I suppose the TL;DR version here would be “I don’t really know”, but I’m going to try to outline some of my thinking on this based on some of my experiences teaching classes at a university and my musings over the years, which are far less ‘evidence-based’ you could say.
I think there’s something to be said for intellectual maturity as well.
In my callow youth, I simply didn’t have what it took to learn math(s) like I thought I wanted to, and it’s easy to blame the appallingly-badly taught fundamentals that I endured. But that, I think, was only part of the problem.
Now nearing retirement, I enjoy, and understand quite well, the mathematics and physics of signal modulation and demodulation and radio transmission, which I know I would have struggled with years ago.
And, even more surprisingly, I get a lot out of Michael Penn’s YouTube channel, understanding (sufficiently to see where he’s going, if not how he’s going to get there) many (but not all) of the really interesting proofs he so expertly and adroitly presents.
I don’t know to what to attribute this pleasant turn of events other than maybe I finally learned, far too late, how to think like an adult.
I was hoping your article might answer my burning question of why the answer to multiples of 9 add up to 9 (or multiples of 9 for the larger numbers) e.g. 6x9=54 - 5+4=9 or 9x235=2115 - 2+1+1+5=9 or 5938x9=53442 - 5+3+4+4+2=18. What is so special about 9? I'm sure there are greater mysteries in mathematics, but my skills don't go much beyond this level.
QUOTE; Question: Is it always true that the sum of the digits in the product of 9 and any other number adds up to 9. Like 9x235=2115 > 2+1+1+5=9. And if so why?
Answer: Yes, it is always true (except for the product of 9 and 0) that the sum of the digits of a product of 9 and any other number, when repeatedly added together until a single digit remains, always adds up to 9. This final single digit is also known as the number's digital root. ...
This property (also known as casting out nines) is a general rule in any number base: multiples of the number one less than the base (e.g., 9 in base 10, 4 in base 5) will have digits that sum to that number (or a multiple of it). UNQUOTE
About note 5: you lay the foundation for developing pattern recognition ability by learning proper spelling, basic grammar and handwriting, simultaneously. By doing this, you notice that some words share -fixes, and from the you figure out that the -fix adds a meaning, and from that universal added meaning of the -fix you can and probably will continue to develop an instinct for understanding your native tongue, and that's pattern recognition at the basic level.
And then you go into semiotics (like me) and really get to sharpen that pattern recognition to ridiculous levels, eventually becoming impossible to hold a normal conversation with because you'll start saying thing like "Dollar? Ah yes, from Old German and Old Nordic 'thaler' which itself is derived from a PIE root-word which..." which makes your company go: "So howabout that weather we've been having lately?"
I'm sure you can relate to that.
To quote the ever-so-woke Harry Potter-lady: "Oh dear, maths". One thing, apart from my own creative laziness when at school, that made it a bit difficult was this obsession with using real-life examples. I did better when it was just numbers and symbols, not Bob and Alice making a fence.
Because growing up part-time in the countryside* I knew how you put up a fence and that practical thinking always got in the way of "visualising" the actual equation or whatever it was.
Same problem when we learned mapping, compasses and pathfinding in 4th grade (age 10); I already knew how to find my way, learned it practical from grandpa when out fishing.
Point of this is, sometimes practical examples make it more difficult. Stick with pure numbers until the numbers stick, then introduce practical applications. Will never forget the poor teacher who tried to teach us equations in 2nd: "Look, if I have a bag of strawberries then I have X strawberries since I don't know how many are in the bag, see? And if I take five out, then I have X-5, see?"
She didn't respond well to "Pour them out and count them. But that's not how you do it, you weigh fruit and berries." (Got the same sour reaction in biology when we were supposed to dissect a fish. My muscle-emory took over and I presented a nicely cleaned and filetted** herring. Angry note to Mom followed, because I was being cheeky.)
There's a third way to get students to put in the work: bribery.
"If you do this and do it right you will be allowed to apply for XYZ" As in stipends, clubs, and so on.
But we are over-educating ourselves to death now. Educations needs this: more basics at the basic levels, applied prep at the prep-levels, and then competing not with grades but in blind and anonymous exams for being allowed entyr into higher education. And teachers ought to do semi-annual exams in their subjects too, to make sure they are up to specs - plus that would introduce an element of firness for students.
If I haven't recommended Daisy Christodoulou's "Seven Myths About Education" and Lynne Truss' "Eats shoots and leaves" I'm doing it now. Both are oh-so-very English takes on education and punctuation, equally fun and entertaining as they are dead serious and inquisitive.
*"Define countryside" reply: "Killing Piers Morgan". Think it was on an episode of QI when Fry was hosting it (the obvious joke being hearing "countryside" as "cuntycide" - there's that pattern-recognition and basic knoweldge at work, right there).
**Filetted and fillet are good words for introducing the weirdness of English spelling and pronounciation to foreign learners. If you just look at the letters and how they are normally pronounced, you don't get the correct sounds. Compare it to Swedish where we spell it "filé" - exact same pronounciation. Verb: filéa/to fillet.
Aaaaand....I reckon there are people with maths brains, and all the rest. If you have a maths brain, whatever that is, then Couchy and Fourier will seem natural to you. There will be more or less effort, but after a while you'll get the picture, like riding a bike. If you don't have a maths brain, then you could spend your entire life looking at those integrals and never develop the faintest idea of what they mean. You'll never ride that bike, ever.
I may be wrong, but I have a university degree in pure mathematics, and a lifetime of observation to back up my hypothesis.
Its an interesting pedagogical argument if nothing else.
Doc, I can understand your Couchy infatuation, but for me, the Fourier transform is the essence of mathematical beauty.
Why? Because I'm not a scientist, I'm a fat fingered engineer who builds things. A glorified tradesman, really, and the FFT is a sharp tradesmans tool.
I spent almost all of my career working in rader system design and development. In my world, the Fast Fourier Transform allows us to convert discrete radar time series data samples to target velocity, track the target, engage it with missiles, and then ascertain if the engagement was successful based on the size and velocity of the particles emitted from the intercept explosion. If necessary, we fire again....all this in the space of a few milliseconds.
The practical application of the most complex mathematics (see what I did there) to build mighty machines that defend my people and my country, while hoping rhose machines are never used in anger.
Anyway...thank you...you inspired me to wax lyrical there, about something that was a large part of my life, but that I couldnt even begin to discuss with anybody outside of work, due to odd looks, rolled eyes etc.
We should get together and discuss group theory over a pint sometime. :-)
> "it’s symbolic of our struggle against symbols."
Arguably -- Loren Eiseley argues in his The Immense Journey that -- symbolic communication goes back to, or undergirds, or is the harbinger of the dawn of consciousness. Several million years ago, though that is maybe somewhat anthropomorphic as "lower" animals have had symbolic communication for much longer.
But, relative to your "motivation" thesis, my, fairly well thumbed, "Matrices and Linear Algebra" [Schneider & Barker, 2nd edition, 1972] asserts, "We have found in teaching this course at the University of Wisconsin [USA ...] that the promise that the subject we are teaching can be applied to differential equations will motivate some students strongly." [viii]
I think there’s something to be said for intellectual maturity as well.
In my callow youth, I simply didn’t have what it took to learn math(s) like I thought I wanted to, and it’s easy to blame the appallingly-badly taught fundamentals that I endured. But that, I think, was only part of the problem.
Now nearing retirement, I enjoy, and understand quite well, the mathematics and physics of signal modulation and demodulation and radio transmission, which I know I would have struggled with years ago.
And, even more surprisingly, I get a lot out of Michael Penn’s YouTube channel, understanding (sufficiently to see where he’s going, if not how he’s going to get there) many (but not all) of the really interesting proofs he so expertly and adroitly presents.
I don’t know to what to attribute this pleasant turn of events other than maybe I finally learned, far too late, how to think like an adult.
I was hoping your article might answer my burning question of why the answer to multiples of 9 add up to 9 (or multiples of 9 for the larger numbers) e.g. 6x9=54 - 5+4=9 or 9x235=2115 - 2+1+1+5=9 or 5938x9=53442 - 5+3+4+4+2=18. What is so special about 9? I'm sure there are greater mysteries in mathematics, but my skills don't go much beyond this level.
Here's the answer from Gemini, Google's AI:
QUOTE; Question: Is it always true that the sum of the digits in the product of 9 and any other number adds up to 9. Like 9x235=2115 > 2+1+1+5=9. And if so why?
Answer: Yes, it is always true (except for the product of 9 and 0) that the sum of the digits of a product of 9 and any other number, when repeatedly added together until a single digit remains, always adds up to 9. This final single digit is also known as the number's digital root. ...
This property (also known as casting out nines) is a general rule in any number base: multiples of the number one less than the base (e.g., 9 in base 10, 4 in base 5) will have digits that sum to that number (or a multiple of it). UNQUOTE
https://www.google.com/search?q=Is+it+always+true+that+the+sum+of+the+digits+in+the+product+of+9+and+any+other+number+adds+up+to+9.+Like+9x235%3D2115+%3E+2%2B1%2B1%2B5%3D9.+And+if+so+why%3F&sourceid=chrome&ie=UTF-8&udm=50&aep=48&cud=0&qsubts=1763696028807&mstk=AUtExfDfJ29huz3V74-T8Vvxw7jrI94T_HgjwZFVHzgXv5uWH4le68r55nH5k-gvd5ooRCkkWA7dWfWNUAPfv3YPyGbUB8ztiKc_C-4LGIPVnfI7fYdWkdMmQW6rHdbtRZyBjkW3ZdiyC811M7uQwRzGsCYYE5afschqMC5UUCzqJwfTCuOKur4PHTGmEQ-GeOrC3mVZlIyNQNqcsUWKUgxO3LPhSIAHRFxbkpnSgH2lh8sIK1unsovRvWzQji2e3zd___sOkt6GJEzQ7ey8xsASALhYsyMQU2i1s88&csuir=1&mtid=ON4faYfNL8eK0PEP-YCV2QI
Meanwhile here in the U.S. I read this in an article about the UC San Diego.
When UCSD's math department assessed Fall 2023 students in remedial courses, the results were sobering: 25% couldn't solve "7 + 2 = ___ + 6,"
Too much to comment on! Argghh! Ahem.
About note 5: you lay the foundation for developing pattern recognition ability by learning proper spelling, basic grammar and handwriting, simultaneously. By doing this, you notice that some words share -fixes, and from the you figure out that the -fix adds a meaning, and from that universal added meaning of the -fix you can and probably will continue to develop an instinct for understanding your native tongue, and that's pattern recognition at the basic level.
And then you go into semiotics (like me) and really get to sharpen that pattern recognition to ridiculous levels, eventually becoming impossible to hold a normal conversation with because you'll start saying thing like "Dollar? Ah yes, from Old German and Old Nordic 'thaler' which itself is derived from a PIE root-word which..." which makes your company go: "So howabout that weather we've been having lately?"
I'm sure you can relate to that.
To quote the ever-so-woke Harry Potter-lady: "Oh dear, maths". One thing, apart from my own creative laziness when at school, that made it a bit difficult was this obsession with using real-life examples. I did better when it was just numbers and symbols, not Bob and Alice making a fence.
Because growing up part-time in the countryside* I knew how you put up a fence and that practical thinking always got in the way of "visualising" the actual equation or whatever it was.
Same problem when we learned mapping, compasses and pathfinding in 4th grade (age 10); I already knew how to find my way, learned it practical from grandpa when out fishing.
Point of this is, sometimes practical examples make it more difficult. Stick with pure numbers until the numbers stick, then introduce practical applications. Will never forget the poor teacher who tried to teach us equations in 2nd: "Look, if I have a bag of strawberries then I have X strawberries since I don't know how many are in the bag, see? And if I take five out, then I have X-5, see?"
She didn't respond well to "Pour them out and count them. But that's not how you do it, you weigh fruit and berries." (Got the same sour reaction in biology when we were supposed to dissect a fish. My muscle-emory took over and I presented a nicely cleaned and filetted** herring. Angry note to Mom followed, because I was being cheeky.)
There's a third way to get students to put in the work: bribery.
"If you do this and do it right you will be allowed to apply for XYZ" As in stipends, clubs, and so on.
But we are over-educating ourselves to death now. Educations needs this: more basics at the basic levels, applied prep at the prep-levels, and then competing not with grades but in blind and anonymous exams for being allowed entyr into higher education. And teachers ought to do semi-annual exams in their subjects too, to make sure they are up to specs - plus that would introduce an element of firness for students.
If I haven't recommended Daisy Christodoulou's "Seven Myths About Education" and Lynne Truss' "Eats shoots and leaves" I'm doing it now. Both are oh-so-very English takes on education and punctuation, equally fun and entertaining as they are dead serious and inquisitive.
*"Define countryside" reply: "Killing Piers Morgan". Think it was on an episode of QI when Fry was hosting it (the obvious joke being hearing "countryside" as "cuntycide" - there's that pattern-recognition and basic knoweldge at work, right there).
**Filetted and fillet are good words for introducing the weirdness of English spelling and pronounciation to foreign learners. If you just look at the letters and how they are normally pronounced, you don't get the correct sounds. Compare it to Swedish where we spell it "filé" - exact same pronounciation. Verb: filéa/to fillet.
PS. This may be related to math-skills dropping:
https://www.psychologytoday.com/us/blog/homo-idioticus/202506/the-rise-of-homo-idioticus-are-we-getting-more-stupid
Aaaaand....I reckon there are people with maths brains, and all the rest. If you have a maths brain, whatever that is, then Couchy and Fourier will seem natural to you. There will be more or less effort, but after a while you'll get the picture, like riding a bike. If you don't have a maths brain, then you could spend your entire life looking at those integrals and never develop the faintest idea of what they mean. You'll never ride that bike, ever.
I may be wrong, but I have a university degree in pure mathematics, and a lifetime of observation to back up my hypothesis.
Its an interesting pedagogical argument if nothing else.
Doc, I can understand your Couchy infatuation, but for me, the Fourier transform is the essence of mathematical beauty.
Why? Because I'm not a scientist, I'm a fat fingered engineer who builds things. A glorified tradesman, really, and the FFT is a sharp tradesmans tool.
I spent almost all of my career working in rader system design and development. In my world, the Fast Fourier Transform allows us to convert discrete radar time series data samples to target velocity, track the target, engage it with missiles, and then ascertain if the engagement was successful based on the size and velocity of the particles emitted from the intercept explosion. If necessary, we fire again....all this in the space of a few milliseconds.
The practical application of the most complex mathematics (see what I did there) to build mighty machines that defend my people and my country, while hoping rhose machines are never used in anger.
Anyway...thank you...you inspired me to wax lyrical there, about something that was a large part of my life, but that I couldnt even begin to discuss with anybody outside of work, due to odd looks, rolled eyes etc.
We should get together and discuss group theory over a pint sometime. :-)
Well, I still don’t like math, but you did make your talk about it interesting!
> "it’s symbolic of our struggle against symbols."
Arguably -- Loren Eiseley argues in his The Immense Journey that -- symbolic communication goes back to, or undergirds, or is the harbinger of the dawn of consciousness. Several million years ago, though that is maybe somewhat anthropomorphic as "lower" animals have had symbolic communication for much longer.
But, relative to your "motivation" thesis, my, fairly well thumbed, "Matrices and Linear Algebra" [Schneider & Barker, 2nd edition, 1972] asserts, "We have found in teaching this course at the University of Wisconsin [USA ...] that the promise that the subject we are teaching can be applied to differential equations will motivate some students strongly." [viii]