Timeline for Natural origins or learned habit: Why do students skip concepts before applications?
Current License: CC BY-SA 4.0
39 events
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| Sep 12 at 3:02 | history | edited | Justin Skycak | CC BY-SA 4.0 |
refine discussion of abstraction ceiling
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| Oct 23, 2023 at 20:02 | history | edited | Justin Skycak | CC BY-SA 4.0 |
tweak grammar
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| Oct 21, 2023 at 14:56 | comment | added | JonathanZ | "everyone has some level of abstraction beyond which they are incapable of engaging in first-principles reasoning." a.k.a. Category Theory. | |
| Oct 21, 2023 at 13:18 | history | edited | Justin Skycak | CC BY-SA 4.0 |
add personal experience
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| Oct 21, 2023 at 12:42 | history | edited | Justin Skycak | CC BY-SA 4.0 |
elaborate on abstraction ceiling
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| Oct 21, 2023 at 1:43 | history | edited | Justin Skycak | CC BY-SA 4.0 |
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| Oct 21, 2023 at 1:31 | history | edited | Justin Skycak | CC BY-SA 4.0 |
simplify grammar
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| Oct 21, 2023 at 0:41 | comment | added | Justin Skycak | Hey @Mahdi Majidi-Zolbanin, just an FYI that I added a lengthy response to your recent addendum here in my answer. | |
| Oct 21, 2023 at 0:22 | history | edited | Justin Skycak | CC BY-SA 4.0 |
add reference to Daniel's answer
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| Oct 21, 2023 at 0:10 | history | edited | Justin Skycak | CC BY-SA 4.0 |
add response to OP's addendum
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| Oct 20, 2023 at 17:37 | history | edited | Justin Skycak | CC BY-SA 4.0 |
tweak grammar
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| Oct 20, 2023 at 17:32 | history | edited | Justin Skycak | CC BY-SA 4.0 |
add restatement of main question at beginning
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| Oct 20, 2023 at 16:55 | history | edited | Justin Skycak | CC BY-SA 4.0 |
tweak addendum 3
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| Oct 20, 2023 at 16:44 | history | edited | Justin Skycak | CC BY-SA 4.0 |
tweak addendum 3
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| Oct 20, 2023 at 16:36 | history | edited | Justin Skycak | CC BY-SA 4.0 |
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| Oct 20, 2023 at 15:40 | comment | added | Justin Skycak | @DanielR.Collins But training in deductive reasoning is entirely the point of math classes. That might be true from the perspective of most pure math classes. But definitely not from the perspective of, for instance, machine learning classes, where the whole point is to use math to build automated systems that use inductive reasoning to make decisions that are correct enough to be useful. Throughout applied math in general, there's a saying "all models are wrong but some are useful." All this just to say that, for most people who use math in their lives, inductive reasoning has its place. | |
| Oct 20, 2023 at 15:27 | history | edited | Justin Skycak | CC BY-SA 4.0 |
add mention of Yakk's answer
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| Oct 20, 2023 at 14:56 | history | edited | Justin Skycak | CC BY-SA 4.0 |
add mention of Yakk's answer
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| Oct 20, 2023 at 12:53 | comment | added | Daniel R. Collins | I'm actually uncomfortable with this answer being supportive of avoiding deductive reasoning, and encouraging inductive reasoning in general. Yes, deductive reasoning is maybe not natural. But training in that is entirely the point of math classes. That topic is usually the first section of any math-reasoning book. C.f., Howard Eves, "The concept of axiomatic development in mathematics [~350 BC] must be ranked as one of the very greatest of the Great Moments in Mathematics." | |
| Oct 20, 2023 at 12:53 | comment | added | Carlyle | It is quite ironic that very advanced mathematicians often use inductive reasoning to make conjectures, and these are a large driving factor of mathematical research, so it too has its place and time | |
| Oct 19, 2023 at 18:08 | history | edited | Justin Skycak | CC BY-SA 4.0 |
mention specific commenters in addendum
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| Oct 19, 2023 at 17:10 | history | edited | Justin Skycak | CC BY-SA 4.0 |
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| Oct 19, 2023 at 16:45 | comment | added | Silverfish | There's been a lot of research into the role of abductive reasoning in mathematics education: a sort of reasoning where students explore and try to form what seems to them a simple, likely model consistent with what they have observed, rather like a mathematical Sherlock Holmes. See eg this study on teaching proof/refutation, or this broader overview (both open-access) | |
| Oct 19, 2023 at 16:10 | history | edited | Justin Skycak | CC BY-SA 4.0 |
add addendum
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| Oct 19, 2023 at 16:04 | history | edited | Justin Skycak | CC BY-SA 4.0 |
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| Oct 19, 2023 at 15:57 | comment | added | Passer By | This makes a whole lot of sense considering people have so much trouble with deductive reasoning, as opposed to inductive thinking, that the scientific method wasn't discovered until the 16th century. | |
| Oct 19, 2023 at 14:48 | comment | added | Pablo H | Perhaps "mathematical reasoning" is just a way to encode inductive reasoning after you've seen enough problematic examples. Perhaps it's enough to pile up example after example until the web of analogies gets so complex that it's easier for them to just use axioms and theorems. :-) | |
| Oct 19, 2023 at 13:33 | comment | added | breversa | ""math people" tend to severely overestimate the degree to which the rest of the population reasons from first principles": so very true. To me, math people have some kind of superpower that allows them to see the fabric of the world. They are the ones who can see ; the rest of the world is blind. | |
| Oct 19, 2023 at 12:22 | comment | added | ryang | Inductive reasoning | |
| Oct 19, 2023 at 11:55 | comment | added | Justin Hancock | @MichałMiśkiewicz In a 2005 paper, "The transition from embodied thought experiment and symbolic manipulation to formal proof," David Tall describes some ways for teachers to help students make sense of formal mathematics. I think analogy is an important tool in mathematics, so I see the question more as how to help students learn and adjust to "formal reasoning." | |
| Oct 19, 2023 at 8:10 | comment | added | Michał Miśkiewicz | I like this answer. Do you have advice on how to help students shift from analogical reasoning to, well, mathematical reasoning? | |
| Oct 19, 2023 at 4:31 | history | edited | Justin Skycak | CC BY-SA 4.0 |
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| Oct 19, 2023 at 3:29 | history | edited | Justin Skycak | CC BY-SA 4.0 |
minor clarifications
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| Oct 19, 2023 at 2:32 | history | edited | Justin Skycak | CC BY-SA 4.0 |
Mention data problem vs model problem
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| Oct 19, 2023 at 1:47 | history | edited | Justin Skycak | CC BY-SA 4.0 |
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| Oct 19, 2023 at 1:39 | history | edited | Justin Skycak | CC BY-SA 4.0 |
add side note
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| Oct 19, 2023 at 1:29 | history | edited | Justin Skycak | CC BY-SA 4.0 |
replace politics example with stereotypes
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| Oct 19, 2023 at 1:22 | history | edited | Justin Skycak | CC BY-SA 4.0 |
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| Oct 19, 2023 at 1:09 | history | answered | Justin Skycak | CC BY-SA 4.0 |