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I am 25 and have been studying mathematics on my own for several years, but I am still between the middle and high school levels.

My main weakness is my dyslexia. I sometimes forget words or confuse them without even realizing it, unless someone points out that I'm talking nonsense.

I also have a big lack of intuition. For example, I know that an equation like $x\mapsto ax+b$ with $a,b\in R$ represents a line. I can prove it, but when faced with a concrete case, I struggle to see the line behind the symbols. It's the same with the set of points $(x,ax+b)$ when $x$ describes $R$.

If I see written $(a+b)^2=a^2+b^2$, I might not necessarily realize the mistake, even though I have known the identity $(a+b)^2= a^2+2ab+b^2$ for years. It's not intuitive for me compared to $1=2$.

If $a>b$ and $c>d$, I know I can't write $a/c>b/d$ but once again it's "by heart". I am perfectly capable of making the mistake and not noticing it despite several reviews. Again, I am capable of proving it's false, but it's not obvious like $1=2$. I've tried to turn it over in my head by visualizing the real line, visualizing a histogram, etc.

Not long ago, I got stuck on a problem because I didn't recognize that $2n + 1$ where $n$ is an integer, is just the sum of two consecutive integers. It's silly, but that's how it is with everything.

Despite my efforts to prove everything, to make analogies, spaced repetitions over time, apply the Feynman technique, and others, I cannot develop a mathematical intuition. I also tried tutoring without noticing any progress.

After reading "The Prince of Mathematics" about Gauss, I tried to construct mathematics by myself and derive new results, but my limited intuition and creativity didn't lead to success.

My thinking is mainly through an internal monologue. I've tried to slow it down to promote more intuitive thinking, in images or sensations, but it wasn't conclusive.

To be precise, I've been back to studying mathematics for four years, dedicating an average of 15 to 20 hours a week to this discipline, but I often revise the same concepts without seeing any evolution. I even feel like I'm regressing at times. I know there are many teachers on this forum. Have you ever encountered a student facing such difficulties? If so, how did you proceed to help them?

Thank you.

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    $\begingroup$ You mention intuition a bunch. But I'm wondering about visualization. I've had students who could not visualize. Some still did well. I wonder if you'd have fun with Art of Problem Solving's problems. Maybe check out artofproblemsolving.com/alcumus $\endgroup$
    – Sue VanHattum
    Mar 22 at 18:28
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    $\begingroup$ @SueVanHattum I wouldn't say that I can't visualize anything, but the majority of my reasoning is verbal. Thank you for the link, I will check it out. $\endgroup$
    – antho
    Mar 22 at 20:05
  • $\begingroup$ For what it's worth: when I see $2n+1$, I immediately think about the general way to write an odd number. The interpretation that it can be written as $n + (n+1)$, being the sum of consequent numbers, is something I don't immediately think about neither (and I have a university math degree). In other words: don't be too hard on yourself! $\endgroup$
    – Dominique
    Apr 30 at 8:00

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This might be an unpopular answer but I'm going to try to be honest and real with you.

Math gets hard and unintuitive for everyone at some point, and that point is different for everyone -- for some people that's algebra, for other people it's calculus, for others it's real analysis, for others it's algebraic topology, for others it's research-level math, ... you get the idea. There's this same gradation even within research-level math problems.

Sometimes people hit that point early due to ineffective or inconsistent practice techniques. But even if you practice effectively and consistently, a ceiling still exists.

It's kind of like sports. Few people practice effectively and consistently enough to reach their athletic potential, but it's just a fact of life that most people could not become professional basketball players even with 100% effective and consistent practice.

Are you at your mathematical ceiling? I guess we can't say with 100% certainty. But if you're practicing effectively and consistently (which it sounds like you are), even sometimes with a tutor, and you're still stuck in a plateau, then it sure seems likely.

In general, when you feel yourself running up against a ceiling in life, the solution is typically to pivot and into a direction where the ceiling is higher.

For instance, the story of many a quantitative software engineer goes like this:

  1. loved math growing up and wanted to be a mathematician

  2. realized during undergrad or grad school that they had lost their "edge" compared to other aspiring mathematicians

  3. also realized that they have a knack for coding and interest in some applied domain, and that the problems that need to be solved there boil down to interesting math that most people in software don't have the math chops for

  4. pivoted in that direction where their ceiling is higher


Addendum. In the comments, OP mentions that their initial milestone is to solve certain problems from the Math Olympiads, and their overall goal is to actually understand mathematics.

One thing that OP might try is learning more advanced math subjects (e.g. calculus, linear algebra, differential equations, etc) as opposed to focusing on problems from Math Olympiads. Competition math is very g-loaded, and the point is separate students who have loads of underlying mathematical talent from students who learned a bunch of math early but don't have as much talent.

Perhaps counterintuitively, it's a lot easier to make progress by continuing into more advanced subjects and layering on top of your content knowledge than it is to make progress by focusing on competition math and hoping that some of the insight you see in the solutions will rub off on you.

Even for measuring progress in general, I don't think Math Olympiad performance is the right metric to use. It's pretty common for people to learn and apply university-level math yet not be able to solve competition problems. You could very well improve your mathematical knowledge and capability to do things with math, and not have it carry over much to competition problems.

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I am a student in mathematics, and I've always been told that mathematics is not for everyone and that our brain is "weird" or works differently so that we can do maths. But as I read maths books or papers for my classes, I've come to a conclusion. Yes, maths is hard, it's abstract and in order to understand difficult concept, you need to build your knowledge and to practice a lot, but mathematician makes it even harder with phrases such as " it follows directly from..." or " it's easy to see that..." well I'm sorry it's not always that easy to see...

I don't consider myself as someone that has a lot of mathematical intuition, when I speak to my peers or with my teachers about maths, I don't understand what they say. I have to write down every concept and dissect them in order to fully understand them. When I see new maths, I don't get it right away. To make an analogy with you, If I see $(a+b)^2$ I don't see right away that this is $a^2+2ab+b^2$, I would have to calculate it. It may take more time, but in the end I can still do the same maths as the people that see it right away.

What I'm trying to say is that If you tend to make a lot of mistakes and that you find it hard to have a "mathematical intuition" then you have to find a method to cope with those "weakness", what I would suggest is that you stop to try to visualize concepts when you can't. Instead, find a method that works for you. For example, when you see an equation like $y=ax+b$ you start to ask you those questions:

  1. Have I seen such an equation before ? --> yes this is a line
  2. What are the properties of a line? --> write down everything that you know about lines
  3. Now that you have all the properties of the lines in front of you, ask yourself which properties might be helpful in the settings of your particular problem?
  4. Resolve your problem!

I use this technique all the time: when I want to resolve a maths problem, I first analyse the mathematical objects of the problem and I write down everything I know about them (for that I take my textbooks or even Wikipedia since I don't remember everything). Then I have all the tools in front of me to resolve my problem, so it's much easier to see what to use and have this "intuition" you're talking about!

Maths is not that hard, it's just a lot of different tools, the hard part is how to use them together to create new tools!

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  • $\begingroup$ That's what I do, and sometimes it works, but often it doesn't. After seeing the solution, I realize I wasn't asking the right questions. You might say that happens to everyone, but I can experience the same blockage just two hours later. At the moment, I think I've understood it well, but in reality, I've retained as much about the problem as the license plate of a car that just drove by. Initially, it's "clear," but then it quickly fades from my memory. One might think it's a memory issue, but I don't believe that's the case. $\endgroup$
    – antho
    Mar 22 at 22:59
  • $\begingroup$ As you mentioned, the problem is connecting everything. When we move on to questions that aren't just applications of what we've learned in class, understanding the whole, or at least a large part of it, is where I get stuck. I'll try to apply this method more thoroughly. $\endgroup$
    – antho
    Mar 22 at 23:00

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