In Mathe bin ich heute besser als gestern.

It means in math / in maths, in the sense of in the school subject mathematics.

German often uses in + subject name to talk about how someone is doing in a subject:

• In Deutsch bin ich gut.
• In Englisch habe ich Hausaufgaben.
• In Mathe bin ich heute besser als gestern.

So in is not really about physical location here. It means in the area of / in the subject of.

Mathe is a very common short form of Mathematik, especially in everyday speech and in school contexts.

So these both work:

• In Mathe bin ich heute besser als gestern.
• In Mathematik bin ich heute besser als gestern.

Mathe sounds more natural in casual conversation. Mathematik is more formal.

German main clauses follow the verb-second rule. That means the finite verb goes in the second position, even if something other than the subject comes first.

Here, the first element is In Mathe. So the verb must come next:

• In Mathe | bin | ich heute besser als gestern.

If you start with the subject instead, that also works:

• Ich bin heute in Mathe besser als gestern.

Both are correct. The original sentence emphasizes in math by putting it first.

Because once In Mathe takes the first position, the verb bin must stay in second position, and the subject ich comes after it.

Compare:

• Ich bin heute in Mathe besser als gestern.
• In Mathe bin ich heute besser als gestern.

This is very normal in German. English usually keeps the subject before the verb, but German is more flexible because of the verb-second rule.

Yes, heute can move.

In the original sentence:

• In Mathe bin ich heute besser als gestern.

heute comes after ich, which sounds natural.

But you could also say:

• Heute bin ich in Mathe besser als gestern.
• Ich bin heute in Mathe besser als gestern.

These all mean basically the same thing, but the emphasis changes slightly:

• Heute... emphasizes today
• In Mathe... emphasizes in math
• Ich... is the most neutral starting point

Because besser is the comparative form of gut.

• gut = good
• besser = better
• am besten = best

This is irregular, just like English good → better → best.

So:

• Ich bin gut in Mathe. = I am good at math.
• Ich bin heute besser als gestern. = I am better today than yesterday.

Guter would be an adjective form used before a noun, for example:

• ein guter Schüler = a good student

It does not fit here.

German uses als after a comparative, like besser.

• besser als gestern = better than yesterday
• größer als ich = taller than me
• schneller als früher = faster than before

This is the normal pattern:

• positive: so ... wie = as ... as
• comparative: ...er / mehr ... als = ... than

Examples:

• Ich bin so gut wie du. = I am as good as you.
• Ich bin besser als du. = I am better than you.

So als is required because besser is a comparative.

Because German, like English, often leaves out words that are understood from the context.

Besser als gestern means better than yesterday, and the full idea would be something like:

• ... besser als ich gestern war.

But that would sound unnecessarily heavy here. The shorter version is natural and standard.

English does the same:

• I’m better today than yesterday.

You do not need to repeat I was.

Because heute and gestern are adverbs.

They already mean:

• heute = today
• gestern = yesterday

So you do not need am, im, an, or an article here.

Examples:

• Heute lerne ich.
• Gestern war ich müde.
• Ich bin heute besser als gestern.

This is different from some noun phrases like:

• am Montag = on Monday
• im Sommer = in summer

Yes, that is also correct and natural.

You may hear both:

• Ich bin heute in Mathe besser als gestern.
• Ich bin heute besser in Mathe als gestern.

The difference is mainly one of rhythm and emphasis. The original version puts in Mathe earlier, so it highlights the subject area sooner.

Both are understandable and idiomatic.

Not exactly, but they are related.

• gut in Mathe sein = to be good at math
• in Mathe besser sein = to be better in math

So in Mathe tells you the area, and the adjective tells you the evaluation:

• Ich bin gut in Mathe.
• Ich bin schlecht in Mathe.
• Ich bin besser in Mathe als früher.

In your sentence, besser does not necessarily mean the speaker is already good overall. It only says they are better than yesterday.