She asked us if the body was accelerating or slowing down, and I immediately said that it was accelerating (because the $a=3>0$). Then she said that I was wrong because the direction of the acceleration vector was the opposite of the direction of initial speed($v_0=-3$). I do not understand why it slows down, because with the passage of time the body moves faster. Can someone give me an explanation?
The initial velocity and acceleration here are in opposite directions. The magnitude of velocity (represented by $S=|\vec v|$) decreases upto a certain instant. (i.e. where $\vec v=0$).
Edit: Also, consider these graphs. ($t^.$ being the time where $v=0$) Note how the velocity increases but the magnitude of it (in the $S$, $t$ graph) decreases till $t^.$.
I understand what your teacher is saying, but I think she's wrong. In my physics classes we were told never to use the word "de"celeration, only acceleration. Why? For elegance reasons mostly.
Acceleration is a vector quantity, therefore its magnitude is always positive or zero with a direction.
The sign (or direction) of the acceleration depends on your reference frame. All inertial reference frames are equal and ultimately arbitrary; therefore to say a certain acceleration is DE-cceleration implies the reference frame you're working with is better, or somehow different, than the one where it would have been a mere ACCeleration. This is not elegant and we strive to have definitions that have elegant results.
For the last point imagine that what you are observing a tennis ball during a match. Then there is another observer observing from the other side of the court (i.e. bleachers on either side of the moving particle). In their reference frame the equation of motion would be: $v(t)=3-3t$. Can the particle really be decelerating, and accelerating? What's so special about her reference frame?
If anything DEceleration is layman's term to mean an acceleration that brings you to rest wrt to the ground. Which is not the case here after t=1
She asked us if the body was accelerating or slowing down
Acceleration is defined as the time rate of change of velocity and, in this example, the acceleration is constant and positive.
So, the full answer is: the velocity of the body is always increasing while the speed is decreasing for $t<1$ and increasing for $t>1$.
In this plot, the velocity $v$ is blue and the speed $|v|$ is red
$$v = -3 + 3t$$
$$|v| = |-3 + 3t| $$
The sense of acceleration has nothing to do with the sense of velocity. Bodies always have negative acceleration due to gravity regardless if they are going up or down. What is important is the convention as to which direction a positive displacement occurs.
In your case all you know that the acceleration vector is in the same direction as a positive displacement ($a=+3$) and your answer is correct.