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.
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.