"I walk to my house."
Here's a good explanation: https://answers.yahoo.com/question/index?qid=20080904112115AAdVVRE#
TL;DR: に would be used if emphasis is the destination (as in answering the question "Where did you go?" with "I went to the house") whereas まで would be used if the destination is not as important as the "going to" is (as in answering the question "Do you go walking?" with "I walk as far as my house.")
According to A Dictionary of Basic Japanese Grammar
Page 225, まで (made) is "a particle to indicate the spatial, temporal, or quantative limit (or an unexpected animate/inanimate object)." Translations include "as far as, till, up to until, through, even."
Page 302 "a particle which indicates a place toward which someone or something moves." Translation is "to, toward."
This is more or less what you said but Japanese (unlike English) doesn't have prepositions but has postpositions aka particles.
I think you should have just used the last para: So, using "まで" implies a meaning more along the lines of "I went as far as the lake" or "I went all the way up to the lake". Instead of answering a question like "Where did you go?", it's answering a question like "How far did you go?".
According to @jugglejunk, に is the direction, まで is the destination, but you said に indicates going to school, doesn't that means school is the destination and not direction?
Also, below, @kai19154 said using に could mean walking to your house but going past it, doesn't that means direction instead of destination?
に is destination/target of action or location of existence, it puts more stress on the location you are going to and usually means you are going somewhere with a specific intention. 学校に行きます You go to the school (and stay there for classes)
へ is direction, 'towards', it puts more stress on the movement itself, focusing on the journey rather than the destination. You may stop other places along the way, you may even continue past the location you mentioned. 学校へ行きます You go to the school (you my stop and get a snack, maybe you'll even go past the school to the park once you get there.)
まで is "until". You will stop doing the movement once you reach this place. 学校まで行きます You go to the school (you stop once you've reached the school. You don't have to go inside or anything, but you don't plan on going any further than this point.)
So the focus: に - the destination へ - the journey まで - the upper limit
うち refers not just to the physical house, but also to the environment, family, stuff, etc. that makes it one's home. It can also be used to refer to something inside your home (ex. うちの台所は狭い to mean "our/my kitchen is small") You can also use it for someone else's home, like うちに帰りなさい (please go home) So when it is いえ it is the physical house. うち is one's physical house and home. Note that you can't say うちに行きます because you can't "go" to your own home, but you can "return" there, so you need to say うちに帰ります
へ and に mean the same thing. The only difference is that へ is used for going from point a to point b explicitly. you cant use へ for things like saying something is inside of something else, for example. 家へ歩いて has the same meaning as this question, but theyre are just trying to teach you a new way to say the same thing. It's important to know both(all of them really).
It does affect the meaning though. 歩くis informal and therefore less polite than 歩きます. Children and elderly people use this plain form, also an older person to a younger person, a parent to a child, a superior talking down to a subordinate - you get the idea. It typically conveys a lower, more familiar level of speech so would be like Duo teaching people to speak in slang.
No, it does not change the meaning, trust me.
"Politeness” is not the word's meaning it is “politeness”
This doesn't have anything to do with politeness though it is often perceived this way by new learners of this language. This has to do with familiarity. The closer you are to someone you would use だ instead of です。Stuff like 謙譲語and 敬語 has to do with politeness.
歩くis not slang or "like slang". Children speak this way to their parents. What I could agree that would "be like teaching someone slang" would be something like ない →ねぇ。 It is not incorrect to translate "I walk to my house" as either家まで歩きます or家まで歩く 。
It is not slang but it is hard to convey the difference in level of speech between 歩く and 歩きます in English or to English speakers who don't have the respectful and highly heirarchical society that Japan has and the many layers of speech that reflect that. Yes, 歩く and 歩きます are both present active verbs but they would only be used with certain people and in certain situations and would conversely not be used with certain people and in certain situations and I would argue that that does affect the meaning as it can reveal the speaker's relationship, position, age and even opinion of the person or people they are talking to. Translating I walk to my house as 家まで 歩く leaves us many questions and things to consider - why is Duo using plain form? Is Duo a child? Is Duo elderly? Is Duo talking to a subordinate or maybe just someone that they don't think much of - ie. are they being rude by speaking in a low form of language to a superior or a senior? With 家まで 歩きます none of these issues or concerns arise because 歩きます doesn't introduce the possibility of those alternate interpretations or meanings.
I understand your point is that it's safe to use ～です ～ます and it makes sense because the foreigner learning this language may not understand these things yet so at the start it would be good to learn to speak using this style so as not to step on other people’s toes inadvertently. I also understand it can help indicate a relationship of the people speaking.
However, the meaning of style difference isn't my point of contention. My point is the that both 歩くand 歩きます have the same meaning. What form you use is contextual but does not change the word's meaning. Remember, we aren't arguing about style we are arguing about meaning. If you refer to my first comment, I explain that I understand the style is different and just that either of the two words -歩くor歩きます-should be acceptable because the meaning is the same. What you can glean about who the person is in in relation to others based on the style does not have any bearing on the meaning of the word. For example if you have the utterance スーパーに歩いたand スーパーに歩きました. For both, while the style may be different, has this person walked in a different manner for either of these sentences? No, I don't think that this person has as the meaning is the same. The verb's meaning does not change thus, for the sentence "I walk to my house" both 家まで歩く and 家まで歩きます should be acceptable.
There are so many people in here pointing out that "ni" would have a subtly different meaning. But elsewhere a lot of lessons seem to accept "ni" with other verbs like "iku" for, e.g., "I go to school". Is this a subtle difference between the two verbs "aruku" and "iku" or are the two lessons just inconsistent?
I like to think まで ("to") has a partner, から ("from"), and together they define your trip: from here to home (ここから家まで). In the Duo sentence, they have omitted the から half (now implied), but the meaning remains: your destination is home.
When に is used, my impression is that you are moving to a more specific destination (学校), whereas へ could be to a broader location (日本). This could be why others describe the difference as emphasis on destination vs. journey. But their nuanced meanings are so subtle that they are often used interchangeably.