Struggling with Math

I might start by trying to give you zero change.

Math is something that takes practice...if you practice a lot, you will get better...and siri can add that up for you, so you just need to repeat the figures into your phone...

Practice will help most but not all. What is unclear in the proposed challenge is must the arithmetic be performed without the aid of any tools (pencil and paper, calculator or Siri) and how many of each of the four teas will be purchased.

Well and then there's the tax or taxes to be subtracted if applicable in some cases....

Intuitive logic suggests that the author intended persons to attempt this using mental arithmetic. To use any form of aid greatly devalues the challenge and the test of this abiilty. So in a way, the verbiage here held not only a test of one's mathematical ability but also one's ability to drawn an inference.

This test of one's ability to infer goes further with the statement, "you purchase four types of tea packs: Chamomile Tea ($4.60), Green Tea ($4.15), Black Tea ($3.35) and Lemon Tea ($1.80)."

It could have been expressed as "you purchase four packs of tea, each of a different type, namely: Chamomile Tea ($4.60), Green Tea ($4.15), Black Tea ($3.35) and Lemon Tea ($1.80)."

I suggest with the first version of the sentence, the reader is required to infer the second. This is therefore not only a test of one's ability to perform mental arithmetic but also one's ability to see through unclear expression and divine the intention of the author or speaker; something, in real life, we often have to do in order to solve problems or interact with people in everyday life.

What if the exact same given conditions were given but an added given condition of the merchant returned $1.50 from the $20. My point here is that as given there are multiple correct answers ($6.10, $1.95, $1.50) depending on how many of each of the four types of teas one purchases. [1 1 1 1] is not the only combination that sums to less than $20.

That not always true,... there are people that can see, recognize the patterns to a point of them being gifted... and not so much as a idiot savant but a savant when it comes to math and still high functional person.

For myself,... in college, I struggled studying for an exam to a point of walking in when its time to take it prepared to write it off as a fail, only to pass quite respectively,... Only then did I learn, to relax and not take math not so much seriously, but not to take it so intensively.

I find it hard to accept the finding that so many people can't do simple math. Math is one thing, clerical ability is another. There are several ways to make mistakes, clerical errors rather than math errors, even when using a calculator.

Also I have doubts about the conclusion that these findings have a catastrophic direct bearing on the ability to make financial decisions. Financial decisions weren't at all tested in the study.

I also pretty much gave up cursive writing in favor of hand printing which I learned in drafting. I barely passed cursive class early on and it was never very legible anyway. That does not mean that it wasn't a valuable skill, especially when it came to reading other's writings in their own hand. And, leroy lettering and eventually CAD, was the saving grace for my producing quality drawings. Unfortunately, now, proper drawing creation is at risk with integrated solid modeling and data transmission between models and machines; CNC and 3D additive manufacturing. Many youngsters consider drafting redundant and, therefore, a waste of their time.

But, all that aside, my musings in this thread have been intended to target skills learned in early primary education. The loss of detail and discipline does not bode well for the future progress of technical disciplines. Over my career of teaching thousands of CAD/CAM related classes I noticed a marked decrease in STEM basic abilities.

I am dismayed at future prospects.

Hooker

Every engineer, regardless of field, at some point, should be required to put pencil to vellum or paper and manually render some objects in three views as well as perspective.

What is learned in that process is invaluable later in life/career.

I too, gave up on cursive as soon as I was permitted in favor of hand printing for the same legibility reasons.

Fundamentals of Spatial recognition and understanding,... on generating or constructing true length, true shape, etc... ga...

Descriptive Geometry. My favorite high school class besides regular drafting. Probably because it's about the only class that I ever aced everything in.

I wonder if they even teach that anymore.

Hooker

Descriptive Geometry.

I don't recall ever having had a class with that name, but I did take four years of drafting/mechanical drawing in high school, where I learned all those concepts, and that experience has served me well for 60 years (and continues to...).

Yep! this June will be the 60th anniversary of HS graduation, and I'm still enjoying working!

In my high school, Descriptive Geometry was 3rd year drafting, available only by recommendation of our drafting teacher. And since we only had 10 thru12 grades we only had three years of drafting. They crammed a lot in those 3 years.

This June is 52 years for me!

Hooker

Exactly... In college, it was only a 2 credit course and on the board. When I took it I was carrying a large class load that semester, full time student is 12 credits, I was taking 22 credits. And at (2) credits I thought it was going to be a cake walk... That course I can easily say, it was sucking up 80% of my time. I had to drop it and take it the following semester.

2nd time around, with a more reasonable class load it was a lot easier... I recall one problem where the construction lines to develop some of the true shape, true lines actually went off the board (drafting machine) and we had to estimate it. For only a (2) credit course, it was tough but it was also very valuable. Especially when I was in the shipyard.

I'll bet it was valuable in the shipyard!! If you had to do any lofting you used those principles!! Great example.

Hooker

I don't believe they do teach it anymore as well as kinematics..

When I was an engineering manager back in the 90's, one of my engineers from Michigan Tech he would say, I did that in college and we had software that did that...

The company I was with, being a smaller fab shop, we didn't have the budget to buy the software, and when he did it by hand it was nothing but an abortion problem.

That's when I began to question any engineers coming out since then if they understand the fundamentals.

(I've have said this, "Sure with technology, we can make a monkey into an astronaut, but that astronaut is still a monkey" ) It was a driving point.

What' next - grammar and sentence structure!! Have you ever read anything (written or keyboarded-texting is another thing) from the generation or 2 behind us in age?? And I am talking about persons with a so called "higher" education as well as those with "lesser" education.

Might just be that the proper grammatical use of the English language is already in the throws of death.

Have you read the grammar and sentence structure from previous generations. When I look at some of the correspondence from the US Civil War I'm embarrassed with my need of a spiel chunker. I'll grant you that the many who couldn't write left no correspondence to compare. My point is that unlike rigid computer languages, English morphs over time.

Not to mention the penmanship of yesteryear

English morphs over time.

well,... we could all learn and stick with Latin....

Morph indeed. It seems that each generation has a different "starting point" in time as a reference. I guess it is just difficult to accept that something as basic as written language, as I (or we) learned it, can change in the space of a generation or two (barring of course the introduction of new words).

After all, being somewhat long in the tooth, I do not know what or how the kids are taught these days unless I pick up and read some article, or follow a thread here on CR4. My only reference is what I was taught/learned.

At least I don't cringe/wince as much any more when I read something written by people many years my junior. Guess I have morphed a little as well to accept what I would call the improper use of language, based, of course, on what I learned.

I agree, but I also doubt my ability to pass standard primary school tests of just 100 years ago. Their attention to grammar and composition and things like English Lit and Shakespeare put us to shame.

Maybe someday the pendulum will swing the other way.

Hooker

throws ≠ throes

Because my daughter has to work late, my wife occasionally picks up our granddaughter after school. On those days I get the "opportunity" of helping her with her STEM math.

Talk about a disaster!! For the life of me, I cannot believe this is what the "great thinkers" of our education system have come up with for teaching math.

Mind you she's only 1st grade, so whatever it is ought to be easy right? Nope - half the time helping her is spent with me trying to figure out what in the world are they trying to do.

It's more like math empathy, "rearrange the problem & feel the correct answer."

That is so sad. I looked once at some early education math curriculum at an online school. It's amazing that a high aptitude adult cannot figure it out without actually going back to first grade.

And these kids still can't balance a check book after high school.

Hooker

I absolutely agree with this. Not only our thought processes and aptitudes are variable, but we also have unique ways of compensating for the specific 'gaps'. Physics and math were among my best subjects in high school, but when we did aptitude tests I scored in the bottom tenth percentile for clerical ability. When faced with tables, ledgers, columns of figures I make more errors than average... a lot more.

I have to do accounting for my business, so I have to use all the compensations for my clerical deficit: no distractions; not tired; extra coffee; double or triple check, and in the first place, keep it simple. All my prices end in a 5 or a 0 and no cents which helps me to get through the tedium of adding up receipts from direct sales events, big time. And I don't use a spreadsheet for accounts at all. I write it all up in a simple text file, where I can actually see what's going on.

OTOH I have no trouble doing the complex financial task of price comparisons on materials, where variables include spot price of metal, markup, dollar exchange rate, difference in shipping, discounts, and whether the tax is 7 or 15%. It's pretty intense, but I like using that part of my brain. It's just not the same part that's required to add columns, fill out forms, tables, ledgers....

I still think the author of the study has made some questionable inferences about financial literacy.

I understand that there are three basic types of thinking: visual, auditory, and kinesthetic. The visual thinker will say "I see what you mean". The auditory thinker will say "That sounds about right". The kinesthetic thinker will say "That feels right to me".

http://www.cleanlanguage.co.uk/articles/articles/113/1/I-See-Hear-and-Feel-What-You-Mean/Page1.html

I see what you are saying, so I suppose we are the same, visual. I am awful at remembering a name, but if I see it written down, I have a better chance of remembering it. I visualize the solution to a problem before I work it out. If I need to explain something, it's second nature to draw a picture.

I'm guessing that a majority of engineers might be visual thinkers.

"I'm guessing that a majority of engineers might be visual thinkers. "

I can't imagine any other way.

Been there and went through it and still no answer ......

"I cannot conceive of thinking of nothing" When I was kid, my brother told me a trick on how to fall asleep at night. You just imagine your brain as a blackboard. And, whenever a thought appears on it, you just erase it. Eventually, you fall asleep.

Sounds like Grannies remedy (Beverly Hillbillies) cure for the common cold....

You take a slug or (2) of Grannies medicine and in a weeks time, your cold is gone.

I play golf at my favorite golf course. And I mean imagining everything from teeing up a ball, to stance, and even hand position and type of swing needed for a particular hole. It is much the same as coaches teach about visualization to get the intended results.

I rarely make it past the 4th or 5th hole before falling asleep so sometimes I start out on the back nine. Works every time at blanking out the nagging worries of the day. My mind is way too active to think of "nothing".

Hooker

I erased blackboards thousands of times, but I can't imagine doing it without an eraser. I used blackboards before the days of electronic ones, where one could press a button to clear it.

https://i.imgur.com/eTWHeaX.mp4

Every time I try to think of nothing virtual particles form in the vacuum and space increasingly expands.

Pretty much the same here. If I'm not thinking of a specific thing, which I usually am, then I'm thinking about how to avoid thinking, and that has never produced any results. I commonly play some simple game of solitaire until my eyes start to droop... to take my mind off everything important.

" claims to be thinking about nothing "

Is it possible that she can control the rate of acetylcholine as a neurotransmitter or her thoughts are being interrupted by anticholinergic substances ?

You're thinking too much.

That was an interesting link. It confirms a few things I have learnt by chance - but a common theme in their presentation was the concept that there was a desire to learn.

That is, the pupil and teacher had a common objective in mind.

But what was not touched on was the fact that in many cases the 'pupil' does not want to know.

I experience this in BSI committee work. Getting a point over (ie. a Yes vote) is a problem enough for those with a genuine interest, but doubly difficult when dealing a subject matter that has to be adapted to suit a diverse audience - especially when objections come from those with their own 'private' agendas.

"..please don't blind us with science..." ..... "...is this going to take long?..."

.......from which the physics and math - the 'engineering' aspect, all-important to understand in support of the case, easily gets stifled - and regretfully passed over.

A GA from me. You have summed up my age related problem with hearing in the speech range, I need sub-titles on the TV.

And you are close with the abstract numbers that to me are a problem. If I can't draw a picture in my head of the relative 'position' and 'relationship' of the numbers when applied to 'things', I have trouble using them - and getting a result.

Since I can't remember facts and formulas - at least not enough to pass an exam - so historically I did not get anywhere in the academic league in my schooling days - was that the teachers fault for not having the ability to exploit the skills I had (It would have helped to have a list of formulas into the exam) - but never mind - I am OK on principles. So nowadays armed with a scientific calculator and books of formulas (the Internet today)- I fare a lot better.

But back to the OP. I can't say what the conclusions are. At least in my case I am 'aware' of the power and 'engineering' of maths to dig up a formula to see if I am being 'cheated' out of my money.

I too think most engineers are visual thinkers.

I agree with DK's thinking. Here’s 2 of my most vivid personal examples. I remember back to my initial horrors in 10’th grade French class (about 100 yrs ago if I remember correctly). The teacher strolls in talking French to us as if it was our 1’st language. She spoke only French for the entire first 2 weeks of the class. Not a single word of English for those 2 weeks. Oh, she was using hand gestures & facial expressions but those didn’t help much. I was REALLY UPSET.

By the end of 2 weeks my mastery consisted of “je m'appelle” & “au revoir”, (“my name is” and “goodbye”). I just couldn’t grasp any of it.

Then on Monday of the 3’rd week, she strolled in as usual, speaking only French as usual, but this time began writing out what she was saying on the board (in cursive I might add). Within a few minutes she also started intertwinning English & French as she spoke & then I finally started understanding French. French only made sense after seeing the words written out with some English assistance here & there. Once I could physically see the words spelled out, I began to understand what I was hearing. Just hearing the words accompanied with hand gestures & facial expressions didn’t work for me.

Many years later my wife & I home-schooled our kids. My wife taught everything except math during the day & I taught the math in the evenings when I got home. My son was never a problem, but my daughter (and wife) was always a challenge. It seemed like for every problem type, my daughter had to be presented with solutions from several viewpoints before one of them “clicked”. But then my wife would be on my case hassling me about, “Why can’t you just teach her the way you’re supposed to in the book!! All these multiple ways you’re using are just confusing her!! How’s she ever gonna learn if you keep confusing her?!#?” Whereupon at this point my daughter would side and chime in with my wife.

(Duh - maybe because the methods in the book weren’t working for her.)

Once we moved from the boonies to civilization, we put our kids into public school. (By that time they were now in their teens, knew everything, were tired of mom & dad as teachers, and actually wanted to go to public school.) However, by the end of the 1’st month my daughter was already complaining about math class; “I don’t understand any of it!!” “They don’t teach like dad does, they only do it one way.” Those were some of the sweetest (re)words. Within just a couple months of public school math, my daughter was on my side, extolling the virtues of “dad’s methods” and mom was off my case.

I continued teaching her math, always using numerous angles of attack; right up through her 2’nd yr of college whereupon her math rqmts were fulfilled (finally!) After each of her college math tests I’d call & ask “What grade did WE get?”

Although I was using complex math every day throughout all of this in my job, I was often surprised how hard it could be to teach simple concepts that I used daily without a thought. Concepts that I had long ago learned, accepted, forgotten, or know the concept but totally forgotten everything leading up to it. Too often we just take things for granted & don’t think about the “why”, instead we just “straight-line” our answer & plod forward.

Most often I had to back up and “think simple” on my quests for multiple math viewpoints. Just because you know something doesn’t mean you understand it anymore. In the process of all this, I learned new math tidbits myself all along the way. And just because you know something doesn't mean you know it for the right reason(s). In my quests to teach my daughter I discovered I had a few of those disconcerts from my childhood past.

It’s given that you don’t really understand something until you truly try teaching it. Try getting inside a teenage girl’s head to teach math! Trying to get inside my daughter’s head taught me how she thinks (for math anyway) & to focus more on developing “whys/ways” of solving problems along the lines of how she thinks.

I know this was long but hope everyone enjoyed it.

I, too, taught one of my daughter's mathematics. Indeed, I insisted she do the subject at high school up to year 12. She resented me intensely for that. In tutoring her, I would always like to go back to first principles in order to explain how a particular problem could be solved. I, personally, had found that was the only way I truly understood and remembered mathematics. She, on the other hand, only wanted to know the answer to how to solve the immediate problem and she would learn that by rote. She scored 97% in her final exams but nevertheless often voiced her dislike of mathematics. She went on to complete a degree in Communications (a journalism degree) and won the University Medal.

This drove home to me that there are, at the extreme, two types of people. Those who read, remember and perfectly regurgitate the knowledge they have learned by rote and there are those who learn by remembering some form of narrative where all the components necessary to achieve or retrieve something are linked together. The former are not good at inductive thinking nor do they cope well when confronted with a new situation about which they have not been instructed.

Unfortunately many of these rote learners end up in vocations like politics and law. ;-(

My struggle with math is that I have trouble remembering formulas. The way I make up for it is that I have a library of math books, reference manuals and other technical literature. My math books range from pre algebra to calculus. When I need to do an equation, I source through the books to find a formula then apply it as needed.

Although math is a struggle, by arranging numbers, letters and symbols, it allows you to make new and interesting discoveries. That is a big part of the reason I enjoy using it and learning.

In grade school, I had to learn math mostly on my own with the help of tutoring from time to time, outside of the class room. Most of my math teachers gave me failing grades, because the way I learned was by using reference guides to perform the work. Many times the teacher would write a big F on my test paper after noticing that i was using a " cheat sheet " while doing tests. Somehow, using a sheet with examples of formulas was cheating. I noticed, early on that learned individuals, used reference materials to solve problems. When I was a kid, I had the opportunity to directly observe all of the different disciplines used on construction sites.

Recently I put up another antenna, this time for a scanner. Now I knew that I had to figure the distance for the guy wire. This meant that I had to figure the hypotenus, the long side of the triangle, so I got out a book and used the information. And I accomplished that task. This approach works for me.

I have met travellers (Romanies) who could neither read nor write but they certainly new how to count money!

Not to question the social point of the 'study' discussed in the blog, which is that financial literacy should be taught and financial instruments should be kept simple (hallelujah to that), however I did read the paper referenced by the author, and I do not agree that the statistics generated are a valid measure of financial literacy.

The author acknowledges that these questions were designed to measure numeracy, not financial literacy. They just happened to be the 4 questions out of 20 which worded a math problem in terms of pocket change or some other (deemed) 'financial' question.

Here is a diagram of the actual test process from which the data was mined.

The "four basic financial questions" came from the numeracy part of a long test of basic skills. Order of testing was varied, also the degree of difficulty of the numeracy questions was varied, using a probabilistic assignment that depended in part on performance in the core test. The questions cited in the article are "mock questions" not the actual questions which varied in difficulty. The degree of fatigue in a long test process was also variable, and an obvious source of error not acknowledged by the author.

The data which was mined in this manner and applied to a question for which it was not intended, also conflicted with PISA data on math skills in a number of countries. The author contends that this supports his contention that "financial" rather than mathematical literacy is being measured in these questions. This belies the conclusion which was made, that a lack of basic numeracy skills is showing a lack of financial literacy. Random error is a more likely explanation, given the conflict with PISA.

Sorry... screen capture.

You can use dropbox for full size stuff....the basic is free, and you can link your image...

The image is from the pdf of the paper, linked in the article from the OP. I doubt there is much chance of improving it, but anyone sufficiently interested can get the pdf and scroll through to find it.

Tx for tips though.

I don't remember the details, but I quit using Dropbox about a year ago, because it started doing things by itself. I think it may have been adding stuff to the Dropbox file that I did not want to share, without asking.

I don't want to provide a link to anything on my computer to anybody I don't know very well (and I've never actually met any CR4 users, although I did mange to meet the wife of one...). I have exchanged emails with a number of CR4 users, most commonly to provide higher resolution copies of illustrations.

When I insert an image using the camera icon, I presume that CR4 (or Globalspec) receives the the image at full resolution, and reduces the resolution to speed up the transfer to users. It's also possible that CR4 could somehow request a lower resolution image from my computer to speed the transfer from my computer to theirs.

It would be perfectly reasonable for CR4 to accept JPEGs and PDFs at full resolution and store them on the CR4 server(s). They could then insert low resolution copies in the blogs, but make it (for example) so double-clicking on a low resolution image would download a full resolution image, either within the CR4 window or in a separate window.

We're collecting suggestions for improvements to CR4. I'll add yours to the list.

I've discovered on my own that images render better when the poster uses the camera icon to search for an image file rather than use a screen capture -- although some scaps work OK. Taking the time to turn an image into a file is a pain though.

Thanks.

Years ago, screen captures did not work at all in CR4, at least on a Mac, so I hadn't tried it recently.

I see it does work now! Thanks for that, too. The pasted image, at least in this case, appears to be enlarged by a factor of two. I did not resize it.

I did sketches, graphs and charts in AutoCAD or MS Excel there appears to be room for improvement.

Unless its the snipping tool that I use that makes the file that comes with MS.

On a Mac, there are at least four screen capture functions built in to the Operating System: Capture full screen to clipboard, Capture selected rectangle to clipboard, Capture full screen to file (PNG), and Capture selected rectangle to file.

There are key combinations to invoke each of these. For example Shift-Control-Command-4 is the Capture selected rectangle to clipboard, which I use a LOT!

I myself, never liked the screen capture functions...

Like I mentioned earlier about the snipping tool I use,... I always save it to a file using a jpeg format,... maybe if I used a different format (bmp or png) it'll be better.

I do a lot of instructional documents; screen captures commonly are very useful for those.

Also, since I started designing machine parts in 3D 5 or 6 years ago, I nearly always rotate the 3D image to an advantageous view angle and copy a bitmap of it for insertion into the part description. The actual part is usually created by CNC from an IGES file, but it's still nice for others to see what the part looks like before it has been made.

I used the MS Snipping tool for basically the same as well as making project files and Operator Manuals in MS publisher, what I like about it is that you can window around just the info you need, without taking a full screen capture.

The earlier MS Publish when you copy clip them in, it wouldn't embed the picture but instead use a pointer to the picture file location. This was bad and it always got 'hosed up'.. but MS improved Publisher that it would embed it in the actual MS Publisher file. That save so much aggravation.

Since humans have five senses, we understand what it means to say, "If it walks like a duck, quacks like a duck, feels like a duck, and smells like a duck, then it probably is a duck". The more senses involved with identifying something, the more sure we are of its truth. It gets more complicated when we encounter a lame duck, a silent duck, a featherless duck, a clean duck, or a dead duck. But what does remain, is the fact that "what a duck is" is easier to remember when more senses are used to perceive/identify a duck. With math, you can also "translate" the concepts of it into visual, auditory, tactile, etc, symbols. That means that you can also decode those symbols/concepts. That's one thing that makes computers so powerful (random access is another). With "object-oriented" coding, you can place complex items into a simple "object". It's easier to remember that simple object and decode it later, than to remember all of its elements every time you try to work out a problem. That's why algebra helps with understanding abstract concepts. Some people know the "object" before they know its elements. And, some people know the elements before they know the object (that's where that saying about the forest and trees has meaning. Some people can't see the object because of their elements; and some can't see elements because of the object that it is.). The concept of algebra helps with translating abstract ideas into real values (and vice versa). If you think of algebra as a concept (object), then it's easier to remember its purpose/function, and therefore its elements. Then, you can translate that to any depth/level (higher or lower) you're comfortable/knowledgeable with. But, if that's not adequate, then you might just have to look it up. But, at least you know what you're looking for.

I like my ducks with big feathers, and spaces in between them. They are easier to look at...and read.

In our great quest to "leave no child behind" we've come up with a program that leaves them all behind, but at least now they're all behind together.

My biggest concern with not being able to do the basic arithmetic is not knowing if there has been a mis-key on the calculator. I, too, struggle with math (not arithmetic). This makes me very aware of what the range of the calculation should be. Most of what I do is electrical, so I try to visualize the correct result and think of other instances where there is a similar circumstance. If the answer is way off (100 amps instead of 10) I think that should jump out at you, just as if you go in to get a bottle of water. If the clerk says that's $20.00, I know something is bad wrong. Same with structure or anything else. Just saying 'that's what the computer said' is not good enough. --IMHO, easy on the H -- JHF

I begin by asking the basic question, "What does a human being need to know now that he/she/it didn't need to know 100 years ago?

Reading, writing, arithmetic, social norms and some sort of cultural context relating to norms of behavior and action. Some history is also necessary to explain why the norms are what they are.

Basic arithmetic gives you the ability to function in a money society and also allows you the ability to do a back of the envelope calculation to do a basic sanity check on problems like, how may boards do I need, or do I have enough cupcakes for the party?

Sadly, learning to GUI code a basic robot or why recyclable paper is needed for the paper mache piñata is considered important, but being able to read and comprehend the instruction manual or follow directions is not. Sadly it is also lacking that modern victims of the system cannot figure out ahead of time that they don't really need 3 metric tons of flour for the piñata. I deal with recent engineering grads and find that few if any can do a back of the envelope calculation or a sanity check. I spend a fair amount of time recovering projects where the start was to throw some parts together and then fix what breaks until something starts working. These are the ones who made it through STEM.

Anyone else run into "Creative Math" yet? The object was to take students and not teach them math, but give them problems and let them invent their own solution method. Doesn't work.

Many years ago I read a SF story set in a future where the basics of math had been forgotten and everyone used a handheld calculator. One fellow rediscovered how math works, but no one would believe him, calling his work just tricks.

Anyone remember that story?

I thought of it a while back when my boss had to multiply 3 by 100...and he grabbed his calculator!

I don't recall your story, but I had a boss, way back when HP came out with their scientific calculators, can't recall the model (45?), who absconded with my unit, sat down, did some key punching, then proceeded to pull out his slide rule to check the result.

I did that a few times myself!

I also continued to teach the use of slide rules well after calculators became available, simply because learning to use a slide rule taught the user to read most any scale correctly. ...and of course because not everyone could afford a calculator in those days.

To me 'those days' were when calculators did not exist - the slide rule was all we had (apart from tables) that also taught how to juggle decimal places - not by strict mathematics - but by 'feeling' or intuition ---- did the answer look right - where should the decimal point be in the answer.

Even today with calculators - apart from the exact numbers - the magnitude of the answer should 'look' right - it should have the right 'feel' to it.

That I suppose this is called experience.

Nerds of the future. ?

This is post #100. I doubt that too many will see, so I'll just note the basics:

Part of some relationships is the synergy when one has someone else to do the math, who may (or may not) contribute to mutual security in other ways.

Fear of math is related to fear of science, which comes from need to believe IN the impossible (magic). Science and math are impersonal, and challenge the distortion of reality that we need to some degree for our sanity.

Re sleeping, the monkey-mind brain can fall asleep just as quickly as the blackboard-erasers.

An understanding of math begins at an early age. When I started school 75+ years ago, addition, subtraction, multiplication and division tables were drilled into us. That made an indelible impression on my brain and as a result, I can do problems in my head, quickly and accurately. The basics once learned and permanently embedded in the brain allows math to be easily learned. I don't know if this basic math is still taught in school.