Learn Hardcore German: Ich notiere jeden neuen Beispielsatz in mein Heft, damit ich meine Lernstrategie später verbessern kann. - I write down every new example sentence in my notebook so that I can improve my learning strategy later.

The noun Beispielsatz is masculine: der Beispielsatz.
Masculine nominative (subject): jeder neue Beispielsatz (every new example sentence).
Masculine accusative (object): jeden neuen Beispielsatz (I note down every new example sentence).

So:

Jeder neue Beispielsatz ist wichtig. – subject → nominative
Ich notiere jeden neuen Beispielsatz. – object → accusative

How can I tell that jeden neuen Beispielsatz is accusative, when meinen, deinen etc. would also end in -en?

With masculine nouns, the article/adjective endings are your main clue:

Nominative masculine:
- der neue Beispielsatz
- dieser neue Beispielsatz
- jeder neue Beispielsatz
Accusative masculine:
- den neuen Beispielsatz
- diesen neuen Beispielsatz
- jeden neuen Beispielsatz

So the change from -er to -en in jeder → jeden shows you it’s accusative.

For possessives (like mein, dein, sein), masculine accusative also takes -en:

mein neuer Beispielsatz (nom.)
meinen neuen Beispielsatz (acc.)

Context then tells you whether a mein-/dein- phrase is subject or object:

Mein neuer Beispielsatz ist lang. (subject)
Ich notiere meinen neuen Beispielsatz. (object)

What is the nuance of notieren compared to schreiben?

Both can mean “to write”, but:

schreiben is the general word: to write.
- Ich schreibe den Satz in mein Heft.
notieren means to jot down / to make a note of, often shorter, purposeful writing, like notes, keywords, example sentences.
- Ich notiere den Satz in mein Heft.

Using notieren in your sentence emphasizes a learning/recording action, not creative or extended writing.

Why is it in mein Heft and not in meinem Heft?

Because in is a two-way preposition (Wechselpräposition) that can take:

Accusative for a direction / movement into something.
Dative for a static location / being in something.

In your sentence, you are putting the sentence into the notebook (a kind of movement):

Ich notiere … in mein Heft.
→ I am writing it into my notebook. (direction → accusative)

Compare:

Es steht in meinem Heft. – It is in my notebook. (no movement → dative)
Ich schreibe den Satz in mein Heft. – I write the sentence into my notebook. (movement → accusative)

So in mein Heft (acc.) is correct here.
In meinem Heft (dat.) would describe something that is already lying/printed in the notebook, not the act of writing it there.

Could I also say auf mein Heft instead of in mein Heft?

Not with the same meaning.

in mein Heft = into my notebook, i.e. on the pages inside.
auf mein Heft = onto my notebook, i.e. literally on the cover or surface of the notebook.

For writing inside a notebook, Germans normally say:

in mein Heft schreiben / notieren

What does damit mean here, and how is it different from just using um … zu?

In this sentence, damit means “so that / in order that” and introduces a subordinate clause of purpose:

Ich notiere …, damit ich meine Lernstrategie später verbessern kann.
→ I note … so that I can improve my learning strategy later.

Difference to um … zu:

Subject
- damit can be used when the subject in the main clause and the subordinate clause is the same or different:
  - Ich notiere das, damit ich besser lernen kann. (same subject: ich)
  - Ich notiere das, damit du besser lernen kannst. (different subjects: ich / du)
- um … zu is only used when the subject is the same in both actions:
  - Ich notiere das, um meine Lernstrategie zu verbessern. (subject “I” for both actions)
Form
- damit
  - full clause with a finite verb at the end:
    - …, damit ich meine Lernstrategie verbessern kann.
- um … zu
  - infinitive phrase:
    - …, um meine Lernstrategie zu verbessern.

In your sentence, damit is perfectly natural; um meine Lernstrategie später verbessern zu können would also work, but sounds a bit heavier.

Why does the verb go to the end in damit ich meine Lernstrategie später verbessern kann?

Because damit introduces a subordinate clause (Nebensatz). In German subordinate clauses, the conjugated verb goes to the final position.

Basic pattern:

Main clause: Ich notiere … (verb in 2nd position)
Subordinate clause with damit:
- damit ich … verbessern kann

Word order in the subordinate clause:

damit (subordinating conjunction)
Subject: ich
Objects/adverbs: meine Lernstrategie später
Verb complex at the end: verbessern kann

So the verb (or verb group) is kicked to the end by damit.

Why is kann at the very end, after verbessern, in verbessern kann?

Because there are two verbs in the subordinate clause:

verbessern – the main verb (infinitive)
kann – the modal verb (conjugated, expressing ability)

In subordinate clauses with a modal verb, the typical order is:

… [other elements] + infinitive + modal.

So:

… damit ich meine Lernstrategie später verbessern kann.

Compare:

Main clause: Ich kann meine Lernstrategie verbessern.
Subordinate: …, damit ich meine Lernstrategie verbessern kann.

Why is kann used at all? Could I just say damit ich meine Lernstrategie später verbessere?

You can say damit ich meine Lernstrategie später verbessere, but:

damit ich … verbessern kann = so that I can improve …
Focus on ability/possibility.
damit ich … verbessere = so that I improve …
More like a simple future outcome / intention, no explicit idea of “being able to”.

In everyday German, damit ich … verbessern kann sounds more natural here, because learners usually talk about getting into a position where improvement is possible.

Your version without kann is grammatically correct but feels a bit more formal or unusual in this context.

Why is meine Lernstrategie not visibly accusative here? It looks like nominative.

The noun Lernstrategie is feminine: die Lernstrategie.

For feminine singular, the article does not change between nominative and accusative:

Nominative: die Lernstrategie
Accusative: die Lernstrategie

With possessives:

Nominative: meine Lernstrategie
Accusative: meine Lernstrategie

So the form meine Lernstrategie looks the same in both cases.
To know it is accusative, you must look at the verb verbessern:

Was verbessere ich? – meine Lernstrategie → direct object → accusative.

Why is Lernstrategie capitalized, and what kind of word is it?

In German, all nouns are capitalized, regardless of where they appear in the sentence.

Lernstrategie is a feminine noun: die Lernstrategie.
It is also a compound word:
- das Lernen / lernen (learning / to learn)
- die Strategie (strategy) → die Lernstrategie = a strategy for learning.

That is why it starts with a capital L.

Why is present tense (kann verbessern) used with später instead of a future form like werde … verbessern können?

In German, the present tense is very often used to talk about the future, especially when the future time is clear from context or an adverb like später, morgen, nächste Woche, etc.

So:

…, damit ich meine Lernstrategie später verbessern kann.
= “so that I can improve my learning strategy later.”

Using Futur I here is possible but sounds heavier and less natural:

…, damit ich meine Lernstrategie später werde verbessern können. – grammatically correct, but overly formal and clunky in everyday language.

So the simple present tense with später is the normal, idiomatic choice.