If you look at this if you use ‘my,’ ‘you (s),’ ‘your (pl)’ and ‘our,’ you don’t need to know the gender of the person the posessive adjective/pronoun is referring to, because you already know it, and to the poles, the most important thing is the gender of the object in these cases.

I found that when talking about swoj to think of it as meaning “my own” and if you start a sentence with ‘I’ and you are talking about something of your own and you need a posessive adjective, it is always ’svoi.” I hope tha thelps.

How can I know when and how to change the end of the the word mapa? What is the name for this, is it a declension?

Other wrong examples maybe:
Masz lampka gratis?
Masz książka gratis?
Masz kubek gratis?
Masz krześło gratis?

Odd questions I know but it is an exercise in grammer.

It’s not so helpful in the first and second persons (where some people are sloppy with it) but it’s worth its weight in gold in the third person.

Spalił swoją książkę.

= He burned his (own) book.

vs

Spalił jego książkę.

= He burned his (someone else’s) book.

I often ask a Pole “Why does Polish say ‘jedna książka jest’, ‘dwie książki są’, but ‘pięć książek jest’ with a different noun ending and back to the singular verb again?” and produce confused surprise and “Wow, you’re right! Why do we do that! I never thought about it before.” I can’t imagine following such arbitrary complex rules automatically and not being aware of them.

Polish computer programmers who have ever had to program a textual user interface seem more conscious of such linguistic hoops that must be jumped through (e.g. an online store that needs to tell you how many books are in your shopping cart). Programming such complex rules forces you to be aware of them.

PS: Basia, you are confusing possessive pronouns (my, your, his, her, etc, which don’t decline in the third person in Polish) with pronouns (I, you, he, she, etc, which do decline in the third person in Polish).

None of the other forms (including swój) are the same as a genitive pronoun.

Of course it’s a little more complicated when you get to the formal pronouns.

Pan has pański (a regular adjective) while Pani has ….. Pani. Once I noticed a woman fumbling in her purse hadn’t noticed she dropped a pen. I picked it up and asked her “Czy to Pani?” which means (simultaneously) “Is this yours?” and “Is this you?”

I was under the impression that jego/jej/ich do decline:

Acc/Genitive: jego/niego/go
Dative: jemu

Acc: ja (can’t type “a” with tail) but elsewhere jej/niej

Dative: im/nim but elsewhere ich/nich.

Lately, I feel like I am making little progress with this language. I think I should resign myself
to speaking English, being labeled as a hopeless monolingual American* (although
I am Canadian) and just put up with the smirks while I’m in Poland.