What is the difference between these two grammar patterns?

>   1. わたしは 大学で **教えに** 英語を 習っています
>   2. わたしは 大学で **教えるために** 英語を 習っています
>
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grammar
What is the difference between ために and -masu root verb+ に?
The verb stem (pre-masu) form + に expresses purpose (~for the purpose of),
however, this construction can only be used with _motion_ verbs, such as 行く,
来る, and 帰る. The plain form + ために also expresses purpose, but unlike verb
stem+に it can be used with a wider variety of verbs. It can also express
reason or cause as well. If you've seen から (because / so) or ので (because /
so), ため can be used there as well, but it sounds a bit too formal for casual
conversation.

>   1. わたしは 大学で 教えに 英語を 習っています
>

Since 教える isn't a motion verb, using に + the verb stem (pre-masu) is
unfortunately ungrammatical in this instance. Nevertheless, I'm fairly certain
people would know what you meant if you said this.

>   2. わたしは 大学で 教えるために 英語を 習っています
>

This is the one you want. Since 教える (to teach) isn't a motion like to go
(somewhere) or to return to (someplace), ため is should be used instead.

ため can also be used with motion verbs, in which case its English translation
is equivalent to verb stem +に construction mentioned above.

For example, 試験の勉強を **するために** 図書館に行きました (I went to the library to study for an
exam) and 試験の勉強を **しに** 図書館に行きました (I went to the library to study for an exam)
are the same. When ため is used with a motion verb instead of verb stem+に, it
expresses a rather important purpose, whereas with verb stem+に, the purpose
does not have to be important.

Here are some sample sentences from _A Dictionary of Basic Japanese Grammar_
if you want some examples:

  1. コーヒーを飲む **ために** 来ました。 (I came **to have** some coffee)
  2. 父が亡くなった **ため** 、学校に来られませんでした。 ( **Because** my father died, I couldn't come to school.)
  3. 雪の **ため** 、電車が遅く来てしまった。 (The train arrived late **because** of the snow)

You may have noticed that に has been dropped in 2 and 3. According to _A
Dictionary of Basic Japanese Grammar_ , に sounds more formal and is can be
added if/when ため modifies a verbal phrase (that's just a fancy way of saying
ため is modifying a phrase that contains a verb). If ため modifies a noun, such as
体育館は、運動する **ための** 建物 then に becomes の. If this rule is confusing, you can
always just drop に and の and simply write ため instead.
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answer
>   1. わたしは 大学で 教えに 英語を 習っています
>

I have never used this. ↑

>   2. わたしは 大学で 教えるために 英語を 習っています
>

I do use this. ↑

**supplement:**  
(e.g.日本人教師の)わたしは (e.g.アメリカの)大学で (e.g.学生に数学を)教えるために 英語を 習っています  
or  
わたしは (大学で 教えるために) 英語を 習っています
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answer
The biggest difference is in the **_range of usage_** that the two expressions
allow themselves.

「ために」 can be used in far more situations than 「に」 because:

> you can only choose from a handful of verbs for Verb B in 「[連用形]{れんようけい} of
> Verb A + **に** + Verb B」

whereas

> choices for Verb B in 「Verb A + **ために** + Verb B」 are unlimited.

Choices for **Verb A** , which expresses the purpose, are literally unlimited
for **both** structures.

Choices for **Verb B** in 「連用形 of Verb A + **に** + Verb B」, however, are
limited to those few that express "motion" such as
[行]{い}く、[来]{く}る、[連]{つ}れていく、[走]{はし}ってくる, etc.

> Your sentence #1 「わたしは[大学]{だいがく}で[教]{おし}えに[英語]{えいご}を[習]{なら}っています。」,

therefore, is incorrect and it makes nearly no sense. Why? Because 「習う」 is not
a motion verb and it cannot be placed there.

> Sentence #2 「わたしは大学で教えるために英語を習っています。」

is correct and it makes sense. It may not be 100% natural-sounding by the
native standard but we will not get into that. Point is it is a grammatical
sentence.
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answer