse to acknowledge that "Computer Science" (as they teach it) and "Software Engineering" are radically different disciplines. The students we hired out of that department tended to suck at programming, mostly because they didn't really like it.

Do I reject the requirement of higher math for a degree in Computer Science? No. But I think the requirement that most programmers need the degree is unnecessary.
My take on the whole thing was that it's a lot easier to teach a decent programmer the math/physics they needed to do their job than it is to teach a physics/EE grad how to code well. OTOH, it really helps if the programmer has at least been exposed to the underlying math concepts, even if the details need explaining to them.

(The other issue was the high percentage of physics/EE PhDs I worked with who thought they knew how to code because they managed to get a 100-line Fortran program to compile without errors. Then they couldn't understand why it took the software people months to wrap that chunk of code in the infrastructure required to perform efficiently in real-time on specialized military hardware that was older than they were.)

[Steve Demuth]

To be fair, Ron, she doesn’t object to algebra being taught. She objects to it being required for certain degrees and certifications. I tend to agree with her to a point. Like I wrote in my original post, I think of things like this as calisthenics for the mind. I made a good living as a computer programmer for thirty years and I don’t think I ever needed any sort of higher math. Unlike many here, I hit a hard wall when I took trigonometry and my math education stopped there. If I had needed trig or calculus as a programmer, I would have known it.

I got into programming through the back door. I hated teaching so I took some night classes at the local jr. College. Then I lucked into a job that couldn’t have been a better situation. My luck held for three decades. My point here is that I don’t have a CS degree and I did well. I did see more than a few people with CS degrees who were hopelessly incompetent. Do I reject the requirement of higher math for a degree in Computer Science? No. But I think the requirement that most programmers need the degree is unnecessary.

There is a very large fraction of the software that runs the world that can be written quite competently with minimal Computer Science background. Most application developers never use the underlying informations theory, complexity theory, compiler theory, mathematics, etc that form the pillars of a computer science curriculum, just as most Civil Engineers will never use any calculus in their day to day work. But alongside that, there are plenty of places where you do need much of that syllabus under your belt, and it shows when you try to use ungrounded programmers to build things where said grounding matters - you get stuff like the "R" programming language - basically created by and for statisticians who understood essentially nothing about language and compiler theory, and which as a result is riddled with flakey semantics, crappy and unpredictable performance, and more than one outright absurdity.

[Dan Friedrichs]

A number of replies suggest teaching statistics or "risk". Steven Levitt (co-author of Freakonomics) has been advocating for this - calling it "data science": https://www.edweek.org/teaching-learning/opinion-data-science-is-the-future-lets-start-teaching-it/2022/01

[Dan Friedrichs]

There is still a great need for actual labor in many disciplines, skilled craftsmen/women. All the math and computer science still can't do what a skilled worker can, and we're losing them with no means of replacing, certainly not at what some are willing to pay. There are simply things that there is no app for.
A college degree shouldn't be the determining factor in wages.

Still only ~35% of young adults earn a 4-year degree - while an all-time high, it is still far from a majority of workers: https://en.wikipedia.org/wiki/Educational_attainment_in_the_United_States

Also consider how innovation has reduced demand for labor. Consider how laborious residential plumbing was even 20 years ago (with cutting copper, sweating fittings, etc) compared to today with pulling PEX and crimping fittings.

[Mike Henderson]

From my observations of jobs and young people entering the market, there appears to be a shrinking number of jobs available for young people without a degree that provide a decent living.

And even for young people with a degree it depends on what the degree is. For college students, there should be more guidance as to what the employment opportunities are for the field they choose. It's nice to say "Follow your dream" but that path may lead to starvation.

Many of the fields with the best renumeration are technical fields (a BS and not a BA).

Mike

[Too many young people want to go into a "glamour" job (example, movie making) but there are too many people in the field and it's tough to break in.]

[David Sochar]

As one that has spent over 50 years in professional woodshops, I eventually learned that I was a problem solver. Often, the work (radius staircases, round and elliptical work, and compound radius work) appeared to be derive from calculus or at least algebra. However, the math was simple, arithmetic, and rigid. As such, solutions were easily at hand, once I realized I merely had to identify the problem.

When in High School (1964 to 1968), I told my math teachers that I did not need to know anything more than minimal math, because these little boxes would be available for us to do basic math upon. Further, I should also have access to the computing machines as they became prolific across the landscape. Popular Mechanics and the like forecast the arrival of such devices. I went so far - too far, but it was high school - as to say that there would be little need for classes in mathematics, and algebra.

[Edward Weber]

Also consider how innovation has reduced demand for labor. Consider how laborious residential plumbing was even 20 years ago (with cutting copper, sweating fittings, etc) compared to today with pulling PEX and crimping fittings.

I don't disagree, but if you have one of the millions of houses with soldered copper plumbing that need repair or maintenance, you need skilled labor to do it.

There is now and always will be a need for skilled craftsmen, no matter how much innovation reduces unskilled labor. The more demand there is for those with specialized skills, the higher the price for the service.

Of course, some may just rip it out, replace with lego piping and sell the copper.

[Mike Henderson]

I don't disagree, but if you have one of the millions of houses with soldered copper plumbing that need repair or maintenance, you need skilled labor to do it.

There is now and always will be a need for skilled craftsmen, no matter how much innovation reduces unskilled labor. The more demand there is for those with specialized skills, the higher the price for the service.

Of course, some may just rip it out, replace with lego piping and sell the copper.

It doesn't take a lot of skill to sweat copper pipe - a lot of average homeowners learn to do it fairly easily. To really have valuable specialized skills, they have to be the type of skills that take a lot of time and effort to learn. Otherwise, others move in quickly and undercut the wage of the "skilled" person.

I don't know what those skills might be, but fixing copper pipe is not one of them.

Mike

[Certain types of welding and the certifications that go with that might qualify as a specialized skill.]

[Perry Hilbert Jr]

Yeah... NO! I was an artsy-engineer type in high school. I was part of an experimental group that studied algebra in 4th grade, geometry in 5th grade and an in depth repeat of them in 6th grade. I started to the see the world in shapes and angles, repetitious patterns, etc. To a large extent I still do. Some people do cross word puzzles, I would pick a large number and calculate the square root. I built sheds, dough boxes with angled sides, etc without a ruler or tape measure. I could calculate the volume in cubic feet or gallons of a large cylinder tank, or cone shaped feed bin. I used those formulas nearly every day on the farm, at the office, even in the kitchen. We have a 13 x 9 baking pan, and a scratch recipe for cake that makes 2 8 inch round layers. How much do you increase the portions? Had to calculate dosing of medication for cattle of various ages and sizes. Mrs. is a nurse and has to calculate medicinal dosing for various patients. (most done now on computers, but not all.)

I have a recipe for a certain soup that makes 3 quarts. What quantities of ingredients to make a 40 gallon kettle for an outdoor fall party? This is all elementary algebra.

When I drive down the road, I see trees by their shapes. Some are giant cones, some ovals, some egg shaped, spruce trees have hanging arched branches with vertically hanging segments, like fringes on a buckskin jacket. all spaced evenly repeated patterns in nature.

[roger wiegand]

Hmm-- I must be missing something, when I went to college the only major for which algebra was a requirement was mathematics. Many other science majors required classes like differential equations (completely useless, I have to say, to me as a biologist!), but algebra was sufficiently abstruse that only the math majors had to take it. The folks who do it for a living (I know several) seem to be kind of proud of its lack of real-world application.

[Steve H Graham]

Given the shocking stupidity of the average American student of 2022, and given the clearly superior performance of past generations, I dismiss anyone who thinks it's smart to be against the old-fashioned methods of teaching by memorization and repetition. That point of view is too ridiculous to give time to.

It is remarkable that we live in an age where people are becoming more stupid and uninformed. Given the resources we have now for spreading information and teaching, there is no excuse for it.

As for unnecessary subjects for higher education, I would put things like English (apart from teaching grammar and writing), history, art history, anthropology, and sociology on the list, and I would keep music, languages, and STEM subjects. With the obscene tuition pointlessly greedy colleges charge these days to build their Musk-like endowments, it is cruel to make a student pay several thousand dollars to learn something he will never need in his career and which he can pick up in a few weeks on his own simply by reading books.

It's hard to learn music, languages, and STEM subjects outside of a school, and these subjects have value in the real world, so they should have priority over what people like me call "basket-weaving" subjects. And kids should have to learn algebra and geometry before college because most high school seniors still don't know what they want to do. They should be prepared.

I got a raw deal when I was a student. I was gifted in languages and writing, and educators with zero common sense encouraged me. Carl Hovde, the head of the English department at Columbia University wrote me an unsolicited letter inviting me to apply. What he didn't tell me was that studying literature and writing was a one-way ticket to becoming a fungible drone in a cubicle farm, or a cab driver.

The world of writing is a club, and talent isn't what gets you in. You need to share the morals and agenda of the people who publish. If you don't, the best you can hope for is to get one of the few niche positions available to tokens. No one told me this when I was young.

I was also pretty good at STEM stuff, and it's what they should have encouraged me to do. A 95th-percentile STEM talent is a gateway to productive work and financial security. A 99.9th-percentile verbal talent is of nearly no value. No one is going to pay you to do crossword puzzles or win spelling bees.

I dropped out of college, but I went back as an adult. By a weird series of events, I ended up studying physics and getting into a good graduate school. I almost didn't make it to my undergrad diploma, though. I had gotten bad grades in math in high school, and instead of trying to teach me good study habits and encouraging me, one of the bigwigs at my prep school suggested I just stop taking math courses. Because I was a big verbal brain.

As an undergrad in physics, I had to take calculus, which meant I had to learn algebra and calculus at the same time. Because I had been encouraged to give up on STEM classes, I didn't have the foundation to understand calculus. Boy, did those creative writing courses help me. Not.

In the end, I got burned out on physics and took the easy way out, becoming a lawyer. Some people are proud to be lawyers. To me, it was a badge of failure, but I had to do something, and law was easy to get into.

I wish I had been brought up in the drill-and-memorize days. I would have had a better STEM foundation and better study habits. I would never have dreamed of wasting my time or my dad's money with liberal arts courses. I don't think it was smart to try to become a physicist, because they tend to be miserable people with horrible social lives, and they generally end up in academic environments where tolerance and freedom of thought are punished. But I would have enjoyed working as an EE or ME.

I suppose it was not a good idea to take 7 or 8 semesters of calculus and tons of pure physics, but on the other hand, it's nice to be able to understand things other people can't get a grip on. A lot of people fall apart when they have to understand things like sines and cosines. Those things are like basic reflexes to me. I have forgotten an awful lot. I don't understand many of my old homework papers. But what I remember is more than most people ever learn.

I think it's a big mistake to throw out what works for most kids in order to cater to the oddballs. We shouldn't sacrifice the welfare of the majority in order to make people with learning problems (and their defensive, emotional parents) feel better about themselves.

Math and science are hard, and they turn kids into better students and better people. My physics homework was so hard I spent about 4 times as much time on it as I spent on math (things like differential equations and complex analysis), and math was much, much harder than tee-ball liberal arts courses. After physics, every other subject seemed like a joke, because it was, and the same principle would have applied had I studied math but not physics.

When I was in law school, I did not study much at all. The first semester, I learned I could live in the library, study all semester, and get an A, or I could study for three days at the end and get a B. I was satisfied with cum laude. You could never do that in math or physics. You have to be responsible. You can't do it in music or languages, either.

I remember being forced to take anthropology as an undergrad. The textbook was like a big coloring book. It was written on a level for children. The professor told us academia's simplistic, absurd theories about humanity, I confirmed what he said on multiple-choice tests, and I got an A-. I did virtually nothing in that course. I also had to take philosophy, and another physics major sat next to me. The poor professor made the mistake of starting to talk about electrons and so forth. We kept raising our hands. "That's actually wrong." "That's actually wrong." It amazed me that I could get 4 credits for that and only 4 credits for things like optics, which actually required intelligence.

Math and physics taught me to think. The liberal arts junk hasn't helped me much at all in life. Whoopee, I know what Immanuel Kant thought. Hooray for me. I read Villon in French. Great. Let's discuss the innumerable times that has gotten me out of a jam. And employers always want to know your take on Villon. "Forget programming languages. Where ARE the snows of yesteryear?"

All of the liberal arts stuff, I could have picked up on my own for almost nothing. I have read more non-STEM books since leaving school than I ever did as a student. Paying to learn about things like history and literature seems ridiculous to me. Some schools are now charging parents (or taxpayers, through loan forgiveness) $2000 per credit to teach kids yoga or the history of porn.

Every year, we think we know more about education than the year before, and every crop of students is more stupid than the last. These things are unquestionably true, but somehow we aren't getting the message.

[Steve H Graham]

Hmm-- I must be missing something, when I went to college the only major for which algebra was a requirement was mathematics. Many other science majors required classes like differential equations (completely useless, I have to say, to me as a biologist!)

That amazes me. One of the first differential equations I was shown was used to predict the growth of a culture of bacteria. I'll bet if I start Googling, I'll find all sorts of uses for differential equations in biology. How can you predict how an epidemic will progress without them? I would think they would be very useful in predicting changes in animal populations. I was able to predict the spread of covid way better than the government estimates for several weeks, and I'm no expert on differential equations.

[Lee DeRaud]

Hmm-- I must be missing something, when I went to college the only major for which algebra was a requirement was mathematics. Many other science majors required classes like differential equations (completely useless, I have to say, to me as a biologist!), but algebra was sufficiently abstruse that only the math majors had to take it. The folks who do it for a living (I know several) seem to be kind of proud of its lack of real-world application.
University-level advanced algebra (e.g. linear algebra, group/ring theory, and the like) is a much different animal than what is normally taught at the middle-school or high-school level: much more of an abstract thing than what is usually used in engineering and physical sciences. It is decidedly NOT something that would be a requirement for HS graduation unless you're running a whole school full of future quantum physicists.
An example: https://www.math.mcgill.ca/darmon/courses/17-18/algebra2/knapp-advanced.pdf
(And quite frankly, much of what that text considers "basic algebra" is still well above anything I'd expect in a normal HS level math course.)

[Mike Henderson]

Hmm-- I must be missing something, when I went to college the only major for which algebra was a requirement was mathematics. Many other science majors required classes like differential equations (completely useless, I have to say, to me as a biologist!), but algebra was sufficiently abstruse that only the math majors had to take it. The folks who do it for a living (I know several) seem to be kind of proud of its lack of real-world application.

Maybe we're talking about different levels of algebra. I view algebra as a high school subject and precursor to classes like calculus. I know there are advanced algebra courses in college but I didn't think we were discussing those. And, yes, only mathematics majors would take those classes.

Mike

[I didn't see your post, Lee, before I did mine. Here's a link to the Basic Algebra book - https://www.math.mcgill.ca/darmon/courses/17-18/algebra2/knapp-basic.pdf
Waaay above high school Algebra.]

[Lee DeRaud]

Maybe we're talking about different levels of algebra. I view algebra as a high school subject and precursor to classes like calculus. I know there are advanced algebra courses in college but I didn't think we were discussing those. And, yes, only mathematics majors would take those classes.

Mike

[I didn't see your post, Lee, before I did mine. Here's a link to the Basic Algebra book - https://www.math.mcgill.ca/darmon/courses/17-18/algebra2/knapp-basic.pdf
Waaay above high school Algebra.]
Yeah, it's only "basic" in comparison to the "advanced". Maybe a better term for what you and I had as middle/high-school algebra would be "elementary". Those look more like courses I had as a sophomore/junior math major.

That said, my knowledge of what constitutes "normal" HS math is over half a century old. But the times (still 20+ years ago) I helped a friend with his child's HS math homework, the thing that struck me wasn't that it was more advanced than what I'd had, so much as that it was less sequential. They seemed to be trying to teach algebra, geometry, trig, and calculus all at the same time, getting to concepts early that they really hadn't laid the foundations for.

[Mike Henderson]

What he didn't tell me was that studying literature and writing was a one-way ticket to becoming a fungible drone in a cubicle farm, or a cab driver.



That's been my observation, also. The joke in college about history majors was that you could teach or sell insurance. Certain other majors had the same problem.

My estate will go to UCLA to help talented young people who could not afford UCLA to get an education. But I've specified that the scholarships will only be available to students who major in certain fields, ones where I think they will be able to get a job - Accounting, engineering, finance, computer science, and a few others.

Mike

[Anuj Prateek]

I studied physics (bachelors) and software (masters) and am software engineer by trade. During bachelors spent time on biology, electrical systems, technical writing, computers and obviously maths (mostly calculous).

I don't use 95% things I studied directly for day to day job. This doesn't mean these courses were useless. Time spent doing those courses shaped how I think or approach a problem. 5% of cases when problem is new, I have various tools available and I use them, after re-reading/refreshing course materials.

Literature (not language) and history are the only two areas that I think added no value.

[Maurice Mcmurry]

My mathematics knowledge has advanced just by following this thread. There is an interesting article in the N.Y.T. It may be behind a pay wall...

Math Scores Fell in Nearly Every State, and Reading Dipped on National Exam - The New York Times (https://www.nytimes.com/2022/10/24/us/math-reading-scores-pandemic.html)

[Stan Calow]

Literature (not language) and history are the only two areas that I think added no value.

Not to get too philosophical about it, but I think the skill that separates the best from the rest in any profession, is imagination. For example, the creativity in woodworking shown by people like Krenov, Maloof, Nakashima, etc., comes from a strong imagination of what could be. And literature, art, history, and even philosophy, can spur imagination, and give you a well-rounded life. Yeah, not necessary to be successful and certainly a luxury not available to anyone, but not a waste of time.

I think the focus on algebra and advanced maths as being irrelevant to most jobs unfortunately gives an excuse to not even try. College profs and even high school teachers i've talked to all say they are mostly doing remedial teaching of these subjects to students interested in technical careers who blew it off early in life as too hard and a waste of time.

[Steve Demuth]

Hmm-- I must be missing something, when I went to college the only major for which algebra was a requirement was mathematics. Many other science majors required classes like differential equations (completely useless, I have to say, to me as a biologist!), but algebra was sufficiently abstruse that only the math majors had to take it. The folks who do it for a living (I know several) seem to be kind of proud of its lack of real-world application.

There are certainly parts of Biology (anything to do with population dynamics, e.g.) that involve differential equations, no?

[Steve Demuth]

University-level advanced algebra (e.g. linear algebra, group/ring theory, and the like) is a much different animal than what is normally taught at the middle-school or high-school level: much more of an abstract thing than what is usually used in engineering and physical sciences. It is decidedly NOT something that would be a requirement for HS graduation unless you're running a whole school full of future quantum physicists.
An example: https://www.math.mcgill.ca/darmon/courses/17-18/algebra2/knapp-advanced.pdf
(And quite frankly, much of what that text considers "basic algebra" is still well above anything I'd expect in a normal HS level math course.)

I agree with this with regard to what was called Abstract Algebra when I was in school, but Linear Algebra is one of the most valuable analytical tools a person can have in their toolbox. I think a basic course in Linear Algebra is co-equal in value to Calculus.

On the other hand, for sheer mathematical fun, Abstract Algebra is hard to beat.

[Lee DeRaud]

I agree with this with regard to what was called Abstract Algebra when I was in school, but Linear Algebra is one of the most valuable analytical tools a person can have in their toolbox. I think a basic course in Linear Algebra is co-equal in value to Calculus.

On the other hand, for sheer mathematical fun, Abstract Algebra is hard to beat.
No argument there, it's just that, fifty-plus years out, my memory of which stuff was in which course is a bit fuzzy.
(Actually, there are a LOT of things from my college days that are a bit fuzzy... :) )

[Lee DeRaud]

The joke in college about history majors was that you could teach or sell insurance.
Oddly enough, during my indentured servitude as a math TA, my best single student, hands-down, was a history major, in my first-year calculus section for no apparent reason.

Not a typical freshman though: he was a 23YO Vietnam vet attending on the GI Bill, who treated college from the POV of a customer rather than a supplicant. Not sure where he went after that, but my suspicion was that he was headed for a career in law or politics, and I have no doubt he was successful at whatever it was.

[Michael Bulatowicz]

. . . I don't think it was smart to try to become a physicist, because they tend to be miserable people with horrible social lives, and they generally end up in academic environments. . .

I've been staying out of this thread so far, but you've inspired me to jump in. What academic institution did you attend for your physics education? Whatever institution you attended seems to have had a rather toxic environment in addition to misleading you on typical career paths.

. . . they tend to be miserable people with horrible social lives. . . This has not been my experience at all nor that of my friends and acquaintances (many of whom are, in fact, physicists with advanced degrees). I have known, interacted with, and worked with a large number of physicists in my career. This, in addition to going back to school myself for an advanced degree in physics after over a decade in industry in a nominally mechanical engineering role that already included a significant quantity of electrical engineering and physics. I can't speak to the general state of theoretical physicists, but the experimentalists (my particular circle) are happy in their careers and quite social--particularly with each other and with people in related disciplines such as engineering, chemistry, and so on. Maybe it's a different story for theoreticians in general, but those with whom I've interacted don't seem the least bit miserable or anti-social.

. . . they generally end up in academic environments. . . Verifiably incorrect; the American Physical Society (APS) and American Institute of Physics (AIP) regularly publish statistics demonstrating that less than half of people with advanced degrees in physics end up in academia (though as I recall a majority who are presently pursuing advanced degrees do indeed desire to pursue careers in academia). Even among physics PhDs (those with the greatest potential for academic employment), only something around 30% end up in academia (https://www.aip.org/statistics/whos-hiring-physics-phds).

Now, a bit more on-topic for the thread. I'll freely admit that I'm an outlier in a lot of ways, and with my career in research and development I have more opportunity than most to put advanced mathematics to use. I use algebra (and calculus, differential equations, etc.) on a regular basis in my career--often every day for weeks on end when working on something entirely new such as putting together a concept for a new development proposal. When running an analysis via computer, for example, I prefer to first generate a rough calculation by hand to help verify the results and make sure I understand the theoretical underpinnings of the calculations and understand the factors that go into the overall behavior of whatever-it-happens-to-be so that I can adjust/optimize as needed. If the hand calculations and computer simulation disagree qualitatively with each other or exhibit a significant quantitative disagreement, I've made a mistake in one or both. Algebra--in particular its application in rearranging/simplifying equations has on a number of occasions provided valuable insight.

I do, however, agree that the liberal arts "general education" requirements for my undergrad degree were almost entirely simply a means for the academic institution to extract more money and to help justify the associated departments and their professors' employment rather than providing any value or any relevance to my degree, let alone my career. The general education classes, for the most part, required almost no real work or challenge but various levels of time commitment and, of course, the required money for tuition, fees, and books. Two full-time semesters (equivalent) of just liberal arts general education classes for a mechanical engineering degree? The ME professors seemed unanimous in agreeing with the students that this was excessive. Some of these classes were simply repeats of what I had learned in high school, and with one exception I can't pinpoint anything I learned in any of them that was in any way useful in my career or my life. The specific exception was a course in English composition that emphasized (and practiced) writing based on a combination of one's own work and research of others' work, with emphasis on structure, organization, and citation. Overall, my classmates and I would have been far better served with classes that were even mildly challenging--particularly if they were challenging classes even mildly associated with the degrees we were pursuing. To be able to get an "A" in a course wherein one never studies and only shows up for the first day and then the scheduled quizzes and tests, while receiving 3 credits for the "effort" is absurd.

A general statement on-topic with regards to Temple Grandin's views on algebra: forcing one's brain to accommodate additional modes of thought and additional approaches to various problems is never wasted effort, even if--especially if--it's a struggle. Research on neuroplasticity confirms this. A caveat is that it must be a struggle with adequate support; what constitutes a struggle with adequate support is going to be different for every individual, and inevitably some will surge ahead on any given topic while others fall behind. While I'm not by any means an expert on the topic, lowering academic standards sounds like a sure way to reduce general levels of achievement and a wonderful way to ensure that even more of the would-be high achievers among us lose their love of learning and underachieve relative to their own potential. The consensus among professors teaching graduate-level physics "core" courses--and I fully agree--is that if they don't leave a significant majority of their students struggling they aren't going fast enough and are doing their students a disservice.

Regarding . . .every crop of students is more stupid than the last. . . my experience suggests there's instead a significant dichotomy among the current crop. Those who are self-motivated and intelligent tend to be extremely high achievers--even more so than what I saw 10-15 years ago (though I may have simply not been in an appropriate position to observe). I have seen again and again that these students learn what they need to learn in order to excel in spite of the education system rather than because of it. I've had the privilege of working with a number of voracious learners still in undergrad or straight out of college--both with and without advanced degrees--who start out inexperienced and yet rapidly build their capabilities to become highly productive and largely independent. I've also worked--briefly--with plenty who very much do not meet that description and are instead happy to twiddle their thumbs unless specifically given assignments for which they will complete the bare minimum required. The latter tend to move on from the research and development end of my field and I typically don't hear from them again. Even among PhD students in physics, I've seen those who are simply there to get a degree and do as little as possible to meet the requirements thereof--all of whom are very different from those who are there to learn with the degree being an inevitable consequence. There simply seems to be fewer people in the middle ground than I had previously observed, though that may be my own bias rather than a reality.

[Steve Demuth]

Regarding . . .every crop of students is more stupid than the last. . . my experience suggests there's instead a significant dichotomy among the current crop. Those who are self-motivated and intelligent tend to be extremely high achievers--even more so than what I saw 10-15 years ago (though I may have simply not been in an appropriate position to observe). I have seen again and again that these students learn what they need to learn in order to excel in spite of the education system rather than because of it.


I'm glad you weighed in on this. I was thinking about how to respond.

I see no reason to believe that the students coming out of schools today are dumber than 20 years ago, or 50 years ago when I graduated. Before I retired a couple of years ago, I had the privilege of working with both graduate students, and many, many recent graduates in companies in tech and biotech. They were, in fact, differently skilled and motivated than the cohort I went to work with, but they were in no way less capable. I worked with one team that put a glucose sensor, a computer, a bluetooth transmitter, and a small computer into the margin of contact lenses to create a continuous glucose monitoring system that required no blood access and was patient replaceable. The only reason it isn't a widely used product for diabetics is that it turns out that glucose concentration in tears doesn't track blood glucose closely enough to be a reliable insulin dose regulator. The same team had another set of contact lenses that coupled the computer, bluetooth and battery to piezoelectric element in the lense that made the contacts magnification-adjustable lens - need a loupe - just whip out your cell phone and dial up the magnification. I expect we'll see these in surgeons toolkit in the not too distant future.

Along the way, these kids (or so they seemed to me), who were mostly baccalaureates, not doctorates, solved a myriad of electrical, chemical and biological challenges.

They are just one of many stories I could tell to illustrate the point. The kids are not dumber.

[Michael Bulatowicz]

They are just one of many stories I could tell to illustrate the point. The kids are not dumber.

I agree--I, too, have many stories. In one example, an undergrad whom I mentored managed, in the midst of completing a double major in physics and aerospace engineering over a total of four years, to learn FPGA programming, mechanical/electrical engineering principles, and the principles of atomic/optical physics well enough to write a closed-loop feedback and data collection system for a magnetic field detector based on spin-polarized atoms and to design new parts for, run, and at least partially tune up the experiment himself with minimal supervision. In the midst of all this, he found ways to innovate on the method and mechanism of feedback--all over the course of 2 1/2 years that I worked with him. If that's stupid, I would really, really like to work with a smart kid--what an amazing experience that would be!

In my experience, people are people--as they have been and will continue to be: past, present, and future generations included. Capability, capacity, potential, intelligence, imagination, etc. know no bounds of age, gender, ethnicity, language, etc. I will refrain from stating my opinion of those who assume that just because someone is (fill in the blank) they are by definition less capable/intelligent. I will, however, say that I wonder how many great minds we (humanity) have missed by turning a blind eye or worse, deliberately working to suppress their potential.

It is always a mistake to conflate naivety, inexperience, or lack of exposure (or opportunity) with stupidity.

[roger wiegand]

[QUOTE=Mike Henderson;3219596]Maybe we're talking about different levels of algebra. I view algebra as a high school subject and precursor to classes like calculus. I know there are advanced algebra courses in college but I didn't think we were discussing those. And, yes, only mathematics majors would take those classes.

Mike /quote

Ah-- the course I took back in high school, and that my kids took more recently, that was called "algebra" was just gussied up arithmetic. Extremely useful stuff to know how to do, but not requiring learning to think outside the realm of everyday reality.

[John K Jordan]

Ah-- the course I took back in high school, and that my kids took more recently, that was called "algebra" was just gussied up arithmetic. Extremely useful stuff to know how to do, but not requiring learning to think outside the realm of everyday reality.

The math I took in high school (in the 60's) seemed fairly intensive. For the advanced college prep path the algebra course was in two parts, Algebra I and Algebra II. This was required before studying trig and calculus. All these counted towards the prerequisites when I got to college, however I did repeat the calculus course then since my understanding in high school was a bit hazy around the edges - I wanted a solid understanding before I went on to other courses. I had fun in all of them except for statistics - ack. I especially enjoyed learning linear algebra (mostly manipulating matrices) which was useful later when I developed 3D display software and my own mini flight simulator. Of course, a half-century later much of this is fading away... I keep some of my textbooks like math and physics handy for occasional reference.

Maybe algebra in HS today is different, dumbed down, I don't know.

[Lee DeRaud]

The math I took in high school (in the 60's) seemed fairly intensive. For the advanced college prep path the algebra course was in two parts, Algebra I and Algebra II. This was required before studying trig and calculus.
It was a bit different where I went (also 1960s), with three different sequences.
The normal sequence was algebra, geometry, and algebra 2 split into two parts. As I recall, only algebra and geometry were actually required for graduation.
The advanced/college-prep sequence was algebra, geometry, algebra 2, and trig.
Then there was a 5-year "nerd" sequence which started with algebra in 8th grade and added calculus in 12th grade. We also ended up being guinea pigs for the prototype SMSG curriculum, oh lucky us.
Out of ~700 grads, only one class of about 30 ended up taking calculus, maybe another 150 in the advanced sequence, and the rest with some flavor of the normal sequence.

[Bill Dufour]

Of course math scores are down the last two years. With Covid forced online only teaching not much got learned.
I bet 50 years ago a much smaller percentage went on to college so I would expect todays larger college bound percentage to not be as smart. I would guess that those who stayed on to graduate from high school in the 1930's were either well off or smart. I do not know numbers but I bet college enrollment dropped a lot in 1931.
Bill D

[Steve Demuth]

That's exactly how it worked in the school (out on the prairie, just on the Minnesota side of the South Dakota border) I attended, except there was no "nerd sequence" and no calculus class. I loved math, and tried to get the school to let me skip Algebra I (in which I was more than adequately self taught) and start with Geometry in ninth grade, but the imagination for such an approach was not to be found. I then dropped out (technically, I was expelled, but I was leaving anyway) at the end of my junior year, and so never even got Trigonometry in High School. I launched into College Calculus and Physics with just my own self-taught trig, and had to burn some serious midnight oil on remedial trig when we got to certain advanced integration techniques and had to use polar coordinates in various Physics courses.

One thing I've realized over the years is that the actual mathematical knowledge you need from High School Algebra, Geometry and Trig for them to be useful isn't that large (that doesn't mean it isn't hard to learn - for those for whom symbolic reasoning doesn't click, it can be quite a heavy lift, and some just never get it). The real work of those courses is developing an intuition for numeric and spatial modeling of the world, and the skill of translating real numerical and spatial problems into the formal language of mathematics, and then applying that limited knowledge to reduce a question to an answer. I know a lot of educators have tried to emphasize that aspect of learning of mathematics, but it's a real challenge to get the modelling and intuition without a fluency in the mechanics of the math, yet teaching the mechanics without the intuition tends turn people off from the math.

[Mike Henderson]

I don't exactly remember, but I think I took Algebra I and II, plus Geometry and Trigonometry. Calculus was not offered. It was a rural school and wasn't focused on college prep very much.

Back in the early 1900's, I expect the only people who went to college were the children of the wealthy.

Mike

[Edward Weber]

Related to this topic, It seems some states are losing more college grads than they are retaining.
https://www.wpr.org/without-more-people-moving-wisconsin-its-workforce-may-shrink-130k-2030

"The economy continues to become more knowledge-focused and requires higher levels of education," he said. "So not being competitive with other states in attracting college graduates is a challenge just from that perspective."

You can have all the "knowledge-focused" jobs you want, if all you're worried about is the economy.
If you find yourself in a situation where your state physically needs something to be repaired or built, (infrastructure) you have no choice but to turn to the hands on trades. The people who are not as "educated".
I just don't see this view being sustainable, there is already a shortage of skilled labor. It doesn't matter how many degrees you have if everything around you is falling apart.

Algebra is a great foundation for just about everything, advanced disciplines of algebra are for those specific subject areas that require it.

[Mike Henderson]

Related to this topic, It seems some states are losing more college grads than they are retaining.
https://www.wpr.org/without-more-people-moving-wisconsin-its-workforce-may-shrink-130k-2030

"The economy continues to become more knowledge-focused and requires higher levels of education," he said. "So not being competitive with other states in attracting college graduates is a challenge just from that perspective."

You can have all the "knowledge-focused" jobs you want, if all you're worried about is the economy.
If you find yourself in a situation where your state physically needs something to be repaired or built, (infrastructure) you have no choice but to turn to the hands on trades. The people who are not as "educated".
I just don't see this view being sustainable, there is already a shortage of skilled labor. It doesn't matter how many degrees you have if everything around you is falling apart.

Algebra is a great foundation for just about everything, advanced disciplines of algebra are for those specific subject areas that require it.

I don't think anyone is advocating having a population of just STEM college graduates. It's just that for the individual, having a STEM degree usually allows you to make more money. Companies who employ those people usually help the local economy because those workers spend most of their money locally.

I expect that if an area only had STEM graduates, there would be a strong incentive for skilled trades workers to migrate to that area.

My advice for the individual is: Get a college degree in a field that is in demand - often a STEM field. You'll make more money during your lifetime and that may provide you with a better life (if you don't screw up otherwise).

Mike

[John Goodin]

If there is a math class that should be required of all majors I would suggest statistics. Not just the calculations but how to apply them, to interpret accurately, and under how statistics can be manipulated and exaggerated.

[Lee DeRaud]

If there is a math class that should be required of all majors I would suggest statistics. Not just the calculations but how to apply them, to interpret accurately, and under how statistics can be manipulated and exaggerated.
Re original topic: not exactly sure how you'd teach statistics without algebra, but so it goes.

[Steve H Graham]

I've been staying out of this thread so far, but you've inspired me to jump in. What academic institution did you attend for your physics education? Whatever institution you attended seems to have had a rather toxic environment in addition to misleading you on typical career paths.

My undergraduate experience was okay. When I got to grad school, I felt sorry for a lot of the people around me. There was virtually no socializing. There were no friendships that I noticed.

When I went to law school, I swam in friends. I always had people to run around with. I had a fiery Brazilian girlfriend. When I was a grad student in physics, I never saw anyone with a girlfriend except myself. Not. Once.

We did have one incredible jerk on the faculty of my undergrad department. His name was Harry Robertson. A true jewel of an instructor. Hostile, arrogant, and passive-aggressive. I will never forget him. After his first exam, all of us thought we were going to fail and have to change majors. All except that one Chinese guy every class has. He made things as hard as he could for us and then tried to tell us it was our fault because we were lazy. Successful physics students who had already proven themselves. We had one guy who was a double major; physics and pre-med. Really lazy! I found an article about Robertson's students rioting because he failed so many of them. And this was back when nobody rioted! I believe Eisenhower was president.


. . . they tend to be miserable people with horrible social lives. . . This has not been my experience at all nor that of my friends and acquaintances (many of whom are, in fact, physicists with advanced degrees). I have known, interacted with, and worked with a large number of physicists in my career. This, in addition to going back to school myself for an advanced degree in physics after over a decade in industry in a nominally mechanical engineering role that already included a significant quantity of electrical engineering and physics. I can't speak to the general state of theoretical physicists, but the experimentalists (my particular circle) are happy in their careers and quite social--particularly with each other and with people in related disciplines such as engineering, chemistry, and so on. Maybe it's a different story for theoreticians in general, but those with whom I've interacted don't seem the least bit miserable or anti-social.

Haven't you heard the famous riddle?

Q: Six physicists are in an elevator; which one is the extrovert?

A: The one looking at OTHER PEOPLE'S shoes.

Maybe you are onto something about experimentalists vs. theoreticians. Theoreticians are smarter, and my experience has been that the more math aptitude a person has, the less room there will probably be in his cranium for things like empathy, warmth, hygiene, table manners, and humor. I'm sure you know I'm not the first person to make this observation. There are hilarious memes about it.

Spending the first 20 years of your life being rewarded by mommy for being the smartest boy in class, and then getting into a top program and being thrown into a room where you are in the bottom 60%, probably doesn't help.

The father of one of my best friends is a great example of the type. He has an IQ of something like 190. Didn't get a single problem marked wrong until he was in college and shamed his son for getting 99% on a test. Made his name in topology. Drawing graphs on doughnuts. Totally warped in his social interactions. He once grounded my friend for an entire year. My friend now thinks his dad is autistic. That would be a nice excuse, if true.

You might want to check out Andrew Dotson's Youtube channel. Funny stuff related to the unnecessary misery inflicted on physics students.


. . . they generally end up in academic environments. . . Verifiably incorrect; the American Physical Society (APS) and American Institute of Physics (AIP) regularly publish statistics demonstrating that less than half of people with advanced degrees in physics end up in academia (though as I recall a majority who are presently pursuing advanced degrees do indeed desire to pursue careers in academia). Even among physics PhDs (those with the greatest potential for academic employment), only something around 30% end up in academia (https://www.aip.org/statistics/whos-hiring-physics-phds).

Well, I stand corrected. I forgot about the guys who end up testing tires and so on. I graduated a long time ago.


I do, however, agree that the liberal arts "general education" requirements for my undergrad degree were almost entirely simply a means for the academic institution to extract more money and to help justify the associated departments and their professors' employment rather than providing any value or any relevance to my degree, let alone my career.

At some point in the past, instructors and institutions had realistic expectations regarding salaries and tuition. Now it's totally different. It's one thing to force someone to take sociology or some other liberal arts mythology-masquerading-as-science class, or classes that are essentially social engineering and indoctrination, when the tuition is $5000 per year. When it's $50,000 before housing and food, it's cruel and wrong.


To be able to get an "A" in a course wherein one never studies and only shows up for the first day and then the scheduled quizzes and tests, while receiving 3 credits for the "effort" is absurd.

Liberal arts people sometimes get very huffy when people like you say things like this. They have no idea what you have been through. They don't realize they're comparing riding an escalator to climbing Mount Everest. You could never make a history major understand. History really IS easy. It's not a prejudice.

I have to take that back. Climbing Mount Everest is MUCH easier than getting a physics degree. Old ladies have done it.


Regarding . . .every crop of students is more stupid than the last. . . my experience suggests there's instead a significant dichotomy among the current crop.

A generalization has to be taken for what it is. Generally, students have gotten dumber. And yes, I use the word "dumber" very loosely. Test scores get worse, and people become more ignorant, which is a remarkable achievement in a world with an Internet. And test scores really do mean something.

The SAT used to be a lot harder. Did you know that? They stupiditified it because it hurt people's feelings. When I was a physics TA, I marveled because one of my students got a 1530. He didn't seem that smart. Later I found out there were lots of 1530's thanks to the new tee-ball participation-trophy SAT.

People said George W. Bush was stupid because he got about 1200 on the SAT. That was actually pretty good back in his day. Friends of mine went to Harvard with scores like that, and they weren't unusual. These days, a guppy could break 1200.

[Tom M King]

Speaking of SAT's back when they used them to separate the cream, my best friend got 1600 in 1967, and three 800's on Achievement tests. Several years before that, he got a free ride through one of the country's top Prep schools from a test in the 9th grade. He's good help working with me fixing tractors, welders, heat pumps, and anything else. He was also one of the lead project scientists on the JWST.

All my closest friends are Physicists. I don't see a socialization problem. The same friend had a 50th wedding anniversary celebration weekend here on the lake a few weeks ago. There were a couple of hundred very smart, mostly couples, that had a big time the whole weekend. They don't mix well with lower intelligence people though.

[Bill Dufour]

Temple is the only person I have heard of who had more then one maid growing up. I am not sure if a silver spoon rich kid taught by private tutors is the best one to give advice for career education needs. Would her books have been published if she could not pay?
BilL D

[Michael Bulatowicz]

Hi Steve,

Thanks for the response.

I'll touch on a few items.

Not even one with a girlfriend (except yourself)? It appears that your grad-school-specific anecdotal evidence and mine may be diametrically opposed, regardless of the reasons why.

So, you forgot about the guys testing tires and so on? Do you have any idea what non-academic physicists do? Maybe you should check the link I posted.

Theorists are smarter than experimentalists?

All one can say unequivocally--as a generalization--is that theorists are better at math. Good theorists pretty much only need to be good at advanced mathematics and at applying that math to the known and/or hypothesized principles of physics. It's entirely possible to be a very good theorist who has no idea how to use a single piece of lab equipment--but I've met exactly one person who fits that description. At a different end of the spectrum, I've also met other theorists who are quite competent in lab work (and experimentalists quite competent in theory). Good theorists tend to be able to formulate creative solutions, testable experimentally, to as-yet-unsolved problems, and be able to analyze and implement them--and use that theory to predict experimental results they would expect to arise should their theory prove valid. Should it not, they must be able to forge ahead down another potentially-valid path.

Experimentalists need to be competent in a much greater variety of topics, but most often don't achieve the same depth in math and its application to physics as theorists do. Good experimentalists need to be competent in physics theory, good at practical application of known/hypothesized principles of physics, at a minimum must be conversant--better yet competent, and better still highly competent--in one or more adjacent disciplines such as electrical engineering (analog and digital electronics, signal processing and analysis), mechanical engineering (apparatus design and analysis), optical engineering (if applicable), software and/or firmware, experimental design/architecture and troubleshooting, interpretation of results, and so on. They need to be good at applying these and working with others in the same and often adjacent disciplines in an experimental setting, producing results. Good experimentalists tend to be able to formulate creative solutions to as-yet-unsolved problems and to be able to analyze and implement these creative solutions in an experimental setting. Further, if the results don't match expectations they must be able to figure out why not and what appropriate next steps to take. Much experimental research is aimed at validating/invalidating theory; I've attended a number of conference presentations that include a slide showing theory branches that have been specifically excluded from possible validity based on the available body of experimental evidence. A great deal of work by theorists went into developing those branches of theory which were later proven not to match physical reality; hence the comment above about theorists applying math to hypothesized physical principles. Much experimental research has also generated repeatable, validated results that aren't/weren't explained by existing theory and aren't more-deeply understood until the theory manages to catch up.

Theory drives experiment and experiment drives theory. It has been this way for hundreds of years and will continue to be so for the foreseeable future.

Who is smarter? That can only be judged--if then--on a case-by-case basis.

Along the lines of one of Tom's comments, high-intelligence people don't tend to mix well with groups of people significantly less intelligent than themselves and generally prefer to interact with other high-intelligence people. There are many reasons for this, one of which is simply the general human tendency to surround oneself with--and interact with--people similar to oneself. This tendency is in no way unique to any particular set of people. People of normal intelligence, as another example, don't tend to mix well with groups of high-intelligence people: it goes both ways (and many more). Introversion or the appearance thereof is largely context-sensitive. As a thought experiment, imagine what would happen were you to take a person of average intelligence and drop them in the middle of a poster session at a physics conference--a highly social event with many highly technical and detailed individual/group conversations and much intermingling. If this person is in any way self-aware, they will almost certainly appear to be an introvert in such a setting even if they may be extroverted among a group of individuals more similar to themselves. For that matter, let's expand the thought experiment: go ahead and place any self-aware individual among a group of very-different people and think about how that individual would behave--particularly if that individual suspects a reasonable likelihood of a negative response of some sort from the group should the extent of the differences become obvious.

[Bill Dufour]

As a rule physicists and mathematicians do their best? work early in life, like before age 40. Other categories for the Nobel prize do there seminal work much later in life. Of course they only get a prize latter in life after people realize the importance of what they did.
Bill D

[Michael Bulatowicz]

Interesting observation, Bill. I've never looked into it. If you're indeed correct, I'd argue instead that they tend to do their most disruptive work early in their careers; much of the later work tends to build upon the early work (particularly if successful) and sometimes the later work contributes to the perception of importance/innovation of the early work. The later careers certainly don't tend to detract from the merit of their earlier work, nor does the later work tend to be of lesser intrinsic merit. Often, the work people do in their later years builds upon the foundation set by their earlier years; this is true across many disciplines. It is not common that someone sets off on a new direction later in life.

Regarding innovation in general, my perspective is that innovation most often comes from a person who doesn't see the world in the same way that others do. Seeing things the same way that others do, how is one to come up with something new?

[Steve Demuth]

A lot of breakthrough math and physics is done by young people, but there are plenty of counterexamples to the rule. Most recently, Yitang Zhang, in his sixties now, published what, if confirmed, will be breakthrough work on the Generalized Riemann Hypothesis - one of the most highly sought after proofs in modern mathematics. Zhang, who was sidelined by Mao's Great Cultural Revolution, didn't even get his PhD in Math until age 36.

[Michael Bulatowicz]

Great example, Steve! I often wonder how many great minds (or potentially great minds) humanity has missed out on due to suppression/oppression/marginalization based on perceived belonging to some particular group, regardless of what that group may be. I've said before and I'm sure I'll say again, in my experience intelligence and talent know no bounds of nationality, race, religion, age, gender, and on and on. Intelligence and talent can come from anywhere, at any time, and ideas and outcomes should always be judged solely upon their own merits: a good idea is a good idea, no matter from whence it came. Similarly, a bad idea is a bad idea, regardless of source.

[Thomas Wilson]

I worked at a national lab most of my career. Most of the technical staff have earned PhD degrees in engineering and basic sciences. I and essentially all my friends have PhDs. The stereotypes do not apply very well. High degree of curiosity, broad range of interests, articulate and well-spoken, compassionate and socially aware, interest in social issues, fun, outdoorsy would be the generalizations that I would cite. They are the best and brightest of a generation, concentrated in a small community. It is a great environment to live in. Schools are great. Civic responsibility exceeds anyone’s level for the best we can be. Community orchestra, band, and theater are skilled at a high level for the size of the community. High ability in math and science correlates to high ability in all areas. They also are great woodworkers.

[Bill Dufour]

I would not consider listening to Paris Hilton for financial advice even though her Grandfather was super rich. I think that skill skipped a few generations. But, she does know how to spend money. Probably never even passed algebra class
Bill D

[Michael Bulatowicz]

Thanks, Thomas, for adding another voice of experience.

While your area no doubt has many highly intelligent people, I will amend your statement, if I may, to say that they are among the best and brightest of a generation. I can say with absolute certainty that no one place has a monopoly on the best and brightest of a generation.

The stereotypes definitely don't apply well. I had formulated (and then deleted) a paragraph regarding the myth of orthogonality of ability; in my view, any attempt to pretend that ability is a unit vector on a scale from social/emotional to intellectual is misguided at best. I would imagine someone at some point has studied the motivations for formulating, propagating, and clinging to such a myth, but all I have at present is my own observation and extrapolation therefrom--guaranteed, I have at best an incomplete picture. I agree that ability in general is closer to a scalar than to a unit vector, but certainly tends to have (usually multi-directional) vector qualities too. Sure, there are individuals who exhibit extreme ability in a relatively narrow set of the possible scope of human ability coupled with extreme inability in one or more other areas, but these are the exception rather than the rule--they're a small minority. They're simply the most obviously different and therefore easy targets for jokes and memes.

Your community sounds like a great place to live and raise a family. Maybe I should take a closer look in preparation for what happens after I finish my dissertation. If you're willing, I'll send you a PM with further questions.

[Lee DeRaud]

Intelligence and talent can come from anywhere, at any time, and ideas and outcomes should always be judged solely upon their own merits: a good idea is a good idea, no matter from whence it came. Similarly, a bad idea is a bad idea, regardless of source.
OTOH, you really do have to apply a "fool me once" (or N times) caveat to judging the sources of ideas.
Life is just too short to indulge every wacky idea that comes down the pike.

[andrew whicker]

Another engineer here:

I agree that calculus is mostly impractical knowledge. I actually picked up my college math book just this week to try and re-remember ODE's and PDE's. There's a lot of arcane knowledge that I have forgotten (esp wrt integral rules).

However, I do think learning calculus while in thermodynamics, systems, etc helped me really understand what was going on. I remember, esp in thermo, that by the end of the semester you could derive your own equations if you forgot one of the equations you needed to solve the problem. From memory, you could write down the PDE's of the thermodynamic laws and basically go where ever you wanted. For me, that pretty much solidified thermo as "I completely understand this topic". I'm not sure if I would have gotten to that level without calculus.

OTH, I've embarrassingly forgotten way more than I expected. I was / still am interested in doing some math tutoring on the side. I was going to sign up for calculus. After all, all of us engineering students basically took 4 years of math class (since every engineering class was also a math class). But after this many years, I have to admit it will take me work to get back to just doing ODE's, let alone PDE's.

Anyway, that's a side discussion I see going on. My long and short of it: without calculus I wouldn't have understood fluids, systems, thermo, dynamics, etc nearly as well (I don't think).

I admit too, that I cannot for the life of me figure out how anyone would go thru life without Algebra! I think a lot of people don't give math problem solving a proper name, but you ARE using Algebra when you do simple ratios (x/y = a/b, solve for the one unknown... a super powerful equation when you're building stuff!). So anyway, learn algebra. Don't worry about calculus unless you have to. If you "hate" math, find a better tutor or teacher. Math is pretty fun and easy... it's just rules and processes.

[andrew whicker]

BTW, just so there isn't confusion going on here wrt the type of math we are talking about. This is from my memory, so feel free to correct me all you grumpy physics and happier engineering majors.

Algebra is learning the system and symbols of math equations to solve for an unknown. Wiki gives the popular example of the quadratic equation to solve A*x^2 + B*x + C = 0. But this could also extend to more complicated terrain such as linear algebra (matrices) and beyond.

Calculus is used when solving FUNCTIONS that are dependent on some independent variable. One of the most popular examples being that if I know an object's position with respect Time, then we can also know it's velocity and acceleration at any give time! If we know it's acceleration at any give time wrt time, then we can possibly determine it's velocity and position at any given time if we know just a few additional data points. Pretty cool and very powerful! You can immediately see how important that would be in physics, engineering, biology, etc. if you open a math book / watch a math lecture on introductory calculus.

I think people here are sometimes getting "advanced math" confused with "calculus". I've been swimming around a YouTube channel called "numberphile" lately which describes fun math problems for us laypeople and it sounds to me like a lot of the problems they work on are super complicated, but are not solved using calculus (but what do I know, I'm not a mathematician). Something has to change wrt to at least one variable for you to use calculus AND you would want to solve (derive or integrate) that something. That's not always true. And even if it is true, you still have to use algebra to get your final answer!

For example, if the Speed of a given point is described wrt Time (solved via calculus) as: t^3 + 5*t^2 + 3*t + 10 = Speed, you would still need to solve for t if you wanted to know at what Time Speed equaled 100 (for example) and that's a difficult equation to solve (100 = t^3 + 5*t^2 + 3*t + 10, solve for t). You got the equation w/ calculus (you knew the point's position wrt time and solved for it's velocity wrt time), but you get the final answer w/ algebra (solve for t). This is a stupid / silly example, but hopefully gets the point across. Because in this case the calculus is way easier than the algebra.

That's how I think of it anyway. In reality, I think algebra probably is the umbrella that ALL math falls under, but this the is way I've decided to separate the math into different buckets.

[Steve Demuth]

Anyway, that's a side discussion I see going on. My long and short of it: without calculus I wouldn't have understood fluids, systems, thermo, dynamics, etc nearly as well (I don't think).

I admit too, that I cannot for the life of me figure out how anyone would go thru life without Algebra! I think a lot of people don't give math problem solving a proper name, but you ARE using Algebra when you do simple ratios (x/y = a/b, solve for the one unknown... a super powerful equation when you're building stuff!). So anyway, learn algebra. Don't worry about calculus unless you have to. If you "hate" math, find a better tutor or teacher. Math is pretty fun and easy... it's just rules and processes.

I agree. You have to understand calculus to understand most of physics, much of engineering, a lot of Chemistry and even Biology (particularly population biology). I would add Statistics to the list. You won't necessarily use DE's, differentiation or integration on a daily basis, because for any standard problem, someone has probably reduced the computation down to relatively basic algebraic and numerical calculation in a standard solution, and even if you do need to do calculus, you can get software to do the hard calculating for you, but if you don't know where the equations come from, and how the respond, you don't know what they mean.

For most of the list above, I would add Linear Algebra to the "must understand" set of prerequisites. Our understanding and ability to manipulate the physical world is pretty much all built on the combination of calculus, linear algebra, and probabality and statistics.

[Bert McMahan]

In the mid-70s, an engineer/physicist coworker of mine saw the way the wind was blowing wrt to computers, and had his high-school freshman son sign up for typing class instead of shop class.
He ended up in a six-month battle with the local school board over it. "What, you want your kid to be a secretary?!?"
(Oddly enough, they would let boys substitute home-ec for shop, but not typing.)

Knowing now how much code I would sling over the course of my career, I certainly wish that had been an option when I started HS.


I know this was from a few weeks ago, but hey- I'm a mechanical engineer who uses calculus in his work :) Now you can say you've met one! Around here, not everyone uses it in the "write calculus on paper" way, but most of us have used it in the "I understand how the math works for this problem" way. It's certainly industry specific.

[Bert McMahan]

BTW, just so there isn't confusion going on here wrt the type of math we are talking about. This is from my memory, so feel free to correct me all you grumpy physics and happier engineering majors.

Algebra is learning the system and symbols of math equations to solve for an unknown. Wiki gives the popular example of the quadratic equation to solve A*x^2 + B*x + C = 0. But this could also extend to more complicated terrain such as linear algebra (matrices) and beyond.

Calculus is used when solving FUNCTIONS that are dependent on some independent variable. One of the most popular examples being that if I know an object's position with respect Time, then we can also know it's velocity and acceleration at any give time! If we know it's acceleration at any give time wrt time, then we can possibly determine it's velocity and position at any given time if we know just a few additional data points. Pretty cool and very powerful! You can immediately see how important that would be in physics, engineering, biology, etc. if you open a math book / watch a math lecture on introductory calculus.

I think people here are sometimes getting "advanced math" confused with "calculus". I've been swimming around a YouTube channel called "numberphile" lately which describes fun math problems for us laypeople and it sounds to me like a lot of the problems they work on are super complicated, but are not solved using calculus (but what do I know, I'm not a mathematician). Something has to change wrt to at least one variable for you to use calculus AND you would want to solve (derive or integrate) that something. That's not always true. And even if it is true, you still have to use algebra to get your final answer!

For example, if the Speed of a given point is described wrt Time (solved via calculus) as: t^3 + 5*t^2 + 3*t + 10 = Speed, you would still need to solve for t if you wanted to know at what Time Speed equaled 100 (for example) and that's a difficult equation to solve (100 = t^3 + 5*t^2 + 3*t + 10, solve for t). You got the equation w/ calculus (you knew the point's position wrt time and solved for it's velocity wrt time), but you get the final answer w/ algebra (solve for t). This is a stupid / silly example, but hopefully gets the point across. Because in this case the calculus is way easier than the algebra.

That's how I think of it anyway. In reality, I think algebra probably is the umbrella that ALL math falls under, but this the is way I've decided to separate the math into different buckets.


If you like Numberphile, try some 3blue1brown as well. He really breaks math down into chunks you can understand. I fully admit, despite a minor in math, that I didn't really "get" matrix transforms until I watched his series on them. Now it makes perfect sense.

[andrew whicker]

I guess I would argue most people on here who went down the calculus rabbit hole in college probably can't actually do a lot of calculus anymore. Integrate 1 / (sqrt(x^2 + 3*x)) if you think you remember how to do calculus (no Googling!). Most of calculus is made up of rules of how to integrate this or that.. very specific and who cares?

What most people w/ calculus backgrounds seem to be saying is that the IDEA of calculus is important and they are happy they understand the concepts. I would suggest that anyone with interest in how the physical world works should watch an intro to calculus lecture. No need to memorize how you integrate such and such with whatever rule or even write down equations.

Statistics is probably one of the most important areas of math ever. I mean in reality, all anyone is doing is estimating real life behavior of a system with equations. Then you test the system as many times as you can in order to modify those equations to reflect reality (statistics). OTH, as an engineering student I never took stats. Stats is complicated! Way more complicated than calculus IMO (at least for me).

Anyway, I think people should literally know how to solve for an unknown or set of unknowns (algebra). I don't think people should literally know how to integrate or derive equations, but it would be cool if more people knew the concepts behind calculus.

[andrew whicker]

I'll check it out. I like nerdy stuff.

[Steve Demuth]

I guess I would argue most people on here who went down the calculus rabbit hole in college probably can't actually do a lot of calculus anymore. Integrate 1 / (sqrt(x^2 + 3*x)) if you think you remember how to do calculus (no Googling!). Most of calculus is made up of rules of how to integrate this or that.. very specific and who cares?


I couldn't recall most integration techniques by the time I left graduate school in physics for "the real world." Doesn't really matter, back in that day you had your CRC handbook, and several calculus texts at your desk, and if you did manage to write down a Hamiltonian for something you thought might have an analytical solution, you went to those for help. Nowadays, you just type the equation into wolframalpha or any of half a dozen other good solvers, and voila, you're on your way. The important thing is that you know why you need that equation, and what it, and the solution, mean. Purists think that's a cheat, but that makes everything in life a cheat. As A.N. Whitehead said: Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

[Lee DeRaud]

In the mid-70s, an engineer/physicist coworker of mine saw the way the wind was blowing wrt to computers, and had his high-school freshman son sign up for typing class instead of shop class.
He ended up in a six-month battle with the local school board over it. "What, you want your kid to be a secretary?!?"
(Oddly enough, they would let boys substitute home-ec for shop, but not typing.)

Knowing now how much code I would sling over the course of my career, I certainly wish that had been an option when I started HS.I know this was from a few weeks ago, but hey- I'm a mechanical engineer who uses calculus in his work :) Now you can say you've met one! Around here, not everyone uses it in the "write calculus on paper" way, but most of us have used it in the "I understand how the math works for this problem" way. It's certainly industry specific.
I think I agree with what you said, but I'm not sure what that quote from me has to do with it.

[andrew whicker]

I agree 100%

Haha, I enjoy math nerding out. I enjoy opening my math book and doing problem sets from time to time. It's just a brain teaser at this point with no applicable value. People do this with Soduku and NY Times crosswords. I don't think math sets are much different. I'm never going to be working on a piece of furniture and think to myself, how can I find the radius vs time if the area is growing at XYZ rate?? : ) But I COULD. Lol.