For example: Say an interested undergrad has just learnt examples of cyclic groups (before the definition). It is reasonable to believe after a bit of fiddling around with n-tuples of elements from the groups he/she will soon be able to land up with an idea of a direct product. And a bit more fiddling around will lead he/she to the result that the direct product is cyclic if the orders of the constituting cycling groups are co-prime, etc.

In sum, IMO a few good examples takes one closer to the math than many a definitions. Moreover, prospective mathematicians shouldn’t get too used to having definitions and theorems served up on a platter.

jel, my preferred order is definition, fifty examples, theorems.

JB, Dedekind domain is a good counterexample. There’s a situation where you’d need to see the examples first. I thought of another example right after I posted: Lie algebras. They seem pretty unmotivated if you’re not already familiar with the relationship with Lie groups.

Definitions and theorems are vital and important — an approach which provides examples (context) first can make it easier to remember the definitions/theorems, because associations with a rich, processed, contemplated context are used. — The context provides something for students to “hang their hats on”. I have seen students in my classes glaze over and have watched definitions and theorems pass straight through their ears when presented without either detailed and careful explanation or some rich, relevant context.

Well - that is just two cents worth…
peace!

That’s your own learning process and not everyone can understand or learn in that way. Not everyone is the same. I am not sure if you are aware of different learning methods by people. For instance you’ll have some people that learn more by pictures, others by reading then you have the ones that enjoy lectures. And as you mention your own where you understand instantly a written definition.

My own experience is that I “learn by example”. I am glad that I stumbled with this post and the link to that guy. Learning by example is not leading someone blindly to the end of the road missing everything else. How I see it is giving me the tool to do something and afterwards I can discover/understand the underlying mechanisms.

You don’t need a thousand PHd’s in Biology, Math, Chemistry, Quantum, Agriculture, etc to understand that an apple is an apple. The path to deeper knowledge can be achieved through different methods.

“being led down a road by someone who already knows the destination, but they won’t tell me where we’re going until we actually get there.”

^^^ That conclusion makes absolutely non-sense. Were you born already knowing what you know? Did you go to university already knowing your subject? And yes, unfortunately ALL professors “already know the destination” when they are teaching students.

Besides a HUGE percentage of our learning process after being born is solely “by imitating”, i.e. “by example”.

Seriously I don’t know why learning “by example” should exclude later a deeper understanding or the theorems.

I think your post has been hastily written without a lot of thinking.

have a nice day.

In other situations, where you’ve already got a good feeling for a subject, just seeing a new definition is enough to cause feelings of pleasure. “Ah yes! Why didn’t I think of that concept?”