Anecdotally I have found this to be the case for the students I tutor. When I introduce a new topic I always start with worked examples, and I find that students are able to learn much more effectively when they have a reference. Poor pedagogy is also one of my biggest gripes with my undergraduate math program too, where the professors and textbooks often included too few worked problems and proofs, and the ones they did include were not very useful. What I found especially frustrating was when a worked example solved a special case with a unique approach, and the general case required a much more involved method that wasn't explained particularly well. Differential equations seems to be a particularly bad offender here, since I've had the same issue with the examples in many texts.
> What I found especially frustrating was when a worked example solved a special case with a unique approach, and the general case required a much more involved method that wasn't explained particularly well.
Amusingly, many people think the solution to this is "abandon worked examples and focus exclusively on trying to teach general problem-solving skills," which doesn't really work in practice (or even in theory). That seems to be the most common approach in higher math, especially once you get into serious math-major courses like Real Analysis and Abstract Algebra.
What actually works in practice is simply creating more worked examples, organizing them well, and giving students practice with problems like each worked example before moving them onto the next worked example covering a slightly more challenging case. You can get really, really far with this approach, but most educational resources shy away from it or give up really early because it's so much damn work! ;)
+100 for "Please, Just Work More Examples, I Swear It Helps".
I don't have nearly as impressive a backstory as you do here, but I did apply spaced repetition to my abstract algebra class in my math minor a few years back. I didn't do anything fancy, I just put every homework problem and proof into Anki and solved/rederived them over and over again until I could do so without much thinking.
I ended up walking out with a perfect score on the 2 hour final - in about 15 minutes. Most of the problems were totally novel things I had never seen before, but the fluency I gained in the weeks prior just unlocked something in me. A lot of the concepts of group actions, etc. have stuck with me to this very day, heavily informing my approach to software engineering. Great stuff.
Eh, I think that’s setting students up for failure once they enter graduate studies or more open ended problems that don’t come from a problem bank. Productive struggle is a perfectly valid approach to teaching, it’s just less pleasant in the moment (since the students are expected to struggle).
> What I found especially frustrating was when a worked example solved a special case with a unique approach, and the general case required a much more involved method that wasn't explained particularly well.
That was the bane of my University degree. "And, since our function f happens to be of this form, all the difficult stuff cancels out and we're left with this trivial stuff" and then none of the problems have these "happy accident" cancellations and you're none the wiser on how to proceed.
The statistics book we used was an especially egregious offender in this regard.
I think often the reason this happens is that the chosen examples[1] are just more advanced topics in disguise. Eg maybe you are given some group with a weird operation and asked to prove something about it, and the hidden thing is that this is a well-known property of semi-direct products and that’s what the described group is.
Two I remember were:
- In an early geometry course there was a problem to prove/determine something described in terms of the Poincaré disc model of the hyperbolic plane. The trick was to convert to the upper half-plane model (where there was an obvious choice for which point on the boundary of the disc maps to infinity in the uhp). There I was annoyed because it felt like a trick question, but the lesson was probably useful.
- in a topology course there was a problem like ‘find a space which deformation-retracts to a möbius strip and to an annulus. This is easy to imagine in your head: a solid torus = S1*D2 can contain an embedding of each of those spaces into R3. I ended up carefully writing those retractions by hand, but I think the better solution was to take the product space and apply some theorems (I think I’m misremembering this – product space works for an ordinary retraction but for the deformation retraction I don’t think it works. I guess both retract to S1 and you could glue the two spaces together along that, or use the proof that homotopy equivalence <=> deformation retracts from common space, but I don’t think we had that). I felt less annoyed at missing the trick there.
[1] I’m really talking about exercises here. I don’t really recall having problems with the examples.
Differential equations seems to be a particularly bad offender here
I think that’s a problem with differential equations as a subject. The only ones we know how to solve are special cases. Solving them in general is an open problem.
My education was basically made of worked examples. What was missing was WHY they were worked THAT way. The thought process of the person solving the problem was missing. Yes, intermediate steps were all there - and still no answer to "why go in THAT direction - from the onset?".
It's not "problem solving", it's the deeper understanding of a discipline which makes the experienced practionner go one way rather than the other.
The downside of teaching using worked examples is that it teaches only one problem solving skill to students: mimicking.
Many students will look only at examples in the textbook and happily ignore definitions, theorems, and proofs. They don't know whether the strategy they picked works, only that it worked on a similar looking problem.
Sure, when (good) teachers explain the example they do go through the effort of referring to the definitions and theorems, but that is not necessarily what the students remember.
The other often-overlooked point is that _memorization_ _itself_ is a skill. You get better at remembering stuff as you keep practicing.
And it doesn't necessarily have to be math. You can also train yourself by memorizing poetry, Chinese characters, foreign language words, and so on. And somehow all of these activities are getting sidelined in the modern education. After all, what use is memorization when you can always look up the answer on a phone?
A lot of people don't seem to understand that fluency in problem solving comes partially from memorization.
Memorizing all of the theorems you need, proofs, and a diverse set of examples is going to make it substantially easier to approach new problems.
I've heard it from people conducting interviews, when we're discussing what we want from candidates: "I'm not looking for memorization, I'm looking for problem solving!" - if you've memorized 1000 problems, you'll be better at problem solving than if you didn't!
Would observe that math is something one should not learn by memorization … rather should try to get to the essence and build it up naturally while at the same time get some sense and intuition of things … so that one would become a natural …
What do you mean by memorizing? Of course, a mathematician remembers things but no mathematician learns a field by first attain the ability to recite the theorems in a book without understanding.
Indeed, learning how to memorize is how I finally got my stride in math. I was already good at proofs and problem solving. But constantly having to dig for stuff was hobbling me.
The older I get the more I believe (realize) the issue with math really is 100% skin-in-the-game. When they're young, I suppose you can force memorization on them, but very quickly: If an individual has no immediate percieved use for the math, they're not going to want or need to learn it. Simple as that.
This really hit me as someone who did the overachievey college math. None of it sticks with me at all unless I can think about "what it's for."
Corollary: When I was a kid, we didn't have the thing we have now which strikes me as the CLEAR USE CASE -- video game development; such a no-brainer for me.
X Y algebra? Oh, you mean making a rainbow in Minecraft? :)
LOL, when I read just the first paragraph of your comment, I immediately thought of computer graphics.
There are other examples. I wouldn't be motivated to analyze some filter circuit's transfer function if it's not related to guitar somehow.
If you are someone who is primarily about Making Stuff, this will resonate.
I think some of the academics in math are not Make Stuff people; they can get motivated by the math itself. Or, well, maybe they are Make Stuff people, but what hey make is the math itself. Their application for something is, oh, I need that to prove this other thing in some structure I'm making.
I've experienced the Make Stuff motivation playing with just math. For instance, in high school, I independently came up with double and triple integration along multiple a, and used that to work out the volumes of common solids (easily verifiable to be right). I was thinking, I'm following this cool idea where we integrate along one variable, to get a formula which we integrate along another; will that work?
Now, that's probably a whole other conversation, given the propensity that "capitalism" or whatever one wants to call it is pretty much dedicated to you and I getting this wrong consistently, but hey.
I don't think that's true for everyone. Your math PhDs and enthusiasts appreciate math as beautiful in and of itself. The disconnect might be that they forget many others do want skin in the game, and that makes the teachers not understand what the students need.
Oh, of course, they exist. But my guess is that's the extreme minority, and I was making suggestions in the realm of "We seem to be doing math badly in general population education, what to do about it?"
And yet, the vast majority of math research serves no direct purpose, and the majority of professional mathematicians, at least in academia, look down on applied mathematics.
You can't take the extreme tail end of the distribution of interests, aptitudes and abilities, that is, people who pursue an academic career in mathematics, for the entire distribution of people who are taught or need to use mathematics at some point in their lives.
Twenty years ago, when I was in college, I remember a classmate had problems installing the particular software we needed to use. The teacher told her that the only solution would be to install Linux on her laptop. All the other students had managed to install that software on their Windows laptops.
The teacher was either one step ahead or 25 steps behind.
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.
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.
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"
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.
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.
It's the "$10,000 for knowing which screw to turn" problem. A non-domain-expert (but good general problem solver) could eventually come up with the solution, but they'll take longer. They have to work out a solution either by trial & error (most common) or from first principles (very rare). Either way takes longer than letting an expert look at it and pull a solution seemingly out of thin air, when the reality is it's the decade or decades of experience looking at similar problems that they draw from.
I meant software engineering as inclusive to computer science.
Computer science/software engineering as taught in school gives a lot of foundational and theoretical understanding. But to apply and practice that knowledge,” then
Math is full of specialized problem-solving skills. Like, "oh, this seems to fit the pattern where we can integrate by parts". If you don't know that, you don't even know that you don't know that; it's an unknown unknown.
General problem-solving skills aren't a substitute for special skills.
General problem-solving skills have limits; one of the outcomes of general problem solving is the conclusion "I don't know how to solve this; it may require someone with special skills".
Without properly honed general skills, you may waste time avoiding this correct conclusion (among other mistakes). General skills let you undestand what the problem actually is, what a solution looks like, and whether you are getting closer.
This is the central claim of E.D Hirsch's Why Knowledge Matters[1] book on educational reform. Hirsch is perhaps best known for coining the term "cultural literacy" in his book of the same name.
A little while back I wrote about cultural literacy in the software industry, following the lead of Hirsch's book.[2]
People like to dismiss memorization because you can only use it to solve very simple problems, but someone once gave me the analogy that to "you can't write a symphony without having memorized all the notes first", and I've found that to be a great analogy. By memorizing the simple stuff, you can tackle the hard stuff.
I depends what you mean by "memorization". All learning requires memory so with an expansive definition all learning is "memorization".
But if one means pure rote memorization, I think the value depends very much on the field. Writing English requires knowledge of the spellings of words since English spellings are fairly arbitrary. A student can benefit from memorizing multiplication table to 10 but they'd do better learning principles than memorization multiplication up 100 or 1000. And many of principles, terminologies and rules of thumb are best remembered in-context.
One thing to consider is that "memory training" approaches can be effectively used to remember long arbitrary sequences of data (the arrangement of a deck of playing cards or whatever) through adding colorful/memorable (but arbitrary) associations.
But such methods are seldom actually used by practitioners of memory intensive fields. Usually such practitioners need to recall facts and ideas in context and so they achieve a high recall naturally, by associations facts and ideas with each other.
Readit News
Amusingly, many people think the solution to this is "abandon worked examples and focus exclusively on trying to teach general problem-solving skills," which doesn't really work in practice (or even in theory). That seems to be the most common approach in higher math, especially once you get into serious math-major courses like Real Analysis and Abstract Algebra.
What actually works in practice is simply creating more worked examples, organizing them well, and giving students practice with problems like each worked example before moving them onto the next worked example covering a slightly more challenging case. You can get really, really far with this approach, but most educational resources shy away from it or give up really early because it's so much damn work! ;)
I don't have nearly as impressive a backstory as you do here, but I did apply spaced repetition to my abstract algebra class in my math minor a few years back. I didn't do anything fancy, I just put every homework problem and proof into Anki and solved/rederived them over and over again until I could do so without much thinking.
I ended up walking out with a perfect score on the 2 hour final - in about 15 minutes. Most of the problems were totally novel things I had never seen before, but the fluency I gained in the weeks prior just unlocked something in me. A lot of the concepts of group actions, etc. have stuck with me to this very day, heavily informing my approach to software engineering. Great stuff.
That was the bane of my University degree. "And, since our function f happens to be of this form, all the difficult stuff cancels out and we're left with this trivial stuff" and then none of the problems have these "happy accident" cancellations and you're none the wiser on how to proceed.
The statistics book we used was an especially egregious offender in this regard.
Two I remember were:
- In an early geometry course there was a problem to prove/determine something described in terms of the Poincaré disc model of the hyperbolic plane. The trick was to convert to the upper half-plane model (where there was an obvious choice for which point on the boundary of the disc maps to infinity in the uhp). There I was annoyed because it felt like a trick question, but the lesson was probably useful.
- in a topology course there was a problem like ‘find a space which deformation-retracts to a möbius strip and to an annulus. This is easy to imagine in your head: a solid torus = S1*D2 can contain an embedding of each of those spaces into R3. I ended up carefully writing those retractions by hand, but I think the better solution was to take the product space and apply some theorems (I think I’m misremembering this – product space works for an ordinary retraction but for the deformation retraction I don’t think it works. I guess both retract to S1 and you could glue the two spaces together along that, or use the proof that homotopy equivalence <=> deformation retracts from common space, but I don’t think we had that). I felt less annoyed at missing the trick there.
[1] I’m really talking about exercises here. I don’t really recall having problems with the examples.
I think that’s a problem with differential equations as a subject. The only ones we know how to solve are special cases. Solving them in general is an open problem.
It's not "problem solving", it's the deeper understanding of a discipline which makes the experienced practionner go one way rather than the other.
Dead Comment
Many students will look only at examples in the textbook and happily ignore definitions, theorems, and proofs. They don't know whether the strategy they picked works, only that it worked on a similar looking problem.
Sure, when (good) teachers explain the example they do go through the effort of referring to the definitions and theorems, but that is not necessarily what the students remember.
And it doesn't necessarily have to be math. You can also train yourself by memorizing poetry, Chinese characters, foreign language words, and so on. And somehow all of these activities are getting sidelined in the modern education. After all, what use is memorization when you can always look up the answer on a phone?
Memorizing all of the theorems you need, proofs, and a diverse set of examples is going to make it substantially easier to approach new problems.
I've heard it from people conducting interviews, when we're discussing what we want from candidates: "I'm not looking for memorization, I'm looking for problem solving!" - if you've memorized 1000 problems, you'll be better at problem solving than if you didn't!
There are lots of folk who can remember all sorts of details but never seem to be able to figure out how to put the pieces together.
This really hit me as someone who did the overachievey college math. None of it sticks with me at all unless I can think about "what it's for."
Corollary: When I was a kid, we didn't have the thing we have now which strikes me as the CLEAR USE CASE -- video game development; such a no-brainer for me.
X Y algebra? Oh, you mean making a rainbow in Minecraft? :)
There are other examples. I wouldn't be motivated to analyze some filter circuit's transfer function if it's not related to guitar somehow.
If you are someone who is primarily about Making Stuff, this will resonate.
I think some of the academics in math are not Make Stuff people; they can get motivated by the math itself. Or, well, maybe they are Make Stuff people, but what hey make is the math itself. Their application for something is, oh, I need that to prove this other thing in some structure I'm making.
I've experienced the Make Stuff motivation playing with just math. For instance, in high school, I independently came up with double and triple integration along multiple a, and used that to work out the volumes of common solids (easily verifiable to be right). I was thinking, I'm following this cool idea where we integrate along one variable, to get a formula which we integrate along another; will that work?
But probably, greed/money would be the other.
Now, that's probably a whole other conversation, given the propensity that "capitalism" or whatever one wants to call it is pretty much dedicated to you and I getting this wrong consistently, but hey.
Twenty years ago, when I was in college, I remember a classmate had problems installing the particular software we needed to use. The teacher told her that the only solution would be to install Linux on her laptop. All the other students had managed to install that software on their Windows laptops. The teacher was either one step ahead or 25 steps behind.
Not true.
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.
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"
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.
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.
You can teach software engineering in school. But you become an expert by reading source code and seeing the many ways to solve a problem.
An expert can intuit a solution because of pattern matching. And their argument is that math is the same.
More so by iteratively building, at least so for me.
Computer science/software engineering as taught in school gives a lot of foundational and theoretical understanding. But to apply and practice that knowledge,” then
General problem-solving skills aren't a substitute for special skills.
General problem-solving skills have limits; one of the outcomes of general problem solving is the conclusion "I don't know how to solve this; it may require someone with special skills".
Without properly honed general skills, you may waste time avoiding this correct conclusion (among other mistakes). General skills let you undestand what the problem actually is, what a solution looks like, and whether you are getting closer.
edit - corrected spelling
A little while back I wrote about cultural literacy in the software industry, following the lead of Hirsch's book.[2]
1. https://hep.gse.harvard.edu/9781612509525/why-knowledge-matt...
2. https://thundergolfer.com/software/culture/2024/01/14/comput...
But if one means pure rote memorization, I think the value depends very much on the field. Writing English requires knowledge of the spellings of words since English spellings are fairly arbitrary. A student can benefit from memorizing multiplication table to 10 but they'd do better learning principles than memorization multiplication up 100 or 1000. And many of principles, terminologies and rules of thumb are best remembered in-context.
One thing to consider is that "memory training" approaches can be effectively used to remember long arbitrary sequences of data (the arrangement of a deck of playing cards or whatever) through adding colorful/memorable (but arbitrary) associations.
But such methods are seldom actually used by practitioners of memory intensive fields. Usually such practitioners need to recall facts and ideas in context and so they achieve a high recall naturally, by associations facts and ideas with each other.