How can I disable ringing when receiving a text message on an iPhone 6s?

I have an iPhone 6s and whenever I receive a text message a ringtone plays and continues for a while. It used to be a simple ding but now it is much more disruptive. Putting the phone on silent has no effect. As far as I know, the only way to make receiving a text not make my phone go off is to power my phone off. But I need a functioning phone to receive reminders and calls, and I need to not disrupt class when I receive a text message. How can I disable playing a ring sound when receiving a text message? I searched through settings and could not find where this setting is and disable it.

How to Change Your iPhone Text Sound

1. Go to Settings
2. Select Sounds & Haptics
   ^{just labelled Sounds on a 6S, no haptic capability}
3. Tap Text Tone.

Under Alert Tones, you can choose one of the options.

4. Tap the tone you like to select it (*If you don't want any tone like you said, just select None)

You can see even more tones and vibrations if you tap Classic.