TFA misses a key difference between apprenticeship and classroom learning. Apprentice training tends to be one-on-one. When classroom instruction is done one-on-one, learning dramatically improves. This is called the "two sigma problem" in the educational literature. Ignoring this aspect gives the other factors discussed in TFA exaggerated significance.
Practice is extremely important, and I don't think its importance is exaggerated at all.
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.
> But tutoring isn't obviously superior to practice.
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.
> Good tutoring will essentially be practice and worked problems with instant feedback
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.
They are related. They are both individual learning.
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?
> Maybe much learning is an individual activity and learning in groups is just ineffective?
Learning in groups is wildly ineffective from the perspective of gaining functional mastery over some subject (whether writing well, solving algebra problems, etc).
However, it does have a lot of unrelated benefits, arguably more important: learning to collaborate with others, understanding how others think and learn, understanding your own skill level by direct comparison with others, competition as a motivator for learning, and more.
I am not convinced that those are realised in real life.
The joy of learning is a better and more sustainable motivator than competition.
Learning to collaborate with others is an important skill, but I am not sure it is particular often promoted within classroom learning. There are lots of things you can do (sports, hobbies, anything that aims at an end in a group) that are better at teaching collaboration.
Spending less time on learning frees up time for other, IMO better, ways of learning all those skill.s
I would argue that learning in groups is potentially exponentially more effective. There is a lot of individual interactions that go on when people are all learning something at the same time. Tidbits that they each share with one another. And emotion that social interactions evoke is a powerful motivator for mental rewiring.
But I don’t want to dismiss your insights. I’m curious what the difference is that I’ve experienced. Certainly just putting random people together isn’t nearly as beneficial as grouping by ability, or motivation to learn the topic is useful. Maybe that is a necessary requirement for effective group learning?
I have studied in streamed classes (in a school that was very selective anyway) so closely grouped by ability.
I did find working with friends in small groups effective at postgraduate level (although we organised it ourselves) and I would have done better to have done more of that. However this was a small group, not a class.
Classrooms and lecture rooms do not promote interactions. On the other hand one to one tuition is continuous interaction, hopefully with someone who is a better model for interaction that other kids, and who is encourages interaction.
I think we are talking about four different things here. Self teaching, 1:1, small groups, and classrooms. 1:1 and learning oneself are far more effective than classrooms. I cannot compare with small groups, and they are used at some universities (e.g. the tutorial system at Oxford and Cambridge), but my feeling is that it will be highly effective for the right people and the right subject. Then again, those universities require a lot of ability and motivation to get into, so maybe that is why it works for them.
As a university level educator that also has assistants that learn through practise I must say I find the question: "Is tutoring better than practise?" useless. Better at what? In which field?Thst surely highly depends at what the goal, the subject, the individual students character, the available time and teaching resources are.
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.
> I find the question: "Is tutoring better than practise?" useless.
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?
> 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?
The abstract concept is meaningless without the concrete examples.
It is only mathematicians, who are accustomed to the abstract theorem being the final goal, who get confused about this. It's only possible to consider the "theorem first" approach as reasonable to the extent that you, or the students, already have the requisite concrete foundations to understand it. Which is to say: to the extent that it is not really "new".
Well yes and no. If I give people a 16 mm camera, a roll of film and a black sack, the likelyhood they will make a positive learning experience is limited. That means some theory is needed first, especially if you have varying levels of knowledge and confidence within a group.
That doesn't mean theory has to be boring or abstract. It just means some things need explaining and some people profit from having had them explain to them. In some cases this may even be a legal requirement, letting a student use a table saw or a turning machine without explaining the ways in which it may kill them can land you in jail.
You could say that all words are abstractions, and therefore my statement is false because almost all teaching uses words... obviously this is not what I meant. Some interpretive generosity about where to draw the lines between what counts as abstraction or concretion in a particular context is required.
With that said, I actually think you could teach someone (say, who didn't speak your language) how to use a film camera merely by demonstration. It would take longer than if you could use words, but would be doable.
And I would argue that this kind of operational ability (with no explicit theory) is usually where the bulk of the actual learning goes on, and is usually what we mean when we talk about competence. Sometimes it's shocking the theory that really talented people don't have in a domain... I'm thinking of things like poker, competitive programming, even competitive mathematics. Obviously such people have some sort of theory, but it may be largely implicit, learned almost entirely by doing, and different-looking from the accepted or academic theories of that domain.
Hmmm, this seems reasonable, but I personally am extremely annoyed when I ask someone "what is thing x?" and they start by saying "let me give you an example" instead of a general description of the thing first.
Is this because I'm starting to think like a mathematician? Or because I'm conflating a deep, theory-first explanation of a concept with a surface-level summary that is then followed by concrete examples? Or something else?
An example would be helpful (ironically), but I think it's because you already have fluency in many abstract concepts (rooted in a lifetime of concrete application and examples), and in this situation the abstraction is usually much faster to work with, and you're annoyed because the example is effectively a long-winded explanation.
I'm not saying a mature mathematician is wrong to ask for an abstract definition first. For them, that might work well. But it's wrong to conclude from that experience that the abstraction is somehow more primary, or that it could exist in isolation. They've just forgotten the process that got them there.
Practicing and getting constant feedback is so important (and sadly underrated in school). It still strange to me how we empathise rote learning in school and have the experimentation away from experts (homework).
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)
This is extremely interesting, because while I'd never heard of the '2 sigma problem' [1] before, one university class I had seems to have been largely modeled on it, but with a very different angle. It was a 'self paced' electrical engineering course where we were given a textbook and free to advance through it at our own pace - kind of farcically, since you needed to complete at least 2 chapters per week to finish by the end of the semester.
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.
I had such a self-paced course in the '70s based on the book "Fundamentals of Logic Design" by Charles Roth, Jr. It should be noted that the book was specifically written for self-paced study, and as such acted as a sort of tutor by carefully laying out a sequence of reading short segments, answering short questions about the material, then doing more involved problems. I found this course to be very effective and motivating for me, especially given the undergraduate class sizes.
Wow, care to share your alma mater? That was the exact book we also used - some decades later, 5th edition for my class! Absolutely wonderful book. Wow, what a wave of emotions I got when looking at that book's cover again!
And yeah that course and book gave me a serious love of electrical engineering to the point I even considered swapping majors (it was part of the CS curriculum for us), and in hind sight I rather wish I did, but hey - wisdom to pass onto the kids.
That was at UT Austin, where Dr. Roth was a professor. Another thing about that course - the problems that were given by the TAs for the 90% proficiency checks seemed pretty challenging and weren't in the book. You really had to know your material, yet there were no big surprises.
A couple things come to mind reading this. Maybe your professor knew the material was engaging in itself or the textbook was exceptionally well written that any added structure on top was likely to complicate it. The second possibility was that maybe they knew it was a fundamental course that students must engage with anyway.
Regarding the lack of feedback, maybe grade was sufficient. Sometimes enough is best.
I feel like whats most important in teaching is that the teacher has integrity. If you can control the teacher in any way, that loses the dynamic. In fact, his idiosyncratic method might indirectly increased his integrity score, which we subconsciously evaluate on teachers before we allow ourselves to engage.
Mastery Learning, which Bloom advocates for in the two sigma problem paper, is an alternative to 1 on 1, not a way to achieve it.
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.
1:1 classroom instruction removes a number of teachers from the labor pool equal to the number of students. Apprenticeships remove only a small fraction of that from the labor pool (because the practitioner spends only part of their time teaching/supervising the apprentice and then makes the apprentice go practice the skill) and partially makes up the lost labor with the labor from the apprentices- apprentices are expected to do actual productive work, not just learn.
> 1:1 classroom instruction removes a number of teachers from the labor pool equal to the number of 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.
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.
Yeah, while the article makes good arguments about learning by doing and context-rich environments, it probably understates how much of the effectiveness comes down to just personalized guidance
If it wasn't for all the pitfalls and hallucinations (and even then there is probably something to be had already) LLMs would be perfect for this. Limitless customizable one-on-one tutoring. I would have killed for something like it when I was in school, instead the choices were expensive tutor (not an option) or else good luck, hope you pay attention in the back of the 30 student classroom.
You seem to be suggesting he's writing from a place of not knowing about the benefits of one-on-one learning and the "two sigma problem" when this is something he frequently writes about.
