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The "worked example effect" they talk about it interesting. The idea that you learn best from worked examples lines up with my experience. However, it seems l

The "worked example effect" they talk about it interesting. The idea that you learn best from worked examples lines up with my experience. However, it seems like higher math abandons this completely. So many math textbooks are just in "theorem, proof" form, with almost no examples or even motivation.

JustinSkycak 2y ago
This is one reason why so many people struggle with higher math. Textbooks & classes are typically not aligned (and often, are in direct opposition) to decades of research into the cognitive science of learning. Not saying that higher math would be "easy" if taught properly. Just that many more people would be able to learn it, than are currently able to learn it. Higher math is heavily g-loaded, which creates a cognitive barrier for many students. The goal of guided/scaffolded instruction is to help boost students over that barrier. Of course, the amount of work it takes to create a textbook explodes with the level of guidance/scaffolding, so in practice there's a limit to the amount of boosting that is feasible, especially if the textbook is written entirely by a single author... but most textbooks don't even come close to the theoretical limit for a single author, much less the theoretical limit for a team of content writers.
- stogot 2y ago
  What is g-loaded?
  - JustinSkycak 2y ago
    Two questions here: 1) What is "g"? "g" is "general intelligence." IQ is a specific measurement of g. https://en.wikipedia.org/wiki/G_factor_(psychometrics) https://en.wikipedia.org/wiki/G_factor_(psychometrics) 2) What is g-"loaded"? This is a good summary: https://www.reddit.com/r/cogsci/comments/j5pug9/comment/g7u4tfc https://www.reddit.com/r/cogsci/comments/j5pug9/comment/g7u4... "it's the degree to which that test correlates with g. A relatively high g-loaded test will have a higher correlation with g, meaning that performance on the test is more indicative of g than performance on a test that is less g-loaded. Often greater complexity or how much mental manipulation a test requires results in a higher g-loaded test. In contrast, higher difficulty (as measured by the percentage of people who fail) does not always mean higher g-loading. For instance, tests of reasoning are generally more g-loaded than tests of rote memorization even when the tests are of equal difficulty." - oscarjeff on Reddit
    - tptacek 2y ago
      You can take this "g" stuff more or less seriously, depending on how much of a math/stat background you have: http://bactra.org/weblog/523.html http://bactra.org/weblog/523.html
will1am 2y ago
It is challenging for learners who benefit from examples
lupire 2y ago
What the higher math classes don't tell you is that you are supposed to study those worked examples an practice problems on your own.
TrackerFF 2y ago
Math progression looks roughly like the following: 1. "Concrete" math, where you learn how to manipulate mathematical constructs, usually guided by worked examples. Little proof involved. (up to advanced HS / junior college level) 2. Proof driven math, use of worked examples becomes more rare (undergrad math) 3. Highly abstract math, where worked examples are more or less entirely abandoned (grad school math) The vast majority of world will never be exposed to math beyond (1), and even people in the STEM field will only be limited to (2). You almost need to study math at a high level, or something very adjacent to math, in order to reach (3). But it should be mentioned that one part of why worked examples diminish as you work your way up, is that you're kind of expected to make your own examples - meaning that you can take highly abstracted mathematical constructs/objects, and relate them to something tangible. Some people have no problem learning math that way, while others struggle. I personally struggled to learn math without any examples, so getting my mind into graduate level math was rough. Luckily there are so many resources to higher-level math, these days. You're not bound to a handful of "bibles" that are filled with "... is left as an exercise for the reader"
- Squeeeez 2y ago
  [Ed. finding these resources is left as an exercise for the reader]
jjmarr 2y ago
The "theorem/proofs" are worked examples in this context since that is what mathematicians do all day. The ubiquitous "existence proof" is just about showing an object satisfying a property exists without actually giving an example. Higher math is a big exercise in shifting symbols around. If you don't have an intrinsic motivation to solve puzzles you will hate higher math.
sno129 2y ago
As a professional mathematician, I strongly disagree with the claim that "higher math" abandons worked examples. Any course or book that does not devote a significant amount of time to examples is a bad course or book. Even Grothendieck, who was famously known for thinking very abstractly and avoiding examples, was motivated by concrete questions (e.g., the Weil conjectures) coming from concrete examples. To me, and most other mathematicians, the whole point of mathematics is to do examples, and theory building or any other abstract nonsense should be motivated by the desire to better understand or unify examples.

JustinSkycak 2y ago

This is one reason why so many people struggle with higher math. Textbooks & classes are typically not aligned (and often, are in direct opposition) to decades of research into the cognitive science of learning. Not saying that higher math would be "easy" if taught properly. Just that many more people would be able to learn it, than are currently able to learn it. Higher math is heavily g-loaded, which creates a cognitive barrier for many students. The goal of guided/scaffolded instruction is to help boost students over that barrier. Of course, the amount of work it takes to create a textbook explodes with the level of guidance/scaffolding, so in practice there's a limit to the amount of boosting that is feasible, especially if the textbook is written entirely by a single author... but most textbooks don't even come close to the theoretical limit for a single author, much less the theoretical limit for a team of content writers.

stogot 2y ago
What is g-loaded?
- JustinSkycak 2y ago
  Two questions here: 1) What is "g"? "g" is "general intelligence." IQ is a specific measurement of g. https://en.wikipedia.org/wiki/G_factor_(psychometrics) https://en.wikipedia.org/wiki/G_factor_(psychometrics) 2) What is g-"loaded"? This is a good summary: https://www.reddit.com/r/cogsci/comments/j5pug9/comment/g7u4tfc https://www.reddit.com/r/cogsci/comments/j5pug9/comment/g7u4... "it's the degree to which that test correlates with g. A relatively high g-loaded test will have a higher correlation with g, meaning that performance on the test is more indicative of g than performance on a test that is less g-loaded. Often greater complexity or how much mental manipulation a test requires results in a higher g-loaded test. In contrast, higher difficulty (as measured by the percentage of people who fail) does not always mean higher g-loading. For instance, tests of reasoning are generally more g-loaded than tests of rote memorization even when the tests are of equal difficulty." - oscarjeff on Reddit
  - tptacek 2y ago
    You can take this "g" stuff more or less seriously, depending on how much of a math/stat background you have: http://bactra.org/weblog/523.html http://bactra.org/weblog/523.html

will1am 2y ago

It is challenging for learners who benefit from examples

lupire 2y ago

What the higher math classes don't tell you is that you are supposed to study those worked examples an practice problems on your own.

TrackerFF 2y ago

Math progression looks roughly like the following: 1. "Concrete" math, where you learn how to manipulate mathematical constructs, usually guided by worked examples. Little proof involved. (up to advanced HS / junior college level) 2. Proof driven math, use of worked examples becomes more rare (undergrad math) 3. Highly abstract math, where worked examples are more or less entirely abandoned (grad school math) The vast majority of world will never be exposed to math beyond (1), and even people in the STEM field will only be limited to (2). You almost need to study math at a high level, or something very adjacent to math, in order to reach (3). But it should be mentioned that one part of why worked examples diminish as you work your way up, is that you're kind of expected to make your own examples - meaning that you can take highly abstracted mathematical constructs/objects, and relate them to something tangible. Some people have no problem learning math that way, while others struggle. I personally struggled to learn math without any examples, so getting my mind into graduate level math was rough. Luckily there are so many resources to higher-level math, these days. You're not bound to a handful of "bibles" that are filled with "... is left as an exercise for the reader"

Squeeeez 2y ago
[Ed. finding these resources is left as an exercise for the reader]

jjmarr 2y ago

The "theorem/proofs" are worked examples in this context since that is what mathematicians do all day. The ubiquitous "existence proof" is just about showing an object satisfying a property exists without actually giving an example. Higher math is a big exercise in shifting symbols around. If you don't have an intrinsic motivation to solve puzzles you will hate higher math.

sno129 2y ago

As a professional mathematician, I strongly disagree with the claim that "higher math" abandons worked examples. Any course or book that does not devote a significant amount of time to examples is a bad course or book. Even Grothendieck, who was famously known for thinking very abstractly and avoiding examples, was motivated by concrete questions (e.g., the Weil conjectures) coming from concrete examples. To me, and most other mathematicians, the whole point of mathematics is to do examples, and theory building or any other abstract nonsense should be motivated by the desire to better understand or unify examples.