<instructions>Your goal is to analyze the following comment and estimate how highly it will be upvoted by the Hacker News community.</instructions><comment><author>bmitc</author><text>The problem exists in the educational methods, in my opinion. It seems most, not just in mathematics, feel that learning is the same as if the student can simply be played a tape that cannot be paused or rewound but can sometimes be sped up. Nearly nothing works like this, and it’s just that the penalties in mathematics are greater.&lt;p&gt;I think of the ideal learning path as a sort of circular, recursive path that bends back and goes back over itself from time to time. Imagine a human painting a canvas. One does not simply do a raster scan, painting pixel by pixel in a linear fashion. By the time the painting is done, the piece has been gone over multiple times with multiple layers of finer and finer detail. A painting is a mishmash of broad and finer strokes, with some layers simply forming foundational layers for the later portions.&lt;p&gt;A pyramid is the wrong metaphor. In my opinion, learning mathematics should be like creating a painting.&lt;p&gt;One of the best examples of this I know of is the book &lt;i&gt;Advanced Calculus: A Differential Forms Approach&lt;/i&gt; by Harold Edwards. The first three chapters introduce the material heuristically. The next three chapters circle back on the material, thoroughly proving everything introduced in the first three chapters. The remaining chapters greet the student with applications and extensions of the material. In the preface, Edwards states that he wanted the book to be able to be opened to any page and be read and make sense. It’s a wonderful goal, and he achieves it.</text><parent_chain><item><author>lmilcin</author><text>I studied theoretical mathematics.&lt;p&gt;The absolutely biggest and most common error and really the only one worth to be on list is not internalizing and understanding entirety of material as the course progresses.&lt;p&gt;Everything else should be basically to achieve goal of understanding the entire course material. It is ok to forget later, as long as you are sure you understood it at least when there was a lot of discussion about that specific part of course material.&lt;p&gt;If you don&amp;#x27;t understand something today, it is very likely the material tomorrow will refer to it and you will not understand that either. Even if you try to make up for it in couple of days, it will be rushed, will require more effort and your brain connections will not be the same quality.&lt;p&gt;Every person will have a different way to achieve that goal. Some people need to ask questions, some like to figure out by themselves, some will want to read, solve exercises for as long as necessary.&lt;p&gt;I have observed many people drop off and the reason was almost invariably the same. A bit of material passes by or maybe the load is too much, and the slippery slope of not understanding starts.&lt;p&gt;Especially when you start to study mathematics, the first couple semesters are foundational and if you don&amp;#x27;t understand something it is like a pyramid, the further in time the more connections with the material that you did not understood and the more trouble you are in.&lt;p&gt;I have seen people drop off because they thought it is the same as in their past or in other study areas. It is not. Mathematics is extremely connected internally and extremely intolerant of ignorance.</text></item></parent_chain></comment><story><title>The Most Common Errors in Undergraduate Mathematics (2009)</title><url>https://math.vanderbilt.edu/schectex/commerrs/</url></story>
<instructions>Your goal is to analyze the following comment and estimate how highly it will be upvoted by the Hacker News community.</instructions><comment><author>grayclhn</author><text>I sort of agree, but your advice is targeted at an entirely different group of students than the article&amp;#x27;s. If someone is confused about how to interpret sin^2(x) vs sin^{-1}(x), there&amp;#x27;s no amount of &amp;quot;understanding&amp;quot; that&amp;#x27;s going to help them (it really is just idiosyncratic, inconsistent notation), but a checklist like the article might clue them in.&lt;p&gt;I think students often underestimate what &amp;quot;understand&amp;quot; means in a math class. The progression you&amp;#x27;re describing (&amp;quot;the first couple of semesters are foundational...&amp;quot;) is definitely not universal in US undergrad programs, though. There&amp;#x27;s just not that much coordination between professors.</text><parent_chain><item><author>lmilcin</author><text>I studied theoretical mathematics.&lt;p&gt;The absolutely biggest and most common error and really the only one worth to be on list is not internalizing and understanding entirety of material as the course progresses.&lt;p&gt;Everything else should be basically to achieve goal of understanding the entire course material. It is ok to forget later, as long as you are sure you understood it at least when there was a lot of discussion about that specific part of course material.&lt;p&gt;If you don&amp;#x27;t understand something today, it is very likely the material tomorrow will refer to it and you will not understand that either. Even if you try to make up for it in couple of days, it will be rushed, will require more effort and your brain connections will not be the same quality.&lt;p&gt;Every person will have a different way to achieve that goal. Some people need to ask questions, some like to figure out by themselves, some will want to read, solve exercises for as long as necessary.&lt;p&gt;I have observed many people drop off and the reason was almost invariably the same. A bit of material passes by or maybe the load is too much, and the slippery slope of not understanding starts.&lt;p&gt;Especially when you start to study mathematics, the first couple semesters are foundational and if you don&amp;#x27;t understand something it is like a pyramid, the further in time the more connections with the material that you did not understood and the more trouble you are in.&lt;p&gt;I have seen people drop off because they thought it is the same as in their past or in other study areas. It is not. Mathematics is extremely connected internally and extremely intolerant of ignorance.</text></item></parent_chain></comment><story><title>The Most Common Errors in Undergraduate Mathematics (2009)</title><url>https://math.vanderbilt.edu/schectex/commerrs/</url></story>