limerott

Math major from Germany, interested in a bunch of stuff. Find me on limerott.com

limerott's Comments

Explaining Visual Thinking

Do you visualize the icosahedron as one object or do you split it up and consider each separately, but reminding oneself that it is actually one object?

My answer to your visual thinking riddle is: breath in through your mouth and breath out through mouth + nostrils. But I can't decipher your anagram!

Explaining Visual Thinking

First of all, if you can solve it without visualization, I think that this is preferable, precisely because it is faster. There is no need to force oneself to visualize everything.

To visualize something, you need to create a map from the formal domain you are studying to visual transformations. In other words, you need to understand "what the formula" mean (or at least one way of looking at them). Do you know what it means visually to multiply one complex number to another? If you don't, you will be stuck doing calculations. If you do, then you can visualize it and quickly come up with the solution.

From my experience, some people naturally tend towards visual thinking, while others don't. But if you consistently try to apply it, it will become natural at some point (it may take some time, don't give up prematurely).

One area where a lot of visual thinking is necessary, but that is relatively easy to visualize, is graph theory. Try to prove that a (connected undirected) graph has an Eulerian cycle (i.e. a cycle that contains every edge exactly once) if and only if all of its vertices have even degree.

Explaining Visual Thinking

Say you have two distinct points x and y. Consider all points whose distance to x is the same as to y. What can you say about the location of these points in terms of the line connecting x and y?

Try to solve any geometry puzzle with only your mind and you will be forced to do visual thinking.

Two explanations for variation in human abilities

I really appreciate you looking this up. Now it is clear that these claims are controversial.

Nevertheless, the question still is what learning ability refers to: Is it the ability to comprehend learning material that explains the topic well, or is it the ability to come up with the simple explanations yourself? It seems that the OP refers to the latter. The first kind probably has lower variation. Also, this means that when measuring learning ability, you should ensure that all involved people have access to the same "source". I would be interested in hearing the OP's thoughts on that.

Two explanations for variation in human abilities

I couldn't find it quickly, but I think that I read this on codehorror.

Two explanations for variation in human abilities

In an experiment, a group of people who have never programmed before have been showed how to code. In the end, their skills were evaluated in a coding test. The expectation was that they would be roughly normally distributed. However, the outcome was that the students were clustered in two groups (within each of which you see the expected normal distribution). The students belonging to the first struggled while the second group fared relatively well. The researchers figured out the cause: the students that did better managed to create a mental model of what a variable is (a container or a box that can hold values) while those that struggled didn't manage to do this. Are the students from the second group inherently less intelligent?

Yes, because they did not manage to find the right model on their own. No, because once they were provided a straightforward explanation, they were able to code just as well as the students in the better group.

A framework for speed reading

Thank you for your kind comment. This is why I wrote this!

Don't depend on others to ask for explanations

And the list is not exhaustive by any means. I really think this is a no-brainer.

Don't depend on others to ask for explanations

My opinion is that, in almost all scenarios, if you have a question, you should always ask it. Why?

  1. If this question occurred to you, it probably has occurred to many other people. Thus, there is likely to be general interest.
  2. The speaker (or author) has the opportunity to share more of his expertise, even if he is not about directly answering the question [this one is very underrated; I've had a lot of interesting conversations develop from seemingly foolish questions]
  3. This gives the author the chance to explain his perspective more clearly. Your question may have implicit assumptions that the author rejects; this gives him the opportunity to state what his assumptions were (and thus, his frame of reference)
  4. It allows a dialogue to happen between author and reader, which may result in even more insights (if you have a format in which this is allowed, such as a discussion board)
  5. You will invariably have a slightly different perspective to the author. Two perspectives clashing result in interesting ideas
  6. If the author forgot to write something down that he wanted to say, it gives him the opportunity to state that
  7. The author may wrongly assume certain background knowledge. If your question indicates that his assumption was wrong, he has the opportunity to back up and explain it.
  8. It makes the whole conversation a lot more interesting and engaging, it emboldens others to join it and contribute as well.
  9. Foolish questions are especially useful, as they address the basic concepts of a subject. This is a lot more important than expert question if you want to get a coherent view of it.
  10. The answer provided by the author will ALSO be helpful and beneficial to the other students / listeners / readers. By making him expand on his topic, you are benefiting everyone [also very underrated!]
  11. Authors are often passionate about the topic they are writing or talking about. Giving them the opportunity to talk more about it, or to challenge them with difficult questions, or to ask things that they have never been asked before is something they, too, deeply appreciate.

As long as you stay within reasonable (context-dependent) limits (for example, you should not be the only one asking basic questions while everybody else looks bored) -- you should ALWAYS ask questions. It is definitely worth overcoming the fear of looking foolish. There is nothing more sad than somebody giving a talk and nobody asking follow-up questions. I mean, why not?

A framework for speed reading

This takes on a higher level view of reading than I was intending to cover here. But nevertheless, this is a valuable resource. It reminds me of Adler's How to Read a Book.

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