https://machinelearning.subwiki.org/w/index.php?action=history&feed=atom&title=User%3AIssaRice%2FTiers_of_learning_in_mathematicsUser:IssaRice/Tiers of learning in mathematics - Revision history2026-10-16T07:47:19ZRevision history for this page on the wikiMediaWiki 1.41.2https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tiers_of_learning_in_mathematics&diff=3245&oldid=prevIssaRice at 06:03, 6 February 20212021-02-06T06:03:27Z<p></p>
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</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tiers_of_learning_in_mathematics&diff=1188&oldid=prevIssaRice at 06:52, 28 December 20182018-12-28T06:52:34Z<p></p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an ''unordered list'' of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does this. || Textbook authors (?). My current feeling is that textbook authors get to this level, and also people who [https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/ specialize in a field]. However, even people at this level might only do this in an ad hoc manner/intuitively, rather than systematically drawing dependency graphs and checking each possible edge and so forth. Even with textbook authors, most of them choose only to write ''one'' path through "theorem space" in the end; they must have alternative paths in mind (some might put these in exercises), but they never seem to talk about how they made their choice of which path to write about and why it's better than other paths. (Note: again, I don't talk to enough mathematicians to know.)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an ''unordered list'' of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does <ins style="font-weight: bold; text-decoration: none;">this. Or in linear algebra, instead of being given definitions for things like linear independence, span, basis, and so on, one knows what one wants to show (e.g. "I know intuitively what a basis should look like") and then comes up with definitions and theorems to prove </ins>this. || Textbook authors (?). My current feeling is that textbook authors get to this level, and also people who [https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/ specialize in a field]. However, even people at this level might only do this in an ad hoc manner/intuitively, rather than systematically drawing dependency graphs and checking each possible edge and so forth. Even with textbook authors, most of them choose only to write ''one'' path through "theorem space" in the end; they must have alternative paths in mind (some might put these in exercises), but they never seem to talk about how they made their choice of which path to write about and why it's better than other paths. (Note: again, I don't talk to enough mathematicians to know.)</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''. I think people at this level might also have a strong historical understanding of math, and be able to visualize how the results came to be proved, but also have "alternate histories" in their mind. || [[wikipedia:John Stillwell|John Stillwell]] :D</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''. I think people at this level might also have a strong historical understanding of math, and be able to visualize how the results came to be proved, but also have "alternate histories" in their mind. || [[wikipedia:John Stillwell|John Stillwell]] :D</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tiers_of_learning_in_mathematics&diff=1070&oldid=prevIssaRice at 20:08, 7 December 20182018-12-07T20:08:29Z<p></p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch || The material is taught using proofs, <del style="font-weight: bold; text-decoration: none;">but </del>the focus is on being able to prove the results oneself, where the exercises ''are'' the proofs of the main results. Example: Tao's ''Analysis''; reading a textbook that gives proofs of main results, but where one tries to prove the theorems oneself first. || Graduate students, most mathematicians (?). My current feeling is that most mathematicians stop here for subfields that they do not specialize in (but I don't talk to enough mathematicians to know).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch || The material is taught using proofs, <ins style="font-weight: bold; text-decoration: none;">and </ins>the focus is on being able to prove the results oneself, where the exercises ''are'' the proofs of the main results. Example: Tao's ''Analysis''; reading a textbook that gives proofs of main results, but where one tries to prove the theorems oneself first. || Graduate students, most mathematicians (?). My current feeling is that most mathematicians stop here for subfields that they do not specialize in (but I don't talk to enough mathematicians to know).</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an ''unordered list'' of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does this. || Textbook authors (?). My current feeling is that textbook authors get to this level, and also people who [https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/ specialize in a field]. However, even people at this level might only do this in an ad hoc manner/intuitively, rather than systematically drawing dependency graphs and checking each possible edge and so forth. Even with textbook authors, most of them choose only to write ''one'' path through "theorem space" in the end; they must have alternative paths in mind (some might put these in exercises), but they never seem to talk about how they made their choice of which path to write about and why it's better than other paths. (Note: again, I don't talk to enough mathematicians to know.)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an ''unordered list'' of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does this. || Textbook authors (?). My current feeling is that textbook authors get to this level, and also people who [https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/ specialize in a field]. However, even people at this level might only do this in an ad hoc manner/intuitively, rather than systematically drawing dependency graphs and checking each possible edge and so forth. Even with textbook authors, most of them choose only to write ''one'' path through "theorem space" in the end; they must have alternative paths in mind (some might put these in exercises), but they never seem to talk about how they made their choice of which path to write about and why it's better than other paths. (Note: again, I don't talk to enough mathematicians to know.)</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tiers_of_learning_in_mathematics&diff=1068&oldid=prevIssaRice at 20:01, 7 December 20182018-12-07T20:01:45Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''. I think people at this level might also have a strong historical understanding of math, and be able to visualize how the results came to be proved, but also have "alternate histories" in their mind. || [[wikipedia:John Stillwell|John Stillwell]] :D</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''. I think people at this level might also have a strong historical understanding of math, and be able to visualize how the results came to be proved, but also have "alternate histories" in their mind. || [[wikipedia:John Stillwell|John Stillwell]] :D</div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==External links==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* https://www.lesswrong.com/posts/MRqnYuCFHW46JPJag/understanding-is-translation</ins></div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tiers_of_learning_in_mathematics&diff=1067&oldid=prevIssaRice at 19:59, 7 December 20182018-12-07T19:59:24Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an ''unordered list'' of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does this. || Textbook authors (?). My current feeling is that textbook authors get to this level, and also people who [https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/ specialize in a field]. However, even people at this level might only do this in an ad hoc manner/intuitively, rather than systematically drawing dependency graphs and checking each possible edge and so forth. Even with textbook authors, most of them choose only to write ''one'' path through "theorem space" in the end; they must have alternative paths in mind (some might put these in exercises), but they never seem to talk about how they made their choice of which path to write about and why it's better than other paths. (Note: again, I don't talk to enough mathematicians to know.)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an ''unordered list'' of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does this. || Textbook authors (?). My current feeling is that textbook authors get to this level, and also people who [https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/ specialize in a field]. However, even people at this level might only do this in an ad hoc manner/intuitively, rather than systematically drawing dependency graphs and checking each possible edge and so forth. Even with textbook authors, most of them choose only to write ''one'' path through "theorem space" in the end; they must have alternative paths in mind (some might put these in exercises), but they never seem to talk about how they made their choice of which path to write about and why it's better than other paths. (Note: again, I don't talk to enough mathematicians to know.)</div></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''. I think people at this level might also have a strong historical understanding of math, and be able to visualize how the results came to be proved, but also have "alternate histories" in their mind. || John Stillwell :D</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''. I think people at this level might also have a strong historical understanding of math, and be able to visualize how the results came to be proved, but also have "alternate histories" in their mind. || <ins style="font-weight: bold; text-decoration: none;">[[wikipedia:</ins>John Stillwell<ins style="font-weight: bold; text-decoration: none;">|John Stillwell]] </ins>:D</div></td></tr>
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</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tiers_of_learning_in_mathematics&diff=1066&oldid=prevIssaRice at 19:56, 7 December 20182018-12-07T19:56:55Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch || The material is taught using proofs, but the focus is on being able to prove the results oneself, where the exercises ''are'' the proofs of the main results. Example: Tao's ''Analysis''; reading a textbook that gives proofs of main results, but where one tries to prove the theorems oneself first. || Graduate students, most mathematicians (?). My current feeling is that most mathematicians stop here for subfields that they do not specialize in (but I don't talk to enough mathematicians to know).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch || The material is taught using proofs, but the focus is on being able to prove the results oneself, where the exercises ''are'' the proofs of the main results. Example: Tao's ''Analysis''; reading a textbook that gives proofs of main results, but where one tries to prove the theorems oneself first. || Graduate students, most mathematicians (?). My current feeling is that most mathematicians stop here for subfields that they do not specialize in (but I don't talk to enough mathematicians to know).</div></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an ''unordered list'' of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does this. || Textbook authors (?). My current feeling is that textbook authors get to this level, and also people who [https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/ specialize in a field]. However, even people at this level might only do this in an ad hoc manner/intuitively, rather than systematically drawing dependency graphs and checking each possible edge and so forth. (Note: again, I don't talk to enough mathematicians to know.)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an ''unordered list'' of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does this. || Textbook authors (?). My current feeling is that textbook authors get to this level, and also people who [https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/ specialize in a field]. However, even people at this level might only do this in an ad hoc manner/intuitively, rather than systematically drawing dependency graphs and checking each possible edge and so forth<ins style="font-weight: bold; text-decoration: none;">. Even with textbook authors, most of them choose only to write ''one'' path through "theorem space" in the end; they must have alternative paths in mind (some might put these in exercises), but they never seem to talk about how they made their choice of which path to write about and why it's better than other paths</ins>. (Note: again, I don't talk to enough mathematicians to know.)</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''. I think people at this level might also have a strong historical understanding of math, and be able to visualize how the results came to be proved, but also have "alternate histories" in their mind. || John Stillwell :D</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''. I think people at this level might also have a strong historical understanding of math, and be able to visualize how the results came to be proved, but also have "alternate histories" in their mind. || John Stillwell :D</div></td></tr>
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</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tiers_of_learning_in_mathematics&diff=1065&oldid=prevIssaRice at 19:54, 7 December 20182018-12-07T19:54:45Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Computation-based || The material is taught, but proofs of the main results are not covered. Example: high school/freshman calculus courses, in most cases.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Computation-based || The material is taught, but proofs of the main results are not covered. Example: high school/freshman calculus courses, in most cases. <ins style="font-weight: bold; text-decoration: none;">|| High school students, lower-level undergraduate students</ins></div></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Proof-based, but focusing on exercises || The material is taught using proofs, but where the student is not expected to prove the main results themselves. Instead, the student is expected to be able to solve (often artificial) problems (including proof-based problems) that make use of the main results. Example: math 13X and math 33X at UW.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Proof-based, but focusing on exercises || The material is taught using proofs, but where the student is not expected to prove the main results themselves. Instead, the student is expected to be able to solve (often artificial) problems (including proof-based problems) that make use of the main results. Example: math 13X and math 33X at UW. <ins style="font-weight: bold; text-decoration: none;">|| Upper-level undergraduate students</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch || The material is taught using proofs, but the focus is on being able to prove the results oneself, where the exercises ''are'' the proofs of the main results. Example: Tao's ''Analysis''; reading a textbook that gives proofs of main results, but where one tries to prove the theorems oneself first. My current feeling is that most mathematicians stop here for subfields that they do not specialize in (but I don't talk to enough mathematicians to know).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch || The material is taught using proofs, but the focus is on being able to prove the results oneself, where the exercises ''are'' the proofs of the main results. Example: Tao's ''Analysis''; reading a textbook that gives proofs of main results, but where one tries to prove the theorems oneself first<ins style="font-weight: bold; text-decoration: none;">. || Graduate students, most mathematicians (?)</ins>. My current feeling is that most mathematicians stop here for subfields that they do not specialize in (but I don't talk to enough mathematicians to know).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an ''unordered list'' of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does this. My current feeling is that textbook authors get to this level, and also people who [https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/ specialize in a field]. However, even people at this level might only do this in an ad hoc manner/intuitively, rather than systematically drawing dependency graphs and checking each possible edge and so forth. (Note: again, I don't talk to enough mathematicians to know.)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an ''unordered list'' of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does this<ins style="font-weight: bold; text-decoration: none;">. || Textbook authors (?)</ins>. My current feeling is that textbook authors get to this level, and also people who [https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/ specialize in a field]. However, even people at this level might only do this in an ad hoc manner/intuitively, rather than systematically drawing dependency graphs and checking each possible edge and so forth. (Note: again, I don't talk to enough mathematicians to know.)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''. <ins style="font-weight: bold; text-decoration: none;">I think people at this level might also have a strong historical understanding of math, and be able to visualize how the results came to be proved, but also have "alternate histories" in their mind. || John Stillwell :D</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tiers_of_learning_in_mathematics&diff=1064&oldid=prevIssaRice at 19:51, 7 December 20182018-12-07T19:51:29Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Proof-based, but focusing on exercises || The material is taught using proofs, but where the student is not expected to prove the main results themselves. Instead, the student is expected to be able to solve (often artificial) problems (including proof-based problems) that make use of the main results. Example: math 13X and math 33X at UW.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Proof-based, but focusing on exercises || The material is taught using proofs, but where the student is not expected to prove the main results themselves. Instead, the student is expected to be able to solve (often artificial) problems (including proof-based problems) that make use of the main results. Example: math 13X and math 33X at UW.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch || The material is taught using proofs, but the focus is on being able to prove the results oneself, where the exercises ''are'' the proofs of the main results. Example: Tao's ''Analysis''; reading a textbook that gives proofs of main results, but where one tries to prove the theorems oneself first.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch || The material is taught using proofs, but the focus is on being able to prove the results oneself, where the exercises ''are'' the proofs of the main results. Example: Tao's ''Analysis''; reading a textbook that gives proofs of main results, but where one tries to prove the theorems oneself first<ins style="font-weight: bold; text-decoration: none;">. My current feeling is that most mathematicians stop here for subfields that they do not specialize in (but I don't talk to enough mathematicians to know)</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an ''unordered list'' of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does this.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an ''unordered list'' of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does this. <ins style="font-weight: bold; text-decoration: none;">My current feeling is that textbook authors get to this level, and also people who [https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/ specialize in a field]. However, even people at this level might only do this in an ad hoc manner/intuitively, rather than systematically drawing dependency graphs and checking each possible edge and so forth. (Note: again, I don't talk to enough mathematicians to know.)</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tiers_of_learning_in_mathematics&diff=1063&oldid=prevIssaRice at 19:43, 7 December 20182018-12-07T19:43:34Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch || The material is taught using proofs, but the focus is on being able to prove the results oneself, where the exercises ''are'' the proofs of the main results. Example: Tao's ''Analysis''; reading a textbook that gives proofs of main results, but where one tries to prove the theorems oneself first.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch || The material is taught using proofs, but the focus is on being able to prove the results oneself, where the exercises ''are'' the proofs of the main results. Example: Tao's ''Analysis''; reading a textbook that gives proofs of main results, but where one tries to prove the theorems oneself first.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does this.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch in parallel || The material is taught using proofs and the focus is on being able to prove the results oneself ''in multiple ways''<ins style="font-weight: bold; text-decoration: none;">. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an ''unordered list'' of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones</ins>. Example: working through multiple analysis books in parallel and [https://gist.github.com/riceissa/d9f191d934c23dc5e42388dfb91b6aca keeping a dependency graph] of theorems in one's mind as one does this.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Reverse mathematics || Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but ''also tracking the strengths of various results''.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Tiers_of_learning_in_mathematics&diff=1062&oldid=prevIssaRice at 19:39, 7 December 20182018-12-07T19:39:56Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 19:39, 7 December 2018</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Proof-based, but focusing on exercises || The material is taught using proofs, but where the student is not expected to prove the main results themselves. Instead, the student is expected to be able to solve (often artificial) problems (including proof-based) that make use of the main results. Example: math 13X and math 33X at UW.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Proof-based, but focusing on exercises || The material is taught using proofs, but where the student is not expected to prove the main results themselves. Instead, the student is expected to be able to solve (often artificial) problems (including proof-based <ins style="font-weight: bold; text-decoration: none;">problems</ins>) that make use of the main results. Example: math 13X and math 33X at UW.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch || The material is taught using proofs, but the focus is on being able to prove the results oneself, where the exercises ''are'' the proofs of the main results. Example: Tao's ''Analysis''; reading a textbook that gives proofs of main results, but where one tries to prove the theorems oneself first.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| Building the theory from scratch || The material is taught using proofs, but the focus is on being able to prove the results oneself, where the exercises ''are'' the proofs of the main results. Example: Tao's ''Analysis''; reading a textbook that gives proofs of main results, but where one tries to prove the theorems oneself first.</div></td></tr>
</table>IssaRice