The drilling is contentious. I'm a strong believer in rote and drill, from bitter personal experience its the innate recall which stays with me, the casually acquired fades. However, this is mainly about fundamentals of arithmetic more than maths, reasoning, logic.
Comprehension of the why in the method comes later, I believe. But there is another school of thought which believes structural discovery of why embeds harder and so young minds are led in this process to invent maths and own discovery of the fundamentals.
I'm not an educationalist. I can't say who is right, I can only share the anecdata that what I keep is the rote and what I struggle with is the invention and discovery.
I don't regard myself as fluent in maths btw. Which, since computer science is a sub branch of maths, is a bit tricky given I'm a computer scientist by degree.
I never took to radians. The approximation inherent in π suits me better, just as log tables do. I never took to group theory, or any of the more arcane underpinning of languages like Haskell which demand a grounding in a particular kind of declarative mathematical notations and what lies behind them. Wish I could.
I have absolutely 0 mathematical intuition or knowledge these days. If I had to estimate how much I’ve retained, I’d say most of elementary school math and a tiny subset of middle school / high school algebra.
I never really failed it, but towards the end of high school in cal us and related courses, I suffered from a “don’t care” attitude combined with teachers who basically just passed everyone.
Now of course I find myself strong mathematical skills to work on the things I want, but the knowledge and intuition just isn’t there. I could likely invest in learning it, but it’s difficult as I not only have to worry about learning the fields immediately applicable to what I want to do, but having to go back through many years of subjects I forgot or was never taught and figuring out what the proper prerequisites I need to relearn. At that point it just seems so far away from my original need and such a time sink I’m not sure I could motivate myself to do it.
And on the intuition side, I am shocked by just how well people around me, even kids are able to grasp the more abstract mathematical concepts around programming. Things like category theory and such. It all sounds like a foreign language when people begin to discuss these things. Not the the more practical things are much better. I have failed to learn digital design a few times now because despite understanding all the primitive Boolean functions I can never on my own figure out how to arrange them into useful functions, simple or complex.
I'm barely 22, even I have started to feel deficiencies in my learning and retaining abilities. I think the main problem seems to be that you "grow" out of your learning phase.
I can't keep myself concentrated on a reading for more than a few minutes anymore, even though I used to be the most voracious reader in my school as a kid who would keep on reading for the whole day if he wasn't disturbed. I can't concentrate on classes or video tutorials as much as I could as a child. Even skills seem to get impacted - I used to do big calculations in my head and retain multiplication tables up to 20 like most other Asian kids, but I can't seem to make myself do that anymore.
I'm much older than you. I grew up without the internet. I found that when the internet became mainstream, my ability to read books and concentrate diminished significantly. Pretty sad about that.
While I'm getting (much) older as well, I have found that I'm much more focused when I want to learn something.
Unlike when I was 16 and essentially behaving like a sponge, I need to decide that I'm going to learn something, and actively go for it. Once I have turned the switch on, I can learn, and more efficiently than when I was younger since I have some kind of plan/target, and years of experience have taught be when I don't understand/think I know but don't and how to fix it.
In other words, I have traded raw natural learning ability for 'I'm slower but more efficient'.
Like a sport, math takes practice. With lessons and enough practice most people can learn the rudiments of golf or softball. With some athletic ability, many people can become good amateur golfers, but this will likely take years of playing a few hours almost every week. Math responds to practice too.
In many of my math courses, I believe I was putting in more time on the material than any of the other students; the result was that I got the grades I wanted. More practice led to better results.
Math though is different than many other subjects. It’s common for math classes to rely heavily on the previous year’s math skills. Calculus depends on a fluency with algebra and trigonometry, trigonometry depends on geometry, geometry depends on algebra 1, algebra 1 depends on fluency with fractions and so on. A bad US history teacher probably won’t affect a subsequent class on ancient history, but a bad math teacher can haunt you for a long time. In my experience, this is a common cause for math frustrations and leads to the feeling that one is “bad at math”.
After having taken over two dozen college math courses, I’ve figured out how to get top grades:
(1) Attend every lecture and recitation. The professor is unlikely to ask for something on a test that isn’t mentioned in class.
(2) Sit in the same seat in the front row in the middle, unless it’s a small graduate level course. The professor will notice any puzzlement in your expression and will often elaborate on a point to ensure that you understand.
(3) Don’t use your phone or laptop at all during math classes. It’s really not practical to take note for math by typing, and you will be less tempted to get distracted—-another reason to be in the front row. Studies show that taking handwritten notes promotes better learning.
(3) Take advantage of office hours to get even the slightest lack of understanding addressed. I always appreciated it when the serious students would come by to see me during office hours myself.
(4) Buy the book. Don’t rely on an electronic or web version. A physical book will be around in a few years for reference in a subsequent class. I still have useful math books from half a century ago. In all of my math courses, I’ve only once had an online book that was as good as the printed copy, and even then it was lost to me a few years later. I’ve often had to flip back and forth between different locations in a math book; this is much easier with a physical book.
(5) Do all the assigned homework, even when not required. Math skills take practice.
(6) Do more than the assigned homework. Math takes practice. Doing all the assigned homework should result in B grades on the tests. To get an A, do addition problems from the assigned sections.
(7) To have the highest grades in the class, do every problem in the assigned sections of the textbook. If there are optional texts, buy those too and do every problem in the relevant sections as well. Any problem that gives you trouble is a good thing to discuss during the professor’s office hours. Putting in six hours of extra time on exercises before the midterm and final can often result in top of the class grades. By keeping up with all the lectures and all the homework, an afternoon of studying before an exam will really reenforce your learning.
> A bad US history teacher probably won’t affect a subsequent class on ancient history
As a historian, I have to disagree here.
The notions that helped shape 1776 and beyond weren't inventions that happened in isolation. On the contrary. They were very shaped by 4.000 years of European history, culture, philosophy and thinking as they were imported directly from the European contintent. For better or worse.
Your understanding of US History most definitely is going to hit a hard limit if you didn't learn about classical antiquity and subsequent European history first.
The problem with school education is history being broken up in discrete fragments and taught in isolation. That's how important pieces of context gets lost. All you're going to learn are discrete events and highlights and not much more. You won't learn how people arrived there or how they influenced subsequent history.
All too often, people voice their opinions online are often based on those isolated bits and pieces they remembered from those lessons in school. This lack of historical awareness becomes astounding when you take a deep dive into history and you learn how many dots you really need to connect to gain somewhat of a broader or a new perspective on things.
You should look up Professor Leonard... he has full lessons all the way from basic algebra through calc 2 and analytic geometry.
Math was always my biggest enemy but I decided to be a stem major.. I’m now going into calc 2 with a 4.0 (not that that matters, but I can’t help but feel good about it when I thought I’d be lucky to even pass) and i never thought that I would find myself saying that math is my favorite class now.. never in a million years.. when I decided that I wanted to pursue a stem major I felt like I was gambling and thought to myself ”how will I ever get through the math” I had never made it past pre algebra and I think I even failed that .
If I can do it then you can too. From my experience it’s all about repetition and working from the ground up, even if you think you have something mastered go through it anyways and once you get to a certain point all of the pieces will fit together. Study as if your life depends on it and you will excel.
The reason I recommend Professor Leonard’s Videos is that I had tried every resource I could find and my professors’ ways of teaching did not work at all for me.. ever since I started following Leonard’s lessons I’ve been acing every exam... I feel that for the most part people who say “I’m bad at math “just never invested enough time or want or focus into learning it... I was that guy... I feel that honing your skills in math is completely worth it because it really exercises your logic and reasoning skills which are incredibly useful in everything we do(mathematics help you to be better at everything...even the things that you would never associate with mathematics) this is why even concepts that people think they will never need to learn or have to use are important..at least that is my opinion.
Honestly, I feel like math is one thing I don't have to practice or drill as it's possible to deduce most things without actually having to drill/learn the concepts - at least on the basic level and if given enough time. This is why I loved math as a kid, because I was too lazy to study and it was one subject I could do well at by nature. So I think the article downplays the genetic influence of it. But of course it's better to think that you can have more effect on your intelligence rather than not psychologically, because otherwise you may end up doing nothing to improve yourself. But I still feel that genetics has very high influence on innate math, abstract logic capabilities. Some people are just addicted to solving abstract problems while others can't force themselves to think about these at all. For instance, if there's an abstract problem I could spend hours and hours without noticing time passing by, and I don't think I ever had any specific raising or environmental influences to be this addicted. There's other issues that I have absolutely zero patience to do, it's as if I have ADHD in some fields and I'm only deeply interested in certain fields. And to me it makes evolutionary sense. It's better in a community to have people specialised to different niches, so people's brains are wired differently to support different niches and specialisations. You can't have everyone thinking about abstract problems all the time - because there would be no one to do the real worldly things any more and abstract thinkers would simply die of hunger or eaten by predators.
11 comments
[ 3.3 ms ] story [ 30.7 ms ] threadComprehension of the why in the method comes later, I believe. But there is another school of thought which believes structural discovery of why embeds harder and so young minds are led in this process to invent maths and own discovery of the fundamentals.
I'm not an educationalist. I can't say who is right, I can only share the anecdata that what I keep is the rote and what I struggle with is the invention and discovery.
I don't regard myself as fluent in maths btw. Which, since computer science is a sub branch of maths, is a bit tricky given I'm a computer scientist by degree.
I never took to radians. The approximation inherent in π suits me better, just as log tables do. I never took to group theory, or any of the more arcane underpinning of languages like Haskell which demand a grounding in a particular kind of declarative mathematical notations and what lies behind them. Wish I could.
I never really failed it, but towards the end of high school in cal us and related courses, I suffered from a “don’t care” attitude combined with teachers who basically just passed everyone.
Now of course I find myself strong mathematical skills to work on the things I want, but the knowledge and intuition just isn’t there. I could likely invest in learning it, but it’s difficult as I not only have to worry about learning the fields immediately applicable to what I want to do, but having to go back through many years of subjects I forgot or was never taught and figuring out what the proper prerequisites I need to relearn. At that point it just seems so far away from my original need and such a time sink I’m not sure I could motivate myself to do it.
And on the intuition side, I am shocked by just how well people around me, even kids are able to grasp the more abstract mathematical concepts around programming. Things like category theory and such. It all sounds like a foreign language when people begin to discuss these things. Not the the more practical things are much better. I have failed to learn digital design a few times now because despite understanding all the primitive Boolean functions I can never on my own figure out how to arrange them into useful functions, simple or complex.
I can't keep myself concentrated on a reading for more than a few minutes anymore, even though I used to be the most voracious reader in my school as a kid who would keep on reading for the whole day if he wasn't disturbed. I can't concentrate on classes or video tutorials as much as I could as a child. Even skills seem to get impacted - I used to do big calculations in my head and retain multiplication tables up to 20 like most other Asian kids, but I can't seem to make myself do that anymore.
Unlike when I was 16 and essentially behaving like a sponge, I need to decide that I'm going to learn something, and actively go for it. Once I have turned the switch on, I can learn, and more efficiently than when I was younger since I have some kind of plan/target, and years of experience have taught be when I don't understand/think I know but don't and how to fix it.
In other words, I have traded raw natural learning ability for 'I'm slower but more efficient'.
In many of my math courses, I believe I was putting in more time on the material than any of the other students; the result was that I got the grades I wanted. More practice led to better results.
Math though is different than many other subjects. It’s common for math classes to rely heavily on the previous year’s math skills. Calculus depends on a fluency with algebra and trigonometry, trigonometry depends on geometry, geometry depends on algebra 1, algebra 1 depends on fluency with fractions and so on. A bad US history teacher probably won’t affect a subsequent class on ancient history, but a bad math teacher can haunt you for a long time. In my experience, this is a common cause for math frustrations and leads to the feeling that one is “bad at math”.
After having taken over two dozen college math courses, I’ve figured out how to get top grades:
(1) Attend every lecture and recitation. The professor is unlikely to ask for something on a test that isn’t mentioned in class.
(2) Sit in the same seat in the front row in the middle, unless it’s a small graduate level course. The professor will notice any puzzlement in your expression and will often elaborate on a point to ensure that you understand.
(3) Don’t use your phone or laptop at all during math classes. It’s really not practical to take note for math by typing, and you will be less tempted to get distracted—-another reason to be in the front row. Studies show that taking handwritten notes promotes better learning.
(3) Take advantage of office hours to get even the slightest lack of understanding addressed. I always appreciated it when the serious students would come by to see me during office hours myself.
(4) Buy the book. Don’t rely on an electronic or web version. A physical book will be around in a few years for reference in a subsequent class. I still have useful math books from half a century ago. In all of my math courses, I’ve only once had an online book that was as good as the printed copy, and even then it was lost to me a few years later. I’ve often had to flip back and forth between different locations in a math book; this is much easier with a physical book.
(5) Do all the assigned homework, even when not required. Math skills take practice.
(6) Do more than the assigned homework. Math takes practice. Doing all the assigned homework should result in B grades on the tests. To get an A, do addition problems from the assigned sections.
(7) To have the highest grades in the class, do every problem in the assigned sections of the textbook. If there are optional texts, buy those too and do every problem in the relevant sections as well. Any problem that gives you trouble is a good thing to discuss during the professor’s office hours. Putting in six hours of extra time on exercises before the midterm and final can often result in top of the class grades. By keeping up with all the lectures and all the homework, an afternoon of studying before an exam will really reenforce your learning.
As a historian, I have to disagree here.
The notions that helped shape 1776 and beyond weren't inventions that happened in isolation. On the contrary. They were very shaped by 4.000 years of European history, culture, philosophy and thinking as they were imported directly from the European contintent. For better or worse.
Your understanding of US History most definitely is going to hit a hard limit if you didn't learn about classical antiquity and subsequent European history first.
The problem with school education is history being broken up in discrete fragments and taught in isolation. That's how important pieces of context gets lost. All you're going to learn are discrete events and highlights and not much more. You won't learn how people arrived there or how they influenced subsequent history.
All too often, people voice their opinions online are often based on those isolated bits and pieces they remembered from those lessons in school. This lack of historical awareness becomes astounding when you take a deep dive into history and you learn how many dots you really need to connect to gain somewhat of a broader or a new perspective on things.
Math was always my biggest enemy but I decided to be a stem major.. I’m now going into calc 2 with a 4.0 (not that that matters, but I can’t help but feel good about it when I thought I’d be lucky to even pass) and i never thought that I would find myself saying that math is my favorite class now.. never in a million years.. when I decided that I wanted to pursue a stem major I felt like I was gambling and thought to myself ”how will I ever get through the math” I had never made it past pre algebra and I think I even failed that .
If I can do it then you can too. From my experience it’s all about repetition and working from the ground up, even if you think you have something mastered go through it anyways and once you get to a certain point all of the pieces will fit together. Study as if your life depends on it and you will excel.
The reason I recommend Professor Leonard’s Videos is that I had tried every resource I could find and my professors’ ways of teaching did not work at all for me.. ever since I started following Leonard’s lessons I’ve been acing every exam... I feel that for the most part people who say “I’m bad at math “just never invested enough time or want or focus into learning it... I was that guy... I feel that honing your skills in math is completely worth it because it really exercises your logic and reasoning skills which are incredibly useful in everything we do(mathematics help you to be better at everything...even the things that you would never associate with mathematics) this is why even concepts that people think they will never need to learn or have to use are important..at least that is my opinion.
https://youtu.be/kenf8E1RuoA