Lower their gaze does not mean to avoid eye-contact or to interact.
Many so-called practising Muslims are uncooperative and unhelpful to members of the opposite gender. They keep away from each other as much as possible. In some cases, if two such people are put in a situation where they have to communicate, one looks into the sky, while the other looks into the ground in order to avoid eye-contact. A person of this breed can ever appear impolite and rude to a person who does not carry this mentality. They back their behaviour using what they have been told about verse 30-31 of Surah An-Nur. This is a misconception that arises purely out of a lack of wisdom and understanding.
"Tell the believing men that they should lower their gaze and guard their private parts..."
(Surah An-Nur : 30)
"Tell the believing women that they should lower their gaze and guard their private parts..."
(Surah An-Nur : 31)
The Quranic meaning that should be derived from the above verses is that men and women who believe in Allah and the Last Day, must behave modestly and carry themselves humbly between one another. Not only is this an interpretation based on reason, it's also supported by Quranic understanding.
We cannot say, by looking at the above expression, the verse means to not speak to the messenger at all just because it says "lower your voice" which compares with "lower your gaze". Rather what is expressed here is that the believers should not have arrogance or haughtiness in their voice and attitude when interacting with the messenger. Likewise, to lower their gaze does not mean to avoid looking at opposite gender, rather to carry a modesty, humbleness when dealing with one another.
Like example, 3 days ago me and my classmate (Riyanti) just took the japanese test in FIB UI that will be determine to allow the next step of scholarship from Embassy of Japan. First step is sending our CV, school grades, and a letter of recommendation that all written in Japanese. We arrived at 9:00 in the morning but the test starts at 10:00. Riyanti sat down beside me. But when the beadles shouted to us to change the seat who is our college-mate, Riyanti took another seat straightaway. And SUDDENLY, someone appears from outside and took Riyanti's seat. He's sat down beside me. He always saw my eyes over and over like he wants to know me more. I felt awkward. He asked me too much questions and I really want to start this test. 10 minutes after that, I completely forgot what's his strange questions is. I tried to focus to finishing all the test in 3 types.
And well, I, a muslimah who is never want to leave my akhwat friend, when I go to college, or taking a test, or want to have a lunch in canteen. But this is what I mean to take control of ourselves to NOT flirting, or so with the non-mahram guy. This is haram.
Because, when you, muslimah, seeing a non-mahram guy each other and cracking jokes or else, satan is the third between you and him.
وَلا جُنَاحَ عَلَيْكُمْ فِيمَا عَرَّضْتُمْ بِهِ مِنْ خِطْبَةِ النِّسَاءِ أَوْ أَكْنَنْتُمْ فِي أَنْفُسِكُمْ عَلِمَ اللَّهُ أَنَّكُمْ سَتَذْكُرُونَهُنَّ وَلَكِنْ لا تُوَاعِدُوهُنَّ سِرًّا إِلا أَنْ تَقُولُوا قَوْلا مَعْرُوفًا وَلا تَعْزِمُوا عُقْدَةَ النِّكَاحِ حَتَّى يَبْلُغَ الْكِتَابُ أَجَلَهُ وَاعْلَمُوا أَنَّ اللَّهَ يَعْلَمُ مَا فِي أَنْفُسِكُمْ فَاحْذَرُوهُ وَاعْلَمُوا أَنَّ اللَّهَ غَفُورٌ حَلِيمٌ
"You (believeng men) do nothing wrong when you announce your engagement to women, or when you keep it between yourselves. Allah knows that you will think of them. But do not make secret appointments to see them, unless you will engage in conscientious conversations. And do not confirm the marital tie until the prescribed (waiting) period (from a previous divorce) is over. Know that Allah knows what is in your minds, so beware of Him, and know that Allah is All-Forgiving, Forbearing."
(Qur'an : 235)