Learn Hardcore Swahili: Kwa ujumla, tunajivunia jinsi watoto wanavyosomea nyumbani. - In general, we are proud of how the children study at home.

It means in general / overall. The comma after an introductory phrase like Kwa ujumla is common but optional. Both "Kwa ujumla, ..." and "Kwa ujumla ..." are fine. Grammatically, it’s kwa (by/with) + the abstract noun ujumla (generality).

How is "tunajivunia" built, and what does "-ji-" do?

It’s a single verb complex: tu- (we) + -na- (present/habitual) + -ji- (reflexive) + -vunia- (verb stem) + final -a.

-ji- marks the action as oriented to oneself; with this stem it yields the meaning to be proud (of).
Note: kujivunia (be proud of, positive) is different from kujivuna (to boast, often negative).

Why is it "jinsi ... wanavyosomea" instead of just "kwamba ..."?

Jinsi means how / the way, so it expects a relative clause marked with -vyo- in the verb: wanavyosomea = “how they study.”
Using kwamba would give a plain complement clause:

Tunajivunia kwamba watoto wanasoma nyumbani = “We’re proud that the children study at home” (statement of fact).
Tunajivunia jinsi watoto wanavyosomea nyumbani = emphasizes the manner/way they study at home.

What exactly is going on inside "wanavyosomea"?

Morphology: wa- (they) + -na- (present/habitual) + -vyo- (relative marker “how/the way”) + -som- (read/study) + -e- (applicative) + -a (final vowel).
So it literally encodes “they-are-how-study-for/at,” i.e., how they are studying (at/for).

Do I need that applicative -e- (as in "somea") here?

Not strictly. Both are acceptable:

wanavyosoma nyumbani (common and perfectly fine)
wanavyosomea nyumbani (adds an “at/for” nuance; can make the location feel more tightly linked to the verb).
In many contexts it’s stylistic; meaning is the same here.

Why is the relative marker "-vyo-" used and not something like "-ivyo-" or "-avyo-"?

With manner words like jinsi, namna, or vile, Swahili uses the general manner relative concord -vyo- to mean “how/the way.”

-ivyo- is tied to demonstratives like hivyo (“thus, like that”), not to jinsi.
-avyo- agrees with certain noun classes (e.g., for class 16/17/18 or with “avyo” after “vyo” in some patterns), but with jinsi the idiomatic choice is -vyo-.

Why "wa-" at the start of "wanavyosomea"?

It agrees with watoto (children), which is noun class 2 (plural for people). Class 2 uses the subject marker wa- (“they”).
Agreement in the relative clause is with the clause’s subject (watoto), while -vyo- is there because of jinsi.

Is "nyumbani" the same as "kwenye nyumba" or "katika nyumba"?

nyumbani (house + locative -ni) = at home / at the house (as a home).
kwenye/katika nyumba = in/inside a house (the building).
Don’t say kwenye nyumbani; nyumbani already has the locative built in.

Can I say "jinsi ambavyo watoto wanasomea nyumbani"?

Yes. You can mark the relative clause either with the free pronoun ambavyo or with the bound marker -vyo-, but not both:

jinsi watoto wanavyosomea ... (with -vyo-, no ambavyo)
jinsi ambavyo watoto wanasomea ... (with ambavyo, no -vyo-)
Using both together is considered redundant.

Could "watoto" mean “our children” here?

On its own watoto means children / the children from context. To make it explicit, add a possessive: watoto wetu = our children. The sentence as given is generic unless context makes it specific.

Where can I place "kwa ujumla" and "nyumbani" in the sentence?

Kwa ujumla is most natural at the start: “Kwa ujumla, ...” You can also place it after the verb phrase, but initial placement sounds best for a sentence-level comment.
Nyumbani typically comes after the verb phrase of the relative clause. You could front it for emphasis: Nyumbani, tunajivunia jinsi watoto wanavyosomea, but that puts focus on the location.

Why is everything glued together in forms like "tunajivunia" and "wanavyosomea"?

Swahili verbs are built as one morphological word:
Subject prefix + Tense/Aspect + (Relative/Object) + Verb root + Extensions + Final vowel.
Hence single-word complexes like tunajivunia and wanavyosomea.

Should there be a "ku-" anywhere, like "kutunajivunia"?

No. ku- is the infinitive marker (dictionary form: kujivunia). In a finite sentence you conjugate and drop ku-: tuna-ji-vunia = tunajivunia.

If I want past time, how do I change the clause?

Switch the tense markers:

“We were proud of how they studied at home”: Kwa ujumla, tulijivunia jinsi watoto walivyosoma/ walivyosomea nyumbani.
Here tuli- (we, past), and in the relative clause wali-...-vyo-... marks past.