Find the time between HIGH state & LOW state of push button(switch)
First of all, frarugi87 is right.
Second, why not use this:

finalMillis = millis() - previousMillis;
previousMillis = millis();


Instead of what you wrote.

if (digitalRead(buttonPin) == LOW) {
    buttonPushCounter++;
    Serial.println("on");
    Serial.print("number of button pushes:  ");
    Serial.println(buttonPushCounter);
    finalMillis = millis() - previousMillis;
    previousMillis = millis();
    Serial.println(finalMillis);
    while(digitalRead(buttonPin) == LOW){}    
  }


It works great.

Hope it helps.

Yoav