Mathematics Paper

[Mike Henderson]

This is really off-topic.

Many years ago, the son of some friends of ours was graduating from high school and heading to Georgia Tech to study engineering. I wrote a paper on some basic mathematics that he would encounter in his engineering studies.

If you have, or know, a child who is heading to college for engineering, mathematics, physics or related subjects, this paper may be of use to him or her. If you find errors in the paper, please let me know.

Mike

[Oops, the paper is larger than the the forum allows for .pdf. You can see it here (http://www.michael-henderson.us/Papers/Differential_Equations.pdf).]

[Matt Day]

I’d prefer not to pass on the joy of diff EQ. Lol

[Bill Dufour]

The high school and I think college miliatary recruiters used to give out nice printed outlines of important formulas and rules etc for chemistry, physics, math and other sciences. Printed on heavy cardstock making nice binder dividers.
Bil lD

[Mel Fulks]

If I knew how to use it ...I would certainly want one! My engineer son gets tired of me needing help with higher math.
Mike, that is most kind to offer that and I'm pretty sure they will become dog-eared treasures.

[David L Morse]

Nice paper Mike. That's a good intro to what's ahead for an engineering student. My favorite: "Get used to radians. You’re going to be using them a lot."

[Rob Luter]

When my Son in Law finished his Masters in Mathematics from the Colorado School of Mines, he showed me his thesis. It was like a small phone book filled with equations. I've been an Engineer for almost 40 years and have done all kinds of math. None of this made the least bit of sense to me. Smart kid. Works on cool stuff now.

[Jim Falsetti]

Mike - thanks for sharing, you obviously put a lot of effort and energy into the paper! Your introductory section, with some life lessons learned, might be the most important part, especially in todays social media driven world.

In the preface, you ask if readers would respond. What feedback have you gotten over the years?

Jim

[David Sochar]

When a high school student, I struggled with math. In one of my numerous meetings with the math instructor, he explained how I would need math in the future.
I explained, with all the assurance a 17 yr old can gather, that I would never be so foolish to pursue a career that required any math ability.

So now I use math everyday, and I enjoy my ability, my facility with it. Granted, I am not working with quadratic functions.

[Frederick Skelly]

Thats quite a paper Mike! Thanks for sharing it. My neighbor kid starts college this Fall (somehow) and I will give him a copy. I agree with your preface - it's always about hard work.

I might have missed it, but have you copyrighted this in some fashion? It's your intellectual property.

Best regards,
Fred

[roger wiegand]

Extraordinarily well written paper, but in a nutshell, why I became a biologist. Even though I got through differential equations in college (and pchem, but that's a different story) that was about the end of my mathematical career; tried algebra (the real thing, not the high school stuff) and ran screaming. I managed a 40 year happy career as a productive working scientist without ever again finding a need to use calculus.

What I wish someone had told me while I was still young enough to absorb it, was how important statistics were going to turn out to be in both my working and personal life. The personal side is obvious, one has only to read the paper these days to see that virtually no one, from the leader of the free world to the morons posting on Facebook has a clue about statistical reasoning, how to interpret the idea of a 30% chance of rain, or why a couple of anecdotes about how Aunt Mary didn't get Covid after standing on one foot while gargling Drano don't constitute data. Needless to say they wouldn't know a Bonferroni correction (https://en.wikipedia.org/wiki/Bonferroni_correction) if it bit them on the big toe. (this lack of understanding, shared by too many scientists, explains a vey large fraction of the hyperbolic headlines that quickly turn out to be wrong).

At work, analyzing genomes, gene expression data, and proteomics on the one hand, deciding where to set cutoff limits in drug screening (and how many replicate assays of what sort might cause you to believe you had a real result), and (pretty relevant these days) whether a drug you were looking at in the clinic was doing anything (good or bad) turned out to be all statistics all the time.

All that said, my friend the mathematician (a leader in algebraic K-theory, whatever that is) would look at any of this and scoff 'that's not math, that's just arithmetic'.

[Bill Carey]

Thanks Mike - well done and very generous to share it. My 12 year old grandson - the one who helps me with programing problems - is reading it. Says it's pretty cool.

[Steve Demuth]

Very nicely written paper.

As someone who originally trained in physics and mathematics, I have always been intrigued by the different kinds of rigor and motivation mathematicians, physicists and engineers each put into the same basic mathematics. When my son became a professional statistician in bio-medicine, I added statistics as a fourth domain of intrigue. The explanatory power of differential and integral calculus in each of these domains is one of the great monuments to human ingenuity, and never fails to settle my mind into a better place ... except when I get to something I want to understand but which is simply too hard for me to penetrate.

[michael langman]

Mike, If only you had been my high school mat teacher.

This paper is such an important aspect of learning mathematics, and should be mandatory course of study in my opinion.

Thankyou for writing it, and posting it here. I will be reading it over and over, as it really is what I needed a long time ago.

[Bob Turkovich]

Did I miss something or are partial diffy Q's coming in the 2nd edition?:p

Great stuff, Mike!

[glenn bradley]

Very cool Mike. The thought and effort you put in to author an assistance paper for your friend's son is to be commended. Much of it is over my head but, the parts I could hang onto are well written and systematically presented. My barometer is generally if something I am unfamiliar with can be presented so that I can follow it, the author must be doing an exemplary job. :D Kudos to you for your thoughtful effort. It is just another example of you sharing your talents; something that I have benefited from myself.

P.s. For others, I am referring to Mike's dovetailing class which I have the pleasure of attending one on one. This really helped me with my dovetail technique.

[Mike Henderson]

Mike - thanks for sharing, you obviously put a lot of effort and energy into the paper! Your introductory section, with some life lessons learned, might be the most important part, especially in todays social media driven world.

In the preface, you ask if readers would respond. What feedback have you gotten over the years?

Jim

Well, I never tried to publicize the paper. I just had it on my web site so it was not something that people would easily find. I also wrote a paper on how logarithms were calculated (http://www.michael-henderson.us/Papers/Logarithms.pdf) years ago and I did get some responses from that one. Surprised me.

And to address Frederick's question about copyrighting it - I wrote it to help young people. If someone wants to steal it and still uses it to help young people, I'm okay with that. But next time I edit it, I'll put a copyright line on it.

A story about people using papers like this: I wrote a tutorial paper on SONET (http://www.michael-henderson.us/Papers/SONET-SDH.pdf) (Synchronous Optical Networks) that was well received. Some years later I found out that the paper had been used as a chapter in a book (https://www.amazon.com/gp/product/1580535283/ref=cm_cr_thx_view). Trouble is, the author had never contacted me to ask if he could use it. I suppose he contacted the company I had worked for (I had left them) and I assume they gave him permission but no one ever contacted me. I don't remember how I discovered the usage but I contacted the publisher and told them that I had never given permission for its use. They did send me a free copy of the book. I didn't mind and would have given permission but I didn't think it was very polite.

And to give a plug of another paper that I wrote (which I think is pretty good) - "DS1/DS3 and E1/E3 Framing and Multiplexing (http://www.michael-henderson.us/Papers/Framing(rev%20b).pdf)". The material is not used all that much today but back in the day it was the major communications technology.

Mike

[For those of you who mentioned statistics: I agree. I didn't take statistics until I was working on my masters but it was really useful throughout my career. I didn't have to calculate statistics but when presented with statistical data I knew how to interpret it.]

[Jim Allen]

Wow, well done. So where did you teach?

[Mike Henderson]

Wow, well done. So where did you teach?

I don't teach mathematics. I'm just a retired engineer. When I retired, I contacted all the high schools in my area (don't remember exactly how many, maybe four). I volunteered to tutor students in advanced mathematics, especially those who planned to go to college for engineering, mathematics, physics, etc. Not one would accept me.

I think what they wanted was someone to tutor the students who were not doing well - they just weren't interested in having someone work with their advanced students. I even volunteered to come in and do a presentation to the mathematics club on something like complex numbers. Again, no interest.

I met one of the mathematics teachers for coffee so that he could see that I wasn't some oddball but I still couldn't get any interest (maybe he did think I was an oddball:)).

I just wrote that paper for the son of some friends and while I was writing it, I decided to target it to a general audience - not just him.

I did a lot of writing in my job so it was fairly easy to write this paper. I had to research some of the mathematics to make sure I was doing it correctly. It's been a long time since I was in college.

Mike

[Jon Nuckles]

And to address Frederick's question about copyrighting it - I wrote it to help young people. If someone wants to steal it and still uses it to help young people, I'm okay with that. But next time I edit it, I'll put a copyright line on it.



Mike, Depending on how many years ago it was that you first "published" this paper, you likely don't need a copyright notice to protect it. My recollection of copyright law is sketchy and faded, so here is what Wikipedia says:

For works first published on or after March 1, 1989, use of the copyright notice is optional. Before March 1, 1989, the use of the notice was mandatory on all published works. Omitting the notice on any work first published from January 1, 1978, to February 28, 1989, could have resulted in the loss of copyright protection if corrective steps were not taken within a certain amount of time. Works published before January 1, 1978, are governed by the 1909 Copyright Act. Under that law, if a work was published under the copyright owner's authority without a proper notice of copyright, all copyright protection for that work was permanently lost in the United States.

Never hurts to include it though!

[John K Jordan]

Your write-up looks interesting and useful. I'll save it for future use. Too bad the teachers were not interested. (Hey, expand that into a book in your spare time!)

I'm no mathematician but when taking college courses I found fellow students often confused about basic math such as trig, even simple derivatives. Meeting at the library for study I found that just showing them how I liked to visualize things with graphs and diagrams helped a lot.

JKJ

[Larry Frank]

Calculus and differential equations were like opening a new world of understanding for me. It tied things together in such an elegant way. People often ask if I speak a foreign language and I reply I speak fluent Calculus and Thermodynamics. I get weird looks. I have very little facility for language but have been proficient in these. The majority of people have no idea the power and usefulness of Calculus.

Since I retired, the use of these is pretty much gone.

[David Powell]

Mike, reading your paper caused me to recall my college days. I enjoyed all my math classes until I got to dify q. I was never so happy when the course was behind me. Because of my results in the prerequisite math classes, I was put into an honors differential equations class with a bunch of math majors. My contribution was to bring the curve down for the class. Never had any reason to apply the fundamentals during my career as a civil engineer.

[Larry Frank]

I did fine with Calculus until the last class I took which was Vector Integral Calculus with triple integrals and grafient, divergence and curl.

For the most part, I have only used Calculus in college where it was used for derivation of formula and ideas.

[Kevin Jenness]

Very generous of you, Mike.

David Sochar's response dovetails with that of my older son. He was a terrible student in high school, didn't graduate in fact. He didn't like to fail so he didn't try to do things that didn't come easily so he spent his time reading and playing video games. He was especially bad in math, even basic arithmetic. When he got interested in timber framing he taught himself enough so that now he runs a successful business, designs his frames in Sketchup, develops accurate cutlists and oversees the bookkeeping. He'll probably never use a differential equation or calculus, but I am sure that if he saw the need he would learn it.