〜ようと思う
be thinking of doing, plan to
Volitional form + と思う / と思っている
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Volitional + と思う says what you're thinking of doing: 夏休みに日本へ行こうと思います, "I'm thinking of going to Japan in the summer holidays". Literally "I think 'let's go'".
と思っています suggests a plan you've had for a while: 来年から一人で住もうと思っています.
Compare its neighbours:
- 行くと思う: "I think (someone) will go", a prediction, not a plan
- 行こうと思う: "I'm thinking of going", an intention still taking shape
- 行くつもりだ: "I intend to go", a firmer decision
It's the natural, slightly modest way to talk about your own plans.
Examples
夏休みに日本へ行こうと思います。
I'm thinking of going to Japan in the summer holidays.
来年から一人で住もうと思っています。
I'm planning to live on my own from next year.
今日は早く寝ようと思う。
I think I'll go to bed early today.
新しいパソコンを買おうと思っています。
I've been thinking of buying a new computer.
週末は部屋を掃除しようと思います。
I'm going to clean my room at the weekend.