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I don't have a degree. I've found that I learn better if I write this site than burning myself over grades and exams. I was enrolled at a very large university pursuing a degree in applied (earth) sciences but it didn't work after long years and a ridiculous quantity of failures. I took the conscious decision to not care about grades or exams, but to care about the applications in life instead. I'd say that way over 50% of all students at university are so much worried with the credits and the grades that learning is left behind. Every semester the same question repeats. Students ask to each other "Professor A is going to teach subject X. Is he/she good? Is he/she friendly? Is he/she rigorous?". See? Almost all students are worried not about learning, but whether they are going to have a hard | easy time with professor A or B. There is also a matter regarding being a scientist vs. being a teacher and not everyone excels at both.
I'm not saying that people should dismiss a degree, but as long as one has the will and access to textbooks, one can learn without having to wait for somebody to teach it for them. Have you ever thought that whoever wrote a textbook, is in fact, waiting for people to open that book and read it? Now there is a whole debate all over the world about what to teach, when, what methodology and so on. What I came to realise is that every degree program has its own pace, own challenges, own order of subjects. Mathematics and every other science that relies on mathematics have the property of being cumulative and more or less linear. One can't understand calculus if one can't get pass basic algebra first. In spite of being linear, some topics can be rearranged. For example Newton's laws of motions rely on calculus, but with both being taught in the first semester, Physics I can't have a pre-requisite of calculus unless it's postponed to the next semester. With multivariable calculus some concepts of linear algebra show up, but calculus doesn't have a pre-requisite of linear algebra. Some people may excel at calculus and fail at linear algebra, or it could very well be the other way around. It's just the natural variance among people.
There is something about asking questions that I noticed in some classes. There are questions that people make that the teacher answers with "What?" or "What you are asking doesn't make sense". When a question doesn't make sense for the teacher it means that you are probably not understanding some concept and the question really doesn't make sense from the point of view of someone who understands that concept. For example: is the tangent function continuous? Because tangent of the right angle doesn't exist. Yes, the tangent of 90° yields a vertical line which extends to infinity. However, the right angle is not part of the domain of said function. When we plot the graph of tan(x) we don't consider the right angle.
When reading the solution for an exercise, try to understand what you were missing or what you were doing wrong. Just reading and then copying it won't make you really learn it. Some authors like to leave proofs to the reader, which is often annoying for some people because they were expecting the author to do it. If you are going to do the proof don't overdo it, they aren't meant to be burdens.
Once upon a time I was talking to a teacher, after he finished giving a calculus lecture, about my struggle with functions. I don't remember what about functions I was talking about. But I remember that he told me that some mathematicians go very deep into functions. They go deeper and deeper, losing themselves and losing the sight of the real world somewhat. As if they "drowned" in mathematics itself and lost the connection with reality. It's rather odd that some teachers do make lots of mistakes during classes and many of them are related to algebraic properties that people often go wrong in exams. I've lost count of how many times I witnessed a teacher erase everything on the blackboard because there was a mistake in a sign here or a misnamed variable there.
The first section in each chapter is dedicated to mistakes. I did that because there is a mistake in the way mathematics and sciences in general are taught. Most of the time teachers focus on what is right, because what is right is a truth that is the outcome of some proof. What about what is wrong? There is the mistake! There is so much emphasis on what is right that we are taught to see the world as either right or wrong. Proofs are seen as undeniable truths, while mistakes are seen as evil. What is wrong is wrong because it is wrong. There is so much focus on proving what is right that the wrong is left unexplained. We are left with a bottomless pit where all the wrong reasoning are discarded because they are worthless. This is exactly where I see an advantage in physics, statistics and computer science. Because if you find a force vector that is pointing in the wrong direction, you can clearly see that something is not right. If you calculated that the average temperature over 100 years is not compatible with the real world data, something is going wrong. If a program is a game and the player is gaining momentum by violating the principle of conservation of energy, there is some mistake with the calculation and to fix it we have to know what the mistake is in the first place.
How does physics evolved over the centuries? There is a certain difference in regards to chemistry and physics in comparison to mathematics. In physics and chemistry we have many experiments that go wrong and because they go wrong we draw conclusions. For example: if we try to apply Newton's laws to the motion of galaxies and other objects we see that the theory predicts something while the observations show something else. That's how scientists developed new theories to explain new phenomena that could not be properly explained with the available models. I see a curious difference in pharmacy and medicine. When something goes wrong or does not work, there is a great effort made in trying to understand why it doesn't work or what went wrong.
I may be stretching too far, but there is a matter of psychology involved in making mistakes. There are certainly teachers and students out there who refuse to admit mistakes for various reasons. Sometimes there are stubborn people who refuse to change their ways or viewpoints no matter what. And there are the extreme cases of being stuck at a mistake for too long or the extreme opposite of ignoring them and never learning in the first place. Science in general is rigorous because it has to be this way. But at the same time people's minds have to be flexible to question theories, to accept that there are different solutions for the same problems. Sometimes we have to think beyond what is right or wrong and have an holistic viewpoint for certain problems.
There is another aspect that is related to beliefs and mistakes contained in textbooks. Lots of typing errors or answer key errors. I've noticed that people go in communities asking for help because they doubt their own answers when compared to the answers provided in the book. I think that very often we believe that books do not lie! We are left in a position that whoever wrote the textbook should be considered to be always right. It happens in exams too. Some teachers even ask students to double check their points because sometimes it's easy to skip steps and go wrong. I've had the experience of pulling my hair out and burning out over exercises which contained errors or the answer was wrong countless of times.
Learning disabilities and mental health
Mathematics have the property of being both concrete and abstract. Some problems arise from geometrical interpretations, while the reverse direction is also possible. Some problems can be translated to geometrical interpretations. It's quite natural for people to find some parts of mathematics easier than others. I mentioned that reading is the source of much of the difficulty to learn math. It can happen that some people are unable to read and yet excel at doing mental calculations. There are also those who are unable to do simple arithmetic operations in their minds.
https://www.researchgate.net/publication/286439968_Dyslexia_in_the_Arab_world
I've found articles that describes that dyslexia is language dependant. There are people who are dyslexic in one language but not in another. I'm only speculating here, but suppose we have a list:
1 - | (one bar)
2 - || (two bars)
3 - △ (three sides)
4 - □ (four sides)
5 - ⬠ or ☆ (five sides)
6 - ⬡ (six sides)
7 - † (the seven deadly sins)
8 - ۞ (two overlapped squares, Rub el Hizb in Arabic)
9 - 🐈 (a cat has nine lives)
0 - Ϲ (crescent moon)
Suppose there are people who are unable to carry out this computation: 1 + 2 = 3. Now if that same person sees "| + || = Δ" they can do it. Is that related to dyslexia? I don't know. For most people arithmetic isn't a problem, but higher level mathematics is. I had a teacher of calculus who mentioned that in an exam some people were writing the partial derivative (∂) mirrored, which for her read as the letter "G". I have no idea if mirroring math symbols is a sign of dyslexia or something else. It could be related to languages such as arabic that are written right to left. I remember that at some point in school I had trouble confusing ">" (greater than) and "<" (less than) for unknown reasons.
https://www.nature.com/articles/news040816-10
https://www.sciencedaily.com/releases/2012/02/120221104037.htm
There is a certain tribe in the Amazon rainforest that lacks words for numbers. How is that possible? I have no idea, but it shows that a life without numbers is possible.
http://www.vega.org.uk/video/programme/301 Emma J. King, some british woman who suffers from dyscalculia and yet has a PhD in astrophysics.
Mathematics have a strong relationship with geometry and visual representations. There are people who excel at calculations and yet fail to trace circles, graphs of sines or draw a cube using orthogonal projection. When we study geometry beyond 3D we are forced to use imagination and for some people this may be hard or impossible to do. To imagine the 4D world may be easy for some people, but not for most people. I'm only speculating but I think that many teachers out there have had experiences with students who fail to trace graphs of functions or draw cubes. And the opposite could also be true, people who at a very young age are able to draw faces, hands and other complex shapes while not doing well with calculations. I'm not saying that being good at one implies being bad at the other, but sometimes such extreme discrepancies do show up in schools.
https://www.frontiersin.org/articles/10.3389/fpsyg.2020.01143/full
https://www.nature.com/articles/s41598-021-96876-6
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4884738/
Because I mentioned reading there comes a question: What if there are people who fail read? I'm not talking about illiteracy, blindness or dyslexia, but another barrier that is about people who are literate and yet, fail to read. They fail to read because they cannot concentrate, cannot focus or even reject reading altogether. That's a very complex set of problems that aren't directly related to failure to comprehend mathematics, but the failure to cope with demands of the educational system, schools or life itself.
What if there are parents who forbid studying mathematics? It may sound absurd but sometimes there are families with beliefs that may include such behaviour. What if there is a fear of making mistakes? What if there is a fear of exams and tests? What if there is excessive perfectionism involved? What if there is the fear of being called out? What if there is a fear of asking questions? What if there is depression? What about traumas? What about ego? What about arrogance? What about pride? What about shame? What about prejudice? What if you are unable to focus on reading for longer than 5 minutes? What if you forget assignments with such a frequency that you often don't remember what you were supposed to do? What if you have a problem with following directions given?
All that aren't directly related to being bad at mathematics, but may be imposing hard barriers that prevent one from learning, practising or even both. I don't have answer for those questions but the answers can be found on many books out there that cover not learning itself, but emotions, psychology, ego and some types of disorders. There are also some highly qualified professionals recording videos about such topics.
There is yet another issue that is related to the previous. All fields of science have a common property that is: one has to process the mathematical thinking by oneself in the sense that the only way to understand a solution to a problem is to do it yourself. However, schools and society itself aren't meant to be filled with loners that do all the work by themselves. There is competition, there is collaboration. How to balance both sides of this equation? I really don't know. I have read some stories that describe how competition can be good in the sense of giving stimulus. But it can also be bad in the sense of creating self defeating behaviours, self blame or other negative thinking. I think there is a myth regarding genius scientists that are loners that work in isolation. That's a very complex issue to solve because there are cultural, social and psychological aspects all involved here.
I can even mention a contradiction that I see in the way faculties and universities are created and managed. There is a lot of research that is the product of collaboration. Yet, professors are often oblivious to one and another. I mean, professor A teaches calculus 1. He or she has his or hers own methodology, pace, assessments. Then comes professor B to teach calculus 2 and there is a completely new methodology, pace, assessments. There is even some prejudice related. I once had a class of linear algebra where the professor openly said that teaching linear algebra for chemistry majors was a terrible experience, with students who were a disaster. Or teachers complaining that the class did not learn the pre-requisites properly. One common complain among students is that professor A is burdening the class with too much work and assignments, as if he or she is oblivious to what other professors are demanding from the same students. I'd be unfair to blame the professors alone though, because many of them wish they could offer something better but they don't know how or what to do.
Physical and social conditions
There is an issue that I cannot argue with. It's virtually impossible to learn without proper conditions. Without books there is no reading. Without pen and paper, no writing. Without a place to sit and study, no classes. Without light, nothing can be done at night. Without fresh water and food, no life. If the country, city or state is under war, what type of learning can you expect? Nothing but survival alone. You just can't expect that a miracle is going to make somebody born under such conditions to defy all logic and become a genius scientist or something.
Politics is an extremely controversial topic. What if the country imposes that some subjects are forbidden? Or the opposite, what if the government pays for people to focus on certain subjects? Look at world war, during that period there were many advances in sciences due to warfare. Libraries may be forbidden to offer certain books, languages or topics. I'm only speculating here, but in the same way kids may have their learning experiences hindered by lack of books, the opposite could mean that out of curiosity, science books could attract more attention simply because those are the books more readily available. Or because some books are designed in a way to attract more attention than others.
Lastly, but not least important, there is gender bias. Some famous universities had had bans for women decades ago. Women were forbidden to study mathematics or physics for example. It's very rare to find textbooks written by women for calculus, linear algebra or physics for example. That's a debate all over the world.
I do not have the power to snap my fingers and change reality itself. Only super villains with super powers can do that. My goal with this site is not to be the definitive source of anything. Is just to have a place that I can store what I learn, what I need and for further referencing.
