https://learning.subwiki.org/w/api.php?action=feedcontributions&user=Issa+Rice&feedformat=atomLearning - User contributions [en]2019-02-17T11:47:23ZUser contributionsMediaWiki 1.29.2https://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=713Combinatorial explosion of questions and errors2019-02-16T00:02:55Z<p>Issa Rice: /* Ways to mitigate */</p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.<br />
<br />
Here is an example of this sort of thing, as related by Nate Soares:<ref>https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things</ref><br />
<br />
<blockquote>The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.<br><br>“Dude. Disjunction. ''Dis''junction.”<br><br>These are the things that eat my days.</blockquote><br />
<br />
==Ways to mitigate==<br />
<br />
For the [[explainer]]:<br />
<br />
* test the explanation on many readers to catch potential errors/questions, so that these can be cached<br />
* leave the explanation for a while to make it fresher in your mind<br />
* move to a more interactive format<br />
* allow comments (if it's a blog post)<br />
* explain more of the background material to "uniformize" the audience, to make the audience more predictable<br />
* stipulate conditions on the audience (e.g. say that X and Y are required background reading, or that this is intended for intermediate students)<br />
<br />
For the [[learner]]:<br />
<br />
* make use of peers/tutors/TAs/teachers/question-and-answer-sites (e.g. if you get stuck reading a textbook, ask a peer)<br />
* [[Learning from multiple sources|make use of multiple explanations]]<br />
<br />
==Related phenomena==<br />
<br />
* the thing where course instructors reuse old exam/homework problems it's hard to come up with new problems (this problem also arises from the static/non-interactive nature of learning material)<br />
* the problem with storing static example problems in [[spaced repetition software]], because the user will just memorize the answer instead of treating it like a new problem (this problem also arises from the static/non-interactive nature of learning material)<br />
* inferential distance/lack of uniform background of learners (this problem also arises from the difficulty of anticipating the identity of the learner or how they will react); see e.g. "“There are 10 pre-requisites for understanding concept X. Most people have 6 or seven, and then I write a blog post for each of the 10. Most people, most of the time, feel like they’re reading a thing they already know, yet I did have to write all 10 to be able to get everyone to take the step forward together.”"<ref>https://www.greaterwrong.com/posts/Q924oPJzK92FifuFg/write-a-thousand-roads-to-rome/comment/zCSb8aGbfuLeKKwBP</ref><br />
* there is a kind of "reverse" or "ironic" problem that happens where the explainer ''did'' correctly anticipate a question, but the reader fails to anticipate that the explainer anticipates this, so the reader stops reading as soon as they become confused, when in fact in the next paragraph (or next section or whatever) the question is answered. Some authors try to prevent this by saying things like "more on this soon" or "see section 4.5.3 for details".<br />
* in arguments/debates (which are a special kind of explanation), there is a combinatorial explosion of ''potential'' arguments each side could make, even if the ''actual'' path of arguments is just a single path through this tree. See https://arxiv.org/abs/1805.00899 for more on this.<br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=712Combinatorial explosion of questions and errors2019-02-16T00:02:11Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.<br />
<br />
Here is an example of this sort of thing, as related by Nate Soares:<ref>https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things</ref><br />
<br />
<blockquote>The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.<br><br>“Dude. Disjunction. ''Dis''junction.”<br><br>These are the things that eat my days.</blockquote><br />
<br />
==Ways to mitigate==<br />
<br />
For the [[explainer]]:<br />
<br />
* test the explanation on many readers to catch potential errors/questions, so that these can be cached<br />
* leave the explanation for a while to make it fresher in your mind<br />
* move to a more interactive format<br />
* allow comments (if it's a blog post)<br />
* explain more of the background material to "uniformize" the audience, to make the audience more predictable<br />
* stipulate conditions on the audience (e.g. say that X and Y are required background reading, or that this is intended for intermediate students)<br />
<br />
For the [[learner]]:<br />
<br />
* make use of peers/tutors/TAs/teachers/question-and-answer-sites (e.g. if you get stuck reading a textbook, ask a peer)<br />
* make use of multiple explanations<br />
<br />
==Related phenomena==<br />
<br />
* the thing where course instructors reuse old exam/homework problems it's hard to come up with new problems (this problem also arises from the static/non-interactive nature of learning material)<br />
* the problem with storing static example problems in [[spaced repetition software]], because the user will just memorize the answer instead of treating it like a new problem (this problem also arises from the static/non-interactive nature of learning material)<br />
* inferential distance/lack of uniform background of learners (this problem also arises from the difficulty of anticipating the identity of the learner or how they will react); see e.g. "“There are 10 pre-requisites for understanding concept X. Most people have 6 or seven, and then I write a blog post for each of the 10. Most people, most of the time, feel like they’re reading a thing they already know, yet I did have to write all 10 to be able to get everyone to take the step forward together.”"<ref>https://www.greaterwrong.com/posts/Q924oPJzK92FifuFg/write-a-thousand-roads-to-rome/comment/zCSb8aGbfuLeKKwBP</ref><br />
* there is a kind of "reverse" or "ironic" problem that happens where the explainer ''did'' correctly anticipate a question, but the reader fails to anticipate that the explainer anticipates this, so the reader stops reading as soon as they become confused, when in fact in the next paragraph (or next section or whatever) the question is answered. Some authors try to prevent this by saying things like "more on this soon" or "see section 4.5.3 for details".<br />
* in arguments/debates (which are a special kind of explanation), there is a combinatorial explosion of ''potential'' arguments each side could make, even if the ''actual'' path of arguments is just a single path through this tree. See https://arxiv.org/abs/1805.00899 for more on this.<br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=711Combinatorial explosion of questions and errors2019-02-15T23:55:09Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.<br />
<br />
Here is an example of this sort of thing, as related by Nate Soares:<ref>https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things</ref><br />
<br />
<blockquote>The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.<br><br>“Dude. Disjunction. ''Dis''junction.”<br><br>These are the things that eat my days.</blockquote><br />
<br />
==Related phenomena==<br />
<br />
* the thing where course instructors reuse old exam/homework problems it's hard to come up with new problems (this problem also arises from the static/non-interactive nature of learning material)<br />
* the problem with storing static example problems in [[spaced repetition software]], because the user will just memorize the answer instead of treating it like a new problem (this problem also arises from the static/non-interactive nature of learning material)<br />
* inferential distance/lack of uniform background of learners (this problem also arises from the difficulty of anticipating the identity of the learner or how they will react); see e.g. "“There are 10 pre-requisites for understanding concept X. Most people have 6 or seven, and then I write a blog post for each of the 10. Most people, most of the time, feel like they’re reading a thing they already know, yet I did have to write all 10 to be able to get everyone to take the step forward together.”"<ref>https://www.greaterwrong.com/posts/Q924oPJzK92FifuFg/write-a-thousand-roads-to-rome/comment/zCSb8aGbfuLeKKwBP</ref><br />
* there is a kind of "reverse" or "ironic" problem that happens where the explainer ''did'' correctly anticipate a question, but the reader fails to anticipate that the explainer anticipates this, so the reader stops reading as soon as they become confused, when in fact in the next paragraph (or next section or whatever) the question is answered. Some authors try to prevent this by saying things like "more on this soon" or "see section 4.5.3 for details".<br />
* in arguments/debates (which are a special kind of explanation), there is a combinatorial explosion of ''potential'' arguments each side could make, even if the ''actual'' path of arguments is just a single path through this tree. See https://arxiv.org/abs/1805.00899 for more on this.<br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=710Combinatorial explosion of questions and errors2019-02-15T23:51:43Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.<br />
<br />
Here is an example of this sort of thing, as related by Nate Soares:<ref>https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things</ref><br />
<br />
<blockquote>The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.<br><br>“Dude. Disjunction. ''Dis''junction.”<br><br>These are the things that eat my days.</blockquote><br />
<br />
Some related phenomena:<br />
<br />
* the thing where course instructors reuse old exam/homework problems it's hard to come up with new problems (this problem also arises from the static/non-interactive nature of learning material)<br />
* the problem with storing static example problems in [[spaced repetition software]], because the user will just memorize the answer instead of treating it like a new problem (this problem also arises from the static/non-interactive nature of learning material)<br />
* inferential distance/lack of uniform background of learners (this problem also arises from the difficulty of anticipating the identity of the learner or how they will react); see e.g. "“There are 10 pre-requisites for understanding concept X. Most people have 6 or seven, and then I write a blog post for each of the 10. Most people, most of the time, feel like they’re reading a thing they already know, yet I did have to write all 10 to be able to get everyone to take the step forward together.”"<ref>https://www.greaterwrong.com/posts/Q924oPJzK92FifuFg/write-a-thousand-roads-to-rome/comment/zCSb8aGbfuLeKKwBP</ref><br />
* there is a kind of "reverse" or "ironic" problem that happens where the explainer ''did'' correctly anticipate a question, but the reader fails to anticipate that the explainer anticipates this, so the reader stops reading as soon as they become confused, when in fact in the next paragraph (or next section or whatever) the question is answered. Some authors try to prevent this by saying things like "more on this soon" or "see section 4.5.3 for details".<br />
* in arguments/debates (which are a special kind of explanation), there is a combinatorial explosion of ''potential'' arguments each side could make, even if the ''actual'' path of arguments is just a single path through this tree. See https://arxiv.org/abs/1805.00899 for more on this.<br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=709Combinatorial explosion of questions and errors2019-02-15T23:47:40Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.<br />
<br />
Here is an example of this sort of thing, as related by Nate Soares:<ref>https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things</ref><br />
<br />
<blockquote>The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.<br><br>“Dude. Disjunction. ''Dis''junction.”<br><br>These are the things that eat my days.</blockquote><br />
<br />
Some related phenomena:<br />
<br />
* the thing where course instructors reuse old exam/homework problems it's hard to come up with new problems (this problem also arises from the static/non-interactive nature of learning material)<br />
* the problem with storing static example problems in [[spaced repetition software]], because the user will just memorize the answer instead of treating it like a new problem (this problem also arises from the static/non-interactive nature of learning material)<br />
* inferential distance/lack of uniform background of learners (this problem also arises from the difficulty of anticipating the identity of the learner or how they will react); see e.g. "“There are 10 pre-requisites for understanding concept X. Most people have 6 or seven, and then I write a blog post for each of the 10. Most people, most of the time, feel like they’re reading a thing they already know, yet I did have to write all 10 to be able to get everyone to take the step forward together.”"<ref>https://www.greaterwrong.com/posts/Q924oPJzK92FifuFg/write-a-thousand-roads-to-rome/comment/zCSb8aGbfuLeKKwBP</ref><br />
* there is a kind of "reverse" or "ironic" problem that happens where the explainer ''did'' correctly anticipate a question, but the reader fails to anticipate that the explainer anticipates this, so the reader stops reading as soon as they become confused, when in fact in the next paragraph (or next section or whatever) the question is answered. Some authors try to prevent this by saying things like "more on this soon" or "see section 4.5.3 for details".<br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=708Combinatorial explosion of questions and errors2019-02-15T23:45:29Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.<br />
<br />
Here is an example of this sort of thing, as related by Nate Soares:<ref>https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things</ref><br />
<br />
<blockquote>The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.<br><br>“Dude. Disjunction. ''Dis''junction.”<br><br>These are the things that eat my days.</blockquote><br />
<br />
Some related phenomena:<br />
<br />
* the thing where course instructors reuse old exam/homework problems it's hard to come up with new problems (this problem also arises from the static/non-interactive nature of learning material)<br />
* the problem with storing static example problems in [[spaced repetition software]], because the user will just memorize the answer instead of treating it like a new problem (this problem also arises from the static/non-interactive nature of learning material)<br />
* inferential distance/lack of uniform background of learners (this problem also arises from the difficulty of anticipating the identity of the learner or how they will react); see e.g. "“There are 10 pre-requisites for understanding concept X. Most people have 6 or seven, and then I write a blog post for each of the 10. Most people, most of the time, feel like they’re reading a thing they already know, yet I did have to write all 10 to be able to get everyone to take the step forward together.”"<ref>https://www.greaterwrong.com/posts/Q924oPJzK92FifuFg/write-a-thousand-roads-to-rome/comment/zCSb8aGbfuLeKKwBP</ref><br />
* there is a kind of "reverse" or "ironic" problem that happens where the explainer ''did'' correctly anticipate a question, but the reader fails to anticipate that the explainer anticipates this, so the reader stops reading as soon as they become confused, when in fact in the next paragraph (or next section or whatever) the question is answered.<br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=707Combinatorial explosion of questions and errors2019-02-15T23:39:51Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.<br />
<br />
Here is an example of this sort of thing, as related by Nate Soares:<ref>https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things</ref><br />
<br />
<blockquote>The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.<br><br>“Dude. Disjunction. ''Dis''junction.”<br><br>These are the things that eat my days.</blockquote><br />
<br />
Some related phenomena:<br />
<br />
* the thing where course instructors reuse old exam/homework problems it's hard to come up with new problems (this problem also arises from the static/non-interactive nature of learning material)<br />
* the problem with storing static example problems in [[spaced repetition software]], because the user will just memorize the answer instead of treating it like a new problem (this problem also arises from the static/non-interactive nature of learning material)<br />
* inferential distance/lack of uniform background of learners (this problem also arises from the difficulty of anticipating the identity of the learner or how they will react); see e.g. "“There are 10 pre-requisites for understanding concept X. Most people have 6 or seven, and then I write a blog post for each of the 10. Most people, most of the time, feel like they’re reading a thing they already know, yet I did have to write all 10 to be able to get everyone to take the step forward together.”"<ref>https://www.greaterwrong.com/posts/Q924oPJzK92FifuFg/write-a-thousand-roads-to-rome/comment/zCSb8aGbfuLeKKwBP</ref><br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=706Combinatorial explosion of questions and errors2019-02-15T23:34:08Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.<br />
<br />
Here is an example of this sort of thing, as related by Nate Soares:<ref>https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things</ref><br />
<br />
<blockquote>The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.<br><br>“Dude. Disjunction. ''Dis''junction.”<br><br>These are the things that eat my days.</blockquote><br />
<br />
Some related phenomena (these problems also arise from the static/non-interactive nature of learning material):<br />
<br />
* the thing where course instructors reuse old exam/homework problems it's hard to come up with new problems<br />
* the problem with storing static example problems in [[spaced repetition software]], because the user will just memorize the answer instead of treating it like a new problem<br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=705Combinatorial explosion of questions and errors2019-02-15T23:29:44Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.<br />
<br />
Here is an example of this sort of thing, as related by Nate Soares:<ref>https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things</ref><br />
<br />
<blockquote>The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.<br><br>“Dude. Disjunction. ''Dis''junction.”<br><br>These are the things that eat my days.</blockquote><br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=704Combinatorial explosion of questions and errors2019-02-15T23:29:21Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.<br />
<br />
Here is an example of this sort of thing, as related by Nate Soares:<ref>https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things</ref><br />
<br />
<blockquote>The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.<br>“Dude. Disjunction. ''Dis''junction.”<br>These are the things that eat my days.</blockquote><br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=703Combinatorial explosion of questions and errors2019-02-15T23:28:39Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.<br />
<br />
Here is an example of this sort of thing, as related by Nate Soares:<ref>https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things</ref><br />
<br />
<blockquote>The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.<br><br>“Dude. Disjunction. ''Dis''junction.”<br><br>These are the things that eat my days.</blockquote><br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=702Combinatorial explosion of questions and errors2019-02-15T23:23:59Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=701Combinatorial explosion of questions and errors2019-02-15T23:21:45Z<p>Issa Rice: </p>
<hr />
<div>Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=700Combinatorial explosion of questions and errors2019-02-15T23:21:26Z<p>Issa Rice: </p>
<hr />
<div>Non-interactive explanations (e.g. textbooks, blog posts, YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=699Combinatorial explosion of questions and errors2019-02-15T23:21:01Z<p>Issa Rice: Created page with "Non-interactive explanations (e.g. textbooks, blog posts, YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinat..."</p>
<hr />
<div>Non-interactive explanations (e.g. textbooks, blog posts, YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Incubation-based_studying&diff=698Incubation-based studying2019-02-14T23:49:53Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>'''Incubation-based studying''' (there might be a better or more standard term) is the idea that one can make more progress on solving a problem/displaying creativity by working on the problem in a concentrated manner, then leaving the problem aside, then coming back to the problem after a break (it is after the break that the problem gets solved).<br />
<br />
The important thing here is that this term should be agnostic about the underlying mechanism (so maybe "incubation" isn't such a good term after all, since it seems to single out the unconscious processing mechanism), or at least there should be a term reserved to refer to the overall phenomenon in a mechanism-agnostic manner. It might be due to "subconscious processing" or it might be due to looking fresh at the problem, or it might be due to something else entirely, or some combination of these things.<br />
<br />
Of course, depending on the mechanism, specific strategies when studying can change. For example, (1) how hard one tries during the initial concentrated session, (2) how long the break is, (3) how many "parallel threads" to have for different problems, and (4) what one does during the break, are all parameters that can be tweaked when one applies this technique, and their optimal values seem to depend on the underlying mechanism. For example, if "subconscious processing" is the underlying mechanism, then presumably one cannot "subconsciously work on" hundreds of problems simultaneously. On the other hand, if the underlying mechanism is that this method gives a "fresh look" at problems, then one might want to attempt as many problems as possible, to "flush the buffer".<br />
<br />
==Notes==<br />
<br />
TODO: process the quotes below into the article.<br />
<br />
In his book ''How to Become a Straight-A Student'' [[Cal Newport]] writes:<br />
<br />
<blockquote>Next, try to solve the problem in the most obvious way possible. This, of course, probably won't work, because most difficult problems are tricky by nature. By failing in this initial approach, however, you will have at least identified what makes this problem hard. Now you are ready to try to come up with a real solution.<br /><br/ >The next step is counterintuitive. After you've primed the problem, put away your notes and move on to something else. Instead of trying to force a solution, think about the problem in between other activities. As you walk across campus, wait in line at the dining hall, or take a shower, bring up the problem in your head and start thinking through solutions. You might even want to go on a quiet hike or long car ride dedicated entirely to mulling over the question at hand.<br /><br />More often than not, after enough mobile consideration, you will finally stumble across a solution. Only then should you schedule more time to go back to the problem set, write it down formally, and work out the kinks. It's unclear exactly ''why'' solving problems is easier when you're on the go, but, whatever the explanation, it has worked for many students. Even better, it saves a lot of time, since most of your thinking has been done in little interludes between other activities, not during big blocks of valuable free time.</blockquote><br />
<br />
In her book ''A Mind for Numbers'', [[Barbara Oakley]] distinguishes between "focused mode" (same thing as [[wikipedia:task-positive network]]?) and "diffuse mode" (same thing as [[wikipedia:task-negative network]]?):<br />
<br />
<blockquote>Focused-mode thinking is essential for studying math and science. It involves a direct approach to solving problems using rational, sequential, analytical approaches. [...] Diffuse-mode thinking is also essential for learning math and science. It allows us to suddenly gain a new insight on a problem we’ve been struggling with and is associated with “big-picture” perspectives. Diffuse-mode thinking is what happens when you relax your attention and just let your mind wander. This relaxation can allow different areas of the brain to hook up and return valuable insights.</blockquote><br />
<br />
In his book ''The Mind Is Flat'', Nick Chater doesn't deny the phenomenon, but disagrees that this sort of incubation-based thinking works due to unconsciously working on the problem:<br />
<br />
<blockquote>Poincaré and Hindemith cannot possibly be right. If they are spending their days actively thinking about other things, their brains are not unobtrusively solving deep mathematical problems or composing complex pieces of music, perhaps over days or weeks, only to reveal the results in a sudden flash. Yet, driven by the intuitive appeal of unconscious thought, psychologists have devoted a great deal of energy in searching for evidence for unconscious mental work. In these studies, they typically give people some tricky problems to solve (e.g. a list of anagrams); after a relatively short period of time, they might instruct participants to continue, to take a break, to do another similar or different mental task, or even get a night’s sleep, before resuming their problems. According to the ‘unconscious work’ perspective, resuming after a break should lead to a sudden improvement in performance, compared with people who just keep going with the task. Studies in this area are numerous and varied, but I think the conclusions are easily summarized. First, the effects of breaks of all kinds are either negligible or non-existent: if unconscious work takes place at all, it is sufficiently ineffectual to be barely detectable, despite a century of hopeful attempts. Second, many researchers have argued that the minor effects of taking a break – and indeed, Poincaré’s and Hindemith’s intuitions – have a much more natural explanation, which involves no unconscious thought at all.<br /><br />The simplest version of the idea comes from thinking about why one gets stuck with a difficult problem in the first place. What is special about such problems is that you can’t solve them through a routine set of steps (in contrast, say, to adding up columns of numbers, which is laborious but routine) – you have to look at the problem in the ‘right way’ before you can make progress (e.g. with an anagram, you might need to focus on a few key letters; in deep mathematics or musical composition, the space of options might be large and varied). So ideally, the right approach would be to fluidly explore the range of possible ‘angles’ on the problem, until hitting on the right one. Yet this is not so easy: once we have been looking at the same problem for a while, we feel ourselves to be stuck or going round in circles. Indeed, the cooperative computational style of the brain makes this difficult to avoid.</blockquote><br />
<br />
http://paulgraham.com/top.html<br />
<br />
==See also==<br />
<br />
* [[Interleaving]]<br />
<br />
==External links==<br />
<br />
* [[wikipedia:Incubation (psychology)]]<br />
* https://www.greaterwrong.com/posts/SEq8bvSXrzF4jcdS8/tips-and-tricks-for-answering-hard-questions<br />
* [[wikipedia:Default mode network]]<br />
* [[wikipedia:Einstellung effect]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Learning_from_scattered_resources&diff=697Learning from scattered resources2019-02-14T04:18:26Z<p>Issa Rice: </p>
<hr />
<div>By '''learning from scattered resources''' I mean the kind of learning that effective altruists or rationalists do to catch up on the state-of-the-art in EA/rationalist thinking, or the kind of learning that Duncan Sabien mentions of how he learned parkour: "Speaking as someone who pieced together the discipline of parkour back in 2003, from scattered terrible videos (pre Youtube) and a few internet comment boards—pulling together a cohesive and working practice from even the best writeups is a tremendously difficult task."<ref>https://www.greaterwrong.com/posts/pjGGqmtqf8vChJ9BR/unofficial-canon-on-applied-rationality/comment/rtmmvSjyDw8EMXRW8</ref><br />
<br />
This kind of learning, where one (1) actively goes searching for many resources (each of which contains only a small amount of information) and (2) receives relatively little feedback from people who know about the topic, seems different from the kind of learning that happens in other situations:<br />
<br />
* in school or apprenticeships, there is usually a teacher/mentor or textbook that contains the vast majority of the information that is to be learned<br />
* when learning on the job, there are again coworkers/bosses, but also there's constant feedback on job performance<br />
<br />
I think maybe a lot of learning that happens in various subcultures is like this. As of 2019, I think learning how to use [[spaced repetition software]] is like this.<br />
<br />
Various tricky things that can happen in this kind of learning:<br />
<br />
* there are multiple people giving little bits of info, and different people can have different opinions and can contradict each other<br />
* depending on the discipline, the majority of the people can be untrustworthy or unreliable in other ways<br />
* the identities of the people giving the info are numerous and difficult to verify, so one must rely on one's assessment of the object-level details (as well as cues like grammar and punctuation)<br />
* many topics are not explained well, and have to be pieced together by the [[learner]]<br />
* there can exist a "grapevine" or various private discussions where people give frank thoughts, in contrast to the public discussions (note: even in more established disciplines this happens, e.g. search "secret paper passing network")<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Learning_from_scattered_resources&diff=696Learning from scattered resources2019-02-14T04:16:22Z<p>Issa Rice: </p>
<hr />
<div>By '''learning from scattered resources''' I mean the kind of learning that effective altruists or rationalists do to catch up on the state-of-the-art in EA/rationalist thinking, or the kind of learning that Duncan Sabien mentions of how he learned parkour: "Speaking as someone who pieced together the discipline of parkour back in 2003, from scattered terrible videos (pre Youtube) and a few internet comment boards—pulling together a cohesive and working practice from even the best writeups is a tremendously difficult task."<ref>https://www.greaterwrong.com/posts/pjGGqmtqf8vChJ9BR/unofficial-canon-on-applied-rationality/comment/rtmmvSjyDw8EMXRW8</ref><br />
<br />
This kind of learning, where one (1) actively goes searching for many resources (each of which contains only a small amount of information) and (2) receives relatively little feedback from people who know about the topic, seems different from the kind of learning that happens in other situations:<br />
<br />
* in school or apprenticeships, there is usually a teacher/mentor or textbook that contains the vast majority of the information that is to be learned<br />
* when learning on the job, there are again coworkers/bosses, but also there's constant feedback on job performance<br />
<br />
I think maybe a lot of learning that happens in various subcultures is like this. As of 2019, I think learning how to use [[spaced repetition software]] is like this.<br />
<br />
Various tricky things that can happen in this kind of learning:<br />
<br />
* there are multiple people giving little bits of info, and different people can have different opinions and can contradict each other<br />
* depending on the discipline, the majority of the people can be untrustworthy or unreliable in other ways<br />
* the identities of the people giving the info are numerous and difficult to verify, so one must rely on one's assessment of the object-level details (as well as cues like grammar and punctuation)<br />
* many topics are not explained well, and have to be pieced together by the [[learner]]<br />
* there can exist a "grapevine" or various private discussions where people give frank thoughts, in contrast to the public discussions<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Learning_from_scattered_resources&diff=695Learning from scattered resources2019-02-14T04:14:07Z<p>Issa Rice: </p>
<hr />
<div>By '''learning from scattered resources''' I mean the kind of learning that effective altruists or rationalists do to catch up on the state-of-the-art in EA/rationalist thinking, or the kind of learning that Duncan Sabien mentions of how he learned parkour: "Speaking as someone who pieced together the discipline of parkour back in 2003, from scattered terrible videos (pre Youtube) and a few internet comment boards—pulling together a cohesive and working practice from even the best writeups is a tremendously difficult task."<ref>https://www.greaterwrong.com/posts/pjGGqmtqf8vChJ9BR/unofficial-canon-on-applied-rationality/comment/rtmmvSjyDw8EMXRW8</ref><br />
<br />
This kind of learning, where one (1) actively goes searching for many resources (each of which contains only a small amount of information) and (2) receives relatively little feedback from people who know about the topic, seems different from the kind of learning that happens in other situations:<br />
<br />
* in school or apprenticeships, there is usually a teacher/mentor or textbook that contains the vast majority of the information that is to be learned<br />
* when learning on the job, there are again coworkers/bosses, but also there's constant feedback on job performance<br />
<br />
I think maybe a lot of learning that happens in various subcultures is like this. As of 2019, I think learning how to use [[spaced repetition software]] is like this.<br />
<br />
Various tricky things that can happen in this kind of learning:<br />
<br />
* there are multiple people giving little bits of info, and different people can have different opinions and can contradict each other<br />
* depending on the discipline, the majority of the people can be untrustworthy or unreliable in other ways<br />
* many topics are not explained well, and have to be pieced together by the [[learner]]<br />
* there can exist a "grapevine" or various private discussions where people give frank thoughts, in contrast to the public discussions<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Learning_from_scattered_resources&diff=694Learning from scattered resources2019-02-14T02:46:54Z<p>Issa Rice: </p>
<hr />
<div>By '''learning from scattered resources''' I mean the kind of learning that effective altruists or rationalists do to catch up on the state-of-the-art in EA/rationalist thinking, or the kind of learning that Duncan Sabien mentions of how he learned parkour: "Speaking as someone who pieced together the discipline of parkour back in 2003, from scattered terrible videos (pre Youtube) and a few internet comment boards—pulling together a cohesive and working practice from even the best writeups is a tremendously difficult task."<ref>https://www.greaterwrong.com/posts/pjGGqmtqf8vChJ9BR/unofficial-canon-on-applied-rationality/comment/rtmmvSjyDw8EMXRW8</ref><br />
<br />
This kind of learning, where one (1) actively goes searching for many resources (each of which contains only a small amount of information) and (2) receives relatively little feedback from people who know about the topic, seems different from the kind of learning that happens in other situations:<br />
<br />
* in school or apprenticeships, there is usually a teacher/mentor or textbook that contains the vast majority of the information that is to be learned<br />
* when learning on the job, there are again coworkers/bosses, but also there's constant feedback on job performance<br />
<br />
I think maybe a lot of learning that happens in various subcultures is like this. As of 2019, I think learning how to use [[spaced repetition software]] is like this.<br />
<br />
Various tricky things that can happen in this kind of learning:<br />
<br />
* there are multiple people giving little bits of info, and different people can have different opinions and can contradict each other<br />
* depending on the discipline, the majority of the people can be untrustworthy or unreliable in other ways<br />
* many topics are not explained well, and have to be pieced together by the [[learner]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Learning_from_scattered_resources&diff=693Learning from scattered resources2019-02-14T02:43:11Z<p>Issa Rice: </p>
<hr />
<div>By '''learning from scattered resources''' I mean the kind of learning that effective altruists or rationalists do to catch up on the state-of-the-art in EA/rationalist thinking, or the kind of learning that Duncan Sabien mentions of how he learned parkour: "Speaking as someone who pieced together the discipline of parkour back in 2003, from scattered terrible videos (pre Youtube) and a few internet comment boards—pulling together a cohesive and working practice from even the best writeups is a tremendously difficult task."<ref>https://www.greaterwrong.com/posts/pjGGqmtqf8vChJ9BR/unofficial-canon-on-applied-rationality/comment/rtmmvSjyDw8EMXRW8</ref><br />
<br />
This kind of learning, where one (1) actively goes searching for many resources (each of which contains only a small amount of information) and (2) receives relatively little feedback from people who know about the topic, seems different from the kind of learning that happens in other situations:<br />
<br />
* in school or apprenticeships, there is usually a teacher/mentor or textbook that contains the vast majority of the information that is to be learned<br />
* when learning on the job, there are again coworkers/bosses, but also there's constant feedback on job performance<br />
<br />
I think maybe a lot of learning that happens in various subcultures is like this. As of 2019, I think learning how to use [[spaced repetition software]] is like this.<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Learning_from_scattered_resources&diff=692Learning from scattered resources2019-02-14T02:41:22Z<p>Issa Rice: Created page with "By '''learning from scattered resources''' I mean the kind of learning that effective altruists or rationalists do to catch up on the state-of-the-art in EA/rationalist thinki..."</p>
<hr />
<div>By '''learning from scattered resources''' I mean the kind of learning that effective altruists or rationalists do to catch up on the state-of-the-art in EA/rationalist thinking, or the kind of learning that Duncan Sabien mentions of how he learned parkour: "Speaking as someone who pieced together the discipline of parkour back in 2003, from scattered terrible videos (pre Youtube) and a few internet comment boards—pulling together a cohesive and working practice from even the best writeups is a tremendously difficult task."<ref>https://www.greaterwrong.com/posts/pjGGqmtqf8vChJ9BR/unofficial-canon-on-applied-rationality/comment/rtmmvSjyDw8EMXRW8</ref><br />
<br />
This kind of learning, where one (1) actively goes searching for many resources (each of which contains only a small amount of information) and (2) receives relatively little feedback from people who know about the topic, seems different from the kind of learning that happens in other situations:<br />
<br />
* in school or apprenticeships, there is usually a teacher/mentor or textbook that contains the vast majority of the information that is to be learned<br />
* when learning on the job, there are again coworkers/bosses, but also there's constant feedback on job performance<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Positive_and_negative_example&diff=691Positive and negative example2019-02-14T02:32:28Z<p>Issa Rice: </p>
<hr />
<div>Both '''positive and negative examples''' are useful in learning.<br />
<br />
Examples:<br />
<br />
* in math: examples and counterexamples of objects<br />
* whitelists vs blacklists<br />
* in "Unteachable Excellence"<ref>https://www.greaterwrong.com/posts/34Tu4SCK5r5Asdrn3/unteachable-excellence</ref> [[Eliezer Yudkowsky]] makes the point that in some cases it is easier to teach what not to do (negative examples; e.g. listing out various failure modes) than it is to teach what to do (positive examples; e.g. verbalizing hard-to-articulate intuitions)<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Positive_and_negative_example&diff=690Positive and negative example2019-02-14T02:32:19Z<p>Issa Rice: Created page with "Both ''positive and negative examples''' are useful in learning. Examples: * in math: examples and counterexamples of objects * whitelists vs blacklists * in "Unteachable Ex..."</p>
<hr />
<div>Both ''positive and negative examples''' are useful in learning.<br />
<br />
Examples:<br />
<br />
* in math: examples and counterexamples of objects<br />
* whitelists vs blacklists<br />
* in "Unteachable Excellence"<ref>https://www.greaterwrong.com/posts/34Tu4SCK5r5Asdrn3/unteachable-excellence</ref> [[Eliezer Yudkowsky]] makes the point that in some cases it is easier to teach what not to do (negative examples; e.g. listing out various failure modes) than it is to teach what to do (positive examples; e.g. verbalizing hard-to-articulate intuitions)<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Learning_through_osmosis&diff=689Learning through osmosis2019-02-13T04:16:03Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>'''Learning through osmosis''' (also called '''learning by osmosis''', '''learning via osmosis''', and '''osmotic learning''') is the idea that one can learn things through a mysterious/not-well-understood method where one immerses oneself in some environment.<br />
<br />
==Notes==<br />
<br />
Is this how people learn their native language?<br />
<br />
In math:<br />
<br />
<blockquote>Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier.<ref>Ravi Vakil. [http://math.stanford.edu/~vakil/potentialstudents.html "For potential Ph.D. students"].</ref></blockquote><br />
<br />
https://www.greaterwrong.com/search?q=osmosis<br />
<br />
https://www.greaterwrong.com/posts/zLZDxXbcXP3hdM3sh/osmosis-learning-a-crucial-consideration-for-the-craft<br />
<br />
https://www.greaterwrong.com/posts/9SaAyq7F7MAuzAWNN/teaching-the-unteachable<br />
<br />
==See also==<br />
<br />
* [[Importance of struggling in learning]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=688Difficulty of learning mathematics2019-02-13T02:39:51Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
[https://files.vipulnaik.com/math-196/linearalgebrabeware.pdf Vipul Naik]: "While linear algebra lacks any ''single'' compelling visual tool, it requires ''either'' considerable visuo-spatial skill ''or'' considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."<br />
<br />
[https://www.quora.com/Was-there-a-certain-age-when-abstract-physics-or-math-concepts-clicked-and-suddenly-made-sense-or-has-it-always-come-naturally-to-you/answer/Brian-Bi Brian Bi]: "Easy stuff (basic algebra, trig, differential and integral calculus) clicked instantly. Slightly harder stuff like multivariate calculus and linear algebra didn't click right away---I had to go through them a few times. And more advanced topics like group theory haven't clicked yet, but I'm confident they will eventually."<br />
<br />
[http://colah.github.io/posts/2015-08-Backprop/ Christopher Olah]: "Derivatives are cheaper than you think. That’s the main lesson to take away from this post. In fact, they’re unintuitively cheap, and us silly humans have had to repeatedly rediscover this fact. That’s an important thing to understand in deep learning. It’s also a really useful thing to know in other fields, and only more so if it isn’t common knowledge."<br />
<br />
https://www.facebook.com/vipulnaik.r/posts/10201718168211884 ; somewhat related are https://www.greaterwrong.com/posts/EByDsY9S3EDhhfFzC/some-thoughts-on-metaphilosophy/comment/Gh7S2NwJMeYjypDnH and https://www.greaterwrong.com/posts/de3xjFaACCAk6imzv/towards-a-new-decision-theory/comment/DRQciHjC8GMonBCFe<br />
<br />
[http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/ Michael Nielsen on Simpson's paradox]: "Now, I’ll confess that before learning about Simpson’s paradox, I would have unhesitatingly done just as I suggested a naive person would. Indeed, even though I’ve now spent quite a bit of time pondering Simpson’s paradox, I’m not entirely sure I wouldn’t still sometimes make the same kind of mistake. I find it more than a little mind-bending that my heuristics about how to behave on the basis of statistical evidence are obviously not just a little wrong, but utterly, horribly wrong."<br />
<br />
https://en.wikipedia.org/wiki/Monty_Hall_problem#Confusion_and_criticism<br />
<br />
https://www.lesswrong.com/posts/2TPph4EGZ6trEbtku/explainers-shoot-high-aim-low<br />
<br />
Bill Thurston: https://mathoverflow.net/questions/38639/thinking-and-explaining<br />
<br />
[[Tim Gowers]]: "''Mathematics becomes hard.'' Every mathematician will be able to tell you rather precisely when it was that they found that mathematics had stopped being an easy subject that they could understand with very little effort and became a difficult subject that they had to struggle with if they wanted to get anywhere. It isn’t necessarily an advantage if this happens to you later rather than sooner. For example, some Cambridge students find the course difficult right from the start, whereas others largely coast through the first year and then find that they can’t coast through the second year. The people who found it hard in the first year may by this time have developed good study habits that the people who found it easy in the first year do not have." [https://gowers.wordpress.com/2011/09/23/welcome-to-the-cambridge-mathematical-tripos/]<br />
<br />
https://news.ycombinator.com/item?id=7331775 "Mathematicians are chronically lost and confused. It's our natural state of being, and it's okay to keep going and take what insights you can."<br />
<br />
Nate Soares: https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=687Difficulty of learning mathematics2019-02-12T23:39:06Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
[https://files.vipulnaik.com/math-196/linearalgebrabeware.pdf Vipul Naik]: "While linear algebra lacks any ''single'' compelling visual tool, it requires ''either'' considerable visuo-spatial skill ''or'' considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."<br />
<br />
[https://www.quora.com/Was-there-a-certain-age-when-abstract-physics-or-math-concepts-clicked-and-suddenly-made-sense-or-has-it-always-come-naturally-to-you/answer/Brian-Bi Brian Bi]: "Easy stuff (basic algebra, trig, differential and integral calculus) clicked instantly. Slightly harder stuff like multivariate calculus and linear algebra didn't click right away---I had to go through them a few times. And more advanced topics like group theory haven't clicked yet, but I'm confident they will eventually."<br />
<br />
[http://colah.github.io/posts/2015-08-Backprop/ Christopher Olah]: "Derivatives are cheaper than you think. That’s the main lesson to take away from this post. In fact, they’re unintuitively cheap, and us silly humans have had to repeatedly rediscover this fact. That’s an important thing to understand in deep learning. It’s also a really useful thing to know in other fields, and only more so if it isn’t common knowledge."<br />
<br />
https://www.facebook.com/vipulnaik.r/posts/10201718168211884 ; somewhat related is https://www.greaterwrong.com/posts/EByDsY9S3EDhhfFzC/some-thoughts-on-metaphilosophy/comment/Gh7S2NwJMeYjypDnH<br />
<br />
[http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/ Michael Nielsen on Simpson's paradox]: "Now, I’ll confess that before learning about Simpson’s paradox, I would have unhesitatingly done just as I suggested a naive person would. Indeed, even though I’ve now spent quite a bit of time pondering Simpson’s paradox, I’m not entirely sure I wouldn’t still sometimes make the same kind of mistake. I find it more than a little mind-bending that my heuristics about how to behave on the basis of statistical evidence are obviously not just a little wrong, but utterly, horribly wrong."<br />
<br />
https://en.wikipedia.org/wiki/Monty_Hall_problem#Confusion_and_criticism<br />
<br />
https://www.lesswrong.com/posts/2TPph4EGZ6trEbtku/explainers-shoot-high-aim-low<br />
<br />
Bill Thurston: https://mathoverflow.net/questions/38639/thinking-and-explaining<br />
<br />
[[Tim Gowers]]: "''Mathematics becomes hard.'' Every mathematician will be able to tell you rather precisely when it was that they found that mathematics had stopped being an easy subject that they could understand with very little effort and became a difficult subject that they had to struggle with if they wanted to get anywhere. It isn’t necessarily an advantage if this happens to you later rather than sooner. For example, some Cambridge students find the course difficult right from the start, whereas others largely coast through the first year and then find that they can’t coast through the second year. The people who found it hard in the first year may by this time have developed good study habits that the people who found it easy in the first year do not have." [https://gowers.wordpress.com/2011/09/23/welcome-to-the-cambridge-mathematical-tripos/]<br />
<br />
https://news.ycombinator.com/item?id=7331775 "Mathematicians are chronically lost and confused. It's our natural state of being, and it's okay to keep going and take what insights you can."<br />
<br />
Nate Soares: https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=List_of_illusions_of_understanding&diff=686List of illusions of understanding2019-02-12T23:30:06Z<p>Issa Rice: </p>
<hr />
<div>An '''illusion of understanding''' or '''illusion of knowing''' is a form of poor [[metacognition]] in which one thinks one knows something that one does not in fact know. An illusion of understanding can happen in both [[explainer]]s and [[learner]]s.<br />
<br />
{| class="sortable wikitable"<br />
|-<br />
! Name !! Description !! Whom does this illusion afflict? !! Example<br />
|-<br />
| [[Illusion of transparency]]/curse of knowledge || || [[Explainer]] ||<br />
|-<br />
| [[Double illusion of transparency]] || || [[Explainer]] and [[learner]] simultaneously ||<br />
|-<br />
| [[Illusion of explanatory depth]] || || [[Learner]] ||<br />
|-<br />
| Illusions of the outsourced mind [https://www.coursera.org/lecture/intellectual-humility-science/illusions-of-the-outsourced-mind-74bDf] || || ||<br />
|-<br />
| [[wikipedia:Illusory superiority|Illusory superiority]] || || [[Explainer]] and [[learner]] (not necessarily simultaneously) ||<br />
|-<br />
| [[wikipedia:Impostor syndrome|Impostor syndrome]] || || [[Explainer]] and [[learner]] (not necessarily simultaneously) ||<br />
|}<br />
<br />
"Rereading a chapter a second time, for example, can provide a sense of familiarity or perceptual fluency that we interpret as understanding or comprehension, but may actually be a product of low-level perceptual priming. Similarly, information coming readily to mind can be interpreted as evidence of learning, but could instead be a product of cues that are present in the study situation, but that are unlikely to be present at a later time. We can also be misled by our current performance. Conditions of learning that make performance improve rapidly often fail to support long-term retention and transfer, whereas conditions that create challenges and slow the rate of apparent learning often optimize long-term retention and transfer." [https://teaching.yale-nus.edu.sg/wp-content/uploads/sites/25/2016/02/Making-Things-Hard-on-Yourself-but-in-a-Good-Way-2011.pdf (Bjork and Bjork)]<br />
<br />
http://johnsalvatier.org/blog/2017/reality-has-a-surprising-amount-of-detail (I think this post reinvents the "illusion of explanatory depth" idea)</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=685Difficulty of learning mathematics2019-02-12T23:24:50Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
[https://files.vipulnaik.com/math-196/linearalgebrabeware.pdf Vipul Naik]: "While linear algebra lacks any ''single'' compelling visual tool, it requires ''either'' considerable visuo-spatial skill ''or'' considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."<br />
<br />
[https://www.quora.com/Was-there-a-certain-age-when-abstract-physics-or-math-concepts-clicked-and-suddenly-made-sense-or-has-it-always-come-naturally-to-you/answer/Brian-Bi Brian Bi]: "Easy stuff (basic algebra, trig, differential and integral calculus) clicked instantly. Slightly harder stuff like multivariate calculus and linear algebra didn't click right away---I had to go through them a few times. And more advanced topics like group theory haven't clicked yet, but I'm confident they will eventually."<br />
<br />
[http://colah.github.io/posts/2015-08-Backprop/ Christopher Olah]: "Derivatives are cheaper than you think. That’s the main lesson to take away from this post. In fact, they’re unintuitively cheap, and us silly humans have had to repeatedly rediscover this fact. That’s an important thing to understand in deep learning. It’s also a really useful thing to know in other fields, and only more so if it isn’t common knowledge."<br />
<br />
https://www.facebook.com/vipulnaik.r/posts/10201718168211884 ; somewhat related is https://www.greaterwrong.com/posts/EByDsY9S3EDhhfFzC/some-thoughts-on-metaphilosophy/comment/Gh7S2NwJMeYjypDnH<br />
<br />
[http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/ Michael Nielsen on Simpson's paradox]: "Now, I’ll confess that before learning about Simpson’s paradox, I would have unhesitatingly done just as I suggested a naive person would. Indeed, even though I’ve now spent quite a bit of time pondering Simpson’s paradox, I’m not entirely sure I wouldn’t still sometimes make the same kind of mistake. I find it more than a little mind-bending that my heuristics about how to behave on the basis of statistical evidence are obviously not just a little wrong, but utterly, horribly wrong."<br />
<br />
https://en.wikipedia.org/wiki/Monty_Hall_problem#Confusion_and_criticism<br />
<br />
https://www.lesswrong.com/posts/2TPph4EGZ6trEbtku/explainers-shoot-high-aim-low<br />
<br />
Bill Thurston: https://mathoverflow.net/questions/38639/thinking-and-explaining<br />
<br />
[[Tim Gowers]]: "''Mathematics becomes hard.'' Every mathematician will be able to tell you rather precisely when it was that they found that mathematics had stopped being an easy subject that they could understand with very little effort and became a difficult subject that they had to struggle with if they wanted to get anywhere. It isn’t necessarily an advantage if this happens to you later rather than sooner. For example, some Cambridge students find the course difficult right from the start, whereas others largely coast through the first year and then find that they can’t coast through the second year. The people who found it hard in the first year may by this time have developed good study habits that the people who found it easy in the first year do not have." [https://gowers.wordpress.com/2011/09/23/welcome-to-the-cambridge-mathematical-tripos/]<br />
<br />
https://news.ycombinator.com/item?id=7331775 "Mathematicians are chronically lost and confused. It's our natural state of being, and it's okay to keep going and take what insights you can."<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Tim_Gowers&diff=684Tim Gowers2019-02-12T23:23:28Z<p>Issa Rice: Redirected page to Timothy Gowers</p>
<hr />
<div>#redirect [[Timothy Gowers]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=683Difficulty of learning mathematics2019-02-12T23:22:54Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
[https://files.vipulnaik.com/math-196/linearalgebrabeware.pdf Vipul Naik]: "While linear algebra lacks any ''single'' compelling visual tool, it requires ''either'' considerable visuo-spatial skill ''or'' considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."<br />
<br />
[https://www.quora.com/Was-there-a-certain-age-when-abstract-physics-or-math-concepts-clicked-and-suddenly-made-sense-or-has-it-always-come-naturally-to-you/answer/Brian-Bi Brian Bi]: "Easy stuff (basic algebra, trig, differential and integral calculus) clicked instantly. Slightly harder stuff like multivariate calculus and linear algebra didn't click right away---I had to go through them a few times. And more advanced topics like group theory haven't clicked yet, but I'm confident they will eventually."<br />
<br />
[http://colah.github.io/posts/2015-08-Backprop/ Christopher Olah]: "Derivatives are cheaper than you think. That’s the main lesson to take away from this post. In fact, they’re unintuitively cheap, and us silly humans have had to repeatedly rediscover this fact. That’s an important thing to understand in deep learning. It’s also a really useful thing to know in other fields, and only more so if it isn’t common knowledge."<br />
<br />
https://www.facebook.com/vipulnaik.r/posts/10201718168211884 ; somewhat related is https://www.greaterwrong.com/posts/EByDsY9S3EDhhfFzC/some-thoughts-on-metaphilosophy/comment/Gh7S2NwJMeYjypDnH<br />
<br />
[http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/ Michael Nielsen on Simpson's paradox]: "Now, I’ll confess that before learning about Simpson’s paradox, I would have unhesitatingly done just as I suggested a naive person would. Indeed, even though I’ve now spent quite a bit of time pondering Simpson’s paradox, I’m not entirely sure I wouldn’t still sometimes make the same kind of mistake. I find it more than a little mind-bending that my heuristics about how to behave on the basis of statistical evidence are obviously not just a little wrong, but utterly, horribly wrong."<br />
<br />
https://en.wikipedia.org/wiki/Monty_Hall_problem#Confusion_and_criticism<br />
<br />
https://www.lesswrong.com/posts/2TPph4EGZ6trEbtku/explainers-shoot-high-aim-low<br />
<br />
Bill Thurston: https://mathoverflow.net/questions/38639/thinking-and-explaining<br />
<br />
[[Tim Gowers]]: "''Mathematics becomes hard.'' Every mathematician will be able to tell you rather precisely when it was that they found that mathematics had stopped being an easy subject that they could understand with very little effort and became a difficult subject that they had to struggle with if they wanted to get anywhere. It isn’t necessarily an advantage if this happens to you later rather than sooner. For example, some Cambridge students find the course difficult right from the start, whereas others largely coast through the first year and then find that they can’t coast through the second year. The people who found it hard in the first year may by this time have developed good study habits that the people who found it easy in the first year do not have." [https://gowers.wordpress.com/2011/09/23/welcome-to-the-cambridge-mathematical-tripos/]<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=682Difficulty of learning mathematics2019-02-12T23:19:23Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
[https://files.vipulnaik.com/math-196/linearalgebrabeware.pdf Vipul Naik]: "While linear algebra lacks any ''single'' compelling visual tool, it requires ''either'' considerable visuo-spatial skill ''or'' considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."<br />
<br />
[https://www.quora.com/Was-there-a-certain-age-when-abstract-physics-or-math-concepts-clicked-and-suddenly-made-sense-or-has-it-always-come-naturally-to-you/answer/Brian-Bi Brian Bi]: "Easy stuff (basic algebra, trig, differential and integral calculus) clicked instantly. Slightly harder stuff like multivariate calculus and linear algebra didn't click right away---I had to go through them a few times. And more advanced topics like group theory haven't clicked yet, but I'm confident they will eventually."<br />
<br />
[http://colah.github.io/posts/2015-08-Backprop/ Christopher Olah]: "Derivatives are cheaper than you think. That’s the main lesson to take away from this post. In fact, they’re unintuitively cheap, and us silly humans have had to repeatedly rediscover this fact. That’s an important thing to understand in deep learning. It’s also a really useful thing to know in other fields, and only more so if it isn’t common knowledge."<br />
<br />
https://www.facebook.com/vipulnaik.r/posts/10201718168211884 ; somewhat related is https://www.greaterwrong.com/posts/EByDsY9S3EDhhfFzC/some-thoughts-on-metaphilosophy/comment/Gh7S2NwJMeYjypDnH<br />
<br />
[http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/ Michael Nielsen on Simpson's paradox]: "Now, I’ll confess that before learning about Simpson’s paradox, I would have unhesitatingly done just as I suggested a naive person would. Indeed, even though I’ve now spent quite a bit of time pondering Simpson’s paradox, I’m not entirely sure I wouldn’t still sometimes make the same kind of mistake. I find it more than a little mind-bending that my heuristics about how to behave on the basis of statistical evidence are obviously not just a little wrong, but utterly, horribly wrong."<br />
<br />
https://en.wikipedia.org/wiki/Monty_Hall_problem#Confusion_and_criticism<br />
<br />
https://www.lesswrong.com/posts/2TPph4EGZ6trEbtku/explainers-shoot-high-aim-low<br />
<br />
Bill Thurston: https://mathoverflow.net/questions/38639/thinking-and-explaining<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=681Difficulty of learning mathematics2019-02-12T23:18:17Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
[https://files.vipulnaik.com/math-196/linearalgebrabeware.pdf Vipul Naik]: "While linear algebra lacks any ''single'' compelling visual tool, it requires ''either'' considerable visuo-spatial skill ''or'' considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."<br />
<br />
[https://www.quora.com/Was-there-a-certain-age-when-abstract-physics-or-math-concepts-clicked-and-suddenly-made-sense-or-has-it-always-come-naturally-to-you/answer/Brian-Bi Brian Bi]: "Easy stuff (basic algebra, trig, differential and integral calculus) clicked instantly. Slightly harder stuff like multivariate calculus and linear algebra didn't click right away---I had to go through them a few times. And more advanced topics like group theory haven't clicked yet, but I'm confident they will eventually."<br />
<br />
[http://colah.github.io/posts/2015-08-Backprop/ Christopher Olah]: "Derivatives are cheaper than you think. That’s the main lesson to take away from this post. In fact, they’re unintuitively cheap, and us silly humans have had to repeatedly rediscover this fact. That’s an important thing to understand in deep learning. It’s also a really useful thing to know in other fields, and only more so if it isn’t common knowledge."<br />
<br />
https://www.facebook.com/vipulnaik.r/posts/10201718168211884 ; somewhat related is https://www.greaterwrong.com/posts/EByDsY9S3EDhhfFzC/some-thoughts-on-metaphilosophy/comment/Gh7S2NwJMeYjypDnH<br />
<br />
[http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/ Michael Nielsen on Simpson's paradox]: "Now, I’ll confess that before learning about Simpson’s paradox, I would have unhesitatingly done just as I suggested a naive person would. Indeed, even though I’ve now spent quite a bit of time pondering Simpson’s paradox, I’m not entirely sure I wouldn’t still sometimes make the same kind of mistake. I find it more than a little mind-bending that my heuristics about how to behave on the basis of statistical evidence are obviously not just a little wrong, but utterly, horribly wrong."<br />
<br />
https://en.wikipedia.org/wiki/Monty_Hall_problem#Confusion_and_criticism<br />
<br />
https://www.lesswrong.com/posts/2TPph4EGZ6trEbtku/explainers-shoot-high-aim-low<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=680Difficulty of learning mathematics2019-02-12T23:16:48Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
[https://files.vipulnaik.com/math-196/linearalgebrabeware.pdf Vipul Naik]: "While linear algebra lacks any ''single'' compelling visual tool, it requires ''either'' considerable visuo-spatial skill ''or'' considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."<br />
<br />
[https://www.quora.com/Was-there-a-certain-age-when-abstract-physics-or-math-concepts-clicked-and-suddenly-made-sense-or-has-it-always-come-naturally-to-you/answer/Brian-Bi Brian Bi]: "Easy stuff (basic algebra, trig, differential and integral calculus) clicked instantly. Slightly harder stuff like multivariate calculus and linear algebra didn't click right away---I had to go through them a few times. And more advanced topics like group theory haven't clicked yet, but I'm confident they will eventually."<br />
<br />
[http://colah.github.io/posts/2015-08-Backprop/ Christopher Olah]: "Derivatives are cheaper than you think. That’s the main lesson to take away from this post. In fact, they’re unintuitively cheap, and us silly humans have had to repeatedly rediscover this fact. That’s an important thing to understand in deep learning. It’s also a really useful thing to know in other fields, and only more so if it isn’t common knowledge."<br />
<br />
https://www.facebook.com/vipulnaik.r/posts/10201718168211884 ; somewhat related is https://www.greaterwrong.com/posts/EByDsY9S3EDhhfFzC/some-thoughts-on-metaphilosophy/comment/Gh7S2NwJMeYjypDnH<br />
<br />
[http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/ Michael Nielsen on Simpson's paradox]: "Now, I’ll confess that before learning about Simpson’s paradox, I would have unhesitatingly done just as I suggested a naive person would. Indeed, even though I’ve now spent quite a bit of time pondering Simpson’s paradox, I’m not entirely sure I wouldn’t still sometimes make the same kind of mistake. I find it more than a little mind-bending that my heuristics about how to behave on the basis of statistical evidence are obviously not just a little wrong, but utterly, horribly wrong."<br />
<br />
https://en.wikipedia.org/wiki/Monty_Hall_problem#Confusion_and_criticism<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=679Difficulty of learning mathematics2019-02-12T23:15:42Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
[https://files.vipulnaik.com/math-196/linearalgebrabeware.pdf Vipul Naik]: "While linear algebra lacks any ''single'' compelling visual tool, it requires ''either'' considerable visuo-spatial skill ''or'' considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."<br />
<br />
[https://www.quora.com/Was-there-a-certain-age-when-abstract-physics-or-math-concepts-clicked-and-suddenly-made-sense-or-has-it-always-come-naturally-to-you/answer/Brian-Bi Brian Bi]: "Easy stuff (basic algebra, trig, differential and integral calculus) clicked instantly. Slightly harder stuff like multivariate calculus and linear algebra didn't click right away---I had to go through them a few times. And more advanced topics like group theory haven't clicked yet, but I'm confident they will eventually."<br />
<br />
[http://colah.github.io/posts/2015-08-Backprop/ Christopher Olah]: "Derivatives are cheaper than you think. That’s the main lesson to take away from this post. In fact, they’re unintuitively cheap, and us silly humans have had to repeatedly rediscover this fact. That’s an important thing to understand in deep learning. It’s also a really useful thing to know in other fields, and only more so if it isn’t common knowledge."<br />
<br />
https://www.facebook.com/vipulnaik.r/posts/10201718168211884 ; somewhat related is https://www.greaterwrong.com/posts/EByDsY9S3EDhhfFzC/some-thoughts-on-metaphilosophy/comment/Gh7S2NwJMeYjypDnH<br />
<br />
[http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/ Michael Nielsen on Simpson's paradox]: "Now, I’ll confess that before learning about Simpson’s paradox, I would have unhesitatingly done just as I suggested a naive person would. Indeed, even though I’ve now spent quite a bit of time pondering Simpson’s paradox, I’m not entirely sure I wouldn’t still sometimes make the same kind of mistake. I find it more than a little mind-bending that my heuristics about how to behave on the basis of statistical evidence are obviously not just a little wrong, but utterly, horribly wrong."<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=678Difficulty of learning mathematics2019-02-12T23:14:47Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
[https://files.vipulnaik.com/math-196/linearalgebrabeware.pdf Vipul Naik]: "While linear algebra lacks any ''single'' compelling visual tool, it requires ''either'' considerable visuo-spatial skill ''or'' considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."<br />
<br />
[https://www.quora.com/Was-there-a-certain-age-when-abstract-physics-or-math-concepts-clicked-and-suddenly-made-sense-or-has-it-always-come-naturally-to-you/answer/Brian-Bi Brian Bi]: "Easy stuff (basic algebra, trig, differential and integral calculus) clicked instantly. Slightly harder stuff like multivariate calculus and linear algebra didn't click right away---I had to go through them a few times. And more advanced topics like group theory haven't clicked yet, but I'm confident they will eventually."<br />
<br />
[http://colah.github.io/posts/2015-08-Backprop/ Christopher Olah]: "Derivatives are cheaper than you think. That’s the main lesson to take away from this post. In fact, they’re unintuitively cheap, and us silly humans have had to repeatedly rediscover this fact. That’s an important thing to understand in deep learning. It’s also a really useful thing to know in other fields, and only more so if it isn’t common knowledge."<br />
<br />
https://www.facebook.com/vipulnaik.r/posts/10201718168211884 ; somewhat related is https://www.greaterwrong.com/posts/EByDsY9S3EDhhfFzC/some-thoughts-on-metaphilosophy/comment/Gh7S2NwJMeYjypDnH<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=677Difficulty of learning mathematics2019-02-12T23:13:40Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
[https://files.vipulnaik.com/math-196/linearalgebrabeware.pdf Vipul Naik]: "While linear algebra lacks any ''single'' compelling visual tool, it requires ''either'' considerable visuo-spatial skill ''or'' considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."<br />
<br />
[https://www.quora.com/Was-there-a-certain-age-when-abstract-physics-or-math-concepts-clicked-and-suddenly-made-sense-or-has-it-always-come-naturally-to-you/answer/Brian-Bi Brian Bi]: "Easy stuff (basic algebra, trig, differential and integral calculus) clicked instantly. Slightly harder stuff like multivariate calculus and linear algebra didn't click right away---I had to go through them a few times. And more advanced topics like group theory haven't clicked yet, but I'm confident they will eventually."<br />
<br />
[http://colah.github.io/posts/2015-08-Backprop/ Christopher Olah]: "Derivatives are cheaper than you think. That’s the main lesson to take away from this post. In fact, they’re unintuitively cheap, and us silly humans have had to repeatedly rediscover this fact. That’s an important thing to understand in deep learning. It’s also a really useful thing to know in other fields, and only more so if it isn’t common knowledge."<br />
<br />
https://www.facebook.com/vipulnaik.r/posts/10201718168211884<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=676Difficulty of learning mathematics2019-02-12T23:12:58Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
[https://files.vipulnaik.com/math-196/linearalgebrabeware.pdf Vipul Naik]: "While linear algebra lacks any ''single'' compelling visual tool, it requires ''either'' considerable visuo-spatial skill ''or'' considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."<br />
<br />
[https://www.quora.com/Was-there-a-certain-age-when-abstract-physics-or-math-concepts-clicked-and-suddenly-made-sense-or-has-it-always-come-naturally-to-you/answer/Brian-Bi Brian Bi]: "Easy stuff (basic algebra, trig, differential and integral calculus) clicked instantly. Slightly harder stuff like multivariate calculus and linear algebra didn't click right away---I had to go through them a few times. And more advanced topics like group theory haven't clicked yet, but I'm confident they will eventually."<br />
<br />
[http://colah.github.io/posts/2015-08-Backprop/ Christopher Olah]: "Derivatives are cheaper than you think. That’s the main lesson to take away from this post. In fact, they’re unintuitively cheap, and us silly humans have had to repeatedly rediscover this fact. That’s an important thing to understand in deep learning. It’s also a really useful thing to know in other fields, and only more so if it isn’t common knowledge."<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=675Difficulty of learning mathematics2019-02-12T23:08:53Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
[https://files.vipulnaik.com/math-196/linearalgebrabeware.pdf Vipul Naik]: "While linear algebra lacks any ''single'' compelling visual tool, it requires ''either'' considerable visuo-spatial skill ''or'' considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."<br />
<br />
[https://www.quora.com/Was-there-a-certain-age-when-abstract-physics-or-math-concepts-clicked-and-suddenly-made-sense-or-has-it-always-come-naturally-to-you/answer/Brian-Bi Brian Bi]: "Easy stuff (basic algebra, trig, differential and integral calculus) clicked instantly. Slightly harder stuff like multivariate calculus and linear algebra didn't click right away---I had to go through them a few times. And more advanced topics like group theory haven't clicked yet, but I'm confident they will eventually."<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=674Difficulty of learning mathematics2019-02-12T23:04:05Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
[https://files.vipulnaik.com/math-196/linearalgebrabeware.pdf Vipul Naik]: "While linear algebra lacks any ''single'' compelling visual tool, it requires ''either'' considerable visuo-spatial skill ''or'' considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof."<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=673Difficulty of learning mathematics2019-02-12T23:01:04Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
NOTE: this section isn't really part of the article; it's a place to throw quotes into, with the hope that the quotes will eventually be smoothed out/summarized and placed in the main article.<br />
<br />
Satvik Beri [https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri makes the case] that "math geniuses" are people who have internalized prerequisite concepts so thoroughly that it has become intuitive to them, which allows them to pick up newer concepts quickly.<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=672Difficulty of learning mathematics2019-02-12T22:55:02Z<p>Issa Rice: </p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects? Why do even some really smart people struggle with math? Why does it take so long to learn math?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=671Difficulty of learning mathematics2019-02-12T22:54:20Z<p>Issa Rice: </p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Difficulty_of_learning_mathematics&diff=670Difficulty of learning mathematics2019-02-12T22:54:02Z<p>Issa Rice: Created page with "This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects?..."</p>
<hr />
<div>This page is about the '''difficulty of learning mathematics'''. What makes it difficult to learn math? Why does math seem more difficult (in some ways) than other subjects?<br />
<br />
==Differences in difficulty between pre-rigorous mathematics and rigorous mathematics==<br />
<br />
==Notes==<br />
<br />
==See also==<br />
<br />
* [[Remembering mathematics]]<br />
* [[Mathematics as isolation of logical difficulties of learning]]<br />
* [[Memorization in mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Learning_from_multiple_sources&diff=669Learning from multiple sources2019-02-12T22:29:23Z<p>Issa Rice: </p>
<hr />
<div>'''Learning from multiple sources''' refers to using multiple source materials while learning a subject. For instance, a student may attend lecture and later watch a YouTube video that explains the same concept at home.<br />
<br />
==Examples==<br />
<br />
* An autodidact reads from multiple textbooks<br />
* A student reads a textbook and a blog post explaining the same concept<br />
* A student encounters a concept in the classroom, then later asks their tutor to explain the same concept<br />
* A college student attends lecture and later watches a YouTube video that explains the same concept<br />
<br />
==Discussion==<br />
<br />
In math, learning from multiple sources can give attention to certain contingencies in the subject (e.g. notation, specific constructions, specific encodings of structures) which may have seemed like necessities. This can make the concepts themselves more robust.<br />
<br />
Since in general two people will learn the same subject from different sources, being familiar with other notation/terminology will help with communication.<br />
<br />
Different sources place emphasis on different parts, and overlap may not be exact (see e.g. [[Discursiveness (explanations)|discursiveness]]), so one may in general learn new things by trying multiple sources.<br />
<br />
Some sources will suit one's background knowledge and thinking styles more than others. Finding sources that one "clicks with" will make learning easier.<br />
<br />
[[Importance of struggling in learning]] discusses one possible downside to finding explanations that are "too easy".<br />
<br />
==Applicability==<br />
<br />
For subjects that have been around for a long time and are learned by many people, there will be many different explanations available (e.g. many books on calculus).<br />
<br />
For less popular subjects, there will sometimes be one or two dominant/obvious choices.<br />
<br />
For new or obscure topics, there may only be a single explanation (e.g. the original paper announcing the discovery).<br />
<br />
==Notes==<br />
<br />
From Peter Smith's Teach Yourself Logic guide: "I very strongly recommend tackling an area of logic by reading a series of books which ''overlap'' in level (with the next one covering some of the same ground and then pushing on from the previous one), rather than trying to proceed by big leaps."<ref>https://www.logicmatters.net/tyl/</ref><br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Learning_from_multiple_sources&diff=668Learning from multiple sources2019-02-12T22:29:06Z<p>Issa Rice: </p>
<hr />
<div>'''Learning from multiple sources''' refers to using multiple source materials while learning a subject. For instance, a student may attend lecture and later watch a YouTube video that explains the same concept at home.<br />
<br />
==Examples==<br />
<br />
* An autodidact reads from multiple textbooks<br />
* A student reads a textbook and a blog post explaining the same concept<br />
* A student encounters a concept in the classroom, then later asks their tutor to explain the same concept<br />
* A college student attends lecture and later watches a YouTube video that explains the same concept<br />
<br />
==Discussion==<br />
<br />
In math, learning from multiple sources can give attention to certain contingencies in the subject (e.g. notation, specific constructions, specific encodings of structures) which may have seemed like necessities. This can make the concepts themselves more robust.<br />
<br />
Since in general two people will learn the same subject from different sources, being familiar with other notation/terminology will help with communication.<br />
<br />
Different sources place emphasis on different parts, and overlap may not be exact (see e.g. [[Discursiveness (explanations)|discursiveness]]), so one may in general learn new things by trying multiple sources.<br />
<br />
Some sources will suit one's background knowledge and thinking styles more than others. Finding sources that one "clicks with" will make learning easier.<br />
<br />
[[Importance of struggling in learning]] discusses one possible downside to finding explanations that are "too easy".<br />
<br />
==Applicability==<br />
<br />
For subjects that have been around for a long time and are learned by many people, there will be many different explanations available (e.g. many books on calculus).<br />
<br />
For less popular subjects, there will sometimes be one or two dominant/obvious choices.<br />
<br />
For new or obscure topics, there may only be a single explanation (e.g. the original paper announcing the discovery).<br />
<br />
==Notes==<br />
<br />
From Peter Smith's Teach Yourself Logic guide: "I very strongly recommend tackling an area of logic by reading a series of books which ''overlap'' in level (with the next one covering some of the same ground and then pushing on from the previous one), rather than trying to proceed by big leaps."<ref>https://www.logicmatters.net/tyl/</ref></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Concept_dependency_tracking&diff=667Concept dependency tracking2019-02-12T22:26:37Z<p>Issa Rice: /* Data structures */</p>
<hr />
<div>'''Concept dependency tracking''' (there might be a more standard term) refers to the tracking of conceptual dependencies (e.g. using a dependency DAG) when learning a subject.<br />
<br />
When trying to learn a concept, there might be several conceptual dependencies (i.e. other concepts you must learn first before learning the desired concept). The nature and structure of dependencies can be simple or elaborate depending on what one is trying to learn:<br />
<br />
* If one tries to do a "deep dive" into a subject by first picking some advanced concept (e.g. "I want to learn about Godel's incompleteness theorems") there might be multiple specific (propositional logic, first-order logic, computability) and general dependencies ("mathematical sophistication") that have some complicated structure.<br />
* If one is following a textbook linearly or has already covered the surrounding material, then the marginal concept won't typically have elaborate dependencies.<br />
<br />
"Every time you encounter a concept you don’t recognize, you need to go back and learn it first. Pretty soon you’re deep in dependency hell, switching between twenty tabs, trying to juggle all the prerequisites of prerequisites, wondering if any of this will actually help you towards your original goal." [https://metacademy.org/about]<br />
<br />
The paper "Retain: Building a Concept Recommendation System that Leverages Spaced Repetition to Improve Retention in Educational Settings" by Shilpa Subrahmanyam also talks about this.<br />
<br />
Several books have a graph near the beginning of the book describing the order in which chapters may be read.<br />
<br />
I think most [[learner]]s don't really pay attention to conceptual dependency tracking (they can just follow along in class/read the chosen-by-teachers sections in the textbook). But tracking conceptual dependencies is important for:<br />
<br />
* [[Explainer]]s who want to produce clear explanations.<br />
* Autodidacts who get to "shop around" for multiple explanations. Different books might cover topics in a different order, so one might try reading up a topic in a different book only to find that this book assumes some other knowledge one does not have. Or the proof in the second book might circularly assume the result because it started from a different place.<br />
* Generalists who jump from subject to subject, becoming a complete novice frequently.<br />
<br />
==Types of dependencies==<br />
<br />
There are different "strengths" of dependencies. For example not understanding high school algebra makes it very difficult to understand calculus, so this might be a "hard" dependency. On the other hand, knowing the construction of the real number system helps to make one's understanding of real analysis solid, but is often considered nonessential during a first pass with the subject, so this might be more of a "soft" dependency.<br />
<br />
Some subjects have feedback loops, where subject X helps shed light on subject Y, and conversely, subject Y cements one's understanding of subject X. For this kind of dependency, a DAG is insufficient. (see data structures section)<br />
<br />
When is a dependency graph cyclic or acyclic? Circular dependencies are kind of weird in the sense that if we increase the resolution of concept space, it seems like we can always get the graph to a point where it's no longer circular. Superficially, we might say <math>X</math> and <math>Y</math> depend on each other, but actually, if you break them down, <math>X</math> has parts <math>X'</math> and <math>X''</math>, and (1) <math>Y</math> depends on <math>X'</math> and (2) <math>X''</math> depends on <math>Y</math>, so at this finer resolution, the dependency has no cycle (the graph looks like <math>X' \rightarrow Y \rightarrow X''</math>), but if you look at the original graph with nodes <math>Y</math> and <math>X=\{X', X''\}</math>, then it looks like there's a cycle. Is there a counterexample to this?<br />
<br />
==Data structures==<br />
<br />
* DAGs seem like the natural way to represent dependencies<br />
* [https://www.drmaciver.com/2016/05/how-to-read-a-mathematics-textbook/ David R. MacIver's Current/Pending system] uses a "current list" to track the current concept plus its dependencies.<br />
<br />
==Related ideas==<br />
<br />
In software engineering, the idea of dependencies is used frequently, e.g.<br />
<br />
* Package managers for programming languages and Linux distributions<br />
* https://en.wikipedia.org/wiki/Dependency_hell - the direct analog of this is probably less problematic in mathematics/the conceptual realm because there aren't "version numbers" on concepts and it is easy to shuffle around ideas. Terry Tao: "I think the basic reason for this is that in the purely theoretical world of mathematics, there is basically a zero cost in taking an argument that partially solves a problem, and then combining it with other ideas to make a complete solution; but in the real world, it can be difficult, costly, or socially unacceptable to reuse or recycle anything that is (or is perceived to be) even a partial failure."<ref>[https://plus.google.com/u/0/+TerenceTao27/posts/Xdm8eiPLWZp "One of the secrets to mathematical problem solving is that one needs to place a high value on partial progress, as being a crucial stepping stone to fully solving the problem."] (Post on Google+). July 22, 2012. Retrieved November 2, 2018.</ref><br />
* https://en.wikipedia.org/wiki/Coupling_(computer_programming)<br />
* Tracing up and down layers of abstraction in object-oriented programming, [[wikipedia:Call graph|tracing function calls]]<br />
<br />
==Interactions with other ideas==<br />
<br />
A concept dependency system can interact with other kinds of "tracking" to obtain a more complete "learning scheduling system":<br />
<br />
* Spaced repetition<br />
* A comprehension score (see Subrahmanyam paper)<br />
* Priority of the subject (e.g. "learning math is more important than learning physics", or "learning linear algebra is more important than learning abstract algebra")<br />
* Current excitedness/curiosity about a subject<br />
<br />
==See also==<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Just-in-time_learning&diff=666Just-in-time learning2019-02-12T22:19:39Z<p>Issa Rice: /* See also */</p>
<hr />
<div>'''Just-in-time learning''' seems to be the idea of looking things up as you need them (while trying to perform some task). For example, googling and reading Stack Overflow answers while trying to program an application.<br />
<br />
"Two Studies of Opportunistic Programming: Interleaving Web Foraging, Learning, and Writing Code" Joel Brandt; Philip J. Guo; Joel Lewenstein; Mira Dontcheva; Scott R. Klemmer. [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.216.682&rep=rep1&type=pdf]<br />
<br />
https://www.google.com/search?q=just-in-time%20learning<br />
<br />
==See also==<br />
<br />
* [[List of learning strategies]] -- maybe just-in-time learning is closest in spirit to project-based learning<br />
* [[Yak shaving]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=User:Issa_Rice&diff=665User:Issa Rice2019-02-12T22:18:01Z<p>Issa Rice: </p>
<hr />
<div>I am [https://issarice.com Issa Rice].<br />
<br />
All my contributions to this wiki are released to the public domain according to the [https://creativecommons.org/publicdomain/zero/1.0/ CC0 Public Domain Dedication].<br />
<br />
Some navigation links I use often:<br />
<br />
* [[Special:AllPages]]<br />
* [[Special:WantedPages]]<br />
<br />
List of pages I want to write:<br />
<br />
* [[Behavior chain]], [[Backward chaining]], [[Forward chaining]], [[total task presentation]] [https://asatonline.org/for-parents/learn-more-about-specific-treatments/applied-behavior-analysis-aba/aba-techniques/behavior-chaining/] [https://www.greaterwrong.com/posts/z79ApTRh7Jhmis2g8/srs-advice/comment/sNHx73SxHnJDn3DrY]<br />
* [[Christopher Olah]], [[Michael Nielsen]], [[Terence Tao]], [[Hans Freudenthal]]<br />
<br />
Things I should go through:<br />
<br />
* https://en.wikipedia.org/wiki/Category:Learning_methods<br />
* https://en.wikipedia.org/wiki/Category:Learning<br />
* https://en.wikipedia.org/wiki/Category:Mathematics_education<br />
* https://en.wikipedia.org/wiki/Augmented_learning<br />
* https://gowers.wordpress.com/category/mathematical-pedagogy/page/2/<br />
<br />
My userspace pages:<br />
<br />
{{Special:PrefixIndex/User:Issa Rice/ | hideredirects=1 | stripprefix=1}}</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Mathematics_via_Primary_Historical_Sources_project&diff=664Mathematics via Primary Historical Sources project2019-02-09T05:33:18Z<p>Issa Rice: /* External links */</p>
<hr />
<div>==External links==<br />
<br />
* https://www.cs.nmsu.edu/historical-projects/<br />
* http://web.nmsu.edu/~davidp/hist_projects/<br />
* https://web.nmsu.edu/~davidp/history/<br />
* http://sierra.nmsu.edu/gbezhani/courses.html</div>Issa Rice