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The best way to learn math?

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Good morning.  I have watched this video before so my comments are not made in a vacuum.  I generally like the content the Math Sorcerer puts out but I would disagree here.  Even in the practice of law, I need the why, not just the process and list of citations for a point.  My career has been much richer for knowing the whys.  For context, I was an aspiring STEM person, did not go that route (tax lawyer) and came back to it at 50.  I would rephrase the suggestion made to not worry about the logic to being one of "don't go down the rabbit hole, but learning 'why' and not just 'how' is a good thing".

I hated the way math was taught when I was coming through pre-college education as well and I even went to a good school.  Last year I worked with a tutor on Spivak's Calculus.  If this person was my professor in college I am sure I would have taken a totally different path.

I think many of us are well served by doing enough repetition to start to see and feel patterns in process and solutions.  Using that bit of understanding to break down the why can be effective.  In my experience, there is no one way to learn, EXCEPT that lots of meaningful practice is required.

If you are inclined to share a bit of information, I would be happy to offer up text or video or website suggestions from my experience and compulsive searches around this topic.

1. Where are you mathematically now?
2. Even for pre-calculus, how much thinking are you willing to do?
3. What is an example of a text or video that resonated with you and helped you "get it" on a topic?
4. If you are past calculus, what are you looking to accomplish/learn?

Just remember, other than deliberate and meaningful practice, there is no one way and that you can find your path to success.

Best,

Totoro
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* The title is stupid click-bait, but it's also not what he says in the video. He's just saying sometimes you can move on.
* The video is clearly aimed at people doing heavier classes with proofs/real problems. (I can tell by the content, you can tell by that being his only example.)
* Don't picks camps. Should you understand? Should you push on? As with all things in life, you should do the right thing for that moment.
* Don't overestimate the depth of what there is to understand. "How do I solve this quadratic equation?" "Use the quadratic formula." "How do you know to use that?" "Because the quadratic formula solves quadratic equations." Is a plenty sufficient interrogatory process. "Why does the quadratic formula work?" "Plug it in and it works." That is why. "Where did it come from?" "You complete the square." Yes, each part represents something interesting about the situation, but you can pick that up over multiple years. Even in the beginning of proofs classes, you should focus on definitions. And the answer to most "why"'s is because that's the definition.
* Having intuition behind stuff is great, but it doesn't help you *execute* stuff. It helps you remember when to use stuff and connect pictures. So when you first see something, try to get the idea. If that doesn't work, maybe it's better for that thing to be learned mechanically first. The notation is the only vehicle you need to explore things. You will find everything has more layers to discover down the line. If it's not causing you trouble, don't sweat it.
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I'm just a hobbiest but this seems spot on to me. Walking away from a tough problem or concept is very often a way more efficient use of your time. There are a few ways things can go from there. Maybe whatever it is doesn't matter downstream, and you really lose nothing by skipping it. Maybe it's not IMMEDIATELY necessary, and you don't lose anything by going back to it days, weeks, months later with more experience. Maybe it's absolutely necessary, but it can still be helpful, after letting it fill up your head space, to get away from it. Working on adjacent stuff or just taking a shower might make it all come clear.
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I always like to use the **Low Resolution** / **High Resolution** analogy.

In the 90s when you were trying to see a picture on a website the picture will be downloaded completely at the only resolution available, therefore you will see the picture being trace line by line, from the top and taking sometimes several minutes to complete. (For just 1 picture) forget about streaming video back then.

But someone got an idea. What if, when there was a picture on a website, the picture will be available on different levels of resolution?

The picture will be downloaded first at a lower resolution, very much pixelated but very quick, and then it will be downloaded over that first impression several times, improving the resolution each time.

The size of the collection of pictures will be larger and the time to download was slightly bigger, but the feeling of watching the whole picture from the beginning and see how the resolution was improving in front of our eyes was a much better approach than seeing the picture being drawn line by line.

This is the approach I like to take when learning maths. Do a first pass learning things at "low resolution" don't go very much into deep. Not trying to understand everything from the beginning. Then go back and try a different but similar problem and try to go a little bit deeper this time. If there is something that you don't understand, don't get stuck in there for so much, give priority to get the problem solved and get the right answer. Write down what you don't understand and keep going.

Each time you do a problem, you'll get a better understanding of the concept, and you'll let less and less room to ignore things that you don't understand.

Then, when you are familiar with the concept, you can have the luxury of downloading the picture at maximum resolution and stop at every detail that you don't understand.

Trying to do this from the beginning will get you stuck with half of the picture, and it will be very difficult to see where are you going and what are you trying to achieve.

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