How is the weather in Japan for winter?

Hello everyone. I will be travelling to Japan on December 2nd for 3 months on a tourist visa. I will be doing 3 separate homestays at different locations. I will be going to Shinjuku and will be spending two nights there, then on December 5th I will be making my way to Nakagawa-ku, Nagoya for the first month. The second month will be in Fukuoka city and my final month will be in Tokyo central.

Now that you know my locations where I will be, my question is how should I prepare for the Japanese winter? Should I bring my big winter jacket? or should I just bring a hoodie and light jacket.
Thanks

http://en.wikipedia.org/wiki/Nagoya#Climate

http://en.wikipedia.org/wiki/Fukuoka#Climate

http://en.wikipedia.org/wiki/Tokyo#Climate

Where are you from? My friend from Canada constantly wears short t shirts until mid-December, and rarely more than a hoodie. In contrast, I'm from Texas. You're lucky if you see any skin at all in winter, including my face.

So if you're the kind who gets cold, bring your big winter jacket. Or just realize that if you're here for three months, you might as well buy stuff here anyway. Japan has excellent warm weather gear.