Mathematical thinking

[Kid Anymore]

I can remember a lecturer asking the room why they thought mathematics was a difficult subject.

Since then I've considered why we end up believing this, in general. Most people would say it's hard to learn. But why is it?

I also recall more or less deliberate attempts to make it hard to understand, or to follow a line of thinking. Why do that?
Unless mathematical skills--the ability to abstract, to think geometrically etc--are meant to be special, only smart people get it.

Anyway, I'm actually interested in whether anyone else here has had ideas about mathematical stuff that they later find out is in use, it's an actual function or something. Anyone?

Like I once had this idea that you could make all the points on a line equivalent by coloring them all the same.
This is abstracting sameness to be an equivalence, a relation between points. When they're all the same color and the line is finite, say a circle, you can contract it to a point, or quotient the colored space. Cool. it is a thing and I independently thought of it before I found that out.

So hi I'm the new Kid, and I want to explore an anthropological aspect of a subject a lot of people learn to hate.

heh heh

 

[billvon]

I can remember a lecturer asking the room why they thought mathematics was a difficult subject.

Since then I've considered why we end up believing this, in general. Most people would say it's hard to learn. But why is it?

Because it's not something we evolved to understand, like spatial relationships. Thus it has to be learned, and that takes effort.

 

[Pinball1970]

Since then I've considered why we end up believing this, in general. Most people would say it's hard to learn. But why is it?

The purpose of STEM subjects is to prepare a student to exist and function in the modern world.
We live in a civilised, numerate, literate technical world.
Beyond that those subjects are methods to explore the world and there will be kids who are able to do that, at different levels.

 

[DaveC426913]

I also recall more or less deliberate attempts to make it hard to understand, or to follow a line of thinking. Why do that?

By whom? When? Why do that indeed.

Unless mathematical skills--the ability to abstract, to think geometrically etc--are meant to be special, only smart people get it.

Only if your above suspicion is true.

Like I once had this idea that you could make all the points on a line equivalent by coloring them all the same.
This is abstracting sameness to be an equivalence, a relation between points. When they're all the same color and the line is finite, say a circle, you can contract it to a point, or quotient the colored space. Cool. it is a thing and I independently thought of it before I found that out.

I can see why some people hate it. It's intimidating.

I have a naturally mathematical mind. Phone numbers and simple math solutions appear in my vision unbidden. I am certain that is improved and honed by affinity and practice. I don't have to actually calculate some solutions each time, if I've done them enough - you know, like 25 squared just pops up 625 - that kind of thing.

But the stuff they do over on Physics Forums - walls of calculations - is beyond my formal education, which ended with high school. So it's intimidating to me, even if the concepts are not.

My sister, on the other hand, despite being a biologist/scientist, is so inept at math she can't multiply two one-digit numbers in her head. She is proud of this, and is only too pleased to demonstrate it by sticking her fingers in her ears. Like me, her proficiency is in a positive feedback loop - only the opposite direction: the more she avoids it the more it eludes her.

 

[Kid Anymore]

Well, the almost-talking-woo-these-days researchers into this that and the other are kind of saying that the idea we don't know what we're capable of is true, or even what our brains do that borders on miraculous, so we are able to do more with the gear upstairs than we get told. How amazing it all actually is and not boring or stuffy or tedious. Interesting instead.

Ho hum

 

[Kid Anymore]

I've been studying math, or algebra and stuff for a while and I pleased with my grasp of this stuff. I've at least been able to reassure myself that "I get it", when I watch decent lectures on youtube or whatnot. It pays to also read arxiv papers and get different approaches to the same kind of goal.

Now it seems it's all about how to construct an algebra, but that's me. Of course that's been true before now.
And I can say that yes, I think I can give an interested person a few tips without them losing the thread.
I think I can do that because I don't believe it's a difficult subject. I don't know about difficulty other than figuring out what you want to learn and how to get there. Difficulties are when someone has questions about what something means, and want a clearer explanation. So giving a less clear explanation isn't helpful, so why do it? I don't want to do that.

 

[DaveC426913]

Well, the almost-talking-woo-these-days researchers into this that and the other are kind of saying that the idea we don't know what we're capable of is true,

We have no idea who this might be. Sounds like it could be someone spouting unattributed rhetoric - possibly a journalist looking for eyeballs - as opposed to qualified researchers making considered statements.

Beware of alarmist statements that aren't backed up by real people. It's easy to start inventing solutions for problems that don't necessarily exist.


My son used to amuse us when he was, like six years old, and used to say things like "People say, that the Green House is bad." He didn't know who, it was always just "I heard..."

 

[Kid Anymore]

We have no idea who this might be. Sounds like it could be someone spouting unattributed rhetoric - possibly a journalist looking for eyeballs - as opposed to qualified researchers making considered statements.

Yes there is the "like and subscribe" type of clickbait woo. However there is also a genuine even a growing discussion about how our brains, the brains of animals in general, might be able to do some pretty woo stuff after all.

Might be able to call it new-woo, the old-woo isn't a thing apparently. I'm talking about the Rainman type of woo--amazing powers of observation coupled with almost incomprehensible mathematical abilities. Inside the mind of a severely autistic man.

 

[DaveC426913]

However there is also a genuine even a growing discussion about how our brains, the brains of animals in general, might be able to do some pretty woo stuff after all.

Perhaps there is. But since it forms the basis of your thread, we kinda need more than your say so.

I have never experienced anyone "deliberately trying" to make math hard, or implying "only smart people get it", so I cannot get on board with your premise as it stands.

 

[exchemist]

I've been studying math, or algebra and stuff for a while and I pleased with my grasp of this stuff. I've at least been able to reassure myself that "I get it", when I watch decent lectures on youtube or whatnot. It pays to also read arxiv papers and get different approaches to the same kind of goal.

Now it seems it's all about how to construct an algebra, but that's me. Of course that's been true before now.
And I can say that yes, I think I can give an interested person a few tips without them losing the thread.
I think I can do that because I don't believe it's a difficult subject. I don't know about difficulty other than figuring out what you want to learn and how to get there. Difficulties are when someone has questions about what something means, and want a clearer explanation. So giving a less clear explanation isn't helpful, so why do it? I don't want to do that.

I think one issue with mathematics is undoubtedly its abstract nature. The teaching world has I think recognised for a while that young people become capable of handling the abstractions more easily when their minds have matured a bit, in some cases only in the mid teens. So trying to force, say algebra, on them too soon can be counterproductive, risking introducing the sense of panic one gets when one has no idea WTF is going on. Once that has happened the damage is done and the child will fear and avoid maths in future. But equally some children are fine from much earlier (I remember - this was in 1966 - we all had to learn how to solve a quadratic equation by "completing the square" at the age of 12. I confess I did it at that stage by rote rather than real understanding, but it did not panic me. I doubt many 12 years olds could do that today.)

Maths definitely can grow on you. When I had - at age 15 - to choose my A levels (for advanced study before UK university) I had to choose maths because I wanted to do physics and chemistry. I was very nervous about whether I could handle it. But after a year, it suddenly started to fall into place and things like conic sections and - especially - complex numbers were really rather fun and amazing. I guess my mind had matured enough by that time, plus I could see applications in science (you need complex numbers for AC circuit theory, for instance.) So I got a B in maths, which I never expected, to go with As in the other 2, and it all worked out fine. I was even emboldened enough to choose the Quantum Chemistry supplementary subject at uni, but there I did find my maths ceiling in the end. (I made it but was conscious of being very much at max altitude, with poor response from the control surfaces.)

As for clear and less clear explanations, there can be a simple but not very rigorous explanation and then a full-blown, rigorous one that is often a lot harder to understand and may seem needlessly complex. I was lucky to be taught maths initially by teachers who had failed at maths degrees and switched to other subjects. They found it easier to see where the problems in comprehension could occur, because they had been there themselves. Our Head of Maths was on a different planet. He would expound some theorem on the blackboard, then gaze at it and say in a sort of wistful ecstasy how beautiful it was. We respected him but thought he was a bit nuts. He was great for the top set, but could cause trouble for us scientists, who were just trying to get a more humble understanding to use in our other subjects.

So much depends on the teaching. You need time for each thing to click and not everybody clicks at the same rate.

 

[parmalee]

My sister, on the other hand, despite being a biologist/scientist, is so inept at math she can't multiply two one-digit numbers in her head.

Personally, I've observed that people are more inclined either towards biological sciences or towards maths/physics. Plenty of people learn both, of course, but it seems that typically either one or the other comes more naturally, and requires considerably less effort to grasp more challenging concepts.

 

[Kid Anymore]

My thoughts today about being taught "how to learn mathematics" when I was younger, is that you may not get to see any of the why it can be elegant, creative, even like a work of art. But not the kind you can sell for millions.

And that it takes a tertiary level was, and remains, to me a bit of a sad commentary on where being taught gets you to.

In that, if you do get to study the subject at university, and that's because you need it to be allowed to study year 2 or 3 stuff, then you might see this creative aspect too. And see that, y'know, God, if he made the first numbers, thinks it's all good.

 

[exchemist]

My thoughts today about being taught "how to learn mathematics" when I was younger, is that you may not get to see any of the why it can be elegant, creative, even like a work of art. But not the kind you can sell for millions.

And that it takes a tertiary level was, and remains, to me a bit of a sad commentary on where being taught gets you to.

In that, if you do get to study the subject at university, and that's because you need it to be allowed to study year 2 or 3 stuff, then you might see this creative aspect too. And see that, y'know, God, if he made the first numbers, thinks it's all good.

Not quite sure what you are saying here. There are many academic subjects where the elegance and artistry only become clear once a good level of proficiency has been reached. Music, for example, or languages. Part of being a school student is having the patience and curiosity to stick with these things in the earlier stages, and make the most of the discoveries along the way.

Sometimes I think modern society encourages false expectations of easy wins which just aren’t there. You look at any successful entertainer, or athlete, and you admire the hours of training and practice they put in to get where they want to be. Yet somehow, people resent having to do the same thing to study a serious academic subject and try to blame the teachers for their own flakiness!

 

[Pinball1970]

And that it takes a tertiary level was, and remains, to me a bit of a sad commentary on where being taught gets you to.

You have to start with the basics and you have to make mathematics real world IMO. Money, pieces of pie type things. One
One of the big complaints in school was “what use is this? What is this used for in work?”

12 or 13 ish.

I kind of bought into that because I found the subject very disjointed and extremely boring.

My friend (who was a genius) showed me simultaneous equations and I liked idea of hunting down an unknown.

Quadratics which Exchemist mentioned also caught my interest, again hunting down “x”

Then Calculus! Wow this is great! Finally, something I can get on board with!

By the time I got to A level (at 16) the pace and volume really took me by surprise and no one there was asking what Mathematics was used for anymore!

If you don’t pass with a good grade you do not go to University!



You have to get past these obstacles before considering what Mathematics actually is.

What is it?


There is no definitive answer to this even from the mathematics community and we are discussing this (to an extent ) in another thread.

 

[Kid Anymore]

I don't know, my recollections of being taught by teachers include the "why can't you understand" approach.
Also a bit of sniggering or sneering, dismissive behaviour; obvious signs of protecting something that they feel is worth all of the clearly Pythonesque theatrics.

Teachers might just be people who embrace a kind of modern insanity; a version of faith or belief in dogma without evidence, without the results.

 

[Kid Anymore]

Music, for example, or languages. Part of being a school student is having the patience and curiosity to stick with these things in the earlier stages, and make the most of the discoveries along the way.

I have negotiated that path myself. I can today say that I have some satisfaction with my abilities to wing it on a keyboard or a guitar, because of what I have learned all by myself. I did this by forgetting everything any music teacher tried to impress on my young impressionable mind.

I went all rock n roll on it. That's guaranteed to tell you you can't get no satisfaction, so it's a win-win.

 

[DaveC426913]

Also a bit of sniggering or sneering,

Your teachers sniggered and sneered at you?

obvious signs of protecting something that they feel is worth all of the clearly Pythonesque theatrics.

Teachers are far more defensive and protective of their classroom authority than they are of their subject matter. They have a job to do, and en entire class to teach and sueprvise. One student can easily disrupt that.

Are you sure they weren't perhaps reacting on a more visceral level to what they perceived as a kid that may have been disrupting the class?

Teachers might just be people who embrace a kind of modern insanity; a version of faith or belief in dogma without evidence, without the results.

In their defense, they're not there to do new science, they're there to teach established science. They knowe the established scicne has evidnce without having to produce it in every class.

To suggest they "embrace dogma" is to misunderstand what their job is.

They are there to teach you how to walk before you run. Walking means learning what is established. Running means challenging existing knowledge. While in school, you are too young and naive to challenge anything, except as a student does (a la "but how do we know for sure 2+2=4? What if it equals five?").

If your teacher got wind of this kind of "getting ahead of yourself", it would not surprise me if they couldn't tamp down the occasional snigger or sneer.

Are you sure you're being objective about your experiences?

 

[Kid Anymore]

Your teachers sniggered and sneered at you?

Yes, not just me though. It was funny to them to see a kid who was supposed to be smart having a bit of trouble reading, from the blackboard.
A jolly jape. Bit like cricket--one of the least fair but gentlemanly games I know about.

 

[DaveC426913]

Yes, not just me though. It was funny to them to see a kid who was supposed to be smart having a bit of trouble reading, from the blackboard.
A jolly jape. Bit like cricket--one of the least fair but gentlemanly games I know about.

How long ago was this? How accurate are your observations? How do you know - without telepathy - what, exactly, they thought was funny?

From what you are saying, it would not be too much of a stretch to wonder if - perhaps - you were one of those kids who thought (or had been told) they were so smart they could challenge the teacher on "dogma".

Were "dogma" and "Pythonesque theatrics" terms that were bandied about in that classroom back then? Perhaps they simply thought it amusing that such grown up terms were coming from a child? (Or have only you invoked them recently, in an attempt to interpret what you experienced?)


To be clear: I'm not dismissing your experience; I'm simply looking for the "lowest denominator" explanation, requiring the fewest assumptions about what other people are thinking and doing - and only dismissing it for a more controversial and dramatic explanation if the simplest is inadequate.

 

[exchemist]

I have negotiated that path myself. I can today say that I have some satisfaction with my abilities to wing it on a keyboard or a guitar, because of what I have learned all by myself. I did this by forgetting everything any music teacher tried to impress on my young impressionable mind.

I went all rock n roll on it. That's guaranteed to tell you you can't get no satisfaction, so it's a win-win.

Well good for you. So you know about the patience, putting in the hours and the setting of incremental goals in order to master something complex. With maths, as it is so abstract, it is probably harder to sustain that approach all by yourself, I suspect. I at least would have found it difficult to do that.