That means “I travel by bike to the school”, right? What about dropping the implied “der”? In English, as I’m sure you know, we could say “I bike to the school” or “I bike to school”.
And there is a case when “Ich fahre mit dem Fahrrad nach der Schule” is almost right: When you mean to say that you ride your bicycle after school. As an evening activity, so to speak. Though changing the order sounds more natural too me: Nach der Schule fahre ich mit dem Fahrrad.
Nach meaning “after” not “to” in that case, I guess.
I’ve found lots of exampled of mit Rad zur Schule online and in quality newspapers as well.
Oct 4, 2010 — Klasse mit Rad zur Schule kommen. Sind ja doch 2 km zu fahren. Eingetragen von eishockeymama am 06.10.2010 um 10:58 Uhr.
I stand corrected. You are correct Eisteinshund and Pardel-Lux. It is incorrect to say “Ich fahre mit Rad zur Schule”. Ich fahre per Rad zur Schule is possible. But then there are the examples I see " *mit Rad zur Schule kommen". That’s where I was confused. It seems possible to apply the Nullartikel in this case. I’m not quite sure why.
It does sometimes mean “overly intellectual”. When people refer to “top-heavy brains” or “top heavy academics” or “top-heavy thinkers” and so forth, that’s what they mean.
I haven’t come across that usage in English, to my memory.
Well, that explains why you thought it didn’t exist!
As other native German speakers have said, much of this is dialectal. A wording that sounds odd in one German dialect might be perfectly idiomatic in another, or in Standard German. This applies to many things, such as inclusion or omission of articles, choice of preposition, or vocabulary. To mention another example, I grew up in a part of Germany where it’s even common to combine proper names with definite articles: “Der Thomas und die Sarah haben gesagt…” - “The Thomas and the Sarah said…”. Absolutely unacceptable in Standard German but very common in many parts of the south of Germany.
I always thought it was correct to use ‘der’ or ‘die’ when referring to a particular individual.
Am andern Tage kam der Peter gerade zur rechten Zeit in die Schule heruntergefahren.
This looks like perfectly good German syntax but I can’t explain the absence of an article before ‘Rad’. Is it a question of style or grammar?
https://yogaistfueralleda.ch/interview-mit-der-autorin-und-musikerin-andrea-bannert-zur-clyatomon-reihe/
Jun 8, 2021 — Wenn es mal nicht weitergeht, wechsle ich den Ort und fahre mit Rad und Notebook an den Starnberger See oder gehe erstmal eine Runde joggen.
This is most probably just an example of lazy and improper online style like also in many (most?) English online conversations. Think of it like old fashioned telegram style omitting words.
I would interpret that you take your bicycle and your notebook and go, perhaps by train, to the Starnberger See. Or maybe you rode/ride your bicycle, not relevant. The important thing is you have it there. It sounds analogous to “mit Sack und Pack” fahren, a quite common alliteration to express to take all your stuff with you. You would not say “mit dem Sack and dem Pack fahren”, sounds much too concrete. What do I care whether you stuff your belongings in one sack or another?
Indeed, it is another word. It is just written the same, to confuse people.
I think you’re right. I haven’t seen any grammar rules to justify it. Thanks Einsteinshund.
Message boards, comment sections and messenger posts are generally not the best sources for proper grammar, in either language. (with the exception of the Dope, of course) 
Yeah, we have that in English, too…
Thanks Pardel-Lux. That makes sense.
Using definite articles together with a person’s name is common in southern German dialects (which includes Swiss German, so I’m not surprised to see that Johanna Spyri does it), but I wouldn’t do it in Standard German except for certain usages such as “the Peter of today is more confident than that of ten years ago”, if you want to say the same person has changed over the course of that time.
doppelpost!
Seems to be a written vs. spoken thing:
You wouldn’t read "Der Hans ist ein Verbrecher" in an e.g. newspaper in the alpine region, either … but spoken, esp. in dialect is fully ok!
and the “radeln” - as mentioned, it seems to be a alpine region thing (s-germ., austria, swiss?) … not sure about the swiss, as they are doing the “velo” thingy IIRC
In den Alpen wird scheinbar viel g’radlt… come to think of it, in german it’s not too common to verbalize a substantive ((Fahr)Rad → radeln)
but isn’t it ok in english to say “I cycle to school every morning”? which is pretty much the same example as “I ride a bike to school every morning”