I would expect students in an environment with a typically high student-to-teacher ratio, but who actually practice what they're being taught, will significantly outperform students who are taught one-on-one by a personal tutor but rarely actually perform the thing that they're trying to learn.
Obviously, "¿Por Qué No Los Dos?" - doing both is even better. But tutoring isn't obviously superior to practice.
As a personal anecdote (not to replace the above general arguments), I've gotten several hundred hours of one-on-one tutoring in an advanced field of physics from a number of experts, and yet I learned significantly less than I have from significantly fewer hours studying a separate (but no less difficult) field of math when I actually worked the problems.
Good tutoring will essentially be practice and worked problems with instant feedback -- not an individual "lecture".
While there is value to being in the forest entirely alone, I think for a motivated student good tutoring will outperform working problems on your own in speed of overall learning. Both are good though, and I agree working the problems out, and working a lot of problems, is the main thing.
Yes, but then we're conflating the two things we're trying to separate - one-on-one instruction, and worked practice.
I was using "tutoring" to mean specifically one-on-ones. I completely agree that a good tutor will have you practice what you're learning, and that's definitely much closer to optimal than the educational mess we're currently in.
One thing I have observed in my own experience (my own, and my, mostly home educated, kids) is that both one to one teaching AND learning on one's own (the amount of it being practice varying with subject) are better than classroom/lecture learning. This is not a statistical sample or a study, but it is three people across multiple subjects, at a pretty full range of levels (from primary school level to postgrad).
Maybe much learning is an individual activity and learning in groups is just ineffective?
That means the question is so context-dependent that any potential answer would only bring insight with that specific context in mind.
That being said, I am a huge fan of practise paired with theory (this is what a good tutor would do). Many people only start to care about theory once they have encountered the problems theory helps with have been encountered in the wild. And getting people to care is one of the first things any educator has to achieve.
There are many who start with the base assumption that theory is worthless, but I'd argue having accurate mental models will greatly improve the speed and quality of the work. Additionally this helps to learn faster, as the question why aomething went wrong in practise can be answered faster and more accurately.
Yes, on further reflection, you're right. My statement was spurred by the claim that practice had "exaggerated significance" in the article relative to practice, which is kind of a hard thing to quantify and argue about.
And I definitely wasn't trying to say that theory isn't important! I love theory - I don't actually like working the problems - and think that it's important, it's just that I've realized that lots of theory is much less effective without practice, even in a highly abstract field like math.
The interplay between abstract (abstract explanation; theory) and concrete (concrete examples in the course of explanation; practice) is fascinating to me.
Based on your experience, do you have any insight for whether, in the course of verbal/written instruction, it's better to start with concrete instances of a concept, and then give the abstract concept itself, or vice versa?
For example in orbital mechanics it was experimentation that got me to actually understand all the retrograde burns, plane changes and Hohmann transfers, almost exactly like the xkcd comic https://xkcd.com/1356/ (though without the job at NASA part of course)
Concur. However, how many teachers and students are willing to engage in candid critique of the student’s work?
https://en.wikipedia.org/wiki/Tutorial_system
Much of graduate education in the US seems similar.
Moving forward to the next chapter required, exactly as described in that paper, the completion of a problem set and then a score of at least 90% on a test demonstrating mastery of the previous chapter, sometimes accompanied by also demonstrating that skill in a lab. But far from 1 on 1, this entire class was effectively 0 on infinity. The teaching assistant/proctors that we engaged with were there only to grade your work and provided minimal feedback.
And indeed it was one of the most educational 'classes' I ever took. But I think this challenges the concept that it has anything to do with 1 on 1 attention. But rather the outcome seems practically tautological - a good way to get people to perform to the point of mastery is to require that they perform to the point of mastery. Of course, at scale, all you're really doing is weeding out the people that are unable to achieve mastery. And indeed that class was considered a weed out course.
What you describe seems to be a very poor implementation of mastery learning. But if the tutor is completely disengaged even 1 on 1 tutoring is unlikely to have good effects.
In music you usually have a small amount of one-on-one instruction and then you practice. In tennis you usually have a small number of one-one-lessons and then you practice and play matches.
You could probably do the same for maths. You're given some problems to try to solve and given two hours, then once you've made a serious attempt you get individual tutoring for an hour, then you go back to solving problems and there's a short one-on-one question session at the end, let's say 30 minutes. Then you have a 5 hour study session with 1.5 hours of teacher time, so he can have around three students.
It does require more teachers, but not 1:1. Students being taught 1:1 learn a lot faster, and can be set work to do unsupervised. From my experience I think less than an hour a seek (sometimes a lot less) of tuition time (plus a bit more for marking, and another few hours of study by the student) is sufficient to cover a subject 1:1 (and it can often me a lot less) for teenagers (specifically for GCSEs - British exams sat in schools at 16).
it does require a significantly higher ratio than classroom teaching usually does, but its a long way from needing 1:1.
a) Group Instruction: Baseline
b) Mastery Learning, ensuring students master the material before moving on: One sigma improvement (outperform 68% of students in group setting)
c) 1 on 1 Tutoring: Two sigma improvement (outperform 98% of students in group setting)
I'm not sure how many of those we have available to us. Many are compromised by politics, funding, or the need to act as a daycare.
I learned a lot at the various schools I went to, but the amount I learned seemed to correlate more directly with how invested I was in learning than how well the school was funded. Plenty of schools with better per-pupil funding had significantly worse student achievement rates than where I was.
The only real exception to that is not all schools offer the same curriculum. Back in my day, not every secondary / high school had someone who could teach calculus, though now there's districts that are getting rid of calculus entirely to promote anti-racism. Honestly, I think learning calculus in high school was good for me, even if I've really only needed to calculate integrals once in my programming career.
At University, things were much the same. Undergrad courses focused a bit more on synthesizing than memorizing compared to high school, but not really by much.
All of this is to say that I'm not really sure it's fair to knock the apprentice program since we don't directly experience optimal pedagogy elsewhere.
Sorry about that. At Caltech, we were never given formulas. Everything was derived from scratch. I never memorized anything (but I found after a while I simply knew all the trig identifies!).
Are you sure about this? Your quoted article only has data from >20 years ago. I wouldn't be surprised if nowadays more people study at university than do an apprenticeship
In 2024, according to the "Bundesinstituts für Berufsbildung" 486,700 people started their apprenticeship [0]. In the same period (2024-2025) 490,304 people started their first semester at university/college, according to the "Statistisches Bundesamt" [1].
So you are right, theres more new students than apprentices, but its not by a lot.
[0]: https://www.bibb.de/de/201811.php [1]: https://www.destatis.de/DE/Themen/Gesellschaft-Umwelt/Bildun...
(It always seemed like I learned it, but when faced with the problem sets I discovered I hadn't learned anything yet.)
It's the same with everything. You can watch a yootoob video on rebuilding a carburetor all day, but you don't know nuttin until you take it apart yourself.
I decided to learn to ride a dirtbike. I took some personal instruction from an expert, and promptly crashed. Again and again and again. Finally, my body figured out how to coordinate the controls.
Can't learn how to double clutch downshift from watching a video, either.
Every time, I have to stop and think through it step by step. My recent rides have all been constantly up and down shifting, in order to get it properly into muscle memory. I was annoyed that my car shifting skills did not transfer.
I did a snowboarding course once, and it was largely useless because they didn’t actually explain any of the mechanics of how the board actually worked beyond seesawing mostly-sideways down the ultra beginner slope. It wasn’t until I had a chance to experiment that I started actually figuring out anything useful.
I absolutely taught myself how to double-clutch from YouTube and Initial D, though. :D (Plus copious practice, of course.)
Another weird thing. I've been using the same text editor for 40 years. I no longer remember what the commands are - but I can still edit files just fine. Sometimes I watch my fingers to see what the command actually is.
I learned how to solve the Rubik cube some years ago and I found the same thing. I instinctively know the sequence of steps but I would find it very hard to actually write it down.
I think this is called the "assumed knowledge problem".
Source: dev who has LARPed as a technical writer.
https://m.youtube.com/watch?v=wn5KqWwP6uQ
Basically lots and lots of lots of practice.
There are reasons why we started sending children to schools rather than businesses for basic education. There is also little need to reach back to medieval times when comparatively less exploitative (but still imperfect) apprenticeship systems are alive and well in the trades today.
One-on-one practical instruction related specifically to what you want to do is awesome, but there are a lot of difficulties in incentivizing people to supply such instruction.
The master very much cares about your quality, because if it doesn't look like his quality nobody will buy it. If the quality goes down to much, there will be complaints to the guilt and he looses its ability to do business.
If you have problems with your master you can look for another one. The good always needed to reject prentices, the bad had nobody showing up. In-fact you were required to stay with multiple masters.
If you complain about them not having an 8-hour day, nobody had that in the middle ages. But tradesman were more of the richer people in a city, maybe behind tradesman.
Also, apprentices duties involved also general housework and pretty much any random thing they told you to do. They would beat you if they thought you do not do what they want and you would be serving literally whole day and that was it. And no it was not whole day of learning. Based on book I read, that frequently involved things wife of the master ordered - they had nothing to do with the trade and that was normal.
With eating, you would wait behind the master while their lunch and tend eat whatever remained.
Yes you were part of the family for better and for worse. I don't see how that is problematic, you weren't working for a different household, you were just part of this household.
> They would beat you
Sure life sucked back than. There is a reason we are not in the middle ages anymore. I would say that was more a problem with the general attitude in society, not specific to masters.
> no you could not just look for another master
But you were required to look for a different master every few years? That claim doesn't make sense to me.
Of course not everything was all roses, it was just different. You didn't had a social net from the state, there wasn't an independent police you could report beatings from the master. On the other hand you had a social security net by the family/household currently living in (meaning your master's in this case) and if you have rows with everyone you can just walk for a day to the next city, where nobody knows you, and start afresh. The latter isn't possible now anymore.
No you was not part of the family. And this was not part of the family thing either. It was more of the sweetshop with workers who have no choice thing.
> But you were required to look for a different master every few years? That claim doesn't make sense to me.
You was not given choice to shop around or change up however you wanted. That was not a thing. Occasionally you could be asked to move to different master at predetermined set points. But not always. I do not even see what is nonsensical or hard to understand about such setup.
> you had a social security net by the family/household currently living in (meaning your master's in this case)
Your master was not your security net. You would be kicked off and expected go back to your actual family.
> if you have rows with everyone you can just walk for a day to the next city, where nobody knows you, and start afresh
That was definitely not a thing either. They would vet you and your character with where you came from. There was a lot of suspicion about new people. You would have quite a difficult time to establish yourself or even find a living. Being banished was a punishment for a reason - and other village you go to will be very aware newcomer might be a troublemaker they did not wanted elsewhere.
> The latter isn't possible now anymore.
That is significantly easier now.
Name it part of the household then. In a modern sweetshop you get some money (or not) generally too less to pay for food and other things. Nobody cares if you die or just don't show up the next day.
As a prentice, you don't get money, but you also don't buy food and you don't pay for a roof. Your master can not afford for you to not show up or be invalid for work, because he can only afford to house and teach so many prentices. He also can't afford you producing low-quality goods, because the goods get sold in his name.
> You were not given choice to shop around or change up however you wanted.
I think this conflicts with that you were required to have served multiple masters as part of your education. When a prentice shows up to a new master, it means the new master gets an already trained workforce while not having to pay for the expenses of the education. Your old master will not be willing/able to afford that loss, so you need to mess up some money before you get kicked out.
> Your master was not your security net. You would be kicked off
When you become invalid indefinitely, yes. When you get sick for some days, your master will get you back to work as soon as possible.
> expected go back to your actual family.
When you show up broke after years, I bet you get kicked out there also. You have better choices moving to a new city or living off the streets.
> That was definitely not a thing either. They would vet you and your character with where you came from. There was a lot of suspicion about new people. You would have quite a difficult time to establish yourself or even find a living. Being banished was a punishment for a reason - and other village you go to will be very aware newcomer might be a troublemaker they did not wanted elsewhere.
Having an influx of new people from the landscape were how the cities operated. Having more people made the city richer and more powerful. It really depended on how skilled you were. If you can claim to be able to do X and you are able to show it, than you have a good chance to find work. If you don't, then good luck dying in the streets.
> That is significantly easier now.
Being a criminal and trying to flee? Good luck with that in the time of surveillance and world-wide police cooperation. Nowadays you can pretty much only double down on being criminal, be convicted or flee to a third-world country, where you will probably die soon/ have a way lower living standard. In the past you also had the opportunity to just switch jurisdiction and stop being a criminal, while not having a different living standard.
The fact that these puzzles can then be used to do cool things is almost just a fortunate coincidence.
I've often learned by recalling the concepts from a lecture, reasoning about the material, and imagining what some of the problems would look like while sketching out solutions in my head. It's not any easier than doing the homework, but it is more convenient and flexible. And it can sometimes help with physical skills.
Theory is still important because it communicates how other people understand what they do. But it's certainly not a replacement for reasoning and experience.
I've found the best model of learning is to... not have a "learning process" in the first place. I try to understand as much as possible from as many angles as possible. This means big concepts, minutae, my ideas, other people's philosophies, imagined scenarios, hands-on-experiences, tangentially related concepts, and so on. Being able to answer questions or do the task is more of a side-effect than the intent.
power in society comes from a knowledge gap, and powerful people have all the incentives to sustain it. consequently education is a battleground, and we, the honest people, have pretty much lost the battles for about a century now.
the OP only makes sense when also considering this aspect of the question.
This is just blatantly wrong. If nothing else I myself have shown dogs how to solve problems, but here's a link to Wikipedia for good measure.
See for yourself
https://www.youtube.com/watch?v=GQwJXvlTWDw
or if you prefer,
https://www.sciencedirect.com/science/article/abs/pii/B97801...
I teach SwiftUI to people. I've written books and teach classes. The books don't work nearly as well (because many people just read it instead of actually practicing SwiftUI). The classes I teach ("workshops") are extremely hands on, I try to defer my explanations to after the exercise as much as possible. The feedback is often very positive, and I can tell afterwards that people have really grasped stuff. I know I'm just trying to confirm my biases here as well, but to me, there's nothing better than doing stuff first and then analyzing it.
What I really hate is explaining the solution before explaining the problem. It's a terrible way to teach and it's quite common. I like to say that there are two bad ways to teach: The cookbook (do this, then do that) and the maths textbook (solutions without problems or context). The good way is a combination of them with some additional things that neither of them has, like motivating examples, relevant anecdotes etc.
Of course the humanities classes were about books, so learning how to study the books themselves was a major part of the practice.
Programming seems to lend itself particularly well to self learning because the computer allows for endless trial-and-error practice.
High-school and undergraduate science classes tend to pair lectures with labs. Practical work is very much the focus of those labs, and the lab instructors work closely with students who need help. And a postgraduate degree typically involves a student working side-by-side with a professor on practical work.
As for the pyramid model, I think the author makes some good points, especially for the grade-school level. However, it's simply a fact that being comfortable with adding comes in handy before moving on to multiplying.
Good teachers find ways to motivate students, and adjust those ways as the years flow by. They know how to do their job, and I trust them to find the best practices.
One thing I've heard from many teachers, especially those who are notably effective, is that teaching theorists are not of much help. And I see that in the silly trends that higher-ups impose on teachers. That way of teaching multiplication that has worked for generations? No good -- we must scrap it. The practice of teaching students to write cursive? So quaint - time to toss that in the trash bin. Years later, I see the results of these trends, when students come to university.
The problem of teaching theorists coming up with silly ideas is a result, I fear, of the system of educating educators. How do you get a PhD in a subject? You have to come up with a new idea. Nobody got an advanced graduate degree in education by writing a thesis that said "teaching is fine as it is." No, that PhD student has to say "this is broken, and here's how to fix it." But some things just aren't quite broken, not really. Sure, some adjustments might be helpful. More one-on-one tutoring would be great. Although then, the non-theorist immediately sees a problem: we don't have enough teachers, as it is.
Our economy changes so fast that we need more generalized skills to adapt. If you were apprenticed as a telephone operator, what would you have done? So we learn math, science, communication, etc.
Kids are absolutely right - much of it you will never use to make money. But if you learn how to learn, then that will help make you successful no matter where you go.
It's not a question of theory or practice; you obviously need both to learn advanced skills.
The idea of church as "someone lecturing you from a book" describes only a few christian denominations, few of which were active/existant in medieval times.
I agree that many churches in the US are "20 minutes singing followed by a 1 hour sermon", which is what you describe, but there are also many denominations where the focus is on the liturgy and the sermon is a side note.
liturgy is basically a spiritual practice you do as a group.
say that week's prayer (from the prayer book)
read the psalm, call-and-response (so the congregation is talking half the time)
say the confession of sins
say the Lords prayer
someone reads 1-2 sections from the bible
a quick sermon
eucharist/communion
Having both is better, but at some point you need to learn the theory.
Hmm .... Something like in the movie "The Hunt for Red October", the US Navy wanted:
(1) Start with recordings by US submarines of underwater sounds, and write software to estimate the power spectra using the FFT (Fast Fourier Transform) and the Blackman and Tukey, "The Measurement of Power Spectra".
(2) Given ships at sea and a war, how long would the US submarines last? Start with some WWII analysis of search and encounters by Koopmans and do a Monte-Carlo Markov process, generate many independent sample paths and average.
Gee, how could I do those without my academic courses in analysis and probability? And there are more examples, including the crucial, original core math in my startup.
But dismissing theories, and just saying “most theories are wrong, anyway” smells too strongly of anti-intellectualism, and it just rubs me in the wrong way. I don't like this trend at all.
Theory is as important as practice. The two depend on each other.
This article starts with the premise that we go to school to learn how to work. In a world where that is the case, yeah, apprenticeships are far better. It happens that many people look at schooling that way, but I don't believe that's even the correct way to think about schooling.
School originally was not about learning to do a job. It was about learning how to learn. That's why writing papers and doing homework used to be such a big deal, because while you might have been stuffing your brain with knowledge about, say the history of bronze-age Europe, what you were really doing was learning how to find facts, how to organize them, and how to take useful notes.
The problem is that in the past 80 years or so, we've started to see school as training to work. Whether it's primary school teaching us to be good factory workers, or college teaching us to be good office workers. College and university came to be viewed as a way for poor children to move up the social ladder. But to do that, you need a good job. And the best way to get that good job is to teach you to do it in university. So you end up in a situation where schools don't teach students how to learn, and since group instruction is a bad way to learn how to do a job, they don't really teach students how to do a job either. And in some countries you pay out the nose for the privilege.
booleandilemma•13h ago
bravesoul2•10h ago
gattilorenz•9h ago
I certainly am skeptical of someone taking Wikipedia as the Truth.