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I'd wholly agree, but emphasize that it's much more important in intro and lower-division texts than in higher levels. And it goes beyond STEM (Yes, Virginia, there is education outside of STEM!) - history, in particular, would be far better taught in a casual, welcoming style.
I believe that one reason why I started thinking about this to begin with was thanks to the character Ian Malcolm in "Jurassic Park";
I liked how he had that extremely casual and mildly sarcastic style that was so far off from the typical idea of what a mathematician was expected to be like, and he acted like some sort of playboy who was on vacation or something without ever really making a fool of himself or anything;
and then I thought to myself "why not?".
To be fair though, I do remember a few teachers from my previous classes who had a similar casual attitude and got away with it;
my teacher in Physics 2 would do random things like showing performances of heavy metal bands while we were waiting for him to begin the lecture or referencing Star Wars whenever he got the chance.
I really like that kind of laidback style, and he was actually really good at explaining things as well and always was serious about his job, so he had the perfect balance between humour and seriousness - just serious enough to come off as professional and respectable, and then the rest was humour and improvisation.
And the fact that he almost always smiled and looked pleased with things was of course a big plus.
By the time you get to calculus, the books are impossible to understand. I'm a visual learner (go Mechanical Universe!), not a linear learner and those books are all linear. My favorite math books were the ones from Lial and Miller - large print, colored ink, lots and lots of examples. User friendly. After flunking out in math all through high school, these books and a good teacher in college were finally able to explain algebra to me so I could understand it. Even pre-calculus was understandable.
Then I got to calculus and it was - WTH? And their examples: "We select 1..." Me: "Well, why 1? Why not 2 or 6 or..." There's virtually no teaching in those books at all. So many times I understood what it was they were trying to teach, but no clue at all as to how to do the problems.
They need to drop all the 'therefores' and 'such thats', at least at the beginning. And for all that's holy, please drop f(x) and just use something else (y worked just fine) until we really get to derivatives. Maybe it was just me, but f(x) drove me bats**t crazy for as long as I saw it (still does).
Even beginning physics books were easier to understand than calculus. I didn't understand how derivatives worked and how to use them until I saw them explained in a physics book. But calculus books? NOT user friendly - no way, Jose!
Quote:
Originally Posted by Quietude
Not much point in using a folksy approach when the training has to mold the student into being able to write that dry, narrow style for the rest of his career.
Which begs the question: What's so great about that dry, narrow style anyway? Maybe it's high time we did away with it.
Last edited by rodentraiser; 12-29-2018 at 03:44 PM..
By the time you get to calculus, the books are impossible to understand. I'm a visual learner (go Mechanical Universe!), not a linear learner and those books are all linear. My favorite math books were the ones from Lial and Miller - large print, colored ink, lots and lots of examples. User friendly. After flunking out in math all through high school, these books and a good teacher in college were finally able to explain algebra to me so I could understand it. Even pre-calculus was understandable.
Then I got to calculus and it was - WTH? And their examples: "We select 1..." Me: "Well, why 1? Why not 2 or 6 or..." There's virtually no teaching in those books at all. So many times I understood what it was they were trying to teach, but no clue at all as to how to do the problems.
They need to drop all the 'therefores' and 'such thats', at least at the beginning. And for all that's holy, please drop f(x) and just use something else (y worked just fine) until we really get to derivatives. Maybe it was just me, but f(x) drove me bats**t crazy for as long as I saw it (still does).
Even beginning physics books were easier to understand than calculus. I didn't understand how derivatives worked and how to use them until I saw them explained in a physics book. But calculus books? NOT user friendly - no way, Jose!
One guy that helped me out a lot with some of my Calculus-related courses was Professor Leonard on YouTube.
My Engineering Physics program in Sweden doesn't really have "Calculus 1/2/3" quite like American universities, although we do have Calculus courses that follow pretty much the same structure.
One difference though was that I started studying the Calculus 2 topic "Sequences & Series" during a course that we call Linear Analysis (which is mostly concerned with Laplace Transforms, Fourier Series and Fourier Transforms - I guess Sequences & Series was brought up partly because it was a natural foundation for Fourier Series), and I found Professor Leonard's Calculus 2 lectures extremely useful back then.
I suggest that you check out his videos if you are still motivated to learn Calculus.
You can instantly tell that he is an excellent teacher, right from the absolute beginning of his first real Calculus 1 lecture.
What you’re noticing is real. The issue doesn’t really have to do with scientists being stodgy and boring. In my experience, professional scientists and mathematicians are less formal and serious than average. People in academia have often never worked in the corporate world where everyone has to be professional all the time.
It has more to do with making any subject interesting and relatable taking a lot of work. For introductory texts, the audience is large enough and the pool of potential authors big enough that you have to have a good textbook to get noticed. As the subjects become more focused, the pool of readers and competent authors gets smaller. I’m sure there are some insightful and plainly written graduate textbooks, but it’s no longer a requirement. Eventually there stop even being textbooks and you have to rely on monographs and reviews to learn stuff (as well as journal articles).
One guy that helped me out a lot with some of my Calculus-related courses was Professor Leonard on YouTube.
My Engineering Physics program in Sweden doesn't really have "Calculus 1/2/3" quite like American universities, although we do have Calculus courses that follow pretty much the same structure.
One difference though was that I started studying the Calculus 2 topic "Sequences & Series" during a course that we call Linear Analysis (which is mostly concerned with Laplace Transforms, Fourier Series and Fourier Transforms - I guess Sequences & Series was brought up partly because it was a natural foundation for Fourier Series), and I found Professor Leonard's Calculus 2 lectures extremely useful back then.
I suggest that you check out his videos if you are still motivated to learn Calculus.
You can instantly tell that he is an excellent teacher, right from the absolute beginning of his first real Calculus 1 lecture.
Thank you, I will check him out. Unfortunately, YouTube wasn't around in the 80s when I needed it. LOL I had hoped eventually to use the math to get a degree to get a better job. At this point, I think working is a long gone idea. I couldn't afford to finish a degree if I lived to be 100. On the other hand, maybe I'll take out some college loans when I'm 95...
Edited to add: OK, I'm not even 10 minutes in and this guy is making more sense than any other calculus teacher I ever had. I can already see the applications for finding a tangent and the area in the use of solving real world word problems.
I'm refilling my water glass and getting an apple so I can get myself comfortable and watch the whole video. Thank you again!
Last edited by rodentraiser; 12-29-2018 at 09:02 PM..
Partially I think it's the higher up the textbooks get, the less the editors can question the wording. The author understands what they think they said, and no one else knows enough to say it didn't make sense. Haliday and Resnick was dry but ok for entry level. But by the time I got to Symon & Goldstein for Mechanics and Lorrain & Carson for Electrodynamics it seemed like gobbledygook.
My biggest pet peeve was the " … the derivation is so simple it will be left as an exercise to the student." Which usually meant it was some unholy mess that no one understood and the author couldn't explain so they left it out.
Quote:
Originally Posted by rodentraiser
... And for all that's holy, please drop f(x) and just use something else (y worked just fine) until we really get to derivatives. Maybe it was just me, but f(x) drove me bats**t crazy for as long as I saw it (still does). ...
Been doing this for nearly 40 years now and f(x) still drives me bats*&t crazy.
Been doing this for nearly 40 years now and f(x) still drives me bats*&t crazy.
Maybe we can start a revolution.
I think some of the other things that we can get rid of in math are things that I don't see a connection to. If I'm wrong, correct me.
Learning sets and subsets in algebra. And always starting off with vectors in physics. I never did get that or understand what vectors had to do with physics until much later in the course. By the time you hit that point, vectors have been forgotten.
It's just there's some things that I think should be shelved in mathematics now. We don't teach students to find the square root of a number anymore - we use a calculator. While it might be important to know how to find the square root of a number without a calculator, the time spent doing that eats into the time spent working on other, more advanced problems. It's a time waster.
How many people would be driving today if we insisted everyone put together the engines to their cars? If you're into telescope making, grinding your own mirror is very satisfying. But if you want to look at the night sky and start learning about the stars and the universe, it might be better and more motivating to buy a commercially made scope and start that way instead. If a person is interested enough, he can always go back and learn how to do the more arcane mathematics if he wanted.
Gilbert Stang is a mathematician from MIT who has written textbooks on calculus & applied math- very pleasant writing style-- reads like he's having a conversation with you& often uses footnotes like the vampire reference.
Higher level textbooks are not really meant to teach, but to serve as a compendium of the current state of knowledge in the field. Because the writing, editing & printing takes so long, new additions are usually out of date by the time they reach the bookstore shelves-- except for maybe Anatomy texts. When was the last time they added a new part to the human body?
Thank you, I will check him out. Unfortunately, YouTube wasn't around in the 80s when I needed it. LOL I had hoped eventually to use the math to get a degree to get a better job. At this point, I think working is a long gone idea. I couldn't afford to finish a degree if I lived to be 100. On the other hand, maybe I'll take out some college loans when I'm 95...
Edited to add: OK, I'm not even 10 minutes in and this guy is making more sense than any other calculus teacher I ever had. I can already see the applications for finding a tangent and the area in the use of solving real world word problems.
I'm refilling my water glass and getting an apple so I can get myself comfortable and watch the whole video. Thank you again!
No problem.
He is a great teacher indeed, I remember stumbling upon his videos when I was looking for some explanations on the "Sequences & Series" part for my Linear Analysis course about two years ago, and I was blown away and thought to myself "who is this guy? I have to watch this", and I have been a fan ever since then.
Several of his teaching skills are relevant for any type of scientific course literature at any level;
for example, he always gives a bunch of example problems fairly early on so that the learners don't get bored, and those example problems are also very simple - this makes it easier for students to make use of small chunks of knowledge at a time, and also focus on the stuff that's truly relevant without getting distracted by a lot of variables and overly tedious algebraic expressions.
Of course, he does derive proofs as well, and proofs are important for lots of reasons (such as justification, intuition etc, and they also allow you to derive the formulas without needing to memorise them), but in my opinion those proofs should always be explained in a way so that there is some form of real-life context;
for example, one good way to justify the "foil method" (which says that [a + b]⋅[c + d] = a⋅c + a⋅d + b⋅c + b⋅d) is to think of it as an area for a rectangle, so that the expression (a + b)⋅(c + d) is a rectangle that has the length a + b and the height c + d;
if you then split up that rectangle into 4 smaller rectangles then it's very easy to see that it does indeed have the area a⋅c + a⋅d + b⋅c + b⋅d, since that's the combined areas of the smaller rectangles.
Some of my math teachers in junior high school skipped those kinds of justifications and just went like "here it is, use it", so that I had to ask for a more detailed explanation.
I'm a little late back here, but one of my favorite teachers never really did teach us anything related to the foil method. He basically said if you look at the problem, you can intuitively figure out what the numbers are. Now I have no idea how the rest of the class handled that, but that was right up my alley. And I never had a problem factoring those little devils out. In fact, in 2009, after not having had any math classes for 20 years, I went in to take the math assessment at a local community college and problem looked like this: (x + 1)/(x² - 1). I panicked and then just like that, I realized I could factor the bottom, just as if I'd done the problem yesterday. Let's hear it for good teachers.
I can see how you're doing it, though, too. The way you describe it is something I can visualize. It's good to understand the basics. I do remember when I was first starting algebra, I had no idea what the teacher was talking about when he started out with telling us about slopes. So I made up my own was of getting the slope. Change an equation into this format: (example) 2x + 3y = 0. Put the y on top of the x and change the sign. Slope of this equation is -3/2. That was great until the teacher actually asked me how I got that number. LOL And that's when class really started.
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