<story><title>Why do many math books have so much detail and so little enlightenment? (2010)</title><url>https://mathoverflow.net/questions/13089/why-do-so-many-textbooks-have-so-much-technical-detail-and-so-little-enlightenme</url></story><parent_chain><item><author>infinity0</author><text>&amp;gt; Most of that foundational information is lost when it&amp;#x27;s not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject.&lt;p&gt;From &amp;quot;A Mathematician&amp;#x27;s Apology&amp;quot;, G. H. Hardy:&lt;p&gt;&amp;gt; Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain.&lt;p&gt;From another angle: when developing new theories or models, your thoughts are all over the place and frankly it&amp;#x27;s &lt;i&gt;boring&lt;/i&gt; to go over your own crappy notes afterwards and try to reconstruct them in a way that others can understand. And much of the time you forget &lt;i&gt;exactly&lt;/i&gt; what happened along the way as well, so any story you reconstruct is going to have some hindsight bias, which defeats the purpose of trying to &amp;quot;teach the story&amp;quot;.&lt;p&gt;Really, from the first answer:&lt;p&gt;&amp;gt; Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through &amp;quot;guided struggle&amp;quot;.&lt;p&gt;This is the only way to &amp;quot;properly&amp;quot; learn mathematics or science. Anything else is only making you &lt;i&gt;think&lt;/i&gt; you&amp;#x27;ve learnt something.</text></item><item><author>jessriedel</author><text>The top-rated answer is either defeatist, or just rationalization for the sentiment &amp;quot;I had to go through this and figure out everything myself, so you should too&amp;quot;.&lt;p&gt;There is a &lt;i&gt;huge&lt;/i&gt; amount of information encoded in the choice of exactly how to define thing, and which theorems people care about. This reflects a long process of trial-and-error as the field was constructed.  For a famous philosophical treatise on this using the  Euler characteristic as an example, see &amp;quot;Proofs and Refutations” by Imre Lakatos&lt;p&gt;&lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Proofs_and_Refutations&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Proofs_and_Refutations&lt;/a&gt;&lt;p&gt;Part 1: &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;math.berkeley.edu&amp;#x2F;~kpmann&amp;#x2F;Lakatos.pdf&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;math.berkeley.edu&amp;#x2F;~kpmann&amp;#x2F;Lakatos.pdf&lt;/a&gt;&lt;p&gt;Most of that foundational information is lost when it&amp;#x27;s not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject.&lt;p&gt;It&amp;#x27;s a huge problem, it applies to physics textbooks too, and it doesn&amp;#x27;t have to be this way.  Unfortunately, the problem has been known for decades and there&amp;#x27;s not much reason to expect things to change.  (Lakatos wrote the above in 1976.)</text></item></parent_chain><comment><author>jessriedel</author><text>&amp;gt;&amp;gt; Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through &amp;quot;guided struggle&amp;quot;.&lt;p&gt;&amp;gt; This is the only way to &amp;quot;properly&amp;quot; learn mathematics or science. Anything else is only making you think you&amp;#x27;ve learnt something.&lt;p&gt;My point is not just that most textbooks make it too hard&amp;#x2F;inefficient to learn this stuff. My point is that most of it is never learned. The Legendre transform connects the Lagrangian and  Hamiltonian mechanics, the two fundamental formulations of both quantum and classical physics, and yet most physicist cannot tell you &lt;i&gt;why&lt;/i&gt; the transform is defined as it is. The reason is they don&amp;#x27;t take seriously the possibility that we&amp;#x27;ll find non-Lagrangian phenomena, and so they have not been forced to consider what observational and theoretical evidence led to it&amp;#x27;s identification in the first place.</text></comment>
<story><title>Why do many math books have so much detail and so little enlightenment? (2010)</title><url>https://mathoverflow.net/questions/13089/why-do-so-many-textbooks-have-so-much-technical-detail-and-so-little-enlightenme</url></story><parent_chain><item><author>infinity0</author><text>&amp;gt; Most of that foundational information is lost when it&amp;#x27;s not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject.&lt;p&gt;From &amp;quot;A Mathematician&amp;#x27;s Apology&amp;quot;, G. H. Hardy:&lt;p&gt;&amp;gt; Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain.&lt;p&gt;From another angle: when developing new theories or models, your thoughts are all over the place and frankly it&amp;#x27;s &lt;i&gt;boring&lt;/i&gt; to go over your own crappy notes afterwards and try to reconstruct them in a way that others can understand. And much of the time you forget &lt;i&gt;exactly&lt;/i&gt; what happened along the way as well, so any story you reconstruct is going to have some hindsight bias, which defeats the purpose of trying to &amp;quot;teach the story&amp;quot;.&lt;p&gt;Really, from the first answer:&lt;p&gt;&amp;gt; Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through &amp;quot;guided struggle&amp;quot;.&lt;p&gt;This is the only way to &amp;quot;properly&amp;quot; learn mathematics or science. Anything else is only making you &lt;i&gt;think&lt;/i&gt; you&amp;#x27;ve learnt something.</text></item><item><author>jessriedel</author><text>The top-rated answer is either defeatist, or just rationalization for the sentiment &amp;quot;I had to go through this and figure out everything myself, so you should too&amp;quot;.&lt;p&gt;There is a &lt;i&gt;huge&lt;/i&gt; amount of information encoded in the choice of exactly how to define thing, and which theorems people care about. This reflects a long process of trial-and-error as the field was constructed.  For a famous philosophical treatise on this using the  Euler characteristic as an example, see &amp;quot;Proofs and Refutations” by Imre Lakatos&lt;p&gt;&lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Proofs_and_Refutations&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Proofs_and_Refutations&lt;/a&gt;&lt;p&gt;Part 1: &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;math.berkeley.edu&amp;#x2F;~kpmann&amp;#x2F;Lakatos.pdf&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;math.berkeley.edu&amp;#x2F;~kpmann&amp;#x2F;Lakatos.pdf&lt;/a&gt;&lt;p&gt;Most of that foundational information is lost when it&amp;#x27;s not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject.&lt;p&gt;It&amp;#x27;s a huge problem, it applies to physics textbooks too, and it doesn&amp;#x27;t have to be this way.  Unfortunately, the problem has been known for decades and there&amp;#x27;s not much reason to expect things to change.  (Lakatos wrote the above in 1976.)</text></item></parent_chain><comment><author>cousin_it</author><text>I learned math through guided struggle. My high school had a kind of macho attitude about it, we were all about math competitions and pushing through university level stuff etc. Today I feel that was a good way to study, but much of the struggle could&amp;#x27;ve been avoided. You must solve problems to progress, but they don&amp;#x27;t have to be hard problems. They just need to be formulated at level n but require solutions at level n+1. Devising such problems is hard and many teachers don&amp;#x27;t bother, instead they give you definitions at level n+1 right away and make you solve problems about those. That&amp;#x27;s the root of the problem IMO.</text></comment>