lǎoshī zài hēibǎn shàng xiě de shíhou, wǒmen dōu rènzhēn zuòbǐjì.

This is a very common Mandarin location pattern:

在 + place + position word

So:

• 在黑板上 = on the blackboard
• 在桌子上 = on the table
• 在书包里 = in the bag

Here, 在 marks location, and 上 specifies the relation on / on top of / on the surface of.
So 黑板上 means on the blackboard, and 在黑板上 means at/on the blackboard surface as the place where the writing happens.

的时候 means when or while, depending on context.

In this sentence, 老师在黑板上写的时候 sets the time background for the main action:

• when the teacher is writing on the blackboard
• while the teacher is writing on the blackboard

So the structure is:

[time/background clause] + [main clause]

Mandarin often puts the when/while part first.

In V + 的时候, the 的 links the action to 时候 and helps turn the whole phrase into a time expression.

So:

• 写 = write
• 写的时候 = when writing / while writing

You can think of 的时候 as a fixed grammar pattern after verbs and clauses.
Without 的, 写时候 would not sound natural.

The subject of the first clause is 老师.

So:

• 老师在黑板上写的时候 = when the teacher is writing on the blackboard
• 我们都认真做笔记 = we all carefully take notes

Mandarin often keeps the subject of each clause simple and clear. Here, 老师 belongs to 写, and 我们 belongs to 做笔记.

Because Mandarin often puts time information and background first.

So it is very natural to say:

老师在黑板上写的时候，我们都认真做笔记。

This is similar to English patterns like:

• When the teacher writes on the board, we all take notes carefully.
• While the teacher is writing on the board, we all take notes carefully.

The first clause gives the situation, and the second clause tells what happens in that situation.

都 means all.

So:

• 我们都... = we all...

It emphasizes that every one of us did the action.

Compare:

• 我们认真做笔记 = we carefully take notes
• 我们都认真做笔记 = we all carefully take notes

So 都 adds the idea of the whole group.

认真 is basically an adjective meaning serious / conscientious, but it is very often used before a verb to mean seriously / carefully.

So here:

• 认真做笔记 = take notes carefully / conscientiously

In formal grammar, you may also see:

• 认真地做笔记

The 地 marks adverbial use more explicitly. But in everyday Mandarin, 认真做笔记 without 地 is very common and natural.

做笔记 means to take notes.

Yes, 记笔记 is also common and natural.

Both can mean take notes, though there can be a slight nuance:

• 做笔记 = make notes
• 记笔记 = write down notes

In many situations, they are interchangeable.

Because 写 by itself is often enough when the context is clear.

Here, since we already know the teacher is writing on the blackboard, it is natural to just say 写.

If you say 写字, it specifically means write characters / write words.
That is possible too, but the original sentence is a little broader: the teacher could be writing words, formulas, symbols, or other things on the board.

No. Mandarin does not mark tense the way English does.

This sentence can mean different times depending on context:

• present/habitual: when the teacher writes on the board, we all take notes carefully
• past: when the teacher was writing on the board, we all took notes carefully

The time is usually understood from context, not from verb tense endings.

Yes, that is grammatical.

It means something very similar, but 写字 is a bit more specific than just 写.

• 写 = write
• 写字 = write characters/words

So the original version sounds a little more general, while 写字的时候 sounds a little more specifically like writing words or characters.

Not exactly.

• 老师写的时候 = when the teacher writes
• 老师在黑板上写的时候 = when the teacher writes on the blackboard

The phrase 在黑板上 adds important location information. It tells you where the writing is happening, not just that the teacher is writing.

The comma separates the background/time clause from the main clause.

So:

• 老师在黑板上写的时候， = when/while the teacher is writing on the blackboard,
• 我们都认真做笔记。 = we all carefully take notes.

This kind of comma is very common in Chinese when a sentence begins with a time-setting or condition-setting phrase.